22
Pergamon Int. J. Engng Sci. Vol. 35, No. 3, pp. 277-298, 1997 {~) 1997 Elsevier Science Ltd PII: S0020-7225(96)00078-X Printed in Great Britain. All rights reserved 0020-7225/97 $17.00+ 0.00 THERMODYNAMIC MODELING OF THE THERMOMECHANICAL EFFECTS OF POLYMER CRYSTALLIZATION: A GENERAL THEORETICAL STRUCTURE MEHRDAD NEGAHBAN Department of Engineering Mechanics and The Center for Materials Research and Analysis, 309 Bancroft Hall, University of Nebraska, Lincoln, NE 68588-0347, U.S.A. Abstraet--A general theoretical structure is developed based on continuum thermodynamics to model the thermomechanical effects of polymer crystallization. This phase transition, seen in many polymers, involves the gradual transformation of the polymer's microstructure from an unorganized amorphous structure to that of a much more rigid semi-crystalline structure. This smooth transition is captured by a set of integral models which obtain the response by averaging the apparent response of the amorphous portion and a continuum of different crystals. A commonly used empirical relation between the extent of crystallization and volume change is imposed as a restriction on the material, and the implication of the entropy production inequality in the presence of this constraint is evaluated. General representations are provided. © 1997 Elsevier Science Ltd. All rights reserved. 1. INTRODUCTION In this article a theoretical structure is proposed to model the thermomechanical response of polymers which can undergo crystallization. Polyethylene, polypropylene, isotactic polystyrene, and natural rubber are examples of such polymers. The process of crystallization in polymers involves a gradual change in microstructure from a disordered amorphous morphology to that of a semi-crystalline morphology, which consists of both amorphous and crystalline regions. This transition is accompanied by several physical changes in the polymer which include a reduction in volume, a pronounced increase in rigidity, and the development of residual strains when crystallization occurs under deformation. Factors which affect this phase transition include temperature and strain. A general background on the phenomenon of crystallization in polymers as it relates to the polymer's thermomechanical response is presented in Section 2. In practical applications crystallization can be responsible for stress concentration around rigid inclusions and voids as shown by Ma and Negahban [1], for example. This can in turn result in damage through void formation or crack propagation. In general, crystallization occurs between the glass-transition temperature and the melting temperature and substantially changes the properties of the polymer. Frequently when processing polymers, the polymer is heated to within the temperature range of crystallization. Therefore, inclusion of the effects of crystallization when simulating these processes becomes essential if one desires to obtain an accurate thermomechanical picture. Examples of manufacturing processes which heat the polymer to within the range of crystallization include injection molding and most extrusion processes. Even though Negahban and Wineman [2] have proposed a mechanical theory for capturing the effects of isothermal crystallization, and this model has shown success in reproducing experimental results, in most realistic processes crystallization is occurring under non-isothermal conditions, and the change in thermal conductivity and the heat of formation associated with crystallization controls the heat transfer, which in turn controls the crystallization process. Therefore, the development of a thermo- dynamic model as proposed in this paper is essential for the study of more practical problems. Section 3 contains a description of the kinematic variables used in this presentation. The equations of continuum thermodynamics and related notation are given in Section 4. 277

Thermodynamic modeling of the thermomechanical effects of polymer crystallization: A general theoretical structure

Embed Size (px)

Citation preview

Pergamon Int. J. Engng Sci. Vol. 35, No. 3, pp. 277-298, 1997

{~) 1997 Elsevier Science Ltd PII: S0020-7225(96)00078-X Printed in Great Britain. All rights reserved

0020-7225/97 $17.00 + 0.00

T H E R M O D Y N A M I C M O D E L I N G OF T H E T H E R M O M E C H A N I C A L EFFECTS OF P O L Y M E R

C R Y S T A L L I Z A T I O N : A G E N E R A L T H E O R E T I C A L S T R U C T U R E

MEHRDAD NEGAHBAN Department of Engineering Mechanics and The Center for Materials Research and Analysis,

309 Bancroft Hall, University of Nebraska, Lincoln, NE 68588-0347, U.S.A.

Abstraet--A general theoretical structure is developed based on continuum thermodynamics to model the thermomechanical effects of polymer crystallization. This phase transition, seen in many polymers, involves the gradual transformation of the polymer's microstructure from an unorganized amorphous structure to that of a much more rigid semi-crystalline structure. This smooth transition is captured by a set of integral models which obtain the response by averaging the apparent response of the amorphous portion and a continuum of different crystals. A commonly used empirical relation between the extent of crystallization and volume change is imposed as a restriction on the material, and the implication of the entropy production inequality in the presence of this constraint is evaluated. General representations are provided. © 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

In this article a theoretical structure is proposed to model the thermomechanical response of polymers which can undergo crystallization. Polyethylene, polypropylene, isotactic polystyrene, and natural rubber are examples of such polymers. The process of crystallization in polymers involves a gradual change in microstructure from a disordered amorphous morphology to that of a semi-crystalline morphology, which consists of both amorphous and crystalline regions. This transition is accompanied by several physical changes in the polymer which include a reduction in volume, a pronounced increase in rigidity, and the development of residual strains when crystallization occurs under deformation. Factors which affect this phase transition include temperature and strain. A general background on the phenomenon of crystallization in polymers as it relates to the polymer's thermomechanical response is presented in Section 2.

In practical applications crystallization can be responsible for stress concentration around rigid inclusions and voids as shown by Ma and Negahban [1], for example. This can in turn result in damage through void formation or crack propagation.

In general, crystallization occurs between the glass-transition temperature and the melting temperature and substantially changes the properties of the polymer. Frequently when processing polymers, the polymer is heated to within the temperature range of crystallization. Therefore, inclusion of the effects of crystallization when simulating these processes becomes essential if one desires to obtain an accurate thermomechanical picture. Examples of manufacturing processes which heat the polymer to within the range of crystallization include injection molding and most extrusion processes. Even though Negahban and Wineman [2] have proposed a mechanical theory for capturing the effects of isothermal crystallization, and this model has shown success in reproducing experimental results, in most realistic processes crystallization is occurring under non-isothermal conditions, and the change in thermal conductivity and the heat of formation associated with crystallization controls the heat transfer, which in turn controls the crystallization process. Therefore, the development of a thermo- dynamic model as proposed in this paper is essential for the study of more practical problems.

Section 3 contains a description of the kinematic variables used in this presentation. The equations of continuum thermodynamics and related notation are given in Section 4.

277

278 M. NEGAHBAN

The commonly made assumption of incompressibility of the amorphous phase and also of the crystalline phase in isothermal processes is presented in Section 5. These assumptions result in a relation that is commonly used to connect the degree of crystallinity to the change in volume of the polymer, based on the known densities of the amorphous and crystalline portions.

Section 6 presents the general structure assumed for the constitutive equations, and also provides the underlying motivations for these assumptions. Section 7 discusses the implications of the entropy production inequality, and provides the expressions for Cauchy stress, entropy, and restrictions on the rate of crystallization and heat flux.

Seciton 8 discusses the implications of the entropy production inequality on the equilibrium states of the material. In this case equilibrium refers to fixed conditions which result in termination of the crystallization process.

Section 9 introduces the restriction of objectivity, and imposes it on the constitutive functionals.

Section 10 presents general representations under the assumption that the initial undeformed amorphous polymer is isotropic. This assumption is consistent with the random microstructure of the initial undeformed amorphous material.

Section 11 introduces a slightly different, yet mathematically equivalent, set of constitutive assumptions which may be more appropriate for dealing with actual experimental results.

2. BACKGROUND

This article is primarily motivated by the behavior of natural rubber as observed in experiments starting in the early 1930s, to experimental results as recent as 1989. Yet the modeling techniques and the thermodynamic results should hold valid for many crystallizable polymers. A general introduction to the mechanics of rubber can be found in the book by Treloar [3]. An overview of the experimental and theoretical research in polymer crystallization can be found in the collected works of Flory [4], the book on Crystallization of Polymers by Mandelkern [5], and the three volumes on Macromolecular Physics by Wunderlich [6-8].

On a microscopic level the process of crystallization is characterized by nucleation, growth and equilibrium. Initially nuclei develop at different locations in the material, either randomly or at precursors. The nuclei either grow past a critical size to yield stable crystals or are destroyed. As the stable crystals are growing, new nuclei are generated in the uncrystallized region, and result in the growth of new crystals. This process of nucleation and growth continues until the growth of crystals is restricted by the crystals impinging on one another, or through any means in which the motion of the polymer chains in the amorphous portion is constrained.

The process of crystallization in polymers normally occurs in the rubbery range, which is characterized at the lower limit by the glass-transition temperature, and at the higher limit by the melting temperature. At either limit the rate of crystallization is very small, reaching a maximum somewhere within this range. At the upper limit the internal energy of the system is too high to allow the formation of stable crystals, at the lower limit the motion of the molecules are too slow for the formation of crystals. For example, for natural rubber the glass-transition temperature is -50°C and the melting temperature is 30°C, with a maximum rate of crystallization at a temperature of -26°C, as shown in Fig. 1. The traditional way to model the kinetics of crystallization is based on the work of Avrami (see Mandelkern [5]), which is a theory for the filling of space through the nucleation and growth of a second phase into the first phase. Avrami's model is based on the equation

~¢ = (1 -xo )~ ; , (1)

Thermomechanical effects of polymer crystallization 279

0.5

0.45 O

0.4

0.35 O

N 0.3

0.25 [. 0.2

• 0.15

0.1

0.05

0 0

0

0

0

0

0

0 | | o n - - - Z

-50 -30 -10 I0

Temperature, C

30

Fig. 1. Rate of crystallization for natural rubber represented by the inverse of the time required to complete one half of the crystallization (from Wood and Bekkedahl [26]).

where xc is the mass fraction of crystallized material , ' .... denotes the time derivative, and x" is the mass fraction of crystalline mat ter if it is assumed that nuclei may be generated and may grow without restriction at all points in the material (even in existing crystals). The expression for x~ is given by

Yo' ' - w(t, r)]V'(r) dr, (2) X c - -

where w(t, r) is the current mass of a crystal formed from a nucleus generated at time r, N ' is the nucleation frequency so that N ' d r is the number of nuclei generated per unit mass in the interval dr. The result of this kinetic theory is an expression for the extent of crystallinity in the

form

] x ~ = l - e x p - w ( t , r ) N ' ( r ) d r . (3)

In this method of analysis one defines the nucleation rate and growth rate of crystals by providing models for w(t, r) and N' . Frequently, after such assumptions one arrives at an equat ion of the form

Xc = 1 - exp[ -k t " ] , (4)

where k is understood to depend on tempera ture and n is associated with the shape of the crystal growth. For example, n f rom 3 to 4 represents three dimensional growth such as spheres, n f rom 2 to 3 represents two dimensional growth such as plates, and n from 1 to 2 represents one-dimensional growth such as rods. This form of the Avrami equation is very attractive for the analysis of experimental data since one can write

ln[ ln(1 _ - - ~ c ) ] = l n [ k ] + n l n [ t ] , (5)

and as a result one can both evaluate the validity of the equation and extract the Avrami coefficients by plotting I n [ I n [ i / ( 1 - x ) ] ] as a function of In[t]. At least three facts must be satisfied for one to use this equation. First, the tempera ture must be constant. Second, the

280 M. NEGAHBAN

B/ c

B C / C ~ j A

Fig. 2. Schematic of a polymer chain anchored in two different crystals (A), two polymer chains in the same crystal (A-B or A-C), and a polymer chain terminating in a crystal (C).

external conditions must be fixed over time (i.e., one must have conditions of constant load, constant strain, or constant pressure, for example). Third, crystallization must be capable of proceding to 100% crystallinity.

Unlike metals, the process of crystallization in polymers usually stops far short of full crystallization. For example, crystallization in rubber normally stops with no more than 20-30% crystallization, even under the most ideal conditions. This termination of the crystallization process before complete crystallization is a result of the long chain-like morphology of polymers. That is, many chain segments need to move for a chain segment to take its place in the crystal structure. Anything which reduces the mobility of chain segments will therefore reduce the possibility of further crystallization. In general, the long chain-like morphology results in many polymer chains sharing in the formation of a single crystal, a single polymer chain frequently spanning over many crystals, and a single polymer chain having both parts which reside in a crystal structure and parts which are in an amorphous structure. Figure 2 shows a schematic of some of these possibilities. Each of these can reduce the mobility of chains. For example, a chain which is anchored in different crystals may have an amorphous part which is not sufficiently mobile to crystallize. Another way in which chain mobility can be reduced is by cooling the polymer below its glass-transition temperature. Normally, a modified Avrami equation is introduced to capture the fact that crystallization rarely proceeds to 100% crystallinity in polymers. This equation replaces (1) and is given by

Xc(~) - x~(t) . f~(t) - ~ ' ( t ) , ( 6 )

xc(~)

where x~(~) denotes the extent of crystallinity at infinite time (equilibrium). Under the assumption of constant terminal crystallinity, xc(~), this equation can be integrated to give

xc(t)=Xc(~C){l- exp[--f0' w(t, z)/'Q'('t') d~]}, (7)

Again, under many of the commonly made assumptions for w(t, r) and N' (see Mandelkern [5]) one arrives at an Avrami type equation

x¢(t) = x¢(~){1 - exp[-kt ']}. (8)

In the analysis of processes such as those observed in manufacture of polymer parts, in which the thermal and mechanical loading conditions are continuously varying, it is frequently impossible to assume a fixed value for xc(~). In such cases, one needs to at least modify xc(~) to include a dependence on the past history of thermomechanical loading.

Crystallization is accompanied by a reduction in volume. This reduction of volume is a result of the higher density of the crystal structure, as opposed to the amorphous structure of the polymer before crystallization. Change of volume is commonly used as a measure of the extent of crystallinity in polymers (see, for example, Gent [9]), even though X-ray diffraction and

T h e r m o m e c h a n i c a l e f f ec t s o f p o l y m e r c r y s t a l l i z a t i o n 281

0.5

0 . o = m " z . = ~ . . • ' A BIB •

~7 -0.5

-1

! -~ -1.5 • • • % >.

- o - Stress for stretch = 3 ~ ] ~ • -2 - e - Stress for stretch = 2 | • i0 A O• • •

--o- Stress for stretch = 1.5 [ • • • • • • Volume change for stretch = 3 [

-2.5 • Volume change for s t r e t ch f f i 2 [

• Volume chaage for stretch =1.5]

-3

1 2 3 4

Logl0 (lime), rain

Fig. 3. Experimental results of Gent [9] for volume reduction and stress relaxation of vulcanized natural rubber at -26°C. The stress relaxation is superimposed on the volume change using a linear relation. In these experiments stresses relaxed to zero at the points where the continuous lines end,

yet volume reduction continued long after the stresses have gone to zero.

calorimetry are also commonly used to characterize crystallinity (see, for example, Goppel [10], Ar lman [11], and G6ritz and MUller [12]). It is common to assume a linear relation between crystallinity and reduction of volume, for natural rubber this would be about 11.7% increase in

crystallinity for 1% reduction in volume [9]. The range and rate of crystallization can be altered by stretching. For example, stretched

rubber at room tempera ture will crystallize and will show noticeable mechanical effects related to crystallization, yet unstretched rubber at room tempera ture will not have any noticeable crystallization. In the case of room tempera ture natural rubber, crystallization under strain will normallly be removed upon unloading. This is not the case for crystallization of rubber at lower temperatures , where crystallization under constant stretch results in stress relaxation, a reversal of stress f rom tension to compression, and permanent residual strains (see Gent [9] and Stevenson [13, 14]). The reversal of stress, f rom tension to compression, indicates that at lower temperatures crystallization does not stop upon removal of the load. The results of experiments conducted by Gent [9] have shown that in stress relaxation of natural rubber there is a linear relation between the reduction of stress and change in volume. Gent showed this by linearly scaling the stress relaxation onto the volume reduction as seen in Fig. 3. For isothermal conditions the axial stress tr under constant stretch can, therefore, be represented by an equat ion of the form

d ~ = ~ o + - - x c , (9)

dxc

where tro and dtr/dxc are constants for a given stretch. Stress relaxation in this case is understood as a process of replacing stressed amorphous material by fairly stress-free crystals, even though Gent ' s results suggest much larger stress relaxation than would result f rom a simple replacement. This is evidenced by the fact that only about 10% crystallization is necessary to fully unload a sample of stretched natural rubber.

Two additional effects of crystallization are an increase in rigidity and an increase in toughness. For example, Leitner [15] has shown that the increase in elastic modulus of unconstrained natural rubber due to crystallization can be up to two orders of magnitude, as is

282 M. NEGAHBAN

120

~ 100 spec.A ? u

80 m Leitner [15] o °

"m--~ "~ oLeitner [15] spec.B o u

60 ~Theore t ica l [21] o o

~ 40

20

0 ' 0 0.5 1 1.5 2 2.5

Increase in density, %

Fig. 4. Experimental results of Leitner [15] for the change in elastic modulus for unconstrained vulcanized natural rubber at 0°C. The continuous line is a fit of this data using the model proposed by

Negahban et al. [21].

shown in Fig. 4. Unlike stress relaxation, the increase of modulus is not a linear function of crystallinity, even under the simplest loading conditions, as is shown in Fig. 4 for unconstrained crystallization. Van Krevelen and Hoftyzer [16] suggest that the shear modulus of a semi-crystalline polymer, G~c, can be obtained from the shear modulus of the amorphous polymer, Ga, and the shear modulus of the crystalline polymer, Go, from the relation

(10)

Unlike stress relaxation, the modulus is a reflection of how the material responds to further loading.

Mechanical effects of crystallization have been characterized by several authors. The work of Ahzi, Argon, Lee, and Parks (see [17, 18]), for example, is based on the characterization of the microstructure of an existing semi-crystalline polymer and providing a method for evaluating the mechanical response of this model microstructure as the polymer is subjected to plastic flow. The models provided by these authors are not in the form of macroscopic constitutive equations, but a set of operations which can be used to evaluate the macroscopic response from a proposed microstructure, knowing microscopic material parameters. The work of Negahban and Wineman [2] has focused on modeling the macroscopic mechanical effects of the crystallization process itself, providing a method of capturing effects such as stress relaxation, changes in residual strains, and changes in mechanical moduli as the process of crystallization proceeds in a mechanically loaded polymer. The basic idea used by these authors to capture the effect of changing crystallinity is a model for Cauchy stress T at the current time t of the form

T(t) = b(t)TA(t) + To(t, s)a(s) ds, s

(11)

where TA(t) is an effective stress for the amorphous material, Tc(t, s) is an effective stress for the crystal created at time s, b(t) is the fraction of amorphous material, a(s) is the rate of crystallization, and & is the time crystallization starts. As is shown in Figs 5, 6 and 4, for natural

Thermomechanica l effects of polymer crystallization 283

16

14

12

~ 8

~ 6

4

2

0 t

1 2 3 4 5 6

Stretch Ratio

Fig. 5. Experimental results of Min [19] for the instantaneous response (one second after loading) of natural rubber at 22°C. ( ) is a fit of these data using the model proposed by Negahban [24].

rubber this type of modeling has been able to capture the instantaneous response before crystallization as observed by Min [19], it has been able to capture the linear stress relaxation observed by Gent [9], and has been able to capture the non-linear increase in elastic modulus observed by Leitner [15]. The work of Flory (see [20]) also looks at the change in mechanical properties during the crystallization process, and is based on a statistical mechanics model of the crystallization process. This work includes the effects of temperature, yet it is currently limited to crystallization under fixed external conditions.

3. K I N E M A T I C S AND N O T A T I O N S

In the following presentation it will be assumed that the polymer body on a macroscopic scale is a continuum, for which one has selected a reference configuration Ko. The motion of this material body is defined by the history of configurations it takes between the time ts when crystallization starts and the current time t. The configuration at any time s between ts and t will be denoted by K(s). The deformation of the body at time s will be described at each material point by the deformation gradient F(s), which compares K(s) to Ko. As shown in Fig. 8, the relative deformation gradient is denoted by Fs(t), and defined by

Fs(/) = F(t)F-'(s), (12)

where the superscript " - 1 " denotes the inverse. For any deformation gradient F, C = FTF will denote the right Cauchy strain tensor, and B = FF T will denote the left Cauchy strain tensor. In the polar decomposition of F = RU = VR, R will denote the orthogonal factor, U will denote the right symmetric factor, and V will denote the left symmetric factor. The velocity will be denoted by v. The velocity gradient will be denoted by L and can be evaluated from L(s) = F(s)F-l(s) , where " ' " denotes the material time derivative.

The volume ratio J(s)= det[F(s)] is the ratio of the volume of the neighborhood of a material point at time s to the volume of the same neighborhood in the reference configuration, and where det [ . ] denotes the determinant operation.

The operator " ' " between two vectors denotes the dot product. The operator " : " between second order tensors A and B is defined as A : B = tr(ABT), where tr(.) denotes the trace operation. The magnitude IAI of a second order tensor A is defined by IAI = ~/A : A.

284 M. N E G A H B A N

2

1.5

[,- 1

3

%.

2.5

0.5

\ \

\

0 1.6

, l!sech 1 Stretch = 2 /

X Stretch=3 /

Stretch = 1.5j

" Stretch=2 ]

'~ ' ~ ~ Stretch=3 ] ; - . . \<,

0.2 0.4 0.6 0.8 1 1.2 1.4

P e r c e n t r e d u c t i o n i n v o l u m e

Fig. 6. Experimental results of Gent [9] for stress realization as a function of volume reduction for natural rubber under constant stretch and at -26°C. The lines indicate the fit of these data using the

model proposed by Negahban et al. [21].

The operator "OA" denotes the partial derivative with respect to A, and is defined such that, for example, for the function ~b(A) of a second order tensor A one has

= ,~Ad~" A. (13)

4. K I N E T I C S A N D T H E R M O D Y N A M I C S

On a macroscopic level the polymer body is assumed to obey the classical balance laws and the entropy production inequality.

The conservation of mass requires that

p(s)J(s) = Po, (14)

Fig. 7. Volume

b = l . . . . . . . . b = 0 . 8 * - - b = 0 . 6 ~ - b = 0 . 4 ~ - b - - 0 . 2 b---O ]

0.98

0.96

~ 0 . 9 4

~ 0.92

0.9

0.88

0.86 i I I i

200 220 240 260 280

Temperature, K

ratio as a function of temperature for natural rubber crystallinities [25].

300

at different constant

Thermomechanical effects of polymer crystallization 285

where po is its density in the reference configuration and p(s) is its density in the configuration at time s.

The balance of linear momentum requires that at each time s and at each material point

div(T) + pb = pi,, (15)

where div(. ) denotes the divergence, T is the Cauchy stress, and b is the body force per unit mass. The balance of angular momentum results in the symmetry of the Cauchy stress.

The balance of energy requires that at each time s and at each material point

p0 = tr(TL) - div(q) + or, (16)

where e is the internal energy density, q is the heat flux vector, and r is the rate of heat generation or radiation density.

The entropy production inequality, also known as the Clausius-Duhem inequality, requires that at each material point one has

(~ ) r (17) Or)/> -d iv + p ~,

where ~7 is entropy density and 0 is temperature. An alternate form of this equation can be obtained after the introduction of the balance of energy and assuming strictly positive temperatures. This form is given as

1 p~b - tr(TL) + 0770 + ~ q . g ~< 0, (18)

where ~/, = e - r/0 is the Helmholtz free-energy, and g is the temperature gradient.

5. R E L A T I N G C R Y S T A L L I N I T Y , VOLUME RATIO, AND T E M P E R A T U R E

As we stated above, the degree of crystallization on a macroscopic level is characterized by change in volume or change in density, since the crystals are denser then the amorphous part. For isothermal conditions, it is common to directly relate the degree of crystallization to the change in volume (see Gent [9]). This implies that there is a relation between the extent of crystallinity, the volume ratio, and temperature. Such a relation will be developed in this section and illustrated for natural rubber.

Before we proceed, it must be noted that such a relation will act as a constraint on the response of the material and, therefore, will introduce an indeterminacy in describing the material response. For example, if the temperature and degree of crystallinity are held fixed, the imposed relation will require the volume to be fixed. Therefore, the material will be incompressible under such conditions. A result of this is an infinite heat capacity at constant volume and constant crystallinity. If this is undesirable, one can avoid this by introducing an additional dependence on pressure. The addition of this dependence will be discussed later. On the other hand, even in the absence of pressure in the constraint, the effect of pressure on crystallization appears in the equation for the rate of crystallization through its dependence on the introduced indeterminate parameter (see Section 6).

The relation between crystallinity, volume ratio, and temperature is developed by assuming each phase of the polymer to be incompressible under isothermal conditions, therefore, allowing changes in volume under isothermal conditions to occur only when matter transforms from one phase to the other. Implied in this statement is the assumption that matter only exists either in the amorphous phase or in the crystalline phase. Let b(t) denote the mass fraction of the amorphous phase at the current time t and a(s) denote the rate of crystallization at any

286 M. NEGAHBAN

time s. The above assumption combined with the assumption that the polymer at time ts is entirely amorphous, and that mass is conserved in the phase transition requires that

and, therefore,

f t b(t) = 1 - a(s) ds, t

(19)

tJ(t) = --a(t). (20)

Next, it is assumed that the density of the amorphous part of the polymer is described by a function RA(0), and the density of all crystals are the same and described by the function pc(0). Adding to this the hypothesis that at each material point the total macroscopic volume is the sum of the volume of the amorphous part and the crystals, one will arrive at the relation

1 1 1 - - - b + - - ( 1 - b), (21) p(O, b) pA(0) pc(O)

where p(O, b) is the macroscopic density. Following the assumption introduced in Section 4, that on a macroscopic level the polymer

must follow the balance laws of continuum mechanics, one obtains, after the use of conservation of mass, that

J (O,b )=- Po b + Po ( l - b ) , (22) pA(O) m.(0)

where J(O, b) is the volume ratio, and po is the macroscopic density at the material point in the reference configuration.

Since J = det(F), equation (22) is a restriction on how F may change. The relation between the rate of change of volume, crystallinity and temperature is, therefore, given by taking the derivative of (22) to get

[pobdpA p o ( 1 - b ) [Po Po]a ) = - [ p 2 d0 + p~- ~PO'I 0 - kpA PC J . (23)

A typical plot of the volume as a function of temperature at constant crystallinity is shown in Fig. 7 for natural rubber using as reference the volume of amorphous natural rubber at 298 ° K. In most cooling processes crystallization does not remain constant and as a result a typical cooling plot would not only involve volume reductions due to cooling, but also volume reductions due to crystallization.

It must be noted that the constraint developed in this section is one of convenience and one which is commonly made to analyse experimental results (Gent [9], Stevenson [13, 14]). Using it reduces the number of experiments needed to characterize the polymer. Like any other

s Crystal formed at time s

F ( s ~ . [ ~ ~ % Fs(t)

Current shape of crystal formed at time s

Fig. 8. Reference configuration, intermediate configuration, current configuration, and kinematical variables.

Thermomechanical effects of polymer crystallization 287

constraint, its use must be evaluated in the context of the type of problems under consideration. For example, one should avoid this constraint in cases where it is essential to allow the volume to change without changes in crystallinity and temperature. Yet the economy of analysis gained by its use has made it an attractive tool in the understanding of many experimental results.

6. THE S T R U C T U R E OF THE CONSTITUTIVE R E L A T I O N S

The basic idea on which the constitutive relations are structured is that the macroscopic response is the sum of the effective response of the amorphous fraction of the polymer and the effective response of a continuum of different crystals, continuously generated over the crystallization period. This type of modeling lets one change the strain or stress during crystallization, it lets one introduce effects of interaction between crystals as the degree of crystallization increases, and lets one capture the strain, and hence stress, developed in old crystals due to the generation of new crystals. The idea behind such a modeling method was probably first introduced by Flory [20], when trying to describe why the stress is larger in vulcanized rubber which has undergone crystallization during stretching, in comparison to the same material in an uncrystallized form, when crystallization at a constant stretch should reduce the stress. Flory describes this "paradox" as follows:

"In attempting to characterize the effects of crystallization on elastic properties of rubber, one is faced with a superficially paradoxical situation. Abundant evidence shows that the tension in a highly elongated vulcanized rubber which has undergone crystalliza- tion during stretching is substantially greater than in another similarly vulcanized rubber (or the same rubber at higher temperature) in which no crystallization occurs at the same elongation. On the other hand, theory leads inevitably to the conclusion that crystalliza- tion should decrease the tension. This can be deduced from a theoretical analysis from thermodynamic (equilibrium) considerations."

Flory's explanation is:

"In the ordinary "non-equilibrium" stretching of rubber, the first crystal nuclei may be presumed to form at an extension a l only slightly beyond the extension for incipient crystallization at equilibrium. Growth of these crystallites may actually reduce the tension somewhat at elongations not greatly exceeding al . . . . . As the elongation is increased to a2, these crystallites, being of sufficient permanence to resist dissolution in favor of a crystalline arrangement more stable at the higher elongation, act in the manner discussed above to increase the tension. However, further crystallization will occur at a2 and thus counteract this effect. The portion of this further crystallization which consists of nucleation and of attachment of chains to existing crystallites (lateral, or apart from longitudinal, growth of existing crystallites) introduces restraints which increase the tension developed in passing to a higher elongation a3; etc. In this way a state is soon reached where the tension rises rapidly with elongation."

In describing the reason for the larger than expected stresses, Flory has also exposed the necessity to consider crystallization in polymers as a continuous process of crystal formation and subsequent straining of formed crystals. Therefore, one cannot consider polymer crystallization as an abrupt transition, but must consider it as a continuous process. In the case of crystallization in the process of uniaxial extension, the stretching of the crystals formed before the completion of stretching is the element which increases stresses above what is seen for similar pure amorphous materials. This is in spite of stress relaxation due to crystallization.

As in [21], in this article time is used as a marker to distinguish between the different

288 M. NEGAHBAN

conditions under which the crystals are formed, and integration is used to obtain the macroscopic effect of this continuum of crystals, each formed under different conditions and subjected to different strains. To thermodynamically characterize the polymer, first a free-energy ~ is selected of the form

J/ t~(t) = b(t)~A(t) + qk-(t, s)a(s) Ch', (24)

where b(t) is the current mass fraction of amorphous material, a(s)ds is the mass fraction of crystals created in the interval of time ~', ~tA(t ) is the current effective free-energy per unit mass in the amorphous part, qJe(t, s) is the current effective free-energy per unit mass in the crystal generated at time s, and t, is the time at which crystallization starts. This model captures the ideas developed by Flory [20] by selecting a continuum of effective free-energy functions qJc(t, s), each describing the contribution of the free-energy of the crystal generated at a given time s on the macroscopic free-energy of the material at the current time t.

The current effective free-energy in the amorphous part is assumed to depend on the current deformation gradient, F(t), the current temperature, O(t), and the current value of the mass fraction of amorphous material, b(t). This relation will be written as

q'A(t)---- ~k[F(t), O(t), h(t)], (25)

where a superscript " t " shall denote the constitutive function or functional for a given physical quantity. The form of (25) suggests that the "actual" free-energy of the amorphous part contributes to the effective free-energy of the amorphous part in relation not only to the macroscopic strain and temperature, but also in relation to the proportion of the amorphous and crystalline phases. This additional dependency on the mass fraction of the amorphous portion is necessary since the macroscopic strains do not capture the actual inhomogeneous nature of the microscopic strains.

The current value of the effective free-energy in the crystal or portion of crystal created at time s will be assumed to depend on the current values of deformation gradient, temperature and volume fraction of amorphous material, and the values of these three variables at time s. This functional relation will be expressed as

~Oc(t, s) = ~b~:[F(t), O(t), b(t), F(s), O(s), b(s)]. (26)

This assumption is reflective of the assumption that the effective free-energy in a crystal created at time s is related to the conditions under which the crystal was generated, in addition to the conditions which are imposed on the crystal after its generation.

In an analogous way, it will be assumed that the Cauchy stress, entropy, and heat flux are given by constitutive functionals of the form

f' T(t) = b(t)TA(t) + To(t, s)a(s) ds, (27)

t .

f' ~(t) = b(t)~TA(t) + ~lc(t, s)a(s) ds, (28)

Ix

S q(t) = b(t)qA(t) + qc(t, s)a(s) ds, (29) Is

where a subscript " A " denotes the effective amorphous contribution and a subscript "C" denotes the effective crystal contribution. The functional dependence of the effective contributions are given by

TA(t) = Tk[F(t), O(t), b(t), p(t)], (30)

7/A(t ) = "0~[F(t ) , O(t), b(t), p(t)], (31)

qA(t) = qk[F(t), O(t), b(t), g(t), p(t)l, (32)

Thermomechanical effects of polymer crystallization 289

and

Tc(t, s) = T*c[F(t), O(t), b(t), F(s), O(s), b(s), p(t)], (33)

nc(t, s) = rt*c[F(t), O(t), b(t), F(s), O(s), b(s), p(/)], (34)

qc(t, s) = q*c[F(t), O(t), b(t), g(t), F(s), O(s), b(s), g(s), p(t)], (35)

where p is a scalar function introduced to capture the indeterminacy introduced as a result of relation (22), the relation between crystallinity, volume ratio, and temperature. In the case of stress the indeterminacy results due to the fact that the material can no longer freely change its volume under the application of a sudden load. For example, a sudden application of load in any isothermal process cannot instantaneously change the volume since the volume change is directly connected to the change in crystallinity, and crystallinity cannot instantaneously change. One is therefore physically motivated to introduce an indeterminate scalar function into the argument of the functional for T(t). The need to also introduce this same indeterminate scalar parameter into the expressions for rt(t) and later into the functional for a(t) becomes apparent when trying to satisfy the ~ntropy production inequality. In anticipation of a potential problem, the same indeterminate scalar function introduced into the expression for the Cauchy stress has also been added to the functionals for entropy and heat flux, and will later be introduced into the rate of crystallization.

Finally, the current rate of crystallization, described by a(t), is assumed to depend on the history of the values of the deformation gradient, temperature, and volume fraction of amorphous material up to the current time, but not on the rate at which these values are currently changing. As stated above, the indeterminate scalar function p also will be added to the functional for the rate of crystallization. This dependence will be written as

[ ' 1 a(t) = a* F(s), O(s), b(s), p(t) . (36) S=t~

The form essentially imbeds in it the assumption that the rate of crystallization at a particular time is defined through the conditions up to and including that time, but not on the rate at which these conditions are changing. It can be seen that the selected form simulates "instantaneous" or "spontaneous" crystallization only as a limit of conditions which make the rate of crystallization become very large. In theory, the equation for the rate of crystallization given by (36) may be given by an Avrami type equation such as that given by equations (6) and (2). In practice, such a selection is excluded by the entropy production inequality, as will be shown later.

7. I M P L I C A T I O N S OF THE ENTROPY P R O D U C T I O N I N E Q U A L I T Y

The Clausius-Duhem inequality (18) can be used to derive constraints on the form of the physical variables. To use (18) one needs to calculate the material time derivative of the Helmholtz free-energy. Following the assumptions given in (24), (25), and (26) on the functional form for the free-energy, one will obtain

f' ~b(t) =/~(t)~OA(t ) + b(t)(OA(t) + qJc(t, t)a(t) + ~c(t, s)a(s) ds, (37)

ts

where

~bA(t) = 0F,)~bA(t) :F(t) + ao,)~bA(t)O(t) + Oh,)d/A(t)lJ(t) (38)

and

~bc(t, s) = ov,)~bc(t, s) : F(t) + Oo~t)tPc(t, s)O(t) + 0b,)~bc(t, s)f~(t). (39)

As noted above, " a " denotes a partial derivative. Substitution of (37)-(39) into the entropy

290 M. NEGAHBAN

production inequality (18), using the relation t r (TL)= (TF-V) :F " and /~(t)= -a(t) , and some reorganization results in

f ' ds -T ( t )F -T ( t ) I : F'(t) lp(t)b(t)Ov~,)qJA(t) p(t) Ovm~,.(t, s)a(s) + s

[ S' 1 + p(t) 71(t) + b(t)Oo(,¢PA(t) + C?o(,~Oc(t, s)a(s) ds O(t) Is

[ - p(t)tOA(t) -- 0c(t, t) + b(t)Oh.~OA(t) + ~h,~Oc(t, s)a(s)

1 + - - q(t). g(t) ~< 0. (40)

0(t) The following lemma may be used to obtain functional relations between the variables. The lemma states that an inequality of the form

b,/3, + d~,,+, ~0 , (41) i I

which must be satisfied for all possible values of/3~, assuming that the /3 are independent and that the ~b do not depend on the/3, results in

~bi=0 for i = l . . . . . n and ~b,+~<0. (42)

Since the components of F" and 0 are related through the constraint (23), the conditions of the lemma are not satisfied. One can work around this restriction by introducing an arbitrary second order tensor F into the problem such that

= F + y J r v, (43)

where 7 is selected so as to impose the restriction given in (23). Introducing the relation J = JF -T : ~" into (23), and use of F" from (43) result in the expression for 7 given as

1 { - J F - " : F [ b dpA 1 - b d p c ] o [ 1 1 1 a Y - j 2 t r ( C -1) -P"Lp~ dO + p2 d 0 J - - P ° k p A - - ~ J J"

(44)

Substitution of the above into the entropy production inequality (40) results in

p(t)b(t)Ov(,)~bA(t) + p(t) Ov(,Bbc(t, s)a(s) ds - T(t)F "(t) + /xF T . F'(t) t ,

+ p(t) t ,

- o ( t ) ~A(t ) -- ~(:(t, t) + b(t)O,..~OA(t) + ~, , .~c( t , s)a(s) Ct~" l ,

[ - ~ o ] ( t ) o~(t) b ~ q ( t ) " g(t) <~ o, (45)

where

/x - t r (C- ' ) p(t)b(t)OF(,~tpA(t) + p(t) ~?v(,)~O¢,(t, s)a(s) cks' -- T(t)F-r(t) : F T. (46) t~

Since ~'(t) and O(t) are independent and can be selected arbitrarily, one must have

[ f ] T(t) = p( t ) i + p(t) b(t)Ov(,~tOA(t) + 0~l,BOc(t, s)a(s) ds F'( t ) (47) t ,

and

f ' . .[ b(t) dpA 1 - b(t)dpc] rl(t)=---b(t)a°"'tPA(t)-- c)°"'tOc(t's)a(s)ds--PU)[p~A(t ) d-O ~- pSc~ -d-O]' (48)

ts

Thermomechanical effects of polymer crystallization 291

where /z has been replaced by the indeterminate scalar p, since p has no specific physical meaning and since/x is a scalar and the only term on the right-hand side which is a function of p. The reader will note that ~bA and ~0c do not contain the indeterminate scalar p; this assumption is critical to the development. The form of equation (48) makes it immediately clear why p must be introduced into the functional for 7/. In addition to (47) and (48), if q is bounded at g -- 0, then (45) also results in

--{~OA(t)--~bc(t,t)+b(t)Oh(,)~OA(t)+ Ii'Ob(Ot~c(t,s)a(s)ds--p(t)[pl(t ) p~(t)]}a(t)<~O (49)

and

-p(t) ~bA(t) -- ~bc(t, t) + b(t)Ob(,)~bA(t) + 8t,(,)~bc(t, s)a(s) ds s

__p(t)[p~(t ) p~t)]}a(t) 1 + q(t) . g(t) 0. (50)

In particular, (49) acts as a restriction on the form of the function a* for the rate of crystallization, as given in (36). In addition, since the term in the curly brackets is a function of the indeterminate parameter p, it is clear that the only way for the function a* to satisfy (49) is if it also is a function of p. In general, p does not represent the hydrostatic pressure, but one can always reorganize the terms in (47), and define the hydrostatic pressure as the indeterminate parameter associated with the incompressibility assumption. Therefore, one can conclude that the rate of crystallization must be a function of the histories of deformation, temperature, and the extent of crystallization, and the current value of hydrostatic pressure. That is, with the removal of compressibility from the deformation, the entropy production inequality has forced us to replace compressibility by an alternate, but equivalent, parameter.

The constraint (49) imposed on the rate of crystallization excludes Avrami type equations such as is given by (6) and (2) from consideration. These types of equations, in general, cannot satisfy this constraint without introducing modifications. This will not be the case for the class of materials which do not have any free-energy, and for which the density of the amorphous and crystalline phases are the same. Even though this is an admissible class of materials for Avrami type behavior, such conditions do not exist in real materials.

8. E Q U I L I B R I U M C R Y S T A L L I N I T Y

Equation (49), in addition to being a restriction on the form of the function for the rate of crystallization a(t), also can be used to identify states of equilibrium. In this context equilibrium refers to equilibrium in the material. That is, fixed macroscopic conditions which stop the process of crystallization and render stress, entropy, and heat flux constant. With the assumption of continuity of the response functional for free-energy with respect to variations in F(t) and O(t) at any fixed degree of crystallinity, one can seek combinations of F(t), O(t), and p(t) which make the coefficient of a(t) in (49) zero. These combinations of F(t), O(t), and p(t) represent points at which a(t) must go to zero. This follows directly from the linear dependence on p(t) of the coefficient of a(t) in (49). Therefore, candidates for points of equilibrium in the sense of termination of the process of crystallization directly follow from setting the coefficient of a(t) equal to zero in (49), yet one must be aware that it is possible to satisfy (49) by having a(t) go to zero, in which case one will obtain termination of crystallization without the need for its coefficient to go to zero. Under the current assumptions the termination of crystallization has not yet been connected to the termination of changes in the stress and entropy. Any further discussion of obtaining points of true equilibrium needs to be accompanied by more specific assumptions on the form of the constitutive functionals.

292 M. NEGAHBAN

9. EFFECTS OF R I G I D BODY MOTIONS

As is clear from the results (47)-(50) for the thermodynamic description of response during crystallization, one need only provide functions or functionals for the Helmholz free-energy ~0, the heat flux q, and the rate of crystallization a. Therefore, the discussion of the effects of rigid body motions will be restricted to these three variables.

Let z (X, s) for s E (ts, t] denote a given motion of the body. Let z*(X, s) = Q(s)z(X, s) + c(s), for every s ~ (t~,t], be a motion obtained from Z by the superposition of rigid body translations and rigid body rotations. In this expression c is a vector and Q is an orthogonal second order tensor (i.e., QQV = I). Let F* designate the deformation gradient associated with the motion Z*, and let F denote the deformation gradient associated with the motion Z. One has the relation

F*(s) : Q(s)F(s) (51)

for every s ~ (t~, t]. In the following, "*" will denote a variable evaluated for the history of (F*, 0) as opposed to the history of (F, 0).

The common assumptions associated with the superposition of rigid body motions (objec- tivity or material frame-indifference) are

a*(t) = a(t), (52)

q,*(t) : q,(t), (53)

q*(t) = Q(t)q(t). (54)

and

Imposing (52) on the functional describing the rate of crystallization yields

[ ' 1 [ ' 1 a* Q(s)F(s), O(s), b ( s ) , p ( t ) --a* F(s), O(s), b ( s ) , p ( t ) , (55) S = t , s = t ~

for every history of the orthogonal tensor Q and any history of F, 0, and b. It directly follows from (52) that b * ( t ) = b(t). Selection of Q ( s ) = RV(s), where R(s) is the orthogonal factor in the polar decomposition of F(s) = R(s)U(s), results in the fact that

[ ' ] a(t) = a* U(s), O(s), b(s) , p ( t ) . (56) S = t ,

The fact that rigid body motions do not influence the rate of crystallization results in the independence of a from the orthogonal part of F. Therefore, an alternate form of the functional for the rate of crystallization can now be introduced as

I ' t a(t) = 6* C(s), O(s), b(s) , p ( t ) , (57)

since C = U 2. The constraint (53) on the functional for the free-energy can be w ~''ten as

f' i' b(t)~,~(t) + 4,~(t, s )a ( s ) ds -- b( t)q,A(t) + 4'c .'. s )a(s ) ds, (58) s t,

where use has been made of b* = b and a* = a, and one has

4,*(t) = q ,k[Q(t )F( t ) , O(t), b(t)] (59) and

q,~(t, s) = q,~[Q(t)F(t), O(t), b(t), Q(s)~(s), O(s), b(s)]. (60)

Thermomechanical effects of polymer crystallization

If Q(s) = RT(s) is selected, for every time s ~ (ts, t], then one can conclude that

~( t ) = b ( t l~A(t ) + ~c(t, s)a(s) d~, ts

where

?~A(t) -- ¢ ,k[u( t ) , o(t), b(t)l = ~k [c ( t ) , o(t), b(t)l and

t~¢(t, s) = ~b*c[U(t), O(t), b(t), U(s), O(s), b(s)] = t~c[C(t), O(t), b(t), C(s), O(s), b(s)].

The constraint (54) on the functional for the heat flux can be written as

b(t)q*(t) + q~(t,s)a(s) ds = O(t b(t)qA(t) + qc(t, s)a(s)ds , ls ls

where

q*(t) = q*A[Q(t)F(t), O(t), b(t), Q(t)g(t), p(t)],

and

293

(61)

(62)

(63)

(64)

(65)

(66) q*(t, s) = q~[Q(t)F(t), O(t), b(t), Q(t)g(t), Q(s)F(s), O(s), b(s), Q(s)g(s), p(t)],

where the relation g*(t) = Q(t)g(t) is used. Selecting Q(s) = RV(s), results in

f 1 q(/) = R(t b(tlrtA(t) + rtc(t, s )a(s) ds , (67) ts

where

rtA(t) = q*a[U(t), O(t), b(t), ~,(t), p(t)] = rtk[e(t), O(t), b(O, ~,(t), p(t)] (68)

and

tic(t, s) = q*c[U(t), O(t), b(t), g(t), U(s), O(s), b(s), ~(s), p(t)]

= {i~[C(t), O(t), b(t), ~(t), C(s), O(s), b(s), ~(s), p(t)],

g(t) = R(t)fg(t) and g(s) = R(s)~(s).

where

(69)

(70)

10. I S O T R O P Y OF THE I N I T I A L A M O R P H O U S M A T E R I A L

The initial material is assumed to be isotropic before deformation and before crystallization. This assumption is consistent with the initial amorphous nature of the polymer, and has direct implications on the form of the response functionals, as will be illustrated in the following presentation.

The reference configuration will be selected such that the symmetry of the material can be represented by the full group of orthogonal transformations. Let ~ denote the group of transformations representing the initial symmetry of the material. The assumption of initial isotropy will therefore be written as ~3 = ~7, where U denotes the full group of orthogonal transformations. In the following presentation H will denote any typical member of the initial material symmetry group ~q.

Let (F, 0) denote any given history of deformation gradient and temperature, and let (~', 0) denote a history derived from (F, 0) by replacing F(s) by F ( s ) - -F ( s )H . Let "^" in the following presentation represent a quantity evaluated for the history (F, 0) as opposed to (F, 0). The statement of initial material symmetry for our thermodynamic model can be written as

a(t) = a(t), (71)

t~(t) = qJ(t), (72)

294 M. N E G A H B A N

and

~(t) =

which must hold for every history of (F, 0), and Imposing (71) immediately requires that b(t)

the rate of crystallization, after the imposition motions, becomes

t

a*[HTU(s)H, Oss(S~, b(s),p(t) ]

q(t), (73)

for a l l H E ~ . = b(t), and the restriction on the function for of the constraints associated with rigid body

= a* U(s), O(s), b(s), p(t) , (74) S ~t_~

for every H e ~. Therefore, it follows that a(t) must be an isotropic scalar functional of the history of U, history of temperature, the history of the extent of amorphous material, and the current value of the indeterminate scalar p. The most general isotropic functional of the history of U can be written in terms of the six invariants tr[U(sl)], t r [U(s0U(s2)] , . . . , tr[U(st) • • • U(s6)], where each si must be varied over the interval between t~ and t.

Even though (72) is the general statement associated with the initial material symmetry, in the following presentation the stronger requirements

~A(t) = ~A(t) (75)

and

~c(t, s) -- qJc(t, s) (76)

will be used. If these requirements are satisfied, it is obvious that (72) will automatically be satisfied. The restriction presented in (75) results in a set of restrictions on the form of the functional for 4JA which are written as

+k[HTU(t)H, O(t), b(t)] = O*A[U(t), O(t), b(t)]. (77)

Therefore, ~bA is an isotropic scalar function of U(t) and can be written as

O A ( t ) = ** 6A[lu(t), IIu(t), lllu(t), O(t), b(t)], (78)

where

1 I U = tr(U), llu = ~ [tr2(U) - tr(U2)], lllu = det(U). (79)

The three invariants 1u, llu, and Hltj can be replaced by 11, 12, and/3 from Table 1. Restriction (76) when imposed on the function for $c(t, s) will result in

q/c[HXu(t)H, O(t), b(t), HTU(s)H, O(s), b(s)] = 6~-[U(t), O(t), b(t), U(s), O(s), b(s)]. (80)

Table 1. Invariants which form the base for the p rope r g roup of or thogonal t ransformat ions for two symmetr ic tensors a and b ( f rom Spencer [22]), and their derivatives. In these express ions set a = U(t ) (or set a = C(t)), and set b = U(s )

(or set b = C(s))

I . 0,,I. ~hl,,

11 = tr(a) 1 0 12 = tr(a 2) 2a 0 13 = tr(a 3) 3a 2 0 14 = t r (b) 0 i (~ = tr(b 2) 0 2b 1<~ = tr(b 3) 0 3b 2 17 = t r (ab) b a Ix = tr(a2b) ab + ba a 2 I v = tr(ab 2) b 2 ab + ba

I.~ = tr(a2b 2) ab 2 + b2a a2b + ba 2

Thermomechanical effects of polymer crystallization 295

Therefore, Oc(t, s) can be written as a function of the three isotropic invariants of U(t); the three isotropic invariants of U(s); the four mixed isotropic invariants of U(t) and U(s); O(t); O(s); b(t); and b(s). This can be written as

Oc(t, s ) = 0*c*[Iu(t), IItj(t), IIIv(t), Iv(s), Iltj(s), IIIv(s),

IVu(t, s), Vu(t, s), VIu(t, s), VIIu(t, s), O(t), O(s), b(t), b(s)], (81) where

IVu(t, s) = tr[U(t)U(s)l, Vu(t, s) = tr[U2(t)U(s)],

VIv(t, s) = tr[U(t)U2(s)], VIIu(t, s) = tr[U2(t)U2(s)]. (82)

It should be noted that these 10 isotropic invariants of U(t) and U(s) can be replaced by/1-I~o given in Table 1.

Following similar assumptions as introduced for 0A and ~bc, it will be assumed that

and

~A(t) = qn(t) (83)

~c(t, s) = qc(t, s). (84)

This will result in the following restriction on the function for qA(t)

l tqk[ItVU(/) l t , O(t), b(t), HVg(t), p(t) l = qk[U(t), O(t), b(t), ~,(t), p(t)], (85)

where g(t) = RV(t)g(t). The results from representation theory (see Spencer [22]) allows one to give a general representation for qA(t) as

qA(t) = qA,g(t) + qAzg(t)V(t) + qA3g(t)vZ(t), (86)

where each qAi is a scalar valued function of the form

qAi(t) = qki[1,(t), 12(t), %(0, g(t)" g(t), g(t)" V(t)" g(t),g(t) " V2(t) " g(t), O(t), b(t), p(t)]. (87)

Following a similar development for the representation of qc(t, s) one will arrive at

= R( t ) [~] q c , ~ ( t ) ( I I , + IIV~) + Z q~:,,~,(t)(®, - OVa) qc(t, s) oL

+ ~ qc2~fg(s)(IL + II~) + ~ q~2~g.(s)(O. - O~V)], (88) cr J

where II~ and O~ are given in Table 2, and qci~ and q*~ are scalar functions of the invari- ants 11 . . . . . 11o; the invariants g(t) . I I . ~(t), ~(s). H . fg(s), ~(t). I I . g(s) + g(s) . H . fg(t), and

Table 2. Integrity base for the full orthogonal group for two symmetric tensors a = U ( t ) (or a=C( t ) ) and b = U ( s ) (or b=C(s)) , and one vector g. The invariants are those in Table 1, and those constructed from this table by the operation ~Hg (i.e., ~ H ~ ) . This table has been

extracted from Spencer [22]

H ®

! a) a 2

b,b 2 ab, a2b, ab 2, a2b 2 ab, aEb, b2a, a2b 2, aEba, b2ab, a:bZa, b2a2b

296 M. N E G A H B A N

fg(t) " O" ~(s) - g(s)" O" g(t) obtained from Table 2; 0(t); 0(s); b(t); b(s); and p(t). Therefore, a typical representation will be of the form

qc(t, s) = 2qcl ,g(t) + 2qcl2g(t)V(t) + 2qc,3g(t)V2(t) + 2qcl4R(t)[g(t)U(s)]

+ 2qc,sR(t)[~,(t)U2(s)] + qc,6R(t){~,(t)[U(t)U(s)

+ U(s)U(t)]} + . . . + q*~,R(t){~,(t)[U(t)U(s) - U(s)U(t)]} + - . . + 2qcl ,R(t)g(s)

+ 2qc~2R(t)[~,(s)U(t)] + 2qc~3R(t)[~,(s)UZ(t)] + 2qc,4R(t)[~,(s)U(s)]

+ 2qc~sR(t)[~,(s)UZ(s)] + 2qc,6R(t){~,(s)[U(t)U(s)

+ U(s)U(t)]} + . . . + 2q~.,,R(t){~,(s)[U(t)U(s) - U(s)U(t)]} + - - . (89)

One can obtain representations based on C and B by replacing in the above representations C(t) for U(t), C(s) for U(s), and B(t) for V(t). These representations may be more convenient due to the ease of calculating C and B, as opposed to U and V.

In the absence of the constraint of incompressibility of each phase of the polymer in isothermal processes, one need only remove the indeterminant scalar parameter p(t) from the expression for a(t) and the expressions for q(t). The expressions for ~0(t) remain unchanged. In the presence of this type of incompressibility, one must note that J(s) = HIu(s) is a function of the extent of crystallinity and temperature. One can, therefore, remove HIu from the argument of the functions without loss of generality. The derivatives of the invariants presented in columns two and three of Table 1 can be used for the construction of representations for Cauchy stress.

11. D E V E L O P M E N T BASED ON F(t) AND F~(t)

An alternate development based on using F(t) and F~(t) as the variables, as opposed to F(t) and F(s), might be more appropriate since one can directly identify F~(t) with the current "effective" distortion in a crystal created at time s. In this case, one may directly use the results stated above and substitute F(s) = Fy~(t)F(t) to obtain a development based on F(t) and F~(t). This method of approach is correct, yet the final representations will contain unnecessary structure left over from the previous assumptions. For example, U(s) will be replaced by RV(s)F~ ~(t)R(t)U(t). To avoid this awkwardness, the following presentation starts by replacing F~(t) for F(s) in the initial constitutive assumptions.

The following assumptions will be made in regards to the functions needed to calculate the free-energy and the heat flux:

~A(t) = ~,[F(t) , O(t), b(t)],

and

g,c(t, s) = ~0~:~[F(t), 0(0, b(t), r~(t), O(s), b(s)],

qA(t) = qk[F(t), O(t), b(t), g(t), p(t)]

(90)

(91)

(92)

qc(t, s) = q~*t[F(t), O(t), b(t), g(t), F~(t), O(s), b(s), g(s), p(t)]. (93)

The same assumptions presented in (27)-(36) can be used for the Cauchy stress, entropy, and the rate of crystallization.

After substitution of these assumptions in the entropy production inequality one obtains

T(t) = p | + O(t) b(t)O~(,)~OA(t) + , [Or(,)Oc(t, S) + Or,(t)Oc(t, s)F-V(s)]a(s) ds FV(t) (94)

and equations (48)-(50) remain unchanged. Using the polar decompositions F ( t )= R(t)U(t) and F~(t)= V~(t)R~(t), one can make the

Thermomechanical effects of polymer crystallization 297

substitution Q(s)= RV(t)Rs(t) to obtain exactly the same result obtained in (56) since Q(t) = RV(t) due to the fact that Rt(t) = I. In a similar manner, one will also obtain an equation identical to (61), where t~A(t) is given by (62), but (63) is replaced by

~c(t, s) = ~Otctt[U(/), O(t), b(t), RT(t)Vs(t)R(t), O(s), b(s)]

= ~**[C(/), O(t), b(t), RV(t)Bs(t)R(t), O(s), b(s)]. (95)

For the heat flux one will arrive at equation (67) with ~la(t) given by (68), but (69) is replaced by

(It(t, s) = qtct*[U(/), O(t), b(t), ~,(t), RT(t)Vs(t)R(t), O(s), b(s), ~,(s), p(t)]

= ~l~**[C(t), O(t), b(t), ~,(t), RT(t)Bs(t)a(t), O(s)b(s), RV(t)Rs(t)R(s)~,(s), p(t)]. (96)

The representations for an initially isotropic material can be obtained simply by using equations (78), (81), (86), and (88) and replacing RT(t)Vs(t)R(t) for U(s). Alternate forms can also be obtained by replacing RT(t)B~(t)R(t) for U(s). The above follows from the fact that R'r(t)~s(t)R(t ) = HVRV(t)V~(t)R(t)H.

12. CONCLUDING REMARKS

A theoretical structure has been developed for characterizing the thermomechanical response of polymers during crystallization. As it currently stands, it is a generalization of the previous work by Negahban and Wineman [2] and Negahban et al. [21]. As a result, the current development is capable of reproducing, in a single model, all the experimental results which were reproducible using the previous mechanical model. These results include: the experimen- tal results presented by Leitner [15] on the change of elastic modulus during unconstrained crystallization; the experimental results of Gent [9] on stress relaxation due to crystallization; and the anomalous response seen in a torsional oscillator made by using a rubber torsional spring and reported by Kolsky and Pipkin [23]. The anomalous response in the torsional oscillator was reproduced in Negahban [24].

The current work represents a theoretical structure consistent with continuum thermo- dynamics and with sufficient generality to model the thermomechanical response seen during polymer crystallization. Characterization of the response of specific materials needs additional specialization of the current structure. This further specialization can be done through identification of continuum parameters with specific microstructural changes, as is done when using statistical thermodynamics, or through a phenomenological study. As an example, elements of a model for natural rubber are presented in Negahban [25].

An important conclusion is that the traditional Avrami type kinetics used to describe polymer crystallization are not, in general, consistent with the entropy production inequality. Also, this inequality establishes both the conditions for the termination of crystallization, and the conditions for the initation of melting of the crystals.

REFERENCES

1. Ma, R.-J. and Negahban M., Mechanics of Materials, 21, 25. 2. Negahban, M. and Wineman, A. S., International Journal of Engineering Science, 1992, 30, 953. 3. Treloar, L. R. G., The Physics of Rubber Elasticity. Clarendon Press, Oxford, 1975. 4. Flory, P. J., Selected Works of Paul J. FIory, Vol. Ill, Part 6, ed. L. Mandelkern, J. E. Mark, U. W. Suter and D. Y.

Yoon. Stanford University Press, 1985. 5. Mandelkern, L., Crystallization of Polymers. McGraw-Hill, 1964. 6. Wunderlich, B., Macromolecular Physics." Vol. l--Crystal Structure, Morphology, Defects. Academic Press, 1973. 7. Wunderlich, B., Macromolecular Physics: Vol. 2--Crystal Nucleation, Growth, Annealing. Academic Press, 1976. 8. Wunderlich, B., Macromolecular Physics." Vol. 3--Crystal Melting. Academic Press, 1980.

298 M. NEGAHBAN

9. Gent, A. N., Trans. Faraday Soc., 1954, 50, 51. 10. Goppel, J. M., Appl. Sci. Res,, 1948, AI, 3. 11. Arlman, J. J., Appl. Sci. Res., 1948, A1, 347. 12. Von Gi3ritz, D. and MiJller, F. H., Kolloid-Z. u. Z. Polymere, 1973, 251, 892. 13. Stevenson, A., Journal of Polymer Science: Polymer Physics Edition, 1983, 21, 553. 14. Stevenson, A., in Handbook of Polymer Science and Technology: Vol. 2--Performance Properties of Plastics and

Elastomers, ed. Nocholas P. Cheriemisinoff. Marcel Dekker, 1989, p. 61. 15. Leitner, M., Trans. Faraday Soc, 1955, 51, 1015. 16. Van Krevelen, D. W. and Hoftyzer, P. J., Properties of Polymers: Their Estimation and Correlation with Chemical

Structure. Elsevier, Amsterdam, 1976. 17. Ahzi, S., Parks, D. M. and Argon, A. S., Current Research in the Therrno-mechanics of Polymer.; in The

Rubbery-Glassy Range, ASME-AMD-Vol. 203. 1995, p. 31. 18. Parks, D. M., Mechanics of Plastics and Plastic Composites, ASME-MD-Vol. 68]ASME-AMD-Vol. 215, 1995, p.

337. 19. Min, B. K., Dynamic behavior of some solids and liquids. Doctoral dissertation, Brown University, Providence, RI,

1976. 20. Flory, P. J., Journal of Chemical Physics, 1947, 15, 397. 21. Negahban, M., Wineman, A. S. and Ma, R. J., International Journal of Engineering Science, 1993, 31, 93. 22. Spencer, A. J. M., Continuum Physics: Vol. l--Mathematics, ed. A. C. Eringen. Academic Press, 1971. 23. Kolsky, H. and Pipkin, A. C., lngenieur-Archiv, 1980, 49, 337. 24. Negahban, M., Journal of Applied Mechanics, 1994, 61, 124. 25. Negahban, M., Current Research in the Thermo-mechanics of Polymers in the Rubbery-Glassy Range, ASME-

AMD-Vol. 203, 1995, p. 45. 26. Wood, L. A. and Bekkedahl, N., Journal of Applied Physics, 1946, 17, 362.

(Received 1 Februarv 1995; accepted 18 July 1996)