Rheokinetics: Rheological Transformations in Synthesis and Reactions of Oligomers and Polymers

A. Ya. Malkin and S. G. Kulichikhin Rheokinetics

A. Ya. Malkin and S. G. Kulichikhin

Rheokinetics

Rheological Transformations in Synthesis and Reactions of Oligomers and Polymers

WILEY-VCH Verlag GmbH & Co. KGaA

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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© 1996 Hüthig & Wepf Verlag, Hüthig GmbH

© 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Printed in the Federal Republic of Germany Printed on acid-free paper

ISBN: 978-3-527-29703-0

5

The idea that rheology is a theoretical base for polymer processing is encountered in many original papers and text-books. However, the field of application of rheology in polymer technology is much wider. The authors of this book suspect that no real technological process in polymer synthesis can be quantitatively described or modelled without a proper description of rheological effects. The rheologic properties of a reactive medium need closer consideration because they reflect the chemical transformation and influence the course of the process.

The logical basis of this book is summarized as follows:

in the course of polymer synthesis and transformations of oligomers and polymers great changes in the molecular structure take place.

these changes lead to tremendous evolution of rheological properties of the sample; i. e. viscosity changes by millions (!) of times. This situation is special for polymer technology because with low-molecular weight products changes in rheological properties of reactants are negligible;

variation of rheological properties can be easily monitored by simple instrumental methods. Therefore it is a convenient method for controlling a technological process;

a reactive system must be sensible to changes of the rheological state because the velocity of molecular movements (and therefore the rate of reaction) depends on viscosity of the medium;

all real technological processes are connected with a flow of reactants. It is important to have a method to influence the hydrodynamic situation in a reactor and to realize the role of deformation in the kinetics of a chemical reaction.

At last, dealing with real technological processes, we must consider the time scale of the process, as it is a problem of productivity, economical factors and finally its competitivity.

So three key words determine the approach of this book

technology - rheology - kinetics

The latter two gave the book its title but in fact its value comes from being a foundation of real polymer technology.

Everybody knows that technology is something highly sophisticated, which was up to now considered as an industrial analogue of the art of cookery. It would be incorrect to disclaim the partial truth of this conception, but it is only half the truth. The more we know about the fundamentals of a technological process the more obedient and controlled it becomes, and the latter means: more safety, more profitability, more reproducibility.

Rheokinetics is one of the keystones to successful of polymer technology. It unites knowledge adopted from various branches of science. It is rather easy to teach students at Universities incorporating it into courses on chemistry, on rheology, on hydrodynamics or

6

similar topics. It is very difficult to combine information and rules coming from different sides but it appears to be necessary when we face real life and technology exists in real life but not in text-books.

The main goal of the authors was to find general rheological roots inside numerous technological processes of polymer syntheses and transformations and to suggest a generalized description of the kinetics of chemical transformations and how to monitor them with rheological methods. This is a rather new approach in polymer technology due to limited data available by publications in periodicals. Certainly the authors are well aware of the fact that this book cannot completely cover the problem. On the contrary we are sure that there are a lot of flaws in the book and the only thing which can excuse us is the fact that we deal with a very lively and rapidly developing field. Therefore any attempt to conceal our knowledge under a book cover opens new unexplored fields. (Once Goethe said that the solution of any problem leads to the appearance of a new one. This is completely true for science as a whole and for rheokinetics in particular).

We express our sincere thanks to our colleagues who read the (Russian-language) version of this book and made useful comments.

Mr. B.M. DuKhan is to be thanked for translating the book in English.

A. Ya. Malkin S. G. Kulichikhin

Moscow, Russia, February 1995

7

Table of Contents

Chapter 1 Introduction

1.1 References for Chapter 1

Chapter 2 Rheokinetics of Linear Polymer Formation

2.1

2.2

2.2.1

2.2.2

2.2.3

2.2.4

2.2.5

2.2.6

2.2.7

2.2.8

2.3

2.3.1

2.3.2

2.3.3

2.3.4

2.3.5

2.3.6

General

Ionic Polymerization

A Direct Rheokinetic Problem

Anionic Polymerization with Varying Amounts of Active Centres

An Inverse Kinetic Problem

Experimental Methods

Anionic Polymerization of Dodecalactam

Increase of Viscosity during Anionic Polymerization with Changing Concentration of Active Centres

Anionic Polymerization of &-caprolactam

References for Section 2.2

Free Radical Polymerization

Rheokinetics of the Process - Theoretical Considerations

Steady Stage of Polymerization - Experimental

Viscosity of the Medium - Its Role.

Gel Effect

Polymerization in Heterogeneous Medium


2.4 Pol ycondensation

2.4.1 Introduction

2.4.2

2.4.3 Experimental

2.4.4 References for Section 2.4

Rheokinetic Pattern of the Process

15

25

27

27

28

28

31

38

40

43

48

53

58

60

60

66 78

82

93

99

102

102

1 04

105

130

8 Table of Contents

Chapter 3 Rheokinetics of Oligomer Curing

3.1

3.1.1

3.2 3.2.1

3.3

3.3.1

3.4

3.4.1

3.5

3.5.1

3.6

3.6.1

3.7

3.7.1

3.7.2

3.7.3

3.7.4

3.7.5

3.7.6

3.8

3.8.1

3.8.2

3.8.3

3.8.4

3.8.5

3.8.6

3.8.7

3.8.8

3.8.9

3.8.10

General principles


Determination of the Point of Gel Formation


Viscosity Increase to the Point of Gel Formation


Microphase Separation in Reactive Systems


Curing After the Point of Gel Formation


Rheokinetic Equations of Curing


Non-isothermal Curing

Causes of Non-isothermal Behaviour

Determination of Kinetic Constants in an Adiabatic Process

Non-isothermal Curing - Calculations.

Heat Transfer from an External Source (Linear Temperature Increase)

Curing at High Shear Rates


Structuring Peculiarities of Different Forms of Oligomers and Polymers

Curing Phenolformaldehyde Resins and Related Oligomers

Production of Polyurethanes

The Curing of Unsaturated Polyesters

Silico-organic Polymers

Amino-formaldehyde Resins

Epoxy silico-organic Oligomers

Rubber Compositions

Thermo-reactive Pol yamides

Epoxy Oligomers


133

133

141

142

145

146

154

156

168

170

176

178

185

186

186

188

192

198

202

206

207

207

216

229

237

248

250

252

258

260

268

Table of Contents 9

Chapter 4 Transformation in Polymeric Systems

4.1 Pol ymer-analogous Transformations

4.1.1

4.1.2 Photo-viscous Effect


Transformations in the Solid Phase

4.2

4.2.1

4.2.2

4.2.3

4.2.4

4.2.5

4.2.6

4.3

4.3.1

4.3.2

4.3.3

4.3.4

Gelation (Curing in Solution)

General Remarks

Antithixotropy During Gel Formation

The Role of the Solvent

Gelation Kinetics

Gelation of Plastisols


Rheolunetics of Phase Transitions

Plotting Phase Diagrams

Crystallization

Phase Transformations in Reactive Systems


273

273

273

274

276

277

277

28 1

283

294

295

299

30 1

301

305

3 10

3 17

Subject index 319

11 ~~

List of Symbols and Abbreviations

a A

[A1 A AN b BM BPO

C D DBP DCDA DMSO DPMDI

E

C

ii

Eeff

ED Eq

f

Ech

EM

t? G’ G”

6“ GO GO3 GPC h H HDMI

[I1 k

k , ki, k,, k , K K 1 ni

M M -

constant, exponent, factor of heat conductivity activator concentration of active centres certain degree of polymerization, Arrhenius constant acry lonitrile constant, exponent buty lmethacry late benzoylperoxide constant of autocatalysis, concentration, heat capacity constant constant, diffusion coefficient dibuty lphthalate dicyandiamide dimethylsulfoxide dipheny lmethanediisocyanate activation energy

dimensionless activation energy activation energy of a chemical process “effective” activation energy “effective” activation energy to achieve some preset degree of conversion “effective” activation energy to achieve a certain level of viscosity ethylmethacry late efficiency of initiation, functionality free fall acceleration rubbery modulus loss modulus

complex dynamic (at a constant frequency) rubbery modulus elasticity modulus equilibrium rubbery modulus gel permeation chromatography height hardness, enthalpy 1,6-hexamethylenediisocyanate concentration of the initiator reaction rate diffusion controlled reaction rate rate constant of the initiation reaction, of chain propagation, of chain termination Kelvin constant length empirical constant, exponent molecular mass, torque

average molecular mass

12 List of Symbols and Abbreviations

M(t) M

[MI

MC

MC

M w

-

-

-

MDI MFR MM MMA MW MWD n N

N -

“3 0 1 PA PA- 12 PA-6 PA-6 10 PAS PB PBMA PBTP PEMA PETP PFR PM PMMA PS PSF PU PVA PVC

4

4 Q Q,, r R R S t

-

time-dependent change of torque

monomer concentration of monomer

critical value of molecular mass

average distance between the points of the network

mass-average molecular weight 4,4’-methylenedi(phenylisocyanat) melaminoformaldehyde resin molecular mass methylmethacry late molecular weight molecular weight distribution empirical constant, material constant quantity of active loci, ionogenic catalyst concentration

number-average degree of polymerization

critical chain length oligo imide pol yamide poly-(e-laurolactam) pol y-(e-caprolactam) poly-(hexamethylene sebacinamide) pol yarylsulfon pol ybutadiene pol ybutylrnethacrylate pol ybutox ytitanphosphoroxane pol yethy lmethacry late pol yethy leneterephthalate phenolfomaldehyde resin pol ymethacry late pol y methy lmethacry late polystyrene poly sulfone polyurethane polyvinylalkohol polyvinylchloride heat

dimensionless heat effect total heat of a reaction integral registered heat effect of a reaction at a given temperature radius, equimolar factor gas constant radius constant, product surface, entropy time

List of Symbols and Abbreviations 13

t

t*

tn * - t n * lac

t C

l c h

l h

rapt

t P

ta T T

Tcure

-

Tf TR

TPh tan 6 TBT TDI TES U V

a a* P P P c 3 P p P, P*,, , P:: Y P F A E

Ei

11 110 11” 11, rl SP K

dimensionless time

time of gel formation, time to reach some critical value, induction period

time before reaching the gel-point in the non-isothermal regime of curing, dimensionless induction period in a non-isothermal process

dimensionless curing time in the non-isothermal regime characteristic time of acceleration time to reach a critical molecular mass characteristic time characteristic time determining the rate of heat propagation optimal curing time time of microphase separation characteristic time for heat transfer with the environment absolute temperature

dimensionless temperature

temperature of curing

temperature of fluidity loss temperature of relaxation transition, glass transition

temperature of phase transition angle of mechanical (or dielectric) losses pol ytetrabutoxytitan 2,4-toluene diisocyanate tetraethox ysilane activation energy of a chemical reaction volume

complex variable, factor of heat transfer

branching factor

degree of conversion

reaction rate

“calorimetric”, “kinetic”, “rheological” degree of conversion

critical values of the degree of conversion

shear strain, rate of temperature increase

shear rate

loss angle, solubility parameter

difference dimensionless shear rate, dimensionless rate of temperature increase

Euler function

viscosity

maximum value of Newtonian viscosity, initial value of viscosity

viscosity of the reactive mass

viscosity of the monomer, microviscosity

specific viscosity

dimensionless energy of activation

14 List of Symbols and Abbreviations

h wavelength

P material constant

AP chemical potential

5

0

Indices 0 ac C

c, cr ch D eff f

g 1

m max or

P r

SP

t W

a P rl m

constant of self-decelaration coefficient of normal stresses, measure of solution elasticity at shear flow dimensionless parameter reflecting the contribution of self-deceleration

concentration of active loci, concentration of polymer, gel concentration

critical concentration of polymer in solution maximum content of filler in a system

coefficient of the time dependency of the degree of conversion

density, share of reactive groups time of relaxation, shear stress

dimensionless time

dimensionless non-insothermal time of curing

shear stress in the region of viscous flow

fluidity limit

constant characterizing the rate of viscosity change, dimensionless temperature characteristic time of solution relaxation

frequency

initial value, value at time t = 0 characteristic value referring to acceleration calorimetric critical value chemical, characteristic diffusion controlled parameter effective refers to fluidity loss refers to glass transition initiation monomer maximum value orientation propagation parameter determined by rheological methods specific at time t ; termination refers to molecular weight refers to heat transfer refers to a certain degree of conversion refers to a certain viscosity final value

15

Chapter 1 Introduction

Numerous operations in the technology of polymer synthesis and processing are connected with the flow of material - caused either by the rotation of the mixer in the reactor, the screw in the extruder or by melt flow filling the form. Evaluations of such operations inevitably require the determination of the properties of a flowing liquid. Experts in lacquers, pigments and polymer solutions came across this problem more than 50 years ago. The determination of the fundamental principle - dependence of material properties on the regime of deformation - gave rise to a new scientific field - rheology investigating anomalous media. This “anomaly” is, of course, only an arbitrary term denoting the fact that the properties of all “rheologically complex liquids” of this kind differ somewhat from those of common low molecular liquids with constant viscosity.

Later the rheology of polymers took the shape of an independent dynamic field with its own approaches, techniques and applications. Most of the results achieved in investigating rheological properties on polymers are presented in [l] which is far from exhausting a long list of innovative articles and treatises devoted to this subject. However, if we try to place the rheology of polymers among fundamental sciences, it can to a certain extent be attributed to the branch of physical chemistry the basic subject of which is the determination of the relationship between the substance composition and its properties, these properties being assumed as constant and inherent in the substance.

At the same time, a somewhat different approach initially existed in the rheology of polymers involving consideration of the time (or kinetic) factor. This factor was introduced in classical rheology in two different ways.

The most obvious are the changes in material behaviour with time caused by relaxation phenomena leading to a number of viscoelastic effects. Investigations abound in this field. However, the variations in material behaviour depending on the time (frequency) scale of observation, are only seeming. Indeed, though a material (at a given temperature) can behave either as a liquid, a rubber or a glass, it is still the same material, possessing a stable structure and properties, the change of the observed behaviour being caused by a spectrum of relaxation times and different correlation between the time scales of the experiment and relaxation.

The time factor plays quite a different role when the structure of the material is being changed. In this case, the observed changes in its behaviour are caused by other reasons and are manifested in a different way. One of the first subjects of rheological studies where structural transformation occurred were thixotropic media, i. e. the materials with properties that change in the course of deformation and relaxation. It is noteworthy, however, that though such media are known to be quite numerous and very important from both theoretical and experimental viewpoints, the level of their investigation remained, to a great extent, merely descriptive. Only a few attempts were made to develop consequent quantitative ideas of their behaviour. One of the main achievements in the thixotropic theory remains the revealing of the kinetic nature of

Rlicokinctics A. Ya. Malkin and S. G. Kulichikhin

Copyright 0 2002 WILEY-VCH Verlag GrnbH & Co. KGaA

16 Chapter I: Introduction

processes, leading to changes of the observed behaviour of the substance, even without going into details of the mechanism of these processes. Therefore time is included into the system of equations describing the behaviour of thixotropic media in an essentially different way than the behaviour of viscoelastic bodies.

The aforementioned examples, however, do not exhaust the most important cases, where the time factor plays a marked role in the rheology of polymers. There is a vast field of considerable theoretical and practical significance, the basic subject of which is the kinetic factor. This field can be defined as “synthesis and transformation of polymers”. Changes in the viscosity of reactive media in the processes associated with the chemistry of low-molecular products do not play any significant role, since the viscosities of initial and final products do not differ appreciably. Quite a different behaviour is observed in polymer synthesis, when the viscosity of a reactive medium changes by several thousand times during the reaction. We deal here not with thixotropic transformations, but with irreversible changes in the material structure (composition). The processes under consideration inevitably reveal their kinetic nature, however, which is similar to thixotropic transformations.

In fact, the term “polymer production and transformation” denotes different processes, the main of which being:

1) formation of linear polymers from monomers;

2) curing, i. e. transition from relatively low viscous solutions of oligomers to three- dimensional net-structured non-fluid materials. The processes of polymer analogous transformations are also covered by this term.

In all mentioned cases a marked change in the whole complex of rheological properties occurs, obviously caused by changes in molecular composition and substance structure. Indeed, the viscosity changes by many decimal orders during polymerization (and polycondensation), rubbery deformation emerges with all consequences inherent in it: large elastic deformations, normal shear stress, and some mechanical relaxation phenomena are initiated within the time scale available in the course of the experiment.

Similarly, viscosity increases unlimited during solidification, i. e. the material loses its fluidity and the rubbery modulus changes by many decimal orders.

The aim of this book is to consider - consecutively and quantitatively - these changes.

The problem of correlating the composition and the properties of reactive systems in the polymer synthesis is closely connected with two fundamental questions:

a) kinetics of occurring reactions;

b) inherent rheological properties of reactive masses.

No matter whether the polymer formation occurs in a solvent or in a block, the reactive medium presents in itself a solution of the forming polymer in its own monomer (oligomer) or in a multi-component solution where this monomer is present. Therefore the first problem to be solved is whether it is possible to identify the reactive mass with the stable polymer solution or, more precisely, where is the limit for this identification. As it will be shown later, the answer to

17

this question depends to a large extent upon the phase of the reactive mass, i. e. whether or not the polymer being formed is soluble in its monomer or in the solvent applied. This idea makes it reasonable to include the diagrams of reactive mass phases into our consideration.

Further consideration depends upon the point, whether we deal with linear polymers or with branched macromolecules or, consequently, with a three-dimensional network. The problem of estimating the rheological properties of a reactive mass for linear polymer synthesis appears to be relatively simple, since the relations describing the dependence of solution viscosity of linear polymers upon the solution concentration and molecular mass of the dissolved polymer are well-known. Some peculiar effects of the solvent nature on the rheological properties of the solution have been also established. In case of branched polymer synthesis and particularly for transition from polyfunctional reactive oligomers to network polymers, such general regulations are unknown which necessitates the establishing of more or less universal relationships describing changes in rheological properties of a reactive mass. As it will be shown further in corresponding chapters, such relationships can be found through phenomenological rheological equations.

Let us examine in more details the principles determining rheokinetics of the synthesis of linear polymers, since they are of general value for further consideration. From the above statements it follows that the relationships of change in rheological properties of a reactive mass during polymer synthesis are determined by the following general factors:

1) the reaction path which establishes interdependence between the degree of conversion p on the one hand, and the length of initiated chains and their content in a reactive mass on the other hand;

2 ) the kinetics of the process and, consequently, the time dependence of the degree of conversion p ;

3) rheological properties of the reactive mass being a dispersion of a certain type and composition.

As an example, let us examine the three simplest patterns of polymer formation simulating the main actual processes [2]. As any model they represent the major features of the phenomenon determining the final result without going into details.

Thus, let the process follow scheme I: Some quantity N of active loci is introduced to the monomer whose concentration in the reactive mass equals cp. These form the basis for the unrestrained growth of molecular chains until the monomer is entirely converted. Then it becomes quite evident that at a degree of conversion p, the content of the polymer will be cpp, and the average molecular mass of the obtained chains is cppMo/N , where M, is the molecular mass of the monomer. Thus, in this case, at a degree of conversion p, the reactive medium represents a solution of cpp concentration with the molecular mass of the polymer cpPM,/N.

Let the polymerization follow scheme I1 in such a way that the chain growth occurs up to a certain degree of polymerization A , beyond which the growth of another chain is initiated, etc. Then, the reactive medium at a degree of conversion p represents a polymer solution with the molecular mass A M , and its concentration is proportional to the degree of conversion p.

18 Chapter 1: Introduction

Type of process:

cp M

I I1 I11

P P 1

P 1 ( l - p ) - '

Hence, an essential difference is evident between the assumed patterns of dependencies cp and @ and, consequently, of rheological properties of a reactive mass on the degree of conversion p for different hypothetical schemes of polymer formation.

Thus, the models given above reveal, in the first approximation, the change of the reactive mass composition depending on the chemical nature of the process with the increase of the degree of conversion P . It is shown in Fig. 1.1, where the differences in anticipations following from the mechanisms denoted as I, 11, I11 are clearly seen.

B Figure 1.1: Pattern of polymer concentration and molecular mass change for three

different reaction paths of polymerization.

The three simplest hypothetical models of polymer formation mentioned above can be compared to the three main mechanisms of polymerization which are considered in the chemistry of polymers.

Thus, the chain growth of active centres the quantity of which remains constant (scheme I) is a model of ionic polymerization, where N is the ionogenic catalyst concentration. The formation in the course of polymerization of macromolecules with constant (on the average) chain length (scheme 11) is inherent in free radical polymerization. And finally, the involvement of all molecules present in the reactive region in the process of polymer formation (scheme 111) is typical of addition polymerization (polycondensation) .

Finally, according to scheme 111, all the molecules of the monomer are simultaneously involved in the process of polymer formation and the mean numerical degree of polymerization ( 1 - p) -', at polymer concentration = 1, corresponds to the degree of conversion p.

Thus, for the three simplest models we can present the dependencies of concentration cp and molecular mass M of the polymer in a reactive mass upon the degree of conversion in the following way:

19

Actually, all these reactions are much more complicated and have a lot of by-effects making the situation more intricate. However, the aforementioned model schemes adequately simulate, in the first approximation, the essential features of the described types of reactions.

They may be applied, therefore (and it will be shown further in subsequent sections of the book), to qualitative rheological analysis of the process of polymer formation taking various reaction paths.

All the above considers the degree of conversion p as the main criterion. The determination of p, however, constitutes a special problem per se, being solved in the analysis of chemical kinetics of polymerization. This problem can be solved with the aid of different physical and chemical methods and the value itself can be determined in different ways, for instance, through concentration changes of reactive groups or of a monomer. It is convenient to reduce the concentration values to limit values reached at the end of the process, formally at t + -. Thus, for instance, if the monomer concentration in the reactive system at the start of the reaction [ MIo, and all the monomer chains are found to be involved into the polymer chain when the

polymerization is over, i. e. the final monomer concentration at t + 03 is zero, the value of p which alters with time may be defined as { [ MIo - [MI ( t ) } / [ MIo where [MI (t) is a current value of monomer concentration in the reactive system which depends upon the time t . Then it is evident that if t = 0, p = 0 and at t + -, [MI = 0 and p = 1 , i. e. p will alter from 0 to 1. Such a reduction of p will be applied further in all cases.

Thus, the rheological analysis of the processes of linear polymer production comprises three points:

a) solving the problem of chemical kinetics to obtain the relation p(t);

0) establishing the reaction nature which permits to relate the composition of a reactive mass to p;

y) independent description of properties of a polymer solvent which permits to interpret the rheological parameters of a material in terms of p and to present them eventually as the time function. The latter presents the solution of a direct rheokinetic problem. This formalism is clearly demonstrated by the scheme in Fig. 1.2.

TH-HTJ u Kinetics reaction o# a reaction

Figure 1.2: Pattern of the solution of a direct problem; calculation of the relationship q( t ) .

20 Chapter I : Introduction

The scheme can be converted, i. e. we can estimate the kinetic constant of the reaction and its origin according to the dependence of viscosity on time q ( t ) .

Figure 1.3: Pattern of the viscosity dependence upon the degree of conversion anticipated by various rheokinetic models.

Indeed, even a qualitative consideration of the relation q ( t ) for the three model patterns discussed (Fig. 1.3.) shows essential differences between them [3]. For more convenience of comparison the chosen starting point is the same in this figure. Its coordinates are q = 1Pa.s at p = 0.1 . The constants for dependencies K , are chosen respectively for the given schemes of polymer formation. Increase in viscosity in these schemes is expressed by the following formulae:

I: q l =

11: r\, = K2Pb

111: q3 = K 3 ( 1 -P) -“ and values a and b are assumed to be a = 3.5. b = 5.0.

If the reaction path is known, the measured dependencies can be quite obviously transformed into kinetic curves for the constants of kinetic equations to be determined. It is natural, however, that such an analysis should be applied to values of p and not to viscosity itself, since formally we can write a “kinetic equation” for viscosity but it is devoid of any physical sense. The possibility for analysis of chemical kinetics of a polymer formation reaction through rheological measurements constitutes a solution of the inverse kinetic problem.

Further considerations of rheokinetic analysis in the processes of polymer formation and conversion make it necessary to consider also non-isothermal phenomena, inevitably concurring these processes [4]. Non-isothermal effects occur either due to some inner reasons, since many chemical reactions run with significant endothermic effects, or due to heat transfer from an external source when the reaction is initiated (“ignited”) in an inert cold system from the reactive mass which is preheated by the walls of the reactor. In any case, it is either practically unfeasible to maintain constant temperature due to large volumes of reactors and low heat

21

conductivity of a reactive mass, or totally unreasonable due to processing problems necessi- tating the optimisation of a process. Therefore non-isothermal phenomena appear to be an inevitable constituent of rheokinetic problems.

The rise of temperature leads to two consequences. First, the rates of all chemical reactions increase and therefore the changes in rheological properties accelerate, particularly, a sharp rise in viscosity is observed. Second, due to increase of temperature the viscosity of a reactive mass decreases. Relative influence of these phenomena upon a general character of changes in rheological behaviour of the medium is different at the start and at the end of the process. Qualitatively it is illustrated in Fig. 1.4, where the plots of viscosity change of a polymerising mass at To = const. and at linearly increasing temperature starting from To are compared. It is noteworthy that the scale of the effects at the start and at the end of the reaction differs appreciably. If we arbitrarily assume the activation energy of viscous flow at the start of the reaction to be 40 kJ/mole, at temperature rise by 10 to 20 K a 2- or 3-fold drop of viscosity is to be expected. For relatively low viscous initial medium this effect appears to be rather pronounced. However, at lo3 to 105-fold increase in viscosity in the course of polymerization, its 2- to 3-fold decrease as a result of the reactive mass heating is an effect of the second order. At the same time, considerable acceleration of polymerization (several times over) caused by the heating appears to be an essential factor of the accelerated increase in viscosity as shown in Fig. 1.4.:

t

ut temperature rise during the reaction (2). Figure 1.4: Pattern of viscosity changes during polymerization at To = const. ( I ) and

Some precise methods of determining non-isothermal phenomena in polymer synthesis and oligomer curing for specific typical situations will be considered further in the corresponding sections of the book as applied to the solution of direct problems. It should also be noted that the analysis of non-isothermal effects may be regarded as basic for the solution of inverse problems as well. This is a more intricate problem. Its analysis, however, may appear to be very fruitful, since it permits to estimate the kinetic constants of reactions in the context of relatively poor experimental information. Some examples of this approach will be given in subsequent chapters.

The analysis of change in rheological properties of a reactive mass in non-isothermal processes of polymer formation and conversion constitutes a direct rheological problem. At the same time the possibility of solving an inverse problem is of great interest, which means to determine the kinetic parameters of a process using the results of measuring the rheological


properties of the reactive mass. The analysis of this far more complex problem is, nonetheless, also attached to rheokinetic problems both in an isothermal and in the more general, non-isothermal case. Some versions of solving inverse problems will be considered further.

In the course of polymer formation and transformation the whole complex of rheological parameters of the reactive mass changes and, generally speaking, the analysis of different properties of the process permits to clarify various points of the process. Therefore the more representative the initial rheological information is, the more fruitful can be the conclusions. Mean- while, it is not exactly the case. Primarily, it should be born in mind that it is very difficult technically to measure the whole complex of rheological parameters in the course of the process, especially if it runs relatively fast. Furthermore, the lack of sufficiently reliable theoretical bases deprives the initial experimental data of their informative value, these data remaining just a list of facts not allowing unambiguous quantitative interpretation. It is preferable, therefore, to restrict ourselves to the employment of relatively simple methods of viscometric measurements and dynamic analysis [5 ] , which constitute the two major groups of methods used nowadays in rheokinetics. The sphere of their application is quite arbitrary and is determined primarily by the properties of the material under study.

If the subject of investigation is a relatively low-viscous liquid, namely, a reactive mass in the process of polymerization, the major role is to be ascribed to the viscometric method, i. e. the monitoring of the viscosity change of the medium in the course of the chemical process.

This monitoring can be carried out through different viscometric techniques available or with the aid of specially designed instruments - capillary or rotational viscometers or vibration instruments - but the parameter to be measured is, in any case, the viscosity of the medium.

The viscometric technique can be unrestrictedly used in the analysis of initial stages of oligomer curing of a very wide range of states - from the initial state of relatively low viscous liquids almost up to the gel point. What is common for all the above mentioned cases is the fact that the range of viscosity changes appears to be very broad and comprises lot less than three to four decimal orders. All that influences the choice of techniques for viscosity measurements.

If the objects of investigation are liquids of high viscosity or cross-linked systems, for instance, gels, cured oligomers or cross-linked elastomers, the major experimental techniques are dynamic measurements. This technique can be applied to the materials in different physical states and, from this viewpoint it is more universal than the viscometric one. Though capable of measuring the range of parameters (rubbery modulus and loss modulus) not wider than four to five decimal orders, this technique possesses two advantages of paramount significance - it is applicable both before and after the transition to the loss of fluidity (i. e. it can be used before the material has reached the gel point as well as at greater degrees of conversion). This technique does not influence the process itself, being “non-destructive”.

The latter is of particular importance, since there is always a danger that the viscometric measurements due to more of less intensive deformations of the medium under study can influence the rate and the degree of a chemical reaction. Dynamic techniques are also indispensable in cases when profound degrees of conversion in rheokinetics of network polymers or conversion in a solid phase are under investigation. If we take into account that the dynamic

23

techniques are also very good in practical applications of rheokinetic investigations, the important role of these techniques becomes quite evident. A certain restriction in their use is associated with the necessity of being aware of viscous properties of the material as they are, which is of paramount significance for technical calculations of output pressure patterns of tubular reactors and pipe-lines, for evaluations of mixers, for determining the mechanism of flow of the resins being cured or the properties of oligomer-based compositions during injection moulding, etc.

The rheological measurements are inevitablely connected with mechanical action on the mass. These inputs can be either instantaneous or continuous in monitoring the state of a substance. In general, three cases should be differentiated:

a) mechanical deformation influences the nature and the rate of achemical reaction in no way;

p) it changes the hydrodynamic pattern promoting the agitation, intensifying the heat exchange;

it plays its own kinetic role influencing the phase state of the system and/or the rates of reactions.

The viscometric measurements should be considered the most “active” ones at large deformations occur in this case. The dependence of the results obtained on shear rate is the criterion of this active role in deformations during viscometric measurements. Therefore when speaking about viscosity change in the course of chemical reactions, the rate of the deformation at which the results have been obtained should be taken into account. It is also important, whether the results depend upon the shear rate. However, even if the influence of shear upon the viscosity change is distinctly pronounced in the experiment, two possibilities should be distinguished.

First, intensive deformation may cause significant heat release and, therefore, the influence of shear rate can be attributed to the non-isothermal pattern of a reactive medium (and the result of measurements here will certainly depend upon the shear rate).

Second, the deformation itself can play a “kinetic” role.

Thus, even if the results of viscometric measurements depend upon the shear rate, the cause of the effects observed should be thoroughly analysed, first of all estimating the role of the dissipative factor, of mixing, of shear rate influence upon the phase state of the system, etc.

The measurements of changes in the rheological properties at harmonic oscillations are less sensitive to the regime of deformation. Two particularly significant situations again can be distinguished in this case:

a) the deformation at which the measurements are taken can turn ”large”, not in the rheological sense when it can cause non-linear phenomena, but in the sense of influencing the structure of a material. The latter is especially typical of highly loaded compositions and heterogeneous compositions of any other type;

p) The results of measurements can be sensitive to the frequency of deformation.

The dependence of relaxation (viscoelastic) properties upon the frequency is in general inherent in polymeric materials, therefore, it can appear in rheokinetic measurements as well. Of particular consideration is the question why the shape of rheokinetic curves depends (if ever

y)


observed in an experiment) on the frequency, i. e. whether it is the consequence of the polymeric origin of the substance, or these are entirely different and, generally speaking, non-equivalent processes observed at different frequencies.

To conclude this chapter, let us establish the scope of the domain which the present book is to cover though it seems clear from the title, the subject needs to specify its place. The main aim of the book is - on the basis of identical approaches - to try to classify and to consider a rather large amount of experimental information obtained by the authors and reported in periodic literature before mid 1989. The choice of data is determined by the task of considering the measurements of rheological properties of reactive masses in the following cases:

a) synthesis of linear polymers;

p) chemical transformations caused by presence of reactive groups in a chain;

y) curing of oligomers leading to network polymer formation;

6 ) gel formation from solutions.

The general treatment of these processes is associated with the fact that the rheological transformations are regarded as the consequence of a reaction with its chemical kinetics inherent in it. This very fact justifies the use of the term “rheokinetics”. Unfortunately, such an approach is not applicable in all cases, since the kinetics of chemical reactions is not always known. It is also rather difficult and sometimes quite impossible to determine the relationship between the composition of a reactive mass and its rheological properties. It is inevitable, therefore, in such cases to restrict ourselves to an exclusively empirical pattern of a process, though it is desirable to present it in a generalised form based on general ideas of relaxation of a material and valid for a wide range of objects.

All the above mentioned specifies an intermediate position of our subject between places the subject of the present book as an intermediate between the rheology of polymers proper and the kinetics of chemical reactions of polymerization constituting its basis. On the other hand, the development of the theoretical considerations and experimental facts presented in the book suggests their use in hydrodynamics of the media possessing variable rheological properties. This is a relatively new domain of investigation per se, falling outside the scope of classical hydrodynamics of viscous liquids which is the basis for synthesis and processing of polymers. This subject is of particular importance for tubular reactor design and for evaluations of instruments with agitators, for estimating the quality of polymer products, for analysis of injection moulding processes, etc.

The latter trend of the rheology of oligomer and polymer transformation is of paramount significance, especially if we take into consideration the new opportunities in new technologies such as, for instance, moulding, injection moulding and resin blend extrusion, various RIM- processes, etc. Some important engineering applications following from the rheokinetic analysis of polymer synthesis and transformation are evident already at this stage of investigation. This trend is worth considering, specially and systematically, therefore, the analysis of hydrodynamic problems associated with the flow of liquids having variable rheological properties and their processing application is also beyond the scope of this book.

1.1 References for Chapter 1 25

Thus, the subject of the present books is, on the one hand, intermediate between the rheology of polymers and kinetics of polymerization, and on the other hand, it is close to hydrodynamics of liquids with variable viscosity. It should be noted that while on the former subjects many studies have been reported, the latter remains “terra incognita” which has been explored by only a few researchers and which is still awaiting those who would create here an appropriate environment.

1.1 References for Chapter 1

1) G. V. Vinogradov, A. Ya. Malkin, Rheology of polymers, Springer; Berlin, 1980.

2 ) A. Ya. Malkin, Rheol. Acta, 1973, 12, 3/4, 4861495.

3) A. Ya. Malkin, Polymer Eng. Sci., 1980,20, N15, 1035-1044.

4) A. M. Stolin, A. Ya. Malkin, A. G. Merzhanov, Polymer Eng. Sci., 1979,19, N 15, 1065-1073.

5) A. Ya. Malkin, A. Ackadsky, A. Chalykh, V. Kovriga, Experimental Methods of Polymer Physics, Prentice-Hall Inc., Englewood Cli$s, N. Z, 1983.

27

Chapter 2 Rheokinetics of Linear Polymer Formation

2.1 General

Changes in the rheological behaviour of reactive systems are determined by two major factors: first, by the kinetic path of the chemical reactions occurring in the course of polymer formation, second, by rheological characteristics proper of the mass being polymerized. These are the decisive factors in the rheokinetic analysis of any process. The problem is to define the way of distinguishing their effects, since the initial experimental material presents a combination of rheological properties and kinetic characteristics of a process.

Thus, the changes in viscosity of a reactive system q(t) are determined firstly by the change pattern of the degree of conversion p(f) and secondly by the dependence of the viscosity on concentration and the molecular weight (MW) of the resulting polymer q(m, c p ) . It is only possible to distinguish participation of these factors in the function q(t) if independent kinetic experiments or rheological measurements with model solutions simulating a reactive system are conducted. For this reason, there are two possible patterns of experimental research in rheokinetic analysis of chemical processes reported in literature.

Some authors stress the necessity of conducting rheokinetic experiments with the kinetic experiments analysing functionality, heat release patterns and molecular weight change to determine the kinetic pattern of the reaction. Even if this approach may appear justified, the rheokinetic method is thereby somewhat restricted, since only a direct problem is under consideration. However, if viscous properties of model solutions q(M, cp) are determined independently and their influence on the experimental kinetic curves of viscosity change q(t) is taken into account, the solution of an inverse problem becomes possible, i. e., we can estimate the kinetic properties of the process using the initial experimental data for viscosity change of the reactive system.

This approach enables us to solve both the direct and the inverse kinetic problems. Investi- gation of rheological properties acquires thereby rheokinetic features. The starting point for obtaining the rheokinetic relationships for polymerizing systems is the assumption that the rheological behaviour of growing and inert macromolecules is equivalent. This assumption enables us to apply the main patterns obtained for polymer solutions to reactive systems. In this treatise we have considered some rheokinetic features of linear polymer formation in the course of ionic and radical polymerization and polycondensation.



28 Chapter 2: Rheokinetics of Linear Polymer Formation

2.2 Ionic Polymerization

2.2.1 A Direct Rheokinetic Problem

Anionic polymerization of caprolactams is a typical example of the process of polymer synthesis running the ionic polymerization path. The reaction path and the kinetics of this reaction change considerably due to initial monomers and activators being used, which presents fruitful opportunities for analyses of major rheokinetic patterns in these processes.

In the simplest case of ionic polymerization, the following kinetic pattern is realized the chain propagation occurs at active centres the concentration [A] of which is assumed to be invariable in the course of the reaction, no chain termination occurring thereby*. The end (limiting) value of the number-average degree of polymerization N for this model pattern equals [ Mo] / [A] . If at a certain moment of time t the current concentration of a monomer is [MI , i. e., the degree of conversion

(2.2.1)

the current value of the number-average degree of polymerization in this case will be

(2.2.2)

The mass share of the polymer being formed represents the ratio of the concentration of monomer chains included into a polymer chain to their total (initial) concentration, i. e.:

(2.2.3)

This simple model pattern corresponds to the case of anionic polymerization, or, at least its first stages when one can neglect the exchange reactions of overamidisation and ignore the possibility of reaction transition from a homogeneous to a heterogeneous nature, even if it is conducted at a temperature below the melting point of the polymer being formed.

In the most general case, the viscosity q of the solution - actually the reactive mass - is determined by the two factors:

1) the polymer chain length,

2) its concentration, i. e. q = f ( N , 9).

* The case when the change in the concentration of active centres occurs in the course of the reaction is considered elsewhere.

2.2 Ionic Polymerization 29

The form of this function may appear different and most diverse analytical representations of the function f(N, (9) are discussed in literature. For instance, the viscosity of a polymer solution can be expressed by the following relationship, which is very convenient for further considerations:

q = KqbNa (2.2.4)

where K, b, a are constants.

Taking into account the relation between the degree of conversion MW of the polymer being formed, formula (2.2.4) for the model of ionic polymerization under discussion acquires the following form [S]:

(2.2.5)

In the case of isothermal polymerization, this relationship determines the pattern of the dependence q(P). In fact, the reactions of polymerization run with significant exothermic effects, so that we cannot neglect the non-isothermality of the process in the most general case. Given this situation, we should write formula (2.2.5) introducing therein a term denoting the temperature dependence of viscosity:

(2.2.6)

where E is the activation energy of viscous flow; R is the gas constant; T is an absolute temperature; K2 is the constant which combines the previously introduced constants.

The possibility of establishing an unambiguous relation between T and p (in the adiabatic regime of the reaction) is of importance for the purposes of the present consideration. Indeed, the final increase of temperature as compared to the initial To is A , at P = 1.

Then

T = To+pA (2.2.7)

By substituting (2.2.7) in (2.2.6) and by using the inequality ( E A P / T i ) << 1 we may represent (2.2.6) as

(2.2.8)

where qb = K2exp ( E / R T o ) .

To obtain the final relationship, that is, the dependence of q on time t , it is necessary to find P(t), i. e., to consider and to solve the kinetic equation for p.


If we neglect some complications occurring at large degrees of conversion [6], then in the simplest case the kinetic equation P(t) is presented in terms of a first-order equation, as for p, but when formulating it one should take into account the basic non-isothermality of the reaction of polymerization,

U - = k',(l -P)exp (--) dt RT

(2.2.9)

where k', is the constant and U the activation energy of polymerization.

To solve this problem we should use formula (2.2.7) and bear in mind that (UAP/R?) Q 1. This permits to present (2.2.9) as

9 dt = k , ( l -p ) ( l + m p ) (2.2.10)

where k, = k'oexp(-U/RT,) is the initial rate of the reaction at T = To and m stands for

the complex UA/(R?).

The solution of equation (2.2.10) allowing for the limiting condition p = 0 at t = 0 has the following form:

exp(kot ( 1 + m) - 1)

= exp(kot (1 + m) + m) (2.2.1 1)

Thus, formulae (2.2.8) and (2.2.11) solve this problem showing the way depends on t [7,8].

We may essentially simplify formulae (2.2.8) and (2.2.1 l), by expanding the exponential terms into series, retaining only linear components. Then formula (2.2.11) at small values of k, will be written as follows:

p = k,t (2.2.12)

and the final formula for q(t) will have the form:

q = q 'ok~+b[A]-Ytnth( l -x t ) EAk,

or, provided k, - [A] [6]

(2.2.13)

(2.2.14)

where q; combines earlier introduced constants.

This formula provides a vivid and sufficiently simple evidence of how basic parameters of a process influence the anticipated pattern of viscosity increase with time, at least at initial stages of the process of ionic polymerization.


Some interesting consequences follow from the results obtained. Thus, let us observe the dependence of q on the initial temperature of the process To and on the concentration of active centres [A].

The value of To is a part of the expressions for q” and k,. Therefore the influence of To on the course of the relationship q(t) cannot be expressed in a simple form. If we neglect ( E A / R c ) k,t (compared with 1) and compare the viscosity values markedly exceeding we

obtain the following formula:

) } t a + b - ( a + b ) U - E

RTO r l - {exp ( (2.2.15)

Then we may determine a certain “effective” value of activation energy, which is defined through the values of viscosity of the reactive mass q* at a certain arbitrarily chosen time of polymerization ( t = const ) . If we plot the relationship q * ( To) on the coordinates In q * - T , E,, will be defined as follows:

Eeff = ( a + b ) U - E (2.2.16)

In case it is of interest to estimate the temperature influence on the duration of the process needed to achieve the preset level of viscosity, then it is obvious from formula (2.2.15) that the corresponding value of activation energy may be written as:

(2.2.17) E

a + b Eef f = u- ~

The value of [Ale influences the relationship q(t), entering into k, and M-. After this manipulation the relationship q( [A] o) is apparently expressed by a power function, as shown in formula (2.2.14). These relationships determine the entire pattern of the relationship q(t) when the main kinetic factors - time of polymerization, the initial temperature of the process and the concentration of the catalyst system - are changed, i. e., these relationships represent solutions of the direct kinetic problem.

2.2.2 Anionic Polymerization with Varying Amounts of Active Centres

To regulate the rate of active centres formation, the so-called indirect activators are used in case of anionic polymerization of lactams [9, 10-121. Their peculiarity is that they react with the components of a reactive mass, giving rise to an active centre where propagation of macromolecules occurs. The use of these activators permits to change the kinetics of polymerization and, consequently, to regulate the rate of viscosity increase of the reactive system. So it becomes possible to effect a specified change in “life” of the polymerizing mass for which it maintains a certain moderate level of viscosity.

Evaluation of “life” and the rate of viscosity growth at initial stages of polymerization are associated with the development of a kinetic pattern of this process, since the form of the function p(t) defines the nature of the relation p(t). This permits to clarify the key parameters of the process.


Here we should consider two consecutive reactions running at comparable rates. The problem is complicated by the necessity of an alternative comparison of two different patterns, since an indirect activator can form an active propagation centre reacting with a monomer, i.e. lactam or its anion. The rate of the second reaction is, evidently, essentially higher than that of the first reaction. However, the lactam concentration is far higher than the anion concentration. For this reason we should consider two possible variants of the kinetic pattern and, having compared the results following therefrom with the experimental data, choose the variant corresponding to the relationships p(t) and q(t) to a greater extent.

If the active centre of chain propagation are formed in a reaction with monomer, the pattern of anionic polymerization may be written as:

A + M

A* + M

A*

P

kl ~

k2 ~

I

where the first reaction represents the formation of active centres A* resulting from the interaction of the indirect activator A with the monomer M, and the second reaction represents the chain propagation (monomer M transforms to polymer P). This notation discards the necessity of analysing in detail the chemistry of the process and a great number of particular reactions occurring in the course of anionic polymerization and permits to operate with measurable values, primarily, concentrations [A] and [MI and reaction rates k, (formation of A*) and k2 (chain propagation). Thus, the notation of the second equation is arbitrary. At more rigorous consideration we should have written the whole chain of reactions from a monomer, via a dimer, trimer, tetramer, etc. (An example of this notation for kinetics of anionic polymerization will be given below).

The system of differential kinetic equations for the rate of concentration change of reactants has the following form for simplified pattern I:

(2.2.18) d

-- “41 = k, [A1 [MI dt

(2.2.20) -[MI d = -k2[A*] [MI dt

with boundary conditions at t = 0, [MI = [MIo, [A] = [Ale and [A*] = 0, where

[MI and [A] are the initial concentrations of the activator and the monomer, and, which is of importance, [A] (< [MI o’

As far as [MI is concerned, equation (2.2.20) of monomer consumption is written directly as a first-order equation,. This possibility of a kinetic description of anionic polymerization has been shown for dodecalactam in [13] and [6]. In all the equations written the concentration is expressed in terms of moV1. Therefore at the limiting degree of conversion (t + m) the number


of the moles in the formed polymer is equal to that of the activator, i. e., it is assumed that one active centre gives rise to one chain. This statement is quite consistent with the results of direct measurements [ 141. It remains valid until the secondary reactions begin to markedly influence the process, for instance, overamidisation leading to the broadening of MW distribution (MWD) up to the values of the most probable MWD. Proceeding therefrom, we may assume in the first approximation that the MW and the viscosity of the melt of polyamides formed according to the anionic pattern are inversely proportional to the concentration of the catalytic system at the ratio of the catalyst and the activator being equimolar (Fig.2.2.1).

a

Figure 2.2.1: Dependence of the intrinsic viscosity in H2S04 (a) and the maximum Newtonian viscosity of the melt at 265 “C (b) on the catalyst concentration for PA-6 ( I ) and PA-I2 ( 2 ) .

The change of [A] at low degree of conversion at the beginning of the process, is described by formula (2.2.2 1), following from solving equation (2.2.18) when assuming that [MI = [MI ,, = const (which is true, since [MI )> [A] ):

[A1 = [Aloexp(-k, [ M I , t ) (2.2.21)


The solution of the linear differential equation (2.2.19) with regard to formula (2.2.21) for [A* ] (assuming again that [MI -- [ MI,) has the following form:

(2.2.22)

The assumption that [MI -- [ MIo means, in fact, that the amount of monomer consumed to form the active centres of macromolecule propagation is small, and polymerization does not yet occur.

For the initial stage of the reaction equation (2.2.22) is simplified to a linear relationship

[ A * ] = k , [A], [MI,t (2.2.23)

provided that ( k , [MI ,t) and ( k 2 [MI ,f) << 1.

Then substitution of equation (2.2.23) into equation (2.2.22) yields the formula describing the change in the degree of conversion p with time during the first stages of the reaction (the consumption of monomer for active centres is again neglected):

(2.2.24)

Thus, according to pattern I the initial stages of the reaction are characterized by a quadratic dependence of the degree of conversion on time. The angle coefficient of this relation is therefore equal to the product of constants k , and k , and to the product of initial concentrations of monomer introduced to the system of an indirect activator.

If the formation of active centres occurs mainly under interaction of the indirect activator with anion Me the equation of active centre formation will take the form:

0 k, I1 A + M * A*

and the chain formation follows the first-order equation as in the first case. A set of kinetic equations in this case has the following form:

d d t -[A1 = k , “41 [MOI (2.2.25)


(2.2.26) d - [A*] dt = k , [A1 [M01-k2[A*] [MI

d [ M ] = k,[A*] [MI (2.2.27) d t

Then, provided that at equimolar ratio [A] = [Me] the formula for [A] will be written as:

r '41 = l + k [ A ] , t

The equation describing the change in the concentration of active centres [A*] (at [MI = [ MIo ) and provided that k, [ AIo t <( 1 takes the following form:

- [A*] d + k , [ A * ] [M],+2k:[A];-k, [A]: = 0 dt

(2.2.28)

(2.2.29)

Its solution, valid at rather small values of p will be written in the following way:

k , [ M ] , t 2k -k M t A l o ( k z [ M ] o t - l + e '[ l o ) (2.2.30) [ A * ] = ~ 1 --

k, [MI 0 k:[Ml;

This expression applied for initial stages of the reaction (k, [MI 0t (< 1 ) is essentially simplified to:

[ A * ] = k,[A]:t (2.2.31)

Finally, the substitution of equation (2.2.3 1) into equation (2.2.27) yields the dependence of the degree of conversion on time for initial stages of polymerization:

p = 2k,k2[A] i t2 1 (2.2.32)

The difference between equations (2.2.32) and (2.2.24) is the structure of the coefficient in the quadratic dependence of the degree of conversion on time p = 8 t : If A * is primarily formed due to interaction of A and M, 8 must be directly proportional to the initial concentration of the indirect activator; if A* formation occurs under interaction of A and Me, 8 is proportional to the square of the initial concentration of the indirect activator.

2

The kinetic analysis considered above pertains to the initial stages of the process of polymerization and permits to choose an optimal kinetic pattern comparing the expressions obtained with the experimental data. At the same time, pattern 11, in its full notation, taking into consideration not only active centre formation but also the formation of dimers, trimers, etc., yields an analytical solution and kinetic description of the entire process. In this case the kinetic pattern has to be written as follows:


Formation of an active centre

and then, chain formation:

The constant of the chain formation rate is assumed to be independent of the length of a

macromolecule. Then the following relationships can be applied:

[A10 = 1 + k , [AIo t

(2.2.33)

n

and by introducing the value of macromolecular ion concentration [ M I = C R i (with

regard to the correlation [ M] + [A* ] = [ A ] - [ A ] ) we obtain an equation describing the

change in monomer concentration:

i

(2.2.34)

After integration, passing to the value of the degree of conversion, we obtain the final expression:

k2

k , I n ( l - p ) = k 2 [ A ] 0 t + - l n ( l + k l [AIo t ) (2.2.35)

By expanding the exponential terms into a series at initial stages of the reaction, equation (2.2.35) transforms into equation (2.2.32).

Thus, the difference in formulae (2.2.24) and (2.2.32) reflects the difference in predictions, following from the choice of the pattern for the process of ionic polymerization under discussion. This makes it possible to discriminate certain patterns experimentally and, finally, describe adequately the relationship P(t) from a quantitative viewpoint. It is of paramount

importance from the kinetic point of view that different relationships of p(t) lead to different relationships q ( t ) .


Finally the dependencies of viscosity on time and activator concentration in this case take the following form [ 151:

For pattern I

= Kk~kg+b[A]~[M]~t"'2b

For pattern I1

= Kkfk~+b[A]~bta+2b

(2.2.36)

(2.2.37)

As it has been noted before, this approach permits to estimate the influence of temperature on the rheokinetics of the reaction. In this case the temperature dependence cannot be described by the value of activation energy alone, since it is a function of two processes: active centre formation and chain growth. The effective activation energy determined at some fixed time in the course of polymerization has the form:

E, = U , + U , (2.2.38)

where U , is the activation energy of the process of active centre formation and U , is the activation energy of chain propagation.

The effective activation energy of the process (i. e., the activation energy necessary to achieve some preset degree of conversion), which determines the temperature dependence of the time of polymerization, is expressed as follows:

(2.2.39)

The effective activation energy of rheokinetics, which determines the temperature dependence of viscosity of a reactive system at a certain time, allowing for the above relationship, is expressed in terms of the following formula:

E, = E - ( a + b ) U 2 - b U , (2.2.40)

where E is the activation energy of viscous flow.

If we define the temperature dependence of polymerization time needed to achieve a certain level of viscosity, the effective activation energy is equal to:

( a + b) U , + b U , - E

a + 2 b E,, = (2.2.4 1)

Thus, the formulae obtained illustrate the difference of final mathematical expressions depending on the type and the chemistry of the reaction under study.


2.2.3 An Inverse Kinetic Problem

An inverse kinetic problem, unlike a direct one, is principally ambiguous. This is characteristic for inverse kinetic problems in their general formulation. To check the validity of their solutions, one would have to compare direct observations with the results of evaluation of an initial experimental function comprising the chosen values of constants [16-181. In a general case, this approximation does not give any direct information concerning the chemistry of the process, and for this reason the choice of an approximative kinetic function must be made proceeding from the requirement of its simplicity. Evidently, using function P(t) in different forms leads to different predictions for the relationship q(t). Therefore let us consider three possible kinetic patterns being of interest as far as the ionic polymerization of e-caprolactam is concerned. We shall compare the relations obtained therefrom with the experimental data, which will enable us to choose the “correct” function (in the sense defined above) P(t) and to estimate the constants in it.

a) Isothermal First-Order Reaction.

The kinetics in this reaction is described in terms of formula (2.2.29). Then, for the initial stage of the reaction the function q(t) is expressed as follows:

(2.2.42)

b) First-Order Reaction with Adiabatic Acceleration

This case has been considered earlier. The final formula can be expressed in the following form:

l + m P In (- ) = ( l + m ) k o t

1 - P (2.2.43)

U where k, = k‘, exp (--) RTO

If we neglect the adiabatic acceleration of the reaction, which is true for initial stages of polymerization, formula (2.2.43) will be reduced to the linear expression

P = kot (2.2.44)

so that the final formula (2.2.42) will keep its value as well.

c) Autocatalytic Reaction with Adiabatic Acceleration

A kinetic equation of this type has the following form:

where c is the constant of autocatalysis.


The solution of this equation may be written as:

( 1 + c p p + m )

( I + m p ) m ( ' + c ) (1 - p) ( c - m ) = exp { ( 1 + c ) ( 1 + rn) ( c - rn) kor} (2.2.46)

or

( I + c p ) c ( ' + m )

( I +m( j )m( '+c ) (1 - p) ( c - m ) In = ( 1 + c ) ( 1 +rn) ( c - rn )ko t (2.2.47)

Provided that autocatalysis is intensive enough (cp N 1) , we may obtain the following expression for the initial stages of polymerization:

l n c p = (c -m)kot (2.2.48)

Then, in conformity with expression (2.2.6) the viscosity increase must be described by the following equation:

(2.2.49) U l n q = A + ( a + b ) (c-m)k,texp(--)

RTO

where the constant A com5ines the earlier introduced ones and is expressed through Inko - ( a - b) lnc.

The formulae were obtained on the assumption that the constant of auto catalysis is sufficiently high. Even for the initial stage of the reaction in the same range of values p which has been considered in the first and the second pattern, function p(t) has an exponential rather than linear shape (as in formula (2.2.48)), however, not in the close vicinity of the initial point. Then, in conformity with expression (2.2.49), the viscosity of the polymerizing system is an exponential function of time and the exponential factor thereby includes rheological properties of the solution of the polymer being formed and kinetic constants of polymerization, i. e., the constants of the reaction rate k,, auto catalysis c and the activation energy of the chemical reaction CJ.


2.2.4 Experimental Methods

Investigations of rheokinetics in the processes of polymer formation and transformation persistently demand the choice of precise techniques of measuring rheological properties of reacting media, taking into consideration the nature of the reaction. For this purpose a number of techniques, based on principles of capillary, rotary and oscillatory viscometry and mechanical spectroscopy were developed [19,20-221.

A non-stationary method of viscosity measurements in polymerizing systems to investigate the initial stages of activated anionic polymerization of lactams has been reported [23,24]. The basic principle of this method is the capillary viscometry and its peculiarity consists in a continuous change in the rate of lowering the level of the liquid (a reactive mass) in a long burette, from which the liquid goes to a capillary. Thereby the pressure determined by the

weight of the liquid column decreases with time and the rate of the lowering of the level - is

reduced not only due to shortening of the column h but also due to the increase of viscosity q of the reactive mass. The final formula for viscosity changing with time has the form:

dh d t

(2.2.50)

I R ~ where A = - p g R 2 and B = are the constants of the instrument expressed in terms of

its geometric dimensions; R is the radius of the burette; r is the radius of the capillary; 1 is the length of the capillary; p is the density of the liquid; g is free fall acceleration.

r B

The errors in the viscosity measurements by this method depend on the accuracy of measuring h(t) , density measurements in the course of the process and by viscosity change due to incomplete isothermal of the reactive mass. This method has some restrictions, however. It cannot be applied, for instance, for liquids with extremely high viscosity due to a dramatic drop

of - which makes it impossible to obtain any reliable values of viscosity at every moment of

time. A real range of reliable measurements of viscosity depends on the capillary size, but it

does not exceed the limit of to 10 Pas. In other words this method is applicable primarily for estimation of the induction period (concerning viscosity increase of polymerization and its initial stages). Another restriction of this method is associated with the rate of viscosity increase. Only cases when polymerization takes not less than several minutes and not more than an hour are convenient for observation.

d h d t

Wider opportunities are offered by a more complex method of studying the rheokinetics of non-isothermal polymerization, reported in [25]. This method is based on simultaneous registration of temperature (with the regime of the reaction close to adiabatic) and viscosity of the reactive system.


The main problem in using this technique is the development or choice of a pattern of viscosity measurements, which does not interfere with the course of the process. For rapidly

performed processes - to which anionic polymerizations pertain - the oscillating methods are

the most appropriate for viscosity measurements [26 ,27] .

For this purpose, a viscometer operating in the range of ultrasonic frequencies* is used. The outlet value of plate current measured by a transducer is proportional to the product of f i (p being the density of the liquid under study). In the process of chemical transformation, when

viscosity changes by several decimal orders, a slight change of density may be ignored assuming p = l g k m . 3

A block diagram of an apparatus for complex investigations of the rheokinetics of non- isothermal processes is given in Fig. 2.2.2. To create a regime of reaction, which is close to adia-

batic conditions, this technique provides automatic equalization of the thermostat temperature and that of the reactor. Temperature and viscosity are registered in a measuring cell throughout the entire experiment.

n

rnr-l--1 I

'(- I

I

Figure 2.2.2: Block diagram of an apparatus for complex investigations of the kinetics of

non-isothermal processes.

I is the measuring cell, 2 is a viscometric gauge, 3 is a temperature-

sensitive element, 4 is a thermostat, 5, 6 are visocisty and temperature

monitors and recorders, 7 is the system of temperature levelling in the cell

and in the thermostat

Comparison of experimental thermometric curves shows that the introduction of a metal viscometric gauge into a thermometric cell leads to a certain increase of heat removal from the reaction volume, which manifests in decreasing of the polymerization temperature (Fig. 2.2.3). The temperature decrease compared to the true value corresponding to adiabatic conditions of the process, does not exceed 5% for the major part of the curve.

* A viscometer of the same type has been used by Shimazaki to investigate the initial stages of epoxy oligomer curing 1281.


f85

T, O C

175

15

3 , Pa.s

40

5

t, min

Figure 2.2.3: The temperature dependence under adiabatic conditions ( I ) and on installation of a viscometric gauge ( 2 ) and viscosity dependence (3) on polymerization time of dodelactam. Initial temperature: I70 "C; catalytic system content: 0.5 mol-%.

Yet, one of the drawbacks of this method may be seen in a relatively narrow range of viscosity measurements (not more than 20-25 Pas) and a very high sensitivity to mechanical effects of the indenture plate made of a magnetostrictive alloy, which impedes its industrial application.

From this viewpoint, the use of rotary viscometry seems more preferable, as this method is effective for a wider range of viscosity measurements in reactive systems. In this case we can measure viscosity up to values of lo5 Pa.s, which practically characterizes the melt of the polymer being formed. However, problems arising here are linked with the need to modify the working unit of the viscometer to provide registration of the reactive mass temperature and to develop a unit for feeding reactive components with minimal time lag [29,30].

Describing the results of experimental investigations of the rheokinetics of anionic polymerization, we did not specify the conditions of the experiment and peculiarities of the experimental techniques, since the data obtained by all these techniques agree satisfactorily with each other.


2.2.5 Anionic Polymerization of Dodecalactam

The main theoretical relations describing the increase in viscosity in the course of anionic polymerization, have been obtained for the first-order reaction against monomer (see Sect. 2.2.1). The anionic polymerization of dodecalactam leading to polydodecanamide formation (PA-12) corresponds to this case. For this reason, the experimental data used for comparison with theoretical expressions for a direct rheokinetic problem, will pertain to this reaction.

Typical experimental data for this system in the form of relationships P(t ) (p being defined through equation (2.2.7)), and q(t) for different regimes of the process are given in Figs. 2.2.4 and 2.2.5. The major interest for the objects of this paper presents the consideration of the influence of [ A ] and To on the rate of viscosity increase. To this end we should, first of all through independent experiments, determine the values of a and b for the system under consideration entering to formula (2.2.14). Since we are considering mainly the initial stage of polymerization, the viscosity of polymers may be assumed as a linear function of their MW, that is, a = 1.0.

Pu.s

I D

t, min

Figure 2.2.4: Viscosiry (a) and degree of conversion (b) change at anionic polymerization of dodelactam at dizerent initial reaction temperatures: 160 "C ( I ) ; I70 "C (2); 180 "C (3); I90 "C (4) .


6 12. 2 4 6 t, min

Figure 2.2.5: The kinetics of viscosity increase in the course of anionic polymerization

of dodelactam at T = 160 OC (a) and 180 “C (b) for different concentrations of the catalytic system: 0.35 ( I ) ; 0.5 (2); 0.75 (3); 1.0 (4); 1.25 (5); 1.5 mol-% (6).

The exponent of power in the concentration dependence of viscosity has been determined through a direct experiment. As it is shown in Fig. 2.2.6, the viscosity of PA-12 solutions in dodecalactam is a power function of concentration with the exponent of a power equal to 2.3, i.e. b = 2 .3 . Thus, the viscosity of the reactive mass is expected to be the power function of polymerization time with a + b E 3 . 3 .

Figure 2.2.6: Concentration dependence of solution viscosity of PA-I2 in it own monomer - dodecalactam at 170 “C ( I ) and 200 “C (2).


Fig. 2.2.7 shows the dependence of the viscosity value achieved for 2 minutes of polymerization on the initial temperature To on Arrhenius coordinates. In this figure one can find

the temperature dependence of viscosity of a 20% solution of PA- 12 in dodecalactam. The slope of these curves permitts to determine the values E, and the activation energy of viscous flow of

the solution E which appear to be equal to 133 and 25 kJ/mol, respectively. If we take into consideration that the activation energy of dodecalactam polymerization in this range of values equals to 50 kl/rnol [6], the values obtained are consistent with theoretically obtained formula

kJ kJ kJ mol mol rnol

(2.2.16) at a + b = 3.3 i. e. E, = 3.3 x 50- - 25- = 140- (the experimental value

is 133 kJ/mol).

Figure 2.2.7: Temperature dependencies of viscosity of the reactive mass: ( 1 ) 2 min after the onset of the reaction and ( 2 ) in 20% PA-12 solution in dodelactam.

The theoretical relations obtained offer an opportunity to estimate the constants a and b

To show this effect we may plot the dependencies q( t ) at [ A ] = const. (Fig.2.2.8) and

directly from the viscosity change in the process of polymerization (using formula (2.2.14)).

q( [ A ] ) at t = const. (Fig. 2.2.9) for initial stages of polymerization.


- 2

Figure 2.2.8: The viscosity dependence on time of the reaction in logarithmic coordinates for anionic polymerization of dodelactam at T=160 "C and 180 "C. For designations see Fig. 2.2.5.

4

0 .

- f .

I I I I

-0,4 0 4 4

&3[A;7, [ [ A L in mat? %?

Figure 2.2.9: Viscosity dependence of the concentration of the catalytic system for anionic polymerization of dodelactam at T = I60 "C (a) and 180 "C (b) and t = const. I ) 0.63 min; 2 ) I min


The exponents for the case of anionic polymerization of dodecalactam appeared to be, respectively, a + b = 2.3 and b = 2.3 at To = 160 "C, i. e., in this case the exponent equals zero; and at To = 180 "C a + b = 3.3 and b = 2.3. Thus, the theory is satisfied for high temperature polymerization To = 180 "C . However, our attention was drawn by the fact of lower rate of viscosity increase in low temperature polymerization at To = 160 "C , since in this case a = 0. This means that the viscosity of the polymerizing mass does not depend on the MW of the polymer being formed. This situation is possible provided the reactive blend presents in itself a two-phase system, i. e., a suspension of macromolecules being formed in the monomer medium, viscosity of which depends only on the colloidal particle concentration of the polymer being formed. In this case the rate of viscosity increase should go down since the reaction of polymerization becomes heterogeneous. The assumption of peculiarities for "low temperature" anionic polymerization agrees with the results of independent experiments which permitt to plot the diagrams of the phase state of PA solutions in their own lactams (see Sect. 4.3.2).

The results obtained show that the rheological method of investigation of polymerizing masses and the theoretical basis of its application, besides determining kinetic properties of reactions permits to determine the phase state of polymerizing masses and, consequently, estimate the homogeneous or heterogeneous pattern of polymer formation.


2.2.6 Increase of Viscosity during Anionic Polymerization with Changing Concentration of Active Centres

As it has been noted above (Sect. 2.2.1), processes during which formation of new centres of macromolecule propagation occurs are particular cases of anionic polymerization. In the considered example of anionic polymerization of lactams reactions with an indirect activator are of this kind [ 10-121. The problem of rheokinetic analysis of these reactions in terms of selecting the "correct" kinetic pattern, consists in the necessity of finding an answer to the question: What is the path of the formation of centres of macromolecule propagation, i. e., whether the indirect activator reacts with the proper lactam molecule or with its anion. As it has been shown in Section 2.2.2, these patterns yield different kinetic and, consequently, rheokinetic predictions.

The two patterns predict occurrence of a quadratic dependence of the degree of conversion on the reaction time. It is confirmed experimentally (see Fig. 2.2.10) for different conditions of the process up to the degree of conversion equal to 0.5 - 0.6.

Figure

sa iao 150, t ,min2

10 20 30 t f min2

2.2.10: The dependence of the degree of conversion p on ? at difSerent temperatures of the process (a) and direrent concentrations of the catalytic system (b), a ) I - 160 "C; 2 - I70 "C; 3 - 180 "C; 4 - 190 "C; b) I - 0.5; 2 - 0.75; 3 - 1.0; 4 - 1.25; 5 - 1.5 mol-%.


As it is clear from formulae (2.2.36) and (2.2.37), the time dependence of viscosity, according to these patterns, must be described by a power function with the exponent equal to a + 2b. For the system under study a = 1 and b = 2.3 (Fig. 2.2.8), i. e., a + 2b = 5.6. These dependencies plotted on logarithmic coordinates for different conditions of the reaction are presented in Fig. 2.2.1 I . The exponent of the power dependence equals 6 which on the whole satisfied the predicted function for q(t) .

A// I I I

Figure 2.2.11: The viscosity dependence of the reactive mass on time in logarithmic

coordinates at the initial temperature of the reaction.

(a): 190 "C (l), 180 "C (2), 170 "C (3), 160 "C (4); A0 = 0.0381 kmoWm3;

and the content of the catalytic system (b) A0 =0,076 (5); 0.0634 (6); 0.051

(7); 0.0381 (8); 0.0254 (9) km0Wrn3 at T = 180 "C.

The choice of one of the formulae obtained may be made by comparing the dependencies w = A [A ] o) and w, = f( [A ] o) where ty and w1 are the coefficients of time dependencies of the degree of conversion and viscosity, respectively.

The dependencies w [A ] and wI [A ] plotted on the logarithmic coordinates are given

in Figs. 2.2.12 and 2.2.13. These data testify to the fact that the coefficient w is a quadratic

function of the indirect activator concentration, whereas a similar dependent for the coefficient

w1 is described by formula w, = K [A ] : 5 , that is, in fact, the exponent of power equals 2. We

may conclude therefrom that, in the example under consideration, active centres of macromolecule propagation are preferably formed under interaction of the indirect activator and the lactam anion rather than monomer.


Figure 2.2.12: The dependence of the coeflicient w on [A] in logarithmic coordinates.

I

Figure 2.2.13: The dependence of the coeficient w1 on [A] in logarithmic coordinates.

It is interesting to consider the data of temperature dependence on rate in this reaction. Here the temperature dependence of viscosity of the reactive system is determined by three values of activation energy: E, U,and U , (formulae (2.2.38) - (2.2.41).

The slope of the dependence of the coefficient on temperature plotted on the Arrhenius coordinates (Fig. 2.2.14) indicats that the effective activation energy is equal to 167 kJ/mol. Since the effective activation energy of the reaction in the presence of direct activator (N-acetyl- E-caprolactam) at not very large values of p is equal to 50 Id/mol[6], we may find the effective


value of the activation energy using formula (2.2.38) by determination of the temperature dependence of the reaction rate leading to the formation of indirect activator from acetyl- diphenylamine (used in these experiments as an indirect activator). The value is 117 kJ/mol.

Figure 2.2.14: The dependencies of w ( I ) and k , ( 2 ) on the initial temperature of the reaction.

The kinetic method under consideration permits to determine not only the effective activation energy but also the product of k , k, , entering into formula (2.2.32). Since the constant k , has been determined independently in [6] by studying the anionic polymerization reaction in the presence of a direct activator, it is very easy to find k , . The temperature dependence of k , values, determined therefrom is shown in Fig. 2.2.14. The activation energy determined by the temperature dependence is 117 kJ/mol.

The set of values obtained for U , , U , and the value of the activation energy of viscous flow E determined independently [9], yield values for E, and E,, which may be evaluated through

formulae (2.2.40) and (2.2.41). Experimentally these values may be defined by plotting logq w l / T o at t = const. and logt vs l / T o at q = const. An example of this plot is shown in Fig. 2.2.15 from which it follows that E,, z 75 kJ/mol and E, = 400 kJ/mol . These values are practically consistent with those calculated theoretically.


\ AO. I I\

2.2 2.3

Figure 2.2.15: The dependence of viscosity of the reactive system at t=const on time of the achievement of a certain viscosity level (2) in the course of anionic polymerization of dodelactam with acetydiphenylamine as an activator.

Thus, the given example of rheokinetic analysis of anionic polymerization with a number of propagation centres shows that this approach makes it possible to define “correctly” the kinetic pattern and evaluate numerical values of constants, even for reactions characterized by complex kinetic parameters


2.2.7 Anionic Polymerization of E-caprolactam

The analysis of different hypothetical patterns of anionic polymerization shows that the predictions of the regular viscosity increase depending of the kinetic peculiarities of the reaction are different. The activated anionic polymerization has been carried out on a complex testing scheme, combining thermometric and viscometric methods of control in the presence of different activators. Description of the plant and activator characteristics are given in Section 2.2.4. Since the activator nature does not influence the general rheokinetic parameters of E-

caprolactam polymerization, but varies only numerical values of kinetic constants, we have chosen N-acetyl- &-caprolactam for our experimental analysis.

To compare experimental data with theoretical dependencies, we require numerical values of the rheological constants a and b . For the initial stages of the reaction, i. e., for regions of comparatively low MW of the polymer being formed, we may assume a = 1. To find the value of b we have to carry out an independent viscometric experiment, the results of which for the solutions of polycaproamide in E-caprolactam are presented in Fig. 2.2.16. It is evident, that the concentration dependence of the viscosity for these solution has a normal form and for the region of solutions with moderate concentration the value b z 6.

- 2

f, 0 f, 5 2nd po o V I 'Pcn . mass % /

Figure 2.2.16: Concentration dependence of the viscosity of PA-6 solution in its own monomer E-caprolactam at 170 "C ( I ) and 200 "C (2).

The viscosity dependence of the polymerizing mass for anionic polymerization of E- caprolactam activated by acetylcaprolactam on the duration of the reaction is presented on the logarithmic coordinates in Fig. 2.2.17. The exponent of power, defined by the slope of this dependence appeared to be close to 13. This is twice the value of a + b = 7, which would be observed, if the first or the second kinetic pattern discussed previously in Section 2.2.2 is satisfied. Using formula (2.2.16) for the temperature dependence of viscosity also yields overrated values of the effective activation energy. We may conclude therefrom, that the two patterns are not adequate to the real kinetics of anionic polymerization of &-caprolactam.


\ %

Q 0 - .$ .cu L "% -1 0, e

-2

-

-

I I I

2 4 6 r 3 t , min

Figure 2.2.18: The dependence of viscosity on time of the reaction in semi-logarithmic coordinates for anionic polymerization of E-caprolactam at the catalyst concentrations 0,s mol-% (a) and 0.7 mol-% (b) for direrent temperatures of the reaction: 150 "C ( I ) , 160 "C (2), 170 "C (3).


The results of data processing by formula (2.2.5 1) are given in Fig. 2.2.18. It is evident that the rheokinetic curves are well described by two exponential dependencies with different values of the constant 8. The change in slope is caused by the reaction transition from the heterogeneous to the homogeneous region. This is consistent with Fig. 2.2.19, where the inflection points of the dependencies lnq vs t (taken from Fig. 2.2.19) are marked on the diagram of the phase state of the system polycaproamide - &-caprolactam (see Sect. 4.3.2). The transition points are located within the region of amorphous incompatibility of phases limited by the binodal. The spread of these points may possibly be explained by variations in MW of the polymer being formed at the moment of transition, and of the high-molecular-weight polymer which has been used to plot the phase diagram.

Figure 2.2.19: The phase diagram of PA-6 solutions in caprolactam. The points under the dome of the binodal are the points of the inflection of the dependencies in Fig.2.2.18.

The activation energy enters the constant 0 and is defined graphically by plotting the dependence of 8 on the initial temperature To on the Arrhenius coordinates (Fig. 2.2.20). The activation energy appeared to be 71 kJ/mol which is practically consistent with the result obtained by other kinetic methods [3 1,321.


[k] . 102mol/s

2.2 2,3 2.4 1oS/r, K - l

From thermometric From rheokinetic data measurements

Figure 2.2.20: The temperature dependence of the coeflcient in formula (2.2.50) for different concentrations of the catalytic system:

0.3 mol-% ( I ) , 0.5 mol-% (2), 0.7 mol-% (3).

~~

4.45

6.23

The rheokinetic analysis permits not only to confirm the qualitative conclusion on the autocatalytic pattern of anionic polymerization of E-caprolactam, but also gives an opportunity to evaluate the main kinetic parameters of the reaction: The rate constant of the reaction k, and the constant of autocatalysis c. The constant k, is determined by formula (2.2.47) at values being very small. The autocatalytic constant may be defined through relationship (2.2.49) for different conditions of the reaction. The results of the constant c evaluation and its comparison with the data borrowed from [32] are presented in Table 2.2.1.

~~

25 22.5

19 15

Table 2.2.1 : The constants of autocatalysis c for anionic polymerization of caprolactam determined by direrent methods.

I 2.57 I 29 I 25 I

The tabulated data testify to satisfactory consistence of the constants obtained by different methods.


Since the activated anionic polymerization of caprolactam is influenced by the autocatalytic effect, then the comparison of the results obtained by different equations with autoacceleration, suggested for the description of the kinetics of this process [31-371, is of major interest. Equations similar to (2.2.35) and (2.2.45) and the expression obtained in [34] on the basis of the kinetic pattern are most commonly used to describe suggested kinetics of anionic polymerization.

(2.2.52)

where [MI is the initial concentration of a monomer; k , is the rate constant of interaction

between the centre of chain propagation and the monomer; k , is the rate constant of the reaction

or polymerization.

The evaluations by equations (2.2.35), (2.2.45) and (2.2.52) have been carried out in conformity with the conditions of the experiment described in [34], and they are presented in Fig. 2.2.21. As it is evident from this figure, the best conformity with the experiment is achieved by formulae (2.2.35) and (2.2.45) obtained on the basis of rheokinetic analysis of the process of activated anionic polymerization of E-caprolactam.

4 2 t , mln

Figure 2.2.21: Comparison of dependencies P(t), calculated by equation 2.2.35 (a solid curve) (2.2.45) (a dotted line) and (2.2.52) (dot-and-dash line). The points are the experimental data from [34]. To = 135.7 "C ( I ) ; 140 "C (2); 145 "C (3); 148 "C (4).


S. D. Lipschitz, C. W. Macosko, J. Appl. Polym. Sci., 1977, vol. 21,8,2009.

S. D. Lipschitz, C. W. Macosko, Polym. Eng. Sci., 1976, vol.16, 12,803.

C. C. Papanicolaou, S. A. Paipetic, J. Appl. Polym. Sci., 1976, vol. 20, 4,903.

C. W. Macosko, Brit. Polymer J., 1985, vol. 17, 2,239.

S . G. Kulichikhin, S. L. Ivanova, M. A. Korchagina, A. Y. Malkin, Dokl. AN SSSR, 1978, vol. 243, 3,700.

A. Y. Malkin, S. L. Ivanova, M. A. Korchagina, Vysokomolekulyarnye soedineniya, A, 1977, vol. 19, 10,2224.

S. G. Kulichikhin, S. L. Ivanova, M. A. Korchagina,V. G. Frolov, A. Y. Malkin, Matemathicheskoye modelirovaniye i apparaturnoye oformleniye polimerizatsionnykh pro- tyessov, Tezisy dokladov, Vladimir, 1979, p. 141.

A. Y. Malkin, S. G. Kulichikhin, S. L. Ivanova, M. A. Korchagina, Vysokomolekulyarnye soedineniya, A, 1980, vol. 22, 1, 165.

S. G. Kulichikhin, S. L. Ivanova, M. A. Korchagina, A. Y. Malkin, Poliamidy-81. Tezisy dokladov, Hlum, CSSR, 1981, p. 194.

S. L. Ivanova, M. A. Korchagina, Plasticheskiye massy, 1980,8,48.

M. A. Korchagina, S. L. Ivanova, Proizvodstvo i pererabotka plastmass i sinteticheskikh smol, 1981,2,9.

T. M. Frunze, S. L. Ivanova, V. V. Kurashev, V. A. Kotelnikov, L. A. Nosova, V. G. Frolov, L. A. Gordeyeva, M. A .Korchagina, L. I. Sherysheva, V. G. Zharkov, Obzol; M., NllTEChlM, 1975, vol.12 (24), p. 28.

A. Y. Malkin, S. L. Ivanova, V. G. Frolov, S. G. Kulichikhin,V. P. Begishev, Plasticheskiye massy, 1980,4,28.

S . L. Ivanova, S. G. Kulichikhin, 0. F. Alkayeva, N. M. Akimushkina,Y. P. Vyrsky, A. Y. Malkin, Vysokomolekulyarnye soedineniya, A , 1978, vol. 20, 12,2813.

S . G. Kulichikhin, M. A. Korchagina, S. L. Ivanova, A. Y. Malkin, Plasticheskiye massy, 1984,2,5.

E. T. Denisov, Kinetika gomogennykh khimicheskikh reaktsiy, Moscow, 1978, p. 367.

A. A. Berlin, S. A. Volfson, N. S. Evikolopyan, Kinetika polimerizatsionnykh protsessov, Moscow, 1978, p. 3 19.

S. A. Volfson, N. S. Enikolopyan, Raschet vysokoeffektivnykh polimerizatsionnykh protsessov, Moscow, 1980, p . 3 12.

G. V. Vinogradov, A. Y. Malkin, Reologiya polimerov, Moscow, 1977, p. 438.

A. Y. Malkin, A. E. Chalykh, Di&ziya i vyazkost polimerov, Metody izmereniya, Moscow, 1979, p. 304.

2.2.8 References


21) A. Y. Malkin, A. A. Askadsky, V. V. Kovriga, Metody izmereniy mekhanicheskikh svoistv polimerov, Moscow, 1978, p. 336.

22) A. Y. Malkin, A. A. Askadsky, V. V. Kovriga, A. E. Chalykh, Experimental methods of polymer physics, Moscow, 1983, p . 520.

23) A. Y. Malkin, B. M. Rendar, S. G. Kulichikhin, L. I. Sherysheva, Vysokomolekulyarnye soedineniya, A., 1978, vol. 20, 8, 1913.

Proizvodsmo i pererabotka plastmass i sinteticheskikh smol, 1978,8,49.

Plasticheskiye massy, 1979, 7,53.

24) B. M. Rendar, S. G. Kulichikhin,

25) M. A. Korchagina, S. L. Ivanova, S. G. Kulichikhin, A. Y. Malkin,

26) A. N. Solovyev, A. B. Kaplun,

27) V. N. Krutin, Kolebatelnye reometry, Moscow, 1985, p . 159.

28) A. Schimazoki, J. Appl. Polym. Sci., 1968, v01.12, 10,2013.

29) S. G. Kulichikhin, V. V. Maizeliya, B. M. Render, L. I. Sherysheva, A. Y. Malkin,

Vibratsionny metod izmereniya vyazkosti zhidkostey, Novosibirsk, 1970, p. 140.

Fizicheskiye svoistva vyazko-uprugikh polimerov, Sverdlovsk, UNTs AN SSSR, 1981, p. 64.

30) S. G .Kulichikhin, V. V. Maizeliya, A. Y. Malkin,

31) A. Y. Malkin, S. L. Ivanova, A. N. Ivanova, Z. S. Andrianova,

Novoye v reologii polimerov, Moscow, INChS AN SSSR, 1981,2, 158.

lnzhenerno-fizichesky zhurnal, 1978, vol. 34, 4,636.

32) A. Y. Malkin, V. G. Frolov, A. N. Ivanova, Z. S. Andrianova, Vysokomolekulyarnye soedineniya, A, 1979, vol. 21, 3,632.

33) A. Y. Malkin, S. G. Kulichikhin, V. G. Frolov, M. I. Demina, Vysokomolekulyarnye soedineniya, A, 1981, vol. 23, 1328.

34) D. I. Lin, J. M. Ottino, E. L. Thomas, Polym. Eng. Sci., 1985, vol. 25, 18, 1155.

35) W. Sibalp, R. E. Camargo, C. W. Macosko, Polym. Proc. Eng., 1983-1984, vol. I , 2, 147.

36) A. Y. Malkin, S. L. Ivanova, V. G. Frolov, A. N. Ivanova, Z. S. Andrianova,

37) Y. A. Kotelnikov, L. B. Danilevskaya, V. V. Kurashov, T. M. Frunze,

Polymers, 1982, vol. 23, I I , 179 1.

Vysokomolekulyarnye soedineniya, B, 1987, vol. 29, 5,365.


2.3 Free Radical Polymerization

2.3.1 Rheokinetics of the Process - Theoretical Considerations

The general rheokinetic principles of free radical polymerization of vinyl monomers are based on the analysis of the major chemical and kinetic principles of this process and on the results of experimental studies of the rheological properties of polymer solutions in their own monomer or in a chosen solvent. We assume immediately thereby, that the dependence of the reactive medium viscosity (and the rubbery modules) on the solution concentration and MW of the growing radical is the same as in the case of its deactivation.

The validity of this assumption has been confirmed experimentally by measurement of reactive solution viscosity, where the chain propagation has been deliberately terminated. Some appropriate investigations have been carried out for certain polymerizing systems. The results of measurements are presented in Fig. 2.3.1 and in Fig. 2.3.2 for solutions of PMMA (1) and PBMA ( 2 ) , respectively. Some experimental data pertaining to the solutions of PS in its own monomer will be considered later. Thus, the above assumption has been confirmed for classical objects obtained by free radical polymerization. Therefore we may regard the reactive mass in the process of free radical polymerization as a solution of the polymer being formed in the solvent (in its own monomer).

Figure 2.3.1: Comparison of rheological (a) and rheokinetic (b) data for

PM solutions in their own monomer. T = 50 "C. MW of the end product (at = 20%) 1.5 x106 g/mol.

The reaction path of free radical polymerization consists of a number of different processes, the main one being chain growth. However, concurring of other reactions is also possible. Nevertheless, it is important to identify the most important path of the process, which determines its rheokinetic regularities. Then the deviations from predictions following from the simplest kinetic pattern may be regarded as independent indications of phenomena not taken into account by the simplest kinetic pattern but essential for real polymerizing systems.

2.3 Free Radical Polymerization 61

I I I I I I

0 48 L6

Figure 2.3.2: Comparison of rheological (a) and rheokinetic (b) data for PBMA solutions in their own monomer. T = 60 “C. MWof the endproduct (MWxlQ-5moWg): 2.6; (1) ; 4.3 (2); 32 (3).

The simplest pattern of chain propagation in the course of free radical polymerization for the steady stage of the process running under homogeneous conditions yields the following well known equation of the polymer formation rate:

-- d [ M 1 - - (k,k?tnk,”*) [ I ] i n [MI dt

(2.3. I )

where k i t k , and k, are the rate constants of the initiation reactions of chain propagation and chain termination, respectively; [ I ] is the initial concentration of the initiator; [MI is the monomer concentration and f is the efficiency of initiation.

The apparent activation energy E of the process of polymerization specifying the temperature dependence of polymer formation rate d [ M I /dt is expressed as:

1 1 2 p 2

E = - E , + E - - E , (2.3.2)

where Ei, E, and E, are activation energies in elementary reactions of initiation, of chain propagation and chain termination, respectively.


This simple pattern yielding two major equations (2.3.1) and (2.3.2) pertains to the steady state, i. e. running at a constant rate, polymerization and it cannot by applied in the following three cases:

At profound degrees of conversion the reaction rate accelerates. This is the so-called gel effect, usually explained by the decrease of the rate constant in chain termination due to diffusion restrictions. The connection of this phenomenon with rheological properties of the reactive mass reflecting the rheokinetics of the process will be discussed later. Here it may be noted that the term “profound degree of conversion” is only arbitrary and means that this effect emerges at a certain achieved stage of the reaction.

The simplest kinetic pattern ceases to be valid for the reaction under heterogeneous conditions. Heterogeneity may initially occur in the reactive system, if, for instance, free radical polymerization runs in the presence of a filler (the pattern of a composite material formation with so-called “polymeric filling”). Heterogeneity may occur due to the fact that the polymer being formed is insoluble in the reactive mass. A case, where the reaction runs in a multicomponent system and phase inversion takes place at a certain degree of conversion, is similar to the described situation. It should be noted that the gel effect is another manifestation of heterogeneity, as it is also associated with the local heterogeneity of the growing chains concentration.

The pattern under consideration cannot be applied at very profound degrees of conversion, when the viscosity of a reactive system is very high, and the reaction passes from the kinetic region to the diffusion one, even if the reactive mass remains homogeneous.

All these “anomalous” effects may be interpreted in terms of equation (2.3. l), assuming that some constants appearing in it depend on the degree of conversion. Therefore the analysis of such “anomalous” cases is also possible if the dependencies of kinetic constants corresponding to more complicated conditions or free radical polymerization on the degree of conversion are known.

Let us express the degree of conversion P and the average length

is defined as:

of the chain of the polymer being formed in terms of the main kinetic constants of homogeneous free radical polymerization. The value of

(2.3.3)

where [MI is the initial monomer concentration, and [MI is the current monomer concentration (depending on time t ) . At t = 0 evidently [MI = [MI0 and P = 0.

Equation (2.3.1) yields the expression for polymerization rate, which permits to find [MI ( t ) and to express P(t ) . The corresponding final equation for P(t) has the form:

(2.3.4)


At a certain relation of constants, namely, at

formula (2.3.4) may be linearized, so that:

(2.3.5)

(2.3.6)

Consideration of the interval limited by equation (2.3.5) is of paramount significance, since it is exactly here that the assumptions used in the theory of homogeneous free radical polymerization are satisfied with the greatest probability. Therefore using formula (2.3.6) in rheokinetic analysis seems to be quite justified, though the limits of its application cannot be always predetermined: The formal limitation is the satisfying of inequality (2.3.5), which is valid if all the kinetic constants entering into it are unchangeable.

The average degree of polymerization N of the polymer being formed is calculated as follows:

(2.3.7)

where p ( t ) is expressed by equation (2.3.4). For values of kit << 1 formula (2.3.7) becomes a linear function and finally yields the following expression for N:

(2.3.8)

Now, in order to proceed to the kinetic equation one should remember that in case of free radical polymerization the concentration of the polymer being formed is equal to the degree of conversion: cp = p and therefore is defined by formula (2.3.6). Thus, there are two initial expressions: For the polymer concentration in the reactive mass and for the average MW of the polymer being formed.

Regarding the reactive mass as a solution, we shall further employ the usual expression for the dependence of viscosity on the concentration and the degree of polymerization:

= KcpbNa (2.3.9)

where K , a , b are specific constants.

As well known from a large number of investigations on rheological properties of polymer solutions, at least two ranges of cp values may be outlined. For low concentrations the value of the constant b is approximately 1 and for high concentrations of polymer solutions with flexible chains b varies between 5 and 7.

If we actually deal with high-molecular-weight compounds, then the constant a = 3.5.


Substitution of the expressions for cp ( t ) and into formula (2.3.9) yields the following final formulae for q ( p) and q (t) :

r l (P) = 8,Pb (2.3.10)

q(P> = e 2 t b (2.3.1 1)

The expressions for the constants 8, and 8, follow from the formulae obtained earlier for 8, and 8,:

a + b

8, = K[M];($? (ki[I] ,)(b-a)/2

(2.3.12)

(2.3.13)

Thus, theoretical considerations based on the analysis of the rheokinetics of the steady stage of homogeneous free radical polymerization, lead to some conclusions accessable to experimental testing. These conclusions are:

a) the dependence of viscosity both on time q ( t ) and on the degree of conversion q(p) is expressed by uniform formulae with the exponent b. In both cases b has the same value as the exponent in the dependence of viscosity on the concentration of the solution;

p) if the duration of the process is invariable, viscosity is proportional to the initial monomer concentration as 3.5;

y) if the degree of conversion is the same (p = item), the viscosity of the reaction mass and if the process

. In other words: With an increase of the depends on the initial concentration of the initiator as [I] duration is the same ( t = item) as [I] initiator concentration the viscosity decreases at p = item and increases at t = item.

( b - a) /2

It should also be mentioned that the formulae presented enable us to calculate directly the viscosity changes during polymerization, if the kinetic constants of the reaction as well as the constants K, a, andb, which enter the expression for the dependence of solution viscosity on concentration and MW, are known.

Now let us consider the temperature dependence of the viscosity of the polymerizing mass. Evidently, it is defined both by the temperature dependence of the proper viscosity of the solution and by the temperature influence on the kinetic constants of the reaction Ei, E,, and Et . All these dependences may be expressed by normal Arrhenius formulae in terms of apparent activation energy. Then we shall write E,, for apparent activation energy and Ech for the activation energy of a chemical process. Precise dependencies of apparent activation energy for the viscosity of the reactive mass follow from formulae (2.3.10) - (2.3.13). Here it should be noted that the pattern of the temperature dependence appears to be different if the comparison is performed at t = item or at p = item.


If we compare the values of viscosity at p = item, it follows from formulae (2.3.10) and (2.3.12) that

(2.3.14)

where complex Ech is defined as

(2.3.15) 1

Ech = E - - ( E i + E , ) . p 2

At t = item the expression for the temperature dependence of viscosity has the form:

E,, + ( a + b ) Ech - bEi I RT rl = KoexP { (2.3.16)

Hence the values of apparent activation energies E p (describing the temperature dependence of the viscosity of the reactive mass at the same degrees of conversion) and E, the same at the invariable duration of the process are found:

E p = E, + aEch (2.3.17)

El ( b - a ) Et = E,,+ ( a + b ) E c h - b E i = E n + ( a + b ) ( E p - - ) +- Ei (2.3.18) 2

Also of interest is the temperature dependence of p and t , corresponding to isoviscous states of the reactive mass. These dependencies are marked by apparent values of activation energies obtained from graphs of In p vs TI (the value E,,, p) or In t vs TI (the value of E,, t). Values for E,, and E,, , may also be obtained from formulae (2.3.10) - (2.3.13), and the final expressions have the form:

1 b

En, , = -- [ E , + ( a + b ) E c h - b E i ]

= - ! [E, ,+ ( a + b ) ( E - - E t ) 1 + - E i ] b - a b p 2 2

(2.3.20)

Thus, we may infer that the notion of apparent activation energy of viscous flow of reactive mass is ambiguous and depends on the method of determining the corresponding value (i. e. on the conditions of comparing viscosity values, taken at different temperatures). Therefore the apparent values of activation energy depend not only on the activation energies of viscous flow and of elementary chemical reactions (initiation, chain growth and termination), but also on the values of the constants a and b, determining the pattern of the viscosity dependence on the solution concentration and MW of the polymer dissolved.


2.3.2 Steady Stage of Polymerization - Experimental

Free radical polymerization has widespread practical applications in manufacturing many

large-scale polymers. Besides, many laboratory syntheses are carried out according to the free

radical polymerization pattern. It is only natural, Therefore that numerous investigations of free

radical polymerization were performed for most diverse monomers and kinetic constants are

known for many reactions. This enables us to implement rheokinetic analysis of the processes

of polymer formation by means of free radical polymerization, although there is not much

experimental data showing the general principles of viscosity change during vinyl monomer

polymerization. Nevertheless, such data are known for a number of basic polymers, and their

rheokinetic analysis permits to compare theoretical predictions with actual observations.

Polystyrene

Free radical polymerization of styrene is a classical example of linear polymer formation,

which is initiated by dissociation of a peroxide compound into two radicals. Numerous results

of viscosity measurements of polystyrene solution in its own monomer, i. e. functions for

q(q, Iv> have been reported [3]. According to the experimental data available, the dependence

of the form (2.3.9) with constants a = 3.5 and b = 5.5 is valid.

Measurements of the dependence q(p) for free radical polymerization of styrene "in block"

were reported in [4]. The experimental data obtained in this paper are presented in Fig. 2.3.3 for

the range of values p up to 65%, where (in the experiments under discussion) steady poly-

merization runs without the gel effect. It is seen that the dependence q(p) plotted on log-log

coordinates is described by a power equation and its exponent turns out to be equal to 5.5, which

is exactly consistent with theoretical predictions expressed by formula (2.3.10).

Figure 2.3.3: The dependence q(p) obtained for block polymerization of styrene.

T = 100 "C.


The shape of the dependence q(p) in Fig. 2.3.3 indicates that in this experiment the region of low degrees of conversion, where this dependence is expected to be not very strong, has not been taken into consideration. As it has been already mentioned it is a well known fact that the viscosity in regions of low concentrations is approximately proportional to q. For free radical polymerization this implies that the viscosity at low degrees of conversion should be proportional to p, followed later by a far stronger dependence pb . This more complex dependence q(p) has been described in [ 5 ] and is illustrated by Fig. 2.3.4. One may distinctly see the transition from the area of low values of p to that of high values; the exponent b changes thereby from 1 to 5 in the medium concentration region, and up to 40 at very high degrees of conversion, when the conditions are close to those of glass transition of the reactive mass and the content of the polymer formed is the solution is high. This effect, i. e., very high values of the exponent b at temperatures approaching the point of glass transition, when the latter is higher than the temperature which causes the viscosity change in the solution, is well known for concentrated solutions of polymers [6]. Fig. 2.3.4 presents, in fact, its rheokinetic analogue.

3

f

- f

-3

Figure 2.3.4: Dependence of the initial Newtonian viscosity on the degree of conversion

during styrene polymerization.

The initiator is BPO. The regulator of the MW is lauryl mercaptan.

MWof the formedproduct ( M W ~ l O - ~ m o l / g ) : 4.4 ( I ) ; 5.0 (2); 5.6 (3).

Viscosity is one of the key rheological parameters of liquids. Indeed, it is viscosity and associated processing parameters of the reactive mass (e. g. agitator torque) that permit to control the state of the reactive mass and the degree of conversion. However, a polymer solution has a wide range of rheological properties and the coefficient of normal stresses 5 as a measure of solution elasticity at shear flow, undergoes the most pronounced changes. Thus, if the viscosity is proportional to qb the coefficient of normal stresses 5 is proportional to q2b. Particularly, for polymer solutions experiments yield 5 - q"" which is a very strong dependence. Therefore measurements of normal stresses (of some other parameters of rubbery properties of solutions) might appear to be a very effective method of rheokinetic control of the processes of poly-

68 Chapter 2: Rheokinetics of Lineur Polymer Formation

merization. However, this approach has not been applied so far for analysis of free radical polymerization, except some qualitative observation associated with the estimation of elasticity reported in [ 3 ] .

One of not very numerous examples of applying the data of dynamic mechanical spectroscopy for rheokinetic investigations of free radical polymerization is presented in [7] (though it does not deal with polystyrene). In this paper the dependencies of the rubbery modulus on the duration of polymerization of ionogenic monomer N,N-dimethyl-N,N-diallylammoniumchloride in water solutions were measured. Typical results of measurements are shown in Fig. 2.3.5. In fact, it is a rather typical example of the rubbery modulus increase during net formation (see Sect. 3 ). In this case we actually deal with a fluctuating engagement net.

E; 20 40 60

t, min

Figure 2.3.5: Dependence of rubbery modulus G' on the time of polymerization of N,N-dimethyl-N,N-diallyl ammonium chloride in aqueous solutions at 333 K. o = 0.63 s-'.

Monomer concentration [moUl]: 3.0 (1); 3.5 (2); 4.0 (3); 4.5 (4); 7.0 (5).

The most interesting result of this study, which reveals new opportunities of the dynamic spectroscopy, is the coincidence of the moment of reaching the maximum of the mechanical loss angle tangent and the maximum in the dependence of the reaction rate on time. The results of measuring G' may be used to estimate the size of a polymer segment between neighbouring engagements. Otherwise, the results of G'(t) measurements do not give direct information on the kinetics of the reaction, though they permit to qualitatively estimate the physical structure of the reactive mass by change in its viscoelastic properties.


Acrylates

Acrylates, especially polymethylmethacrylate (PMMA), have been the subject of investigation in a number of studies which permitted a wide and systematic rheokinetic analysis of the formation processes of these polymers.

Of particular significance is the fact that independent measurements of viscosity for acrylate solutions in their own monomers have been conducted (e. g. see Fig. 2.3.1 and 2.3.2). This yields a possibility of quantitative rheokinetic analysis of free radical polymerization. For instance, for the solutions of PMM in its monomer in regions of sufficiently high concentrations the following expression for the function q(q, in a precise form representing formula (2.3.9) has been obtained:

logq = -31.0 + 12.8log~+3.410gM (2.3.21)

The “free” constant here pertains to 50°C, and the exponent values are: a = 3.4, and b = 12.8. The viscosity values, evaluated by this formula are denoted by poise Pas.

Experimental data obtained in the studies of polymerizationof PMMA “in block” at different contents of the initiator are presented in Fig. 2.3.6 [9]. It is seen that the slope of the straight lines plotted on log - log coordinates equals 13 which is in good conformity with the coefficient in formula (2.3.21) and the ramp of the straight lines does not depend on the initiator concentration, which also agrees with theoretical predictions.

4 r

Figure 2.3.6: Dependence of the initial Newtonian viscosity on the degree of conversion during MMA polymerization. T = 70 “C. Concentration of the initiator in wt-%: 0.02 (1); 0.05 (2); 0.75 (3); 0.1 (4); 0.15 (5).

As has already been discussed for polystyrene, the invariability of the exponent b values is valid only for solutions of high concentrations, i. e., for comparatively profound degrees of conversion. At the same time, if a wider range of conversion degrees and polymer concentrations in a reactive medium is under consideration, the dependence q(p) really appears to be more complicated, as it is shown in Fig. 2.3.7 for PMMA and in Figs. 2.3.8 and 2.3.9 for another


acrylate - PBMA (Fig. 2.3.8 [9] and Fig. 2.3.9 [lo] under different conditions of poly-

merization). The identity in viscosity change with increase of the degree of conversion is also

confirmed by the data obtained for the other two polymers, given in Fig. 2.3.10 [ 1 13.

-3

-0.4 0 44 48 42 46

h+p [wf.%?

Figure 2.3.7: Increase of initial Newtonian viscosity (the viscosiv of monomer q, is excluded) during MMA polymerization. T = 70 "C.

The ratio of concentration of initiator/regulator of the chain lengh (wt.%): 0.2/0.3 ( I ) ; 0.2/0.2 (2); 0.2/0. I (3); 0.2/0 (4).

Figure 2.3.8: Increase of initial Newtonian viscosity of the reactive mass during BM polymerization, T = 70 "C. Initial concentration of the initiator [I10 in wt %:

0.05 ( I ) ; 0.1 (2); 0.2 (3); 0.8 (4); 1.0 (5).


a 2 %

a

- 2

Figure 2.3.9: An example of the division of the dependence obtained during kinetic investigation of BM polymerization, into three parts with diserent values of index b. T = 60 "C. MW values of the polymer formed (MWxlCi6moVg): 0,12 (I); 0.29 (2); 0.5 (3); 0.8 (4); 1.3 (5); 1.6 (6); 5.0 (7).

Figure 2.3.10: Dependence of the initial Newtonian viscosity on the degree of conversion during polymerization of pyrrolidone (A) and octyl methacrylate (curves 1-3). T = 70 "C. MWof the formingpolymer (MWxlC'moVg) : polypyrrolidone: 2.10; polyoctylmethacrylate: 4.0 (1); 5.7 (2); 7.0 (3).


Polymer

PMMA

Evidently, in Figs. 2.3.7 - 2.3.10 we observe a general case of concentration dependence of viscosity, this concentration comprising two regions:

1.

2.

with low concentrations, when p = I and

with high concentrations, and p far exceeding 1.

Below we have tabulated the values of p defined by rheokinetic measurements (the first figure of the fraction) and by measuring viscosity is stable solutions (the second figure of the fraction [9]):

b values for regions of

low concentrations high concentrations

1.011.0 12.8113.00

I PEMA I 0.81- I 4.21- I I PBMA I 0.910.85 I 4.013.9 I I PORMA I 0.810.8 I 4.514.4 I I ps I 0.810.8 I 5.515.5 I

It is obvious that in all cases the rheokinetic data are in good conformity with the results obtained from independent investigations of viscous properties of corresponding polymer solutions.

Let us now consider the effect of the initial concentration the initiator on the viscosity change of the reactive mass. According to theoretical considerations the dependence q( [I] o) at p = item should be a power function with the exponent equal to - a /2 , i. e. -1.7. The corresponding experimental data are presented in Fig. 2.3.1 1 for two reactive systems. As one can see, the slope of straight lines is actually close to 1.7, that is, the theoretical predictions are satisfied here, and the rheokinetic exponent agrees very well with that obtained from the dependence of solution viscosity on the MW of the polymer.

4 r

2

-2 - { 0

eo2 D10 imp. %I Figure 2.3.11: Dependence of the initial Newtonian viscosity on the initial

concentration of the initiator during polymerization of MMA at T = 7 0 "Cand p = 16 wt-% ( I ) and BM at T = 60°C and p = 26 wt-% (2).


The final group of experimental data, consistent with general theoretical considerations, comprised the results of measuring the temperature dependence of viscosity of the reactive mass.

fo5/T, K-'

Figure 2.3.12: Dependence of MW of the forming PM on temperature of polymerization

The free radical polymerization theory permits to define E,, through temperature dependence of MW of a forming polymer. The corresponding experimental data for PMMA are given in Fig. 2.3.12 [9]. From these data follows:

Ech = Ep - - ( Ei + E,) = -48 kJ/mol

The following values entering E,, are reported in the literature [ 121:

Ei = 130kJ/mol E - - E =20kJ/mol

[I10 = 0.2 wt-%.

1 2

1 p 2 '

which finally yields a value, close to Ech . Fig. 2.3.13 shows an experimentally measured dependence

nates at p = 18 % w/w. - -8 ?

$ 3 -&

6-

which finally yields a value, close to Ech . Fig. 2.3.13 shows an experimentally measured dependence

nates at p = 18 % w/w. - -8 ?

$ 3 -&

6-

of the Arrhenius coordi-

2. a 3.0 x 2

10 T CK-']

Figure 2.3.13: Dependence of the initial newtonian viscosity of the reactive mass at p =I8 wt-% on temperature during polymerization of MMA. [I30 = 0.2 wt-%.


This permits to define the value of Ep which, according to the plot, is 145 kJ/mol. The value for PMM solutions in its own solvent in the mean composition region according to direct calculations is 20 kJ/mol. Hence, calculations by formula (2.3.17), using the value of Ech borrowed from literature and equal to 45 kJ/mol, yields Ep = 173 kJ/mol, which agrees qualitatively with the value defined by direct measurements.

The dependence is plotted in Fig. 2.3.14, where p values have been chosen at the same viscosity level (in this case 500 Pas ). Direct measurements yield the value E,,, = 12 kJ/mol

whereas direct calculations of this value by formula (2.3.19) lead to the value of 13 kJ/mol. This is sufficiently good consistency to confirm the validity of theoretical formulae for the apparent activation energy.

Figure 2.3.14: Temperature dependence of the degree of conversion corresponding to the

same level of viscosity of the reactive mass during polymerization of MMA.

[I]() = 0.2 wt-%.

When discussing the experimentally observed dependence q(p) for styrene poly- b merization, we have noted that this dependence, expressed in terms of power equations q - cp

may consist of three regions:

a) when the degree of conversion is low, b, = 1,

p) when the degree of conversion is profound, b = 5 ,

y) when approximating the glass transition point, b rises drastically.

The latter case is possible if the polymerization temperature is higher than the glass transition point of the forming polymer. From the kinetic viewpoint this pattern is entirely identical to that of concentration dependence of viscosity of polymer solutions [6] . According by, we may determine “critical” (limiting) values of conversion degrees p:, and pz:, separate regions of the dependencies q(p) with different values of the exponent b for rheological curves as well [ 101. In fact, these regions correspond to different structures of solutions and, consequently, to different manifestations of the whole complex of their rheological properties.


Interconnection between the MW of the forming polymers and p:, and p,*:: values for polymerizing acrylates is shown in Fig. 2.3.15 [ I 11. As in case of polymer solutions, the interconnection between M and p,, may be expressed in terms of formulae of an exponential form: p,, Ma = const, values of a depend thereby to a certain extent on the nature of the polymerizing monomer, that is, this formula is not universal.

I 45 r.0 f. 5 2.0

Figure 2.3.15: Interconnection between p*,, (a) and p:: (6) and MW, established in the

course of rheokinetic anlysis of polymerization of methyl - ( I ) , butyl- (2) ethyl. - (3), hexyl- (4), octylrnethacrylates (5).

The preceding considerations proved the theoretical conclusions concerning some general principles of free radical polymerization to be true. That permits to state a general problem on evaluation of the whole viscosity dependence on time q( t ) or on the degree of conversion q(p) according to formulae (2.3.9) - (2.3.13). The results of such evaluations for MMA polymerization "in block" are presented in Fig. 2.3.16 and 2.3.17 [9]. The efficiency of the initiator decomposition was assumed to be equal to f = 0.8 for all values of [I10 which bears some uncertainty, it possibly being the reason for a significant (but not drastic) deviation of experiment and evaluated curves in Fig. 2.3.16 for q(p).


f6 r

4

t . 10-3, s

Figure 2.3.16: Comparison of calculated (dashed lines) and experimental (solid lines) dependences q ( t ) during polymerization of MMA. [I10 wt-% : 0.02 (1); 0.05 (2) 0.075 (3); 0.1 (4); 0,15 (5).

n b 20

/3, W t %

Figure 2.3.1 7: Comparison of computed (dashed lines) and experimental (solid lines) dependences q( p) during polymerization of MMA. Labelling of the curves as in Fig. 2.3.16.


Another example of rheokinetic analysis application is shown in Fig. 2.3.18, where the results of theoretical calculations are compared with experimental data for EM. As one can see, on the whole, theoretical evaluations satisfactorily agree with experimental data in all the cases considered. It should be noted that such an agreement is observed up to the most profound degrees of conversion, of about 65%. Thus, the entire range of p values may be regarded as a homogeneous steady stage of polymerization.

6

4 8 f2 i6 20 p , w t . %

Figure 2.3.18: Comparison of calculated (dashed lines) and experimental (solid lines) dependencies q(p) during polymerisation of EM. T = 70°C. MW of the formingpolymer ( M W ~ l Q - ~ r n o V g ) : 3.4 ( I ) ; 4.5 (2); 8.0 (3); 10 (4); 20 (5).

Finally, we may state that the rheokinetic approach to the analysis of viscosity change under homogeneous free radical polymerization provides correct correspondence to the anticipated pattern of viscosity change and permits to predict its rise due to main determining factors - time, monomer and initiator concentration, temperature. This enables us to solve not only the direct, but also the inverse problem, i. e., to estimate kinetic parameters of the process on the basis of measurements of rheological properties of the polymerizing mass.

The presented approach to the evaluation of rheokinetic curves is completely based on formula (2.3.9). However, as we have already indicated, considering a wide range of conversion degrees (and solution concentrations) one should take into account numerical change of constants at the critical point. At the same time, the choice of the initial rheological correlation is not critical. In [ 131 it is reported that to evaluate rheokinetic curves of viscosity increase in the course of free radical polymerization a more general relation for the dependence was used which is effective for the entire concentration range. This relation permits to take into account the change of rate constants of elementary reactions at higher degrees of conversion. This approach permits to obtain the dependence q(t) for the entire range of values and to use experimental curves for q ( t ) as a starting point for the solution of the inverse problem, that is, for determining the kinetic constants of elementary reactions.


2.3.3 Viscosity of the Medium - Its Role.

In the experiments described above the viscosity of the reactive mass changed on an enormous scale although free radical polymerization occurred under homogeneous conditions and the viscosity increase up to rather high degrees of conversion was described by simple rheokinetic equations. All this means is that, for the cases considered, the values of the main kinetic constants did not depend on the viscosity of the medium. Generally speaking, one cannot expect this to be valid for any conditions of polymerization, even if the process remains homogeneous and the gel effect is not observed. The validity of rheokinetic constants is limited by diffusion, which is the most natural boundary in the situation, when the diffusion rate of a radical of monomer is commensurable with or less than the rate of an elementary event of the reaction, causing it to pass to the stage controlled by diffusion.

There is a well-known rough estimate of boundaries of the diffusion-controlled phases, based on analysing the behaviour of interacting particles, which are assumed to be globules. This limiting value of the rate constant defined by the transfer is expressed as:

k, = 8RT/(3x1O4q) (2.3.22)

where k, is expressed in - * R = 8.31~10 - erg , and viscosity q in Pas. mobs ' K.mol

The monomer viscosity is customarily close to 5 ~ 1 0 - ~ P a s (0.5 op). Then k, is of the order of 10" (mol.s), which is a span that exceeds by several orders of magnitude the common values of rate constants for chemical reactions for free radical polymerization. This means that diffusion limitations are insignificant at the beginning of the reaction which proceeds in a purely kinetic regime. However, when the viscosity rises by 4-5 decimal orders in the course of the reaction and a corresponding decrease of k occurs , the situation may totally change and it is the diffusion transfer that becomes the limiting phase of k, the process on the whole.

When discussing this problem one should distinguish macroviscosity (or viscosity) of the reactive mass, measurable by any viscometric methods, and micro-viscosity, characterizing the transfer rate of molecular-size particles. From a simple molecular-kinetic model for low molecular liquids it follows that the viscosity q, is connected with the diffusion coefficient D by a well known relation:

6kT rl, = 7 (2.3.23)

Even if to take into account that this formula is true only for globular particles and to generalize it for anisodiametric particles, only the numerical coefficient in formula (2.3.23) will change, and the linearity of the relation between q, and D-' will remain. The situation changes drastically with high-molecular compounds the viscosity q of which is determined by the movement of the whole chain, whereas for diffusion of low molecular compounds movements of chain segments is sufficient. Therefore in the case of polymers the notion of micro-viscosity q, - defined by the coefficient of diffusion and its transformation by formula (2.3.23) into q, - and that of macro viscosity q are at variance with each other.


1 7 F .h

3 4 5 6 eo3 M

Figure 2.3.19: Comparison of macro- (q) and microviscosity (11,) underflow and

diffusion of equiconcentrated solutions of B in N-methyl pyrrolidone.

T = 25°C. Polymer content in the solution, %:

1 ( 1 ) ; 5 (2); 10 (3); 30 (4); 50 (5); 90 (6); 100 (7).

A vivid example illustrating this statement is the result of parallel measurements of 11 and q, in PB samples with narrow MWD and different MW (Fig. 2.3.19 [14]). In this case the following dependencies are valid:

q = K,[MIa’

where constants K, - K, and a, - a2 do not coincide. Especially pronounced is the difference in values of a, and a2: while a, = 3.5, as is usually observed for polymers, a2 is only a little more than 1, that is, not only values of q and qm differ, but also their dependence on the chain length.

qm = K2[MIaZ

This distinction leads to a situation that - provided for instance measurements of reactive mass viscosity are taken by classical rheological methods - the change in the run of the dependence q(q) always occurs at a certain concentration of the solution qcr, whereas no critical point exists in the concentration dependence of microviscosity. This fact has been established, for instance, by measuring microviscosity through fluorescence in the course of VT polymerization [ 151. General ambiguity of the term “microviscosity“, which depends on the method of measurement and particularly on the size of the “sample” added to the solution (microviscosity of the solution is estimated by movement of this “sample”), has been reported in several works [ 14-16]. However, the question of which kind of viscosity determines the rate of elementary processes during free radical polymerization remains without a definite answer.

80 Cliapler 2: Rheokinetics of Linear Polymer Formation

At the same time, some systematic studies have been carried out to compare directly the viscosity of the reactive solution (macroviscosity) with elementary kinetic constants. Given the ambiguity of choosing a viscosity predominantly influencing the rates of elementary chemical reaction, the results reported should be rather regarded as qualitative estimates illustrating the role of viscosity of the medium.

In the survey of earlier studies (before 1972) it has been postulated [17] that the viscosity of the medium may influence all elementary constants of free radical polymerization. Thus, viscosity influences the constant of initiation rate ki and the efficiency of initiation$ This effect is associated with diffusive (i. e. caused by viscosity) hindrance during decomposition and consequent free movement of formed radicals in the reactive medium.

The data of [I81 may be referred to as an example illustrating influence of the reactive medium viscosity on the initiation rate. Peroxides of different structure were used as initiators. It has been shown thereby that due to the nature of peroxide the reduction of both the decomposition rate constant and the efficiency of initiation are observed with the medium viscosity increase. Even a slight increase in viscosity led to more than two times reduction of the rate constant of benzoyl peroxide decomposition [ 171. Though initiators of one type have been used (peroxides) the effect of the medium viscosity manifested itself in different ways. Later it has been pointed out [18,19] that the possibility of the viscosity influence on the rate of initiation is associated with the particular mechanism of radical formation. Thus, the viscosity of the medium did not influence the decomposition rate of azobisisobutyronitrile which is frequently used for model reactions to study free radical polymerization, but it influenced significantly the rate of free radical formation during decomposition of N-nitrophenylazotriphenylmethanate [ 191. It should be noted that the reduction of the rate constant occurred to a lower extent than the viscosity increase of the reactive mass. This demands the use of the conception of viscosity as the factor determining the diffusion transfer in a viscous liquid.

Later, however, many authors dealing with the dependence of kinetic constants of elementary chemical reactions on viscosity of the reactive medium came to a conclusion that the rates of initiation and chain propagation (i. e. constants ki and k p ) do not depend, in a general case, on viscosity, whereas the constant of termination rate k, decreases starting from a certain level of viscosity [21,22]. Thus, it has been found in [21] during MMA polymerization in various solvents, the viscosity of which changed by two orders, that k, decreases in inverse proportion to the viscosity of the mediumq so that the product of k,q remains constant. It would be only natural to infer that k, is proportional to the diffusion coefficient D (since D - q-'), and diffusion mechanism determines the termination rate of the growing chain from the very beginning of the polymerization.

In a more general case, however, it is difficult to imagine that diffusion hindrance plays a significant role in the initial low viscous medium, so the case when k, starts decreasing after reaching a certain level of the viscosity of the reactive medium seems more probable.

Indeed, in a systematic investigation of the rate of MMA polymerization initiated by azobisisobutyronitrile, in media of different viscosities, it has been shown that at starts decreasing with viscosity increase only at q > 10-2Pas [22]. The result is still more interesting


due to the fact that, first, all the experiments have been carried out within the medium of the same composition, and the viscosity has been varied by introducing high-molecular PMMA into the reactive medium, that is, the chemical composition of the medium was the same and could not influence, per se, the rate of the reaction. Second, it has been vigorously shown for the investigated system that both ki and k , remain constant throughout the varied range of viscosity values, thus the observed changes in the reaction rate are due to the decrease of the constant k , . In the cited paper it has been shown that k, remains constant at viscosity variations in low concentration regions up to 100 times (from up to 10-2Pas). At further increase of viscosity by 1000 times, k, decreases only by 30 times, that is, to a far smaller extent than it should have been observed according to the results presented in [21], which suggested that k, was to decrease to an equal extent with the increase of the medium viscosity.

A group of Japanese researchers have carried out a systematic investigation on the effect of the medium viscosity on the rate of polymerization (both thermal and photosensitization polymerization) of MMA [23]. They varied the solvent in the system monomer/solvent, varying thereby the viscosity. As in other investigated cases the constants ki and k , remained unchanged, whereas the constant of the termination rate k, changed approximately proportionally to inverse values of viscosity of the system monomer/solvent. This is a rather convincing result though the data cited earlier from Brooks on MMA polymerization in the presence of a polymer in the initial region covers a range of viscosity change by 100 times, i. e. the range is far wider than that reported by the Japanese authors. No conformity is observed between these results and the data discussed in Section 2.3.2, pertaining to homogeneous polymerization of MMA, where no specific effects caused by viscosity increase of the medium due to PMMA formation are observed.

Inconsistency of the reported experimental data concerning the viscosity influence on the rate of free radical polymerization may probably be explained by the fact that variation of the solvent nature may affect the elementary constants of chain propagation and, especially, termination rate not only due to viscosity changes, but also due to other factors, e. g. formation of complexes of reagents with each other and with the solvent, formation of hydrogen bonds, etc. All this is most characteristic for polar monomers [24]. Therefore the true reasons for the observed change in polymerization rate and for the viscosity change compared with the simple anticipated model of a free radical process are not always evident. Obviously the viscosity increase of the reactive mass is not in itself indifferent to the kinetics of the process and must lead, in a general case, to some deviations from the simple relations, described in Section 2.3.1. If this does not occur, it may mean that the corresponding effects are insignificant and may be ignored. However, if systematically advancing deviations from the relations anticipated by a simple rheokinetic model are detected, we can estimate therefrom the role of the viscosity of the medium in elementary processes of free radical polymerization in various cases.

In fact, most of the inconsistencies of the experimental results discussed above might be discarded, if the difference between micro- and macroviscosity are taken into consideration. Naturally, it may be done only qualitatively, since nobody has actually measured the micro-


viscosity of reactive media, and, specifically, with regard to the estimation of the mobility of reactive particles. In terms of qualitative evaluation it should be borne in mind that the relation between microviscosity q, and the viscosity of the medium q is approximately q, - qln [20].

Then, provided that q - [MI 3’5, we obtain that q, increases a bit faster than M’ which agrees with the experimental data, presented, for instance, in Fig. 2.3.19. It follows therefrom that the diffusive mobility decreases approximately proportionally to the chain length.

Although the diffusion hindrances, caused by viscosity increase, may influence various elementary reactions of free radical polymerization, it is not clear in advance, whether this effect is to be observed or not in each particular case, and what is the relation between the viscosity change of the reactive mass and its chemical composition with regard to various peculiarities of particular chemical reactions.

2.3.4 Gel Effect

It was noticed long ago that a sharp increase of the free radical polymerization rate is observed at a certain degree of conversion. This phenomenon has been called “gel effect”, which means that at a certain level of polymer concentration, gel appears in the reactive mass. In reality, however, the gel effect never concurs the formation of the real gel, that is, of the non- soluble fraction of a cross-linked (network) material, which may be separated from the rest of the mass. At the same time, reaching the gel point is by all means connected with certain structural transformations of the reactive solution. Rheological properties of the solution are very sensible to these transformations. It would be only natural, therefore, to connect the changes in rheological properties of solutions with the moment of reaching the gel point.

It has been suggested in [25-271 that reaching the gel point is caused by fluctuation network formation in the reactive solution, which restricts the mobility of macromolecules. This solution structure appears in an equiconcentrated solution of reaching a certain critical chain length N,, or in the solution of a given polymer with the increase of concentration cp up to a certain critical threshold qCr.

Reaching boundary conditions at rheological or rheokinetic measurements is expressed in a sharp change of the viscosity increase rate, from the form close to linear to the power form N a at cp = const or cp at N = const, where a = 3.5 and b = 5 to 7. This form of the dependence q ( N , cp) is well known in the rheology of solutions and has been observed not once in rheokinetic investigations (as was discussed earlier). It is essential, however, to compare the observed course of the dependence q ( N , cp) with the conditions of reaching the gel region estimated independently.

b

It should be noted that the problem of correlating the shape of the rheokinetic curve and the conditions of the gel effect development is complicated by the fact that in reality the rheokinetic dependence q(p) - equivalent to the dependence q(cp) for many polymers obtained by free radical polymerization - appears to be more complicated than anticipated by a simple model of fluctuation network entanglement formation. It has been already shown above in Figs. 2.3.7 to 2.3.10, and we have indicated that a sharp increase in viscosity occurs when index p )> 5, corresponding to the conditions of glass transition. Therefore the dependence q(p) may be


Polymer

PS

presented in a general form as it is shown in Fig. 2.3.20 on the curve consisting of three sections separated by “critical values” of P*,, and Pz:. Characteristic values of index b in the three parts of the curve may serve as an example for some typical polymer solutions in their own monomers.

Values of b at: Reference

P < P * P* < P q : P>P,*;F = 1 5.0-5.5

Figure 2.3.20: Schematic illustration of a typical dependence q(P) or q(q) with two

critical points.

PBMA

Table 2.3.1: Values of index b in separate sections of the dependence q(P)

1.1 f 0 . 3 3.2 k 0.5 1 2 k 3 I ~ 9 1

PM I 0.7kO.l I 3.5k0.5 I > 7 I P S I I


centrations and of the points p:, and pz: depend to a great extent on the MW of the dissolved polymer (or a polymer being formed in the course of polymerization), so there are no absolute values available. Furthermore the viscosity values at critical points may be quite different, since their values are connected not with a certain level of viscosity, but with the formation of some macromolecular structure in the solution.

Taking all these considerations into account, we may attempt to compare the characteristic points of the rheokinetic curve with the moment of the appearance of the gel effect. Consecutive comparison of the rheokinetic data with the polymerization rate has been accomplished in the studies of MMA and BM polymerization [27]. In the case analogous to that presented in Fig. 2.3.9, varying the initiator (azobisisobutyronitrile) concentration made it possible to alter the conditions of gel effect appearance and the position of the critical point systematically. As it has been shown, the value of the degree of polymerization and of the degree of conversion are related as:

(2.3.24)

where K and a are the constants characteristic for the given system “polymer/solvent”.

It is essential to know that the values of N and cp are related in the same way (and by the same constant values) at the point of transition from dilute solutions to those, where a fluctuation engagement network emerges, i. e., it is claimed that the gel effect is associated with the concentration q* (or with the degree of conversion p* ). It is quite obvious from Fig. 2.3.9 that the values of p* may be reached at different levels of viscosity, so that the gel effect cannot be compared with the isoviscous state of the reactive medium. The same is shown is a more vivid form in Fig. 2.3.21, where the dependencies of viscosity at critical points on the solution concentration at these points are presented for polymers of different Mw.

Figure 2.3.21: The values of initial Newtonian viscosity of PBMA solutions in their monomers corresponding to critical points of the rheokinetic curve.


The results presented above seem to refute the idea of the isoviscous state being the general condition of gel effect appearance. However, other data are also reported. Thus, in [30] systematic and rheokinetic investigations of MMA suspension polymerization initiated by BPO in an aqueous medium have been carried out. Varying the M W regulator content (dodecyl mercaptan) was successfully used to change the rheokinetics of the polymerization process. The characteristic results of this investigation are graphically shown in Fig. 2.3.21 in the form of kinetic curves.

The experimentally observed deviations of the solid lines from the dotted line, which corresponds to the kinetics of polymerization with unchanged values of elementary reaction constants, present in themselves the auto acceleration effect ("gel effect"). As it is shown in Fig. 2.3.22, lower concentration of the chain growth regulator and higher MW of the formed polymer, initiate earlier autoacceleration (i. e. at a lower degree of conversion p).

t Figure 2.3.22: Schematic illustration of the time dependence of the degree of conversion

for a reactive system with different contents of the regulator of the chain length. The arrow shows the direction of the increase of concentration of the regulator. The dashed line is a theoretical dependence corresponding to the path of homogeneous polymerisation.

Discussing the data summarized in Fig. 2.3.22, it should be borne in mind that the rate constant of polymerization in the initial phase (before the solid lines start deviating from the dotted line) was the same both for block and suspension polymerization, i. e., it is reasonable to regard each suspended particle where polymerization takes place, as a micro block reactor. Therefore instead of measuring the viscosity of each polymerizing particle, which is hardly possible, we may measure the viscosity of the medium on the whole. This enables us to estimate the values of viscosity q at degrees of conversion p, corresponding to the deviation or solid lines from the dotted line in Fig. 2.3.22, or to the transition of the system to the regime of autoacceleration.

g

The values q, have been determined after stopping of the reaction (by adding hydroquinone), which permitted to obtain rheokinetic information on the polymerization run. The direct comparison of the dependencies q(p) and the polymerization kinetics have demonstrated that, when altering the concentration of dodecyl mercaptan a rather wide range (from 0 up to 1 .O weight percent), the viscosity values in the critical point remained practically constant, equal to q, = 60 f 5 mPas (at 25 "C and the shear rate being lx104s-').


Although this experiment may be subjected to criticism, for instance, the equivalence of

block and suspension polymerization and application of a rather high shear rate at viscosity measurements may seem doubtful, yet in a general case, the interpretation of the results presented above is problematic due to more general considerations. This is due to weaknesses of the idea concerning the comparison of the kinetics of a chemical reaction with the microscopic rheological properties of the reactive medium. As it has been already discussed, the predominant role here belongs to microscopic mobility, only indirectly associated with the viscosity of the solution. It is probably the difference in micro- and macroviscosity that explains the discrepancy of the conclusions in the cited papers concerning the problem whether the occurrence of the gel effect corresponds to the isoviscous state of the reactive mass or not.

Naturally, the factors promoting the formation of intermolecular contacts, must lead to intensified gel effect on to its occurrence at lower degrees of conversion. This idea was confirmed in [27] by kinetic data on MMA polymerization carried out in the presence of ZnC1,. Adding this salt led to the establishment of coordination bonds between macromolecular chains.

This suppressed segmental mobility of chains and led to a decrease in values of p at which the gel effect was observed. Thus, in principle, the way of restriction of macromolecular mobility is not essential. This is probably the manifestation of the similarity in the rheological behaviour of solutions, where bonds of different types are formed and it is essential here that the molecular mobility decreases to an equal extent due to macromolecular interaction [31].

Although qualitatively the correlation between the degree of conversion corresponding to gel effect occurrence and characteristic peculiarities of rheological (or rheokinetic) curves looks rather obvious, it would be useful to discuss in detail some particular figures. It appears necessary to specify the sense of the critical points p:, and p::. In estimation of the moment of gel effect emergence some uncertainty is possible, since the gel effect development is not dramatic. Thus, it is suggested in [32] to regard the minimum point in the dependence of the reduced rate of polymerization on the degree of conversion as the moment of gel effect formation. We shall label this point by p:. Another definition of the moment of gel effect formation is associated with the transition to the increasing part of the concentration dependence of polymerization rate. This point will be labelled by py. It is evident thereby that

Py > P*k. Now let us compare quantitatively the critical degrees of conversion, obtained by rheo-

logical (p: and pT*) and kinetic ( p: and fit*) measurements. The corresponding data, based on the results of investigations in [ 101 are presented in Table 2.3.2 for a few methacrylates. It is evident, that a rigorous equality between p, and pk is not valid for all cases. It is difficult to say to what extent the deviation between some particular rheological and kinetic values p* and p** is associated with the object of our considerations and to what extent it is caused by the ambiguity of estimates of corresponding parameters. No doubt, however, that the correlation between p: - p; and p:* - &* actually exists. This confirms the accuracy of the rheokinetic analysis of the gel effect during free radical polymerization.


Methyl-

Table 2.3.2: Critical Degrees of Conversion Obtained by Rheological p, and Kinetic p, Methods for MMAs (w/w)

0.5 6.2/ 2.9 11.4 195

0.32 8.2/ 4.4 15.8 I 14.1

Polymer: Mw

8

5

. ~~

13.5 I 9.7 35.0/28.2

14.0 / 12.8 40.0 I 34.3

I 2.0 I 9.51 5.0 I 18.4/ 15.1 1 I 1.7 I 10.21 5.2 I 19.21 15.8 -1

13.71 6.9 20.8 I 19.1

19.9 /14.5 41.2 136.3

Butyl 8.01 7.4 28.0 / 20.0

I I 8.5 I 9.5 I 30.5 /25.3 1 I 13 I 11.01 9.6 I 33.0/27.3 I

The idea of structural transformation at the critical point as the cause of gel effect is connected with the experimental data discussed above, according to which the termination rate constant k, decreases with the increase of reactive mass viscosity. Indeed, if we suppose that restriction of molecular mobility is the cause of a decrease of k, then it would be natural to associate this phenomenon with the engagement network formation, and the latter - with reaching fl,, or qcr - determined by (2.3.25). This approach has been presented in the form of the consecutive quantitative theory of gel effect, which may be called a “diffusion model” [3 11. An essentially new element of this model, as compared to the classical theory of free radical polymerization, is the assumption that in the region before the critical point (reaching of which


is determined by relation (2.3.25)), k, = const, and after the critical point k, decreases proportionally to the density of the engagement network, which, in its turn, is proportional to Ncp. This theory also took into account polydispersity, since critical conditions are simultaneously reached not for all the growing chains.

The idea of k, decreasing at high degrees of conversion naturally explains the gel effect occurrence. The quantitative anticipations, however, depend on initial assumptions concerning the form of the dependence on the degree of polymerization provided both radicals have N > N,, and on the rate of interaction of radicals, for one of which N > N,, and for the other N < Ncr.

Actually a universal dependence of k, on Ncp hardly exists, since such factors as viscosity of the medium and the nature of the solvent in which polymerization takes place, may play a certain role.

The diffusion-kinetic model of the process developed in [34] differs from that described above, first of all, in the assumed form of the dependence of the rate termination constant on the composition of a reactive system. In the cited work they obviously take into account the existence of a number of concentration intervals, where the structure formed by macromolecules appears to be different. Thus, the “first” critical concentration cp* corresponds to equality of the k, values, which are determined by segmental and transversal mobility of the chain. The “second” critical concentration (p** corresponds to the situation under which mobility restrictions become essential due to reptational movements of the chain (in accordance with the molecular views of De Gennes). The molecular behaviour at cp > cp** appears to be most significant. As it follows from the molecular model, the diffusion coefficient is determined in this region by factor ( N2cp’.75 ), and therefore k, - (N cp ) . The region cp > cp** is called “pseudo-gel”, stressing the interconnection of the phenomena under discussion (gel effect) with the structure of the reactive system, which indeed due to emerging restrictions in molecular mobility, resembles gel. Intermolecular contacts in the solution are, however, not stable though at cp > cp** ; their lifetime becomes comparable with the time of an elementary reaction of growing chain termination. It is due to this factor that k, decreases in the course of the reaction.

The model under discussion looks very promising also due to the fact that the estimates of dependencies of degree of conversion and MW of the formed polymer on time, following from this model, are in a rather good agreement with the experimental data already known. This may be regarded as an independent “kinetic” fact, testifying to the physical reality of the reptation model (according to De Gennes) of molecular chain motion in the solution. Besides, the diffusion model naturally explains the discussed cases of correlation between the kinetics of free radical polymerization in the gel effect region and the change in rheological properties of the reactive media in the course of the reaction.

The idea of diffusion control of the reaction of chain termination being the main cause of gel effect is widely accepted and may serve as the basis of a quantitative model of this phenomenon [35]. As it has been already discussed earlier, the basis of any model of that kind must proceed from the dependence of the rate constant k, on the degree of conversion. In [35], in contrast to other papers cited, this dependence is introduced by using the most general ideas of macromolecular dynamics in concentrated solutions and namely using the idea of relatively free volume being the main factor to determine the mobility of macromolecules.

2 1.75


The main role here belongs, however, to the suggested dependence of the effective constant of termination rate on the product of radical concentration CR Rj . Since the co-factors and the sum decrease with the increase of the degree of conversion, the constant k, must also decrease. The quantitative analysis of the system of kinetic equations with the variable k , introduced in this yields a prediction adequately reflecting the main principles of free radical polymerization in the gel effect region. One of the most interesting theoretical anticipations is the fact that the distribution of radicals along the chain length in the gel effect region should differ significantly from the Flory distribution.

Further development of the diffusion model of the gel effect is associated with generalization of experimental data by using dimensionless variables. The degree of conversion p is considered to be a natural kinetic parameter, and dimensionless time is introduced in the terms of the following formula [35]:

(2.3.25)

where, as noted earlier, k i , k , and k, are the rate constants of elementaq initiation reactions, of chain propagation and chain termination, respectively; [ I ] is the initial initiator concentration,fis the efficiency of initiation, t is time. The value of k, is determined for the region before the gel point, that is k, is assumed to be constant in determining.

According to the theory of homogeneous free radical polymerization we may plot the dependence p(z). Experimental points agree with the theoretical dependence up to some critical degree of conversion p*. It is the moment of deviation of experimental points from the theoretical curve that the value of p* is determined from. We may further analyze the dependence of p* on the parameters of the reactive system. Thus, in [34] a uniform dependence of p* on the number-average degree of polymerization was established N, for the whole set of data (known in literature) for MMA block polymerization.

The dependence of p* on Nn was expressed by the power law: p* - Nt with the exponent x = 0.24. Although the form of the dependence p* on N,, corresponds to the discussed rheological conditions of reaching the critical state of the solution, when change in its structure occurs, the value of the exponent x = 0.24 appears to be queer, since for critical dependence on concentration (p* on N the power law is valid with the exponent ranging from 0.5 to 1.0. Possi- ble explanation of this discrepancy may be found in the fact that concentration dependencies of micro-and macroviscosity are non-equivalent, which was repeatedly discussed above. It is quite possible that the critical concentration (p* (or p*) which gives rise to the gel effect, determined by microviscosity of the solution, actually changes depending on N somewhat different from macrorheological characteristics of this solution.

Development of methods of gel permeation chromatography for a complete analysis of polymerization products permitted to estimate the constants of elementary chemical reactions from a new quantitative viewpoint [36-381. We acquire thereby an opportunity of determining all the constants without applying to any a priori stated forms of their dependence on p. According to the results obtained [37], in the course of block polymerization of styrene initiated


by BPO, the constant of the rate of thermal initiation increases several times with the increase of the degree of conversion and, consequently, of the viscosity of the reactive medium which usually is not taken into account in the theory of the gel effect in free radical polymerization, and factor (k:/k,) decreases essentially. In the considered reaction path the value of k, consisted of the disproportioning rate constant and the double value of the recombination rate constant.

The rate constants of other elementary reactions have been determined. It has been found that the values of reaction rate constants of chain transfer to the initiator and the monomer (pertained to the constant k p ) up to the value p r= 0.7 decrease only by 20%. However, as it has been demonstrated by direct measurements, a rather wide range of variations in rate constants of all elementary reactions depending on is observed, which should be borne in mind when developing a complete quantitative theory of gel effect. Nevertheless, the change of factor (k? / k p ) is of predominant importance.

Determination of all the constant of elementary reactions, as well as measuring the dependencies [ I ] and [MI on p and parallel determination of the dependence of N on p permitted to quantitatively describe the entire kinetics of free radical polymerization, including the gel effect region [35]. This complex approach shows that gel effect emerges due to interaction of many factors. This approach permitted, therefore, without explaining the reasons of change in the constants of elementary reactions and using their values only, not only to describe the change differences resulting reactive medium.

in polymerization rate at p = p* but also to take into account from the variation of the initiator nature and the composition of

f0

6

2

O f 2 3

Figure 2.3.23: Dependence x (0 [q] )for PM at 70 "C ( I ) and 90 "C (2) and PS at 70 "C (3) and 90 "C (4).

the the

Since factor (k:' /kp) plays a leading role in the quantitative description of a process, it is important to find its dependence of the parameters of the reactive solution. A rather impressive answer to this question is given in Fig. 2.3.23 (according to [36-381). It shows that the change of factor (k:' / k P ) turns out to be the function of a uniform argument, the product of the degree


of conversion p and the intrinsic viscosity [q]. Generalizing of the experimental data in Fig. 2.3.23 achieved by using parameter x as a function presenting in itself a relative change of factor (k: ’ /k , ) , which is expressed as:

(2.3.26)

The subscripts 0 and p indicate that the corresponding values pertain to the reaction initiation and to the state of the reactive mass at a certain current (variable) degree of conversion p.

As was pointed out above, for free radical polymerizations p = cp i. e. the argument p [q] may be substituted by the product cp [q] or (bearing in mind a relatively subtle change of the density of the reactive mass during polymerization) c [q] , where c is concentration in g/crn3, and the product c [q] presents in itself a dimensionsless argument, i. e. dimensionless concentration. As it is well known from the theory of solutions, such a value may serve as a measure of filling the volume of the reactive medium by the formed macromolecular balls.

It is useful to remember that the argument c [q] also determines the viscous properties of polymer solutions with different MW in a wide range of concentrations [6, 261. Therefore the data in Fig. 2.3.23 reflect to a certain extent the effect of the solution viscosity on the change in the complex ( k:’ / k p ) . The data of Fig. 2.3.23 pertain to the reactive media with the invariable solvent (for each polymer). If we proceed with the analogy between the parameter x and viscous properties of solutions in different solvents, it would be necessary, according to [6,39,40], to introduce the Martin constant K , (or the Huggins constant, which is nearly the same) to the argument. Therefore it may be anticipated that in a general case (that is, on generalizing the kinetic data obtained for different solvents), x will be determined by the argument KMP [q] (or by KMc [q] . Thus, the results of [36-381 yield a quantitative interpretation of the idea of diffusion restrictions being the cause of gel effect, although the final dependencies obtained (particularly those shown in Fig. 2.3.23) are of empirical nature.

The dependence of the factor (k:’ / k p ) on the degree of conversion may be expressed, for instance, by the following empirical formula [4 11:

(2.3.27)

where a l , a2 and a3 are the coefficients found by statistical processing of the

According to the data available [41], factor ( k : ’ / k p ) may decrease up to 100 times as

The generalized dependence of this factor of the product p [q] is described by the

experimental data.

compared with the initial value corresponding to the steady region.

following empirical formula [39]:

In(k: ’ /kp) = 4.2p [q] . (2.3.28)


This empirical expression contains a uniform factor as an argument, which appeared to be sufficient for some systems. The complete generalization of experimental data is achieved, however, if same other factors influencing the coefficient of macromolecule self-diffusion are taken into account. A generalization of this kind has been reported in [42], where the parameter x, determined in terms of formula (2.3.26), was compared with the complex variable a, expressed as:

(2.3.29) 95 AT

a = - 3 + - + O o . 7 6 ( p [ ~ ] )

The coefficients (-3; 95 and 0.76) are chosen here by processing the whole bulk of experimental data available, and the difference AT = T - Tg presents in itself the distance in the state of a reactive mass from the glass transition temperature. Thus, in addition to formula (2.3.28) the value AT is taken into account. The extent in conformity on this approach with the experimental data available may be estimated from Fig. 2.3.24. It is evident, that the whole set of available data for different polymers with varied M W , obtained by using different initiators may be described uniformly by the given method.

Figure 2.3.24: A generalized representation of experimental data, showing the change in factor ( k : 5 / k p J in the course ofpolymerisation.


2.3.5 Polymerization in Heterogeneous Medium

Free radical polymerization is often carried out in heterogeneous systems. Even if we ignore microheterogeneity associated with the gel effect, three situations may be distinguished when heterogeneity of the system takes place.

First, free radical polymerization is often carried out in emulsion or in suspension. In this case the process in each droplet may be regarded as a micro block polymerization, but one can by no means neglect the transfer through the interface.

Second, the polymer formed in the course of polymerization may turn out to be insoluble in the reactive blend, though the initial system may be homogeneous. Dissociation of the reactive system to phases emerging at a certain degree of conversion affect inevitably the kinetics and rheokinetics of the process of polymerization.

Third, free radical polymerization may run in the presence of the solid phase of a filler, and polymerization filling takes place. The presence of the solid phase on the surface of the particles where the reaction takes place naturally influences the kinetics of polymerization.

It should be noted that there are no general approaches to the description of peculiarity of free radical polymerization in a heterogeneous medium, although a number of experimental observations are reported pertaining to this problem. We may suppose in advance that not only rheological properties of the reactive medium but some physical-chemical effects must influence the kinetics of the process in the cases mentioned above, these effects being associated with the peculiarities of interaction between the polymer and the solvent; the solution and the solids.

As a typical phenomenon illustrating the role of the solvent nature, the following experimental data may be presented [43]: Adding “good” solvents to styrene during its emulsion polymerization leads to reduction of the polymerization rate. The higher the concentration of the added solvent, the more pronounced is this effect. If we take the solvents with a lower thermodynamic affinity to the forming polymer, adding thin (“bad”) solvent influences the rate of emulsion polymerization of styrene to a far lesser extent. Evidently, we deal here not with a simple “average” concentration effect, but rather with re-distribution of local concentration of the polymerizing monomer in microvolume (“microreactions”) of emulsion, and this effect is caused by the peculiarities of thermodynamic interaction between the components in the system.

Of interest are also the results obtained in the cited paper [43] relating to the role of the solvent viscosity, added to the emulsion. If the viscosity of a solvent is high, its adding to the reactive mass accelerates emulsion polymerization of different monomers. In the cited paper this is explained by the hindrance in the reaction of recombination of macroradicals. This explanation is easily associated with that described in the previous section, concerning the deduction of the termination rate constant in the region of gel effect during free radical polymerization in a homogeneous system.


It is known that the kinetics of free polymerization of acrylates running in an emulsion, depends on the deformation rate [44]. Evidently, the thinning effect is connected with peculiarities of distribution of local concentrations in heterogeneous polymerizing systems, which is influenced by deformation (mixing) of the emulsion. This explanation permitts to understand the role of deformation rate. Indeed, at the beginning of the process there is a significant amount of monomer in the system and consequently the dispersed monomer-polymer particle retain fluidity and may be distorted under some external action. That is why the kinetics of polymerization at the initial stage of the process essentially depends on the deformation rate. As soon as the degree of conversion becomes higher, the monomer in the polymerizing particles disappears, and they transform into quasi-solid globules. Otherwise, viscosity of the material in emul- sified particles appears to be very high in comparison with the average viscosity of the medium. Therefore it is the low viscous medium that deforms under mixing rather than the polymerizing particles. So deformation does not influence any more the rate of polymerization at this stage of the process.

The non-monotone nature of the effect the deformation rate has on the kinetics of polymerization observed experimentally is explained by two concurrent processes:

1. formation of new microdrops in a monomer emulsion of the polymerizing monomer and

2. coalescence of particles which impedes the previous process because the acceleration of defonnation leads to more frequent collisions of particles.

A direct experiment has shown that the average size and the size distribution of latex particles really depends on the deformation rate (at shear rates of lo3 s-') [44].

The experimental data described show that the predominant role during emulsion (and, presumably, suspension) polymerization belongs to colloidal rather than rheological effects associated with the stability of particles and deformation effects as well as surface phenomena of transfer through interface.

Essentially different is the role of rheological factors and the pattern of rheological transformations in the systems where heterogeneity appears in the course of the reaction itself, when the forming polymer is not soluble in its own monomer (or, in a more general case, in the reactive mass). At the initial stages of the process conventional homogeneous polymerization takes place, regulated by the principles described in Section 2.3.1. However, at the moment of phase separation as the formed polymer is isolated from the solution, the viscosity of the remaining solution drops drastically (Fig. 2.3.25 [45]). Naturally, phase separation depends on the solvent composition. It is illustrated in Fig. 2.3.25, where two rheokinetic curves obtained for different solvents are compared. The effect of varying solvent quality has been achieved. through variation of the ratio solvent (toluene)/precipitator (cyclohexane).


4 5 r

Figure 2.3.25: Increase of relative viscosity during polymerisation of 50%-solution of MMA in the medium precipitator/solvent with ratios: 18/82 ( I ) ; 56/44 (2); 60/40 (3); T = 40 "C.

As the process advances, the viscosity of the reactive mass increases again, so the complete dependence q(p) turns out to be rather complicate, as it is demonstrated in Fig. 2.3.26 [9,46], where thermodynamic properties of the medium varied through change in the ratio of precipi tator/solvent.

f00 c3 ' 60 4

20

3

2 4

6

d5 ' 0 2 0 - 4 b 6 0 8 0 0 20 40 60

hap fwf. Figure 2.3.26: The dependence ~ ( p ) during polymerisation of the 50% solution

of MMA with varying solvent compositions (a) of the MW of the polymer being formed (b). The ratio precipitator/solvent is: 30/70 (1); 60/40 (2); 65/35 (3); 75/25 (4). MW of the forming polymer MW~lO-~rnoVg is: 0.63 (5); 2.5 (6).

The pattern of viscosity change of the reactive medium for a system, in which in the course of polymerization separation of phases takes place, as shown in Fig. 2.3.26 is, in fact, similar to the extreme change of viscosity in a multi-component system, used for manufacturing PS with high impact strength, which was described in [47] rather long ago. In the process of obtaining


PS with high impact strength, at a certain degree of PS conversion, phase transition takes place from the system where a homogeneous solution of rubber in styrene is the continuous phase, to the one, where rubber is released in the form of a dispersed phase in the solution of PS in styrene. This solution forms a continuous phase where the solubility of rubber disappears as the concentration of the PS formed exceeds a certain threshold.

The phase inversion is concurred by the change of the whole complex of rheological properties of the reactive mass. For instance simultaneously with the viscosity leap (or with its extreme change) equally sharp changes in elasticity of the medium and normal stresses as its manifestation are observed. It is a well known fact that the conditions of phases separation in the course of manufacturing PS with high impact strength and other multi-component systems influence to a great extent the properties of the end product. Therefore the kinetic analysis appears to be very helpful in controlling the production process.

The change in rheological properties of the reactive mass at the moment of phase transition significantly influences the hydrodynamic situation in the reactor, heat- and mass change and, finally, the kinetics of the reaction and the properties of the end product. All this necessitates a special attention towards the problem of constructing phase plots of reactive systems.

3 r

4 8 12

Figure 2.3.27: Viscosity increase during polymerisation of AN in an aqueous solution of sodium rhodanide. The salt concentration in water, wt-%, is: 54 ( I ) ; 51.5 (2); 45(3); 43.5 (4); 38 (6). T = 80 "C.

However, practically no quantitative data are available concerning the phase state of the reactive systems or polymers obtained by free radical polymerization. Partially it is connected with a basic difficulty of separating the colloid and phase events, since a net phase is often formed first as a steady particle of colloid size, i. e., in the form of some microgel particle. Finally it may lead not to polymer release in the form of an independent phase with depletion of the remaining solution (as was shown in Fig. 2.3.25 and 2.3.26), but to gelation of the solution [45,48]. An example of this behaviour is presented in Fig. 2.3.27 for the system of polymerizing acrylonitrile (AN) in aqueous solution of NaCNS. It is seen that at high concentrations of the salt (curves 1-4) the viscosity in the course of polymerization increased monotonically and gradually. However, with the decease of sodium rhodanid concentration, the situation changes


drastically. The obtained polymer forms a gelatine-like system, and the lower the content of the salt in water, the lower is the polymer concentration, that is, the quality of the solvent used becomes worse (curves 5 and 6). This effect is of a pronounced threshold nature and a sharp rise of viscosity is really associated with a phase transition. The latter is evident from Fig. 2.3.28 where coincidence of polymer concentrations is shown at which unlimited rise of viscosity is observed and the transparency of the solution z drops sharply.

2 *

2 c o c , A f 00

p,wt . %

Figure 2.3.28: Change in viscosity (q) and turbidity (z is translucence) during polymerisation of AN in a 38% aqueous solution of sodium rhodanide. T = 80°C.

Another problem to be considered here is the situation when two components are simultaneously present in the solution forming a blend of incompatible polymers during polymerization. A typical example of this system applied in industry is polymerization of PMMA in the presence of small amounts of PS. This process is applied in manufacturing of opaque glass. Another system of interest is MMA polymerizing in the presence of PBMA being incompatible with PM [49]. MMA polymerization takes place in the homophase regime. A heterophase system is formed in the presence of PM at a certain degree of conversion which is confirmed by turbidity emergence.

As was pointed out previously, a monotone increase of viscosity is observed during PM formation (Fig. 2.3.1). A complicated behaviour of the dependence of viscosity on the degree of conversion is observed in the presence of PM, which is similar to the situation described above, when the transition from the homo- to heterophase system occurs (Fig. 2.3.29). This effect is thereby more pronounced the higher the MW of PM dissolved in the initial reactive mass.


Q

8 Y -

Figure 2.3.29: Viscosity change during MMA polymerisation depending on the degree of conversion in the presence of PBMA. PBMA content, in wt.-%: 0 (1); 0.5 (2); 1.5 (3); 2.0 (4); 2.5 (5). MW of PBMA 6 . 0 ~ 1 0 ~ g/mol. Temperature 70 "C. The arrow indicates the moment of phase separation due to turbidity occurence.

With regard to phase separation during polymerization one should be especially carefully estimating the deformation rate. The point here is that deformation affects the path of phase transition in system "polymer-low-molecular solvent", and it may be of thermodynamic nature (that is, determine the condition of equilibrium) and therefore influence the kinetic of phase transion [49]. Although this assumption may relate to the systems where the polymer formation follows the radical path with removal of the end product from the reactive system, practically no experimental data are available in this domain. Moreover, even in cases when the effect of the shear rate on the rheokinetics of polymerization has been obviously demonstrated, the interpretation of the physical sense of these observations has been far from evident.

Thus, for instance, in [38] it has been reported that in the course of MMA polymerization initiated by BPO increase in the shear rate up to 350 s-' led to the 10% increase of the polymerization rate. This result was explained by the effect of deformation on the constant of initiation efficiency. It might have been also explained by the influence of shear rate on the conditions of gel effect emergence, if we ascribe the role of structure formation to deformation. However, a more common explanation is not excluded. Thus, it is rather evident that for the reaction rate to increase by 10% at a normal activation energy value in the order of 80 kJ/mol, it is sufficient for the temperature of the reactive mass to rise only by 1 K*. This effect might be

E * Indeed, let the reaction rate be described by the Arrhenius equation k = A exp(-RT) .

Then the relation o f the rate constants at two temperatures T I and T2 ( T = 320 K ) differing by 1 K,

80'103'1 ) - 1 . 1 , i . e., the increase of temperature by 1 K really leads to a is evaluated as - = exp (-

10% increase of the polymerization rate.

kl k2 8.320.320


successfully achieved at the shear rates used as the consequence of dissipative heat release in a viscous medium (in spite of the fact that, according to the regime of the experiment the temperature was maintained constant with an error up to +1 K). This real effect of non- isothermality and its influence on kinetics is discussed in detail in Section 3.7.

This example vividly points out the difficulties of the reaction, which are more complicated under conditions of phase separation of the polymerizing mass.

2.3.6 References

1.

2.

3.

4.

5 .

6.

7.

8.

9.

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I. V. Smetanina, D. N. Yemelyanov, Fiziko-khimicheskiye osnovy sinteza i pererabotki polimerov, Gorkovsky Gos. Universitet, 1980,56.

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10.B. A. Korol, ev. Reologich. Thesis, MGU im Lomonosova, Moscow, 1987.

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12.G. H . Bamford, W. G. Barb, A. D. Genkins, P. E. Onyon, The Kineticsof Vinyl Polymerization by Radical Mechanism, Butterworths, London, 1958.

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14.A. E. Chalykh, L. V. Titkova, A. Ya. Malkin, V. E. Dreval, I. S. Pronin, Vysokomolekulyarnye Soedineniya, A, 1974, vol. 16, 8, 1844.

15.P. D. Neilson, I. Soutar, W. Steadman, Macromolecules, 1977, vol. 10, 6, 1193.

16.V. A. Bagdonaite, S. S. Yushkevichute, Yu. A. Shlyapnikov, Polymer, 1981, vol. 22, 2, 154.

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100 Chup fer 2: Rheokinetics of Linear Polymer Formation

18.S. S. Ivanchev, L. P. Skublina, E. T. Denissov, Vysokomolekulyarnye Soedineniya, B, 1967, vol. 9, 9,706.

19.M. G. Kulkarni, R. A. Mashelkar, L. K. Daraiswamy, Chem. Eng. Sci., 1980, vol. 35,823.

20.M. G. Kulkarni, R. A. Mashelkar,A. I. Ch. E. Journal, 1981, vol. 27,716.

21.5. P. Fisher, J. Mucke, G. V. Schulz, Ber. Bunseng. Phys. Chem., 1969, Bd. 73, 157.

22.B. W. Brooks, Proc. Royal SOC. A, 1977, vol. 357, 183.

23.Toh. Yamamoto, Td. Yamamoto, Tosh. Yamamoto, M.Hirota, J. Chem. SOC. Japan, 1980, vol. 101,4,618.

24.V. P. Gromov, N. I. Galperina, T. I. Osmanov, P. M. Khomikovsky, A. D. Abkin, Europ. Polymer J., 1981, voLI6,529.

25.R. A. Symonyan, V. A. Kassaikin, M. B. Lachinov, V. P. Zubov, V. A. Kabanov,

26.H. B. Lee, D. T. Turner, Macromolecules, 1977, vol. 10,226;

Dokl. Akad. Nauk. SSSR, 1974, vol. 21 7, 3,63 1.

A. C. S. Polymer Preprints, 1977, vol. 18,539; 1978, vol. 19,603.

27.M. B. Lachinov, R. A. Simonyan, T. G. Georgieva, V. P. Zubov, V. A. Kabanov, J. Polymer Sci., Polymer Chem. Ed., 1979, vol. 17, 2,613.

28.A. Ya. Malkin, Rheol. Acta, 1973, vol. 12,314,486.

29.N. M. Biturin, V. I. Genkin, V. P. Zubov, M. B. Lachinov, Vysokomolekulyarnye Soedineniya, A , 1981, vol. 23, 8, 1702.

30.P. R. Dvornic, M. S. Jacovic, Polymer Eng. Sci., 1981, vol. 21, 12,792.

3 1 .Yu. N. Panov. Europ. Polymer J., 1979, vol. 15,395.

32.5. M. Dionosio, H. K. Mahabadi, K. F. O’Driscoll, E. Abusin, E. A. Lissi, J. Polymer Sci., Polymer Chem. Ed., 1979, vol. 17, 1891.

33.5. Cardenas, K. F. O’Driscoll, J. Polymer Sci., Polymer Chem. Ed., 1976, vol. 14,883; 1977, vol. 15, 1883 and 2097.

34.T. J. Tulig, M. Tirell, Macromolecules, 1980, vol. 15, 1501.

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39.V. E. Dreval, G. 0. Botwinnik, A. Ya. Malkin, A. A. Tager, Mekhanika Polirnerov, 1972,6, 11 10.

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41.V. P. Budtov, N,G. Podosenova,

42.V. P. Budtov, N. G. Podosenova,

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Polimerov, pod Red. Malkina i Papkova, Moscow, Khimiya, 1980,9 1.


2.4 Pol ycondensation

2.4.1 Introduction

Essential distinctions in mechanisms of polymerization and polycondensation must lead inevitably to the change of rheokinetic principles of these processes. Therefore the rheology of reactive systems in the processes of polycondensation is marked by several distinguishing features and requires a separate consideration.

According to the functionality of the monomers and oligomers used, two types of polycondensation may be distinguished: linear or three-dimensional [ l , 21. In this section linear hetero-polycondensation is under consideration, i. e., polycondensation of bifunctional compounds which yields a polymer of linear structure.

The main distinguishing feature of polycondensation which is of importance for the present considerations consists probably in the fact that the polymer MW increase involves simultaneous interaction of all the reactive centres of the system reagents and, with the progress of the reaction, the amount of such centres decreases. The reactive system following such a path will not contain monomer molecules as early as after the first stages of polycondensation.

Assuming, as it is customarily done with regard to the processes of polycondensation, that the reaction runs by macromolecule duplication and, consequently, the number-average degree of polycondensation 3 is expressed as the ratio of initial and current reactive group concentrations, the dependence between p and 3 is expressed by the following relationship:

N = ( l - p ) - ' (2.4.1)

Here it is essential that the concentration of the forming product (polymer and oligomer) in the reactive system does not change in the course of the process. Then, proceeding from general rheokinetic principles, we may assume that the viscosity of the reactive system must be proportional only to the MW of the forming polymer.

The assumption of an unambiguous connection between the reactive mass viscosity and the degree of polycondensation, independent of the peculiarities of the process, has been confirmed by direct experiments when synthesizing poly-(hexamethylene sebacinamide) (PA-610) [3,4].

The process has been carried out under conditions corresponding to the actual processing pattern of PA-610 manufacturing, where it was successively heated with a pressure increase up to 1.8 MPa followed by a pressure drop in the course of removal of the vapour of condensed water. As pressure decreased, the reactive mass was sampled at certain time intervals.

Viscosity of the diluted PA-610 solution q,, and the maximum value of Newtonian viscosity of the reactive system qo regularly increase with the increase of retention time (Fig. 2.4.1). The two dependencies qsp(t) and qo(t), thereby, are of the same shape. These functions may be approximated by exponential dependencies q = 8 exp(t/to) with similar values of the exponent to. For q - t o = 10 min and for q,, - t o = 1.1 x102 min. The similarity of the dependencies q,,(t) and qo(t) is expressed by a simple linear relationship of the form qo = Kqsp + A .

2

2.4 Polycondensation 103

ii Pa. S'

t, min

Figure 2.4.1: Change of specific viscosity of PA-610 solution in 96% sulphuric acid (1) and the maximum Newtonian viscosity of PA-610 melt (2) in the process of polycondensation in linear (a) and logarithmic (b) coordinates.

In this case, as apparent from Fig. 2.4.2, q,, of the 1% solution of PA-610 in sulphuric acid is in fact a linear function of its average MW [5]. This simple relation between the reactive system viscosity and the specific viscosity of the diluted solution of products being formed at different stages of polycondensation indicates that qo is determined by a single variable which increases in the course of the process, namely by the average MW of the forming polymer. As it is seen, the conditions of the experiments correspond to actual technological processes and may be pertained to the diffusion-controlled stage of the reaction, which makes it impossible to implement the rheokinetic analysis of the process.

Figure 2.4.2: Relationship between characteristic and specific viscosity of a 0.5% solution of PA-610 in concentrated sulphuric acid.


2.4.2 Rheokinetic Pattern of the Process

Let us consider the form of the rheokinetic relations for polycondensation processes, e. g. take the simplest reaction of bifunctional compounds, which is not complicated by diffusion impediments. The kinetics of this reaction is described by a second-order equation on the basis of functional groups, which, at equal concentrations of reacting groups, is written as follows:

dx - - - kx2 d t

(2.4.2)

where x is the concentration of functional groups, k is the rate constant of the reaction.

After the equation is integrated using assumptions usual for polycondensation processes, it can be easily shown that at xokt D 1 (where x,, is the initial concentration of functional groups) the change in the numerical mean degree of polycondensation with time is described by the linear expression:

= xOkt (2.4.3)

Combining formula (2.4.3) with a conventional power dependence of viscosity on MW we shall obtain an expression describing the change in the viscosity of the reactive medium in the course of polycondensation [6] :

?l = K(xokt)' (2.4.4)

where K and a are the constant factor and the exponent in the dependence of viscosity on MW or the degree of polycondensation.

a thereby is close to 1 at M <a, ( t < t * ) and is equal to 3.4 at is the time needed for the MW to reach the critical value of a,.

with different exponents

Consequently, the viscosity of the reactive system is a power function of time, the exponent 2 M, ( t > t * ) , where t*

The constants k and K entering formula (2.4.4) are exponential functions of temperature

E RT

K = Koexp (-)

U RT

k = koexp (--)

(2.4.5)

(2.4.6)

where E and U are activation energies of viscous flow and of the chemical reaction of polycondensation, respectively. Generally speaking, both values may depend on the degree of polycondensation, which, as will be shown further, is in agreement with particular reactions under consideration.

At t = const the dependence of the viscosity of the reactive system on temperature is expressed by the relation

1 ?l-eXP(- E - a U

RT (2.4.7)


and consequently the “apparent” activation energy of the viscosity increase in the course of

polycondensation, which is determined by plotting Inq vs (1/T) at t = const, equals:

E, = E - a U (2.4.8)

If the viscosity level of the reactive medium is kept constant q = q* and the time needed

to achieve this is considered depending on the temperature of polycondensation, the “apparent”

activation energy of this process equals:

E 9 a

E = u - - (2.4.9)

All the expressions written previously according to a normal rheokinetic pattern may be

examined experimentally from the viewpoint of solving both direct and inverse rheokinetic

problems. It is noteworthy that the formulae obtained are only valid for the reactions running

without diffusion hindrances. For polycondensation processes this is a rather essential limita-

tion, since the rate of the process is in many cases limited by the removal of the low-molecular

product of the reaction.

2.4.3 Experimental

We have chosen the reaction of the sodium salt of diphenylene propane with 4,4’-dichlor-

diphenylsulphone in DMSO-solution, which results in PSF formation, as an experimental

model to test the obtained expressions.

0 - 1

Sodium chloride is a low-molecular product of this reaction, the major amount of which is evolved at the first stages of the reaction, and which precipitates without distorting the results

of viscometric measurements.

Typical experimental data describing the viscosity increase in the reactive mass in the

course of PSF formation in the 50% solution of DMSO at different temperatures are presented

in Fig 2.4.3:


Figure 2.4.3: Dependence of viscosity of a reaction system in a 50% solution of dimethylsulphoxide on the time of polycondensation at 150 "C ( I ) , 160 "C (2), I70 "C (3) and 180 "C (4).

As it has been shown previously, the viscosity dependence on time on log-log coordinates must be represented by two rectilinear sections with different slope coefficients a . Thereby the numerical values of the coefficient can be determined independently with model systems in a rheological experiment. The results of this experiment in the form of a dependence q(M,) are presented in Fig. 2.4.4:


QP 0

Figure 2.4.4: Dependence of viscosity of a 50% solution of polysulphone in dimethylsulphoxide on the numerical average of the molecular mass at 150 "C ( I ) , 160 "C (2), I70 "C (3) and 180 "C (4).

From the given data it follows that for PSF solutions in DMSO within the experiment

1.7 M<M, a = {

3.4 M 2M, At the same time it is seen from Fig. 2.4.3 that, except for a short initial period of the

reaction, the value of the index a is indeed 3.4. A slower increase of viscosity in comparison with that anticipated by the dependence q(t) at initial stages of the reaction must be ascribed to more complicated kinetic processes of the first events of the reaction, which, in a general case, are not described by equation (2.4.2). The results of NMR spectroscopy of initial stages of the reaction indicate that there are actual complications associated with the change in reactivity of functional groups at the initial stages of the reaction [7,8-111.

Experimental measurements of the viscosity increase in the course of polycondensation permit to directly register the increase of the number-average degree of polycondensation, since viscosity increase unambiguously indicates the increase in MW. The linearity of the dependence N ( t ) , observed thereby, testifies that polycondensation follows the simplest second-order pattern. The dependence plotted in that way for the reaction of PSF formation in a 50% solution of DMSO is presented in Fig. 2.4.5. The linear shape of the dependence #(t), plotted in Fig. 2.4.5 in the range of N values from 10 to 50 corresponds to formula (2.4.3), which enables us to determine the value of the constant of the rate of chain growth k. The dependence of the obtained constant values on temperature is shown in Fig. 2.4.6:


20 t, min

Figure 2.4.5: Dependence of the numerical averaged degree of polycondensation on the process duration at 150 "C (I), 160 "C (2), 170 "C (3) and 180 "C (4).

Figure 2.4.6: Dependence of the constant of polycondensation rate (I) and the time needed to reach the viscosity of 1 Pa-s (2) on temperature.

The values of the rate constants at different temperatures may assist in determining the activation energy of the reaction of polycondensation U, which appeared to be equal to 69.6 HI mol. The temperature dependence of time necessary to reach the viscosity level of 1 P a s in the


course of polycondensation is shown in the same figure. Hence we may determine the “apparent” activation energy E,, which is equal to 50 kJ/mol. The value obtained corresponds to formula (2.4.9) at activation energy of viscous flow E = 32 kJ/mol, which agrees with the value E determined in an independent experiment [12].

Thus, the combination of numerical values of the obtained kinetic and viscometric constants corresponds to derived rheokinetic relationships. This confirms the validity of these expressions as well as the correctness of the kinetic pattern of PSF synthesis in DMSO solution for the main part of the process (except for the initial stage of oligomerization) until the end product is obtained.

To get a complete picture, the rheokinetic investigations have been carried out parallel to a study of the process with the aid of NMR-spectroscopy. These methods are complementary, since rheokinetic analysis alone does not describe initial stages of the reaction due to insignificant change in the viscosity of the reactive system during transition from monomers to dimers and tetramers, which is in fact within the error of a viscometer. The initial moments of the reaction have been investigated in ‘3C-NMR-Fourier-spectra, permitting to identify the composition of the reactive system [ l 1 , 131.

The ‘3C-NMR-Fourier-spectra (22,63 MHz) of the reactive blends have been recorded by a Brucker Spectrometer H-90. Methylene chlorided2 has been used as a solvent, hexamethyl disiloxane served as internal standard. Signal identification was made on the basis of the data from [9].


After treatment with HCl the reactive blend contained the following monomer compounds:

and condensation products:

-*-@{-W--

(dimer)

(Dend)

(Dmid)

I3C-NMR-spectra may be used to determine relative concentrations (fractions) of free monomers D (yl), P (ys), chloroformiate end groups Dend (y2) and hydroxyl end groups Pend

(y6), involved in the chain Dmid (y3) and in Pmid (y7), and of the “dimer” ( ~ 4 ) .

The concentrations of y to y7 have been determined at different stages of the reaction and thus NMR-spectroscopy measurements yielded the kinetic dependencies y ~ ( t ) to y7 (t) .

In the formulation of a kinetic model it is of interest to discuss two problems:

the order of the reaction;

and the change of the rate constants with time (or depending on the extent of the reaction).

1.

2.


According to the results reported in [S], the reaction of alkali phenolates and halogen- phenyls is of second-order. The change of the constants of the reaction rate may be caused either by an increase in the viscosity of the reactive mass, a change in ionic force, or by a change of interaction between reactive centres in the course of the reaction. As this section of the book deals with polycondensation in a diluted solution, the viscosity change and its effect on the run of the process may be ignored, since the reaction runs in a kinetic rather than diffusion controlled region. The ionic force of the solution (0.49 g-eqL) practically did not change in the course of the reaction. As for the mutual influence of reactive centres it is the vital problem to be considered. This factor - within the scope of the approach under discussion, as will be shown later on -is taken into consideration by introducing various rate constants for free and terminal links.

Taking into account that both comonomers have dependent reactive centres [S] and assuming that at the amount of monomer links (P + D) in the oligomer m > 2, the effect of long- range order does not manifest itself, we may describe the process of nucleophilic substitution for the system sodium - salr/P/D - by the following system of differential equations [ 1 11:

(2.4.10) dY I - = - k l I 1 Y5 - k21a0Y I Y6 - k21a0Y I Y4 d t

dy 6 2

dy I 2

- d t = k12aOY 2y5 + k12aOY4Y 5 + k22aOY4 - k21a0Y l y 6 - k22a0y 2y6

- dt = k 2 l a O ~ 1 ~ 6 ~ k 2 l a O ~ l ~ 4 ~ k 2 2 a O ~ 2 ~ 6 ~ k 2 2 a O ~ 4 ~ 6 ~ k 2 2 a O ~ 2 ~ 4 ~ k 2 2 a O ~ 4

As independant constants, four values of kij are introduced here:

k , I is the constant of interaction rate between P and D;

k,, is the constant of interaction of P and Dend and P and dimer;

k,, is the constant of interaction rate of Pend and D and of dimer and D;

kZ2 is the constant for Pend and Den,+ for dimer and dimer and Dend,

and for dimer and dimer.

The value of a. is the initial concentration of monomers (in this case a0 is the same

for P and D).


Model

M 9

M 4

The system of differential equations (2.3.10) together with the equation of material balance

Y,+Y2+Y3+2Y4+YS+Y6+Y, = 2 (2.4.1 1 )

constitutes a four parameter model of the process, the parameter of which are the required values of kij .

Beside the four parameter model a nine parameter model (M 9) was also developed, taking into account the effect of a long range order up to m = 3 with aid of the constants k13, k31, k23, k32, and k33 (Table 2.4.1). We have also considered two "reduced" two parameter models (M 2a and M 2b). The parameters of the four models are given in Table 2.4.1.

Table 2.4.1 : Reaction Rate Constants (T = 120 "C, cp = 0.49 g-equivk).

Rate constants (L/g-equiv-min)

kll k12 k13 k2 1 k3 1

2.43 f 0.1 0.44 f 0.08 0.73 f 0.08 1.26 f 0.06 0.97 f 0.3

k22 k23 k32 k33

kll k12 k2 1 k22

0.14 f 0.2 0.43 f 0.2 0.09 f 0.004 0.028

2.35 f 0.1 1 0.56 f 0.05 1.22 f 0.09 0.12 * 0.01 0.036

I 0.86fO.l 0.43 f 0.08 0.13

k2b

0.087 f 0.01 0.068 2b I :::6 f 0.05

The analysis of the results presented in Table 2.4.1 enables us to draw some conclusions:

The set of experimental data used for evaluations is not suffcent for a reliable determination of all the parameters in model M 9 (particulary, we failed to determine the constants k22 and k23);

The four parameter model is a good approximation both to experimental data (Fig. 2.4.7) and to the model M 9: The constants of the reaction rate between monomers ( k l l ) in models M 4 and M 9 are the almost the same, the values of other constants of model M 4 (k12, k21 and k22) are within the value ranges of corresponding constants of model M 9. The scope and accuracy of the set of experimental data are sufficient for a reliable determination of the parameters of this model;

The two parameter models are a much rougher approximation than the four parameter models. In the framework of these models one cannot obtain a true idea of the reactivity of the components. A better approximation to the experiment is provided by model M 2b, where one of the rate constants (klb) describes all the reactions involving monomers and the other (k2b) describes all the reactions of oligomers (starting from dimer).

Therefore the results obtained in the present work are discussed on the basis of model M 4.


0.6

i7 43

0 0 30 60 90

t (min)

Figure 2.4.7: Variation with time of fraction yi of the free (A, E). end (B, F ) and bonded (C, G) dichlordiphenylsulphone (A-C) and diphenylolpropane (D-G), as well us dimer (D) in the course of the condensation reaction in the sodium salt of diphenylolpropane - dichlordiphenylsulphone at 120 "C. The experimental points (yi values) are determined from 13C-NMR spectra. A-G are the calculations from equation ( I ) and (2).


In the course of PAS synthesis interaction between nucleophilic (phenoxide ions) and electrophilic centres occurs:

As it is seen from the results obtained, the activity of the second electrophilic centre is reduced two times, as soon as the first centre enters the reaction kll ( 2 k12 = 2.08). Evidently, this is caused by the electron donor properties of P residue in dimer and in Dend [8].

When the first nucleophilic centre starts reacting, the activity of the second centre practically remains unchanged (kl1/2 k12 = 0.96). This reflects a poor delocalization of the phenoxide charge in the dimer and in Pend.

First of all, note that the results obtained by NMR described earlier refer to dilute solutions, so the kinetic stage of the process is discussed disregarding diffusion limitations.

This is also confirmed by rheological data. Therefore a slight difference in values of cp must not affect the values of kinetic constants, which justified comparing of the values discussed below. The reaction rate determined by NMR depends on the chain length of the oligomer needed to form tetramers, the constant of interaction rate between the least active (the second nucleophilic and the second electrophilic) centre, k22 appears to be 0.25 L/mol.min at 120 "C. The rheokinetic analysis yields the constant k, corresponding to the slowest stage of synthesis. Recalculation of this constant at 120 "C yields the value of 0.30 L/mol.min, and consequently it is evident that k is sufficiently close to k22. This result is of great interest, since it demonstrates the opportunities of rheokinetics, as far as the investigation of kinetic principles of processes of polymer formation is concerned [7].

The examined reaction represents in itself a rather simple case of the polycondensation process. From this viewpoint it is interesting to see what amount of information may be obtained by rheokinetic methods with complications arising in the course of the polycondensation process. One of the main complications of the process, causing deviations of kinetic parameters from their theoretical prediction, is the possibility of phase decomposition and of the transition of the reactive system to a heterogeneous region. As is shown in Section 4.3, rheological properties of the polymerizing systems are rather sensitive to transitions of this kind, which immediately alter rheokinetic correlations. In polycondensation reactions the main compli- cation might be the possibility of reaching the diffusion-controlled region.


If we now consider PSF synthesis in the solutions of higher concentrations it may be principally expected that at lower contents of the solvent in the reactive system the transition of the process to the diffusion-controlled region may be possible. In this case the increase in viscosity of the reactive mass may lead to a deceleration process. This statement is clarified by Fig. 2.4.8, where the dependencies of viscosity of the reactive system on the duration of the reaction are shown at various temperatures and solvent concentrations.

f 2 poS t,(min)

Figure 2.4.8: Change of viscosity in the process of PSF synthesis pegormed in solutions with concentrations of the forming polymer in a system 20% ( I ) , 60% (2), 70% (3) and 80% (4).


cp

Evidently the change in the shape of the dependence q ( t ) is observed right at cp = 60% and at 150 "C and 160 "C, i. e. the rate of viscosity change decreases, which reflects the reduction of the reaction rate constant (Table 2.4.2), evaluated on the basis of the previously obtained rheokinetic relations. At 170 "C and 180 O C the rate constants for 50%- and 60%-solutions remain practically unchanged (Table 2.4.2). This proves that under these conditions the process obviously runs in a kinetic region. However, at lower temperatures and higher solution concentrations a somewhat different situation is observed.

k N k N k N k N U E

50

150 "C 160 "C 170 OC 180 "C

1.44 68 a 2.56 96 1 3.6 68 * 4.8 84 * 69.6 31.9

60 I 1.125 90 I 2.0 96 I 3.75 108 I 5.0 102 I 65.4 31.9

70

80

0.39 77 0.79 100 0.92 108 1.32 110 33.6 31.9

0118 8 0.03 14 0.95 28 073 44 77.7 77.7

a. These values do not correspond to N-, since the reaction was interupted deliberately on reaching the given degree of polycondensation to determine MW.

The change of with time correlates with the data of viscometric measurements (Fig. 2.4.9). Increase of cp up to 70 - 80% leads to deviation of the dependence R(t ) from the linear and the whole process is inhibited at the initial stage, if cp = 80% and at T = 150 "C, so the number-average degree of the product of polymerization obtained does not exceed 9 - 10. The obtained values of rate constants k, values of the polymerization degree of the end product N , the activation energy of the polycondensation process U and the activation energy of viscous flow E of PSF solutions of respective concentration are tabulated in Table 2.4.2.


f0U

80

180°

so ioo 150 so fuo /so t, min t , min

Figure 2.4.9: Dependence of the mean numerical degree of polymerisation on the duration of the process at 150 (a) , 160 (b), 170 (c) and 180 “C (d).

To estimate the reaction path in concentrated solutions, let us consider the temperature dependence of the total rate of polymer formation. It is characterized by the apparent activation energy U corresponding to the section of the dependence m(t) on q ( t ) , where the reaction is described by a second-order equation, and its rate is presented by the value of the constant k. The values of U and E are compared in Fig. 2.4.10 for solutions of different concentrations:


I ' I I I

50 60 70

Figure 2.4.10: Dependence of effective activation energy of the polycondensation process

( 1 ) and viscous flow of solutions (2) on PSF concentration in DMSO.

For 50%- and 60%-solutions the activation energy practically does not depend on concentration (at U > E). Reduction of the activation energy of the chemical reaction to the value of the activation energy of viscous flow at transition to the 70%-solution and the subsequent equivalent increase of U and E for the 80%-solution reflect the radical change in the course of polycondensation. The increase of activation energy of viscous flow as the polymer concentration in the solution rises is quite typical of the rheology of solutions and the coincidence of U and E may be interpreted as a manifestation of the fact, that the rates of viscous flow and the chemical reaction are controlled by the same factor. This factor is probably diffusion transfer.

This change in the kinetics of the reaction due to emergence of diffusion hindrances may be associated with different reasons. However, the simplest explanation - that diffusion limitations are caused by viscosity increase and the corresponding growth of the diffusion coefficient of the reactive medium - appears to be insufficient. The determining factor here is, probably, the approximation to the region of transition in the system PSF-DMSO, because with the increase of concentration or temperature decrease the system approaches the conditions of glass transition in the solution. This is demonstrated by the phase diagram of the system PSF-DMSO shown in Fig. 2.4.11. The shaded section in the diagram corresponds to the temperature-concentration range of PSF synthesis considered in the present work. The high- concentration systems (9 = 70 - 80%) are very close to the concentration dependence of the glass transition temperature, which significantly affects the reaction rate of polymer formation.


f00

I I I I 1

za GO 60 80 'p, % mass

Figure 2.4.11: Diagram of the phase state of the system PSF-DMSO:

I - binoidal, 2 - temperature of solution vitrification, 3 - the region where polycondensation is efsected.

The effect of the concentration of the reactive system on the transition of the reaction to the diffusion-controlled region, and inversely, is confirmed by the following especially conducted direct experiment [ 141. The solvent was added to the working unit of the rotation viscometer (in which the polycondensation reaction was effected at cp = 70%) in the amount necessary to decrease the concentration of the polymer cp to 60%. The change of viscosity and degree of polycondensation before and after introduction of the solvent is shown in Fig. 2.4.12:


3 - 6 I Q

P

2 : J : 1

0 4

N

/“ !j / , I

I I I

t, min 1 2

Figure 2.4.12: Change of viscosity (a) and the mean numerical degree ofpolymerisation (b) in the process of PSF formation at sharp concentration changes in a forming polymer in a reaction ranging from 10 to 60%. A - The moment of introducing a solvent into a reaction solution, B - dependence N(t) typical of the kinetic region (cp 560%).

It can be seen, that the dilution of the solvent (i. e. concentration decrease of reactive molecules) leads to sharp acceleration of the polycondensation process due to the adding of the solvent to the system. It is caused by the fact that the decrease of concentration and consequently of viscosity in the reactive system leads to the transition of the reaction from the diffusion- controlled region to the kinetic region. It is essential here that quantitative “recovery” of the system takes place: the numerical values of the rate constants after the dilution fully coincide with the constants typical of the process running in the kinetic region [6, 15, 161.

Thus, it may be regarded as confirmed that transition from the 60%- to 70%-solution and then, to a greater extent, to the 80%-solution, leads to sharp reduction of the polycondensation rate due to diffusion limitation. Generally, two causes giving rise to these limitations may be distinguished:

1. increase of viscosity and

2. approaching of the system to the glass transition temperature and reduction of the molecule mobility.

The viscosity, per se, does not imply the transition from the kinetic to the diffusion- controlled region. Probably, the mobility of the growing molecules depending on their 8 must be also taken into account. An attempt of such an account is the use of the complex parameter r\N as an argument, when the isothermal constant of reaction rate is considered. A value of this kind in rheology and in polymer physics determines relaxation time and, consequently, the rate of the diffusion transfer. The results of the corresponding processing of experimental data are presented in Fig. 2.4.13 in the form of dependencies of k on r\N.

2.4 Polycondensution 121

C

s

Figure 2.4.13: Dependence of the macro kinetic rate constant of PSF formation on the complex parameter qN (viscosity is expressed in Pa.s) at polymer concentrations in a solution 20% (I), 60% (2), 70% (3) and 80% (4).

A rather typical and important fact reflected here is that initially (at low values of this parameter), the value of the reaction rate constant does not depend on the parameter qN, then it drops, and the dependence k on q N appears to be common for the solutions of different concentrations. This confirms the determining role of the complex parameter in the form of qN and means that the increase of the parameter qN, i. e. eventually of the relaxation time, determines the change in the mechanism controlling the reaction [6, 141.

Thus, the employment of rheokinetic data permits to establish the precise limits of the kinetic region of the reaction and, consequently, the optimal temperature-concentration regime of the production process.

The change in the relaxation state of the reactive system in the course of polycondensation may result from approaching to the region of relaxation transition with elongation of the reactive molecules. This situation is realized in the course of the synthesis of oligoimide based on isocyanate and dianhydride [17].

According to the results of IRS investigations, the rate of the consumption of isocyanate groups exceeds by 10% the rate of imide fragments formation in the course of oligoimide. This fact probably shows that a certain amount of isocyanate groups may be consumed in the course of the process. For this reason we used the values of concentration of imide fragments 1 as major initial data to estimate the kinetics of imidization. The viscosity of the samples of the end oligoimide product chooses in the course of synthesis was measured by the cone-and-plate rotation viscometer “Rheotest-2”. To reduce the release of carbon dioxide which might affect the results of the measurements, the viscosity estimate was taken at 160 “C.

Interaction of BTDA with MDI in the presence of a catalyst is a complex multi stage process which yields various by-products [ 18 - 201. At kinetic simulation of these processes, if there is no need to investigate the mechanism of the reactions, it is reasonable to the phenomenological approach permitting to establish the major quantitative principles of the


formation of the end product of the reaction. The time-dependence of the mean degree of polymerization a of the forming oligoimide, evaluated from the concentration of imide fragments, is presented in Fig. 2.4.14:

Figure 2.4.14: Change of the degree of polymerisation in the process of oligoimide formation at 170 "C ( I ) , 180 "C (2), 190 "C (3) and 200 "C (4).

Two facts are of interest here:

1. The first is that the values of are rather small and not exceeding 2 even at the final stage of the reaction. It means that a large amount of initial products remains in the reactive mass, and condensation actually does not go beyond the formation of products with a low degree of polymerization, whereas products with the highest degree of polymerization are practically absent. This fact has been also observed in a direct experiment, where the MWD of the reaction products at different stages of the reaction has been investigated by GPC [21].

The second fact is that the obtained dependencies of polymerization degree on time were of linear nature which is characteristic for a kinetic equation of the second-order reaction.

The numerical values of the constant k = kxo entering formula (2.4.3) have been determined at different temperatures based on the data obtained. The corresponding experimental data are given in Fig. 2.4.15 on the Arrhenius coordinates. The value of the apparent activation energy evaluated by this dependence appeared to be 153 kJ/mol. This result evidently means that the multistage process of 01-formation is actually limited by a certain second-order reaction, maybe by interaction of the products of hydrolysis between diisocyanate and dianhydride [19,20].

2.


Figure 2.4.15: Dependence of the rate constant k on temperature.

Of interest is the possibility of the evaluation of viscous properties of the reactive system using the correlations obtained for the polycondensation processes, taking into account the data of IRS investigations of the process under discussion, and the comparison of the data of this evaluation with the results of direct measurements.

The viscosity changes of the reactive mass at the initial stages of polycondensation are described by the following formula:

- _ - ( l + k t ) a q 0

(2.4.12)

However, according to the calculations following from the major rheokinetic relationships for polycondensation processes, formula (2.4.12) does not describe the kinetics of viscosity increase during synthesis of oligoimide resins at any reasonable values of index a . Since the numerical value of the constant k entering this formula is obtained on the basis of IRS measurements, the emerging discrepancy may be associated with the fact that the experimental methods used permit to estimate various elements of the process of 01-formation. The change in the concentration of the functional groups (of imide fragments) reflect only the chemical path of the reactions, whereas the macroscopic properties of the reactive system which are characterized by its viscosity are, to a great extent, determined by the physical-chemical events concurring the process of chemical transformation of functional groups. An increase of the polyimidization degree leads to a regular increase of the fluidity temperature up to the value of the actual temperature of the process (Fig. 2.4.16). Limitation of the mobility of the growing macromolecules as the reactive system approaches the temperature of the fluidity loss (T') causes, in its turn, an increase of viscosity at T = T' and q + OQ.


T I , O

200

150

f00

50

Figure 2.4.16: Change offuidity temperature of a reaction system in the process of

forming oligoimide resin at T = I70 "C ( I ) , 180 "C (2), 200 "C (3).

Therefore, although the process is carried out at a constant temperature, the equivalent state of the material is not provided due to the change of the distance from transition temperature. It is necessary, therefore, to introduce some corrections, accounting for the distance of the system from the transition temperature, into the initial rheokinetic correlation. Since the process is characterized by the existence of a specific point Tp where q + 03, it is reasonable to introduce a correction factor f in the form of the Scaling relation, obtained for gel systems [22,23]. Then we may write:

b t t*

f = ( 1 - - ) (2.4.13)

where t* is the time during which the fluidity temperature of the reactive system becomes equal to the temperature of the process, b is a certain constant which is generally not known in advance.

Then the viscosity change may be presented in terms of rheokinetic relation (2.4.12) customary for polycondensation reactions, with a correction in the form of the factor f in formula (2.4.13). Then we may write the following generalized expression, accounting for the influence of different factors on viscosity [24].

(2.4.14)


To reduce the amount of “free” constants, we assumed that a = b. This assumption, generally speaking, is not well-grounded. It may be confirmed or refuted by precise experimental data. Formula (2.4.14) may be regarded in this sense as semi-empirical.

Anamorphism of the experimental data of the coordinates of formula (2.4.14) is presented in Fig. 2.4.17. It is seen, that the values of viscosity of the reactive system obtained at different process temperatures may be presented by the same straight line which proves that formula (2.4.14) actually describes the kinetics of viscosity increase in the course of 0 1 formation in the given case. The value of index a appeared to be independent of temperature and equal to 3.4.

T 1 +kt Figure 2.4.1 7: Dependence of log - on log ~

t q0 1 - _ t*

0 200 “C A 190 “C 0 180 “C 0 170 “C.

The equality of the derived value of the constant a to the known “universal” value of the exponent in the formula for the dependence of viscosity of polymers on their MW, is, evidently, incidental taking into account that the value Nof the process under study is not very high. More- over, for polymers with increased rigidity of chains the value of index a in the dependence of q on M W may be significantly higher than 3.5, e. g. even for relatively diluted solutions of poly- p-benzamide in the concentration region before the formation of LC-phase the exponent is 6 - 8 [25].

Comparison of experimental and evaluated (by formula (2.4.14)) dependencies ~ ( t ) ,

shown in Fig. 2.4.18 demonstrates their practically complete coincidence throughout the process up to the moment of fluidity loss of the reactive system.


4

2

Figure 2.4.18: Experimental (dots) and calculated (solid lines) changes of viscosity with time at different temperatures of process performance. Symbols are similar with those in Fig. 2.4.14 (0 200 "C A 190 "C 0 180 "C 0 170 "C). 2' - Calculation by formula (2.4.12) for T = 180 "C.

Formula (2.4.14) permits to compare and estimate the effect of different factors determining the viscosity of the reactive mass, since its numerator represents the viscosity increase caused by elongation of the reactive molecules (that is, by the degree of polymerization), and the denominator represents the reduction of molecular mobility with approaching to the temperature of fluidity loss. A comparable effect of these parameters is observed only at initial stages of the reaction; then the major increase of viscosity occurs due to the reduction of the molecular mobility and relative importance of this factor increases continuously throughout the process. For instance, after 400 minutes of the reaction performed at 180 "C the value calculated from formula (2.4.12) constitutes only 20% of the value of viscosity of the reactive system experimentally observed or evaluated from formula (2.4.14).

It should be pointed out that, according to formula (2.4.14), the kinetics of polycondensation - complicated by relaxation transition - is also determined by a complex parameter being the product of the viscosity of the reactive medium and the mobility of reactive centres:

(2.4.15)


Although in this case the mobility of reactive molecules is determined not by their size, as

The results obtained show that in certain cases the anticipation of the property change of the reactive systems, based on investigation of kinetic parameters only, e. g. the change in the concentration of functional groups on the determination of the reaction rate constant, may lead to incomplete of even incorrect conclusions concerning physical-chemical properties of a material, since the properties of the reactive mass are often affected by secondary (as far as the chemical reaction is concerned) phenomena, for instance, phase or relaxation transitions.

In the case of polycondensation accompanied by a release of a low molecular product of the reaction, removed from the reactive zone, the value of polymerization degree is determined by the relation:

is the case of PSS formation, but by the distance from the region of relaxation transition.

(2.4.16)

where K is the equilibrium constant and [MI is the concentration of the low-molecular product .

Evidently, to obtain high-molecular products, the released substance should be removed fast and effectively, the rate of removal being limited by its diffusion through the layer of the reactive mass. Thus, except for very thin layers, the polycondensation reaction of this kind proceeds in the diffusion region [26-281.

The reaction of PETP formation occurs in the kinetic region only at the vacuum value of 0.1 Torr and with the thickness of the layer of the reactive mass not less than 0.2 mm [27]. If the thickness of the layer increases the process passes to the region controlled by diffusion of the released DEG. For this reason, when the process is conducted in sufficiently large reactive volumes, practically the entire reaction of polycondensation is likely to proceed in the diffusion region. As was demonstrated at the beginning of this section in the case of PA-610 synthesis, the process of viscosity increase in the reactive system is satisfactorily described by an exponential dependence. Fig. 2.4.19 shows the curves of viscosity increase in the synthesis of two laboratory samples, the linear and branched PETP of different structure in an autoclave [29]. These dependencies are also approximated by the formula of the form:

(2.4.17)

where q,, is the initial value of viscosity of the reactive system, and the constant k, may be regarded as the effective “rheological” constant of the rate of PETP formation.


hl- Q 3 I

9 Z - 2

1

30 60 t, min

Figure 2.4.19: Change of viscosity in the process of forming linear ( 1 ) and branched ( 2 ) polyethyleneterephthalate (PETP) at To = 275 "C.

For the present considerations these experiments are of interest in terms of clarifying the problems to what extent rheological properties of the reactive systems are affected. Whereas curve 1 in Fig. 2.4.19 corresponds to the process of synthesis of normal linear PETP, curve 2 characterizes viscosity increase during formation of PEPR of branched structure. Emergence of branches is explained by adding of a modifier, promoting the formation of additional hydroxyl groups in the middle of the chain at the stage of interesterification [30]. Emergence of a secondary hydroxyl group leads to a significant acceleration of the process of viscosity increase - k , in formula (2.4.17) increases from 0.05 to 0.19 min-'.

Another essential problem is interconnection of the constant k, with the kinetics of polycondensation, i. e. with the kinetics of increase in MW of the forming polymer. Combining formula (2.4.17) with the known exponential expression of the dependence of melt viscosity and polymer solutions of MW, we obtain the following expression:

(2.4.18)

I/a V O There il?, is mass-average MW, 0 is the constant equal to (-) k

The rate of MW increase is determined by values of k, and the exponent a, which in this case is to be regarded as a structure-sensitive parameter, since it is known to vary with the polymer structure [31]. Formula (2.4.18) permits to implement direct evaluation of the MW change in the process of polymer formation on the basis of the data of rheokinetic measurements and takes into account the effect of polymer structure on the kinetics of its formation. The comparison of the results of the evaluation of this kind with MW values determined by sedimentation in [32] is given in Fig. 2.4.20:


20

I 31

t, min

Figure 2.4.20: Experimental ( I , 2 ) and calculated ( I ’, 2 ’, 3) by formula (2.4.18) dependencies of average molecular mass in the process of forming linear (I) and branched (2, 3) polyethylenterephthalate: 2’ - M,(t) calculated at a= 1.9, 3 - M,(t) calculated at a = 1.6.

The value of index a was determined independently, and for linear PETP in the given range of MW it appeared to be 1.6. As the data presented in Fig. 2.4.20 show, total consistency of experimental and theoretical values of M, for linear PETP and significant deviation of these values for PETP with the branched structure of macromolecules is observed. This difference is caused by a change of index a for polymers with branched structure, associated with strengthening of intermolecular contacts in branched PETP. The increase of index a to 1.9 leads to practically full coincidence of experimental and calculated values of MW (curve 2 in Fig. 2.4.20). According to the experiments, the dependence q(M,) for the given samples is, indeed, characterized by this index value [29]. These results show that the rheokinetic approach is sensitive to the change in structure of the forming polymer and may provide grounds for preliminary quantitative estimates of the degree of branching of polymers directly in the course of their formation.


1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1 1 .

12.

13.

14.

15.

16.

L. B. Sokolov, Osnovy sinteza polimerov metodom polikondensatsii, Moscow, Khimiya, 1979, p. 264.

M. I. Siling, Polikondensatsiya. Fiziko-Khimicheskiye osnovy i matematicheskoye modelirovanie, Moscow, Khimiya, 1988, p. 256.

G. I. Demina, S. G. Kulichikhin, E. S. Bokareva, A. Ya. Malkin, Proizvodstvo i pererabotka plastmass i sinteticheskikh smol, 1978,8,44.

G. I . Demina, S. G. Kulichikhin, E. S. Bokareva, A. Ya. Malkin, Plasticheskiye massy, 1980,1,57.

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S. G. Kulichikhin, V. A. Kozhina, L. M. Bolotina, A. Ya. Malkin, Vysokomolekulyarnye soedineniya, 1982, vol. 24,4,309.

A. Kh. Bulai, V. N. Klyuchnikov, Ya. G. Urman, I. Ya. Shonin,L. M. Bolotina, V. A. Kozhina, M. M. Golder, C. G. Kulichikhin, V. P. Begishev, A. Ya. Malkin, Polymer, 1987, vol. 28, 7, 1349.

V. V. Korshak, I. P. Storozhuk, A. K Mikitaev, Polikondensatsionnye Protsessy i Polimery. Nalchik, KBGU, 1976,40.

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V. A. Kozhina, L. M. Bolotina, A Kh. Bulai, V. N. Klyuchnikov, I. Ya. Slonim, Ya. G. Urman, Matematicheskoye modelirovaniye: apparatumoye oformleniye polimerizatsionnykh protsessov. Tezisy dokladov, 1983,60.

V. N. Klyuchnikov, A. Kh. Bulai, Ya. G. Urman, I. Ya. Slonim, L. M. Bolotina, L. E. Reitburd, M. M. Golder, Vysokomolekulyarnye soedineniya, A, 1984, vol. 26, 8, 17 18.

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133

Chapter 3 Rheokinetics of Oligomer Curing

3.1 General principles

Numerous and varied applications of polymer materials are based on the technology of oligomer curing (structuring). This process allows the formation of an article with a specific shape when the material is still a relatively low-viscous liquid. It enables to impart the manu- factured articles with complex and sometimes quite elaborate shapes to effect the flow of the material in the presence of large quantities of a solid filler or fabric impregnation, etc. Then, due to chemical processes, three-dimensional-networks of chemical bonds are formed and the liquid loses fluidity turning into an infusible and insoluble state. The forming polymer acquires strength, rigidity and other valuable properties of a construction material. Here it is essential that the entire technological process is time-dependent and a certain level of rheological properties is essential for each particular stage of the process, because both an excessively fast viscosity increase and an excessively long retention of fluidity may have a negative effect on both the productivity of the technological process and the properties of the end product.

Thus, changes of rheological properties in the course of transition from liquid to solid state are the very essence of the technological process to produce articles from reactive oligomers and, consequently, its course is determined by curing rheokinetics. To some extent curing of reactive oligomers is similar to curing rubber compounds. Features common for both processes are the formation of a three-dimensional network of chemical bonds and the appearance of material which has lost fluidity. The differences are caused by the nature of the initial substance: For oligomer curing it is a relatively low-molecular product and for elastomer curing it is a high-molecular compound. Furthermore, in the former case a polymer with a thick network in a glass-like state is formed, in the latter case the network is rather rare and the formed polymer is valuable for its ability to retain high elasticity. Besides, rheokinetics of both processes has much in common. Therefore many theoretical considerations and experimental methods developed for oligomer curing find ample use in the studies of elastomer curing. The similarity of the processes is emphasized by the fact that presently applications of molding technologies for the production of industrial rubber articles and even tires from reactive oligomers are used, which, however, yield rubbers rather than glassy products in the course of polymerization. Here, naturally, general rheokinetic methods may be fully applied.

Generally, the process of curing may be divided into two stages:

At the first stage branching of macromolecules increases but no continuous network is formed which would cover the whole bulk of the material:

1.

2. This network is formed at the second stage.



134 Chapter 3: Rheokinetics of Polymer Curing

In this respect, it is possible to assume a certain critical degree of conversion p* corresponding to the transition from the 1st to the 2nd stage of the process. However at p < p* the formation of more or less extensive regions of interlinked chains, which may be considered a microgel, is also possible. Whether a continuous network of covalent bonds is formed at some “critical” moment or the process preferably develops by the formation of closed cyclic structures of different size which later merge to form a common network - in general - the particular topological forms appearing during condensation of polyfunctional molecules depend on the nature of reacting substances and the rate ratio of different elementary reactions. It determines the specific value of p* corresponding to the loss of fluidity, i. e. the moment designated by different terms - “structuring”, “gelation”, “gel formation”, etc. implying the same phenomenon of fluidity loss.

In the classical statistic theory of gel formation developed by Flory it was shown that when no cyclic structures are formed and all the reactive groups are assumed to have equivalent chemical reactivity irrespective of their environment, the critical value of the branching factor a* corresponding to the point of gel formation is

(3.1. I )

where f is the arithmetic mean of functionality of the molecules forming the branches.

Correlation between a and the conversion degree of the groups of A type (PA ) and B type ( p, ) is expressed as

(3.1.2)

where r is the equimolar factor (total number of groups of A type vs. those of B type). p is the share of reactive groups belonging to polyfunctional molecules in the total number of these groups in the system.

At equimolar concentration of groups A and B the factor r equals 1 and p, = 0, and, respectively, the critical value of a* is related to a critical value of the conversion degree p* = PA = p, . Thus, if interaction of two three-functional compounds has the reaction path

A3 + B3, then f = 3 and a* = 0.5.

In this case p = 1 and from (3.1.2) it follows that p* = @ = 0.707. In other situations the critical values of p* may be quite different. Moreover, from equation (3.1.2) it follows that gel formation can occur only with a certain correlation between r and p.

3.1 General principles 135

According to the statistic model of gel formation, changes in the polymer yield and product viscosity are represented schematically as shown in Fig 3.1.1 where p* designates the moment of time corresponding to the point of gelation.

Figure 3.1.1: Theoretical prediction for time-dependent changes of viscosiv, the rubbery

modulus G‘ and the gel fraction content g according to the statistic model

of gel formation.

Later, repeated attempts were made to achieve quantitative agreement between theory and experiment using considerably improved models of structuring (gel formation) which could take into account the fact that the formation of a three-dimensional network in the general case necessarily implies the formation of a cyclobranched structure, i. e. ring formation before the point of gel formation is reached. If intramolecular ring formation is also considered critical conditions of gel formation for a* differing from equation (3.1.1) can be formulated. This approach was developed by different authors in various forms when an additional parameter considering the probability of intramolecular reactions was introduced into the model (e. g. see [3]). Depending on the adopted scheme of the process, expressions for a* and p* appear to be different. Therefore Flory’s model may be regarded as the first simplified approximation to the real situation when critical structuring conditions are attained.

Flory’s scheme and, accordingly, the nature of changes in the rheological properties of the cured oligomer (Fig. 3.1.1) are quite representative but they do not reveal two important circumstances. The first is the formation of microgel before reaching the gel-point. The very notion of a microgel is rather arbitrary. In fact, microgel is a large extensively branched macromolecule which has reached a colloidal size. But this definition is quantitatively ambiguous, and therefore the evaluation of the moment depends on the research method used.

Second, before the moment of fluidity loss the already formed macromolecules can reach a sufficient size to show flexibility and a set of relaxation properties. Therefore the behaviour which is actually observed will differ in some essential details from that shown in Fig. 3.1.1. This is illustrated by Fig. 3.1.2 where the dashed line represents the dependencies of Fig. 3.1.1 and the solid lines show what usually occurs in reality:


Figure 3.1.2: Act 1 pattern (the solid line) of change in gel fraction content (g), viscosity and rubbery modulus G’ in the process of oligomer curing.

The difference, however, caused by the fact that the rubbery module G’ at a certain frequency and the equilibrium modulus of high elasticity are used for comparison, is of no principal value, as usually the elasticity of the forming polymer is evaluated from the values of G’ .

Changes of rheological properties in the course of structuring depend on specific details of topology of the forming branched (solid fractions) and cross linked structures (in gels). In general they remain unknown. Therefore we can refer only to extreme model cases. Corres- ponding calculations were made by Kastner [2] who regarded the relaxation spectrum as a sum of spectra corresponding to gel and solid fractions. According to his calculations, which should primarily be treated as model-illustrative ones, in the course of structuring before reaching p* the long-term portion of the relaxation spectrum is expanded towards longer relaxation times. Herewith, the relaxation spectrum in the region of long times has a V-shape and its length depends on p whereas the angle of the “V” slope depends on the nature of branching during the growth of the macromolecules.

Further we shall consider the most general qualitative regularities of changes in rheological properties of structuring systems at different stages of structuring depending on correlation between the determining factors.

Initial oligomers are relatively low-viscous liquids where cross links are formed due to reactions of structuring (usually of the polycondensation type) and eventually a rigid material with a rather dense three-dimensional network of chemical bonds is formed. If the process is carried out at a temperature exceeding the temperature of vitrification of the material, the end product will be an elastomer, and the differences between oligomer curing and rubber cross-linking (curing) will disappear. Therefore classification of conditions for oligomer curing should be based on the physical (relaxation) state of initial and end products. An important

3. I General principles 137

factor here is the change in the position of the temperature of relaxation transition (vitrification) Tg as the conversion degree of the cured composition increases. This is shown schematically in Fig. 3.1.3 where the curve shows the dependence T,(p) (the dashed line shows the critical degree of conversion p* corresponding to the moment of network formation) and four marked zones which correspond to various physical states of the material. Considering the dependence Tg(p) and the diagram shown in Fig. 3.1.3 it is important to introduce another determining parameter: The temperature at which the reaction of curing is conducted.

Figure 3.1.3: Dependence of the glass transition temperature upon the degree of conversions: I liquid: II elastomer (cured rubber);

III glassy liquid; IVpolymeric glass.

General ideas concerning possible correlations between the change of vitrification temperature in the course of curing Tg@) and the temperature of the process Tcure were formulated in the works of Gillham [3-51. Therein a basic role is played by the moment of fluidity loss t*, where two principally different cases are possible:

The first case: Fluidity is lost at Tg(r*) > T,,,. It means that elastomer is formed and

further chemical reactions proceed in a state of high elasticity. Then the moment t* corresponds to the first relaxation transition (from melt to the high elasticity state) and at a certain degree of curing at the moment tg > t* due to increasing density of the space network the condition

Tg,r = = Tcure is reached and the second relaxation transition occurs - from elastomer to glass.

Both of these transitions are expressed, like any relaxation transitions, in the form of the maximum tangent of the angle of mechanical (or dielectric) losses tan8 as its time dependence is measured in the process of curing.

6

The second case: Fluidity disappears due to vitrification of the reaction mass where the

possibility of further curing is by far not exhausted, i. e. Tg,t = < Tcure at tg < t* . In fact, it

means that the flow ability does not disappear due to formation of a three-dimensional network, but due to vitrification (very intensive viscosity growth) of the reaction mass. If the temperature

T,,, is increased the reaction will continue and again either the first (fluidity loss at t*) or the

second case (fluidity loss at tg) will be possible. Naturally, vitrification of the reaction mass, like

any relaxation transition, is determined by the maximum of tans.

g


Gillham generalized the above mentioned qualitative considerations to T-T-T-diagrams (time - transformation - temperature) [3,4] which show correlation between temperaturehime regions corresponding to different chemical and relaxation states of the system. These diagrams may be plotted also for non-isothermal regimes of curing with the known law of t-dependent changes of Tcure. In this case both the position of “technological” regimes and the properties of the end product certainly depend on the rate of changes of the temperature TCure. However, it should be emphasized that Grillham’s diagrams refer to one-phase systems which do not undergo phase separation in the process of structuring.

Thus, curing of reactive oligomers necessarily has the time region where the material retains fluidity (at t < t* or at t < t g ) and a region where the material becomes solid (rubber-like or glassy). Accordingly, two principally different methods of investigation are possible - viscometric and dynamic.

The first consists in measuring time dependence of the viscosity q at different temperatures - the method of “q-t-T-diagrams”.

The second method consists in measuring the complex dynamic rubbery modulus 6“ (at a constant frequency) and may be called the method of “@ -t-T-diagrams”, a particular version of which is measuring tans.

These measurements provide correct selection of the technological regime of curing compositions based on different oligomer binders. A separate task here remains the determining of a correlation between the chemical nature of the oligomer (or its reactive groups) and the curing kinetics found with direct chemical or structural methods (i. e. by IRS), and the observed changes in rheological properties of the material. Research of that kind can be done both with pure oligomers and with filled compositions based on them. Although in the first approximation we may assume that the introduction of a filler does not influence the kinetics of curing, the form of the curing diagram and the position of characteristic lines of gelation and vitrification; for more complicated cases this is not true since an active filler, particularly if introduced in large quantities, may have a considerable influence on the regularities of curing and the position of characteristic points of the discussed diagrams. The role of the filler in the process of curing is studied on the basis of the same principles which led to the plotting of curing diagrams. However, the studies of the effect the filler (or generally the composition of the cured material) has on the curing diagram are a separate task which is important in terms of optimisation of compositions for different uses and the selection of the most appropriate technological regime for molding articles from cured oligomers.

In the technological practice it is usual to measure the dependencies q and G’ separately, the modulus being measured at a general fixed frequency (most often - lh). Meanwhile, the material response to different frequencies yields independent valuable information about the behaviour of the material. It is still more useful for cured compositions because repeated measurements of viscosity and modulus (at different frequencies) dependencies reveal notable variations of properties and behaviour even in apparently uniform samples which impedes


comparison of the obtained experimental results. Therefore particularly promising for this research is the method of multifrequency measurements, or the method of mechanical fourier-spectroscopy suggested in [6] and, independently, in [7-91.

The main point is the use of a non-sinusoidal effect on the material to initiate, respectively, its non-sinusoidal response. Then both the agitating signal and the response are expanded into a series of Fourier-equations, i. e. are represented as a set of harmonics having different frequencies. For each of them the time dependence of the modulus and tan6 are found, i. e. changes of different relaxation modes in the course of curing are analysed simultaneously.

It is well known that a complete characteristic of viscoelastic properties of the material is provided by its relaxation spectrum H(0) connected with the frequency dependence of the complex dynamic rubbery modulus 6" (0) by the known correlation:

(3.1.3)

where G, is the equilibrium rubbery modulus, AG is the relaxing component of the modulus.

Consequently, if we measure the dependence &(a) in a wide frequency range, it will enable us to trace the evolution of the relaxation spectrum in the course of curing. However, to obtain a complete pattern it is necessary to analyse a very wide frequency range covering several decimal orders. Unfortunately, this is not feasible in a single experiment. However, it is reasonable to study changes of viscoelastic properties in the course of curing measuring the parameters of viscoelastic behaviour of the model with a limited set of constants making a relatively narrow frequency range sufficient for analysis.

In this respect it is convenient to use a four-element model of the viscoelastic liquid - the so-called Burgers' model. This model consists of a viscous and a viscoelastic (Calvin-Feucht) element connected in series. The viscosity at the stationary flow is qo. The relaxation time of the viscoelastic element T, = q,/AC, where AG is the relaxing modulus and q, is the internal viscosity. For the sake of complete consideration, the possible existence of an equilibrium (residual) rubbery modulus G, is taken into account. If we consider how the parameters of this model change in the course of curing, two principal effects will be immediately seen:

at t < t * G, = 0,andat t 2 t * G,#O.

The use of mechanical Fourier-spectroscopy, or multifrequency analysis, must reflect the above mentioned effects provided all four constants are measured simultaneously (in a single experiment). Fig. 3.1.4 shows an example of results for measurements of time dependencies of the model parameters qo, T, G, and AG obtained from the analysis of curing rheokinetics of a polyurethane composition [6]. It is seen that the results of the measurements are in perfect agreement with the general scheme of changes in the relaxation parameters of the material in the course of curing a reactive composition, at least, for the transition from liquid to solid at t* :


Figure 3.1.4: Changes in the parameter values for the model of viscoelastic behaviour of a cured polyazethane composition measured by the method of multi-frequency (Fourier) spectroscopy.

This transition is also reflected in the appearance of a loss maximum of the time

dependence tan6 (Fig. 3.1 S). The method of mechanical Fourier-spectroscopy also allows to detect the appearance of a loss maximum at different frequencies, and it is very important

methodically that the time dependencies tan6 are obtained from an experiment performed with a single sample:

tuns t d

6

-t t*

Figure 3.1.5: Typical time dependencies of mechanical losses observed in the course of curing oligomers at diflerent relative positions of experimental and glass temperatures.


The data in Fig. 3.1.4 shows the presence of another phenomenon which was mentioned above in the discussion of Fig. 3.1.2, namely, the presence of relaxation (viscoelastic) phenomena at t < t* since there are non-zero values of relaxation time and AG in this region. This phenomenon is evidently caused by the formation of a fluctuation network of engagements in the region where a continuous three-dimensional network of chemical bonds has not yet been formed, the formation of the latter leads to appearance of not only AG but also G,. Therefore the discussed method makes it possible to observe changes of two networks having different nature-physical bonds (fluctuation network) and chemical bonds (stable network).

Thus, the given example clearly demonstrates the advantages of the method of multifrequency (Fourier) spectroscopy for contemporary rheokinetic analysis of forming and developing networks of different types in the course of curing reactive compositions (e. g. elastomer curing).


1.

2.

3.

4.

5 .

6.

7.

8.

9.

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A. Xu Guohu, Zhu Ying, Thermochim. Acta, 1987, vol. 123,s 1.


Determination of the Point of Gel Formation

The point (time) of gel formation t* is one of the most important kinetic characteristics of

curing since it characterizes reaching of some critical degree of conversion corresponding to the

transition from the first to the second stage of the process. In the classical statistic theory of gel

formation developed by Flory [ l ] the moment of gel formation is characterized by the

appearance of a macromolecule in the reactive system, having an infinitely large molecular

mass M, + 00 [2]. Naturally, this condition requires viscosity to be turned to infinity.

These general considerations imply the possibility of experimental determination of the

point of gel formation as the moment of fluidity loss in the reactive system. The application of rotational viscometers in this field results in separation of the cured composition from the work-

ing surfaces of the device. The break-off moment naturally depends on the intensity of material

deformation when viscosity changed and, the higher the shear rate the earlier the flow of the

cured material is interrupted. Therefore the error in determining the point of gel formation is related to the regime of viscometric measurements. As a rule, it is possible to obtain a complete

curve of viscosity changes in the process of gel formation of the shear rate changes in the range

of several decimal orders, and the moment of reaching the gel-point is determined at the mini-

mum shear rate. Strictly speaking, the most precise determination of t* with the viscometric

method is only possible provided the experiments are made at several shear rates with subse-

quent extrapolation to the zero value. It should be stressed that the effect of shear rate on the

separation of the reaction mass from the working surfaces of the viscometer should not be iden-

tified with the problem of influence, the mechanical field has on the very kinetics of viscosity

growth in the processes of curing. This problem is of general value for rheokinetics of polymer

formation and conversion processes and is considered in Section 4.3.3.

In some cases, a reactive system retains fluidity until the upper limit of the effective range

of the viscometer is reached. Then the point of gel formation is often determined by extra-

polation of the dependence of the reverse viscosity value on time at the final stages of gel for-

mation to zero - l / q + 0 [3 - 61. This approach implies a rather strict determination of the

gel-point as the time of reaching infinite viscosity although it assumes some ambiguity of extrapolation. An example of this kind is shown in Fig. 3.2.1. According to some estimates, the error in determining the gel-point by extrapolation does not exceed several per cent [4]:

Another possibility to determine the point of gel formation by means of rheological

methods is provided by the use of dynamic mechanical spectroscopy. Analysis of changes in the

dynamic mechanical characteristics of reactive systems shows that the criterion of reaching the point of gel formation may be the specific time of the process at which tan6 or the loss modulus

G" goes through a maximum [7 - 91. It is of interest to consider the intersection point of the

curves corresponding to the changes of the rubbery modulus and the loss modulus, i. e. the moment of time at which tan6 = 1 treated as reaching the threshold of gel formation [ 10 - 121.

3.2 Determination of the Point of Gel Formation 143

- I I T (.,

. 0, f5 e L .

0,fO

0.05

2 3 4 s 6 t , min

Figure 3.2.1: Determination of the point of gel formation by extrapolation of the reverse viscosity value to zero. Curing of silico-organic oligomer at T = 190 "C ( I ) , 200 OC (2), 220 "C (3).

However, theoretical calculations showed that the conditions of gel formation could only comply with the intersection point of the rubbery modulus and the loss modulus if the relaxation behaviour of the material obeyed a definite law and the coincidence of the moment when G' = G" with the gel-point is a particular case [ 13 - 15 1. Fig. 3.2.2 shows changes of viscosity of the components of the dynamic modulus in the course of thermal treatment of fusible polyetheramide [16]. The equality of values G' and G" is reached with polyetheramide preserving its viscous-flow state at a sufficiently high level of viscosity, whereas the point of gel formation registered when the flow actually stops, corresponds to a much longer duration of the process.

20 GO 60 80 f, min

Figure 3.2.2: Change of viscosity q rubbery modulus G' and loss modulus G" in the process of heat treatment of fusible polyetherimide T = 350 "C.


The most general approach to the determination of the point of gel formation with rheological methods is most probably comparison of viscometry with reaching the maximum tangent of the angle of mechanical loss. Fig. 3.2.3 shows an example of determining the position of the point of gel formation from the change of tan6 and viscosity in the course of curing epoxy silico-organic oligomer. The time of reaching the maximum tan6 practically coincides with the moment of fluidity loss and corresponds to the conditions of gel formation as a relaxation transition from the viscous-flow to the high-elasticity state. Similar results were obtained for structuring systems of various nature [9, 17 - 231:

fan$ 9 20

0. fa5

OI io

0, 0s

Figure 3.2.3: Change of viscosity and mechanical loss angle tangent in the course of curing epoxy silico-organic oligomer. T = 180 "C (I, l'), 160 "C (2, 2'), 140 "C (3, 3') and 120 "C (4, 4').

It is necessary to note that in dynamic mechanical experiments the point of gel formation is assumed as reaching both the maximum value of tan6 and the loss modulus. However, the positions of extreme on the time dependencies of tan6 and G" always have minor inconsistences caused by a peculiar nature of relaxation behaviour of viscoelastic bodies.

Besides, deviations are possible between the position of the maximum G" and turning of viscosity to infinity as it was observed in the structuring of silicoorganic rubbers [24]. The difference between these two points amounts to 7% of conversion which let to the assumption that the identification of the point of gel formation with dynamic mechanical measurements is impossible [4]. Apparently, this conclusion is not quite grounded and the observed divergence in a particular system may be connected with structuring peculiarities of a given material and underestimated effects of deformation intensity when the point of gel formation is determined on the basis of viscometric measurements.

The high content of information of dynamic mechanical characteristics used to register points of gel formation is demonstrated by the results obtained for the kinetics of forming gels of a physical nature (products of phase disintegration of polymer solutions) investigated with rheological methods [ 151. In this case changes in microscopic properties of the material are similar to gel formation in the course of chemical reactions of polyfunctional compounds. When the system "gelatine-water" is turning to a gel, viscoelastic properties change from characteristics of a viscous liquid (G' = 0) to those of a completely elastic body (G" = 0), and the extreme change of the loss modulus indicates the existence of a specific point characterising the transition from a viscoelastic liquid to a viscoelastic solid body [25]. Thus, comparison of the results of viscometric and dynamic mechanical measurements is a physically justified way to determine the points of gel formation in cured compositions.

3.2 Determination of the Point of Gel Formation 145


I . P. J. Flory,

2.

3. 4. 5.

6. 7. 8.

9.

10. C.-Y. M. Tung, P. J. Dynes, J. Appl. Polym. Sci., 1982, vol. 27, 2, 569. 1 I . H. H. Winter, F. Chambon, J. Rheology, 1986, vol. 30, 2,367. 12. F. Chambon, H. H. Winter, PolymerBulL, 1985, vol. 13,499. 13. H. H. Winter, Colloid Polymer Sci., 1987, vol. 75, 104. 14. H. H. Winter, Polym. Eng. Sci., 1987, vol. 27, 22, 1698. 15. E. E. Holly, S. K. Venkataraman, F. Chambon, H. H. Winter,

16. M. M. Koton, S. Y. Frenkel, Y. N. Panov, L. S. Bolotnikova, V. M. Svetlichny, I. A. Shibayev,

Principles of Polymer Chemistty, Ithaca, New York, Come11 University Press, 1973,672. S. I . Kuchanov, Protsessy studneobrazovaniya v polimernkhsystemakh, Mezhvusovsky Nauchny Sbornik, Saratov University, €? 1,1985,61.

V. M. Conzales-Romero, C. W. Macosko, I. Rheology, 1985, vol. 29, 3, 259. C. W. Macosko, Brit. Polymer. J., 1985, vol. 17, 2,239. S. G. Kulichikhin, A. S. Reutov, M. S. Surova, E. V. Osipova, A. Y. Malkin, Plast. massy, 1988,5,43. D. S . Lee, C. D. Han, Polym. Eng. Sci., 1987, vol. 27, 13,955. I . K. Gillham, Brit. Polym. J., 1985, vol. 17, 2,224. S. G. Kulichikhin, Problemy teplo- i massoperenosa v topocknykh ustnoistvakh, gazogeneratorakh i khim. reaktorakh, Minsk, ITMO AN BSSR, 1983,88. P. A. Astakhov, S. G. Kulichikhin, L. I. Golubenkova, V. A. Kozhina, E. I. Chibisova, Y. P. Chernov, A. Y. Malkin, Vysokomolekulyarnye soedineniya, B, 1984, vol. 26, I I , 864.

J. Non-Newt. Fluid. Mech., 1988, vol. 27, I , 17.

S. G. Kulichikhin, E. E. Krupnova, A. S. Reutov, I. I. Ushakova, Vysokomolekulyarnye soedineniya, A, 1988, vol. 30, I I, 2425.

17. S. G. Kulichikhin, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina,. I. Golubenkova, A. Y. Malkin, Protsessy perenosa v strukturiruyushchikhsya zhidkostyakh, Minsk, ITMO AN BSSR, 1985,63.

18. A. Y. Malkin, S. G. Kulichikhin, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina, L. I. Golubenkova, Mekhanika Kompozitnykh Materialov, 1985,5,878.

19. S . G. Kulichikhin, G. I. Shuvalova, V. A. Kozhina, Y. P. Chernov, A. Y. Malkin, Vysokomolekulyarnye Soedineniya, A, 1986, vol. 23, 3,498.

20. L. A. Rodivilova, L. A. Korolyova, N. V. Yurchenko, S. G. Kulichikhin, V. A. Kozhina , Z. D. Abenova,Y. P. Chernov ,Y. A. Dubrovsky ,V. A. Safonova, Polyamidnye Konstruktsionnye material) M, N I1 TEChlM, 1986.58.

21. S. G. Kulichikhyn, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina,L. I. Golubenkova, A. Y. Malkin, Vysokomolekulyarne soedineniya, A , 1986, vol. 28, 10,2112.

22. S. G. Kulichikhin, Z. D. Abenova, N. I. Bashta, 0. P. Blinkova, G. S. Matvelashvily, A. Y. Malkin, Plast. massy, 1988,3,51.

23. S . G. Kulichikhin, Y. P. Chernov, V. A. Kozhina, A. S. Reutov, I. I. Miroshnikova, A. Y. Malkin, Mekhanika Kompozithnykh materialov, 1988,2,350.

24. E. M. Valles, C. W. Macosko, Macromolecules, 1979, vol. 12, 4,673. 25. M. Djabourov, J. Maquet, H. Theveneau, J. Leblond, P. Papon,

Brit. Polym. J., 1985, vol. 17, 2, 169.


3.3 Viscosity Increase to the Point of Gel Formation

One of the main problems in studying kinetic regularities of viscosity change in the processes of curing is the effect of deformation on viscosity of reactive systems. This problem is of principal importance since any technological operation at the stage of gel formation is linked with certain mechanical effects on the cured composition.

If the curing process proceeds without distortions of isothermal nature the deformation regime, in general, has no apparent effect on the kinetics of viscosity change in the curing of a homogeneous system. Absense of shear rate effects were registered in specially performed experiments with reactions involving totally different (in their chemical nature) substances forming cross linked materials. This is shown in Fig. 3.3.1 where typical dependencies of viscosity on time are given for the curing process of epoxy silico-organic oligomer [ 11:

0

4

0-1050 A-226

-

3 3 I5 t" t, min

Figure 3.3.1: Viscosity values in the course of curing epoxy silico-organic oligomer

obtained at diFerent share rates. T = 150 'C.

Viscosity values obtained with shear rate changes by several decimal orders lie within the same dependence. Since analogous results were obtained for various objects independent of whether the curing process is polymerization [2,3] or polycondensation [4,5], it is possible to conclude that the given fact is of a rather general value.

Viscometry is one of the most common methods to investigate the initial stages of curing processes yielding information which is significant for technological applications. For this reason, extensive experimental material concerning regularities of viscosity change at the initial stages of forming cross-linked materials exists. To describe the kinetics of viscosity increase in the process of gel formation, empirical formulae with an exponential co-factor are often suggested [6-81:

3.3 Viscosity Increase to the Point of Gel Formation 147

rl = 170exp ( W (3.3.1)

where qo is the initial viscosity value in the reactive system, 8 is the constant quantitatively characterising the rate of viscosity change.

Formulae of this type are only used, as a rule, for purely practical needs because, when dealing with applied tasks, they provide sufficiently precise predictions of viscosity values for the reactive system as the temperature or the composition of the cured material changes [9-131. A disadvantage of these formulae is that they provide no formal reference to the gel-point at the moment at which q + = since, according to 3.3.1, viscosity is limited at any moment of time t . This consequence is, to a certain extent, formal since the gel-point may be identified with a certain level of viscosity values, e. g. q = lo3 or lo4 Pas, then t* is the time needed to reach this level. Attempts were reported to relate the nature of time dependence of viscosity in the course of curing to the reaction kinetics. Thus, in the course of curing diglycol ether of bisphenol A by triethanolamine the curve of viscosity growth q( t ) was approximated by two linear fragments [14]. The appearance of a bend point was related to the formation of an engagement net, and the linear dependence q(t) in the first fragment was reportedly caused by the fact that the reaction of curing was described by a zero-order kinetic equation.

A formally more strict and, to a certain extent, alternative approach to calculation of viscosity growth is based on changes in molecular characteristics of a cured oligomer.

In a general case polymer viscosity is connected with molecular mass by the following dependence [ 151:

q = K M a (3.3.2)

where K is the constant depending on temperature and polymer nature, a is an exponent which has the “universal” value a = 1 at M < M, and a = 3.5 at a > M, . M, is the critical value of molecular mass corresponding to change in the nature of intermolecular interactions in polymers.

Determination of interrelation between rheological and molecular characteristics of cured materials is very difficult due to formation of complex branched molecular structures during interaction of functional compounds. The most detailed and systematic investigations in this field were carried out with structuring of polyurethanes [16 - 191 and siloxane oligomers [20-23]. In these processes the molecular mass of an oligomer at different times was calculated from changes in concentration of functional groups, e. g. isocyanate groups for structuring of polyurethanes.


As a result, the following correlation was derived for polyurethane based on &-caprolactam and hexamethylenediisocyanate, which reflects the interrelation of viscosity, temperature and molecular mass [23].

( C / R T + S ) -

q = Ae (3.3.3)

where A, D, C and S are the constants, MWo is the initial molecular mass, R is the universal gas constant.

In a general case, formula (3.3.3) is the consequence of formula (3.3.2), but it includes a great number of constants characterising changes of viscosity and molecular mass of an oligomer with temperature.

Generally, the temperature dependence of viscosity in reactive systems is determined both by the activation energy of a process of the chemical reaction itself and that of viscous flow [24]. Therefore the effect of temperature on viscosity of reactive systems is ambiguous, and identifying contribution of different components depends on the reaction mechanism [25 - 271. To simplify the situation and to exclude the temperature dependence of viscosity, a relative variation of this parameter q/qo is considered, where q, is the initial viscosity value of a reactive system. This assumption is valid provided the activation energy of viscous flow remains constant throughout the state of gel formation, since in a general case E should change with the degree of conversion.

It is necessary to note that the change of viscosity in the process of polyurethane structuring even on the basis of bifunctional compounds often reflects formation of branched and, eventually, cross-linked structures which is evidently connected with the formation of a physical network the points of which are sufficiently strong specific interactions like hydrogen bonds [28]. An example of this process is interaction of macrodiisocyanate with 3,3-dichloro- 4,4-diaminodiphenylmethene [ 171.

Polycondensation kinetics of two bifunctional monomers at their equimolar ratio is usually described by a second-order equation from concentrations of functional groups which after integration yields a linear dependence of the mean polymerization degree on time (2.4.3). An example illustrating the change in polymerization degree during interaction of macrodiisocyanate with diamine is given in Fig. 3.3.2. Here the dependence N(t) is approximated not by one, but by two linear fragments with different slopes which indicates a change in the constant of reaction rate in the course of the process. Similar conclusions showing the impossibility of describing even initial stages of curing by a single value of the rate constant were drawn from the analysis of results of isothermal calorimetry [ 171:


2- f I .5

f 2 3 4 5 t, min

Figure 3.3.2: Dependence of the mean numerical degree of polymerization on time in the process of polyurethane fomzation at 60 "C ( l ) , 70 "C (2), 80 "C (3), 90 "C (4), 100 "C (5).

Viscosity changes with increasing degree of polymerization in the course of the discussed process are also characterized by two linear fragments (on logarithmic coordinates) with the exponent varying from 1 to 4.6 (Fig. 3.3.3). If the value a = 1 is typical of a polymer with a small molecular mass, the value 4.6 exceeds the "universal" value of this index, which is 3.5. Consid- eration of the temperature dependence of viscosity of a reactive system also reveals the validity of formulae (2.4.8) and (2.4.9) only at the index value 4.6. This index value is not typical of linear polymers but rather common for branched macromolecules.

o,f 42 0.3 eo3 tii Figure 3.3.3: Viscosity dependence of polymerization degree during polyurethane

formation on the basis of macrodiisocyanate and 3.3'-dichlor-4.4' -diaminodiphenylmethane at 60 "C ( I ) , 70 "C (2), 80 "C (3), 90 "C (4).


The described peculiarities of the process are explained by the fact that after the very first stages of the reaction the growing chains acquire branches being the nuclei of the insoluble fraction [ 171. This assumption is confirmed by data shown in Fig. 3.3.4 comparing curves of viscosity growth in the process of structuring polyurethanes based on bi- and polyfunctional diamines. The analogous nature of q(t) dependencies and equal indices a = 4.6 in both cases testify that the compared processes are identical in their physical nature.

f, 5 2.0 t , (min)

Figure 3.3.4: Viscosity change in the process of polyurethane formation: 1 - bifunctional curing agent, 2- polyfunctional curing agent.

The given example shows that the nature of viscosity changes in the process of polyurethane structuring reflects the effect of formation of cross-linked products in spite of using bifunctional monomers.

The discussed formulae assume the existence of a point where q + m since a, + 00

when the threshold of gel formation is reached. Since molecular mass is a function of the conversion degree (allowing for the existence of the point of gel formation) [5] suggests an expression for the dependence with regard to the conditions at a = 1, q/qo = 1 and at a = a*, q/qo + 00. This formula has the form:

(3.3.4)

where a is the conversion degree, a* is the conversion degree at the point of gel formation, A and B are constants equal to 3.5 and - 2 for different polyurethanes.

In its structure formula (3.3.4) is close to expressions of exponential type derived with the use of the percolation theory for measuring rheological characteristics in close proximity to the point of gel formation [29, 301:


q - (a* -a)-S (3.3.5)

where S is the constant, the theoretical value of which is 0.7 k 0.07 [29].

Applicability of formula (3.3.5) is, in general, limited by the narrow interval a when approaching the point of gel formation. If we assume that in this narrow interval a - t it will be possible to apply the equivalent time dependence of viscosity:

q - ( t* -qS (3.3.6)

< A t / t * < 5 ~ 1 0 - ~ the expression (3.3.6) can be well applied to describe the kinetics of viscosity change with the value of the constant S approaching its theoretical prediction.

Each of the formulae discussed above reflects a certain stage of structure formation in reactive systems. In a general case, the nature of viscosity increase is determined by a set of physical-chemical phenomena accompanying the process, of chemical conversion properties and change of molecular characteristics of a curing oligomer. The main stages of gel formation are most clearly seen on the curves of viscosity change in the process of three-dimensional polycondensation. The kinetic curves of viscosity change of logarithmic coordinates are approximated by linear fragments, each corresponding to a certain stage of the process of gel formation [31]. As is seen from the scheme shown in Fig. 3.3.5, at the initial stages of gel formation the exponent of the dependence q(t) increases from = 1 ( t < t , ) to = 3.5 ( t > t c ) which corresponds to increasing intermolecular interactions in a reactive system with increasing molecular mass of the forming polymer. This dependence is evident since in polycondensation processes the time dependence of viscosity adequately fits the dependence of viscosity on molecular mass [27,32] and the reaction time t , characterizes the reaching of a critical molecular mass [33]. In fact, if curing proceeds in an ideally homogeneous manner, this law of viscosity change should work until gel formation is completed, as is the case with increasing molecular mass of linear polymers of different structure to very high values [15]. In rheokinetic studies of polymerization (Sections 2.2 and 2.3) and linear polycondensation (Section 2.4) it was shown that viscosity changes in reactive systems to large degrees of conversion obey this exponential law provided the gel effect and diffusion limitations are absent.

The experiment cited in [29] showed that in the time interval


Figure 3.3.5: Viscosity growth during gelation (scheme).

In Fig. 3.3.5 a hypothetical viscosity change under this law is shown by a dashed line. In a real curing process the rate of viscosity growth increases at a certain critical moment t p and the exponent rises to very high values. Correlation of times t , and t p depends on the structure of reacting components and composition of the cured material. For instance, enhanced functionality of a curing agent in the process of curing derivatives of epoxy oligomers results in a regular reduction of the linear part of the dependence q(t) [ 11.

The appearance of another fragment of the viscosity/time dependence ( q ( t ) ) can be hardly attributed to the nature of changes of molecular mass in reactions forming polymer material, since these high values of the exponent are not found in the dependence q(M).

Typical experimental data corresponding to the given scheme are shown in Fig. 3.3.6 for curing of epoxydiane resin with diamine. The kinetic curve of viscosity change clearly shows three sections with different exponential indices (0.8,4 and 35) separated by two characteristic points t , and t p .


4

2

Figure 3.3.6: Kinetics of viscosity growth in the process of curing epoxydiane resin by diamine.

An attempt to explain a sharp increase of viscosity as the point of gel formation is approached was made with the conception of levelling the curing and vitrification temperatures in a reactive system [3,34,35 1, as it is well known that Tg increases in the course of the process [36, 371. Using this assumption Grillham developed a model assuming viscosity change to be determined by two factors - increasing molecular mass of an oligomer and growing temperature of vitrication. Contribution of vitrification temperature to the value of viscosity of a reactive item is estimated from the known WLF correlation. In this case the general formula to calculate viscosity of a cured oligomer has the form [34]:

(3.3.7)

where c , and c2 are the constants of the WLF equation, To is the reduction temperature, determined from the following correlations: To = Tg + 50 at T < ( Tg + 50) and To = Tat T 2 (Tg+50) .

Calculation of viscosity by formula (3.3.7) is possible if the following dependencies are predetermined:

- relationships between molecular mass and conversion degree a,,,@) ;

- relationships between vitrification temperature and conversion degree Tg(p);

- changes of conversion degree with time p(t).

In this case the formula would give an unambiguous dependence q [ p( t ) ] , i. e. describe the course of viscosity growth.


It is quite natural, that formula (3.3.7) satisfactorily describes kinetics of viscosity change when a reactive system loses fluidity due to reaching the vitrification temperature [38] irrespective of whether the forming polymer is linear [39] or crosslinked [34]. On reaching the threshold of gel formation above the vitrification temperature, i. e. during transition from the viscous flow to the viscoelastic state, the last term in the formula (3.3.7) turns to zero and does not affect the value of viscosity in a reactive system even if in close proximity to the point of gel formation.

Since vitrification is a rather particular case of curing reactive masses and a sharp viscosity increase when approaching t* is a general case, it seems reasonable to trace changes in the nature of viscosity growth at the point tp. This may be caused by formation of a new phase, which will be discussed in the following section.


1.

2. 3. 4.

5. 6.

7.

8. 9.

10. 11. 12. 13. 14. 15. 16. 17.

18.

19. 20.

21.

S. G. Kulichikhin, A. S. Reutov, A. Y. Malkin, Polikondensatsionnye protsessy i polimery, Nalchik, 1988. V. M. Gonzales-Romero, C. W. Macosko, J. Rheology, 1985, vol. 29, 3,259. D. S. Lee, C . D. Han, Polym. Eng. Sci., 1987, vol. 27, 13,955. S. G. Kulichikhin, A. S. Reutov, M. S. Surova, E. V. Osipova, A. Y. Malkin, Plast. massy, 1988,5,43. J. M. Castro, C . W. Macosko, S. J. Perry, Polymer Commun., 1984, vol. 25, 3,82. R. P. Jr. White, Polym. Eng. Sci., 1974, vol. 14, 1,50. M. B. Roller, Polym. Eng. Sci., 1975, vol. 15, 6,406. M. R. Kamal, S. Sourour, Polym. Eng. Sci., 1973, vol. 13, I, 59. H. Schwesig, C. Heimenz, W. Milke, G. Menges, Kautsch. und Gummi Kunstst., 1980, vol. 33, 1, 15. Y. M. Lee, Y. S. Yang, L. J. Lee, Polym. Eng. Sci., 1987, vol. 27, 10,716. V. Liska, Po1ym.-tworz. sielkoczasteczk, 1980, vol. 25, 6-7, 219. V. Liska, Progc Org. Coat, 1983, vol. 11, 2, 109. W. J. Lee, A. C. Loos, G. S. Springer, J. Compos. Muter:, 1982, vol. 16, 6, 510. F. G. Mussati, C. W. Macosko, Polym. Eng. Sci., 1973, vol. 13, 3, 236. G. V. Vinogradov, A. Y. Malkin, "Reologiya polimerov", Khimiya, Moscow, 1977,438 S. D. Lipschitz, C. W. Macosko, Polym. Eng. Sci., 1976, vol. 16, 12,803. A. Y. Malkin, V. P. Begishev, S. G. Kulichikhin, V. A. Kozhina, Vysokomolekulyarnye soedineniya, A, 1983, vol. 25, 9, 1948. V. P. Begishev, S. G. Kulichikhin, A. Y. Malkin, Pervy Vsesoyuzny simpozium PO makroskopicheskoi kinetike i khimicheskoy gazodinamike, tezisy dokladov: Chernogolovka, 1984,8,27. S. D. Lipschitz, C. W. Macosko, J. Appl. Polym. Sci., 1977, vol. 21, 8,2029. E. Y. Naroditskaya, I. D. Khodzhaeva, S. G. Kulichikhin, V. Y. Pozdnyakov, I. N. Yunitsky, V. V. Kireyev, A. Y. Malkin, Vysokomolekulyarnye soedineniya, B, 1985, vol. 27, 9,713. A. Y. Malkin, S. G. Kulichikhin, E. Y. Narodistkaya, V. Y. Pozdnyakov, I. N. Yunitsky, V. V. Kireyev, Vysokomolekulyarnye soedineniya, A, 1985, vol. 27, 10,2040.


22. F. M. Valles, C. W. Macosko, Macromolecules, 1979, vol. 12, 4, 673. 23. F. M. Valles, C. W. Macosko, W. J. Hickey,

Amer. Chern. SOC., Polymer Prep., 1979, vol. 20, 2, 153. 24. A. Y. Malkin, S. G. Kulichikhin, "Reologiya vprotsessakh obrazovaniya iprevrashcheniya

polimerov, M., Chimia, 1985,240. 25. S . G. Kulichikhin, A. Y. Malkin,

Vysokomolekulyarnye soedineniya, A, 1980, vol. 22, 9,2093. 26. A. Y. Malkin, S. G. Kulichikhin, V. G. Frolov, M. I. Demina,

Vysokomolekulyarnye soedineniya, A, 1981, vol. 23, 6, 1328. 27. S. G. Kulichikhin, V. A. Kozhina, L. M. Bolotina, A. Y. Malkin,

Vysokomolekulyarnye soedineniya, B, 1982, vol. 24,4,309. 28. L. M. Sergeyeva, Y. S. Lipatov, N. I. Binkevich,

Sintez ifizikokhimiya poliuretanov, Kiev, Naukova dumka, 1967, vol. 4, 13 1. 29. M. Adam, M. Delsarlti, D. Durand, G. Hild, J. P. Much,

PureAppl. Chem., 1981, vol. 53, 12, 1489. 30. D. Stauffer, A. Coniglio, M. Adam,Adv. Polym. Sci., 1982, vol. 44, 103. 3 1. S. G. Kulichikhin, Problemy teplo- i massoperenosa v topochnykh ustroistvakh,

gazogeneratorakh i khim. reaktorakh. Minsk, ITMO AN BSSR, 1983,88. 32. A. Y. Malkin, S. G. Kulichikhin, V. A. Kozhina, L. M. Bolotina,

Vysokomolekulyarnye soedineniya, A, 1987, vol. 29, 2,418. 33. S. G. Kulichkhin, V. A. Kozhina, L. M. Bolotina, A. Y. Malkin,

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3.4 Microphase Separation in Reactive Systems

Since peculiarities of viscosity change q(t) when approaching the point of gel formation after some time are not described in terms of homogeneous chain growth, we should probably refer to more general structural conceptions reflecting the mechanism of net polymer formation [ I ] . Fig. 3.4.1 shows curves of viscosity growth in the process of curing epoxydiane resin by 3,3’-dichlor-4,4’-diaminodiphenylmethane in the presence of various quantities of the chemically inert plasticizer dibutylphthalate. Attention should be drawn to the following peculiarities of viscosity change in the process of gel formation in these plasticized systems:

different viscosity values at the moment preceding gel formation;

a very weak (or practically no) dependence of time needed to reach the point of gel formation on the content of plasticizer.

4

2

Figure 3.4.1: Viscosity change in the process of curing epoxydiane resin by 3,3‘-dichlor- 4,4’-diaminodiphenylmethane in the presence of dibutylphthalate: 1 - 0% DBP, 2- 20% DBP, 3- 40% DBP.

Apparently, a common explanation for these and other data given below is the assumption that the curing process proceeds non-uniformly and local concentrations of functional groups do not correspond to an average oligomer concentration in a system. Otherwise, at a certain stage of curing a new phase may be formed with fragments of cross-linked structures at its growth points [ 2 - 61.

3.4 Microphase Separation in Reactive Systems 157

Figure Depenc

t , mi0

!ncies of viscosity, mechanical loss angle tangent and optica density on curing time of melaminofomaldehyde resin in a solution and in

a block; T = 80 "C. Solution concentrations are 40% ( I ) , 50% (2), 57% (3).

Quite characteristic in this respect are the results shown in Fig. 3.4.2. It presents dependencies of viscosity, optical density and angle tangent of mechanical loss in the process of curing melaminoformaldehyde resin in aqueous solutions of different concentration and in a block. Time needed to reach the point of gel formation does not change with varying concentration of polyaminoformaldehyd resin in a solution and coincides with time needed to reach the maximum tan6 in the course of curing anhydrous resins. Fig. 3.4.3 shows the dependence of inverse time of gel formation l / t * on temperature on Arrhenius coordinates.


-3,s

3/ T, K -

Figure 3.4.3: Dependencies of the time of gel formation found with viscometric and dynamic mechanical methods, the point of phase separation and the

constant of reaction rate on temperature in the course of MFR curing.

Time values of gel formation found with the viscometric method in the course of curing in solution, and with the dynamic mechanical method in the course of curing in a block are described by a single exponential dependence.

A sharp increase of optical density, as shown in Fig. 3.4.2, is observed long before reaching the point of gel formation. A registered leap in optical density is linked with separation of a reactive system and appearance of a new phase in the solution which consists of fragments of branched and cross-linked molecules. These structural forms are known, their existence has been long discussed in literature, and they were called "microgels" [ 1 - 111.

Investigations have shown that the moment of reaching the point of phase separation is related to the appearance of a dependence of system viscosity on applied stress, which is in agreement with general foundations of rheokinetics of polymerization processes (Section 4.3.3).

From this point of view it becomes clear why viscosity values are different at the moment preceding separation of a reactive system from the effective surfaces of a viscometer, and the viscosity value in this period regularly increases with oligomer concentration in a solution. The reason is that before the moment when a continuous phase is formed from fragments of cross- linked structures the flow is determined by a dispersed medium, the viscosity of which depends on solution concentrations.


Information on the nature of a forming structure resulting from phase separation in a reactive system is provided by direct microscopic analysis of the process of curing epoxy oligomer with diamine. Fig. 3.4.4 shows micrographs of a suspension of forming microgels made at different moments of time. Micrographs register the moment of microgel formation and their subsequent growth until their complete coalescence.

Figure 3.4.4: Pictures of microgel suspension at different moments of epoxy oligorner curing by diamine. Time [min]: 160 ( I ) ; 220 (2); 280 (3).


The temperature dependence of the point of microphase separation tp (Fig. 3.4.4) has the same factor as the temperature dependence of the point of gel formation t* and the rate constant of a curing reaction (see Section 3.6). Equal values of activation energy for processes of microphase separation, gel formation and curing show that all these phenomena are governed by kinetics of the same chemical reactions which eventually yield a cross linked product.

Thus, the discussed results show that before the point of gel formation in cured systems phase disintegration results in the formation of microgels - cross-linked and branched molecules reaching the size of colloidal particles. After the point of phase separation the curing process proceeds in a two-phase region. As a rule phase separation characterizes initiation of intensive viscosity growth in a filled system to which a reaction mass turns due to the increased amount of a “filler”, i. e. a microgel.

The fact that a viscosity increase after phase separation (at t > tP) is connected with this mechanism is confirmed by the data of gel-chromatography of the sol fraction in the process of structuring low-molecular siloxane rubbers (Fig. 3.4.5). In the course of the reaction molecular mass of the sol fraction decreases and consequently, viscosity of a dispersed medium decreases too, though the entire viscosity of a reactive system increases due to the concentration of the second phase.

Figure 3.4.5:

I

5 7 9 V, me Gel chromatograms of the sol fraction in the process of structuring low molecular siloxane rubbers. Processing time = 1 (l), 1.5 (2), 2, 3, 4, 5 and 24h (3).


A4 +a,- Network I I I I I

r

d* d Figure 3.4.6: Representation of theoretical estimations of molecular mass and sol-

fruction variations in the process of three-dimensional addition [12, 131.

Fig. 3.4.6 schematically shows changes of molecular mass and content of the sol fraction calculated on the basis of the branching theory in the process of three-dimensional polycondensation of functional compounds [ 12, 131. According to these calculations, the mean molecular mass of a curing oligomer rises monotonically to the point of gel formation where it turns to infinity. At the same point a gel fraction appears in a reactive system. Real information of change in the same characteristics obtained from gel chromatography of curing epoxydiane resin by diamine, is quite different (Fig. 3.4.7).

2

M s tp 40 5

20

t, min Figure 3.4.7: Changes of viscosity, molecular mass and sol-fraction content in the

process of curing epoxydiane resin by diamine.


The molecular mass of the forming polymer changes extremely, the maximum value of molecular mass being observed at the point of phase separation tp. At the same moment rather than at the point of gel formation with retained possibility of flow for the entire product, a gel fraction appears in a reactive system. It is interesting to note that a similar situation with the extreme change of molecular mass was observed long ago for a totally different system: polybutadiene in the process of curing [ 141.

Thus, the observed regularities in the change of viscous and molecular characteristics of a reactive system are explained by the existence of another specific point besides the point of gel formation in curing processes - the point of phase separation, which has a decisive effect on regularities of gel formation. After formation of a two-phase structure molecular mass of a dispersed medium decreases since the most high-molecular fractions contribute to formation of the second phase.

From the above considerations (if the process is viewed physically) the point of gel formation may be determined at the moment of reaching phase inversion in a reactive system. Fragments of cross-linked structures from a dispersed phase form a continuous dispersed medium which causes changes in a relaxation state of the system and a transition from a viscous-flow state to high elasticity and, eventually, to a glassy state.

Otherwise, if we consider a reactive system in proximity to the point of gel formation as a filled composition, the point of phase inversion corresponds to the maximum possible concentration of a filler. This transition enables us to describe q ( t ) dependencies using concepts based on the rheology of filled polymers.

Various analytic and empirical expressions are suggested for filled systems which describe viscosity dependence on filler concentration.

Most common are the expressions of the kind [ 15, 161:

cp --c - - (1---) q0 qmax

(3.4.1)

where qmaX is the value of maximum possible concentration of a filler in a system depending on the way of packing and c is a constant.

Since a filler in a reactive system is generated in the course of the reaction, assuming that cp - t , we may obtain an expression for the kinetics of viscosity change in the processes of gel formation analogous to the scaling formula (2.3.6). However, the described approach has a significant difference as in this case viscosity of a dispersed medium qo is not a constant, but it varies in the course of the process. Then change in viscosity may be expressed as:

(3.4.2)

where q(t) is the average viscosity of a reactive system; qo(t) is the viscosity of a dispersed medium or a matrix, t* is the time needed to reach the point of gel formation and C is the constant.


Formula (3.4.2) was tested experimentally for a number of quite different systems cured both in solution and in block - epoxy, epoxy silicoorganic and silicoorganic oligomers, melaminoformaldehyde and carbamide resins and derivatives of furane resins [ 17 - 351. Typical results of processing experimental data on the coordinates of formula (3.4.2) are given in Fig. 3.4.8, After phase separation (at t > t p ) experimental dependencies q(t) are fully described by formula (3.4.2). Calculated on the basis of this formula the viscosity of a dispersed medium, i . e. the function of its molecular mass, changes significantly and the position of the maximum corresponds to the appearance of a new phase in a reactive system - a gel fraction.

* t D

P P

( p 0

0 0

oO tp = i6Omin

I I I I I I I I I I I I -LL

1

41/q 2

ioo 1sotp t+ t, m i n

2

Figure 3.4.8: Viscosity dependence on time in the process of curing epoxy resin by diumine on the coordinates of formula (3.4.2).

The ratio t p / t * does not depend on the temperature at which the reaction is effected (Fig. 3.4.3) and is determined by the concentration of a reactive oligomer in a system. Fig. 3.4.9 shows data indicating change in the ratio t p / ? , i. e. relative time between formation of a micro- or a macrogel depending on the concentration of a cured (melaminoformaldehyde) resin in an aqueous solution. These results show that the higher the oligomer concentration in a reactive system is the closer is the point of microgel formation to the point of gel formation.


*

60

bD

Ip, mass, % Figure 3.4.9: Point of gel formation and change of t p / t* -ratio depending on

concentration of melaminoformaldehyde resin in the solution.

Thus, a combination of the results given above shows that the nature of viscosity change at the stage of gel formation is determined, first, by the change of molecular characteristics of an oligomer and, second, by microphase separation of a system near the point of relaxation transition, namely, gel formation.

In general, this complex mechanism of viscosity change is reflected by the following formula:

(3.4.3)

where f(q) is the function determining viscosity change due to formation of local networks in the reactive system, structures which may be regarded as a filler.

Both factors in formula (3.4.3) can be calculated from measurements of molecular characteristics of a cured oligomer in the course of gel formation.

Contributions of different components to viscosity change in the process of curing an epoxy oligomer are shown in Fig. 3.4.10. As it is obvious from the results shown in this figure, the viscosity change fully corresponds to increasing molecular mass of the oligomer at the initial stages of curing. On reaching a certain time, the viscosity calculated from changes in the molecular mass of a dispersed medium becomes lower than the viscosity of the entire reactive system.


z .

so f00 t , min

Figure 3.4.10: Time dependencies of viscosity in a reaction system (1) and viscosity

calculated from MW change ( 2 ) by formula (3.3.2) in the course of curing epoxy oligomer ED-7 by diamine. T = 150 "C.

After that moment, the role of the third factor of formula (3.4.3) determining contribution of local network structures in a reactive system to viscosity change becomes increasingly important. With a general sharp viscosity increase, the relative contribution of the molecular component is constantly decreasing as well as its absolute value. When gel formation is completed the difference between these components reaches several decimal orders. Taking this into account the full equation of viscosity change in polycondensation processes of curing in a dimensionless form is expressed as follows:

(3.4.4) t fa t -b

t* t* 2 = ( A + - ) ( 1 - - ) %I

where a and b are constants. a r 1 at t < t , and a = 3.5 at t 2 tc. a changes its sign at the point

of microgel formation due to decreasing molecular mass; A = ( k t * ) - ' ; q,= q,, (kt* )" ;

k is the constant of the reaction rate.

For three-dimensional polymerization, formula (3.4.4) will apparently retain its structure, but the limiting values of index a will be different since in polymerization processes it reflects not only changes in molecular mass but also in concentration of the forming product [36].

The first co-factor on the right side of formula (3.4.4) characterizes the viscosity change caused by the change in molecular mass of a dispersed medium. The sign of the contribution of this factor viscosity is determined by the position of the point of microgel formation vs the point of gel formation. Fig. 3.4.11 shows results of theoretical calculations of viscosity change at different values of the ratio t p / t * and the value of the constant 6:


0.6 47 a,a 4 9 t/ t *

Figure 3.4.11: Various pattern of viscosity change at b = 1 and t p / t * = 0.6 ( I ) , 0.7 (2), 0.8 (3), 0.9 (4), 1.0 (5); b = 2 and t p / t * = 0.7 (6).

These calculations show that depending on the value of the ratio t p / t * quite different forms of the dependence q(t) may be observed. The usual form of these curves with monotone viscosity growth is only observed when t p / t * 2 0.9 which is generally typical of curing in a block. Reducing t p / t * , e. g., during transition from curing in a block to reaction in a solution, gives a more distinct plateau of the dependence q(t) or even leads to a certain viscosity drop. The nature of this part of the dependence is also determined by the value of the exponent in the second term of formula (3.4.4).

From this point of view, it seems possible to understand curves q(t) with clearly observable sections of decelerating rate of viscosity growth near the point of gel formation described in some works [ 37-39]. As a rule, in this case a slower rate of viscosity growth was attributed to a slide effect as the threshold of gel formation is reached. However, for certain systems this sufficiently long section of the dependence q(t) may be experimentally verified. Fig. 3.4.12 shows viscosity and tana dependencies in the process of curing a silico-organic oligomer. When phase separation occurs (at the moment of t 2 t,,) viscosity of a reactive system decreases up to a certain degree. But thereafter its monotone increase continues until the point of gel formation is reached which can be registered by reaching the maximum value of tana. Similar changes in rheological properties with a plateau on the time dependence of components of the dynamic modulus was registered in the process of curing epoxy resins by diamines [40] and a similar effect of viscosity drop in a reactive system at the moment of phase separation has been observed in the process of linear radical polymerization [41,42].


.

2

4

0

20 40 60 t , min

Figure 3.4.12: Change of viscosity (I), tan6 ( 2 ) and sol fraction content (3) in the process of curing silico-organic oligomer at T = 120 "C.

Microphase separation, as it was mentioned above, may give rise to the dependence of the kinetics of viscosity growth on the applied stress or shear rate [3 11. This effect can be registered experimentally if the section of retained viscous flow is sufficiently long, i. e. when the ratio t p / t * is relatively small, e. g. in the course of curing melaminoformaldehyde resins in solution [43, 441. In the same case, when t p / t * is close to 1 this effect is often difficult to detect experimentally.

A combination of the results discussed above reveals the complex nature of changes in viscous properties of reactive systems at the stage of gel formation since they are caused by a set of heterogeneous physical-chemical phenomena developing in the process of chemical transformation of an initial oligomer to a cross-linked material. However, the rather simple formulae obtained as a consequence of certain structural views on cured compositions demonstrate a possibility of a reliable description of the kinetics of viscosity change in the process of gel formation.



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3.5 Curing After the Point of Gel Formation

Although gel formation is the most important stage in terms of both physical chemistry and technology, it is only part of the entire process of curing. Rheokinetics, however, is aimed at a description of the whole process.

For this reason, a characteristic feature of rheological studies of curing compositions reported recently is the introduction of a new parameter -

the “rheological” degree of conversion 0,

reflecting kinetics of growing cross-links in a material. In this case, the main variable used for evaluation of the curing process is the rubbery modulus which is generally proportional to the density of the network formed by chemical bonds and physical engagements. The latter assumption was tested in special experiments determining the density of networks from cross-linking low-molecular polydimethylsiloxane rubbers by swelling according to the dependence suggested in [l]. A comparison of the kinetics of growing cross-links calculated with the swelling method and from measurements of the rubbery modules of cured compositions is shown in Fig. 3.5.1:

2

0

t , hours

Figure 3.5.1: Change of network density in the process of structuring low molecular

silico-organic rubber at 20 “C calculated from the rubbery modulus (1) and equilibrium swelling in toluene (2).

Curves of q, change are symbiotic, i. e. such a macroscopic parameter as the rubbery modulus reflects kinetics of growing cross-links [ 2 ] . A minor difference in the values of q, reaching 20% at the end of the structuring process is apparently connected with contributions of a physical engagement network to the rubbery modulus. This contribution, naturally, cannot be established with the swelling method, but does have a notable effect on the value G‘ of a cross-linked material in the state of high elasticity [3].

3.5 Curing Afrer the Point of Gel Formation 171

It should be stressed that these results were obtained with materials with cross-linked ends of oligomer chains and producing network with preset (and controlled) structures: number of points and, respectively, M, chains between the points - M, [4,5]. In this case the value of M, at the end of structuring is equal to the mean MM of the initial sample. In the formation of real tightly bonded systems based on reactive oligomers this ideal quantitative pattern is not likely to occur, however. The use of the rubbery modulus as a parameter reflecting general structural changes in a reactive system seems quite reasonable.

Among numerous physical-chemical methods used to study curing processes, rheokinetics is the closest to calorimetric also yielding integrated characteristics of process dynamics made up of a combination of separate chemical reactions. For this reason, a comparison of quantitative curing parameters obtained by rheological and calorimetric methods was carried out. Fig. 3.5.2 shows time dependencies of “rheological” p, and “calorimetric” p, degrees of completion of curing epoxy silicoorganic oligomer with polybutoxytitanphosphoxane:

loo 200 300

t, min

Figure 3.5.2: Dependencies of the degree of conversion p (open points - fir, solid points - p,) of the system epoxy silico-organic oligomer/PBTP (composition 100 : 7.5) on time at curing temperatures 140 “C ( I ) , 160 “C (2), 180 “C (3), 193 “C (4), 200 “C (S), 220 “C (6).

It is obvious that for similar process conditions p, and p, practically coincide [6 ,7 ] but it might be that the obtained correlation is not universal, since, in a general case, relationships between rheokinetic and calorimetric parameters of the process will depend on thermal effects of particular reactions occurring in a material as only some of them promote formation of cross-linked structures [8]. Yet, the discussed data confirm a possibility to compare kinetic parameters of curing processes calculated on the basis of rheological and calorimetric measurements.


The comparison of rheology with other experimental methods which register changes in certain functional groups is a far more complex task, as inadequate changes of rheological characteristics and concentration of functional groups are observed. For example, Fig. 3.5.3 shows time dependencies of the degree of conversion calculated from concentration changes in hydroxyl groups and rubbery modulus variations in the process of curing a silico-organic oligomer [ 1,9]. Considerable differences in the kinetics of changes in the mentioned parameters are apparently connected with a variety of chemical interactions resulting in the formation of a net structure when cross-linking is preceded by consecutive and parallel reactions of various functional groups.

3 Y \

0 * 2

1

- 8 * 2'

-4

I

I

400 800 t, min

Figure 3.5.3: Change in relative content of hydroxyl groups ( I , 2 ) and rubbery modulus

(1 2') in the process of curing polyphenylmethylsiloxane oligomer at temperatures 170 "C (1: I ) and 190 "C (2: 2).

There is a separate group of experimental methods which register mobility of molecular structure, particularly, pulse NMR spectroscopy. This method proved to be highly informative in the studies of gel-forming systems [ 101 and valid for direct measurement of the density of cross-links in net materials [ 111. Using pulse NMR spectroscopy to study kinetics of curing it is possible to obtain kinetic equations similar to those formulated on the basis of calorimetric and rheokinetic studies of analogous systems [ 121.

Microphase separation and formation of a micro-heterogeneous structure, naturally, influence curing kinetics. Polymerization reactions leading to formation of a cross-linked structure are not reactions of separate macromolecules and therefore cannot be regarded separate from structural changes in a reactive system. It can be concluded from the above that here we also have an inverse relation, i. e., micro heterogeneity appears due to the very mechanism of a net-forming reaction. Given all this, kinetic and rheokinetic regularities should be analysed allowing for the possibility of inhomogeneous curing of oligomer compositions as a result of phase separation processes accompanying the reaction.

Numerous experiments show that curing kinetics cannot be described by a phenomenological equation of nth-order at any sensible values of n [9, 131. But in a majority of cases kinetics of curing processes are satisfactorily described by a phenomenological equation reflecting the effect of self-acceleration

9 dt = ( k , +k2p'n) ( I - p ) " (3.5.1)

where k , and k , are the rate constants and m and n are empirical constants.

3.5 Curing After the Point of Gel Formation 173

The given equation (3.5.1) is certainly of a rather general value since it contains many

empirically found constants which at the same time complicates its use in the analysis of

experimental data. Therefore curing rheokinetics of various systems is often described using a

different version of the phenomenological equation including self-acceleration [ 151:

- = k ( 1 - p ) ( l - cp ) dt (3.5.2)

where p is the rheological degree of conversion, t is the time, k is the constant of the initial

reaction rate equal to curing rate at p + 0 and c is the constant characterizing the

self-acceleration effect.

A similar mathematical description of curing kinetics of agents varying in their chemical

nature, obtained with totally different experimental methods, makes it possible to assume that

the effect of self-acceleration is common for processes forming materials with cross-linked

structures in a block. It should be stressed that the effect of self-acceleration should not be

identified with auto catalysis of the reaction of epoxy monomers interacting with diamines

which was thoroughly studied on the basis of model compounds [ 16, 171, although these effects,

as it is suggested in [IS], may be interconnected. For each process of curing particular chemical

agents, different variants of the self-acceleration mechanism are suggested: chemical auto-

catalysis [ 191, presence of proton-donor functional groups [20], emergence of local inhomo-

genities [21], influence of gel effect [22], parallel occurrence of catalytic and non-catalytic

reactions [23]. Probably the above-mentioned phenomena are more or less involved in particu-

lar processes and contribute to self-acceleration of a curing reaction. Still, they cannot be surely

called decisive. Here it is important to note that the phenomenon of self-acceleration is common

for various systems. The above enables us to assume that the most important factors in produc-

ing material with net structure are structuring peculiarities. In this case, every subsequent reac-

tion run considerably limits the system mobility leading to phase separation and localization of

the further reaction course in limited regions. The separated phase of fragments of branched and

cross-linked molecules which have reached the size of colloidal particles are containing

unreacted functional groups. This generally results in a reaction taking a parallel course in dif-

ferent coexisting phases.

We may assume that kinetics of isothermal processes in homogeneous media are described

by the following equation:

9 = k f(p) dt

(3.5.3)

where f(p) is the kinetic function, the form of which depends on the particular mechanism of

the curing process.


Since concentration of a separating phase is proportional to the conversion degree, the general reaction rate, if it has a parallel course in different phases, can be written as a sum [9]:

9 = k f(p) + k f(p)c dt

or (3.5.4)

9 = k f(p) (1 + cp) dt

where c is the constant reflecting the differences of reaction conditions if the reaction exhibits coexisting phases.

The presence of the second co-factor in the obtained equation (3.5.4) indicates the appearance of self-acceleration in a curing process irrespective of the type of the kinetic function f(p). The method of deriving equation (3.5.4) shows clearly that the mechanism of self-acceleration results from phase separation and reaction acceleration due to increased concentration of reactive groups in a separated microgel phase. Thus, the analysis of process kinetics in terms of structural peculiarities of a cured composition proves that curing accompanied by phase separation and leading to micro heterogeneity should really exhibit self-acceleration. A reason of initiating self-acceleration, besides redistribution of the concentration of functional groups during phase separation, may be a local deviation from isothermal nature of the process in a separated phase. It can not be observed as a notable violation of the isothermal condition for the whole reactive mass since microgel is surrounded by a large amount of a solution.

Methods of analysing experimental data by equation (3.5.4) and determining its constants for different variables of the kinetic function f(p) are discussed in [24]. The integral of equation (3.5.4) for the first-order reaction, i. e. if f(p) = ( 1 - p) , has the following form:

1 +cp

1 - P In- = ( I + c ) k f

or

exp [ ( 1 + c) kt] - 1 p = e x p [ ( l + c ) k t l + c

for the second-order reaction, i. e. if f(p) = ( 1 - 0) *, the integral of the equation is:

p c 1 + c p

I - p I + c 1 - p -+- In- = ( I + c ) k t

(3.5.5)

(3.5.6)

(3.5.7)

A practically complete coincidence of experimental and calculated (from equation (3.5.4)) dependencies of the degree of conversion on time was obtained for curing epoxy [25], epoxy silico-organic [6, 7, 26-29], silico-organic oligomers [9, 30, 3 13, low-molecular silico-organic rubbers [23, 32, 331, unsaturated polyethers [24], melaminoformaldehyde [34, 351, methylolpolyamide [36] and carbamide resins [35].


Equation (3.5.4), being theoretically derived in that way, should agree with experimental data starting from the point of phase separation and formation of a two-phase system. However, the results given above confirm its real validity throughout the entire curing process. This is caused by the fact that, as was shown above in section 3.4, phase separation occurs at the initial stages of the process, and the time interval between the beginning of the reaction and the point of phase separation is small compared with the general time of the process.

In fact, equation (3.5.4) reflects formation of a two-phase structure in a reactive system by the mechanism of nucleation and growth, provided that concentration increase in the second phase is determined by the course of chemical reactions. A physical analogue of this process may be found in crystallization. In [37, 381 a new model of crystallization kinetics was developed which can be reduced to an equation of auto-catalytic nature close to equation (3.5.4), and it was reported to describe adequately crystallization processes of most varied polymers.

In the cited works crystallization kinetics was mainly registered using classical calorimetric data. It is none the less interesting, in terms of rheokinetic methods discussed here, to consider one more example of describing crystallization kinetics (shown in Fig. 3.5.4) giving a comparison of experimental (from [39]) and calculated (by formula (3.5.6)) dependencies of the relative rubbery modulus on time in the process of crystallization of cis- 1 ,4-polybutadiene:

w

0 - f o t 2

tty t , (min)

Figure 3.5.4: Change of the relative rubbery modulus in the process of crystallization of cis-1,4-polybutadiene [39/. Points correspond to experimental data, the line is calculated by formula (3.5.6).

Good agreement of experimental and theoretical data shows a possibility to describe crystallization kinetics whatever measuring techniques are used with formulae derived by integrating equations of a (3.5.4)-type. Thus, a combination of the described results shows that kinetic phenomenology of curing reactive compounds reflects some regularities of both chemical reactions proper and phase phenomena accompanying the process of chemical transformation. As a consequence, a self-acceleration effect is observed in the processes of oligomer curing irrespectively of a particular reaction mechanism.



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A. Y. Malkin, S. G. Kulichikhin, P. A. Astakhov, Y. P. Chernov,V. A. Kozhina, L. I. Golubenkova, Mekhanika kompozitnykll materialov, 1985,5,878.

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25. A. Y. Malkin, S. G. Kulichikhin, V. P. Batizat, Y. P, Chernov, I. V. Klimova,

26. P. A. Astakhov, V. A. Kozhina, Y. P. Chernov, S. G. Kulichikhin, A. Y. Malkin,

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29. S. G. Kulichikhin, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina, L. I. Golubenkova, A. Y.Malkin, Vysokomolekulyarnye soedineniya, A , 1986, vol. 28, 10,2115.

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39. M. Gesari, L. Gargani, G. P. Giuliami, G. Perego, A. Lazzetta,


3.6 Rheokinetic Equations of Curing

The previous analysis showed that a characteristic feature of curing rheokinetics is the effect of self-acceleration. As its consequence, the rheokinetics of curing, in a general case, includes a co-factor providing a description of extreme change in the process rate with increasing conversion degree. However, in some cases interesting deviations from this general rule can be observed.

From this point of view the most interesting results are those obtained from investigating curing regularities of systems with reactive groups of similar nature but different functionality [ 11.

Figs. 3.5.2 and 3.6.1 show P(t) dependencies for curing epoxy silicoorganic oligomer by compounds having identical reactive groups but different functionality - tetrabutoxytitan (TBT) and polybutoxytitanphosphoroxane (PBTP).

0, 4

200 600

t. min

Figure 3.6.1: Time dependence of fractional conversion for the system epoxy silico-organic oligomer - TBT. Temperatures: 170 "C (I), 180 "C (2), 190 "C (3), 200 "C (4).

For the curing agent of low functionality - TBT, general regularities of curing rheokinetics discussed in Section 3.5 are realized. As to the curing agent of high functionality - PBTP, p(t) dependencies have a totally different form, namely, they have no apparently S-shaped curve and at relatively low temperatures (< 200 "C) the curing process ends at p < 1 .

The conclusion that at t + 00 situations are possible when p < 1 is based on considering a relation of the rubbery modulus G', in a reactive system not to GL ( t + -) at a given temperature, but to the maximum possible value of GIm obtained in a high temperature region. Here it is important that the observed temperature dependence of G' is in no way related to the

3.6 Rheokinetic Equations of Curing 179

temperature dependence of the limit modulus. In fact, direct measurements of G’ at various temperatures for a sample completely cured at high temperature, did not reveal any appreciable temperature dependence of G’ [2]. Therfore in the discussed case we certainly deal with incomplete conversion (p < 1 at t + -).

As to the absence of an apparently S-shaped curve for the dependence P(t), this may be simply linked with the form of presenting experimental data, i. e. with the initial apparent “induction” part of the dependence P(t) being too small to discern on the selected time scale.

The possibility of incomplete conversions can be seen from the results of computer simulation of curing processes with certain limitations applied to the mobility of elements in a reactive system [3], and from an experimentally observed phenomenon called the effect of self-deceleration [4,5].

Phenomenological kinetics of these processes can be described by the following equation:

d t 33 = k ( 1 - p ) ( l - t p ) (3.6.1)

where 5 is the dimensionless parameter qualitatively reflecting the contribution of self-deceleration to the kinetics of the process.

The constant 6 has a physical sense essentially different from the value c in equation 3.5.4. The value 5 reflects acceleration of mobility in a system causing diffusional limitations for continuation of the reaction. In some cases these limitations may be eliminated by a simple temperature rise or introduction of a plasticizer into a reactive system, i. e. by direct influence on the mobility of reacting molecules. The realization of these two possibilities is illustrated in

i

1 I I I I

too 200 300 100 t, min

Figure 3.6.2: Dependence of the degree of conversion on time for the system epoxy silico-organic oligomer/PBTP (100 : 7.5) ( I , 2) and the system epoxy silico-organic oligomer/PBTP + 20% mass. DBP (3). Temperatures 220 “C ( I ) , 180 “C (2, 3).


A temperature rise (transition from the initial curve 1 to 2) or introduction of a plasticizer (transition to curve 3) results in self-deceleration of degradation in the course of curing epoxy silico-organic oligomer. A certain drop in the process rate with introduction of a plasticizer is obviously linked with the effect of dilution. The temperature rise results in a consecutive growth of the parameter 5 until it reaches 1. Evidently, at 6 = 1 equation (3.6.1) is transformed to a kinetic second-order equation. Taking into account the physical sense of the constant 5 as a value of P-' (i. e. maximum admissible degree of curing) at t -+ m and using experimental data it is easy to find the values of the self-deceleration constant for each particular temperature of a reaction. The integral of the equation (3.6.1) has the form:

e x p [ ( I - t ) k t ] - 1 = exp [ ( l -5) k t ] -5

or another form more convenient for analysis:

In- I - " - - ( 1 - c ) k t 1 - P

(3.6.2)

(3.6.3)

Thus, determining the value of 6 provided 6 = P i ' by plotting P(t) dependencies on the

coordinates In ~ -" vs t we may verify the applicability of the equation (3.6.1). 1 - B

100 200 300

t , min

P Figure 3.6.3: Time dependence of In ~ - " (1-3) and ~ (4, 5) for the system 1 - P 1 - P

epoxy silico-organic oligomer/PBTP (100 : 7.5) at curing temperatures 140 "C ( I ) , 160 "C (2), 180 "C (3), 200 "C (4) and 220 "C (5).


Fig. 3.6.3 presents experimental data on the coordinates of formula (3.6.3) for curing temperatures below 200 "C and on the coordinates of the second-order equation ( ( 1 - p) vs t

for curing temperatures of 200 "C and 220 "C. Linear dependencies plotted on the given coordinates confirm the validity of equation (3.6.1) for low-temperature curing and its applicability for rheokinetic description of these processes [4, 51. Besides, in the high temperature region the process goes up to the end. Therefore its kinetics is described by a simpler second-order equation (i. e. where 5 = 1 and n = 2). Equations of this type (with incomplete conversion) describe kinetics of curing with curing agents of high functionality [2]. In these systems gel formation occurs very quickly and probably a macrogel is formed beyond the stage of microgel formation. However, decreased functionality of the curing agent initiates self-acceleration and the lower functionality the higher is the value of the self-acceleration constant [2].

The discussed cases are extreme variants of self-acceleration and self-deceleration effects in curing processes of reactive compounds, where one of them completely suppresses the other. However, a variant of their mutual superposition and manifestation is also possible. Fig. 3.6.4 shows time dependencies of the rubbery modulus in the process of curing melaminoformaldehyde oligomer [6]:

I I I I I

t, min

60 f60 300 420

Figure 3.6.4: Change of rubbery modulus during MFR curing. T = 140 "C (I), 130 "C (2), 120 "C (3), 110 "C (4) and 100 C (5).

The obtained results show that if the process of curing is effected below 130 "C a typical incompleteness of the reaction is observed the ultimate value of the rubbery modulus being a function of temperature. Since equation (3.6.3) cannot be used to describe curing kinetics of the given material it is necessary to compile an equation allowing for a mutual effect of self-acceleration and self-deceleration on the phenomenological pattern of curing [2]:

dt = k ( 1 - b ) ( I + c P ) (1 -Qi ) (3.6.4)


When the curing temperature rises to 130 "C, then 5 = 1 and equation (3.6.4) turns to a kinetic second-order equation with self-acceleration:

9 = k ( 1 - p ) 2 ( 1 +cP) dt

(3.6.5)

Integration of equation (3.6.4) with regard to the boundary condition p = 0 at t = 0 is results in the following formula:

or in another form more convenient for calculations:

(3.6.7)

Formula (3.6.7) allows to compare experimental and calculated dependencies of P(t) which are compared in Fig. 3.6.5. The results presented show that formula (3.6.7) obtained from the kinetic equation (3.6.4) satisfactorily describes changes in rheological properties of a reactive system in the process of curing melaminoformaldehyde resin [6]. An increase in the process temperature up to 130 "C results in practically complete degradation of self-deceleration (5 = 1) and the dependence P(t ) is described by the formula obtained by integrating a conventional kinetic second-order equation:

I I $+ 120 240 360 540

t, min

Figure 3.6.5: Dependence of rheological degree of conversion P on time in the process of curing MFR. Notation is similar with Fig. 3.6.4. The points are experimental and the line is calculated by formula (3.6.7).

3.6 Rheokinetic Eauations of Curing I83

It is interesting to note that formula (3.6.7) is of a rather general nature and fits well to describe curing kinetics of totally different systems, e . g., unsaturated polyether resin-benzoil peroxide. This is shown in Fig. 3.6.6 giving comparison of experimental data for the dependence of conversion degree of time obtained in [7] and curves calculated by formula (3.6.7). It can be seen that there is a good agreement of experimental data and theoretical predictions. Similar results were obtained using expressions (3.6.6) and (3.6.7) to describe curing kinetics of chemically different reactive compounds.

P

4 5

Figure 3.6.6: Experimental (points) and calculated by formula (3.6.7) (solid lines) dependencies of fractional conversion on time on semilogarithmic coordinates for the system "polyester resin-benzoyl peroxide". Curing temperatures: 90 "C ( I ) , 85 "C (2), 80 "C (3).

We may assume that the universality of the mathematical description of phenomenological kinetics of curing various systems reflects the common nature of physical-chemical phenomena in the process of forming materials with a cross linked structure.

Due to different physical states of reactive systems at the stages of gel formation and curing after the gel-point, these are studied using different experimental methods - viscometry and dynamic mechanical spectroscopy - which is reflected in the structure of the present Chapter 3. Here a question arises about the common nature of processes occurring at different stages of curing and about comparability of results obtained by these methods.


Formulae (3.4.4) and (3.6.7) should have a point in common which is the point of gel formation. For the condition of reaching the threshold of gel formation formula (3.6.7) is transformed in in the following way:

(3.6.8)

where P* > 0 is the value of the rheological degree of conversion at the moment of reaching the point of gel formation.

The derived expression indicates the existence of proportionality between the inverse value of the time of gel formation and the constant of the reaction rate. The relationship of l / t * and the constant of the reaction rate found via a rheokinetic equation for curing melaminoformaldehyde oligomer is shown in Fig. 3.6.7:

0, 5

Figure 3.6.7: Correlation of l/t* and the constant of curing rate for melaminoformaldehyde oligomer.

The value of the proportionality factor determined from data in Fig. 3.6.8 coincides with the prediction from formula (3.6.8). Particularly, in the studies of melaminoformaldehyde resins formula (3.6.8) acquires the form:

(3.6.9)

The determined relationship between t* and k shows that the rate of gel formation and the nature of change in rheological conversion degree are governed by a single process. Generally, the constant of reaction rate may be introduced into formula (3.4.4) in its apparent form and


determined on the basis of viscometric measurements. For instance, a calculated formula allowing to predict kinetics of viscosity growth in the reaction of epoxy oligomer with aromatic diamine near the point of gel formation has the following form [8]:

k -1.2 - = ( l - 2 t ) rl

rl0 (3.6.10)

where k is the constant of reaction rate found from measurements of the concentration of functional groups.


1. A. S . Reutov, S . G. Kulichikhin, A. Y. Malkin, III Vsesoyuznaya nauchno-tekhnicheskaya konferentsiya "Kompozitsionnye polimernye materialy - svoistva, proizvodstvo i primeneniye. ' I Tezisy dokladov, Moscow, 1987, 1 13.

S . G. Kulichikhin, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina, L. I. Golubenkova, A. Y. Malkin, Vysokomolekulyarnye soyedineniya, A , 1986, vol. 28, 10,2112.

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S . G. Kulichikhin, C. A. Astakhov, Y. P. Chernov, V. A. Kozhina, L. I. Golubenkova, A. Y. Malkin, Protsessy perenosa v strukturiruyushchikhsya zhidkostyakh, Minsk, ITMO AN BSSR, 1985,63.

A. Y. Malkin, S . G. Kulichikhin, P. A. Astakhov, Y. P. Chernov, V. A. Kozhina, L. I. Golubenkova, Mekhanika kompozithnykh materialov, 1985,5,878.

S . G. Kulichikhin, Z . D. Abenova, N. I. Bashta, 0. P. Blinkova, G. S. Matvelashvili, A. Y. Malkin, Plasticheskiye Massy, 1988,3,5 1.

C. D. Han, K. W. Lem, Polym. Eng. Sci., 1984, vol. 24, 7,473.

S . G. Kulichikhin, L. G. Nechitailo, I. G. Gerasimov, V. A. Kozhina, E. P. Yarovaya, Y. S . Zaitsev, Vysokomolekulyarnye soedineniya, A, 1989, vol. 31.

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3.7 Non-isothermal Curing

3.7.1 Causes of non-isothermal behaviour

From a researcher’s point of view, which is aimed at establishing quantitative kinetic regularities of chemical processes, temperature change in the course of the reaction is an undesirable complementary factor imparing the treatment of results obtained. However, in technological practice the non-isothermal nature of the process is often its inherent peculiarity which is extremely difficult, if at all possible, to remove. Moreover, temperature change in a reaction mass in the course of curing may prove an indispensable factor for a technological process ensuring production of a high quality end product. Therefore analysis of curing peculiarities under non-isothermal conditions is necessarily included into consideration of the entire problem.

Furthermore it is noteworthy that also in terms of laboratory studies of a curing process analysis of a reaction taking place in the non-isothermal regime has certain advantages. In fact, this analysis performed in a single experiment permits evaluation of reaction kinetics over a wide temperature range in spite of the complicate mathematical analysis in the processing of experimental data.

There are three sources of non-isothermally in the course of a curing reaction:

exothermic nature of the chemical reaction,

external heating and

dissipative heat release under reaction mass flow.

Heat release is always accompanied by heat exchange with the environment as well as propagation of heat flow within the material (article) cured. For this reason generally inhomogeneity in the temperature field appears in the article causing

a) different reaction times in different parts of the product and

b) possible volumetric inhomogeneity of product properties (quality) including initiation of considerable inner tension.

To evaluate inhomogeneity of the temperature field in a product (and all consequent undesirable effects) the problem of heat conductivity in a material should be analysed based on the equation of heat equilibrium. However, from the viewpoint of curing rheokinetics it is not of primary importance since heat propagation is not directly related to reaction kinetics “at a point” (although this problem may appear undoubtedly decisive for technological practice). Therefore below we shall consider only adiabatic processes for which the entire received (or generated) heat remains in the material leading to temperature rise and reaction acceleration.

If we assume uniform heat release throughout the volume of the product, temperature change is evidently expressed as:

(3.7.1)

3.7 Non-isothermal Curing 187

where p is the density, c is the heat capacity, Tis the temperature, q is the amount of heat and t is the time.

However, it is important to evaluate under what conditions the process may be considered adiabatic and inhomogeneity of temperature field throughout the volume of the product may be neglected. For that purpose, it is necessary to compare characteristic times of simultaneous processes of heat release and heat removal since they are both time-dependent.

Characteristic time of a chemical reaction may be denoted as tch . It is determined by the kinetics of the chemical reaction and depends on temperature. Nevertheless, even for nonisothermal reactions we may introduce a certain characteristic time tch , e. g. referring it to the initial temperature.

The characteristic time determining the rate of heat propagation in a material, t h , is

expressedas: th = 12/a

where 1 is the characteristic size of a product (e. g. thickness of a flat or ball radius ),

and a is the factor of heat conductivity: a = - h

(CP)

Then the condition of homogeneity of a temperature field in a product which have the form of sharp inequalities:

tch N th or tch (< th

The former means that the reaction is much slower compared with temperature levelling throughout the volume of the product, and the latter indicates that the reaction is so fast (and uniform throughout the volume) that heat transfer is negligible.

Further, if there is heat exchange with the environment this process also has it characteristic time t , expressed as

where a is the factor of heat transfer, S is the product surface (through which the exchange takes place), and V is the volume of the product.

Here again is the condition of homogeneity (isothermality) of the temperature field in the product, e. g. under external heating, expressed in the form of a sharp inequality:

‘ch ’’ t~ and the reaction is adiabatic if

t << t,. ch

If the above conditions (inequalities) are accomplished, inhomogeneity of the temperature field in the product is negligibly small. If it is not so, equation (3.7.1) should be replaced by the equation of heat conductivity which can yield (by means of analytical or numerical methods) true temperature distribution in a product. In numerous cases the kinetic equation for a non-isothermal process can be written down as:


(3.7.2)

where p is the conversion degree representing the depth of the reaction course, 6 is the reaction rate, f(p) is the kinetic function, the form of which does not depend on temperature, K is the constant, and E is the activation energy. The function f(p) is by far not always independent of temperature, particularly, it is never the case for reactions with self-deceleration (see Section 3.6 ). However, in many cases the assumption that the form of the function f(p) is independent of temperature represents a real situation.

Then the integral of the kinetic equation of a non-isothermal process has the following form:

(3.7.3)

where p is the current conversion degree at the moment of time t .

The left part of the equation is, as a rule, easily integrated since the kinetic function f(p) is expressed by rather simple analytic expressions, and the possibility of calculating the right part of the equation depends on the law of temperature change T(t).

For this change two cases are possible:

a temperature rise caused by internal reasons (heat effect of an exothermic curing reaction or dissipative heat release);

and a temperature rise in a reaction mass caused by inflow of heat from an external source.

Below these cases will be given a more detailed consideration in the analysis of

1.

2.

non-isothermal kinetics of curing.

3.7.2 Determination of Kinetic Constants in an Adiabatic Process

Equation (3.7.1) is valid provided there is no heat exchange with the environment. The value of heat release dq is proportional to the degree of conversion dp since the change of p is the heat source. The correlation between dq and dp has the form

where Q is the total heat effect of the reaction.

Then

dT - Q dp dt cp d t '

- - - -

(3.7.4)

(3.7.5)

Q . CP

The ratio - IS the maximum temperature gain which is possible in an adiabatic curing

process ATmax = T,- To, where T , is the maximum temperature, and To is the initial

temperature of the reaction mass.


Then it is easy to derive the following correlations:

and

(3.7.6)

(3.7.7)

Now, if we substitute p for Tin equation (3.7.2) we have an equation including two kinetic constants K and E.

This provides an experimental method of determining these constants if the form of the kinetic equation f(p) [ 1,2] is already known.

Thus, let the kinetic function f(p) be expressed by an n-order equation f(p) = (1 - p)". Substituting expression (3.7.6) in equation (3.7.2) for this kinetic function yields:

If a thermometric experiment is performed, i. e. the dependence T(t) is measured, this experiment would also yield the value of T , and the function T(t).

1 I

n '' T The last equation is presented on the coordinates In ( T - - T>

where the value of index n is chosen so as to attain the maximum possible linearization of experimental data. A straight line is obtained, which allows easy determination of values for the activation energy E and constant K (both from the slope) as well as constant K (from the line intersection with the Y-axis at T' = 0). Naturally this linearization of experimental data combined with determination of the optimal value of index n and subsequent calculation of the kinetic constants E and K may be easily automated and computerized.

As is seen from the above, a thermometric experiment makes it possible - in a single test performed in adiabatic regime - to find all the constants describing reaction kinetics over an arbitrary temperature range.

However, this general method of processing non-isothermal kinetic data has a serious disadvantage due to which it has to be regarded with precautions. The fact is that, as is seen from the above given procedure, the use of this method requires a priori knowledge of the form of the function f@), e. g., in the form of the exponential law. If the function f(p) has a different form, its incorrectly defined form leads to principal errors in treatment of experimental results and false estimates of activation energy values. This was demonstrated for example in [3,4], where it was shown that the direct application of the thermometric method to kinetic analysis of a non-isothermal reaction of nylon-6 formation with the assumption that f(p) is expressed by


the exponential function gave a value for the activation energy E of 200 kJ/mol, whereas in reality the function f(p) for this reaction has an auto-catalytic nature, and the correct value of activation energy is - 65-70 kJ/mol.

A more general method to analyse thermometric data obtained in the studies of non-isothermal kinetics implies performing a series of measurements at different values of the initial temperature To and comparing values of f (and, thus, values of p) at points where p = item [ 5 ] . This is illustrated by Fig. 3.7.1, the left part of which shows a series of initial experimental curves T(t) at different initial temperatures To, and the right part shows recalculated (non-isothermal) dependencies p( t ) . In the same plot the dashed line shows a certain value of pi and corresponding reaction rates pi. Coming from the right part of the figure back to the left one, we find for each value o f t a corresponding temperature value:

J

t t

Figure 3.7. I : The thermometric method of determining kinetic constants in experiments with direrent initial temperatures of the reaction: TO,^ > To,* > To,,. Determination of p(7) dependence.

Therefore for each value pi = item a value of activation energy may be found and, thereby,

the factor exp (--) . Then, by plotting a dependence of the complex on 6 exp (--) we

find the unknown function KO f(p), and with the normalising condition f(p) = 1 at p = 0 a

separate evaluation of the factor K , is attained.

E E RT RT


!(PI 7

6

5

4

3

2

I

0

0 a

I 0 f59 f70

0

\ \ \ \ \ \ \ \

o \ 0 1

\ \ \ \

Figure 3.7.2: Determination of the kinetic function f (p) by the the thermometric method using various initial temperatures To. The dashed line represents the function f(p) = ( 1 - p) ( 1 + 22.5 p) . Experimental data are obtained from the investigation of non-isothermal kinetics of nylon-6 production.

An example of the function f(p) thus evaluated is shown in Fig. 3.7.2. A dashed line shows the precise form of this function found with more rigorous methods [ 3 ] . It is obvious that the form of this function is really far from the exponential equation at any value of n. Therefore this somewhat complicated variant of the thermometric method can yield true information about kinetics of a non-isothermal process since it is not related to the a priori choice of the form of the function f(p). However, it is based on using an enlarged body of initial experimental data (tests should be performed at several values of To).


3.7.3 Non-isothermal Curing - Calculations.

As it was mentioned above, in practice curing always takes place in a non-isothermal regime since reactions of curing always exhibit a considerable exothermal effect. Analysis of non-isothermal curing is based on considering a kinetic equation where the law of temperature change itself is determined by reaction kinetics.

Let us suppose that the kinetic equation (3.7.2) is satisfied. Temperature included in this equation is related to p by formula (3.7.7). Therefore by eliminating temperature we may obtain an equation with p as the only variable:

This equation may be integrated numerically

(3.7.8)

to find the non-isothermal dependence p(t) , but for this purpose it is necessary to know the form of function f(p).

We may demonstrate the possibility of practical application of this equation both for qualitative evaluations and for precise quantitative calculations, e. g., let the kinetics of an

isothermal process be described by the first-order equation, i. e. f(p) = (1 - p). Evidently measurement of P ( r ) in the course of isothermal curing here is expressed by the

exponential function

p ( t ) = 1 - e-Q (3.7.9)

where KO = K exp (--) is the value of the initial rate of an isothermal reaction at E temperature To. RTO

For a qualitative (but realistic) evaluation of the dependence P(t) in an adiabatic regime we may use the following approximate expression:

r 1

The latter expression was obtained by the following manipulations:

r 1 r 1


These approximate correlations are valid provided the inequalities (QP) / (cp To) << 1 and

(EQP) / ( c p R c ) i( 1 are accomplished which is always the case if exothermic heating is not

too extensive or at least at initial stages of the reaction. The dimensionless factor

(EQP) / ( c p R c ) in formula (3.7.10) is designated as a.

Equation (3.7.10) is easily integrated by quadrature yielding

1 -exp [ -Kot( l + a ) ] (3.7.1 1)

Now, comparing formulae (3.7.9) and (3.7.1 1) it is easy to evaluate the exothermal part in the dependence P(t) by plotting the dimensionless time t = Kot vs P (Fig. 3.7.3). The parameter a characterises the deviation from the isothermal regime of curing:

' ( t ) = 1 + a exp [-KO' ( 1 + a) 1

d= 0

4 E I 2 3

Figure 3.7.3: Time dependence of the degree of conversion (on dimensionless coordinates) at direrent values of the parameter a reflecting the effect of non-isothermal self-acceleration of the reaction.Curves are calculated from eq. (3.7.1 1). The curve for a = 0 corresponds to isothermal conditions.


Evidently the boundary case a = 0 manifests a return from formula (3.7.11) to the isothermal situation expressed by formula (3.7.9). The criterion a as is described by the kinetic equation (3.7.10) shows the effect of curing acceleration caused by an exothermic nature of the reaction. This is also seen from Fig. 3.7.3 where it is possible to detect accelleration when the role of non-isothermality becomes significant. In fact, the higher a the more pronounced - and starting from smaller time 7 - is the role of non-isothermality. As we see, there are situations possible when exothermic effects lead to a rather strong acceleration of curing.

The problem may also be solved numerically if precisely formulated, i. e. without the approximate correlations used above, and calculations may be performed for functions f(p) of different types. To this effect equation (3.7.8), after quite evident transformations, can be written down as follows:

(3.7.12)

where the function f(p) is represented by the exponential expression (1 - p)" and the following dimensionless parameters are introduced:

- E dimensionless time t = Kot = K t exp (--)

RTO

Q - AT,,, dimensionless heat effect 4 = ~ - - C P T o To

- E dimensionless energy of activation E = ~

This equation may be represented using the earlier introduced complex a as in fact

RTQ

a = E q . This will provide a visual comparison of precise and approximate solutions.

After introduction of complex a equation (3.7.12) acquires the form:

(3.7.1 2a)

In addition to the paramater a the complimentary parameter 4 is introduced, which in

Now, according to equation (3.7.12), we calculate the dependence PG) at different values of the dimensionless criteria a and 4 for any values of n. This was done for several sets of constants, and the results are given in Fig. 3.7.4 for n = 1 and n = 2. The obtained results of calculations evidently show the role of major factors determining non-isothermality in processes of curing as well as the reaction acceleration due to heat effects compared with - the initial case of isothermal curing. It is obvious that the determining factor is the value a = Eq, whereas the factor s taken separately plays a subsidiary role. This is seen from a comparison of the upper and lower plots corresponding to quite different values of G , (0 and 0.5). These plots also show the effect of the reaction order on the dependence p(7).

equation (3.7.11) equals zero. Its role will be seen from the results of calculations.


V I 1 I I I

42 44 4 6 4 8 f

I-

42 0.4 46 0,s

r Figure 3.7.4: Calculated curves for kinetics of adiabatic curing allowing for the heat

effect of the reaction. The solid lines correspond to the second order equations, and the dashed lines correspond to the first-order equations.

Similar calculations, if required, can be performed for kinetic functions the type of which is different from the exponential equation.

In many cases the main interest is not the whole course of the kinetic function f@) but concerns the evaluation of a certain critical point reached in the course of curing. In this respect the gel-point t* is of special importance. It can be reached both in the isothermal and in the adiabatic regimes of curing. The central criterion for this evaluation is reaching the gel-point in the non-isothermal regime of curing.


The criterion of non-isothermal curing was formulated in [3] as a law of integrating contributions of reactions taking place at different temperatures. Analogous to the known criterion of Beily describing conditions for mechanical destruction at the variable loading regime, the criterion of non-isothermal curing may be written down in the following form:

(3.7.13)

where tit* is the time before reaching the gel-point in the non isothermal regime of curing, B is the constant.

Let the temperature dependence of isothermal curing time be commonly expressed by means of Arrhenius equation:

E t* = Bexp(-) RT (3.7.14)

It is convenient to exclude the constant combining the two last formulae and assuming t* = t$ at some initial temperature To. Then

or passing to dimensionless variables:

-* 7

j e x p (ET)at = 1 o

where the dimensionless values are defined as follows:

(3.7.15)

dimensionless non-isothermal time of curing i* = t,* / t t ;

dimensionless time (here) 12 = t / t : ;

- E dimensionless energy of activation E = - ; RTO

- (T-To) dimensionless temperature T = ~.

TO -* . .

To determine T it is necessary to obtain a combined solution of the kinetic equation (3.7.2), e. g. in the particular form (3.7.12) and the criterion of non-isothermal curing (3.7.15).

3.7 Non-isothetmal Curing 197

The results of a combined solution of these equations are shown in Fig. 3.7.5 as a dependence ?* (4) at different values of the constants E and = Kot: for the two values of the exponent n = 1 and n = 2 [71. From this plot it can be concluded that the greater the energy of activation the faster the reduction of curing time with increasing heat effect. The physical nature of this effect is quite obvious. However, it is essential that Fig. 3.7.5 gives a strictly quantitative expression of this effect.

0. f

a

I

2

3

Figure 3.7.5: Dependencies of curing times on heat effects of the reaction (in dimensionless variables) with various values of activation energy: E + I0 (I), 30 (2), 50 (3). Values of initial reaction rate KO = 0.1 (a) and I0 (b). The solid lines correspond to the second order equation, the dashed lines correspond to the first-order equation.

-

It should be noted that the form of a kinetic equation (i. e. the value of the exponent n in this case) eventually plays a subsidiary role and its effect on the course of dependencies ?* (4) is not too pronounced although the differences can be seen, especially, if the initial reaction rate is great, i. e. at large values of K O . It is also seen in Fig. 3.7.5 where the results of calculations


are given for two widely differing values of KO. In the lower part of the figure the data obtained for first and second-order equations coincide. Thus the general approaches above enable us to give a complete analysis of non-isothermal curing problems - from calculating the whole kinetic curve P(t) up to evaluating the duration of curing in the non-isothermal regime r,*.

To conclude this section of the book, it should be noted that the equations discussed above, primarily the general kinetic equation (3.7.2) and the criterion of curing (3.7.13) make it possible to give a solution for any non-isothermal (homogeneous) problem obeying the arbitrary law of temperature change. This may be, for example, heat transfer from an external source the temperature of which varies by a given law. This case will be discussed in the next chapter. In the general case, however, the arbitrary nature of temperature change does not provide universal solutions which are possible to obtain from analysis of exothermic (adiabatic) curing with an internal source as it was shown above.

3.7.4 Heat nansfer from an External Source (Linear Temperature Increase)

Let the exothermal effect of a reaction be small and the temperature of the reaction mass be determined by the ambient temperature. Thereby, as we did earlier, we shall assume the temperature distribution in the product (sample) to be negligible. Besides, we shall consider a reaction with a rather slow rate so that the temperature of the reactive medium has sufficient time to “adjust” to the ambient temperature. Then the given law of ambient temperature change will determine the dependence T(t) entering the non-isothermal kinetic equation.

Now we shall consider the kinetics of non-isothermal curing with programmable linear change of temperature with time, i. e. we shall consider T(t) as follows:

T(t) = To+ar (3.7.16)

where To is the initial temperature, and a is the rate of temperature growth.

arization of the exponential function which were employed in Section 3.7.3. Then we obtain: To illustrate the role of non-isothermality we may use the same approximations for line-

E K exp (-

This means that the general kinetic equation (3.7.2) will acquire the form:

p = KO f(P) 1 +- ( ;;I

(3.7.17)

(3.7.18)

i. e. non-isothermality leads to self-acceleration of the process with time, the intensity of which

is determined by the factor ( W E ) / (RT;) which has the dimension of a reverse time (i. e. the

characteristic time on the scale of which the effect of self-acceleration is observed equals

fa, = (RT;) / ( a E ) ) . This is the time to be compared with the characteristic time of the

chemical reaction t C h . If t,,, = t,, , the non-isothermality caused by heating the reaction mass

from an external source is negligible, and vice versa.


If we avoid linearization, integration of a non-linear (non-isothermal) kinetic equation may be completely performed by numerical methods provided the kinetic function f( p) is unknown. Let us consider, for example, the situation when f(p) is expressed by a second-order equation, i. e. f(p) = ( 1 - p) *. Then the integral in the left part of the equation (3.7.3) will acquire the form p/( 1 - p) and we obtain:

I P E

1-p = 6" [-R (To + a t ) ] " or, if we denote the right part of the formula as I we get:

I 1+I

p = - .

(3.7.19)

(3.7.20)

It is convenient to make calculations in a general form using dimensionless variables, namely:

a t TO

dimensionless time introduced here in the following way T = - ;

dimensionless energy of activation E = ~.

Then integral I will be expressed as follows:

- E RTO

a 0

(3.7.21)

It can also be expressed by the Euler function E~ tabulated in all mathematics reference books as follows:

Then, if we know the constants E and K and have the preset parameters of the non-isothermal regime To and a, using equation (3.7.20) and the above expression for integral I we can calculate the course of the non-isothermal dependence P(t) .

The obtained formulae make it also possible to consider the reverse problem, i. e. use of the measurements of the dependence p(t) for determining the kinetic constants K and E. This needs employing equation (3.7.19) the left part of which may be different if we take up a different (not second) order kinetic equation.

Further, equation (3.7.19) is presented as

log (p) = logk+ F(T) 1 - P

(3.7.22)

7 - E

l + T where k = T o K / a and F(T) = l o g b p (-)dz.

0


The experiment gives the dependence P(t) which enables us to find the left part of equation (3.7.22). In Fig. 3.7.6 the function F(z) is plotted. Then, shifting the experimentally derived dependence log [ p/ ( 1 - p) ] from t along the y -axis until it coincides with one of the curves plotted in Fig. 3.7.6, we may find k and E

- 1

30 35

- I3

- f 7

4 2 46 f,O B

Figure 3.7.6: Function F(T) included in equation (3.7.22) for calculation of kinetic constants at linear temperature increase with time.

In principle this method provides a solution for the reverse kinetic problem. However, its successful application requires experimental covering of a wide range of argument z variations from 0 to 1 . Otherwise, the experimental curve may well coincide with different calculated curves in Fig. 3.4.1 1, i. e. the described method, though generally effective in obtaining kinetic constants, lacks sensitivity. Therefore it is not very reliable. Besides, its use requires the form of the kinetic function f(p) to be predetermined, which further limits its potentialities.

Using a linear mode of temperature increase leads to a rather simple quantitative correlation which, nevertheless, successfully conveys the role of the rate of temperature increase.

We may really use the general criterion of non-isothermal curing (3.7.15) and introduce the following law of temperature change into it:

T = T,+yt where y is the rate of temperature increase.

When the general criterion will acquire the form: *

/ exp[-R(T,,+yt)] d t = B (3.7.23)

The general temperature change is assumed to be not very large, namely, the condition y t /T , (( 1 is realized (e. g., if curing starts at 400 K, this condition is fulfilled with a temperature gain not exceeding 50 to 60 K). Then it is possible to linearize the exponent included in equation (3.7.23) similar to the above mentioned procedure when the kinetics of adiabatic curing under the impact of only internal heat sources (exothermic reaction effect) was discussed.

3.7 Non-isothermal Curing 20 1

This enables us to calculate the integral included in equation (3.7.23) which results in:

where, again, to is the value of t* at T = To (in the isothermal process).

Hence - * 1 t , = - In ( 1 + E )

E (3.7.24)

- * where t ,

E = Eyt; / R e is the dimensionless rate of temperature increase.

It is evident that with increasing E the values of t , dependence I,* (E) in equation(3.7.24) is shown in Fig. 3.7.7.

= t,*/tO is the dimensionless curing time in the non-isothermal regime and

- * decrease. The general nature of the

- 3 - 2 - f a f 2 3 4 &

Figure 3.7.7: Plot of the dependence t , on E calculated by formula (3.7.24).

It is seen that an appreciable drop of t , compared with the isothermal value of to* = 1 begins approximately at E = 0.1 whereas at E 40 the value of t , goes down compared with to* by about a decimal order.

For better orientation consider the following numerical example based on ordinary values of items constituting the complex, e. g. let E = 80 kJ/mol, to* = 240 s (4 min) and To = 350 K. Then the value of E = 40 kJ/mol is reached at the rate of temperature increase y = 2 Ws.

The general criterion of non-isothermal curing (3.7.15) as well as its specific form (3.7.24) corresponding to a linear temperature increase have the common value disregarding the nature of non-isothermality. Thus, the temperature may rise due to heating by an external source. The temperature increase due to the exothermal nature of a curing reaction, as would be seen from the preceding section, displays on the whole,a sharper temperature increase with time than the linear one. However, there exists another notable internal source of heat release causing a practically linear growth of temperature. This is the dissipative heat release appearing under flow of a cured liquid. This problem will be dealt with in the following section.

- * - *

202 Chapter 3: Rheokinetics of Polymer CurinR

3.7.5 Curing at High Shear Rates

If curing is effected at high deformation rates this may result in a non-isothermal process. In fact the flow of a viscous material is connected with dissipative losses realized as heat release and consequently resulting in a temperature increase. This may considerably accelerate curing compared with the induction period to* measured in a static experiment (i. e. without high deformation rates). Practically it may cause clogging up of the gating system and other undesirable consequences.

The intensity of heat release q under simple shear flow is expressed as:

q = z j where T is the shear stress, and 7 is the shear rate.

(3.7.25)

If flow (when the induction period is determined off-line by the viscometric method or in the course of pressing or moulding) occurs at j = const, the temperature growth causes a drop in the shear stress z, whereas the stress due to non-Newtonian properties of cured polymer compositions may also depend on the shear rate.

Therefore in a general case the temperature gain with time is expressed by the equation:

Y EV dT = A ( - ) exp{-}d C P RT

(3.7.26)

where c the heat capacity of the reaction mass, p is its density, E is the activation energy of

viscous flow calculated, at j = const., and A is the factor of proportionality in the formula

expressing the temperature dependence of shear stress z = A exp { ( E j ) / ( R T ) } .

Hence, in the linear approximation (if the temperature gain AT << To) the temperature

change is expressed as follows:

(3.7.27)

E?j RTO

where zo = A exp { -} is the tangent stress acting at the initial temperature To.

Substitutin this expression in 3.7.13 or 3.7.15 and performing corresponding calculations results in the t , dependence on T.

To obtain a clearer result, it can be assumed that the temperature growth with time is not described by a logarithmic law as it follows from (3.7.27) but by a simpler linear law [8]. Because of an extremely small value of E . it is rather often possible to regard the change of z induced by temperature growth as negligible and assume that z = T~ = const. up to the end of the induction period.

- %

707 Then T = To+-t = To+Ct

“P (3.7.28)

i. e. the temperature growth can be described by a linear law at the rate 5 = - . CP

3.7 Non-isotheimal Curing 203

This makes it possible to employ the results following from formula (3.7.19) given above. Then the final formula (3.7 29) for t,* is also valid. The parameter E in this case has the meaning of a dimensionless shear rate and is written in the following way:

(3.7.29)

Formula (3.7.24) and its plot shown in Fig. 3.7.7 reflect the dependence of the dimensionless induction period in a non-isothermal process tn* on the dimensionless shear rate E expressed by (3.7.29). Apparently, at sufficiently high values of 7 a rather notable drop of t* may be expected compared with to" corresponding to the initial temperature To.

The problem of the effect of deformation rate on the induction period can be considered in a stricter formulation if it is impossible to use the limitation used before. In its full formulation the problem of t,* dependence on the dimensionless shear rate E is written as two equations:

. *

(3.7.30)

The first equation describes the law of temperature change with time, and the second equation expresses the criterion of non-isothermal curing.

The solution for these equations is the dependence of the non-isothermal induction period tn* on the shear rate at different values of activation energy. To obtain a general solution it is convenient to represent the system of equations (3.7.30) in dimensionless variables, namely:

(3.7.3 I )

It employs the same dimensionless values as before:

the dimensionless time 7 = t / t $ ,

the dimensionless non-isothermal induction period t,* = tz / t $ ,

the dimensionless energy of activation K = RTo/E ,

the dimensionless temperature 0 = - ( T - To) , E

RTO

E 2oj t * the dimensionless shear rate E = - - .

RT; cp

The solution for the system of equations (3.7.31) was obtained and discussed in detail in [9]. This solution is the dependence t,*(&, K), which can be determined when the variable 0 is excluded. An additional factor considered negligible in the approximate theory in the system of equations is the variable K which affects the course of the function t , * ( ~ ) . Yet particular calculations show that reasonable changes of K do not alter the course of the function In*(&)

considerably. Therefore the role of the shear rate, although in the first approximation, is suffi-


ciently well expressed by function (3.7.24) as seen from a comparison of theoretical and experimental data in Figs 3.7.8 and 3.7.9 where the function t,* = E-’ In ( 1 + E ) is compared with the results of t,* measurements performed at different shear rates.

An essential feature of the experimental data is that these were obtained for two principally different materials: phenolformaldehyde resin (Fig. 3.7.8 with “natural” coordinates used for better illustration) and silico-organic oligomer (Fig. 3.7.9):

Figure 3.7.8: Dependence of the induction period on shear rate in the course of curing phenol resin. The curve is calculated, the error bars correspond to experimental data.

Figure 3.7.9: Effect of shear rate on the value of the induction period in the course of curing silico-organic oligomer for various initial temperatures To (dimensionless coordinates).


The chemical processes occurring in the course of curing differ considerably. The role of the shear rate, however, appeared to be the same for both materials and can be described with high precision by equation (3.7.24).

This is a convincing argument in favour of the idea that the effect of high deformation rates on the reduction of the induction period is confined to the growth of temperature due to dissipative heat release.

In fact this mechanism is of universal value for any material. However, in a general case the role of shear rate may be different. Intensive deformation itself may activate a chemical process initiating both structuring and mechanical destruction of macromolecules and the final result may be different depending on the nature of the material.

The most important effect connected with the influence of shear rate on the rate of a chemical process is the dependence of conditions of phase transitions on the mode of deformation. This problem applied to the process of polymerization was considered in Chapter 2.

It appears to be much more difficult to obtain equally definite results for curing processes as here the criteria of phase transitions are by far not always defined. There is no doubt, however, that also in these cases the deformation rate affects the curing time by changing the conditions of phase separation. Apparently this phenomenon should become appreciable after the formation of a microgel when the two phases are really present in the system being cured - a homogeneous cured mass and newly-formed colloidal particles of a microgel.

An example illustrating the effect of shear stress on the rate of viscosity growth in the course of oligomer curing is shown in Fig. 3.7.10. Here the moment of time up to which changes in shear stress do not affect the nature of viscosity growth is evident. Starting from this moment the course of the dependence q(t) is determined by the shear rate. Direct experiments (optical observations) have shown that a microgel is formed at this point or, if we specify it, growing particles of a new phase reach colloidal sizes. In such a two-phase system we observe the effect of shear rate on curing rate (up to the macroscopic gel-point). In this case the effect of shear rate is not connected with non-isothermal effects [lo].

I f 2 3 4 5 6

0 80 t, min

Figure 3.7.10: Effect of shear stress on the pattern of viscosity change in the course of curing a 57% solution of melaminoformaldehyde oligomer at 80 "C. Values of z (Pa): 0.7 ( I ) , 1.5 (2). 2.5 (3), 9.9 (4), 20 (5), 80 (6).



1 .

2.

3.

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A. Ya. Malkin, A. E. Teishev, Ing-Phyz. Zh., 1989, vol. 56, 4,591.

A. Ya. Malkin, Plastmassy, 1982,4,47.

A. Ya. Malkin, V. P. Begishev, Polymer Process Eng., 1983, vol. I , 83.

4.

5.

6.

7.

8.

9.

10. A. Ya. Malkin, G. Z. Shuvalova, Vysokomolekulyarnye soedineniya, B, 1985, vol. 27,865.

Structuring Peculiarities of Direrent Forms of Oligomers and Polymers 207

3.8 Structuring Peculiarities of Different Forms of Oligomers and Polymers

3.8.1 Curing Phenol formaldehyde Resins and Related Oligomers

Phenolformaldehyde resins (PFR) are the oldest thermo-reactive oligomers used in plastics industry to produce items by forming a relatively low-viscous melt with subsequent transition to a cured (non-fluid) material. The process productivity and, to a great extent, the quality of products obtained (particularly, feasibility of thin walls manufacturing) depend on the fluidity of the initial material and its curing rate. Therefore rheological peculiarities of curing PFR and compositions on their basis determine the technical properties of these materials.

PFR and related materials can be used as moulding compositions (for production of monthing powders), impregnation (binding), compositions glues, lacquers, etc. Recently rapid advance is seen for the method of producing articles from PFR by injection moulding. All these applications demand peculiar rheokinetic characteristics of oligomer curing which necessitates special control of composition content and needs reliable methods to evaluate their properties.

The formation of cured materials from PFR is effected by polycondensation of phenol with formaldehyde. This process comprises a great number of various chemical reactions. Their

relative rate and value depending not only on temperature but also on a particular composition. These factors also determine the curing rate. The dominating (though not the only) reaction

resulting in a non-fluid end product is the formation of methylene links between phenol units. Instead of phenol and formaldehyde other compounds with similar functional groups may be

used. For instance phenol may be replaced by cresol, xylenol, resorcine, aniline, etc., and formaldehyde by acetaldehyde, phurphurol, etc.

In industry, it has been old practice to use a number of standardized methods determining technical properties of PFR which imply evaluation of their fluidity [ 11. Thus a rather wide field of application is observed for a method of evaluating PFR fluidity called “the method of a glass” (described, e. g., in USSR State standards for materials, British standard BS2782, method 105B, American standard ASTM D 731-67). This method implies analysis of filling a standard form shaped as a shallow glass with thickened edges. The mass of a product (“a standard glass”) man- ufactured in this form is 2 - 2.5 g. The temperature varies with adopted standards: 163 f 1 “C for BS, 150 “C, 155 f 1 “C or 165 f 1 “C for ASTM. In the US standard the characteristic of material fluidity is assumed to be the minimum load which enables a quality item to be formed within 10 - 20 sec; e. g. the material “moulding index” 650018 means that at the minimum load 6500 kg the glass formation is completed within 18 sec.

In the UK standard the fluidity characteristic is defined by time needed to fill the form with the standard force 100 f 5 kN. A similar measurement of material properties is acquired in the FRG standard.


Another widespread method of determining technical properties of thermo-reactive materials, particularly PFR and compositions on their basis as well as aminoplasts, is evaluating the fluidity index according to Rashig*. With this method, a sample under a given bad is pressed into a hot moulding form shaped as a channel with the cross section reducing along its length. The channel dimensions are exactly specified. The material is ejected in the channel under pressure which reaches a preset level within 20 sec. The characteristics of fluidity is assumed to be the length of a moulded rod, i. e. the depth of the channel reached by a material before it completely solidifies and loses fluidity.

These methods (“method of a glass” and “Rashig’s method”) provide only an arbitrary technical characteristic of fluidity and retention time of the PFR viscous-flow state fit to control reproducibility of material properties in industrial batch production but inapplicable for broad comparison of different materials and quantitative evaluation of rheokinetic characteristics of various cured compositions, the latter being particularly important for moulded materials. Therefore various viscometric techniques were developed to provide a complete pattern of PFR curing including evaluation of the induction period t* .

Justifying the choice of the viscometric method of controlling processes of curing PFR- based reactoplasts, it is necessary to take account of the main characteristic features of rheological properties of moulded compositions and compositions for injection moulding. These features are the following:

low viscosity of a binding agent (i. e. PFR), (by several orders lower than that of the composition),

existence of fluidity limits, i. e. stresses below which the material does not flow (the latter is evidently caused by forming the filler structure and is typical of dispersed systems),

possibility of phase separation under flow occurs over the low-viscous layer of the binder.

These peculiarities of rheological properties of compositions should be borne in mind when evaluating technical properties of these materials.

The peculiarity of rheological properties of materials based on PFR and related oligomers impedes (and possibly excludes) applicability of conventional viscometric instrumentation for their analysis and requires special semi-technological methods to be developed. These should ensure generation of rather high pressures and shear stresses as well as rapid heating of a sample to a given temperature.

These requirements are most efficiently embodied in the method of “plastometric tests” performed with a specially designed viscometer - Kanavets plastometert. The principal scheme of its measuring uni t is shown in Fig. 3.8.1. The sample is pressed by a punch under pressure p between the fixed die and the rotating rotor. The encased measuring unit is placed in the heating chamber. The rotor is rotated by a mechanic drive. This brings about a torque which causes

* This method is described, e. g. , in the USSR state standard 9359-80 “Moulding carbamido- and melami noformaldehyde masses”. This apparatus is produced in the USSR in two versions labelled P-1 and BP-I, it is also commercially available from Hettfert, FRG, as “Kanavets plastometer”.


twisting of the torsion, its twisting angle being a measure of torque and, thus, of shear stresses acting in a studied sample is measured by a turn angle meter. Test data are registered by a recorder as dependencies of shear stress on time in the way shown in Fig. 3.8.2.

3

4

7

Figure 3.8.1: Kanavets plastometecl - punch, 2 - die, 3 - case, 4 - rotor, 5 - torsion, 6 - torque meter, 7 - sample.

I

t *

Figure 3.8.2: For determination of the induction period t*: Nature of viscosity change with time in the course of isothermal oligomer curing.

In order to eliminate an undesirable effect of rate on the value of the induction period (see Section 3.7), the standard deformation rate is assumed to be 0.015 s-' although the drive of the apparatus is able to develop deformation rates up to 15 s-l. As it was shown above (Section 3.7), tests, performed at high deformation rates simulating industrial flow conditions during injection moulding of reactoplasts, result in a sharp reduction of the induction period.


Plastometric tests make it possible to determine two main characteristics of PFR-based thermosets -the values of t* and shear stress z in the region of viscous-flow state zs . Evidently, the latter value determines the material viscosity q = ~ ~ / 0 . 0 1 5 .

Characteristic data of low-rate plastometric measurements for a number of Soviet PFR- based thermosets are shown in Fig. 3.8.3 (from [2]). Crosses in these diagrams correspond to t* .

0.4 I

I I I I

2 4 6 t, min

Figure 3.8.3: Plastograms of some PFR-based reactoplasts at 120 “C: I - K-18-24; 2- 015-010-75; 3- K-18-2; 4- K-18-28 [2].

4 8 f2

t, min

Figure 3.8.4: General diagram characterizing low-grade (v = 0.015 s-’) plastometric properties of PFR-based reactoplasts at 120 “C 121.

The values of T~ and t* determine the way of classifying PFR-based thermosets dividing them into “press” and “mould’ types. However, the boundary between the recommended methods of processing is arbitrary, and typical press-materials (e. g., K-218-24 and 015-00-75 sorts) may be successfully processed by injection moulding provided a correct technological regime is employed. Nevertheless, there is a diagram available shown in Fig. 3.8.4 which should be used for material evaluation. Practically all commercially available PFR-based reactoplasts (as well as aminoplasts) keep within a range of viscosities (at 120 “C) approxi-

Structuring Peculiarities of DifSerent Forms of Oligomers and Polymers 21 1

mately from lX105 to 6 ~ 1 0 ~ Pa.s and values of t* vary from 70 to 700 s. The suggested diagram enables us to give the following classification of materials according to recommended processing methods. If material characteristics lie within the region between lines 2 to 4, these materials may be processed by transfer moulding, preferably with high-frequency electric heating (the range of viscosities is 1 x107 to 5 ~ 1 0 ~ Pas, the range of t* values 120 to 170). The region between lines 3-4 (the range of viscosities is 1 x107 to 2 . 5 ~ 1 0 ~ Pas, the range of t* values 180 to 700 s) corresponds to materials processed on moulding machines with screw plasticators and with a maximum loading capacity in the machine cylinder of up to 250 cm3. Materials corresponding to the region between lines 4 and 5 (viscosity values from 1 x105 to 1 X107 Pa.s, t* values from 180 to 700 s) are low-viscous long-livedcompositions. They should be processed on machines having a device to prevent material leakage through the norele during plastification and a reverse valve excluding material backflow during injection. The region between lines 1 and 2 corresponds to high-viscous short-lived materials, which should be processed by pressing. This can certainly be applied to materials whose characteristics lie below line 2.

Behaviour of reactoplasts at high shear rates, as was shown in Section 3.7, is largely determined by their characteristics measured in low-rate standardized tests (i. e. at ? = 0.015 s-'). Therefore using the formulae above, we can calculate t* values in real moulding conditions from the results of low-rate tests. This will evidently show that upward and rightward drifts in the diagram shown in Fig. 3.8.3 result in intensifying the t* dependence on moulding conditions and thus, generally, impairing moulding properties of the material. Therefore the best moulding parameters are displayed by materials plastograms which, in low-rate tests, fall between lines 4 and 5 in Fig. 3.8.4.

Usually plastometric tests of PFR-based reactoplasts (as well as aminoplasts) are performed at 120 "C. Of general and practical interest is, however, a possible extrapolation of these test data on other temperatures. This proves the necessity of kinetic studies of PFR curing at various temperatures.

Kinetics of PFR curing is rather complicated which is determined by a whole set of simultaneous reactions. For this reason it is not obvious a priori which of them is a limiting one and governs changes in rheological properties of a cured composition. Besides, depending on reaction conditions (e. g. medium acidity) curing kinetics may vary radically due to a particular reaction which becomes limiting, being characterized by its kinetic equation. As a rule, kinetics of PFR curing is evaluated from concentration changes in particular agents (loss of urotropine amount of released ammonia, gain of insoluble gel-fraction). Evidently, these methods do not yield adequate results as they reflect the course of different simultaneous reactions. An essential consideration for using these methods is also a correct account of macroscopic factors like effects of diffusion phenomena and reactions of non-isothermal nature on the observed changes in the parameters measured.


Using the calorimetric method (in the form where the measured parameter was the temperature change in a non-isothermal process) it was demonstrated [3] that the curing kinetics of PFR type K-18-2 is perceived as a single-stage reaction and described by the equations:

U 9 = k(T) ( 1 - p)" k(T) = ko exp (--) dt RT (3.8.1)

where p is the conversion degree (a relative value), ko, n and U are kinetic parameters which, in our case, appeared to be: n = 1.42 and U = 96 kJ/mol. Apparently we can assume equation (3.8.1) with universal values of n = 1.42 and I!/ = 105 f 10 kg/mol to have general relevance for PFR curing.

A possibility of describing kinetics of PFR curing as a single stage process the depth of which is characterized by one kinetic parameter - the relative concentration - enables us to assume that also viscosity change with time will be unambiguously defined by this parameter. In fact, as was shown by varying conditions of sample preparation [4], dependence of PFR viscosity q on temperature T and parameter p is described by the formula:

(3.8.2)

where the effective energy of viscous flow E appeared to be 34.5 kJ/mol and practically independent of parameter p throughout its possible variation range. The form of the function (13) is shown in Fig. 3.8.5:

I

0 0,4 0,8 '' 488 496

P Figure 3.8.5: Viscosity growth in the course of PFR curing (from [3, 41)

The dependence q,(p) in the course of PFR curing can be approximated as follows:

(q',, = const at p 50.837

(3.8.3)

where q', is the initial value of the function q,(p) which remains practically constant up to a rather high degree of conversion (0 = 0.837). At p + 1 an unlimited viscosity growth is observed, i. e. PFR loses fluidity.

Structuring Peculiarities of Different Forms of Oligomers and Polymers 213

Here it should be noted that a sharp viscosity increase starts only at a high conversion degree although intensive ammonia release is detected at much smaller values of the parameter p. This deviation of the intensity maximum in the course of the reaction detected by chemical methods or from viscosity changes is typical of polycondensation processes (see Chapter 2.4) and is connected with the fact that viscosity is far more sensitive to the same relative change of molecular mass of high-molecular products rather than oligomers (e. g., in the same chemical fusion of two similar molecules oligomer viscosity changes approximately 2-fold, that of a high-molecular linear product approximately 23.5 = 10-fold, and in three dimensional condensation leading to formation of a network of cross links viscosity may grow practically beyond limits).

Using formula (3.8.3) it is essential (which was confirmed by direct measurements [4]) that viscosity dependence on the degree of conversion p is unambiguous, i. e. independent of temperature - time prehistory of reaching the value of p at which the material viscosity is determined.

A comparison of Fig. 3.8.5 and the nature of the dependence q ( t ) following from equation (3.8.1) clearly shows that the presence of a certain apparent induction period on the dependence q ( t ) does not reflect the true pattern of the kinetic curve having no period of the kind.

The problem of presence or absence of an induction period in a reaction of PFR curing was repeatedly discussed in literature. Nevertheless, this problem seems to have a formal nature and its solution depends on the choice of a parameter for studying the process kinetics, and on the acquired method of processing experimental data. If we base our research on kinetic equation (3.8.1), it should be assumed that no induction period exists in the reaction of PFR curing.

Thus, the description of time dependence of viscosity in the course of PFR curing may be based on the same rheokinetic principles which were the reason for considering regular changes of viscosity in various paths of polymer formation and conversion, namely, using equation (3.8.1) to determine the degree of conversion in time and the rheological equation (3.8.2) describing the viscosity dependence on the degree of conversion.

As an alternative method to evaluate kinetics of viscosity growth in the course of PFR-curing, the results of direct measurements of the dependence q ( t ) are used sometimes in a certain form of its analytic description. Thus, the increase of tangent stresses ‘t (as a function of viscosity) with time t until they reach a plateau prior to PFR curing may be approximated by an empirical expression of the type [5]:

1 - K ( f - to)

‘t = T , , , ~ ~ [ 1 - e (3.8.4)

where K , and to are the constants, and K is being understood as a measure of rate of viscosity growth enabling us to evaluate apparent activation energy of the process Eef. Direct measurements show that the value of Eef depends strongly on the type of material employed for testing varying from 27 to 76 kJ/mol. However, it should be noted that the given value is of equivalent nature and has no immediate physical sense unlike values of activation energy in chemical processes of structuring and viscous flow used above in equation (3.8.1) and (3.8.2) in considering a rheokinetic model of PFR curing.


Plastometric tests used to evaluate kinetics of PFR curing involve deformations applied to a reactive material. However (as is seen from quantitative estimates discussed in detail in Section 3.7), shear deformation within a sensible rate range is not likely to be a kinetic factor, i. e. shear itself does not change the process kinetics. Its role is confined only to heat release accompanying the flow of a high-viscous liquid. At the same time, application of high-frequency vibrations to PFR being cured may have a definite kinetic effect. Thus, intensive pretreatment of a PFR oligomer at 120 "C (prior to curing initiation) results in acceleration of the structuring process and enhancing physical-mechanical properties of phenoplasts [6], as is the case with epoxy resins (see Section 3.5.2). This effect was explained by enhanced structural regularity of the initial oligomer [7].

The dynamic method as a tool for non-destructive control was rarely used in kinetic studies of PFR curing in spite of its rare prospects. Nevertheless, the potential of this method was vividly demonstrated, for instance, in the kinetic studies of changes in the dynamic modulus G' in the course of curing a PFR by hexamethylenetetramine at high temperatures [S]. This yielded dependencies of G'( t ) which are characterized by an initial (however very short) section where the modulus does not change appreciably. This section corresponds to the initial viscous-flow state, followed by a section of a sharp modulus growth by several decimal orders and by a final plateau where the modulus reaches values of the order of 10' Pa practically independent of temperature at which curing is effected. The whole process takes about half an hour. In fact, measurements of the dynamic modulus appeared to give very useful and visual information about the course of PFR curing.

An original version of the rheological method of studies of curing phenol resins is the use of sample hardness as a measured parameter 191. Hardness is a complex characteristic of a material's mechanical behaviour reflecting indirectly its viscoelastic properties. Therefore the nature of change in hardness may be indicative of curing kinetics. In fact, in the cited works it was shown that there is a correlation between certain definable chemical parameters of curing depth (namely, the fraction extracted by acetone) and hardness after Rockwell. Although this method provides no data or absolute values of kinetic constants, it is very convenient and demonstrative as a comparative means of evaluating the course of curing and allows, particularly, to determine the optimal curing time, t o p t . The value of topt is obtained from the curve of the hardness dependence H on time H(t) as shown in Fig. 3.8.6 for a series of curves determined at various temperatures:

It is essential that the value of topt thus obtained agrees with the optimal curing time determined from strength properties of the material being cured. Although this method has certain technological advantages (e.g., simplicity of measurements), it is not deprived of some inherent drawbacks. Thus, the obtained results are very sensitive to surface condition (although authors of original works where the method is applied pointed out that this disadvantage may be overcome by replacing a sphere penetrator with a conical indenter). Besides, the method being indirect, one cannot establish a sufficiently simple and evident correlation between the results of measurements which it yields and real kinetics of material curing.


H I

Time, t Figure 3.8.6: Change of hardness in PFR samples with time at various temperatures.

The arrow indicates direction of temperature rise. The shaded region corresponds to optimal duration of structuring.

Kinetics of viscosity change in the course of curing PFR and related oligomers (up to the gel point) is often described by empirical equations similar to those used for other polymer systems with variable viscosity. Exponential functions are particularly convenient in this respect. Usually q(t) dependencies of double logarithmic coordinates can be represented by two sections of straight curves with different slopes. Thus, in a typical case of studying structuring of a phurphurolacetone monomer used industrially as a binder for polymer - concrete mixtures, it was shown [lo] that the dependence q(t) may be represented empirically as:

k , t P n ’ rn, = l - 2 a t t < t o

k2tm2 m2 = 4 - 6 a t t > t O q-qo =

where qo is the initial value of viscosity, ki and mi are the empirical constants. The position of the characteristic point depends on catalyst concentration (benzenesulfuric acid was used as catalyst) and temperature. When evaluating technological properties of a material, special concern should be given to the moment to up to which composition viscosity grows relatively slowly. Therefore the time to may be identified with the material’s “life span”. As was shown in the cited work, to linearly decreases with increasing catalyst concentration and temperature. These qualitative observations and applied empirical methods of describing the observed regularities are typical of various cured compositions with oligomer binders where at t < to to the function q(t) may even be weaker that a linear one, i. e. the exponent m may be smaller than 1. In this case, particularly distinct is the presence of the induction period preceding intensive composition curing with viscosity retained practically constant or weakly changing.


3.8.2 Production of polyurethanes

Synthesis of polyurethanes (PU) consists of the formation of urethane bonds -NHCOO- during the process of polymerization (or polycondensation). PU may be produced by different reactions in the form of monoliths or foam plastics. A typical interaction is that of reactive -OH and -NCOgroups; since these groups are found in compounds of most various types this enables a broad variety of PU having different structure and properties to be produced. If the initial compounds include two reactive groups each, this leads to production of linear PU. If their number is more than two, three-dimensional structures are formed including cross-linked polymers.

Thus, PU synthesis may occur in such a way that in the production of linear products a material retains its viscous-flow state until high degrees of conversion are reached or, when network polymers are produced, the process can end in the conversion of the material either into a highly elastic or into a glassy state. A variety of options predetermines the nature of changes in the rheological properties of a polyurethane-forming material depending on peculiarities of the chemical structure of initial products.

Additionally, the use of rheological methods makes it possible to monitor variations in the technological behaviour of compositions depending on fine features of their content and structure.

5 T = 353K

4 I

Q x

1;

0 2 4 6

Figure 3.8.7: Growth of molecular mass of linear PU with time when oligomer polybutadiene (a, = 5900) with hydroxyl end groups ( I ) and/or HMDI ( 2 ) or TDI (3) are used as initial components.

The formation of PU is a typical bimolecular reaction which is reflected in a linear growth of the mean molecular mass with time M,( t ) . This is shown in Fig. 3.8.7 demonstrating (from experimental data of [ 1 11) the nature of time dependence M, for PU synthesized from low molecular polybutadiene with hydroxyl end groups and either 1,6-hexamethylenediisocyanate

Structuring Peculiarities of Digerent Forms of Oligomers and Polymers 217

(HMDI) or 2,4-toluene diisocyanate (TDI). In both cases a linear PU is produced. The constants of reaction rate for 353 K are 1.2 and 5.6 cm3/rno1.s for HMDI and TDI respectively [ 1 11. It is also interesting to note that the linear nature of the time dependence of a, is retained when the trifunctional I ,2,3-propanetriol is introduced into the reaction mixture in various quantities. This gives rise to branches the concentration of which depends on the amount of trio1 introduced, but in all known instances the reaction remained biomolecular until a rather high degree of conversion was reached, corresponding to 87.5% conversion. It should also be noted that the constant of polymerization rate depends considerably on the nature of reacting substances, i. e. the nature of the remaining part of the molecule bound with reactive hydroxyl- and isocyanate- groups is relevant for the reaction of PU formation.

The latter circumstance also affects the nature of the dependence of viscosity of a reaction mixture on the molecular mass of a forming PU. In fact, peculiarities of viscosity change in the course of PU synthesis largely depend on the structure of initial products, particularly on whether PU with linear or branched structures are produced. If we refer again to the work published in [ 1 11 we see in Fig. 3.8.8 that during the interaction of the oligomer polybutadiene with hydroxyl end groups and bifunctional isocyanate - if the synthesis results in a linear polymer - the dependence of its Newtonian viscosity q on a, appears to be the same for PU produced both with HMDI and TDI. This dependence can be described sufficiently by an “universal” exponential law with the exponent 3.5:

c

Figure 3.8.8: Viscosity dependence of forming PU on their molecular mass in the synthesis of linear PU based on HMDI ( I ) or TDI (2), and branched PU based on HMDI (3) or TDI (4), and oligobutadiene with hydroxyl end groups (5).

If propanetriol was introduced additionally into a reaction mass causing formation of branched products, the situation altered sharply. This is also shown in Fig. 3.8.8, vividly demonstrating not only the difference in the dependence I@,) for PU with various compo-


sitions, but also a distinct non-linear nature logy dependence on log M, with accelerating rate

of viscosity growth as the molecular mass increases. A very sharp viscosity growth leads eventually to fluidity loss (“gelation”) due to the formation of a three-dimensional network of chemical bonds. It is also typical that, with growing molecular mass, the viscosity of a branched PU starts to exceed considerably that of the corresponding linear products. This effect is especially evident for TDI-based PU having a more rigid chain compared with PU in which a relatively “soft” hexamethylene unit (in HMDI) is included into its macromolecule.

Absence of the effect of the chemical structure of diisocyanate on viscosity of linear PU and differences in the viscous properties of branched PU based on HMDI and TDI can be qualitatively explained in the following way. In linear TDI-based PU, rigid TDI segments show no specific effects as the whole chain appears to be quite long and consequently has a sufficiently high kinetic flexibility. Therefore the dependence q@,) is common for both types of PU and similar to any polymer with flexible chains. Besides, in strongly branched PU produced with triol TDI chains with relatively short chain segments are found to have no kinetic flexibility, and here their rigidity becomes a significant factor.

On the whole, the discussed experimental data illustrate quite evidently the role of both the chemical structure of initial components and the three-dimensional structure of PU macro-

molecules. Accordingly the rate of viscosity growth is determined both by the kinetics of the polyaddition reaction, and by the structure of the polymer formed. Therefore, e. g. from Figs. 3.8.7 and 3.8.8, it is clear that the dependencies y( t ) should differ for linear PU based on HMDI and TDI. This was confirmed in the original work by comparison of time dependencies of the loss modulus G” (G” is directly proportional to viscosity if measured at a constant frequency), and according to Fig. 3.8.7 the loss modulus appeared to grow more rapidly for HMDI-based PU compared with PU produced by condensation of oligodiole with TDI. Measurements of the loss modulus and viscosity are equivalent up to high degrees of conversion. But after a continuous three-dimensional network is formed viscosity measurements are no longer possible, and the values of the rubbery modulus reach a constant level determined by the density of a network of chemical and effective physical bonds (engagements). Further measurements of PU rheological properties become inefficient in this case since no relevant changes in the components of the dynamic modulus occur.

Non-linearity of logy dependence on log M, for branched PU shown in Fig. 3.8.8 can be connected with the choice of methods to evaluate M, of branched PU. Thus, for triol-based PU (produced from E-caprolactone) and HMDI-based PU the dependence (M, ) appeared to be of exponential nature for viscosity change by 4 decimal orders (Fig. 3.8.9) if a, is calculated from the chemical scheme of triol reaction with HMDI using the observed degree of conversion [ 121. An unambiguous relation between M, and p can always be established for a polycondensation product from molecular masses and concentrations of initial components. This shows that measuring the dependence P ( t ) (where p may be found for any reactive groups) gives an immediate estimate of M, change with time.

Structuring Peculiarities of DifSerent Forms of Oligomers and Polymers 219

Figure 3.8.9: Viscosity dependence on the molecular mass of branched PU based on E-caprolactone and HMDI (from [I21 ).

It is typical, however, that the exponent in the dependence q on M, first decreases from about 2.6 to 2.1 as the temperature rises from 45 to 65 “C and second is far below the common “universal” value of 3.5, which was kept in the course of linear PU formation (see Fig. 3.8.8). In general we can hardly speak of any “universal” value of this index in the course of branched PU formation. Thus, when HMDI was replaced by 4,4-diphenylmethanediisocyanate (DPMDI) in case of oligomer products the index went up sharply reaching the values of 3.4 to 6.7 and for M, > 1 . 5 ~ 1 0 ~ it became constant and practically equal to the universal value [13].

This discrepancy in the course of experimental dependencies (a,) for different PU-forming systems shows that, keeping a general approach to plotting rheokinetic models for the process of PU formation based on the analysis of reaction kinetics and on the dependence q(aw), it is not only necessary to use particular kinetic equations but also a particular dependence q(@,) for each type of system. Then, as for any other system, viscosity change with time q(r) in an isothermal process can be represented as:

(3.8.5)

where A is the constant, U the activation energy of viscous flow, and the dependence M,(t) must be derived from a kinetic equation relating p (and accordingly a,) to time.


P

I

200 400

Figure 3.8.10: Change of viscosity (a ) and degree of conversion (b ) in the formation of net PU based on E-caprolactonetriene and HMDI (from [12]) .

The nature of the isothermal change of the viscosity in the reaction mass with time during the formation of branched PU is typical of processes, where condensation leads to the formation of cross-linked products. Fig. 3.8.10 clearly shows the main rheokinetic peculiarities of this process: Inversion of q(t) dependencies with growing temperature (caused by the fact that with rising temperature the viscosity of initial products decreases, while chemical reactions are accelerated resulting in intensive viscosity growth), and a sharp viscosity growth up to q + 00 as the gel-point is approached [ 121. It is typical here that the gel-point is related to the degree of conversion calculated from concentrations of reactive groups, approaching p = 0.7 1 which agrees with the theoretical value of p when a continuous network of chemical bonds is formed and the system loses fluidity (to be exact, the gel-point in the cited work was assumed as the viscosity level of lo5 Pas with the corresponding p = 0.707, see Fig. 3.8.10).

The nature of dependencies q(t) and p(t) shown in Fig. 3.8.10 is of general relevance for PU formation. This is illustrated by Fig. 3.8.11 presenting data for the reaction of PU formation from polyisocyanate and polyene for a wide temperature range (after [ 141):

Structuring Peculiarities of Different Forms of Oligomers and Polymers 22 1

0.8

Figure 3.8.11: Change of viscosity (a) and conversion degree (b) in PU formation at 23 C" ( I ) , 30 C" (2), 50 c" (3), 60 C" (4), 70 C" (5), 80 "C (6) (from [14]).

This, too, clearly shows the main peculiarities of the course of compared dependencies: A monotonic nature of temperature effect on relative positions of q(t) dependencies and intersection of P(t) curves, attained at different temperatures. The reason is that at small times temperature is the factor decreasing viscosity, whereas at long reaction times the predominating factor is that the temperature increase results in accelerating polymerization and medium viscosity growth. It is essential that here, too, it appeared possible to represent viscosity as an unambiguous function of the degree of conversion evaluated from concentration of isocyanate groups:

where the constant a does not depend on P and can be measured for p = 0 (if we take into account that a can be represented as E B , it is possible to calculate the activation energy of the viscous flow of PU which appeared to be 41 kJ/mol). The function f(p) can also be represented by the empirical correlation:

where K = 35 and the exponent m is expressed as follows:

1 . 7 2 ~ 1 0 - ~ T - 3.81 at T < 338 K 2 at T > 338 K

m = {


If we know the dependence p ( t ) and it is determined by an equation of chemical kinetics of PU formation, which - as usual for polycondensation processes - is second order with respect to p, then the given correlations or any similar dependency provide a complete solution of the problem concerning the course of viscosity growth in the process under consideration.

i - 1 ,

I 2 3 4 5

f, min

Figure 3.8.12: Dependence of the mean numerical degree of polymerization on the duration of PU formation at 60 c" ( I ) , 70 c" (2), 80 C" (3), 90 C" (4).

A difference in the nature of viscosity change during formation of linear and branched (and eventually cross-linked) products in the course of PU synthesis has been mentioned above. Ty- pical in this respect are the results of rheokinetic studies of polycondensation of polyfunctional macro-isocyanate and diamine, which nevertheless lead to formation of a network product. This apparently takes place due to intensive intermolecular interactions caused by the formation of a large number of hydrogen bonds [ 151. Kinetic studies of this process by the time dependence of the polycondensation degree N (see Fig. 3.8.12) show that it consists of two stages characterized by different values of the kinetic constant. Consideration of viscosity dependence on condensation degree (Fig. 3.8.13) shows that it also consists of two sections with different slopes on double logarithmic coordinates. The very fact of slope change is quite trivial and as always reflects a transition from oligomer (low molecular) compounds to polymer products. However, the value of the slope tangent for curves in the region of high degrees of conversion is not quite usual 4.6 which is not typical of linear products but rather common for branched chains. It is interesting to note that in the case of using a trifunctional curing agent, i. e. when the formation of branched macromolecules was predetermined [ 121, the slope of logq dependence on log t , as in our case, was 4.6. As was mentioned in Chapter 2.4, the similarities of slopes of logq dependencies on log N and log t is typical of polycondensation processes.

Structuring Peculiarities of DSfSerent Forms of Oligomers and Polymers 223

Figure 3.8.13: Viscosity dependence on polymerization degree in PU formation based on macrodiisocyanate and 3,3'-dichlor-4,4'-diaminodiphenyEmethane at

60 C" (I), 70 C" (2), 80 c" (3) and 90 "C (4).

The situation under discussion is rather typical of PU where not only chemical but also weaker intermolecular interaction should always be taken into account. In this respect, PU are characterized by emergence of associates caused by strong intermolecular interactions, being largely determined by the presence of urethane groups. The rheological method provides a highly sensitive control of these interactions as they result in an apparent (effective) increase of the molecular mass of the units revealed by rheological measurements. Thus in a model experiment comparing viscosities of samples obtained on the basis of block copolymer of butadiene with isoprene having hydroxyl end groups and TDI at varied ratios of oligomer diisocyanates (i. e. the ratios [NCO]/[OH] from 2.0 to 1.0) [16] a rather peculiar viscosity dependence on MW at 11 "C and 25 "C was observed: the dependence curve acquired a minimum (see Fig. 3.8.14).


h

Figure 3.8.14: Viscosity dependence of linear PU on their molecular mass at I1 c“ ( I ) , 26 c“ (2), 50 c“ (3), 80 “C (4) (from [Is]).

This is caused by the predominant effect of association since a low-molecular sample was correlated with the maximum value of the ratio [NCO]/[OH] which means that it is associated to the greatest extent. For this reason its “apparent” molecular mass is considerably elevated compared with the actual one and viscosity appears to exceed its theoretical value. With a temperature increase, the stronger heat movement destroys intermolecular associations and the dependence q(M,) acquires its common linear form (on double logarithmic coordinates).

Most of all PU viscosity is influenced by interaction of reactive (functional) end groups. Thus, in polyesterurethane oligomers, for instance, viscosity increases 100 times when hydroxyl end groups are replaced by NCO-groups, and 10 times more when carboxyl groups are introduced as end groups [ 141.

A relatively new technological trend in PU synthesis requiring rheokinetic analysis is PU production from oligomers based on macroglycoles of various structure forming “soft” blocks in the structure of the polymer chain. An example of these oligomers is polytetramethylene- adipat glycole forming PU in the reaction with diisocyanate (e. g., 4,4 diphenylmethanediisocyanate) and 1 ,Cbutandiol [ 181. Although polycondensation of these components yields linear chains, generally the product is a structured (“quasi-crosslinked”) network due to the forming of aggregates of “rigid” chain segments stabilized by hydrogen bonds of urethane groups and n-electrons of aromatic nuclei, etc. which, as was mentioned earlier, is generally typical of PU and their intensive intermolecular interaction. In [ 181 viscometric studies of structuring rheokinetics were reported for this system (in a melt, without introducing a solvent) using the Brabender’s technological device “Plasticorder” with the time-dependent change of the torque M(t) as the measured parameter. This investigation revealed with high sensitivity the course of the M ( t ) dependence and the limiting value of the torque moment versus the composition content and, particularly, concentration ratio of reactive groups [NCO]/[OH], macroglycole acidity, introduction of a stabilizer or a catalyst into a reactive mass, temperature effect, etc. The


accompanying comparison of rheological and calorimetric kinetic curves showed a far higher sensitivity of the viscometric method in terms of peculiar features of the composition content. Therefore (although the referred work contained no proper quantitative kinetic analysis of the obtained data) it should be assumed that the use of rheological methods is a very efficient means of controlling composition quality for PU production, their comparative analysis and predicting a technological process.

Since the formation of cross linked (network) PU always follows the formation of branched prepolymers, the problem of correspondence between changes of rheological properties and expanded branching is of importance. Meanwhile, this problem is very complex due to difficulties and ambiguity in evaluating the extent and nature of branching in the initial chain, the chain-lengths of side branches, their distribution along the chain, the formation of secondary, tertiary, etc. chains. Therefore it seems that the only possible way to evaluate branching at present remain indirect methods. In this respect it is of interest to analyse data obtained from parallel studies of molecular characteristics and viscous properties of oligomer polyether- urethanes resulting from interaction of excessive amount of diisocyanate and polyoxypropylene glycole [ 191.

The formation of branched products during the process was controlled by the change in the concentration of isocyanate groups A[NCO]. As shown in Fig. 3.8.15, there exists a single correlation between A[NCO] and viscosity, on the one hand, and between A[NCO] and the molecular mass of the polymer, on the other hand, both not depending on the synthesis temperature. This means that changes in the molecular mass are fully determined by a uniform process (at different temperatures) of consuming isocyanate groups which, in its turn, determines changes in product viscosity.

I 1 t I I

Figure 3.8.15: Change of molecular mass M and viscosity of polyetherurethane prepolymer in the course of isocyanate group consumption at 80 “C ( I ) , 100 “C (2) , 120 “C (3) Cfrom [19J).


Closer consideration of Fig. 3.8.15 also reveals a change in the nature of the dependencies M(A[NCO]) and q(A[NCO]) at some critical point marked by a dashed line. Apparently, presence of this point can be linked with the transition from branched single chains to microgel formation, i. e. with integration of separate molecules branching in the course of synthesis into “double”, “triples”, etc. macromolecules and, eventually, resulting in the formation of a three- dimensional network. This interpretation of the nature of the bend in the dashed line region in Fig. 3.8.15 agrees with the changing nature of the effective viscosity dependence on shear rate.

Thus, up to this critical point the prepolymer behaves like a Newtonian liquid whereas on transcending the point viscosity becomes clearly anomalous which is likely to be connected with the emergence of a certain “structure” in the material, possibly of a relaxation nature.

Like in other cases, the process of PU formation is greatly influenced by heat release accompanying the reaction of curing. Therefore it is of general interest to answer the question of possible transition from the results of isothermal rheokinetic measurements to predicting viscosity changes in the reaction mass with time at arbitrary non-isothermal conditions when the heat-up of the reaction mass and heat removal into the environment take place or, in particular, in the course of adiabatic curing. The latter is typical of the formation of bulky items when the process of PU formation is accomplished so quickly that heat removal is practically irrelevant. This transition is possible from the general equation (3.8.5) if we know the constituent constants and the type of the dependence M , ( t ) for a particular situation.

The temperature influences a number of parameters which are clearly or not clearly included into the equation:

First, the temperature influences the first exponential co-factor in this equation.

Second, the temperature is included into the kinetic equation thus determining the course of M, change with time.

It is essential that - as was shown in Section 3.7 for adiabatic curing - a simple linear relation is observed between the temperature gain T and the degree of conversion p so that T = T(P). Therefore the kinetic equation:

‘ f P = p7 T(P)

contains only one variable P. This allows the solution of the equation (precisely or approximated by any numerical method) to be completed, thus obtaining a clear dependence P(t) and, consequently, the dependence M , ( t ) for any known composition of the reaction mass. The latter enables us to calculate viscosity change with time using equation (3.8.5) where both T(t) and M , ( t ) are known.

Quantitative calculations reported in [ 131 for the system triole-DPMDI fully confirmed applicability of this method to the real process of adiabatic PU formation.

The cited works [ 12, 131 also reported a sharp increase in normal stresses in the course of PU formation, especially as the gel-point was approached. However, it appeared to be rather difficult to interpret these results as the integral effect is constituted of components connected not only with the shear flow but also with volume changes in the course of polymer formation. At any rate, it was noted that normal stresses connected with volume changes do not relax on


completion of flow. Although the changes of normal stresses under the shear flow near the gel- point are very sharp and their measurements could be, in principle, a very sensitive tool for evaluating rheological transformations during the formation of network polymers, there is no evidence at present when the use of viscometric control, which is much simpler and more available than the measurement of normal stresses, would be insufficient for this purpose.

Various techniques can be applied to the rheokinetic control of processes of PU formation and curing. For example the use of the method of non-destructive acoustic control employing the measurement of time-dependent changes in oscillation intensity at 150 kHz applied to the sample [20] is rather convenient.

The signal is agitated by a generator and converted into mechanical oscillations. These go through the tested sample and are received by a meter which again converts the mechanical signal into an electric one. The intensity of the received signal serves to measure the extent of PU foaming and curing processes.

0 20 W t, min

Figure 3.8.16: Change of intensity of mechanical oscillations and temperature of the reaction mass in the course of PU foaming and curing (from [20]).

Four sections may be discerned of the diagram (Fig. 3.8.16 from [20]) showing a change of the signal intensity I with time. In section I the signal intensity decreases which is caused by growing viscosity of the reaction mass retaining its “life capacity”. The region of transition to the plateau section I1 corresponds to foaming of the mass accompanied by intensive heat release in the sample. This process constitutes section I1 on the diagram. In section I11 the density of formed PU is stabilized, heat release gradually stops and an increase of the signal intensity I occurs due to a continuing process of PU curing. The process is completed in section IV of the diagram. Thus, the use of the rheokinetic method controlling the formation of foam PU in its acoustic implementation ensures a complete monitoring of all process stages and is convenient for industrial application of the method.

Rheokinetic studies of PU formation and structuring, as well as other cured oligomers, were mainly carried out under static conditions. Meanwhile, it is interesting to investigate the role of deformation in the process of polymerization. This problem becomes vital due to development of technological methods of chemical moulding, i. e. obtaining end products directly from monomers (of low-viscous oligomers) in the course of a chemical reaction in the moulds. In this case, the process of filling the mould, as a rule, is aligned with the initial stages of the chemical reaction. Therefore deformation inevitably overlaps the course of the reaction. Even if possible

228 Chapter 3: Rheokinetics of Polymer Curing ____

heat release in the process of rapid flow is not taken into consideration, the answer to the question of whether the role of deformation is active or passive depends on specific peculiarities of a polymerizing system.

In relation to polyurethanes this problem was considered in [21] where besides oligourethane a second model object - oligobutylmethacrylate - was studied. Tests were performed with a composition based on oligodiethyleneglycol adipinate (with a molecular mass of 1.75 x103), toluene diisocyanate and a binding agent (a mixture of tri- and diethylamines), the

ratio of the three basic reagents in the composition being 1 : 2.05 : 0.66.

40 w no t , min

Figure 3.8.1 7: Kinetics of viscosity growth in the process of oligourethane structuring perjGormed at 20 "C with various shear rates [s- '1 0.038 ( I ) , 0.087 (2), 0.38 (3), 8.78 (4), 25.4 (5) (from [21]).

The initial composition typically displays the phenomenon of anti-thixotropy, i. e. a certain growth of effective viscosity q with the growing shear rate y in the region of relatively low values of 1' at high shear rates the phenomenon of viscosity anomaly common for polymers, i. e. a drop of effective viscosity with growing shear rate was observed. This original course of the dependence q(y) of the initial composition accounts for the positions of initial points on the y-axis in Fig. 3.8.17 showing typical results of studying q(t) dependencies observed at the initial stages of curing an oligourethane composition at a single temperature. Similar results were also obtained for other temperatures. Discussing the data given in Fig. 3.8.17 it should be taken into account that as polymerization proceeds the dependence q( j ) may change its form so that the points on the dependencies q( t ) at 1' = const. may correspond to different regions of flow curves. Yet, irrespective of that, the supplied figure clearly shows a very pronounced effect of deformation rate on the rate of viscosity growth. It is seen that as the shear rate grows the rate of viscosity growth reflecting the course of polymerization increases, this effect is intensified with temperature growth. In the original work [21] this effect was correlated with the effect of a more regular relative position of chains induced by the flow and change in chain conformation in the flow on the rate of a chemical reaction. Evidently, in this case deformation plays a rather active role in the kinetics of the structuring process.

The described case is one of very few known instances of the very strong effect deformation has on the rate of a chemical process in a homogeneous region, where a trivial reason for this effect - linking it with intensive heat release under shear - evidently should be excluded.


3.8.3 The Curing of Unsaturated Polyesters

The curing of unsaturated polyester resins (polyalkyleneglycol malcinates and fumerates) is widely used in modern technologies for production of potting compounds, lacquers, glues and high-filling compositions like premises and prepregs. In all cases the process of manufacturing products from reactive polyesters may be represented as a two-stage process where the boundary between the stages is determined by the nature of changes in the rheological properties of the material. At the first stage a forepolymer (oligomer) with double bonds is produced which is then cured by means of copolymerization with olefins by the radical mechanism or otherwise. The double bonds are opened enabling cross links and, consequently, formation of a three dimensional (network) structure. Accordingly rheokinetics of both stages of the process - forepolymer preparation and curing - should be considered.

Curing kinetics of unsaturated polyester resin employing the radical mechanism were repeatedly investigated with conventional methods like calorimetry and IR-spectroscopy [22-241. However, the data reported in these works are controversial even in treating the principle regularities of this reaction, namely, whether the process is diffusion-controlled or not and what the extent of reaction completion is when no more changes in material properties are observed. At the same time, this reaction was successfully described macrokinetically as a nonisothermal process obeying a first-order equation in the degree of conversion, initiated after completion of certain induction period t* [25]. This made it possible to provide a correct description of temperature distribution along the thickness of a sheat workpiece (prepreg) from a filled composition based on polyester resin and measurement of its change with time in the course of curing. It is also essential to note that the simple kinetic scheme yields correct results at t > t* but until the degree of conversion p = 0.4. At higher p-values the reaction apparently shifts to a diffusion controlled region (at 107-127 "C used for the experiments). This phenomenon (corresponding to the intersection with the vitrification line or at least an approximation to it) is accompanied by a sharp viscosity growth of a reaction mass and should always be taken into account in kinetic or rheokinetic studies of compositions subjected to curing.

Despite the wide industrial availability of reactive unsaturated polyester resins and compositions based on them as well as a gradual expansion of the manufacturing of articles from them by casting moulding, very few rheokinetic studies of prepolymerization and curing are known, where a systematic comparison of the process chemistry (and kinetics) with the change in rheological properties of the composition would be drawn. Undoubtedly there is a correlation between the course of the chemical process of change in the degree of conversion with time p ( t ) and changes in rheological properties of the material. Thus, it was shown [25] that the rubbery modulus starts to change on completion of the induction period, i. e. at t > t* it occurs in the same time interval as the growth of p and is completed when p + 1. Moreover, we may assume that the observed macrokinetic pattern of change in the rheokinetic properties of compositions based on polyester resins of different content is quite uniform. Therefore it will be illustrated by several characteristic examples. Besides, the general approach to characterizing the rheokinetic process of prepolymerization and curing of polyester resins is not fundamentally different from the methods used for this purpose with other oligomeric materials.


This is clearly confirmed by the data obtained from investigating prepolymerization of the monomer-oligomer system "Rolivsan MV- 1" which is a mixture of unsaturated ethers and oligoethers [26] . The viscosity of the initial product is as low as 1.9 Pa.s at 30 "C and drops down to 1 x I O - ~ Pa.s at 140 "C.

It was peculiar for the performed investigation of thermal prepolymerization of this monomer-oligomer mixture that the reaction was effected at an elevated temperature (140 "C) whereas the viscosity change was controlled by its measurements at 30 "C (the samples were cooled sharply, which caused interruption of the reaction and yielded stable products of prepolymerization). The time-dependent viscosity change measured in this way is shown in Fig. 3.8.18. The viscosity data referr practically to the region of non-Newtonian flow:

300

200

t , min

Figure 3.8.18: Time dependence of viscosity taken at 30 "C for the products of

prepolymerization of a monomer-oligomer polyester composition

pe$ormed at 140 "C.

Evidently, the course of the dependence q(t) is quite similar to that observed for many other cured oligomer compositions, i. e. there exists an apparent induction period of practically constant viscosity, the transition zone and a region of sharp viscosity growth. A certain elongation of the transition zone is connected with the above-mentioned peculiarity of the experimental method of monitoring the course of the process. If the measurements of the dependencies ~ ( t ) , or shear stress T dependencies on t (which is the same provided the shear rate in these experiments is constant and non-Newtonian effects are negligible) are performed directly at the same temperature as that used for prepolymerization, then the curves will be steeper although the general nature of these dependencies will certainly remain unchanged. This is clearly seen in Fig. 3.8.19 showing dependencies ~ ( t ) for various temperatures measured directly in the course of prepolymerization. Here, too, the induction period can be clearly separated and its temperature dependence enables the effective energy of activation to be derived for processes eventually resulting in a sharp viscosity growth. In this particular case the activation energy thus obtained appears to be 100 kJ/mol.


so

t, mil? Figure 3.8.19: Change of tangent stress with time during prepolymerization of a

monomer-oligomer polyester composition pegomzed at 140 "C ( I ) , 150 "C (2), 160 "C (3), 170 "C (4) and I80 "C (5).

The rheokinetic curves shown in Figs. 3.8.18 and 3.8.19 reflect the initial stage of prepolymerization. Further control of this process by viscometric methods is not feasible. However, like in other curing processes, starting from the moment of a notable increase in material viscosity, it becomes efficient to employ a dynamic observation method which, in this case, allows to monitor changes in the dynamic rubbery modulus from 10 to -- lo5 Pa, and the curing process proceeds well beyond this point if we assume it from the continuing change of the rubbery modulus. Besides the temperature dependence of the initial rate the modulus increase can be used to define the effective energy of activation in the process of gel formation, and it can be assumed that both the viscosity growth in the transition zone and the modulus increase reflect the same process of gel formation, it is quite understandable therefore that the activation energy calculated from the initial rate of the modulus increase appeared to be the same as that evaluated from the temperature dependence of the induction period.

In existing industrial environments, unsaturated polyester resins are used not in a pure form but as binders in high-filling compositions. This causes frequently problems during viscometric analysis of initial stages of curing. However, this relates to other high-filling cured compositions as well. To overcome these difficulties there was suggested a method of testing compositions with the polyester binder based on sample compression between flat parallel plates [27]. According to this method, disc-shaped samples are placed between two parallel plates. A pack- age thus prepared is fixed between the clamps of a testing apparatus, e. g., "instron", and a constant rate of movement is imparted to one of the clamps, i. e. one of the plates, in the vertical direction; at the same time the dependence of arising effort on time F(r) is being measured. Although this method is well-known in viscometry, its essential peculiarity when used for filled compositions is the choice of a rheological equation of state for a studied material and integrating an appropriate equation for particular conditions of deformation in the course of testing. Thus, it was shown [27] that an adequate description of rheological behaviour of the studied compositions can be achieved with the following equation linking the tangent stress T, deformation y and deformation rate v:

where zo is the fluidity limit of a high-filling composition, p, n, and K are material constants where p and n reflect peculiarities of viscous, and K of elastic properties of compositions.


By integrating this equation it is possible to calculate the time dependence of the compressing force F( t ) at a preset initial sample thickness H when it is placed between the plates, and time dependent changes in the distance between the plates h(t) which, in its turn, enables calculated results to be compared with experimental data.

The results of these measurements allow a relatively simple interpretation for low-temperature testing when no changes in the rheological properties of the binder and the whole composition occurs. Testing of that kind is convenient for comparative evaluation and control of materials in a technological process. But if the compression is accompanied by sample heating with its simultaneous curing (taking place under non-isothermal conditions), the quantitative interpretation of compression curves, i. e. F(t) dependencies, at various rates of plate movement is no longer possible. These data can only be employed for qualitative observations, comparative sample testing and finally for modelling a real industrial process of prepreg pressing. Therefore these data can hardly be used for physically unambiguous rheokinetic analysis of the curing process. The latter seems possible only by using the results of dynamic testing performed with specially prepared samples so as to avoid any undesirable and uncontrolled non-isothermal effects.

The process of prepolymerization shown in Fig. 3.8.18 is rather slow. Therefore in existing technological conditions this material may be processed into end products practically in the isothermal regime, and then its rheological characteristics, namely isothermal dependencies q(t) or T ( t ) , are sufficient for evaluation of material behaviour in the course of moulding. How- ever, it is very often the case, that the technological process is very rapid and proceeds in the non-isothermal regime. This brings about a question of possibility of a general approach to the analysis of non-isothermal processes on the basis of material characteristics obtained at various constant temperatures as it was discussed in Section 3.7.

A specific analysis estimating the possibility of considering a non-isothermal process of prepolymerization (and thickening) for unsaturated polyester resins was carried out for example with polyethylenepropyleneglycol maleinate phthalate thickened by magnesium oxide [28]. Isothermal curves of viscosity growth for this material at various temperatures are shown in Fig. 3.8.20. A peculiar feature of this process is that it occurs in a heterogeneous system.

t , min Figure 3.8.20: Kinetics of viscosity growth of unsaturated polyester resin thickened by

M g O at 40 "C (I), 50 "C (2 ) . 60 "C (3) and 70 "C (4).


The description of isothermal dependencies of viscosity q on temperature T and the degree of conversion may probably be provided using a formula of the type:

q = q ' e x p [ z + B P ] (3.8.7)

where E,, is the activation energy of viscous flow and B is the constant characterizing the

effect of the extent of a chemical reaction on the rate of viscosity growth.

According to the general approach to rheokinetics of a chemical process, the dependence (3.8.7) should be considered together with the kinetic law describing the change of q with time. For instance, we may assume that the increase of P(t) obeys the simplest kinetic law of a zero-order reaction, i. e. in isothermal conditions p = const., and in the general case:

Ep p = KO exp (--) RT

where KO is the pre-exponential factor, and E p is the effective energy of activation of resin

Then

thickening.

+ BKof exp (--) RT 1

or, in isothermal conditions:

( t ) = ?lo exp ( A + K t ) (3.8.8) q~ = const

where A and K are the constants.

Formulae of this type for the isothermal function q ( t ) were often suggested with respect to the task of describing viscosity growth during prepolymerization and curing of reactive oligomers. As it is seen from Fig. 3.8.20, it does efficiently describe experimental data which allow a satisfactory approximation by a curve of the coordinates log q - t .

However, the simplest rheokinetic analysis shows that a formula of the type (3.8.8) in fact represents no universal law, because it is a consequence of two assumptions: The exponential nature of the dependence q(p) and the zero order of the prepolymerization reaction. If the reaction were of second-order with respect to p, i. e. under isothermal conditions expressed as

6 = k , ( 1 - P I

it could be easily seen that here the dependence qT= const(t) would be completely altered, namely:

Evidently this function does not agree with the data in Fig. 3.8.2, at least unless only the initial region specified by the condition k,t B 1 is considered.


&!

c 4-

C *- 1

*t7 \ 0 Dcl. / --

Figure 3.8.21: Comparison of theoretical (dashed lines) and experimental (solid lines) dependencies q ( t ) in the thickening of unsaturated polyester resin by MgO in the non-isothermal regime with fast ( I ) and slow (2) temperature increase.

Calculations performed for the non-isothermal regime of prepolymerization (or thickening) of polyester resin by the general scheme revealed in Section 3.7 - allowing for the rheokinetic law expressed by formula (3.8.8) - made it possible to obtain the dependence q [ t , T(t)] for functions T(r) of any type [39]. The agreement of theoretical and experimental data for two different temperature regimes is shown in Fig. 3.8.21 which also includes T(t) dependencies approximated by two straight sections. It is seen that in both cases the dependence T(t) shows a minimum whose position depends on the rate of temperature increase. This corresponds to the general qualitative scheme shown in Fig. 3.8.22 according to which the decisive role is at first played by a viscosity drop due to rising oligomer temperature as reflected in the first item in the exponent of formula (3.8.8), and then a predominant role passes over to the process of thickening described by the second item in this formulae. The obtained data show that theory and experiment are not only in good qualitative, but also in satisfactory quantitative agreements.

Figure 3.8.22: Nature of viscosity change during non isothermal oligomer curing (the solid line). The dashed line indicates terms of formula (3.8.8): viscosity drop caused by temperature increase ( I ) and viscosity growth caused by the developing process of curing (2 ).


The described approach apparently is of general value for prepolymerization and curing of various unsaturated polyester resins, as well as other reactive products. In this case, like with polyesters and other cured oligomers, it is convenient to approximate the observed dependencies with appropriate analytic expressions. This facilitates further calculations and provides a convenient basis for mutual comparison of the materials. Viscometric studies of curing unsaturated polyester resins performed on this basis show that their viscosity increases exponentially with time so that the viscosity logarithm increases linearly with time. This is seen from Fig. 3.8.20 and is reported for other polyester resins of different composition in [30]. Using the exponent to describe the rate of viscosity growth vividly demonstrates the process intensity but does not allow an unambiguous determination of the gel point as the moment in the vicinity of which q + 00. Instead, the gel-point is assumed to be the moment of time when viscosity attains some conditional level, e. g. 32 Pa.s or 320 Pas [29]. This approach is very practical as it permits to compare curing rates of polyester resins with various compositions and, particularly, to study the effect of concentration of an initiating system (accelerator) and a catalyst on curing rate understood either as the rate of viscosity growth or as the duration of the process needed to reach a preset conditional viscosity level of the reaction mass.

Rheokinetics of curing unsaturated polyester resins was also studied by the dynamic method [30] which made it possible to determine typical peculiarities in the changes of the rubbery modulus G' and the loss modulus G" with time depending on the main determining factors. The characteristic results of the performed measurements are shown in Fig. 3.8.23 illustrating first the course of the dependencies G'(t) and G"(t ) and second, the role of initiator concentrations namely that of methylethylketoneperoxide. Similar dependencies were obtained at other temperatures and frequencies.

In general, the obtained results may be summed up as follows:

there is a distinct maximum on the dependence G"(t). Its position on the time axis to precedes the moment when high values of G' reach the plateau, and approximately corresponds to the middle of the region of measurable values of G', i. e. the maximum of G" does not indicate the beginning of the region of the glassy state;

the point to shifts towards lower values as the initiator concentration in a system rises;

the position of the point to is to some extent dependent on frequency: to is inversely proportional to the frequency at which the dynamic properties of a cured system are measured;

as temperature rises, to shifts towards lower values. The effective reactive activation energy of the structuring process calculated from the temperature dependence of t o , is 40 & 8 kJ/mol.

Its essential to compare the observed pattern of changes in the dynamic properties of a material with the chemical processes that occur in it. The time of gelation, i. e. the moment of fluidity loss due to formation of a three-dimensional network throughout the bulk of the material (according to the data provided by the producer) approximately corresponds to the point at which G' becomes measurable, i. e. the first points in Fig. 3.8.23, although it was the maximum of G" that was treated as the point of gelation in the referred work. It is typical that the maxi-


mum tan6 is attained earlier than that of G“ and apparently it would be more correct to evaluate the moment of gelation, namely, from the maximum tans. At the same time, data presented in Fig. 3.8.23 (as well as other similar data given in the work) provide to grounds for evaluation of the point of vitrification.

4 e a

;f. L_ I f i4 2 3

I

P L

a

Figure 3.8.23: Time dependencies of components of the dynamic modulus in the course of curing polyester resin in the presence of I % (solid lines) or 2% (dashed lines) methylethylquitone peroxide at 25 “C and 110 Hz.

It has been mentioned above that, as a general rule, not polyester resins themselves but their compositions with various fillers are of practical interest. Presence of the second component in a reaction mass can have a dramatic effect on material “lifetime” and curing rates. Until now systematic studies of this problem are still missing. Only some results of occasional qualitative observations were reported. Thus, it was found for maleinateacrylate phosphorus containing polyester [ 3 1 ] that the lifetime of its compositions with glass or phenolformaldehyde spheres considerably depends on the nature of the filler: for the two mentioned filler types it varied with the ratio 3 : 5. This effect was linked with the inhibiting impact of OH-groups of the phenol filler interacting with active functional groups of polyester. Similar effects are likely to be expected whenever a non-inert filler is used. Whether a given filler is “inert” or not depends not only on its nature but also on peculiarities of the composition being cured which makes the division of fillers into “inert” and “active” arbitrary. Therefore the boundary between them is determined by fine peculiarities of formulations used.


3.8.4 Silico-organic Polymers

The application of silico-organic oligomers with different functionality is very convenient in the studies of the effect of branching on the nature of changes in the rheological properties of a material during polycondensation. In terms of the reaction model, the situation here is very close to that described for polyurethanes (Section 3.8.3), where the functionality of reagents is also easily variable. Accordingly changes are brought into the course of viscosity growth as well as the form of the correlation between viscosity and molecular mass of the product obtained at different stages of the process. The analysis of curing silico-organic resins (oligomers) performed with interaction of bifunctional polydimethylsiloxane with vinyl end groups and bi-, tri- and tetra-functional silane (systems of the type A2 + B3 + B2 orA4 + B2) revealed the role of the functionality of component A (monomer) [32]. For the studied systems it is typical that they are not involved in side reactions and do not form rings, i. e. are convenient as model compounds for the analysis of growth and branching of chains leading to the formation of net products if in A, n = 3 or 4. For these products the conversion degree p* corresponding to reaching the gelpoint is: p* = 0.7 for the system A, + B2 and p* = 0.58 for the system A4 + B2.

The experiment showed perfect agreement with this theoretical evaluation of p* which confirms the possibility of applying these products as ideal models of structuring - eventually leading to formation of a network of chemical bonds - since the use of the bifunctional component with removed end groups practically eliminates the formation of closed rings.

Of principal interest is the comparison of viscosity dependencies of the products formed in the course of curing and their molecular mass. If we compare viscosity values qo for samples with the same molecular mass, the increasing functionality of monomer A,, will naturally result in a drop of viscosity since at the same average value of the molecular mass n the size of the macromolecular coil decreases with the increase of n in A,. However, if @ we replace n by the parameter ML = g M where g is the radius of inertia taking into account the effect of branching on the size of the macromolecular coil, it appears that products of different structure obey a universal common qo dependence on ji?, . This dependence has two regions, and when sufficiently high values of ML are reached the dependence q0(@,) appears to be exponential with an universal value of the exponent equal to 3.4 if qo changes by more than factor 300.

Thus, results of the mentioned research - as a means to plot the universal dependence qo(ML) with the generalized parameter ML and the universal value of the exponent in the dependence qO(ML) - are common for most diverse polymers, e. g. linear and branched poly- ethylenes. It is convenient to use this correlation in rheokinetic studies and the time dependence of @, can be determined from the data of kinetic measurements. Then the use of the universal function for yo@,) makes it possible to find viscosity values of a cured system up to the gel- point. In addition, the slope of the curves on the coordinates log qo - log @ (i. e. if @ is not normalized by the values of the g-factor) for systems of the type A3 + B2 and A4 + B2 appeared to be smaller in the high-molecular region than the universal value 3.4. Thus, for the system A3 + B2 it is 2.6, and for the system A4 + B2. This is a rather unusual result which is evidently seen from comparison of the obtained data with the discussed experimental results for polyfunctional polyurethanes. However, the very fact of deviation of the dependence qo(M) from universal specified for polyfunctional monomers (oligomers) is sufficiently demonstrative for the data of rheological measurements to reveal principal peculiarities of particular reactions resulting in the formation of either linear or branched chains or eventually a continuous three-dimensional network of chemical bonds leading to the loss of fluidity.


Characteristic results forming a picture of the process mechanism are obtained in rheokinetic studies of curing phenylmethylsiloxane oligomers [33].

The main experimental data in the form of dependencies of the rubbery modulus G', the mechanical loss angle tangent tan6 and the viscosity q of a cured oligomer on time are shown in Figs. 3.8.24 and 3.8.25. It is seen that the temperature increase results in a regular acceleration of the curing reaction and a more rapid approach to the point of gel-formation ("gel-point").

t, #?in Figure 3.8.24: Change of the rubbery modulus G' in the process of curing

polyphenylrnethylsiloxane oligomer at various temperatures. T = 170 "C ( I ) , 190 "C (2), 230 "C (3) and 250 "C (4).

t, m;R

Figure 3.8.25: Change of mechanical loss angle tangent tan6 (1-4) at T = I70 "C ( I , I f ) , 190 "C (2, 2'), 230 "C (3) and 250 "C (4).


The position of the point of gel-formation in a general case is determined by the kinetics of the curing process. The temperature dependence of the time needed to reach the point of gel formation t* (or inverse value l/ t*), plotted on the coordinates In t* - 1/T can be used to calculate the effective (apparent) energy of activation, i. e. to obtain the kinetic characteristics of the process important in terms or technological applications since it characterizes sensitivity of a system to temperature changes. The majority of authors [34 - 381 studying rheology of cured compositions did not go beyond that (alongside with gaining experimental data) in their rheokinetic analysis.

Considering dependencies of the rheological degree of conversion p on time provides a quantitative evaluation of the kinetics of curing processes occurring above the vitrification temperature of cured composition. The task is to select and to justify the form of the kinetic equation describing the observed dependencies P-t and then to solve the inverse problem, i. e. to determine the constants of the phenomenological model from experimental data.

Since the considered curing process occurs in the course of interactions between hydroxyl groups, it may probably be described by a second-order kinetic equation. As was shown by spectroscopic measurements [39], in a certain range of curing degrees this reaction characterized by a decrease of hydroxyl groups can be described in this way. However, in the given system IR-spectroscopic measurements should be interpreted with care as, first, we find it rather difficult to reliably identify hydroxyl groups with continuous water release in the process of the reaction and, second, a decrease of hydroxyl groups may be connected with elongation of oligomer chains rather than with the structuring process.

Nevertheless, first of all we verified the assumption of a possibility to describe the considered process by a kinetic second-order equation. The dependencies given in Fig. 3.8.26 show that experimental data are not described by linear dependencies in the coordinates P/ ( 1 - P) following from the second-order equation.

Figure 3.8.26: Dependencies of p ( 1 - P) (a ) and In [ P/ ( 1 - P) ] (b) on the duration of

the curing process. Signs are similar with Fig.3.8.24.


The processing of experimental data performed according to the autocatalytic equation does not yield any positive result either (Fig. 3.8.26), i. e. the points on the coordinates In [ p/ ( 1 - p) ] - t do not comply with the curves. This is a typical deviation in the behaviour of polysiloxane resins from what is known for a number of other oligomers.

Based on the assumption that the curing of phenylmethylsiloxans oligorners obeys the kinetic second-order equation with self-acceleration in [34], the following integral expression was obtained:

c l + c P p I + c 1 - p 1 - p

~ In- +- = ( 1 + c ) k t (3.8.9)

which, provided that c H 1 and cp >) 1 , is transformed to the formula:

(3.8.10)

vs t . The inter- Formula (3.8.10) is a curve equation on the coordinates In __ + ~

section of this curve with the y-axis gives the value of In c and the slope tangent the product ( 1 + c) k . Thus, in this case it is possible to determine numerical values of the constants k and c.

P P 1 - p 1 - p

200 COO t , min

on the duration of the curing process. Figure 3.8.27: Dependence of In __ + __ P P

I - p I - p Symbols see Fig. 3.8.24.

Fig. 3.8.27 shows anamorphosis of experimental data according to formula (3.8.10). The results given in this figure show that the experimental data are really straightened on the given coordinates which confirms the validity of the selected kinetic equation. In this case, the function P-t presents an S-shaped curve with the bend point corresponding to the condition

c/'P = c dt2

at 30% conversion.


Tin "C

kx104, min-'

170 190 230 250

1.7 6.8 25 36

C - 6 2 4 Y 6 - 8

u 2,o 2,2

=I'I 1-4 C

j ; -6

f03 / r K- '

Figure 3.8.28: Dependencies of the constant of reaction rate k ( I ) and inverse time of gel formation I/t* ( 2 ) on curing temperature (k, min-', t, min).

The amount of the effective energy of activation calculated with these two parameters coincides and equals 67 kJ/mol ( 1 6 kcal/rnol). This is an interesting result showing that here the activation energy of the initial stage of curing (Kcharacterizes the initial reaction rate) coincides with the activation energy of a deeper curing process.

For the ultimate conclusion on the validity of the selected rheokinetic scheme and the calculated values of the constants it is necessary to compare experimental and calculated dependencies P(t) . As is seen from Fig. 3.8.29, a practically complete coincidance is observed between the experimental and calculated dependence of the conversion degree or time at curing temperatures 190, 230, and 250 "C. Besides, at 170 "C with sufficient correlation of the experimental and calculated functions P(t) at initial stages of curing, their deviation is observed after reaching 20% conversion (400 min from the start of the reaction). This point coincides with the second maximum on the dependence tan5 vs t (Fig. 3.8.25) which physically corresponds to vitrification of the reaction system as shown in [37]. Upon transition of the vitrification temperature the reaction rate decelerates sharply. This is expressed in the deviation of the experimental and calculated curves P-t, the latter exceeding the former.

I

2aO GOO boo 800 t, min

Figure 3.8.29: Experimental (dots) and theoretical (curves) dependencies of P(t).


An interesting example of using the rheokinetic method for silico-organic compounds is the comparison of the rate of viscosity growth q = d q / d t of polyorganosiloxane rubbers in the presence of tin-organic compounds [40]. Cross linking was made with compounds of the general formula (EtO),,-Si-R, (where n = 1 or 0) in which the nature of the radical R was varied. Depending on the structure of R the curing period (characterized by the time before reaching a certain viscosity) changed from = 10 to > 1000 min, i. e. more than two decimal orders. If the growth of viscosity q is described (as suggested in [40]) by an equation of the type 11 = Kqm where (as it was found) the average rn = 0.84, the growth of viscosity occurred almost proportionally to t 6 . The great significance of the exponent is apparently linked with simultaneous occurrence of chain growth and side branching (and cross-linking). If m = const, constant K is convenient as a comparative measure of the process rate. The constant K, i. e. the coefficient of relative activity of binding compounds is connected with the nature of the radical R. It is essential that the series of activities plotted on the values of the "rheological" constant K coincides with the series of changes in electron acceptor properties of the radical R. It provides direct confirma- tion to the physical sense of using the rheokinetic method to evaluate, the rate of structuring (and accordingly the technological properties) of polyorganosiloxanes.

The studies of structuring rheokinetics of this group of materials were discussed in more detail in [41, 421. Industrial batches of siloxane rubber containing 4 mass fr. of tetraethoxysilane, 0.14 mass fr. (composition I) on 0.5 mass fr. (composition 11) of tin octoate were analysed [43].

The process of curing low-molecular silico-organic rubbers in the isothermal case is described by the equation with self-acceleration. Following from the scheme given above, the constants k and c of this equation - characterizing the structuring rheokinetics of these compositions - were determined. The dependencies of the values of k and c on the amount of curing agent at 80 "C are shown in Fig. 3.8.30, and the temperature dependence of these constants at cp = 4 mass fr. is shown in Fig. 3.8.31. The temperature dependence of the constant of reaction rate r is of common Arrhenius nature, and the value of activation energy is 23.5 kJ/mol for composition I and 30 kJ/mol for composition 11. A higher value of activation energy reported in [44] may be caused by the fact that in the given work this value was determined for a model reaction and in a very narrow temperature range (20-26 "C).

mass. ;C J

Figure 3.8.30: Dependence of the constants of reaction rate k (1) and autocatalysis c (2) on the content TEP of curing agent for composition I at 80 "C.


-2,2

-2,4 5 c Qw

f4

40

-2,8

Figure 3.8.31: Temperature dependence of the constants k ( I ) and c (2) for composition I at cp = 4 g w/w.

The value of the constant c is independent of the temperature at which the process of

The constants k and c are inversely dependent on the content of tetraethoxysilane (TES) in a composition. As the content of TES increases, the value of k increases as well, whereas the value of c decreases.

structuring is performed.

These dependencies are characterized by the presence of the saturation section, i. e. the increase of TES content in a composition above 4 mass fr. does not result in further changes of the constants. We may assume that this amount of a curing agent in a composition arbitrarily corresponds to equimolarity with actual concentration of rubber hydroxyl groups capable to react with TES. Then the rheokinetics of the structuring process is described by the equations:

9 = 0.015(1-p) (1+1 .8p) d t

cp = 4 - 10 mass fr.

d p - = O . O O ~ ( P ' . ~ ( 1 - p) ( 1 + 13.2cp'.4) cp cp < 4 mass. fr. dt

The comparison of experimental and calculated values of the conversion degree in the structuring process is shown in Fig. 3.8.32. There is a notable deviation between the calculated (curves 2'-5') and experimental (curves 2-5) dependencies P(t) at the initial stages of curing. Since these differences are observed only at elevated temperatures of a reaction, whereas at T = 20 "C the calculated and experimental curves practically coincide throughout the process, it may be assumed that this phenomenon is connected with a transitory process - heat-up of the reaction system as the cell of the apparatus is plunged into the thermostat.


so fUU t, min

Figure 3.8.32: Experimental (dots) and theoretical (curves) dependencies of conversion degree in the process of curing composition I (a) and II (b). 2-5 is calculation ignoring heat-up of the reaction mass a ) T = 20 "C ( I ) , 50 "C (2) , 60 "C (3) and 80 "C (4); 6) T = 20 "C ( I ) , 35 "C (2), 53 "C (3), 70 "C (4) and 90 "C (5).

Under these conditions at the initial stages of the process a marked non-isothermality is observed, and the calculated formula should be written in the following way:

where T(t) describes the temperature increase in the reaction mass in a cell, ko is the pre-exponential factor in the temperature dependence of the constant of reaction rate.


Consequently, if we know the function T(t) it is possible to calculate the value of the I

integral jexp (-x) dt and (since K can be found from the analysis of the isothermal part of W t )

0

the reaction) to introduce a correction for the non-isothermality of the initial stages of the process. However, when the processes are performed in a laboratory or industrially, it is usually hard to formulate a definite law of temperature change. In these cases, the required information may be obtained from appropriate processing of experimental data with regard to changes in the rubbery modules and the "rheological" degree of conversion. Plotted dependencies of the function

on time (Fig. 3.8.33) show that under isothermal conditions of the reaction per- ~ ln-

formance this dependence is linear (curves 1,2), whereas a deviation from isothermality brings about considerable non-linearity (curves l', 2'):

1 1 + c p l + c 1 - p

20 40 60 t, min

on time in the course of Figure 3.8.33: Dependence of the function - In - 1 I + C P

I + c 1 - p

isothermal ( 1 and 2 ) and non-isothemtal(1' and 2') curing composition 11

at 35 c" ( I and 1 ') and 53 "C (2 and 2').

Then, if we know the values of ko and CJ, it is possible to calculate the temperature change in a reaction mass during its heat-up.

The comparison of theoretical and experimental dependencies T(t) is shown in Fig. 3.8.34, where their good convergence can be seen. The conclusion based on the these results may be that under the given conditions of the experiment at the initial stages a heat-up of the reaction mass is observed causing a change in the reaction rate. Allowing for this consequence, the experimental and theoretical dependencies p(t) in Fig. 3.8.32 fully coincide.


60 T,o

$0

20

- 2 . .

! 1; - cI)-o-

- I I

Figure 3.8.34: Experimental (dots) and theoretical (curves) dependence T(t) at 35 "C (I) and 53 "C (2).

Changes of some structural parameters of the network in the process of curing were calculated on the basis of the sol-gel analysis. A relative change in the mole number of chain fragments between the points of the network in the course of curing was determined similarly with the "rheological" degree of conversion. The comparison of these parameters is shown in

Fig. 3.8.35. The values of the function In - calculated from the rubbery modulus and the

equilibrium swelling of rubber in toluene are similar, and so are the values of the rate constant k calculated on the basis of this function :

I - P

k , = 1 1 xlOP3 and k , = 0.9x103min-', the values of the constant c coincide. c

--+w.- I I I ' 5-

24 t, 4 8

Figure 3.8.35: Change of MMfiagments of the chain between the points of the network M, (1) and the rubbery modulus G' (2) in the process of curing PDMS at 20 "C.


Another peculiar case of structuring in silico-organic polymers is copolymerization of spyral and cyclic siloxanes since the introduction of spyrocycles leads to formation of three-dimensional polymers [45]. This process has a definitely expressed induction period during which the network is not yet formed, then a sharp increase in gel-fraction release follows. The process slowly ends in reaching the asymptotic equilibrium content (about 80%) of the cross-linked product. It is typical of this process that there is no single gel-point since the content of the gel-fraction gradually increases in the course of the process. Nevertheless qualitative changes in the rheological properties of a reaction mass are the same as in other structuring systems. Thus, time dependencies of the components of the dynamic modulus permit clean determination of the induction period as well as the full duration of the process up to the equilibrium gel-fraction content. The maximum tangent of the mechanical loss angle corresponds to the moment at which the gel-fraction, i. e. net structure, formation is initiated [45]. Measurements of the rubbery modulus provide qualitative evaluations of the nature of a forming network depending on the composition of a reaction mass.

Thus, even in this rather specific case of copolymerization with silico-organic compounds of different types the rheokinetic analysis (in its dynamic variant) gives useful information on certain regularities of forming a three dimensional structural network, though its direct comparison with the course of the chemical process is difficult due to the complexity of occurring reactions and the lack of data of their kinetics.


3.8.5 Amino-formaldehyde Resins

Typical experimental data of dependencies of the rubbery modules and the mechanical loss angle tangent on time [46-481 are shown in Fig. 3.8.36 and 3.8.37. When ammonium chloride is introduced into the resin, its curing is notably accelerated which is indicated by a decrease of time needed to reach a stationary value of G' and a maximum for tan6 (Fig. 3.8.36 and 3.8.37). The position of the maximum on kinetic curves, as a rule, corresponds to the completion of gel formation in curing compositions and the development of a continuous network through the entire volume of the sample.

As the curing temperature increases the time of gel formation regularly decreases both in the presence of ammonium chloride and without it. However, without ammonium chloride the eventual value of G' is a function of temperature (see Fig. 3.8.36) whereas the eventual values of G' at various temperatures practically coincide with the introduction of NH4CI.

The nature of the dependence of p on the curing time for resin containing NH4CI is the same as in all reactive oligomers. The curing rheokinetics of this resin obeys the equation with self-acceleration. This equation cannot be used to describe the curing kinetics of melaminoformaldehyde resin containing no ammonium chloride because it does not take into account the possibility of reaction incompleteness at a relatively low temperature. Here the equation with self-acceleration can be used. The values of k (see the table) calculated from this equation show its validity as a tool to describe the curing kinetics of resin containing no NH4C1.

As ammonium chloride is introduced into the resin its curing rate increases about 100 times and the activation energy does not change appreciably being equal to 65 kJ/mol. The constant from these resins is 148 and 67 respectively and does not depend on temperature either. The value of 5 for a resin without a curing agent drops from 1.67 down to 1.16 as the temperature increases from 100 to 110 "C. Here, in the presence of NH4CI the effect of the process self-deceleration at low temperatures is not detected and the reaction proceeds until a practically constant value of G' at any temperature is reached. When 1% NH4CI is introduced into the resin the reaction rate grows to the same extent as with the curing temperature increase to 170-180 "C.

6 -

.-

a 6

f20 240 360 t20 260 c I I I I

t , min t, min Figure 3.8.36: Change of the rubbery modulus in the course of curing melamino-

formaldehyde resin: a ) without a curing agent, T = 140°C ( I ) , 130°C (2), 120°C (3), II0"C (4), 100°C (5); b) with ammonium chloride, T = 100°C ( I ) , 80°C (2) . 60°C (3), 50°C (4).

Structuring Peculiarities of Diyerent Forms of Oligomers and Polymers 249

f, min

Figure 3.8.37: Change of tan6 in the course of curing melaminoformaldehyde resin a ) without a curing agent, T = 140 "C ( I ) , 130 "C (2), 120 "C (3), 110 "C (4) and 100°C (5); b) with ammonium chloride, T = 100°C ( I ) , 80°C ( 2 ) , 60°C (2) and 50°C (4).

Tab. 3.8.1: Values of the rate constant for curing melaminoformaldehyde resin.

Resin

without a curing agent

with a curing agent

Temperature, "C

100

110

120

130

140

50

60

80

100

k . f i n - '

1.4

2.7

4.4

6.5

13

4.9

12

41

94


3.8.6 Epoxy silico-organic Oligomers

The curing of epoxy silico-organic oligomers is characterized by peculiar features (mentioned in Section 3.6), according to which at process temperatures below 200 O C the equilibrium value of the rubbery modules and complete amount of released heat depend on temperature [49, 501. Conse- quently, the kinetics of curing this oligomer with a high-functionnal element -organic curing agent - polybutoxytitanphosphoroxane is described by an equation with self-deceleration. Temperature dependencies of the constant k and the constant of self-deceleration 5 on Arrhenius' coordinates are given in Fig. 3.8.38:

Figure 3.8.38: Dependencies of the constant of reaction rate k and the constant of self-deceleration 5 on temperature.

The activation energy of curing determined conventionally from the dependence Ink - 1 / T is 75 kJ/mol. The activation energy of the process of reaction deceleration characterized by the constant 5 appeared to be 10 kJ/mol. The derived value is comparable with the activation energy of diffusion processes. This is likely to be the evidence of the fact that the observed deceleration is connected with limited mobility of oligomer molecules in the formation of net structures. This may be caused by two reasons: the approaching of a reaction system to the vitrification temperature or topological complexity of the structure of a forming material making some functional groups unable to enter the reaction. However, taking into account that the whole curing process occurs with retained high elasticity of the system (the vitrification temperature in a completely cured oligomer is 120 "C) we may assume that the process deceleration is determined by structural topological phenomena developing in the course of curing a given system. Probably, at high functionality of the applied curing agent [ I41 after the first stages of the reaction a continuous network (macrogel) is formed which limits the mobility of oligomer molecules.

Decreasing functionality of the curing agent brings about changes in the form of the rheokinetic equation and initiates the effect of self-acceleration. The dependencies P(t) for low-functional curing agents of various chemical nature (tetrabutoxytitane and dicyandiamide) are shown in Fig. 3.8.39. Rheokinetics of these processes are described by an equation with self-accelelation despite of the total difference of the chemical nature of curing agents [51]. The reported results show that the functionality of the curing agent determines the structure of the second co-factor in the rheokinetic equation. The increase of functionality leads to the coefficient decrease at the conversion degree varying from positive (5 > 0) to negative values (5 < 0). As a result, changes in functionality cause qualitative changes in the curing process - self-acceleration degrades, the process is apparently incomplete and a temperature dependence of the constant 5 appears.


Figure 3.8.39: Dependence of the conversion degree p on time for the system epoxy silico-organic oligomer/TBT (a) and epoxy silico-organic oligomer/DCDA (b). Temperature: a ) I70 "C ( I ) , I80 "C (2), I90 "C (3), 200 "C (4); b) I50 "C ( I ) , I60 "C (2), I70 "C (3), I80 "C (4), I90 "C (5).

It is of interest to investigate curing rheokinetics at various ratios of epoxy silico-organic oligomer and polybutoxytitanphosphoroxane. The dependencies P(t) are similar in all cases although the increased amount of a curing agent results in regular reaction acceleration. However, irrespective of the amount of the introduced curing agent the process at T = 200 "C comes to an end before complete conversion is reached, i. e. the equilibrium degree of reaction completion is mainly determined by the temperature of curing. The eventual value of the conversion degree with changing cp varies within a rather narrow range, e. g., at T = 180 "C P, P 0.82 although the constant of reaction rate with concentration of the curing agent increasing from 2.5 to 15 mass fr. increases almost 5 fold (Fig. 3.8.40). This is indicated by the fact that at sufficiently great amounts of the curing agent (Acp > 7.5 mass fr.), the value of 6 is practically constant. A certain increase of the self deceleration constant as the amount of the curing agent decreases (Acp c 7.5 mass fr.) may be connected with the emerging lack of reactive groups of the binders.

5 fa '5 p. Figure 3.8.40: Dependencies of the constant of reaction rate k and the constant of

self-deceleration 6 on the concentration of a curing agent cp, T = 180 "C.


3.8.7 Rubber Compositions

The rheological control of rubber compositions curing is one of most vivid examples of using methods of measuring mechanical characteristics to monitor the course of chemical transformations in polymers. Besides, application of this method to elastomers has always been limited by practical considerations. Therefore, on the one hand, rheological methods of determining curing characteristics of rubber blends - due to their wide use and practical importance - have given rise to standardized techniques. On the other hand, they are still staying mere descriptions of observed patterns and, as a rule, are never compared with the kinetics of chemical transfoma- tions in a cured material.

At present two methods of measuring rheological parameters of rubber composition are extensively applied: Torque measurements at a constant deformation rate (employing one of the most common technological device, the Mooney plastometer) and measurements of torque amplitude values for assigned harmonic oscillations with constant deformation amplitude (this scheme is used, e. g., in the apparatus “Rheometer-100” produced by “Monsanto”, USA).

Time-dependent changes of torque (M) measured with any of these devices occur as shown schematically in Fig. 3.8.41 borrowed from GOST (State Standard) 12535-78 which is in good agreement with the requirements of the international standard I S 0 3417-77. In addition, the standard differentiates sample testing in two types of apparatus: in one of them employing the method “A” oscillations are produced by flat parallel movement of an oscillating plate, in the other employing the method “B” a cone rotor is used as a source of oscillations. The standard does not demand a definite frequency of oscillations to be observed. The same applies for deformation amplitude and temperature since these conditions are determined by specifications of particular materials.

Below we shall consider possible types of curing curves and the physical sense of marked characteristic points for the more promising method “B” in detail . As it is seen from Fig. 3.8.41, curing curves differ in the form of their final sections as the torque change may either end in reaching a plateau, where the torque acquires a constant value M,, (Fig. 3.8.41 curve a) or, on passing through a maximum M,, , the torque begins to decrease (Fig. 3.8.41, curve b), or, finally, the section AB corresponding to a constant rate of the torque increase is reached (Fig. 3.8.41, curve c). For all the three curves the following characteristic parameters of the curing process are determined. The most significant rheological characteristic of a material is the minimum torque value M , representing the properties of a non-cured material. The maximum torque value for cured material is easily determined from the curves of the first two types as M,, or M,, and is arbitrarily found for a certain testing period, which was chosen from the curve of the third type as M , depending on the nature of a tested material. The moment of time at which curing begins is assumed to be at the point t , where the torque increases by 0.1 HM (compared with M,). Thus, t , defines the material’s “life span”, i. e. the time during which it retains the state of a relatively low-viscous liquid. Certainly, the choice of torque shift from M , by 0.1 H, is arbitrary and admits options in evaluating t , .


Extremely important is the evaluation of curing time which is characterized by t; (X) corresponding to the point of reaching the torque equal to M , = M , + X ( M H - M , ) where M , denotes one of the values M,, , M,, or M , . X is the current value of curing extent. The “optimum curing time” t: (90) (see Fig. 3.8.41) is assumed to be the duration of the process allowing the torque to reach the value M,, which is derived as M,, = M , + 0.9 ( M , - M,) .

Finally, one more kinetic characteristic of the curing process is the parameter V , expressed as V , = 100/ [ t ; (90) - t s ] , i. e. V, is actually inversely proportional to the duration of the process between its two characteristic points. Thus, the curing curve provides rather extensive rheological information enabling qualitative comparison of different rubber blends and control of raw materials as well as of technological processes.

I L L .-- Figure 3.8.41: Typical curing curves obtained in testing rubber blends to GOST 12535-78,

method B: a ) curing end when plateau is reached, b) curing with reversion, c ) on completion of the main curing period the modulus rises continuously. See details in the text.


For further considerations we shall use two characteristic parameters - the values of process duration corresponding to the conditions X = 0.5 (T~,) and X = 0.9 ( t : (90)or T~,,).

The dependence M(t) in the main curing period, i. e. in the region from the minimum torque value M , up to M , (where M , , depending on the type of the curing curve acquires the form M,, or M H ) , can be described with fair approximation by an exponential function corresponding to the apparent first-order reaction from the torque (which, of course, does not actually characterize the first order of a chemical reaction since no linear relation should necessarily exist between the torque and the degree of conversion). Then

M - M ,

M H - M L = 1 -exp [ ( -k , ) ( t - T , ) ]

where k , is the characteristic time of the process and 7, is the induction period of the curing

Using the results of measurements of the standardized parameters T~, and zg0, it is easy to

reaction.

calculate the time constant k , which is equal to [52]:

1.6 k , =

('90 - ' 5 0 )

This value is in fact similar to the standardized parameter of curing rate V , . The process rate in the induction period, i. e. from the beginning of pre-curing up to the

moment T ~ , is determined by its kinetic constant k , . Taking into account that k, is determined by the difference ( T~~ - zs0) and k , by t , , it may be assumed that the correlation of constants k , / k , is expressed by the rheokinetic characteristics of the curing process, i. e. the parameter ( T~~ - T 5 , ) / t , , which is in itself a combination of standardized parameters derived from the curing curve. It is essential that this parameter is connected with the formula of the rubber blend, and, as was demonstrated by direct measurements [45]. The value of (z9, - ~ ~ , ) / t $ depends considerably on the sulphur content in the rubber blend but is independent of the concentration of sulphonamide accelerator. Therefore measuring standardized parameters on the basis of the curing curve immediate control of sulphur and accelerator feed (the latter is effected from the values of k , ) may be accomplished industrially. This approach was put into practice by designing an automated scheme of operational control for processes producing rubber blends for tyres with varying sulphur and accelerator concetrations [53].


Other Polymers

The discussed examples do not exhaust all the possibilities of net polymer formation, and the results of studying this process by rheological methods. However, whereas the described polymers are considered in extensive literature allowing a critical approach to available data, the information involving other objects is in many cases occasional. Nevertheless, to provide a complete picture it is reasonable to present here the available data on various polymer materials.

Network Formation in the Course of Radical Polymerization

Radical reactions in net polymer formation are typical of the above considered process of curing unsaturated polyester resins copolymerized with styrene, methylmethacrylate and olefins. Beside that, it is convenient to consider this reaction for the model system as an example. Thus, a typical example of net polymer formation in the course of radical polymerization is styrene copolymerization with divinylbenzene. Studies of this process initiated by azobisisobutyronitrile revealed typical changes of viscosity in the reaction mass with time [54]. The main object of the research was to determine the point of gel formation t* in the transition from the fluid to the rubbery state of the system. This was achieved by extrapolation of the experimentally obtained dependence q(t). It was found that this dependence in the vicinity of t* can be described by the formula of the type:

-S q = K [ y ] (3.8.1 1 )

where K and S are constants, S = 0.79 f 0.02.

This dependence is totally different from the exponent since, according to (3.8.1 l), at r + t* viscosity shows unlimited growth. Selecting the constant values having the best agreement with experimental data, provides a quite reliable means of deriving t* . At any rate, evaluations of t* cited in the original work boast of an error not exceeding 0.1%.

Another example of a curing process occurring by the radical mechanism is polymer formation on the basis of diallylphthalate [55] . In the course of prepolymer synthesis, 70% double bonds were involved in the reaction which is close to the gel-point theoretically reached at 78% double bond conversion. Under these conditions the prepolymer remains a viscous liquid and only some time after acquires the high-elasticity state. The calorimetric analysis of the process showed that structuring in this case is described by a first order equation. The activation energy of this process in the temperature range 400-480 K drops from 100 down to 75.5 kJ/mol as the catalyst tert.-butylperbenzoate concentration increases with a consequent decrease of the pre-exponential factor in the given temperature range by more than three decimal orders [55] .

The transition from a prepolymer to a cross linked product occurs in a very narrow range of changes in the conversion degree p evaluated from the content of unreacted double bonds in the prepolymer. Therefore a slight change in p should result in a very sharp increase of the molecular mass of intermediate products and of their viscosity. In fact, it was found in [56] that as p increases from 0.712 to 0.754 the molecular mass of the oligomer increases from 33 X103 to 31 x104. This is related to the change of the rubbery modulus, dynamic viscosity and charac-


teristic time of relaxation representing the position of a material's relaxation spectrum on the frequency axis, also by about a decimal order, i. e. a practically linear correlation exists between the given rheological characteristics of the prepolymer and the values of p. This fact indicated absence of any appreciable kinetic flexibility of the prepolymer despite a rather high molecular mass of intermediate products since the formation of intermolecular engagements (entangle- ments) should have led to a far greater degree of dependence of rheological characteristics on molecular mass. This is apparently caused by considerable branching of intermediate oligomer products so that, despite rather high values of molecular mass, overall dimensions of the macromolecular coil remain relatively small.

Measurements of changes in the components of the dynamic modulus with time show that reaching the gel-point (indicated 1 min after the process has begun) is manifested by a maximum on the dependence of the loss modulus G" on time t similar to that described for the general scheme of changes in the mechanic properties of an oligomer in the course of its structuring (see Fig. 3.1 S). Of particular interest is the attempt to set up a correlation between the rheological and the kinetic data. It is based on the assumption that in the region of high elasticity which the material acquired on reaching the gel-point, the linear correlation between the rubbery modulus G' (thus identified with the equilibrium modulus of high elasticity in the plateau region of the rubber-like state) and the density of cross links is X. Then, if the change of X with time is described by a kinetic first-order equation, the dependence G'( t ) (after the gel-point) should also be represented by the same equation.

Hence the formula for G'(t) :

where the index g denotes values referring to the gel-point, k , is the kinetic constant, Gb,

This formula, in fact, reflects accurately the nature of changes in G' at t > tg (i. e. after the gel-point), which confirms the principal assumption of the type of a kinetic equation describing the process of forming a network of three-dimensional links. Comparing values of the dynamic viscosity q' before reaching the gel-point depending on temperature at t = item has shown that the effective energy of process activation in the network formation is 106 to 105 kJ/mol. This value, as was shown in the analysis of radical polymerization, should not be identified with the effective energy of activation of a chemical process (see formula 2.3.18). Therefore it is natural that it does not coincide with the activation energy found with the calorimetric method. Appli- cation of formula (3.8.12) to the given process is also arbitrary since the topology of branched and cross-linked structure formation is unknown but expected to be more complex and diversi- fied.

is the limit value of G' corresponding to complete conversion of double bonds.

Studying the process of cross-linking diallylphthalate resin makes it possible to confirm some conclusions which are of general importance for investigating rheological properties of oligomers in the course of their curing. First, this is the necessity to use various techniques to control the process before and after the gel-point, i. e. for treating fluid and solid materials. Second, this is the difficulty of direct comparison of chemical and rheological kinetics in the

Structuring Peculiarities of Dizerent Forms of Oligomers and Polymers 257

course of curing, which makes both methods indispensable and complementary. Third, it is the opportunity to describe changes of the rubbery modulus in the course of curing by a simple "quasi-kinetic" equation which implies possible extrapolations of measured results and quantitative comparison of different rheokinetic situations. Finally, the performed research has shown that passing through the loss maximum at the gel-point is a general regularity of the curing mechanism irrespective of a particular chemical scheme of the reaction.

Cross-linking of Polyethylene

Structuring reactions also include the formation of rate cross-links between polyethylene chains when peroxide is added to its melt and the formation of relatively few branches and/or to cross-links results in a considerable modification of a material's rheological properties. The kinetic of this reaction is rather hard to describe. For this reasons the researchers investigating the problem used to employ methods traditional for rheokinetics: they used an appropriate empirical equation to represent a certain rheological parameter of a polymer, like it was described above for diallylphthalate structuring.

For instance, the results of studying changes in the dynamic rubbery modulus G' with time f can be represented by formula [57]:

G'(t) = GIp, - c , e -k,(Vr

where GIp, is the value of G' in the plateau region of high elasticity on completion of cross-linking, ci is an empirical constant, k , is the principal kinetic characteristics of the process depending on temperature, GIp, and ci depend on peroxide concentration and temperature.

Processing of experimental data shows good effectiveness of the given equation in describing measurement results for the dependencies G'(t) in the temperature range 160 to 210 "C and peroxide concentrations of 0.15 to 0.5% (mass). The constant k, equals 3.5 ~ 1 0 , where the activation energy E = 156 kJ/mol, the rate constant k , determined rheokinetically is close to the rate constant of thermal decomposition of peroxide. The latter circumstance indicated a direct correlation between the results of rheokinetic analysis and the data obtained in direct kinetic studies of the reaction.

16 - E / R T e


3.8.8 Thermo-reactive Polyamides

Aliphatic high molecular polyamide are widely used in industry: machine-building, electrical engineering, instrument-making, etc. This is caused by a set of valuable properties inherent in the group of polyamides and net structure formation.

Particular importance is attributed to increasing use of high molecular polyamides as glues, binders, impregnating glue and film materials. Application of glues in various machines, instruments, apparatus is in many cases the only feasible way of joining to give a required construction strength. At present film glues are gaining particular importance in designing automated production lines. The properties of thermoreactive polymers are ultimately formed in the process of curing. For that reason, the studies of curing kinetics is of prime importance in terms of investigating the mechanism and main kinetic peculiarities of this process as well as its temperature-time parameters.

The main experimental data in the form of time dependencies of viscosity (11) and rubbery modulus (G' ) at various curing temperatures are shown in Figs. 3.8.42 and 3.8.43. As it is seen from Fig. 3.8.42, the approaching of the system to the point of gel formation is characterized by intensive viscosity growth, which changed by several decimal orders within a relatively short period of time. At the moment when the gel point is reached (Fig. 3.8.42) the reaction system loses its ability to develop irreversible deformations and breaks off from the effective units of a rotational viscometer.

8 C

-cI.

-&

'Y

G

4

3

2

60 80 t min

Figure 3.8.42: Change of viscosity of epoxy methylenepolyamide with time of curing at temperatures: 220 "C ( I ) , 200 "C (2), 180 "C (3).

Structuring Peculiarities of DiFerent Forms of Oligomers and Polymers 259

t, min

Figure 3.8.43: Change of the rubbery modulus G' of epoxymethylenepolyamides with time

Representing the experimental data on the coordinates In - ' vs t following from formula (3.5.5) we may determine the values of constants at different curing temperatures. Since the value of the self-acceleration constant does not depend on the reaction temperature, the activation energy of the curing process is determined by the temperature dependence of the rate constant and equals 109 kJ/mol.

at temperatures: 220°C ( I ) , 200°C (2), 190°C (3), 180°C (4), 160°C (5).

1 - P

The validity of the given model is confirmed by a satisfactory agreement of experimental and calculated dependencies (Fig. 3.8.44). This figure shows that disagreement of theoretical and experimental values never exceeds 3%.

t, min

Figure 3.8.44: Experimental (dots) and theoretical (curves) changes of the conversion degree with time in the course of curing epoxymethylenepolyamides. T = 220 "C ( I ) , 200 "C (2 ) , 190 "C (3), 180 "C (4) and 160 "C (5).


3.8.9 Epoxy oligomers

Compounds (monomers and oligomers) containing epoxy groups /"\ have come to

wide use in modern polymer technology. High polarization of the bond C-0 in this tense cycle enables various chemical reactions with compounds of different types leading to the formation of materials with diverse technological and physical-mechanical properties. Therefore epoxy oligomers are used as binders in compositions of most varied compositions and applications - glues, lacquers, enamels, reinforced construction materials, etc. The main requirement for application of epoxy compounds is the possibility of forming polymer materials with three-dimensional cross linked structure, i . e. polymer networks. Varying the chemical composition of epoxy resins and the composition of cured materials makes it possible to control their technological properties in a very wide range.

/H+H

The literature on this subject contains rather rich (although often quite controversial) material on kinetic and rheological studies of the processes of curing epoxy resins and compositions based on them.

The diversity of results obtained from the studies of kinetic regularities of curing epoxy resins is caused by three factors:

1. the use of a great number of compounds of various composition containing, reactive epoxy groups

2 . the use of curing (structuring) agents of different types, and

3 . the diversity of research methods applied which give different contributions of separate reactions, which are eventually leading to the formation of cross-linked structures.

The most widely used epoxy resins are diane compounds with oligomer charms containing aromatic nuclei and hydroxyl groups and epoxy groups at the end of chains. These compounds are highly reactive enabling the curing process to be performed at high rates and relatively low temperatures. Aliphatic or aromatic amines combined with acid anhydrides are most widely used as curing agents. New metal-organic curing agents have recently come into wide use which also effect macrokinetic regularities of the curing process and the properties of obtained products. Nevertheless, owing to similarity of general methodology and many principal results of rheokinetic studies of curing epoxy resins for compounds of different structure, we will discuss the problem of the effect the structure of components of a cured composition has on the observed rheokinetic regularities of the process.

The mechanism of curing epoxy oligomers has been given sufficient consideration in the literature (e. g. [%I). Although there is no universally accepted view of the curing mechanism and topological peculiarities of forming net structures at present, a general and rather simplified approach to the analysis of the main reactions, which occurr in the course of curing epoxy resins by most common curing agents - amines, yields the following scheme.


OH

OH I

CH2 <H*

R-N\ + /"\ R N \ (11) k2 ~

H

CH2 -CH* I

OH

CH2 --CH t CH2 - C H M

I OH

Even from these rather simple formulae it is easy to notice that the structural peculiarities of reacting components and their ratio in a reactive system may greatly influence the kinetics and the mechanism of a curing reaction as well as the structure of the forming product. In fact, if reaction (11) is sterically impeded and its rate is low, reaction (I) will run to the end first, i. e. at the first stage of the process the molecular chain of oligomer is elongated and linear products are formed. The possibility of this course of a curing reaction is confirmed by the results of independent determination of the number of epoxy and primary amino groups involved in the reaction [59]. In this case equimolar amounts of epoxy and primary amino groups are consumed at the initial stages of the reaction and secondary amino groups enter the reaction only when primary amino groups have been exhausted. The point of gel formation in this reaction is reached when 5560% of the total amount of amino groups have been consumed. Other types of correlation are also possible between changes of epoxy and amino groups which reflects different correlations between the kinetic constants of interaction between epoxy groups and primary and secondary amino groups k, and k2. Hence we have the possibility of controlling the structure of a forming product, the kinetics of curing and the position of the gel point. The available literature gives different values for the ratio kl/k2, e. g. 10 [59], 2-4 [60] and = 1 [61].

This divergence is apparently caused not only by application of various epoxy oligomer and diamine derivatives but also by different experimental conditions, i. e. the use of model or actual compositions, the performance of the reaction with an inert solvent or in a block, etc. A rather scrupulous analysis of model compositions [6 11 revealed minor difference in the reactivity of \

N-H bonds in primary and secondary amines. This is probably caused by the fact that a / decrease in the activity of the secondary amine due to steric impedance is compensated by the electron donor effect of the substituent [61].

Considering the interactions of epoxy and amine groups it is necessary to take into account the effect of hydroxyl groups forming in the course of the reaction.

The presence of hydroxyl groups results in activation of the carbon atom in the epoxy cycle to the nucleophylic attack by amine [62]. Therefore curing epoxy compounds by amines may have an auto catalytic nature determined by accumulation of hydroxyl groups in the course of the reaction. In this case the rate of a curing reaction may be described by equation [63]:


where k is the constant of reaction rate, [A], [El and [Z] are the concentrations of amine, epoxy and hydroxyl groups, respectively.

A more detailed consideration of the main reactions allowing for the effect of hydroxyl groups given the following system of differential equations describing concentration changes in the main reacting components [61].

where [B] and [C] are the concentrations of secondary and tertiary amino groups, respectively.

A still more complicated picture results if we take into account the reactions of intramolecular ring formation which alter the topology of the forming net structure [64, 651. In this case, the regularities of network formation can be sufficiently described only before the gel-point is reached. After the gel-point, especially at the final stages of the reaction, the analysis of curing kinetics is hardly feasible. The process enters a diffusion-controlled region due to limited mobility of reaction groups localized in different parts of the network [66,67]. Here the constants of reaction rate decrease. The dependence of the rate constant k2 on the conversion degree of epoxy groups E during interaction of diglycidyl ether with 4.4.1-diaminodiphenylsulphide was reported in [%I. In this case, the reaction proceeds in the kinetic region up to 70% after which a sharp decrease of the rate constant is observed.

The above given examples demonstrate evidently the rather complicated nature of the process of curing epoxy resins and the necessity of taking into account a great number of various parameters in experimental and theoretical studies of the process.

Speaking about experimental analysis of the process of epoxy oligomer structuring, it should be noted that the direct and continuous monitoring of changing concentrations of reactive groups is only possible with the method of IR-spektroscopy [60, 68-70]. However, in a number of cases it may be extremely difficult to apply this method due to overlapping of absorption lines corresponding to different functional groups and consequently ambiguity in the interpretation of the obtained results.

Therefore in the studies of the entire curing process using the minimum number of macrokinetic parameters, calorimetric and rheological methods appear to be the simplest and most informative. Here the rheological approach offers greater advantages since it links macrokinetic description of a reaction directly with practically significant changes in the physical-mechanical


properties of a reaction system such as viscosity and rubbery modulus. The rheological method also provides a sufficiently simple determination of the point of gel formation from the change of the system viscosity and the mechanical loss angle tangent (see Section 3.1).

In macrokinetic description of the curing process based on calorimetric and rheological data, the first problem to arise is that of determining a correlation between the depth of the chemical transformation and the observed change in physical-chemical properties. With the more common calorimetric analysis of the process the “calorimetric” degree of curing p, is determined from the relative amount of heat released at the moment t [71,72]

where Q(t) is the amount of heat released under the isothermal flow of the reaction for the period t. Q, is the integral registered heat effect of the reaction at a given temperature.

It is important that, in the general case, Q, depends on temperature as the attained depth of curing differs depending on reaction conditions.

This representation of calorimetric data appeared to be quite useful and permitted to suggest models of relatively simple description of the process of curing epoxy compositions based on data of calorimetric measurements. However, analysing the data reported in literature one pays attention to significant differences in interpretation of calorimetric (and, generally, other) measurements and, consequently, different suggested variants of macrokinetic description of curing epoxy oligomers. This concerns primarily the form of the kinetic equation. In many works attempts were made to describe the macrokinetics of curing epoxy oligomers by a simple kinetic equation of the n-th order with one kinetic constant , i. e. by an equation of the type

p = k ( 1 - p ) ” (3.8.13)

where k is the constant of reaction rate.

It has been noted, however, that the apparent reaction order, i. e. the indicator II in equation (3.8.13), may be zero, one or some fractured value [73]. Particularly changes of “n” from 0.7 to 2.1 and the activation energy of the curing process from 50.8 to 100.8 kJ/mol were described for the system diglycidyl ether or bisphenol A - dicyandiamide [73], which were linked with changes in the size of dicyandiamide particles, i. e. the reaction was assumed to proceed in the heterogeneous region. This change of constants makes it extremely difficult to process and to interpret experimental data. On the whole, activation energies of curing epoxy oligomers found by different authors are in rather good agreement and keep within 50.4 to 71.4 kJ/mol. The following table shows the values of activation energy for curing epoxy oligomers by amine curing agents obtained with different methods.


Activation energy in kcal/mol

51.7

Tab. 3.8.2: Activation energies of curing epoxy oligomers.

Parameter used to determine the activtion energy by its temperature dependence Literature

Consumption rate of reaction groups 68

62.2

58.8 58.8-67.2

Consumption rate of reaction groups

Resistance 74,78

71

Gel-point 75

50.4-54.6

58.8-67.2

58.8

58.8

46.2

1 50.4 I Dynamic mechanical properties I 82 I

~

Viscosity 76

Gel-point 77

Consumption rate of reaction groups 79

Gel-point 80

IR-spectroscopy 81

I 71.4 I Constant of reaction rate 183 I It is noteworthy that activation energy is found from changes of indirect parameters rather

than the temperature dependence of the constant of reaction rate. This is apparently connected with certain problems emerging in the course of precise determination of the constant of reaction rate since equation (3.8.12) cannot describe the process of curing epoxy oligomers with sufficient precision [71, 73, 741. For this reason a more general equation was suggested to describe macrokinetics of curing epoxy oligomers, namely:

6 = ( k , +qn) ( 1 - P I " (3.8.14)

where kl and k2 are rate constants, rn and n are empirical constants.

The written equation ( 3. 8.14 ) is certainly of great generality as it contains many empirically selected constants, but at the same time it complicates (and possibly makes ambiguous) its application for analysis of experimental data.

Since epoxy compositions are widely used as binders in the production of reinforced materials we should - at least briefly - dwell upon filler effects in the curing process. There are qualitative evidences of filler effects on the curing kinetics, the nature of which depends on the nature of the tiller. A general equation describing the curing kinetics of filled epoxy system is suggested in [83]:

B = k ' J , ( c ) ( l -P)"+k'2f2(c)Prn( l - P ) " (3.8.15)

where c is the filler concentration, f , (c ) and f2(c) are functions reflecting the effect of the filler, k' , and k ' , are rate constants of the curing reaction in the presence of the filler.


The values of the constants k', and k', are connected with kl and k2 by the following correlation:

(3.8.16)

For an unfilled system equation (3.8.15) is reduced to equation (3.8.14). The relevance of the filler effect on numerical values of the constants of reaction rate can be seen from curing of epoxy compositions filled with technological carbon. In this case the functions f,(c) and f2(c) can be represented by polynomials of the following type [83]:

f,(c) = 1 - 0 . 2 2 ~ + 7 . 8 2 ~ 1 0 - ~ ~ ~ - 6 . 4 9 ~ 1 0 - ~ ~ ~ (3.8.17)

f2(c) = 1 -o.14c+5.10x10-2c2-4.35x10-3c3 (3.8.18)

provided that 0 I c I 6 (the concentration c is expressed in % (w/w)).

Thus, in a number of cases filler effects on the curing kinetics should not be considered negligible. Moreover, formulae (3.8.17) and (3.8.18) show that changes in the kinetic constants may differ depending on filler concentration.

Formulae (3.8.14) and (3.8.15) are presently the most complete macrokinetic description of the process of curing epoxy oligomers.

It was shown previously, that calorimetric and, to some extent, chemical and spectroscopic methods yield analytic expressions describing the curing kinetics of epoxy resins. However, a problem remains as to what extent these expressions correspond to the results obtained by rheological and rheokinetic methods.

As it was mentioned in Chapter 2 in the course of linear polymer formation the viscosity of polymerizing systems is unambiguously related to the conversion of reactive groups and can serve a measure of the conversion degree. The situation for cured compositions is somewhat different. In fact, at the gel point q + 00 long before the curing reaction is completed. The position of the gel point is related to different values of the conversion degree depending on the structure of reacting compounds, their composition and the reaction mechanism. Nevertheless, attempts were reported to link the nature of the time dependence of viscosity in the process of curing epoxy oligomers with the reaction kinetics [84-891. Thus, in the course of curing diglycidyl ether of bisphenol A by triethanolamine the curve of viscosity growth q(t) was approximated by two linear sections [84]. The emergence of the bend point has been linked by the authors with the formation of an engagement net (which is typical for linear flexible chain polymers but greatly impeded for systems having no high-molecular linear chains). Linearity of the dependence q ( t ) in the first portion was explained by the fact that the curing reaction is described by a zero-order kinetic equation.

In general, the nature of viscosity growth in the process of curing epoxy resins is much more complex. The peculiarities of viscosity growth are determined by changes in the molecular mass and molecular structure of a cured oligomer (see Section 3.3).


It is rather typical of the curing process that the activation energy of viscous flow increases with the growing degree of conversion [ 12,71,90]. As reported in [91], in the course of curing the activation energy of viscous flow E, increases from 3 1.5 kJ/mol to 44.1 kJ/mol at p = 0.26 and further to 205 kJ/mol at p = 0.37. It apparently reflects intensification of intermolecular interactions in the reactive system with increasing degree of conversion.

In the studies of viscosity growth in the process of curing epoxy oligomers it is of great importance to state precisely the moment of fluidity loss in a reaction system. With rotational viscometers (commonly used for these measurements) the reaction mass breaks off from the working surfaces of the apparatus at this point. This moment approximately corresponds to reaching the gel-point.

In principle the time when the gel-point is reached can serve as a characteristic kinetic parameter, the change of which may indicate the effect of various factors such as temperature, the curing agent concentration, etc. on the constants of reaction rate. In [33,84,92] it was generally assumed that direct proportionality should exist between ( t* ) -' and the constant of reaction rate. This assumption is confirmed in [82] in rheokinetic studies of curing epoxy oligomers by amine curing agents. The curing rheokinetics of epoxy resin by a conventional curing agent of the amine type dicyandiamide is described by a kinetic equation with self-acceleration. The results of comparing experimental values of the conversion degree with those calculated by formula 3.5.2 at different curing conditions are shown in Fig. 3.8.45. The dependencies of the obtained values of reaction rate and self-acceleration constants on temperature and the concentration of the curing agent are shown in Fig. 3.8.46:

t , min t, min

Figure 3.8.45: Experimental (points) and calculated (lines) time dependencies of degree

of conversion at curing agent content (a): 5 (I), 6 (2), 7 (3), 8 (4), 8 (5), 2 (6), 13 (7) and at different temperatures (b): 140 "C (I), 150 "C (2), 160 OC (3), 170 "C (4), 180 "C (5).


c

- 2

- 3

6 I 0.7

-2,5

0.5

-3,s

-bo

- 30

- 20

Figure 3.8.46: Temperature (a) and conversion (b) dependencies of the rate constant k ( I ) , self-acceleration constant (2) and gel-time (3).

As the content of dicyandiamide increases in a reaction system, the constant of reaction rate k increases. However, the increase of dicyandiamide concentration above 9 mass fr. does not change the rate constant since at this concentration the ratio of reactive groups o f oligomer and the curing agent is close to equimolar. The effect o f self-acceleration with the concentration increase of the curing agent up to 9 mass fr. degradates and the value of the constant decreases linearly with the growth of cp. However, at cp 2 9 mass fr. it also becomes constant. Typically the temperature influences only the value of the constant k rather than c (Fig. 3.8.46).

The dependence In k (1/T) was used to derive the activation energy of curing which appeared to be 70 kJ/mol. This value agrees with the data obtained by other authors (see Table 3.8.9). The same value of activation energy is obtained from the temperature dependence of inverse time of reaching the point of gel formation ( t * ) - ' (Fig. 3.8.6 ), which proves the existence of direct correlation between ( t * ) -' and the constant of reaction rates. Therefore, provided the kinetic equation applied is true, only one constant of the two entering the kinetic equation ( k and c ) is temperature dependent, namely k. This is the exact dependence which determines the change o f t * with the change of the temperature of curing.



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46. S. G. Kulichikhin, Z. D. Abenova, N. E. Bashta, 0. P. Blinkova, G. S. Matvelashvili, A. Y. Malkin, Plasticheskiye massy, 1988,3,5 1 .

47. A. Y. Malkin, S. G. Kulichikhin, Z. D. Abenova, V. A. Kozhina, N. I. Bashta, L. A. Kuzmina, 0. P. Blinkova, Y. P. Brysin, N. M. Romanov, G. S. Matvelashvili, Vysokomolekulyarnye soedineniya, A, 1989, vol. 31, 8, 1716.

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Structuring Peculiarities of Diyerent Forms of Oligomers and Polymers 27 1

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273

Chapter 4 Transformation in Polymeric Systems

4.1 Polymer-analogous Transformations

4.1.1

Polymer-analogous transformations observed in the solid phase are most difficult to analyse. In this case, the use of rheological techniques offers a unique opportunity to control the kinetics of these processes. Naturally, the only rheological method applicable for investigating reactions which occur in the solid phase is the mechanical dynamic spectroscopy where the greatest resolution is obtained with the measurement of the temperature dependence of the rubbery modulus G' and, to a still greater extent, of mechanical loss, namely tan6. The validity of this method is also related to the fact that the dynamic method for solids is rather simple to apply with available devices and is relatively easy to automate, etc.

Transformations in the solid phase

Herewith, it is possible to realize two quite different approaches to the use of mechanical dynamic spectroscopy in the solid phase [ 11. First, it provides direct observation of the processes occurring in a homogeneous material and causing changes in its mechanical properties. Then it is possible to study reactions in a polymer introduced into a solid matrix which is practically invariant within the given temperaturehime range. The component introduced may also interact with the matrix. It is convenient to make that sort of matrix from heat-resistant polymers where no appreciable relaxation processes occur in a wide temperature range and no changes are observed either in G' or tan6.

The effectiveness of this approach was clearly demonstrated in the studies of network structure formation due to interaction of the heat-resistant aromatic polymer polyphenylquinoxaline into which a low-molecular chemically active add-on was introduced. Changes in the dynamic properties of this system over a wide temperature range clearly indicated that a structural transformation was occurring in the solid phase during interaction of the add-on with the polymer and specified the temperature region where the reaction of structuring takes place, etc. [2].

One more very important example of rheokinetic studies of transformations occurring in the solid phase is ring formation in the main chain. Although, as a rule, the reaction of structuring continuing in the glassy state weakly affects the mechanical properties of the material and is extremely difficult to detect with rheological methods, yet if profound chemical transformations take place within the glassy state, it leads to appreciable changes in the rheological properties of the polymer. It is essential, however, to choose a parameter to be the most sensitive to occurring alterations of polymer structure.

Thus, measurements of the time dependence of tan6 of the polyetheramide film based on anilinephthalein and 2,5-dicarbomethoxy terephthaloylchloride with a strictly stable heating rate revealed changes in tan6 occurring due to ring formation and correlating with the growing rate of ring formation [3]. It proved particularly efficient, in this respect, to compare the area S



274 Chapter 4: Transformations in Polymeric Systems

occupied by the dependence plot tan6 vs T with the degree of ring formation enabling the latter to be determined with an error not exceeding f 3%. The corresponding calibration plot for poly- imide obtained through ring formation in polyamidoether based on the above-mentioned compounds is shown in Fig. 4.1.1. Polyimides of different chemical structure are likely to have their own correlations of the same type.

I

Y

60

40

7

Figure 4.1.1: Dependence of the area S (in rel. units) upon degree of cyclization X .

4.1.2 Photo-viscous Effect

Changes in the rheological properties during polymer-analogous transformations are quite originally manifested in the photo viscous effect caused by the cis-trans-isomerization of the photochrome groups present in the backbone or side chain.

The photo-viscous effect belongs to the group of photomechanical effects. Another typical case is the change of sample size under the action of light observed, for example, in the network polymer obtained by copolymerization of ethylacrilate with the bifunctional azoaromatic binding agent 4,4-dimethacryloylaminoazobenzene [4]. In both cases, the photomechanical effect is caused by the change of the macromolecule length due to cis-trans-isomerization of the chromophore as the length of the chain fragment in trans-configuration is greater than that in the cis-form. This is shown in the scheme (from [4]):

--R

4.1 Polymer-analogous Transformations 275

The transition from the trans-form to the cis-form is initiated by radiation of the wavelength h = 350-380 nm (maximum absorption by the trans-form is at 365 nm): the reverse transition takes place at the radiation wavelength h = 425-450 nm (max. at 436 nm). Similar observations were also made in the studies of the photomechanical effect in spiropyrans [5] though in this case the transformation mechanism is different compared with azoaromatic chromophores.

The photomechanical effect is not only an interesting manifestation of relationships between rheological properties and fine peculiarities of the polymer structure, but also an important example of direct conversion of light energy into mechanical work. Length changes in the single chains containing photochrome groups caused by radiation are directly reflected in the results of measurements of the relative viscosity q, in diluted solutions. These measurements were made for different polyamides where azobenzene groups in the main chain were added in the meta- or para-position [6,7].

The measurements [6] showed that the q, values determined in the light were lower than the q, values found in darkness, the difference being more pronounced in case of the para- polymer. This effect is linked with the linear structure of the para-polymer whereas the meta- polymer has a spiral structure. The radiation results in trans-cis-transition reducing the chain length, whereby the relative change is greater in the flat rather then in the spiral conformation.

A typical feature of the photo-viscous effect is its complete reversibility depending on radiation conditions. This is shown in Fig. 4.1.2. where q, changes are compared with the relative content cp of transforms of azobenzene groups in the chain of aromatic polyamide (from [7]). It evidently shows a correlation not only between the wavelength and the solution viscosity, but also between viscosity and cp.

-I . U 50 f0D f50 200

t , min

Figure 4.1.2: Cyclic variation of the relative content of cis-azobenzene groups in the chain (a) and viscosig of polyamide solution in dimethyl acetamide (b) Ultraviolet ( I - A, = 350-410 nm) and visible (II - ;2, > 470 nm) light. T = 20 "C. Concentration of the solution 0.9 g/dl[7].


In all cases, the isomerization kinetics are described by a first-order equation, and the temperature dependence of the transition rate constant can be characterized either by the amount of activation energy or by the WLF equation [4] if we assume isomerization to be a relaxation process the rate of which is determined by “free volume”. It is essential that the constants of rate and activation energy are practically independent of the nature of the polymer chain on the whole and polymer concentration in the solution; moreover they remain the same both for the polymer and for its low-molecular analogue [7]. However, the absolute value of the photo-viscous effect naturally depends on the chain structure as it affects the relative content of isomer- izing (and changing their length) photochrome groups.

The relation between the change in the solution viscosity and the kinetics of the configuration transition in the chain is determined by a direct correlation between the chain length and the solution viscosity. Since the chain length depends on the relative shape of transformed photochrome groups, the solution viscosity is expected to change as well. However, the relation between the chain length and viscosity is non-linear which complicates the interpretation of the results of rheological measurements. Meanwhile, if we know the analytic form of this relation (possibly from the general theoretical pattern of correlation between the chain length and the polymer viscosity), it will enable us to obtain a quantitative kinetic interpretation of time- dependent viscosity changes under the photo-viscous effect in terms of a kinetic equation determining the rate and the extent of the configuration transition.

The photo-viscous effect can apparently be observed in polymers (rather copolymers) of most different structure provided photochrome groups are introduced into the macromolecule. Thus, the viscosity change caused by azo-group isomerization during UV radiation was registered, for example, in the copolymer of maleinic anhydride and styrene containing photochrome groups in the side chains or in the cross links forming the net (in the latter case, it refers to gels) [8]. The rate of photochrome transformations depends obviously on the system nature - the reaction kinetics conditioned by steric obstacles, i. e. by the chemical structure of the main chain, environment viscosity, the photochrome group position in the macromolecule, etc. In this respect, the analysis of viscosity changes is found to be a convenient and sensitive tool to compare isomerization kinetics in polymers of different structure. It should be noted that the studies of the photo-viscous effect is a rather new trend in rheokinetics and here unexpected phenomena may be detected giving rise to new ways of their practical applications.


1.

2.

3.

4.

5.

6.

7.

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4.2 Gelation (Curing in Solution) 277

4.2 Gelation (Curing in Solution)

4.2.1 General Remarks

The term “gelation” was used above to denote the process of fluidity loss in oligomers due to formation of a network of chemical bonds. In scientific and technical literature this term is also used in a different though related meaning, namely, to describe fluidity loss in a solution or a plasticized melt independent of its reasons. It is this meaning of the term gelation that will be used in the present section of the book devoted to the results of rheokinetic studies of fluidity loss in polymer systems containing a solvent. Also, the initial systems are homogeneous, but in a number of cases they disintegrate into phases which, for example, are accompanied by the release of the excessive amount of solvent incompatible with the polymer after the completion of curing.

In earlier literature sources, especially those dealing with technological problems, the term “solution gelation” sometimes meant a simple rise of solution viscosity, e. g. due to increased polymer concentration resulting from solvent removal and not connected with any physical transformations and/or chemical reactions. In fact, there is no fluidity loss in this case, and no gel is formed. Therefore this process cannot be regarded as gelation.

The mechanism of the solvent gelation (fluidity loss) can be linked with various phenomena. In general, we may distinguish two fundamental reasons for this phenomenon:

First, in the course of chemical reactions the polymer may acquire a three-dimensional network of chemical bonds which will eventually lead to the formation of a swollen elastomer or an extensively plasticized structured oligomer. Practically this mechanism, except in the presence of large or small amounts of a low-molecular solvent, is quite similar to the one described in Chapter 3 for oligomer curing which in our case occurs in a solution.

Second, the fluidity loss can be caused by the formation of strong physical bonds. They include crystallization which may occur in small volumes and form microcristals acting as stable points of the three-dimensional network. This network may also be formed due to appearance of clusters where macromolecules are linked with strong bonds, e. g. hydrogen bonds.

Thus, we can distinguish the “chemical” and the “physical” gelation mechanisms having wide variations in the strength of bonds at the point of the structural network formed in the course of gelation and in their lifetime. Different mechanisms and a variety of possible bonds specify both the gelation rheokinetics and the properties of the gels formed.

An unambiguous interpretation of the gelation mechanism is by far not always feasible. Nonetheless, the analogy of the processes occurring in the course of gelation of different systems and the similarity of properties in the obtained products allow the sol-gel transition to be treated on the general rheokinetic foundations considering gelation as one of the possible structuring ways. In many cases it enables us to ignore a particular gelation mechanism and to describe this process using the common methods of rheokinetic analysis and the techniques of qualitative evaluation of the obtained results.


The initial gel-forming system is a solution where macromolecules can be linked by a fluctuation network with limited lifetime of the bonds. This system flows at extremely small stresses and so can be considered a real liquid. Like any other polymer solutions, it typically shows the effect of non-Newtonian flow. As stronger bonds are being formed at the initial gelation stages, the fluidity limit T~ appears and the medium viscosity in its flow increases. Thereby flow is possible at stresses exceeding the fluidity limit (T > zY ). In terms of rheology, this system should be regarded as viscoplastic. Further structuring eventually leads to complete loss of fluidity and to gel formation. A typical example of changing rheological properties of a solution turning to a gel is shown in Fig. 4.2.1 (from [l]) for the system polysulfonamidedimethylacetamide. It is seen that at the initial gelation stages, viscosity increases sharply in the region of low shear rates, whereas in the region of high stresses destroying the forming structure it increases negligibly. Thereby, the problem of the moment since which the fluidity limit can be definitely detected still remains disputable (namely, dependent on precision of measurements and techniques applied). At any rate, if we use Fig. 4.2.1 curves 2 and 3 (for gelation time up to 32 h) do not seems to show it, but starting from curve 4 (at t > 48h) it is definitely discerned and its absolute value evidently goes up with gelation time.

O f 2 3 4

Figure 4.2.1: Change of rheological properties of a 12% solution of polysulfonamide in dimethylacetamide containing 6% water in the process of gelation at ambient temperature. Time (h) < 22 ( I ) , 29 (2), 32 (3), 48 (4), 53 (S), 73 (6).

Thus, the lower limit (in terms of low shear stresses) for the fluidity of a system turning to a gel is related to the fluidity limit T ~ . However, the very notion of a fluidity limit has a kinetic nature, i. e. the structure destruction impeding the flow is time dependent as the structural bonds possess some specific "durability". Therefore defining the fluidity limit is linked not only with the absolute stress level but also with the time during which the stresses are applied. In this sense, the zY values evaluated by low vertical regions of the curves in Fig. 4.2.1 constitute the upper limit for values of destructive stresses. As it is shown by direct observations [2, 31, the flow may actually develop when the stresses applied are below i y . It takes the course shown in Fig. 4.2.2 (see examples with numerical data in [2]). The higher the stress zi (zl < T* < T~ < z4)


the earlier the flow begins (z* > Z* > T* > z*J , i. e. irreversible deformations y, start to develop. On exceeding the fluidity limit, viscosity, i.e. the value (z i / ( d y f / d t ) ) may remain constant or decrease with growing shear stress.

2s 75 125 t, min

Figure 4.2.2: Dependence of irreversible deformations yf (of viscous flow) on time at dlfSerent tangent stresses zxlO-*, [Pa]: 2 ( I ) , 5 (2), I0 (3), 20 (4).

Loci tT , i. e. the portions cut by yf(t) dependence plots on the abscissa, are the stress time (at different stresses), for which no flow is observed, i. e. all deformations are reversible, and the material should be regarded as elastic and not liquid. The value tT evidently means system structure durability, the destruction of which is necessary for flow to begin. In this respect, t* in Fig. 4.2.2 is similar to durability of solids. The difference is that the solids, when their lifetime has elapsed, disintegrate whereas the structured gels begin to flow, i. e. a transition of the gel into the sol takes place. From the above said it is clear that this transition is time dependent, i. e. it has a kinetic nature.

The dependence t* (z) for gels appears to be very weak. It is well-known that for very rigid plastics the dependence is exponential: t* - e ; for elastomers it is weaker and of exponential nature: t* - z” (where a and n are constants). An example of the t*(z) dependence for a gel is shown in Fig. 4.2.3 [ 2 ] , which demonstrates its logarithmic nature: t - a In T (a is the empirical constant). The form of the t* (T) dependence reveals an evident difference between the “rigid” structure of plastics and the “soft” structure of gels.

ar

C

*G * ‘ -&

60

GO

2 0

Figure 4.2.3: Dependence of lifetime of asphaltene structural network in the region of gel-sol transition on shear stress.


Thus, on the “7-t” coordinates the gel-sol transition is limited from below by the zy(t)

dependence and underlying narrow region of kinetic structure destruction. Besides, it seems possible to evaluate a certain stress z0 smaller than 7 below which no sensible stress times lead to destruction of the structure and the gel-sol-transition as its reflection.

Y

However, the flow of a solution at z > zy is possible at any stress values. It is well known [4] that at high deformation rates a relaxation transition from the fluid to the rubbery state occurs due to the processing time (at high deformation rates) lagging behind the lifetime of fluctuation bonds of macromolecular chains in the solution.

The criterion for this relaxation transition is attaining a critical value of Debora’s criterion

De = v e (where ’f is the shear rate, 0 is the characteristic time of solution relaxation). A different form of this criterion is the relation of shear stress to rubbery modulus De* = z* / G ‘ = const., where T* is the stress resulting in the transition from the fluid to the rubbery state. In the course of gelation, i. e. with increasing density of bond network, the rubbery modulus also in-

creases and, since De* = const., the critical stress T* must increase as well.

All what was said above concerning the critical stresses limiting the lifetime of different solution states during gelation is shown for one definite instance in Fig. 4.2.4 using the same

initial experimental data as shown in Fig. 4.2.2. This scheme does not indicate the region of

small times: The z* ( t ) line is easy to extend to t = 0 as it is not difficult to determine the De* value for the initial solution. But it is quite obscure how the z ( t ) and zO(t) lines look at small

t values since their position depends on the quality of the measuring devices, which always has a limited sensibility in the range of low stresses. For this reason, the entire diagram in Fig. 4.2.4. is plotted for the region t > 40 min.

Y

t, min

Figure 4.2.4: Change of characteristic (critical) stresses in the course of gelation of a

10% solution of polysulfonamide in the system “dimethylacetamide-water ”

(94 : 6) at 22 “C.

4.2 Gelation (Curing in Solution) 28 1

Prime attention in the diagram of the physical states of the solution turning to a gel is drawn to three main states denoted in Fig. 4.2.4 as I, 11,111, the transition region (section-lined) and a specially marked upper critical point - pole at t t where the three plots meet.

Let us specify the physical sense of these regions.

Region I is a gel-like solution state where the structural network is retained however long.

Region I1 is the state of a fluid solution.

Region 111 is a rubbery solution state.

Section-lined is the transition region where, depending on the combination of stress level and duration, the system has either a solution (i. e. fluid) or a gel (i. e. reversibly deformable) state. A specific sense of the critical point tt is that at t = t$ complete gelation occurs, i. e. the solution turns to a gel, and the sol-gel transition is no longer feasible under any conditions.

In fact, the point tt does not differ from the gel-point in oligomer- and, even more so, elastomer-curing. The attempts to exceed the stress limits result not in destruction of some inner bonds thus enabling the material to flow but in a cohesion break (theory admits a situation when instead of the cohesion break an adhesion separation of the sample from the wall of the measuring device takes place, but in any case it will be a macro break and no breaks of molecular bonds which provide the possibility of irreversible deformations - the flow).

In the gel-solution transition region the destruction degree of initial substance structure may vary widely. Consequently, the volume flow of the “gel with a partially destroyed structure” or the “partially structured solution” is always accompanied by sliding along the walls, the flow- slide ratio depending on the degree of destruction of the material structure.

The above-said evidently shows that the idea of a “gel” refers not only to a substance state but also very much to its behaviour, which is not always the same. Moreover, the boundary between the states where a two-component system can be defined as “a solution” or “a gel” also depends on the time of forming a structural framework and on the duration of the stress applied capable to destroy the structural framework (a three-dimensional network of bonds of various nature). This reflects a situation common for polymer materials when a substance in one physical state behaves as if it were in another state (e. g. rubber at high deformation rates behaves like a glassy body, and a melt like a cross-linked elastomer).

In all cases the determining factor is the relaxation mechanism of polymer deformation quantitatively expressed by the value of Debora’s number.

4.2.2 Antithixotropy During Gel Formation

The sensitivity of region boundaries for different states of the two-component system “polymer-solvent” to the deformation regime and, particularly, the dependence of the sol-gel transition conditions on shear stress and/or on rate reflects a more general effect than the intensity of the mechanical action has on the phase transition. This effect is possible due to (slow) deformation of the solution causing changes in its macromolecule conformations and, eventually, in the entropy and internal energy of the system. This manifests itself in accumulation of reversible deformations during the flow and the growing elastic properties of the medium.


Consequently, from common thermodynamic grounds specifying phase transition

first, the equilibrium temperature of the phase transition and,

second, the transition kinetics at a given temperature,

sirice deformation affects its difference from the equilibrium (under selected deformation

conditions, the deformation should influence

conditions) transition temperature.

Here we should distinguish two quite different situations. The first has been considered above. It implies that deformation “liquefies” the system up to gel-sol transition with a subsequent viscosity drop in the non-Newtonian flow region. At relaxation the initial material structure is restored and the solution returns to the gel state. Otherwise, here typical thixotropic phenomena occur.

However, a totally different, quite opposite situation is possible. If macromolecules in the solution allow specific interactions, deformation intensifies this effect due to unwinding molecular coils and enhancing accessibility of the groups forming strong intermolecular bonds. All this is especially typical of the polymers capable to form salt and hydrogen bonds. In this situation, deformation causes solution thickening or a sol-gel transition. This case can be defined as antithixotropy leading to gel formation.

Apparently, the most typical objects revealing these “antithixotropic” properties are the solution of polymethacrylic acid and its derivatives for which these effects were discovered long ago [5] and have been observed by many researchers. Among other cases of the similar behaviour of polymer solutions, the most important is probably the phenomenon of mechanical denaturation of solutions of biopolymers, however, the most systematic observations were made with solutions of polymethacrylic acid.

The manifestation of “antithixotropic” behaviour depends on conditions of observation, namely, on whether the viscosity change and other rheological parameters of material properties are considered at a constant stress (z = const) or at a constant shear rate (7 = const.). Thus, it was shown [6] that during shear deformation of a relatively diluted (1.4%) solution of polymethacrylic acid at z = const a rather durable increase in the relative viscosity q,, is observed, which eventually reaches the plateau before the viscosity goes down.

Different changes of qrel were observed for deformation with y = const. In this case a gradual continuous growth of qrel was observed with time t . For a rather long period of time qrel changed proportionally to t l ,*. Naturally, with growing solution viscosity the condition 7 = const. could be only realized with a monotone increase of shear stresses. This effect is strongly dependent on deformation rate growing weaker with increasing 7 i. e. relatively low shear rates have the most efficient thickening effect [6].

The described phenomena are clearly related to structure formation in the solution. The determining role of solution concentration and polymer molecular mass is seen from the results of such observations: It was shown that the longer the chain the lower the critical concentration of the solution, from which the deformation thickening could be detected [6]. The quantitative regularities of this phenomenon resemble conditions for the formation of the “engagement net”


in the transition from dilute to concentrated solutions which is one of the possible forms of structure formation in a system. It is also confirmed by the influence of certain low-molecular add-ons introduced into the solution which either promote or impede macromolecule structuring. Specific mechanisms of structure formation in the solution can be quite different - from forming a fluctuation network with relatively durable points in the engagement net, with a relaxation nature of their formation and destruction, to the formation of a high order phase in the form of a solution in a liquid-crystal state. Accordingly, external manifestations of rheological properties of systems in the region of phase transitions andor intensive structure formation are quite varied.

4.2.3 The Role of the Solvent

Analysing the rheokinetics and rheological properties of gels it is necessary to take the role of the solvent into account since the formation possibility and properties of gels are largely determined by specific interactions “polymer-solvent”.

Here two factors are essential: Concentration and nature of solvent.

Although it is possible to provide analytic descriptions for concentration dependence of rheological properties of gels in most different ways, we refer, as a very typical example, to the experimental results of rheological investigations of gels based on polydimethylsiloxane rubber plasticizer by an oligomer silicone liquid with very close chemical compositions and affinity. The stationary state of this system is that of gel and a mechanical action enables it to perform a gel-sol transition. Therefore the rheological behaviour can be conveniently represented by three parameters: The fluidity limit z the high elasticity modulus Go at low stresses (z < z ) and the viscosity q at high stresses (z > z ). The dependencies of these three parameters on the gel concentration illustrate explicitly the role of a low-molecular (or oligomer) solvent. The following dependencies were established [7]:

Y ’ Y

Y

zty = A e x p [q(l.O-O.l58In q,)]

q = B exp [q ( 1.73 - 0.28 In q,) ]

G = Cexp [ q ( l . l -0.18In q,)]

where A, B, C are constants and qo is the solvent viscosity.

Evidently, all the three functions are expressed by exponential dependencies:

zy - A exp (1.09)

q - B exp (1.739)

G - C e x p (1.19)

(4.2.1)

(4.2.2)

In some literature sources, exponents are used for some of these dependencies rather than exponential functions. This replacement is of no considerable significance. Yet it should be considered that the effect of concentration on the rheological properties of gels is very strong.


Analysis of the gel structure and properties should provide an answer to the following question: what is the role of the solvent nature in a particular reaction? The most convenient objects to clear out the situation are regular networks obtained by side bonding of reactive oligomers. These networks approach homogeneous ones most closely. In [8] it was reported that the networks were obtained from 2-13,w-di hydroxy-(oligodimethylsiloxane) bound with tetraethoxysilane. The synthesis was carried out in two solvents: Toluene and low-molecular polydimethylsiloxane. Both of these solvents are rather similar in quality and present 0-solvents for the oligomer. As a result, the solvent turned out to have no effect on gel properties. The initial concentration played a certain role, as it determined the average distance between the points of the network - M , . The M, value almost corresponding to the chemical structure of the initial oligomers was attained only if the gel synthesis was carried out in a 20% solution. In all other cases deviations from the theoretical value of M , caused by faults in the network structure were observed (e. g. incomplete conversion of hydroxyl end groups formation of macromolecular cycles).

The effect of concentration of the initial solution on the properties of the forming gel was vividly demonstrated in [9], where the networks were obtained by polycyclotrimerization of diisocyanates in solutions. It was found that as the initial concentration decreased, an increase in gelling and a decrease of the rubbery modulus of the forming networks were observed. This effect reflects growing probability of formation of closed macromolecular cycles in the microgel form (i.e. transition to a heterogeneous mechanism of network formation) in the course of diluting a solution. These macro cycles are not capable of being points of a continuous network, though at the finishing binding stages they may enter the whole net. Thus, this example demonstrates clearly that the concentration of the initial solution during gelation influences the degree of heterogeneity of the process and, eventually, the topology (and properties) of the forming networks.

The nature of the solvent used for gel synthesis also plays a considerable role. The peculiarities of gel topology (and properties) are seen from sharp differences in the nature of the concentration dependence of the rubbery modulus for the gels produced in different solutions [lo]. For networks produced without a solvent (and probably in the 0-solvent) the gel elasticity

modulus is proportional to the polymer concentration as 1/3: G - Vi’3 which agrees with the

statistic theory of high elasticity. At the same time for gels produced with large quantities of a

good solvent G - Vi.7 . Evidently, we have a striking difference in exponents (0.33 and 1.7). This

difference should be linked with network topology, namely, with the growing role of intramolecular cyclization in dilute solutions.

It should also be emphasized that, i n all these cases, attention should be drawn not to the solvent nature itself, but to the solvent’s influence on rate interaction of competing reactions of network formation and intramolecular cyclization. An example illustrating this argument is shown in Fig. 4.2.5 (from [ 1 11). The first case refers to siloxane networks. Here intramolecular cyclization is enhanced in poor solvents. Therefore introduction of acetonitril impairing solvent quality into the reaction mixture results in a drop of the rubbery modulus due to increased average distance effective between the network points - M, (Fig. 4.2.5 a). The second case refers to poly- cyanurates where the effect of macroring formation, on the contrary, increases with the improve- ment of solvent quality. Therefore, as is seen from Fig. 4.2.5 b, using poor solvents results in brittle gels whereas good solvents (N-methylpyrrolydol) yield gels capable to develop large deformations.


G, KPa 0. 8 r U

2oc* % f0 0

Figure 4.2.5: Effect of solvent nature on rheological properties of gels; a ) reduction of high elasticity modulus of siloxane gels with increasing the content (c., %) of acetonitril in the reaction mixture; b) creep ( I ) of polyisocyanurate gels produced with bad - anisoE(1) and good - N-methylpyrrolidone (2) solvents.

One of the most disputable points concerning gel formation is the question of whether this process is homogeneous or heterogeneous. The latter situation, as was shown with a number of oligomers (see Sections 3.4 ), is rather typical of their structuring. This is indicated e. g., by the independence of gelation time t* from the “solution” concentration in curing melaminoformaldehyde resins (Section 3.4).

Curing heterogeneity of these resins can be caused by their high polyfunctionality. Therefore it is interesting to examine the other extreme case, i. e. gel formation from regular networks. These networks are produced by bonding chains having functional groups only at their ends. An often cited example are the interaction products of a,w-dihydroxy-(oligodimethylsiloxane) and tetraethoxysilane. As is shown in Fig. 4.2.6, the kinetics of viscosity growth in this case largely depend not only on bound agent concentration (tetraethoxysilane) but also on the initial solution concentration. Thus, it is quite evident from comparison of curves 2 and 4, or 3 and 5 where bound agent concentrations are close whereas the oligomer content in the solution is about 10 times different. Reducing oligomer concentration evidently leads to a sharp increase of induction time.

Figure 4.2.6: Rheokinetic curing curves for a, w -dihydroxy(oligodirnethylsiloxane) without a solvent (1-3) and with a solvent (4-7), with oligomer content in the solution = 0.094 v01.-%. The content of binding agentper 1 mole oligomer: 0.9 ( I ) ; 1.7 (2 ) ; 3.4 (3); 2 (4); 5 (5); 8 (6), 16 (7) moles.


It is essential to note that considering the effect of concentration and solvent nature on gelation rheokinetics and properties of the gel formed, a decisive role is to be attributed not to concentration itself, but to the structure existing in the initial solution. In this situation, naturally, the concentration effect can not be gradual, i. e. the process rate should not be expected to vary with concentration. On the contrary, as shown in previous sections of the book (see Section 2.3), the concentration dependence of rheological properties of polymer solutions always has a critical point (or a narrow region) where a physical engagement network appears. It is natural to expect this effect to influence gelation rheokinetics as well.

In this respect, rather typical are the data obtained for the system “polyvinyl alcohol-water- salt (NaCI)” with a polymer content of 5-15% [ 121. The basic experimental results are shown in Fig. 4.2.7 where data are compared for the solutions with cp = 9 and 12% and with different salt content. This relatively small difference in the solution concentration invokes a sharp change both in gelation kinetics and in the salt role. 9% solutions appear to be practically stable and show no tendency to form a gel. Addition of NaCI, in this case, accelerates the process in no way though the presence of a salt should undoubtedly enhance hydrophobic interactions between macromolecules.

P0.S

40 80 t , days

Figure 4.2.7: Change of maximum Newtonian viscosiq in a 9% (1-3) end 12% (4-6) water-salt solution of polyvinyl alcohol in the course of ageing. NaCl content, [mass-%]: 0 (1, 4) ; 10 (2 , 5); I5 (3, 6). Temperature 25 “C.

For 12% solutions the situation alters conspicuously intensive structure formation is observed with a substantial viscosity growth and an added salt accelerates this process. Besides it should be noted that 9% solutions are not capable of large reversible deformations whereas 12% solutions show considerable rubbery elasticity.


The observed principal difference in ageing rheokinetics of 9% and 12% solutions is naturally explained by the fact that cp = 9% < cp* and cp = 12% > cp* where cp* is a critical concentration corresponding to a sharp change in the nature of the dependence q(cp) due to formation of the network of intermolecular functional bonds in the solution. A solution with cp < cp* is not capable to form a gel at all (as it has no continuous network) and a solution with cp > cp* turns to a gel in the course of ageing. Respectively in the first case adding NaCl leads to enlarging molecular aggregates and possibly to their enhanced stability as colloidal particles. But, because of an insufficient number of contacts between these aggregates no gel is formed. In the second case, adding NaCl promotes structuring in the system as a whole.

Stating the mechanism of structure formation in concentrated (cp > cp*) polymer solutions makes it possible to indicate the way of system stabilization, which is the removal of intermolecular contents by introducing a diphilic solvent. In the referred work this was demonstrated by introducing isopropyl alcohol and isobutyric acid into the solution. These two substances improve thermodynamic properties of the solvent (water) and accordingly reduce macromolecule interactions in polyvinyl alcohol which results in sharp inhibition of solution gelation.

The problem of intensifying structure formation in a solution with cp < cp* is solved by introduction of salts of polyvalent metals, e. g., chromium acetate (111). At its certain concentration cpcr a solution with cp < cp* turns into a gel. Evidently, the lower the polymer concentration (p the higher should be the concentration qCr. This effect is shown in Fig. 4.2.8, where gelation at a certain critical concentration of chromium acetate cpz, is vividly seen [12]. It was found that for cp ranging from 5 to 8% a critical value for cp:, at which the solution gelation starts is empirically expressed by the formula cp:, # 19.1-1.84 reflecting a drop in cpzr with growing cp.

7

5 f

3

I

- I

Figure 4.2.8: Dependence of maximum Newtonian viscosiv in solutions of polyvinyl alcohol containing 15% NaCl on concentration cpz, of chromium acetate. Content ofpolyvinyl alcohol in the solution [wt-%]: 5 ( I ) , 6 (2), 7 (3), 8 (4). T =25 "C. NaCl content 15% of the weight of polyvinyl alcohol.


The considered experimental results show clearly the effect of the initial solution structure on gelation rheokinetics.

While studying gelation of polyvinyl alcohol solutions (aqueous, with different salt content) special consideration was given to the problem of a phase separation mechanism in this system [13]. The experimental data shown in Fig. 4.2.9 are of great significance for under- standing the process mechanism. The most important fact reflected in this figure is the presence of maxima in the temperature dependencies of viscosity in solutions turning to gels.

I I I I I

50 60 70 80 90 T o C

Figure 4.2.9: Temperature dependence of viscosity of aqueous (1) and water-salt solu-

tions of polyvinyl alcohol. Salt content 8 mol-%. Salts: KI ( 2 ) or NaCl(3,4). Polyvinyl alcohol content (wt-%): 18 (lh 16 (2), 15 (3), 12 (4).

1 Vertical line marks temperatures of phase transition. Shear rate = 0.19 s- .

The presence of these maxima is known to be linked with phase transitions (the phase transition temperature Tph is marked with the vertical dashed line in the figure). Maxima of this type were reported long ago for the region of amorphous separation.


An analogous effect caused by melting of the liquid-crystalline phase is also known to scientists. Can it form occur in systems based on polyvinyl alcohol? Using direct structural methods gave no evidence of any traits of the crystalline phase. However, its content may fall beyond the limit of method resolution. Therefore, although the existence of phase transition at Tph is doubtless, the problem of its nature remains unsolved.

Naturally, the gelation process may occur only in the region below the separation temperature on the phase diagram for the system of the given composition. In this respect, the very fact of constancy and variability of rheological properties of two component systems at a given temperature reflects its phase state. Meanwhile, the phase transition and subsequent thermodynamic processes can be quite complex which will manifest itself in different rheokinetic peculiarities of the material behaviour. Investigation of results for phase separation kinetics in the system “cellulose triacetate-nitromethane” turning to a gel [ 141 are characteristic for this behaviour. In the region of compatibility, this pair of substances forms stable solutions with a time-constant viscosity dependence on shear rate q(y) which, for different concentrations, obeys regularities common for polymer solutions. However, as the temperature decreases (and transcends the binodal), the q(v) dependence changes with time as is shown in Fig. 4.2.10 for a 5% solution. The transition from curve 1 to curve 6 is the same as that shown in Fig. 4.2.1 (although in [14] z was not reached in the low stress region nor was the fluidity limit shown in Fig. 4.2.1 definitely stated). Nevertheless, in subsequent thermosetting a new effect was found: Viscosity drop during transition from curve 6 to curve 7 as shown in Fig. 4.2.10 by a dashed line, i. e. the q ( t ) dependence at different shear rates is of extreme nature.

Y

f- f 2 3

p.8 (r/p0) Figure 4.2.10: Change of viscous properties during phase separation of a 5% solution of

cellulose triacetate in nitromethane at 40 “C with time 0.25 (I), 1 (2), 2 (3), 3 (4), 4 (5), 5 (6) and 6 h (7) afier the beginning of measurements.

Generally, these viscosity measurements are connected with the mechanism of incomplete phase separation governing the gelation process. Yet there is a possibility of another far more effective mechanism of viscosity reduction determined by the transition to a high-order (liquid- crystalline) solution state as a result of deformations in cooling, because viscosity of an anisot- ropy solution is much lower than that of an isotropic one [15].


Apparently, this mechanism can provide a real explanation for the change of the q(v) dependence during gelation of the 7% solution of cellulose triacetate in nitromethane at 35 "C shown in Fig. 4.2.11 [ 141. This case seems to correspond to a state located farther from the binodal compared with the 5% solution at 40 "C with the phase transition rheokinetics shown in Fig. 4.2.10. In Fig. 4.2.1 1 during phase separation the fluidity level first goes slightly down as the values T in this system are small. Thereby, however, a sharp drop in viscosity is observed so that viscosity measured at 35 "C appears to be even a little lower than at 70 "C. The latter is also an evidence of the system transition from the isotropic to the anisotropic state.

Y

Figure 4.2.1 I : Change of viscous properties during phase separation of a 7% solution of cellulose triacetate in nitromethane at 35 "C with time 0.25 ( I ) ; 0.5 (2); 1 (3); 2 (4), 3(5); 4(6) and 5 h (7) after the beginning of measurements.

Attention should once again be drawn to a rather sharp difference in the nature of the curves in Fig. 4.2.10 and 4.2.11 indicating a difference in the processes causing the observed rheokinetic phenomena during phase separation of polymer solutions- These phenomena typically occurr in systems consisting of the same components with minor composition and temperature variations. Thus, practical time-dependent changes in rheological properties of systems turning to gels can be caused by different overlapping mechanisms resulting from occurring physical- mechanical processes.

The above mentioned data characterizing the gelation process of a solution were obtained with the method of continuous shear deformation. As immediately follows from the meaning of the data discussed, this test procedure tends to destroy the material structure which is already enough to affect the gelation kinetics. Therefore it is natural to try a dynamic method of investigating rheological characteristics of a system as a means to study this process (as well as other structuring processes). However, this attempt is impeded by the fact that even at very small

4.2 Gelation (Curing in Solution) 29 1

deformations a solution turning to a gel behaves as non-linear viscoelastic medium. It means that fairly small deformations destroy the material structure, i. e. non-linearity is caused by a "physical" (rather than geometric, i. e. the very fact of creating large deformations) mechanism. Nevertheless, if we still try to undertake these measurements and find a way (see [16]) to determine the rubbery modulus G' and the loss modulus G" , it will be possible to observe changes of these values with time reflecting the gelation kinetics.

The characteristic results of these measurements are given in Fig. 4.2.1 comparing time dependencies of stress and the dynamic modulus components G' and G" for solutions of polysulfonamide in dimethylacetamide with different water content in the system. A certain correlation is clearly observed between the course of the functions z(t) , G'(t) and G"(t) with the analogous shift of all the there dependencies for different water content in the system.

fao zoo 300 t , hours

Figure 4.2.12: Change of rheological parameters at T at v= 0.1 s-' and G', G" at o = 0.63 s-' (all values in Pa) during gelation ofpolysulfonamide solutions in dimethylacetamide with d$ferent water contents: 0 ( I ) , 3 ( 2 ) and 6% (3).

The kinetic curves enable us to distinguish three stages of gelation:

the initial induction period when the solutions remain viscous Newtonian liquids with practically constant viscosity,

the stage of the intensive gelation process with a rapid increase of all the measured parameters;

and the final period of reaching the limit values of T, G' and G" where the forming gel is stabilized.

Respectively, we can define time constants characterizing these stages

to - the end of the induction period,

t , ,* - half-period of gelation,

t- - complete time of the process.

The agreement of kinetic parameters of the gelation process found with different methods

1.

2.

3.

is seen from comparing the time constants given in Table 4.2.1 (from Fig. 4.2.12.).


Tab. 4.2.1 : Time constants of the gelation process, obtained with different rheological parameters

Evidently, different measurement methods yield, on the whole, comparable results. There- by it should be taken into account that the curves in Fig. 4.2.12 are plotted with an error up to 15% as the reproducibility of kinetic curves cannot be complete for casual reasons. The use of integrated time characteristics of the discussed process (i. e. lo, t,,2, t , ) is reasonable because its mechanism and, accordingly its kinetics are unknown. At the same time, the values to, t,,* and t_ quite vividly render the main peculiarities of the gelation process, namely the existence of an autocatalytic induction period of viscosity growth and a tendency to saturation - reaching the limiting state of the gel.

It is rather useful to examine the kinetic curves in Fig. 4.2.12 since the evaluation of the integrated parameters to, t , ,2 and tm gives rise to some important considerations concerning the process under discussion. Thus, Fig. 4.2.12 demonstrates clearly both the role of water in gelation kinetics for the cited example and that the limiting characteristics of the formed gel do not depend on water concentration in the system. On the basis of similar measurements we can analyse the role of the determining factors in the process of gelation, such as the composition of the initial solution, the temperature, etc.

The general methodological approach to the rheokinetics analysis of the gelation process based on experimental studies of time dependencies of rheological properties remains applicable to any gel forming systems. The most typical situation here is similar to that shown in Fig. 4.2.12 with three conspicuous regions

the induction period (constant rheological properties)

the gelation (S-shaped curves of the modulus increase),

and the plateau of properties of the completion of the process.

However, this typical example reflects only one of the possible cases where all the process stages are distinctly separated.

The reality is more variable and gives a rather wide range of possible situations differing in depth of structuring processes. Thus, i t is possible to consider the case when the process never comes to an end (the formation of a non-fluid gel) but results in raising viscosity of the initial solution to a new level [ 171. This case should be described more correctly not as gelation (since the fluidity of the solution is preserved) but as structure formation within the solution state.


An example of a more profound course of the process was considered above when a sol-gel transition was possible depending on the deformation regime, i. e. the bonds forming the gel structure were not very strong.

Finally, the other extreme case is possible: The process of gelation does not stop at the moment of fluidity loss and continues within a gel-like state. In fact, that is how the curing of binders occurs if the gel point lies below the glass transition temperature (Section 3.1). In this case the interaction possibility for reactive groups is preserved and new intermolecular bonds are formed beyond the gel point. It is all the more typical of unvitrified gels. In this case the rubbery modulus of the solution gradually increases for a very long period of time. An example of this gel "maturing" is shown in Fig. 4.2.13 (from [ 181) for a technical product based on gelatine. The change of the rubbery modulus G' in concentrated (6 and 205) gels appears to have two stages: initially G' shows a rather rapid growth, then the growth rate of G' decreases, i. e. the process mechanisms at the beginning and at the end of gelation are quite different. We can assume that a bend in the dependence G'(t) corresponds to the moment of forming a continuous network of chemical bonds excluding the flow and inhibiting a further course of the chemical reaction. It is also possible that in the region of the bend the network density reaches the value when the system undergoes a relaxation transition limiting the mobility of macromolecules which contain reactive groups. However, any of these (or other) assumptions would require more independent research.

T I

I I

r / 1

t\

4 8

5

0

t. hours

Figure 4.2.13: Change of the rubbery modulus G' in the course of aging gelatine gels. Polymer content in the systems: 2 ( 1 ) ; 6 (2, 2') and 20% (3). Curves 1 and 2 were obtained at 20 "C, 2 and 3 at 25 "C. Curve 2' refers to the system containing additionally 4% surfactant.

Many of the above-given experimental data prove high sensitivity of the rheokinetic method not only to solution concentration and selection of solvents but also to the presence of certain ingredients even if their amounts very small. This is especially important for structures forming low-molecular add-ons (salts) and surfactants. In this respect, the application of rheokinetic methods may prove a very useful technological procedure to select an optimal composition for different uses and to control the process of gelation.


4.2.4 Gelation Kinetics

Quantitative descriptions of the gelation process in available publications were based on purely empirical approaches, which were not connected with evaluating the rate of inducing and accompanying chemical reactions. Generally these were formal descriptions of time-dependent changes of some measured characteristics of the rheological behaviour of a material - viscosity, rubbery modulus, etc. Certainly the gelation kinetics are determined by the rate of the reaction leading to the formation of a three-dimensional network. As it was mentioned above, it is by no means clear whether this network is formed for physical reasons (crystallization) or due to a chemical reaction. Therefore a convenient way of quantitative description of gelation kinetics is in this case the application of a rather "flexible" equation related to no particular mechanism of the process.

Accordingly the gelation kinetics can be conveniently described by Avrami equation which reflects any transformation linked with the formation and growth of a new phase. Then the time dependence of the measured parameter x (usually the rubbery modulus) is expressed by the formula:

(4.2.3)

where xo is the initial, x, the final and x the current value of the measured parameter, K and n are empirical constants. The constant n has a certain physical sense connected with the growth geometry in a new phase according to the original theory of Avrami.

The description of kinetic curves with Eq. (4.2.3) was successful, e. g., for gelation of solutions of polysulfonamide in dimethylacetate [ 11, aqueous solution of polymethacrylic acid [5] and is apparently possible in many other cases.

A similar task was discussed for the description of gelation kinetics of oligomer binders (Section 3). It was shown that a successful description of numerous kinetic curves was achieved with equations of auto catalytic type (3.5.2), the integral of which has the form:

(4.2.4)

where co and c , are empirical constants.

Here it should be stressed that Eqs. (2.4.3) and (4.2.4) describe the curves with almost identical forms, so the same experimental data are approximated equally well and practically with the same precision both by Eq. (4.2.3) and Eq. (4.2.4). Moreover, a rather strict quantitative correlation may be stated between pairs of constants K and II, on the one hand, and co and c , on the other hand. As was shown in [ 191, the closest correlation of curves described by Eqs. (4.2.3) and (4.2.4) is achieved with the following correlations between the constants:

(4.2.5)

4.2 Gelution (Curing in Solution) 295

Therefore in all cases when the Avrami equation can be applied, the equation of the autocatalytic type is valid too, and vice versa. All this is true for solution gelation (with gel formation) and oligomer binding which confirms natural affinity of the systems belonging to the both types. The above-said also raises doubts concerning possibility to reveal direct correlations between experimentally obtained constant values in Avrami equation and a specific mechanism of a new phase growth during gelation. At least, nowadays we can definitely speak only about empirical quantitative description of the process and a convenient way of comparing gelation constants for the values of experimentally found constants in the above cited equations.

4.2.5 Gelation of Plastisols

It is of great interest and practical importance to consider another field within the scope of

This process is similar to all the above considered cases being an instance of transformation

problems discussed here - gelation of polyvinylchloride (PVC) pastes (plastisols).

rheokinetics in the initially two-component (and probably two phase) system

I .

2.

- polymer-plasticizer.

The difference is that

the system in its initial state does not present a true solution (PVC is in the form of undissolved microparticles consisting of great numbers of macromolecules) and

the final state corresponds not to the formation of a non-fluid gel but, on the contrary, due to PVC gelation a homogeneous technologically uniform mass is formed, which is processable, i. e. can flow under the action of sufficiently large stresses.

However, the rheokinetic approach to gelation of PVC pastes has much in common with the analysis of any other gel-forming systems. Like in the other abovementioned cases, the studies of the process kinetics with rheological methods proved to be a rather fruitful and technically convenient technique to analyse gelation of PVC plastisols.

PVC-based plastisols exhibit very complex rheological behaviour. Thus, for instance, the dependence of the effective viscosity on shear stress may have both the “dilution” region caused by structure destruction during deformation and the “hardening” region treated as dilatance, i. e. shear-induced structuring. Still more complex is the behaviour of PVC-plastisols in transient deformation regimes when rheokinetic effects are clearly seen. In this respect a very characteristic example is provided by the data shown in Fig. 4.2.14 for different ranges of shear rate [20].


cy' 49

20

a 4

2Lu 0 - G a 20 66 IOD

8, retunit*

Figure 4.2.14: Evolution of shear stresses depending on deformation of 65% plastisol of polyvinylchloride in dicetylphthalate for regions of low-rate, 7 = 1,3 s-', thixotropic ( I ) ; dilatant 7 = 2.86 s-' (2) and high-rate, 7 = 5.1 s-', thixotropic (3) behavioul: The temperature is 25 "C.

At low shear rates (curve 1) we observe thixotropic behaviour typical of polymer solutions. As the deformation develops the stress goes through a maximum corresponding to the structure strength, then the stress goes down to the stationary value denoting viscosity of the stabilized flow (destroyed structure). Rapid subsequent deformation gives no maximum of this kind. (since the structure has been destroyed by the previous deformation) and during relaxation the sample gradually recovers (its structure is "healed).

At moderate shear rates (curve 2) a typically dilatant behaviour is observed, i. e. the shear stress gradually goes up to reach the stationary value.

Finally, at high shear rates (curve 3) the system thixotropy shows again, which also manifests itself in a sharp maximum before reaching the equilibrium values. Apparently this stress maximum is determined by destruction of the strongest structural elements which at smaller shear rates move as an integrity.

This complex thixotropic dilatant behaviour of PVC-plastisols subjected to rheokinetic studies is evidently caused by their multi-level structure which actually determines the peculiarity of their technological properties. When PVC-plastisols turn to gels, this is also accompanied by complex changes in their rheological properties due to the processes reflected by the curves in Fig. 4.2.14. Thus, structural changes in the course of gelation of PVC-plastisols are clearly observable from the nature of changes in the components of the dynamic rubbery modulus [21]. This is schematically shown in Fig. 4.2.15. It is seen that the whole process may be


divided into two stages. At the first stage the rubbery modulus increases more than 100 times against the initial value, whereas at the 2nd stage it decreases reaching the values at which PVC- plastisols become processable.

Ti me Figure 4.2.15: Nature of changes in the components of the dynamic modulus G’ ( I ) and the

storage modulus G” (2) during gelation of polyvinylchloride compositions.

A notable difference of Fig. 4.2.15 from the model curves given in Fig. 4.2.14 is that the technological studies of plasticising PVC-pastes are carried out under non-isothermal conditions. Thus, increasing G’ and G“ at the first stage of the process takes place at the temperature increase approximately from 100 “C to 150 “C. In fact, the “gelation” process starts near 75 “C. From this temperature on, the plasticiser is gradually combined with PVC forming a gel, i. e. the modulus increase reflects the transition from a dispersion of non-reactive particles to a gel. This transition leads to a viscosity increase appr. from 20 P a s (measured with a “Brookfield” viscorneter) to the values of dynamic viscosity exceeding 104 Pas. The system obtained here is similar in its characteristics to any other gels or, generally, to all structured systems, the role of points in the structural network of the PVC gel being played by microcrystallites.

However, it should be noted that a “gel” of that kind can probably flow at sufficiently high stress, i. e. like in the above mentioned case of gelation of the polysulphonic solution, the boundary between the gel and the solution is determined not only by temperature but also by the exponential factor. Moreover PVC itself without a plasticiser is known to undergo irreversible deformations below the temperature of plasting for its crystalline phase is effected by plastic deformations of crystallites and the flow of the surrounding amorphous matrix.

During further temperature increase the gel turns to the solution (i. e. plasticized PVC is formed). The maximum temperature attained under equilibrium conditions is 195 “C. The process is accompanied by a sharp viscosity drop which is abnormal for a conventional one- phase system, but quite natural here since decreasing q’ and rubbery modulus reflect the process of microcrystallite melting in the two-component system. It is also typical that the process is more vivid in terms of G‘ rather than G” .


Thus, the rheological method in its dynamic variant enables us to follow all the stages of PVC-plastisol formation (in the course of gelation) and melting. Since they are multicomponent systems with complex composition and structure, this method appears to be unique both for comparing compositions with different component content and for selection of appropriate time and temperature regimes to turn them to goods in industrial manufacture.

A similar approach can be applied for any gel-forming systems with different composition. Thus, [22] gives a detailed rheokinetic study of the sol-gel transition in the plasticized copolymer of methylmethacrylate with methacrylic acid at variable acid content (5-12%). The obtained results are shown schematically in Fig. 4.2.16 where the whole region of change in the dynamic modulus G' and the mechanical loss factor tan6 is divided into five characteristic zones corresponding to different structural processes taking place in the course of gelation. According to the approach suggested in the cited work and based both on qualitative considerations concerning the nature of the sol-gel transition and on some direct structural observations, in region I the system remains practically unchanged. In region I1 the aggregates of latex particles disintegrate which is indicated by the results of direct measurements of particle sizes in the material. As a consequence of the enlarged particle surface, the penetration of the plasticiser and their swelling increase, which is seen in region 111 in Fig. 4.2.16. Polymer plasticizing causes a drop in the temperature of glass formation, which is reflected by the appearance of a specific maximum and then leading to subsequent intensive modulus growth in region IV due to active structuring (the formation of a physical network). The gelation process finishes in region V with stabilizing the gel structure, i. e. with formation of an unchangeable material structure.

n

tan

Ti me

Figure 4.2.16: Diagram illustrating the general nature of changes in viscoelastic characteristics of acrylic copolymers in the process of gelation.

The location of extension of certain regions in the diagram naturally depends on the composition of the copolymer which governs the rate of gelation and the induction period (i. e. the extent of region I in Fig. 4.2.16). Evidently, a direct correlation should be observed between t*


determined from the results of dynamic measurements and from the lifetime ti of the plastisol evaluated viscometrically. Indeed, as is seen from Fig. 4.2.17 (from [22]), this correlation really exists for copolymers with different composition forming gels at different temperatures.

x t ,min

50

0

0 t i , min I200

is0

ioo 8 r2

Figure 4.2.Z7: The effect of content of methacrylic acid 'p,,, in its copolymers with methylmetacrylate on the ratio between the induction period ti ( I ) and lifetime t* (2) ofplastisol at 50 "C ( I ) and 60 "C (b).

To sum up, the given examples referring to gels based on polyvinylchloride and a copolymer of methylmethacrylate with methacrylic acid, clearly illustrate the possibilities, resolution capacity and practical perspectives for application of the rheokinetic method in the analysis of gelation processes of plastisols having different compositions.

4.2.6 References

1. A. Y. Malkin, L. P. Braverman, E. P. Plotnikova, S. G. Kulichikhin, Vysokomolekulyarnye soedineniya A, 1976, vol. 18, 11,2596.

A. Y. Malkin, 0. Y. Sabsai, E. L. Verebskaya, V. A. Zolotarev, G. V. Kolloidny Zhurnal, 1976, vol. 38, 1, 181.

2. 'inogradov,

3. 0. Y. Sabsai, M. P. Lukianova, A. Y. Malkin, G. V. Vinogradov, K. A. Chochua, V. P. Azovtsev, Vysokomolekulyarnye soedineniya, 1980, vol. 22, 5,384.

G. V. Vinogradov, A. Y. Malkin et al., J. Polym. Sci, A, 2, 1972, vol. 10, 6, 1061.

J. Eliassaf, A. Silberberg, A. Natchalsky, Nature, 1955, vol. 176, 11 19.

S. Ohya, T. Matsuo, J. Colloid Intei$ace Sci., 1979, vol. 68, 3,593.

A. Y. Malkin, 0. Y. Sabsai, L. P. Lukianova, K. A. Chochua, V. P. Azovtsev, Vysokomolekulyarnye soedineniya, 1976, vol. 18A, 9,203 1.

4.

5 .

6.

7.


8. L. Z. Rogovina, V. G. Vasilyev, G. L. Slonimsky, Vysokomolekulyarnye soedineniya A, 1982, vol. 24, 2,254.

L. Z. Rogovina, V. G. Vasilyev, T. M. Frenkel, V. A. Pankratov, G. S. Slonimsky, Vysokomolekulyarnye soedineniya A, 1984, vol. 26, I , 182.

Vysokomolekulyarnye soedineniya B, 1985, vol. 27, 11,803.

Mekhanika kompositnykh materialov, 1988, I , 72.

Vysokomolekulyarnye soedineniya A, 1984, vol. 26, I I , 2259.

9.

10. L. Z. Rogovina, V. G. Vasilyev, G. L. Slonimsky,

11. L. Z. Rogovina, V. G. Vasilyev, G. L. Slonimsky,

12. N. K. Kireyeva, Y. D. Semchikov, D. N. Emelyanov,

13. N. K. Kireyeva, R. A. Kamsky, D. N. Emelyanov,

14. G. N. Timofeyeva, V. M. Averyanova, Kolloidny Zhurnal, 1980, vol. 42, 2,393.

15. S. P. Papkov, V. G. Kulichikhin, V. D. Kalmykova, A. Ya. Malkin, J. Pol. Sci., Polym. Phys. Ed., 1974, vol. 12, 9, 1753.

16. A. Ya. Malkin, A, A. Askadsky, A. E. Chalykh, V. V. Kovriga, Experimental methods of polymer physics, Prentice-Hall Inc., Englewood Cliffs, N. K , 1983.

17. S. E. Shalabi, L. A. Nazaryina, L. Z. Rogovina, G. L. Gabrielyan, Vysokomolekulyarnye soedineniya A, 1979, vol. 21, 5, 1153.

18. S . D. Nagaslayeva, A. A. Trapeznikov,

19. A. Y. Malkin, I. A. Kipin, S. A. Bolgov, V. P. Begishev, Inzhenernofizichesky zhurnal, 1984, vol. 46, I , 124.

20. A. A. Trapeznikov, A. A. Frolova, Kolloidny zhurnal, 1987, vol. 49, 11,386.

21. N. Nakajima, D. W. Ward, E. A. Collins, Polymer Eng. Sci., 1979, vol. 19,210.

22. Y. G. Yanovsky, A. V. Vasin, G. Y. Vinogradov, A. V. Bulynko, Y. V. Sorochkin, M. A. Fioshina, Vysokomolekulyarnye soedineniya A, 1982, vol. 24, 12,2563.

Vysokomolekulyarnye soedineniya B, 1988, vol. 30, 3, 193.

Vysokomolekulyarnye soedineniya A, 1979, vol. 21, 4,836.

4.3 Rheokinetics of Phase Transitions 30 1

4.3 Rheokinetics of Phase Transitions

Using rheological methods for studying phase transitions implies two possible approaches:

1. measurements of dynamic characteristics (storage modulus and loss modulus) with small amplitude harmonic oscillations and,

2. viscometric measurements.

In the latter case, deformation is both the factor affecting the process kinetics and the basis for the control method. The application of either method is defined from the physical state of the material under study. The analysis of elastomer crystallization is often conducted with measurements of dynamic mechanical characteristics [l]. In the studies of phase transitions in linear polymers, viscometric measurements proved to be rather useful both for plotting phase diagrams and for identifying the initial moment of crystallization, i .e. duration of the induction period in proportion to the rate of nucleation [2,3-51.

4.3.1 Plotting Phase Diagrams

Viscous characteristics were used to plot phase diagrams of polyamide solutions with overlapping of different types of separation [6,7]. As a rule, approaching the temperature of amorphous separation leads to a considerable viscosity increase with its subsequent drop after the critical temperature is reached [S]. However, a possibility remains for phase separation not to be completed if the test time is limited, which is particularly true for concentrated solutions, and viscosity may not change appreciably. In these cases the phase state of a system is evaluated from the change in the temperature dependence of the solution viscosity [2].

The position of transition loci corresponds to the change in the nature of the temperature dependence obtained in the course of gradual cooling of the solution, since in the transition region changes occur in the flow mechanism and, consequently, in the amount of activation energy of the viscous flow E. A notable advantage of the viscometric method is that the parameter, the change of which determines a phase state of the solution and whose value is of special interest for calculations of technological facilities, is viscosity.


The principle of processing viscometric data is shown in Fig. 4.3.1, which includes available examples of different variants of logq dependencies on ( 1/T) experimentally observed for the systems discussed. Thus, PA are readily crystallized polymers and Fig. 4.3.1 a illustrates a sharp transition from a melt (or a concentrated solution) into a crystalline state, i. e. the point A belongs to liquidus. In region I (Fig. 4.3.1 a) flow of the one-phase solution is observed where the value of E is characteristic for polymer solutions. For the studied solutions, this value depending on concentration is 25-40 kJ/mol. Crystallization of a concentrated solution results in an unlimited viscosity growth indicated by the vertical starting at A.

b

I I

I / T

Figure 4.3.1: Direrent variants of dependencies logq on ( 1 / T ) a ) a crystallizing solution, b) a solution with amorphous separation, c ) a solution with LCTM and UCTM, For details see the text.

The concentration dependence of crystallization temperature exemplified in Fig. 4.3.1 a, and phase diagrams thus obtained have a very simple form: Two regions, separated by a liquidus curve. Similar diagrams are observed for systems PA-IZDMAA, PA-12-DMSO (Fig. 4.3.2).

T,

460

42 0

Q

20 60 f00 9, mass

T,

170

450

430

tJ

I I I

20 60 fa0 $0, mass

Figure 4.3.2: Phase diagram of the systems PA-I2-DMAA (a) and PA-12-DMSO (b).

4.3 Rheokinetics of Phase Transitions 303

The other more complex variant of q(T) dependence is shown in Fig. 4.3.1 b. In this case, besides the crystallization point A and the region of the one-phase solution I the dependence has a bend at B and the flow region P where E is reduced to a value practically equal to E for pure solvent (about 8- 13 kJ/mol). We may assume that at B amorphous separation of the solution occurs and then, until the crystallization point is reached, the flow of this system obeys regularities for flow of the dispersed phase, i. e. practically pure solvent. We may also assume that B lies on the binodal. This assumption for B as belonging to the binodal is confirmed by the method of optical interferometry. That is why a transition to dependencies shown schematically in Fig. 4.3.1 b corresponds to a more complicated form of the phase diagram where the binodal appears in addition to the liquidus curve. Similar phase diagrams were obtained in the range of the systems under study for solutions of PA-12 in dodecalactam (Fig. 4.3.3). Curve I in Fig. 4.3.3. is the line of the liquidus and curve 2 is the binodal. An alternative explanation may be suggested according to which components of the mixture form a chemical compound. However, this explanation is not likely to be valid as is seen from the interferometric data [ 101.

fa0

170

f60

20 60 ioa 9, moss, %

Figure 4.3.3: Phase diagram of the system PA-12-dodecalactam: 1 ) liquidus, 2 ) binodal.

The third, most complex variant of q(n dependence is shown in Fig. 4.3.1 c. Here, as the polymer solution is cooled, the flow region of type I1 appears twice in different temperature ranges and two transitions are observed respectively: From region I1 to region I at B and from region I to region 11; point B marks the temperature of this transition.

This variant of the phase diagram was obtained for the system PA-I2/caprolactam and is shown in Fig. 4.3.4. We may assume that in this case the solution goes twice through the region of amorphous incompatibility, i. e. this system (Fig. 4.3.4) is characterized both as UCTM (curve 2) and as LCTM (curve 3). As before curve 1 corresponds to the line of the liquidus. The right branch of binodal2 is not shown in Fig. 4.3.4 because of experimental problems in reliable determination of the points in a narrow concentration/temperature range. The presence of LCTM in the system PA- 12/caprolactam is connected with destruction of specific intermolecular interactions between PA-I2 and caprolactam above 120 "C. We could assume that LCTM also exists in the system PA-6/caprolactam, but for this pair the given region should lie at higher temperatures and are not realized practically due to reaching the boiling temperature of caprolactam. The presence of specific interaction in PA solutions is indicated by the fact that solubility of these polymers cannot be described with the known conception based on similarity of values for the solubility parameter 6 as a criterion of component compatibility.


T,O

fW

too

I

20 60 f0ff

V, mass Figure 4.3.4: Phase diagram of the system PA-I2/caprolactam:

I ) liquidus, 2, 3) binodals corresponding to UCTM (2 ) and LCTM (3).

Temperature dependencies of viscosity for the solutions studied characterized by concentration dependencies of the activation energy E(q) are shown in Fig. 4.3.5. This figure is indeed analogous to Fig. 4.3.1. It also contains the points of leap-like changes in the values of E corresponding to bends in Fig. 4.3.1. The location of bend points corresponds to the crystallization point in Fig. 4.3.5 a and to the binodal points in Fig. 4.3.5 b, c respectively.

a

A

6

1 6 I I I I I

I-A

1 6 I I

I I

I I

I I I I 2 - A

Figure 4.3.5: Dependence of activation energy of viscous flow of systems “polyamide- solvent” under diflerent conditions of phase separation. Signs are the same as in Fig. 4.3.1.


4.3.2 Crystallization

The use of rheological methods for the analysis of phase transformations in polymer systems is, as a rule, inseparably connected with the studies of orientation processes in polymer solutions and melts [2]. Increase of shear stress on completion of the induction period during crystallization of a polymer melt undergoing deformation at p = const, is caused by two reasons: Viscosity increase during transition from a true solution to a colloidal one and then to a suspension of crystallites, and/or increase of apparent shear rate due to precipitating of the crystallized polymer on the surface of one of the cylinders and reducing the working clearance [ 1 I]. These two variants can be observed for various configurations of the working unit of a rotational viscometer - “cylinder-cylinder’’ or “cone- and -plate”.

Temperature dependencies of viscosity of PA-6 solutions in caprolactam at different concentrations are shown in Fig. 4.3.6.

2

4.0

30 20

10

Figure 4.3.6: Temperature dependence of viscosity for solutions of dzrerent concentrations (denoted near the curves in wt- %)

for the system “PCMcaprolactam”.

In the flow region solutions behave like Newtonian liquids and therefore viscosity does not depend on shear rate. When the temperature of phase separation is reached the viscosity of the material increases sharply. The exact location of the transition temperature depends on shear rate and we see a region of phase transition shown as a space between the arrows for each concentration (Fig. 4.3.6).


Figure 4.3.7: Phase diagram of the system “PCNcaprolactam ”: 1) binodal of amorphous separation, 2, 3) curves of liquidus for deformable and undefotmable solutions respectively (0 - viscometric method, A - optical method).

Fig. 4.3.7 shows the phase diagram for the system “PAd/caprolactam” plotted on the basis of viscometric measurements with regard to the deformation effect [ 101. The crystallization temperature was assumed to be the temperature at which crystallization started within 2-5 hours after deformations began [ 121. The obtained results were confirmed by independent experiments determining temperature-concentration regions of different phase stated by means of optical interferometry [ 151 and polarized microscopy [ 141. Unlike conventional uses of the interferential micromethod to identify amorphous separation, here the introduction of a polariz- er and an analyser into the optical system of the interferometer made a diffusion test capable to yield information not only about the curve of concentration distribution near the phase boundary of mutual solubility of components but also about the region of temperatures and compositions visible on the distribution curve within which a crystalline phase of the polymer exists, i. e. it becomes possible to specify the location of the liquidus line [6].

The experimental data shown in Fig. 4.3.7 indicate a shift of the liquidus curve under shear deformation [ 151. Within the shaded area crystallization temperatures of the solutions v“y with slow conditions. Thus, in the rotatary viscometer deformations caused polymer crystallization at temperatures far above its crystallization temperature in a stable state [16]. The melt of PA-6 could be crystallized at T = 235 “C which exceeds its equilibrium melting temperature, defined with different methods (223 “C) [17].


A rise in the melting temperature of an oriented material follows from thermodynamic conditions of the phase transition and is expressed as:

AS0r T = To m AS',,, - ASor (4.3.1)

where Tk and A S m are temperature and entropy of melting, respectively. ASor is the change of entropy caused by orientation.

The increase of melting temperature and consequently of the extent of melt or solution over cooling results in the increase of nucleation rate. If crystallization take place near the melting temperature the Turnbull-Fischer equation has the form:

where N is the rate of nucleus formation, Nc , w, a are the constants.

Assuming for simplicity that a = 1 and substituting (4.3.1) into (4.3.2) gives:

(4.3.2)

(4.3.3)

Using the above-given correlations and having the values of thermodynamic constants of PA-6 melting [ 171 we can calculate the change of the equilibrium melting temperature and the rate of nucleation depending on molecular orientation in the polymer (Fig. 4.3.8). If the value A T monotonically increases with growing entropy orientation, the rate of nucleation - which increases considerably at the initial stages of orientation - then depends weakly on entropy changes. Besides, at high deformations the rate of nucleus formation is practically independent of crystallization temperature. In this case, the orientation of macromolecules is so high that a change in crytallization temperature by a few degrees leads to the effect which is negligible compared with the role of overcooling extent (T, - T'), gaining importance due to orientation.

0

9 4 4 8

A Sor, c o t " / m o ~ o ~

Figure 4.3.8: Variation of equilibrium temperature of melting and rate of nucleation at diflerent temperatures (near the curves) depending on melt PA-6 orientation.


In reality the application of shear flow to a crystallising polymer or to its solution results in a non-monotone increase of the melting temperature (Fig. 4.3.9) has a complex effect on the duration of the induction period of crystallization (Fig. 4.3.10). The equilibrium melting temperatures used to plot Fig. 4.3.9 were calculated on the basis of kinetic data [5, 161.

T,

260

240

Figure 4.3.9: Dependence of equilibrium temperature of melting and entropy change in the process of orientation on shear rate under the flow of melt PA-4.

Figure 4.3.10: Dependence of induction period of PA-6 crystallization from the melt (a) and its 30% solution in caprolactam (b) at direrent temperatures (shown near the curves).

Quantitative evaluations of the dependence of crystallization rate on deformation regime are quite varied including data both of accelerating [18, 191 and decelerating [4, 201 effects of shear rate on crystallization of linear polymers or of the change of sign of the dependence with the change of the polymer temperature of M [20]. As is seen from the data shown in Fig. 4.3.10, this contradiction is apparently explained by relatively narrow ranges of shear rates used for the analysis.


At low shear rates the rate of crystallization increases with the shear rate to the exponential dependence t i - f” where the exponent n increases with temperature. This causes the kinetic data on the logarithmic coordinates to acquire the form of some converging lines which in extrapolation cross at a point corresponding to the conditions under which the rate of crystallization is practically independent of the temperature. This experimental fact is in good agreement with the above given analysis confirming the existence of this region (Fig. 4.3.8). Practically, this region is not reached since at high shear rates the rate of crytallization starts to decrease. This effect is observed both during crystallization of PA-6 solutions in caprolactam and during crystallization of PA-6 melt, because by the fact that the flow with a high shear rate may result in destruction of nuclei having the critical size and reducing time of contact with nuclei formed earlier. This ‘‘negative” effect of shear deformation on crystallization kinetics is most clearly seen at high temperatures when the size of the critical nucleus and the rate of macrornolecular relaxation increase.

It is necessary to note that the kinetic curves of crystallization include regions where a 2 to 3-fold change of shear rate causes changes in the rate of crystallization by a decimal order [2]. It shows that undermining the phenomenon of orientation crystallization and using the results of its analysis unaffected by external disturbances under real technological conditions may provoke considerable errors and even principal mistakes.

If the orientation of macromolecules during phase separation of crystallising polymers always results in rising temperature of phase transition, this dependence during amorphous separation is of a more complex nature.

Changes in the temperature of phase separation as the polymer is oriented in the solution can be illustrated on the basis of general thermodynamic relationships. Here the following variants of changes of the chemical potential in the course of dissolution are possible.

A. A p , < 0 ; A H , > 0 ; AS , > 0 ; ITAS,[ > IAH1l

B. A p , < 0 , A H , < 0 ; AS , < 0 ; IAH,I > ITAS,/

where A p , , A H , and AS, are changes of the chemical potential, the partial molar enthalpy and entropy of the polymer in the course of dissolution.

The entropy change in the course of dissolution is: AS, = S , - Sy

where S , and Sy are polymer entropy in the solution and before dissolution, respectively.

Polymer orientation during flow causes its entropy to decrease to the value SI, or c S, . Then the temperature of phase separation T f in the solution of a given concentration can be determined from the following expressions:

[AH11 = ITAS11 = ITorSi,orl

Further we should consider the change of the co-factor A s , / A S i , or for different variants of changes in thermodynamic functions in the course of dissolution.

3 10 Chapter 4: Transformations in Polymeric Systems

Variant A:

Then Tf, or < T f . In this case polymer orientation in the solution causes the temperature of phase separation to increase. Consequently the region of incompatibility on the phase separation diagram for the system “polymer-solvent” is expanded.

Variant B:

Then Tf, or < T f . In this case the temperature of phase separation during orientation of polymer macromolecules decreases. However, this correlation between enthalpy and entropy changes agrees with the existence of the lower critical temperature of mixing in the phase separation diagram. Here application of stresses to the polymer solution also expands the region of incompatibility.

As was demonstrated in the experimental studies, this rather simple analysis gives correct qualitative predictions for behaviour of polymer solutions in the region of phase separation when a mechanical field is applied [lo, 12,21,22].

4.3.3 Phase Transformations in Reactive Systems

Phase separation in reactive systems can lead to a sharp drop of viscosity due to isolation of polymer from solution. The flow of the formed colloidal system is determined by flow regularities of the low viscous solvent filled with particles of the polymer phase. However, appreciable separation is not often the case when the boundary of phase separation is formed in reactive systems. In some processes thermodynamic conditions for the formation of a two-phase system are accomplished in a kinetic situation which is unfavourable for phase separation, e. g. at relatively high viscosity of the reaction medium. This case should be regarded as microphase separation since phase disintegration practically stops at the state of forming nuclei of a new phase.

An example of this process is anionic activated block polymerization of lactams, when under certain effective conditions the forming polymer becomes insoluble in the initial monomer. This causes changes in basic rheokinetic relationships, because of the disappearance of the co-factor characterizing the viscosity dependence of the molecular mass of the forming polymer [23]. In the two-phase region of the process viscosity is determined only by concentration of the polymer phase growing in the course of the reaction.

Temperature changes in the reaction mass where the polymer concentration increases during polymerization (“the path of the reaction”) if the molecular mass of the forming polymer is assumed constant are shown with dashed lines on the phase diagram of the system “PA-12ldo- decalactam” (Fig. 4.3.1 1). As is seen from Fig. 4.3.1 1, at To = 160 “C “the path of the reaction” at initial reaction stages goes through the binodal and enters the region of amorphous incompatibility of phases, enriched with polymer and monomer, respectively. As to the reaction path at To = 180 “C, it is all located above the binodal in the region of the one-phase solution.


20 40 60 80

'f, moss % Figure 4.3.11: Variation of temperature and composition of the reaction mixture

allowing for its phase state: 1) binodal, 2 ) curve of the liquidus, 3) reaction course at To = 160 "C, 4) reaction course at To = 180 "C.

Another important consequence of the phase disintegration in the reactive system is the emerging dependence of kinetics of viscosity growth on deformation intensity [24, 251. This phenomenon is illustrated in Fig. 4.3.12 showing dependencies q(t) under various process conditions. It is seen from the figure that at 190 "C the change of the shear rate has no appreciable effect on the rate of polymerization whereas at 170 "C and 135 "C an increase in deformation rate leads to a regular and rather significant deceleration of viscosity growth. Viscosity of the reaction mass during polymerization at 135 "C and 170 "C with a fixed process time in the given range of shear rates are described by the formula [24,25]:

8 q = 7

Y + C

(4.3.4)

where 0 and c are the constants, and it is essential to note that in the studied range of shear rates )> c.


% 4 to

. -h

6

0

t, s Figure 4.3.12: Kinetics of viscosity growth in the process of anionic polymerization of

caprolactam at difSerent temperatures of pe~orming the reaction: 190 OC (a), 170 OC (61, 135 "C (c). Figures near the curves denote deformation rate in s- I .

On the whole these results are similar to the earlier discured dependencies of the rate of anionic polymerization of dodecalactam on the phase state of the system. If polymerization is carried out in the region of relatively low temperatures ( T < 190 "C), the reactive system enters the region of amorphous separation where the moment of reaching the binodal depends on deformation intensity, or more precisely, the deformation intensity affects the location of the binodal [2].

These data enable us to assume that if polymerization is carried out at 190 "C the observed spread of points in the measurements of q(t)-dependence in Fig. 4.3.12 is linked with different manifestations of the effect the deformation rate has on the location of the upper critical temperature of dissolution (UCTD) for the system "polycaproamide-caprolactam". This assumption is confirmed by the diagram of the phase state of this system shown in Fig. 2.2.29 where it is seen that under similar polymerization conditions the temperature of the reactive system lies very close to UCTD.

Rheological regularities described in Chapter 2.3 for radically polymerizing masses refer to reactive systems retaining their single-phase nature throughout the process of polymerization. A totally different situation is observed when the solvent quality is changed as the monomer is used up and a phase transition occurs in the course of chemical transformations. These studies were carried out using polymerization of methylmethacrylate in the mixture of the solvent - toluene and the precipitant - cyclohexane [26-281 as an example. The solvent quality was varied by changing the ratio of toluene and cyclohexane. During the process in the course of decreasing the concentration of the monomer, which is a good solvent for polymethylmethacrylate in the conversion region 25-30%, the reactive system acquires a two-phase structure due to separation of a polymer enriched phase form the mixture solvent [26]. This transition is characterized by


sharp turbidity and a drop in viscosity of the polymerizing mass because viscosity of the dispersed medium - here it is the polymer-enriched phase - decreases. The forming polymer particles of the disperse phase have a globular structure [28]. With subsequent growth of transformation depth, viscosity begins to grow due to enlarging of the forming globules and their aggregation into more complex structures.

The extreme viscosity change is a typical feature of the moment of phase inversion taking place in the production of stress proof polystyrene [29, 301. On reaching a certain degree of transformation corresponding, as a rule, to 9-12% styrene conversion in solutions containing 8% of elastomer [3 11, a transition is observed in the polymerizing mass. The system rubber solution in styrene as a continuous phase changes to the system where rubber becomes a dispersed phase in the newly formed dispersion medium - polystyrene dissolved in styrene. This moment correlates with the viscosity drop of the polymerizing mass, since in the range of molecular masses of polymers used to produce stress-proof polystyrene, viscosity of the elastomer phase is higher than that of the equilibrium polystyrene phase [32]. Viscosity changes in the model mixtures of the solutions of polystyrene and polybutadiene in styrene show that phase inversion occur in a relatively narrow range of styrene concentrations and is a reversible process [33].

The effect of viscosity drop at the moment of phase inversion in the production of stress- proof polystyrene is expressed so vividly that it is offered as a method of technological control for polymerization processes [34]. Besides, phase transition in a polymerizing system is not necessarily accompanied by a drop of viscosity [35]. In some cases, due to a variety of physico- chemical phenomena accompanying phase separation in the reactive system, an opposite effect may be seen. Unlike polymerizing alkylmethacrylates, isolation of the forming polymer in the form of a separate phase during polymerization of alkylonitrile in an aqueous solution of NaCNS leads to a viscosity increase due to gelling of the reactive system [36]. However, this effect is not likely to be of general value and reflects specific features of the given system.

Reactions of polymer-analogous transformations result in a considerable change in the whole set of properties of polymers and their solutions [37]. When this kind of process is carried out in a solution, the rheological properties of the reactive system may change not only due to formation of new polymer fragments, but also due to a shift of phase separation lines with “qualitative” changes in the solvent vs the newly formed polymer. In the extreme case, changes of the solution properties may be accompanied by phase disintegration and the formation of a two-phase system. An example of this process is acidic saponification of polyvinylacetate in the medium of ethanol during the formation of polyvinyl alcohol [38,39]:

C,H,OH -CH-CH,- * -CH-CHZ-

I - CH3COOEt I ? OH

c=o I CH3


The viscosity change 11 in the studied solution and the change of the transformation degree due to reaction course at a single shear stress z (here z = 8 Pa) are shown in Fig. 4.3.13. It is seen that as alcohol groups are formed in PVA macromolecules, interaction between the polymer and the solvent becomes weaker and the solution viscosity gradually decreases until the minimum point is reached. This extreme dependence q(t) may be connected with two reasons -change of the solvent quality vs the forming copolymer or vs the phase transition "solvent-gel". However, the end of the reaction at the moment of reaching the minimum value of viscosity and quick discharge of the viscometer enabled visual observation of a two-phase system appearing in the form of a rubber-like gel.

0

120 2 40 360 t, min

Figure 4.3.Z3: Viscosity change of a reaction solution (a) and change of conversion

degree (b) during acidic saponification of PVA with time at 500 "C ( I ) ,

550 "C (2) , 600 "C (3), 650 "C (4), 70 "C (5).

With increasing temperature of the reaction the time for reaching the point of phase transition t* is reduced from the change of t* depending on the temperature T. Calculations were made for the effective activation energy of the reaction of acidic saponification of PVA which was found to be 48 kJ/mol. This value practically agrees with the energy of activation determined by chemical-analytic investigation of saponification kinetics [40].

Comparing the values of p and q at the moments of time corresponding to the minima on the curves in Fig. 4.3.13 a we find the values of p* corresponding to the points of the transition "solution gel". Unlike the curing process, here the degree of transformation at the moment of gel formation p* is a function of temperature, and the transition points were obtained at different temperatures. The results of that kind are of specific interest as a means to use rheological methods for plotting diagrams of the phase state of systems undergoing polymer-analogous transformations where the role of precipitant is performed by the concentration of substituted groups. A fragment of this phase diagram for the system PVA-PVC-ethanol is shown in


Fig. 4.3.14. In fact, a phase diagram of this system must be three dimensional. Therefore Fig. 4.3.14 presents only a section of the full phase diagram since the experimental results were obtained in the tests conducted with a constant concentration of ethanol.

Figure 4.3.14: Fragment of a phase diagram of the system “PVA-PVS-ethanol”

showing the dependence of temperature of phase transition on hydroxyl

group concentration in the copolymer c O H .

Polymer concentration in solution is 20% mol.

Attention should be paid to an original peculiarity of the obtained phase diagram - it is characterized by a practically symmetrical position of the binodal. It is well known that a specific feature of phase diagrams for “polymer-solvent’’ systems is their great asymmetry caused by different sizes of polymer and solvent molecules [41]. This is not observed here, because the fragments of polymer molecules dissolving in ethanol are likely to have sizes approaching those of insoluble blocks.

Another moment of rheokinetics of polymer analogous transformations deserves detailed discussion - mechanical impacts affecting this process. Fig. 4.3.15 shows the curves of viscosity changes during acidic saponification of PVA in the solution of ethanol at different shear stresses T. In this figure the nature of viscosity changes in the course of the reaction before reaching the minimum of the curve q ( t ) does not depend on the value of applied stress, so in this region mechanical action has no effect on the course of the chemical reaction. Besides, the position of the point of the phase transition evaluated from the moment t* of transition from a descending to an ascending branch (Fig. 4.3.15) is to a considerable extent determined by the value of applied stress. Here we can also trace the regularity which was observed earlier, namely, if a chemical reaction is effected in the homogeneous region (preceding the transition point), deformation does not affect its rate and, if it is complicated by the process of phase separation, we should expect a notable effect of the intensity of the mechanical action on the change of rheological properties.


IOU 200

t , min

Figure 4.3.15: Viscosity change of a reaction solution with time of 70 "C and at shear

stresses: 0.4 ( I ) , 0.76 (2) , 1.1 I (3), 1.53 (4), 3.05 (5) and 8.08 Pa-s (6).

It is necessary to note that in the studied process phase separation in a reactive system has a negligible effect on the nature of the dependence P ( t ) . Possibly this is explained by the structure of the forming gel, which is formed with retained segmental mobility of macromolecules and without apparent difficulty for molecules of the low molecular agent to come into contact with the functional groups of the polymer.

I, Pa

Figure 4.3.16: Dependence of time of reaching the point of phase transition on shear stress at 70 "C.

The t* dependence on the applied shear stress is shown in Fig. 4.3.16. At low stresses mechanical loading results in reducing time of transition to a gel. Apparently the change in macromolecule conformations accompanying the flow process facilities orientation of chain fragments where the reaction of substitution has taken place causing micro separation of the system. A further increase of stress in the flow results in disintegration of these microgel nuclei of a new phase, which leads to a considerable weakening of the dependence of gel formation rate on deformation intensity. This nature of the dependence of the location of the phase transition point on the intensity of applied mechanical action, which reflects different aspects of the influence of the mechanical field, is of a rather general value since it is observed for various systems disintegrating by different mechanisms.

4.3 Rheokinetics of Phase Transitions 3 17

4.3.4 References

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18. R. S. Zubaryan, V, G. Baranov, S. Y. Frenkel, Mekhanika polimerov, 1974,2,35 1.

19. R. R. Lagasse, B. Maxwell, Polym. Eng. Sci., 1976, vol. 16, 3, 189.

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24. A. Y. Malkin, A. G. Merzhanov, T. M. Funze, S. P. Davtyan, S. G. Kulichikhin, A. M. Stolin, V. V. Maizeliya, T. V. Volkova, P. B. Shleifman, V. A,Kotelnikov, V. V. Kurashev, Dokl. Akad. Nauk. SSSR, 1981, vol. 258, 2,402.

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3 19

Subject index

Absolute temperature I3 Acceleration, adiabatic 38 Acetaldehyde 207 N-Acetyl-e-caprolactam 50 Acidic saponification 3 I5

Acrylates 69,94 Acrylonitrile 1 1 Activated lactams 40 Activation energy 1 I , 203 - -. apparent 109 - -, effictive 1 I , 37 Activator I 1 - concentration 37 -, indirect 35,48 Active centres 28 - -, concentration of 1 I , 3 1 - - formation 31 - - -, rate of 31 Active loci 12, 17 - -, concentration of 14 Adiabatic acceleration 38 -process 188 - regime 192 Affinity, thermodynamic 93 Ageing rheokinetics 287 Aliphatic high molecular polyamide 258 Alkylonitrile 31 3 Amine groups 26 1 Amino-formaldehyde resins 248 Aminoplasts 2 I0 Analysis, kinetic 35 -, rheological 19 Anhydrous resins I57 Aniline 207 Anilinephthalein 273 Anion 32 Anionic activate block polymerization of

lactams 310 Anionic polymerization 3 I , 43,47f, 53,56 - -, pattern of 32 Antithixotropy 228 - during gel-formation 28 I Apparent activation energy 64, 109 Arrhenius constant 1 1 -coordinates 45.55, 157,241,250 Asphaltene 279 ASTM 207 Auto acceleration effect 85 - -, constant of 57 Autocatalysis 38, 242

--OfPVA 314

Autocatalysis, constant of I I , 56 Autocatalytic pattern 56 -reaction 38 Avrami equation 294 Azoaromatic chromophores 275 Azobisisobutyronitrile 80, 84, 255

Behaviour, rheological 27 Benzoylperoxide 1 I Bifunctional compounds 102, 104 - diamines I50 - polydimethylsiloxane 237 Bifunctionality 21 7 Binding 207 Binodal 55 - of amorphous separation 306 - phase state 3 1 1 Bisphenol 263 Block polymerization 86,89 BM-polymerization 70 BPO 85,90 Branching 225 - factor 13, 134 Brookfield viscometer 297 BTDA 121 Burgers’ model 139 1,4-Butandiol 224 Butylmethacrylate 1 I , 87

Calorimetry 21 2 -, isothermal 148 Caprolactam 28, 3 12 &-caprolactam 38,53,55f, 148 - polymerization 53 &-caprolactonetriene 220 Carbamide resins 174 Carboxyl groups 224 Catalyst concentration 33, 54 Catalytic system 48 Cellulose triacetate 290 - --nitromethane system 289 Chain termination 28 Chains, macromolecular 86 Change of conversion degree 314 - - -, time-dependent 252 --torque 12 - - viscosity 27. 220 Changes in the components 256 Chemical autocatalysis 173 Cis-I ,4-polybutadiene 175 Cis-azobenzene 275



Subject index 320

Cis-trans-isomerization 274 Coalescence of particles 94 Colloid 96 -events 96 Compositions glues 207 -, long-lived 21 1 -, short-lived 21 1 Compounds, bifunctional 102 Compressing force 232 Concentration 28,32 -, activator 37 -, catalyst 54 -, initial 35 - of active centres 1 1 - of active loci 14 - of monomer 12 - of polymer 14 - of rubber hydroxyl groups 243 Cone-and-plate rotation viscometer I21 Constant of autoacceleration 57 - of autocatalysis I 1.56 - of initiation rate 80 - of self-decelaration 14 Constants of reaction rate 242 -, kinetic 188 Conversion degrees 74 -, critical degrees of 87 -, degree of 13,27,43 -, rheological degree of 182 Coordination bonds 86 Copolymerization 255 Cresol 207 Critical chain length 12 - degrees of conversion 87 -point 85 Crystallization 305 - of a polymer melt 305 Curing 16, 133 - after the point of gel formation 170 - at high shear rates 202 -, equations of I78 -, initial stages of 231 - in solution 277 -, non-isothermal 186 - of oligomers 24 - of unsaturated polyesters 229 -, temperature of 13 Cyclic siloxane 247 Cyclohexane 94,312

DBP 156 Debora’s criterion 280 Decomposition rate 80

Deformation 15 - rate 203, 252 Deformations, irreversibility of 279 -, irreversible 297 -, reversibility of 279 -, reversible 281 DEG 127 Degree of conversion 13, 17, 20,43,88,266 - - -, time dependence of the 193 G*-t-T-diagrams I38 Diallylphthalate 255 - resin, cross-linking 256 Diamines 166,222 -, bifunctional 150 -, polyfunctional I50 Dianhydride 121 f, Dibutylphthalate 1 1, 156 2,5-Dicarbomethoxyterephthaloylchloride 273 Dicetylphthalate 296 3,3’-DichIor-4,4’-

Dichlordiphenylsulphone 1 13 4,4’-Dichlor-diphenylsulphone 105 Dicyandiamide I 1,263 Diffusion 78 - coefficient 1 1 -model 87 Diffusion-controlled phases 78 Diglycidyl ether 263 2-a,w-Dihydroxy-

Diisocyanates 122,225,284 Dilatant behaviour, thixotropic 296 Dimethylacetamide 291 --water 280 N,N-Dimethyl-N,N-

Dimethylsulphoxide I I , 106 Diphenylene propane 105 Diphenylmethanediisocyanate 1 1 4,4-Diphenylmethanediisocyanate 21 9,224 Diphenylolpropane 1 I3 --dichlordiphenylsulphone I 13 Direct rheokinetic problem 28 Dispersed phase 3 13 Dissipative heat release 99 Divinylbenzene 255 DMSO-solution I07 Dodecalactam 42f, 45 Dodecyl mercaptan 85 DPMDI 219 Dynamic mechanical spectroscopy 68, 183 Dynamic modulus 256

diaminodiphenylmethane 149, 156

(oligodimethylsiloxane) 284f,

diallylammoniumchloride 68

Subject index 32 1

ED-7 by diamine 165 Effect, exothermal 198 Effective activation energy I 1, 37 Effects, endothermic 20 -, exothermic 29 -, non-isothermal 20,205 Efficiency of initiation 1 I , 80 Elasticity modulus 1 1 Elastomer 137 Electrophilic centre I14 EM 77 Emulsion polymerization 93 Endothermic effects 20 Enthalpy 1 1 Epoxy 174 - oligomers 152, 163,260 Epoxy silico-organic oligomers 144, 146, 163,

Epoxydiane resin 153, 156 Epoxymethylenepolyamides 259 Equations of curing I78 Equilibrium rubbery modulus I I , 139 Equimolar ratio 35 Ethylmethacrylate I 1 Euler function 13, 199 Exothermal effect 198 External source 198

Filler 162 First-order-reaction 38 Flory’s model 135 Flow, non-Newtonian 230 Flow, viscous 21 Fluctuation network 82, 141,278 Fluid solution state 281 Fluidity 22,94 - loss, temperature of 13 -, characteristics of 208 Forepolymer 229 Formaldehyde 207 Four parameter model I I2 Fourier-spectroscopy 139 Free radical polymerization 60,66,93 - - - rate 82

G*-t-T-diagrams 138 Gelation 134, 277 - kinetics 294 - of plastisols 295 Gel-chromatography I60 Gel effect 62, 82,85,93 - formation 24, 134, 142, I64 - formation, antithixotropy during 281

174,250

Gel formation, point of 146 --like solution state 281 - maturation 293 - permeation chromatography I 1 --point 195,238 Glass transition point 74 - - temperature 120 Globules, quasi-solid 94 GOST 252

GPC 122 GOST 12535-78 253

H2S04 33 Hardness 1 1 -change 215 Heat conductivity 1 I - transfer 13, I98 Hetero-polycondensation 102 Heterogeneous medium, polymerization in 93 - region 55 Heterogenity 62 Heterophase system 97 Hexamethyl disiloxane I09 Hexamethylenediisocyanate 148 1,6-Hexamethylenediisocyanate 1 I , 216 Hexamethylenetetramine 214 High shear rates, curing at 202 HMDI 217,220 Homogeneous free radical polymerization 89 - region 55 Homophase system 97 Huggins constant 91 Hydrodynamic pattern 23 Hydrolysis 122 Hydrophobic interactions 286 Hydroquinone 85 Hydroxyl groups 261

Impregnation I33 Incompatibility of phases, amorphous 55 Increase, linear temperature 198 Indirect activator 35 Induction period 203,299,308 -, non-isothermal 203 Industrial rubber articles 133

Initial concentration 35 - oligomers I36 - reaction temperatures 43 - stages of curing 23 1 Initiation, efficiency of 80 - rate, constant of 80 Initiator 83

322 Subject index

Injection moulding 207f Insolubility 93 Instron 231 Interferential micromethod ' 306 Intramolecular cyclization 284 Intrinsic viscosity 33 Inverse kinetic problem 38 Ionic force 1 1 1 - polymerization 28 Ionogenic catalyst concentration 12 Irreversibility of deformations 279 IRS 121, 138 IR-spectroscopic measurements 239 IR-spectroscopy 229 Isobutyric acid 287 Isocyanate I21 Isomerization 274 Isopropyl alcohol 287 Isothermal 40 - calorimetry 148 - polymerization 29 - rheokinetic measurements 226 Isothermality 245 Isoviscous state 65. 85f

Kanavets plastometer 208 . Kinetic analysis 35 - constants 188 -equation 20 - problem, inverse 38 Kinetics of viscosity 228 -, gelation 294 KM 91

Lactam 32 -anion 49 Lactams reactions 48 Latex particles 94 Lauroyl mercaptan 83 -peroxide 83 LCTM 303 Lifetime 299 Linear chains 224 - - production I9 - polymer formation 27,66 Linear polymers 17,24 Linear temperature increase I98 Liquid, low-viscous 22 -, Newtonian 226 Local networks, formation of 164 Loci, active 12 Long-lived-compositions 21 1 Loss angle 13

Loss modulus 1 I , 143 Low temperature 47

Macro kinetic rate constant I21 Macro-isocyanate 222 Macrodiisocyanate 149 Macrokinetic pattern 229 Macromolecular balls 91 -chains 86 - mobility 86 Macromolecule conformations 31 6 - propagation 48 Macromolecules, orientation of 307 -, polymerisation of 18 Macroscopic gel-point 205 Macroviscosity 78, 86 Maleinateacrylate phosphorus containing

polyester 236 Martin constant 91 Mass, reactive 49 Material, network 82 Maturation, gel 293 MDI 121 Measurement techniques 40 Measurements, isothermal rheokinetic 226 -, viscometric 22 Mechanical deformation 23 - loss angle tangent 238 - loss factor 298 Media, anomalous 15 Melaminoformaldehyde 174, 184, 248,249 - oligomer 18 I - resins 12, 157, 164, 167,285 Meta-polymer 275 Method of a glass 208 Methylene chloride-d2 109 4,4'-Methy lenedi(pheny lisocyanat) 1 2 Methylethylquitone peroxide 236 Methylmethacrylate 12, 87,298,312 Methylolpolyamide 174 N-Methylpyrrolidone 79,284 MFR 181 -curing 158 Micro block polymerization 93 - separation 316 Micro-viscosity 78 Microdrops 94 Microgels 135, 158 Microheterogeneity 93 Micromethod, interferential 306 Microphase separation 156 Microreactions 93 Microviscosity 79, 86

Subject index 323

MM 171 - fragments 246 MMA 85 - polymerization 70, SO Mobility 120 -, macromolecular 86 Molecular mass I2 - weight 12, 27 Molecule mobility I20 Monomer 12 - consumption 32 Monomer-polymer particle 94 Monsanto 252 Monthing powders 207 Mooney plastometer 252 Multi-component systems 96 MW 47,53,60,83 - distribution 33 MWD 33,79,122

NaCNS 96

A[NCO] 225 NCO-groups 224 Network, fluctuation 82 - formation 255 -material 82 -, three-dimensional 133 Newtonian liquid 226 - viscosity 33, 67, 84, I03 N-Nitrophenylazotriphenylmethanate 80 '3C-NMR-Fourier-spectra I09 NMR spectroscopy 107, I72 Non-isothermal curing I86 - induction period 203 - phenomena 20 - polymerization 232 -regime 13 - regime of curing 195 Non-isothermality 30. 194 Non-Newtonian flow 230 Nucleophilic centre I14 Nylon-6 formation I89

NH,CI 248

Octylmethacrylate 87 01-formation I22 Oligo h i d e 12 Oligobutylmethacrylate 228 Oligodiethyleneglycol adipinate 228 Oligodiole 2 I8 Oligomer 102 -curing 133 -, epoxy 152, 163

Oligomer, epoxy silico-organic 144, 146,

-, melaminoformaldehyde I8 I -, silico-organic 143, 163, 204 - structuring I33 Oligomerization 109 Oligomers I74 -, phenylmethylsiloxane 238 -, structuring of 207 -, thermo-reactive 207 Oligourethane 228 One-phase-systems 138 Orientation of macromolecules 307 Oscillations, harmonic 23

163,250

PA solutions 47

PA-6kaprolactam 306 PA-6 33,53

PA-I2 33,43,44 PA- 12-DMAA 302 PA- 12-DMSO 302 PA- 12-dodecalactam 303

- synthesis I27 Para-polymer 275 PAS synthesis I14 Pattern, autocatalytic 56 PB 79 PBMA 60,70,84 PBTP 178 PCNcaprolactam 305 PEMA 72 PEPR 128 Peroxides 80 PETP 128 PFR 207 --based reactoplasts 21 I - viscous-flow state 208 Phase diagrams, 301 -events 96 - separation of polymer solutions 290 -state 55 - transformations in reactive systems 310 - transition, temperature of I3 - transitions, rheokinetics of 301 Phases, diffusion-controlled 78 Phenol filler 236 Phenolformaldehyde resin 12,204, 207 Phenomena, non-isothermal 20 Phenylmethylsiloxane oligomers 238 Photosensitization polymerization 81 Photo-viscous effect 274f Phurphurol 207

PA-610 102

324 Subject index

Plasticized copolymer 298 Plasticorder 224 Plastisols 295 -, gelation of 295 Plastometric properties 210 - testing 210 -tests 208 Plastometry 2 I4 PMM 69 PMMA 60,69 Point of gel formation 142, 146, 238 Polyalkyleneglycol malcinates 229 Polyamide 12 Polyamides, thermo-reactive 258 Polyaminoformaldehyd resin I57 Polyarylsulfon I2 Polyazethane 140 Poly-p-benzamide I25 Polybutadiene I2 Polybutoxytitanphosphoroxane 12, 178,250f Polybutoxytitanphosphoxane I7 1 Polybutylmethacrylate 12 Poly-(E-caprolactam) 12 Polycondensation 18, 27, 102f, 213,216 - process I16 - rate 120 -, three-dimensional 15 1 Polycyclotrimerization 284 Polydimethylsiloxane 284 - rubbers 170,283 -, bifunctional 237 Polydodecanamide 43 Polyene 220 Polyester resin, thickening of unsaturated 234 - resin-benzoyl peroxide 183 - resins, unsaturated 255 Polyesters, curing of unsaturated 229 Polyetheramide 143 Pol yethers, unsaturated I74 Polyetherurethane prepolymer 225 Polyethylene, cross-linking of 257 Polyethyleneterephthalate 12 Polyethylmethacrylate 12 Polyfunctional diamines 150 Poly-(hexamethylene sebacinamide) 12, 102 Polyisocyanate 220 Pol ymer-analogous transformation 273 Polymer chain 62 --length 28 -, concentration of 14 - formation 27 - production, linear 19 - solutions, phase separation of 290

Polymer chain synthesis 16, 21 Polymeric filling 62 - systems, transformation in 273 Polymerization 27 -, anionic 28,40,43,48,53,56 - in heterogeneous medium 93 -, isothermal 29 -, non-isothermal 40, 232 - of macromolecules 18 -, photosensitization 81 - rate, reduction of the 93 -, steady stage of 66 -, thermal 81 -time 44 Polymerizing mass 21 Polymers, rheology of I5 -, silico-organic 237 -, structuring of 207 -, three dimensional 247 Polymethacrylate I2 Polymethylrnethacrylate 12, 69 Polyorganosiloxanes 242 Polyoxypropylene glycole 225 Polyphenylmethylsiloxane 238 Polyphenylquinoxaline 273 Polystyrene I2,66 -, stress proof 3 13 Polysulfonamide 29 I Polysulfonamidedimethylacetamide 278 Polysulfone I2 Polytetrabutoxytitan I3 Polyurethane 12, 148 Polyurethanes (PU) 216 Polyvinyl alcohol solutions 288 Polyvinylalkohol 12 Polyvinylchloride 12,295 PORMA 72 Premis 229 Prepolymer 225 Prepolymerization 232 Prepreg 229 Production of polyurethanes 216 I ,2,3-Propanetriol 21 7 Properties, plastometric 210 -, rheological 21 Proton-donor functional groups 173 PS 60,72,96 Pseudo-gel 88 PSF 105, 107

--synthesis I I6 PU foaming 227

--DMSO 118

Subject index 325

PU production from oligomers based on

PulseNMR 172 PVC 295 --plastisols 296

macroglycoles 224

Quasi-crosslinked 224

Radical polymerization 255 Rashig’s method 208 Rate constant of the reaction 56 Reaction, autocatalytic 38 -, first-order 38 - paths 18 -rate I I , 13, 32 -, rate constant of the 56 Reactive mass 49 - - composition 18 - - viscosity 87 Reactive system 19, 50 - -, phase transformations in 3 I0 Reactoplasts 209,210, 21 1 Reduction of high elasticity modulus 285 Regime of curing, non-isothermal 195 Region, homogeneous 55 Relaxation 15 - spectrum 136, 139 -transition 127 - -, temperature of I3 Reptation model 88 Resin, amino-formaldehyde 248 -, anhydrous 157 -, carbamide I74 -, melaminoformaldehyde 157, 164, 167, 285 -, phenolformaldehyde 204 -, polyaminoformaldehyd I57 -. silico-organic 237 -, unsaturated polyester 229 Resorcine 207 Reversibility 275 - of deformations 279 Rheokinetic models 20 -problem 28 Rheokinetics of phase transitions 301 Rheological analysis 19 - degree of conversion 170, 182 - properties 21 - - of moulded compositions 208 Rheology 15 Rheometer-I00 252 RIM processes 24 Ring formation I35 Role of intramolecular cyclization 284

Rotary viscometry 42 Rotation viscometer, cone-and-plate 121 Rubber blends 252 - compositions 252 - cross-linking I36 - hydroxyl groups, concentration of 243 Rubbery modulus 1 I , 60,68, 135, 143, 18 I , 238 - solution state 281

Saponification kinetics 3 14 Second relaxation transition I37 Self-acceleration 240, 248 - constant 259 -, non-isothermal 193 Self-deceleration 250 Shear rate 13,86,98,203,205 - strain 13 Short-lived compositions 21 1 Silico-organic oligomers 143, 163, 174,204,237 Siloxane networks 284 - rubbers 160 -,cyclic 247 -,spiral 247 Sodium rhodanide 97 Solid phase, transformation in the 273 Solidification 16 Solution concentration 60 - viscosity 64 Solvent gelation 277 -, role of the 283 Spectroscopy 40 Spiral siloxone 247 Spiropyrans 275 Spycrocycles 247 Stable network 141 Steady stage of polymerization 66 Stress proof polystyrene 3 I3 Structure formation within the solution

state 292 -, cyclobranched I35 Structuring I33f - of oligomers 207 - of polymers 207 Styrene 66,89,93 - polymerization 67 Suspension polymerization 86 Systems, reactive 27

TBT 178 TDI 217 T-T-T-diagrams I38 Technological carbon 265 Temperature, absolute I3

326 Subject index

Temperature dependence 64 - -of viscosity 50 -ofcuring 13 - of fluidity loss 13 - of phase transition 13 - of relaxation transition 13 Termination rate constant 87 Tert.-butylperbenzoate 255 TES 243 Tetrabutoxytitan 178 Tetraethoxysilane 13, 242f, 284f Tetrameters 109 Thermal polymerization 81 Thermo-reactive oligomers 207 - polyamides 258 Thermo-reactivity 258 Thermodynamic affinity 93 Thermosets 2 10 Thickening of unsaturated polyester resin 234 Thin walls manufacturing 207 Thixotropic dilatant behaviour 296 - media IS - phenomena 282 - transformations 16 Thixotropy 296 Three-dimensional network 133 - polymers 247 --polycondensation IS I Time 13 Toluene 94,284 - diisocyanate 228 2,4-Toluene diisocyanate (TDI) 13, 217 Torque amplitude 252 - measurements 2.52 Total heat of a reaction 12 Trans-configuration 274 Transfer moulding 2 I I Transformation in polymeric systems 273 -, polymer-analogous 273 Transition points 55 Translucence 97 Turbidity 97

Turnbull-Fischer equation 307 Two parameter model 1 I2 Two-phase system 3 I0

UCTD 312 UCTM 303 Unsaturated polyester resins 229,255 Unvitrified gels 293 Unwinding of molecular coils 282 Upper critical temperature dissolution 3 12 Urethane 216

Vibration instruments 22 Vinyl monomers 60 Viscometer 22, 142 -, rotation 1 19 Viscometric analysis 23 1 Viscometry 40, 146, 183 -, capillary 40 -, rotary 42 Viscosity 13, 16,20f, 28 -, change of 220 - dependence 20 - increase 48,81, 146 -, kinetics of 44,228 - measurements 40 -, Newtonian 33, 84, 103 - of the medium 78 -, pattern of 21 -, reactive mass 87 -, temperature dependence of 29 Viscous-flow state 210 Vitrification 137, 153 - temperature 250

Weight, molecular I2 WLF 276 - correlation IS3

Xylenol 207

Documents

Rheokinetics: Rheological Transformations in Synthesis and Reactions of Oligomers and Polymers