Studies of spinodal decomposition in a ternary polymer-solvent-nonsolvent system

Studies of Spinodal Decomposition in a Ternary Po lym er-So Ive n t- Nonso Ive nt System

RAHUL SAXENA' and GERARD T. CANEBA*

Department of Chemical Engineering Michigan Technological Unwersity

Houghton, MI 4993 1

Much of the work in modeling and computer simulation of spinodal decomposition has been done for binary systems. This work attempts to carry out the analysis of spinodal decomposition in ternary polymer-solvent-nonsolvent systems, where the solvent is the monomer used to produce the polymer and the nonsolvent is the major component. Various experimental methods are used to determine values of the parameters of the ternary version of the Cahn-Hilliard equation of spinodal decomposition, such as cloudpoint experiments, time-resolved light scattering in the ternary system, and morphological development of polymer membranes formed during the early stages spinodal decomposition. The combination of these experimental methods and computer simulation work shows the validity of the assumptions made in characterizing spinodal decomposition in ternary polymer systems of interest.

INTRODUCTION

n the past, we have reported results of modeling, I computer simulation, and experimental studies of spinodal decomposition in binary polymer-solvent systems (1). For the most part, the target application has been the thermally induced phase separation (TIPS) process in polymer membrane formation (2), wherein phase separation is made to occur by dropping the temperature of a polymer-solvent film below the upper critical solution temperature (UCST). In membrane formation via the phase inversion process, a non-solvent is used to effect the phase separation of a polymer-solvent system (3) at a constant temperature. Its analysis involves the kinetics of phase separation of polymer/solvent/nonsolvent systems. Another inter- esting application of phase separation kinetics in polymer/ solvent/nonsolvent systems is when a monomer polymerizes in the presence of a nonsolvent. In partic- ular, we have been studying a free-radical polymerjza- tion process that is accompanied by phase separation above the lower critical solution temperature (EST) (4). Since we found that this polymerization process results in relatively narrow molecular weight distribu- tion products, its macroscopic analysis is that of a ternary polymer-monomer-nonsolvent system.

The theory for spinodal decomposition was initially developed by Cahn and Hilliard (5-8) to study the

+Resent Address: S. C. Johnson and Son. 1525 Howe st. &cine, WI 53403. *Corresponding author.

phase transformation observed in metal alloys and was later extended to polymeric systems (9-12). The result of Cahn's analysis was a diffusion equation which showed an interplay of bulk free energy, gradient energy and strain energy on the kinetics and morphology of transformation. The spinodal decomposition is distinguished from nucleation and growth by the absence of an energy barrier, i.e., it occurs spontaneously. For a system that phase separates via spinodal decomposition, the flux of molecules is against the direction of the concentration gradient. Thus, spinodal decomposition is often characterized by uphill diffusion with a negative mutual diffusion coefficient. It is this negative mutual diffusivion coefficient that causes the small, random fluctuations to grow. In this process we expect a rapid formation of extremely small clusters that are spread periodically in space ( 1, 13). As time increases, concentrations of domains of the new phases increase and lead to a more uniform structure. This type of structure evolution applies only to the early stages of spinodal decomposition. However, numerical solutions to the Cahn- Hilliard equation (1, 13, 14) show that the distances between clusters established in the early stages con- tinue to exist even at the late stages of spinodal decomposition.

In this paper. we carry out mathematical modeling and computer simulation of the early stages of spinodal decomposition of ternary polymer-monomer-nonsohrent systems above the LCST. Time-resolved light scattering and morphological experiments are performed in

POLYMER ENGINEERING AND SCIENCE, MAV2002, Vol. 42, No. 5 1019

Rahul Saxena and Gerard T. Caneba

order to obtain some values of the model coefficients for the 7 Wt0/0/3 wto/oO/90 wt% poly(methacrylic acid)- methacrylic acid-water system at 80°C. Since the binary poly(methacry1ic acid-water system has been known to phase separation above the LCSX of about 50°C (15), the given ternary system is expected to exhibit above- LCSX phase behavior at the operating temperature of 80"C, as it was shown in our prior work (16). In order to determine the inter-domain distance, polymer membranes from the phase separating system are formed and their structure studied. To complete the experimental data needed, we use our prior results from phase equilibria work (16).

THEORY

Figure 1 represents a ternary phase diagram for a polymer-monomer-nonsolvent system. The line AB represents the conditions of a deep quench process with the phase separation expected to occur by spinodal decomposition. This system is unstable towards infinitesimal concentration fluctuations. The system, shown to undergo a process represented by line A' B' in Fig. I , phase separates by the conventional nucleation and growth mechanism (7). Cahn and Hilliard (6, 7, 17) derived the following expression for a binary system to describe the change in concentration as a function of time during the early stages of spinodal decomposition.

IvleLa; Reg ibll

ac a t - = DBV2c - 2M K V c

Note that the binary diffusion coefficient, DB (sub- script "B" stands for "binary"), is proportional to the second derivative of the free energy, f; and is given by:

where M is the diffusion mobility. One of the conditions for the phase separation to occur by spinodal decomposition is when a2f/dc2 < 0, where the system is unstable towards infinitesimal concentration fluctuations. Because mobility is always positive, DB takes its sign from the sign of d2f/ac2. Then, for spinodal decomposition, the right hand side of Eq 2 is less than zero. This implies a negative diffusion coefficient, (DB < 0). This mass transport process is termed "uphill diffusion" because the direction of material transfer is against the concentration gradient. The quantity K is the so-called gradient-energy coefficient, which is the proportionality factor for the dependence of the total potential of the system to the concentration gradient.

Cahn and Hilliard (6. 7, 17) obtained the general solution to Eq l to describe the change in concentration as a function of time during a spinodal decomposition process:

S

Region Homogeneous \ / K Phase

stable / A X I _ - _

... \ /

/ Binodal curve

Spinodal curve

Tie Line

Q. 1. Representative tenuuy phase diagram of apolymer P)/soluent (S)/non-soluent (NS) system.

1020 POLYMER ENGINEERING AND SCIENCE, MAY 2002, Vol. 42, No. 5

Studies of Spinodal Decomposition

where c is the local composition, c, is the overall composition, q is the wavenumber of the sinusoidal composition fluctuations, t is the time variable, r is the position variable, and R(q) may be thought of as a first-order rate constant describing the phase separation process.

Cahn (6) in his linearized theory has shown that only fluctuations with wave vector q smaller than some critical value 4, will yield a positive growth rate. Fur- ther, R(q) has a sharp maximum, R(q,), and we will assume that only fluctuations with a wave vector close to qm contribute significantly to the phase separation process. Then

(4) R(q,) = -0.5 DB 4,' For an early stage of phase-separation by spinodal decomposition, it has been shown that (18-21)

QSALW 0: exp[2~(qrn)tI (5) where Q%(t) is the total integrated intensity of the scattered light. A plot of h~[Q%(t)]"~ vs. time t should yield a straight line whose slope is equal to R(q,). The parameter qm is determined from the angular dependence of the maximum point in the light scattering pattern (18, 19):

qm = (4a/A) sin(8/2) (6)

where A is the wavelength of light and 8 is the scattering angle in the medium.

The multicomponent diffusion equation has been derived by de Fontaine (13) for spinodal decomposition in isotropic metal solutions. We transformed it in nondirnensional form; thus,

where

r = D1 lt/A$ akJ = Dkj/Dll

6, = 2% Kij/D11~?' q = x/X, (8)

Here, A, is a kind of characteristic length, also defined as interdomain distance. Also, u, = ck - Ck", where Ck" is the initial or overall composition of the kth component.

Equation 7 reduces to the following pair of equations for a ternary system (k = 3). Note that there are only two equations due to the Gibbs-Duham relation, and that Component 3 is taken as the reference material.

- arllV2ul + ul2V2u,- (s,,v~u, + 612V4u,) a u1 a7

__ -

For the ternary system, we designate poly(methaqlic acid) as Component 1; methacylic acid as Component

2: and water as Component 3. The reason for this des- ignation is that water is the major component in the mixture, accounting for 90 wt% overall composition: thus, we can assume that the multicomponent system has a constant density and that all phenomenological coefficients can be referred to water. This extends the applicability of J3q 7 to our polymeric system. Based on Eqs 8 and 9, we normally have four diffusion coefficients: D,,, L&,, D,,, and Q,. In addition, we have four mobilities Mo,, and four gradient energy coefficients K ~ . Since there are only two liquid phases in such a system that involves complete solubility between methacrylic acid and poly(meth-acrylic acid) and between methacrylic acid and water, it is reasonable to expect only a single interdomain distance, A, for each of the components.

The analytical solution of Eq 9 was done by Morral and Cahn (22), where they showed that the spinodal decomposition behavior of ternary systems can be similar to that of binary systems, especially if a single dominant composition wavelength exists. Specifically, pseudobinary equations similar to Eq 1 are applicable to both PMAA (Component 1) and MAA (Component 2). In this case, pseudobinary coefficients (D,, D2, K ~ .

K2) are related to the phenomenological coefficients (D11. 012. 91. I&. K ~ ~ . K~,, K ~ ~ , ~ ~ 2 ) in ways that can only be decoupled numerically even though linear al- gebra provides a compact pseudobinary formulation of the multicomponent system.

In order to fit limited experimental data with theoreti- cal equations, we will take advantage of the pseudobinary approach that was proven to be valid by Morral and Cahn (22). This will reduce the number of parameters we have to determine experimentally from 9

K,. K,, A,). We can also assume (Dll. Diz, Dzl. 4 2 . K11. K12, K21, Kzz. Am) to 5 (D1. Dz,

which is valid only for a true binary-i3 system (23). This may be valid in our ternary system because both Components 1 and 2 are dilute in water (Component 3). Thus, the number of parameters that must be determined from ternary experiments reduces to three

We will experimentally determine the interdomain distance, A , through morphological studies. The two other parameters, D, and 4. will be determined from time-resolved light scattering studies and from the rel- evant tie line in the ternary phase envelope.

(D1, Dz- Am)*

EXPERIMENTAL

Sample Preparation

Slightly polydispersed polymethacrylic acid (PMAA) samples were used in this study. The PMAA sample had a number-average molecular weight, &, of -2 X lo5 g/mol, and a polydispersity index, MJMn, of 1.40. Methacrylic acid was purchased from Aldrich Chemical Company. For time-resolved light scattering

POLYMER ENGINEERING AND SCIENCE, MAY 2002, Vol. 42, No. 5 1021


experiments, ternary so1utions of 7vo P w / 3 % m/ 90% Water, were prepared in quartz cells with an in- terior thickness of 1 mm. The cells containing ternary solutions were sealed and rotated at room temperature to provide mixing. The initial solutions were mixed for at least two days in this fashion to ensure a homogeneous system.

Time-Resolved Light Scattering Work Light scattering techniques have been developed to

determine the mobility, diffusion coefficient, and gradient-energy coefficient for phase separating binary polymer systems. Several published studies (10, 18, 19, 24-31) have shown that the above-mentioned quan- tities can be obtained through intensity measurements and/or observation of the scattering pattern during the initial stages of spinodal decomposition.

In order to make rapid measurement of scattered light intensity at various scattering angles, an instru- ment was constructed using an Oriel photodiode array detector with 512 pixel elements QNSTASPEC Model 77140). Figure 2 depicts the schematic diagram of the optical system. A vertically polarized He-Ne laser with wavelength, X = 632.8 nm, was used as the incident lght beam. The lenses were arranged to make the scattered light parallel and focussed on the photodiode

array detector. This detector system made it possible to measure scattering intensity at different angles simultaneously, shortening the time intervals of con- secutive measurements. Because the light focused at the detector is coming from different parts of the scattering sample, desmearing of the scattering data is avoided using the lens system. Stray light is kept to a minimum by blackening all other surfaces and by run- ning the experiment in a dark room. Data at different times and angles were collected and stored onto diskettes. The phase separation in our system was extremely fast, requiring minimal scan time. For our system, most of the data collection was done at the rate of 2 scans/sec. The relationship between the pixel elements, X, and scattering angle 8, was obtained by first allowing the laser to fall directly onto the detector through lenses, and measuring the angle of the peak for a pixel element. Then the detector was rotated to obtain 8 for different X. A plot of pixel elements vs. scattered angle was obtained for up 8 to 45".

Figure 3 shows the schematic diagram of the flow- through heat transport setup. The system consisted of two hot water loops. The open hot water loop was for the top chamber of the temperature enclosure, where good temperature control was not necessary. Temper- atures below 50°C were maintained in this chamber,

1022

m. 2. schematic diagram of the optical system used in the time-resolwd laser light scattering work

POLYMER ENGINEERING AND SCIENCE, MAY 2002, Vol. 42, No. 5


Fig. 3. Schematic diagram of thefiw system used in heahg and cooling the sample ceU used in the light scattering work.

where the solutions were homogeneous. For the bottom chamber of the temperature enclosure a closed loop system was used as shown in Fig. 4. The cell was transferred from the oven to the top chamber where it remained homogeneous. At time t = 0, the cell was pushed to the bottom chamber where a constant temperature of within tO.l"C of the set point was maintained. A detailed diagram of the cell chambers is shown in Rg. 5. A reference scan was taken before the sample was quenched. The reference scan was as- sumed to consist of all the stray light entering the detector, and hence it was subtracted from intensity data collected for the phase-separating system. Thus, a reference scan was taken for each trial and subse- quently subtracted, allowing us to assume that the scattered light at all angles before the cell is plunged to be zero. The two chambers were separated by a

DuPont Teflon piece with a rectangular hole having the same cross-sectional dimensions as the cell, allowing the cell to be pushed through with the plunger (see also Fig. 4). This permitted the sample to be quenched from the top chamber to the bottom chamber without any intermixing of water between them.

Morphological Studiea

Membranes were prepared from the ternary 7% PMAA/3% MAA/SO?h water system by first separating the system via spinodal decomposition and then freeze-drying to remove water and methacrylic acid without changing the polymer structure. After the freeze-dried membranes were formed, their pore mor- phologies were analyzed using a Scanning Electron Microscope.



Standby Position Phase Separation Fosition Flg. 4. Cutout views of the upper standbg chamher and lower quench-ing chamber of the cell enclosure that is used to provide a step heating of the polgymer solution inside the celL

Procedure The ternary polymer/solvent/non-solvent solution

was prepared at room temperature, containing 7% PMAA, 3% MAA, and 90% water. This was the same ternary solution used in the time-resolved light scattering experiment.

A small glass bottle with flat bottom was immersed in a hot water bath such that about 1/2 to 3/4 of the bottle was inside the water (see Rg. 6). The water temperature Tp was selected as the temperature at which phase separation was desired. It was maintained to within 20.2"C of the set point. The bottle was allowed to equilibrate in the hot water bath for a minimum of 10 minutes. This ensured that the inside walls of the bottle were at the same temperature as the water. This means that that the walls of the bottle form almost an ideal quench temperature condition. Any liquid that is in direct contact with the bottle wall will, almost immediately, reach the wall temperature. However, as one moves away from the wall, due to the heat capacity of the liquid, there will be a temperature gradient inside the sample. There will be a de- lay, depending on the sample size, before the entire mixture reaches equilibrium. For this reason alone,

when investigating membrane structure using SEM, we looked primarily at that cross section of the membrane that was adjacent to the wall of the bottle.

About 0.4 ml of the homogeneous polymer solution (in this case prepared at room temperature where it is outside the phase envelope) was added to the bottle (Step 1 in FLg. 6). This resulted in a thin film of the polymer solution at the bottom of the bottle. The bottle was immediately covered with a rubber stopper containing a glass rod to minimize any solvent loss (Step 2 in Rg. 6). The timer was started. The system was allowed to phase separate for a predetermined length of time tps. At the end of phase separation time, the bottle was immediately removed and immersed in liquid nitrogen (Step 3 in Rg. 6). Note that the time it took to transfer the sample into liquid nitrogen bath in most cases was less than a second. Once the entire system was frozen (it took maximum up to a second for the polymer solution in contact with the glass surface to freeze), solidified MAA and water were removed from the system via freeze-d-g. Since the removal of sol- vents using this process took several days, we used the following cold finger technique to increase the effi- ciency of the freeze-drymg procedure.



Freeze-Dry ing

Once the polymer system was frozen, the rubber stopper was replaced with another stopper containing a glass tube that provided an opening into the bottle. While the bottle was still inside the liquid nitrogen bath, the other end of the glass tube was connected to one leg of a vacuum manifold (e.g. port # 1 in Fig. 6). The vacuum manifold valve was opened and the system was allowed to pump-out for few minutes. Swiftly, the bottle was transferred to a dry ice bath that was maintained at or below -10°C at all times (Step 4 in Fig. 6). This ensured that both MAA and water would not melt during drying. (The dry ice bath was prepared by slowly adding solid CO, (dry ice) pieces to 2-propanol until the mixture temperature was about 20-30°C below zero. An well-insulated system will maintain this temperature for several hours. In our case, it took almost one day before the dry ice bath warmed up to -10°C).

The bottle was left under vacuum and inside the dry ice bath for 40 hrs when all MAA and water was removed from the system. This can be denoted as the drymg time td. After the drying period, the bottle was removed from the dry ice bath but hrther maintained under vacuum at room temperature for two additional days. This was done to ensure that all traces of solvent had been removed and that the membrane was absolutely dry.

Ij19.5. Actual wnstruction of the cell assembly used to bring the cell containing the polymer solutionfrom a lower temperature to its quenching temperature of 80°C.

To \'actaunt !'tinip

Fig. 6. Schematic of the method used to prepare polymer membranes for mrphologicaI ana[ysis.



Sample Preparation for SEM binary diffusion coefficient is that for the polymer

Samples for the Scanning Electron Microscope (SEM: Model Jeol JXA-8600) were prepared by first immers- ing the membrane into liquid nitrogen. This made the membrane brittle. The membrane was broken into small pieces and some of the pieces were mounted on the metal cylinder that formed part of the stage inside the microscope. The membrane was mounted such that its cross section (or the broken face) was perpen-

(Component 1) in the ternary mixture, because it is the polymer structure that scatters the type incident light used in the experiment. Thus, a pseudobinary diffusion coefficient of D, = -5.84 X 10-" cm2/sec was obtained for the poly(methacry1ic acid) in the ternary system. Note that it should be a negative number, ir- dicating that uphill diffusion occurs during spinodal decomposition.

Morphological Studies

w e 9 shows SEM micrographs of the PMAA membranes formed via thermal inversion process start- ing with a 7%/3%/90% mixture of PMAA/MAA/water at 80°C. We should note that morphological evolution of spinodal decomposition in polymer/small molecule systems starts with the formation of interconnected

dicular to the electron beam. This way we were able to study the pore s t r u c k of the membrane directly. Our focus was to study the edge that was in contact with the bottom wall of the bottle. All membranes were sputter-coated with carbon to make them conductive prior to any microscopy work.

EXPERIMENTAL RESULTS

The-Resolved Light Scattering Work

The early stage of phase separation by spinodal decomposition for the ternary 7% PMAA/3% MAA/QO% water system at 80°C is characterized here by the a single scattering peak (Q. 7). Values of the maxima in Fig. 7 is in the range of qm = 4 - 6.5 X lo4 cm-'. Based on qm = 2a/hm, this corresponds to the interdomain distance, A , to be in the 0.97 - 1.6 km range. Figure 8 also shows a plot of lnIQw(f)]1/2 vs. time, and it should have a slope of R(qm) according to Eq 5. In turn, the diffusion coefficient can be obtained from R(qm) and qm through Eq 4. The calculated psuedo-

networks, followed by the formation of cells due to coarsening (1, 32). Then, we have the formation of closed cells due to the so-called hydrodynamic flow mechanism (33). The interdomain distance during the early stages of spinodal decomposition is the pore size of the initial interconnected network. When coarsening starts to occur, some of the nodes of the initial network structure disappear to form open cells that are interconnected by pores. The interconnecting pores are usually the surviving pores from the initial network structure. Thus, the task of obtaining the interdomain distance from the micrographs translates to the measurement of the smallest pore in the membrane

Flg. 7. Raw scattering data lintensity, If@ us. q] for the ternary 7% PMAA/3% 1MAA/9096 water system at 80°C. showing the existence of a dominantpeak for each of the cures thatpertain to &s&rentphase separation times.



7%PMAA/3%MAA/SO%Water System at 8OoC 4.5

4

N . - 3.5

.cI Y 4 a

E 3 a" Y

J

2.5

2

0 6 12 18 24 30 36 42 48

Time, sec Flg. 8. Plot of the logarithm of the total integrated intensity of scattered light (lnQ& us. time, based on data from Q. 7. The slope of the plot gives the quantity R(q,,J, according to Eq. 5.

cross-section. This investigation reveals that the smallest structure or pore size is in the order of 1-2 pm. This is consistent with the result obtained from Rg. 7, where maxima in I(4-q data at different times result in an interdomain distance of 0.97-1.6 pm. In the subsequent computer simulation work, we decided to use an interdomain distance of A, = 1 pm.

COlyIpuTER SIMULATION OF SPINODAL DECOMPOSITION FOR TERNARY SYSTEMS

We will primarily focus on solving Eq 9, which is in spatial domain, by converting it into reciprocal domain. This is done by multiplying Eq 7 by ( l / V ) exp(4E.d dr: and integrating over the domain volume V.

1% exp(-i _ _ P'.r')dr' - = v ar V

In the above equation, k (unless specified other-wise) varies from 1 to e l , and n is the number of components. Note that in Eq 11, J.X' is described as g-mk,/ An and r' is the space vector (in one dimension L + x) . Both and are made dimensionless by defining:

where X, is the interdomain distance. Note that in one-dimensional form of r is given by I = q = x/A,. Then, the non-dimensionalized form of the product g.f in Eq 1 1 is 8.r.

Based on a similar derivation procedure outlined in the literature (13), the one-dimensional form of Eq 11 reduces to

where

and

kl = --m ,..., -l ,O,l , . . . , - r + q = x/A,

- P + T k l L / (15) Equation 12 is sohred using an Euler algorithm (13), and the composition profile is obtained from

n

k,=l % (q* '1 = (Bk.k, cOs(pq) - %k1 sin(pq) (16)

where

The extent of phase separation, pk. is defined as



(c) (d) Fig. 9. Scanning electron micrograph of the fractured s d m e of the cross-section of a phase separated PMAA membrane that was castfrom a 7% PMAA/3% MAA/9096 water solution. The top portion of the micrograph is the side of the membrane that is exposed to the opemting temperature of 80°C. Phase separation times are: (J 30 sec, @) 60 sec, (c) 90 sec, and (d) 120 sec.

where If Eq 20 is applied to the poly(methacq4ic acid), then the resulting degree of completion u1 value should also

uz,k = (4 - - 4) (19) coincide with the scattering function obtained from the time-resolved light scattering data for the ternary system. This is valid because poly(methacrylic acid) (Component 1) is the only component that will exhibit scattering of light at the frequency used in the experiment (A = 632.8 nm). Also, Eq 1 4 actually defines Ak&) to be of the same form as that of the scattering

ponent 1 (k = 1) as a means of matching computer simulation results with experimental data.

Here, cg and c! are the equilibrium compositions in the polymer-rich phase (a) and the polymer-lean phase (p), respectively, and egis the initial composition. The u k can be calculated from Eq 1 7 as follows (13):

n R function (Qd(t)) in Eq 5. Thus, we use J3q 20 for com- a“k = c IAkk1/’ = c (B&+ ‘z.k1) (20)

k , = l k l = l



COMPUTER SIMULATION RESULTS We performed a series of numerical experiments

using the model described by Eq 9, and implemented through Eq 14. In these runs, the interdomain distance h, = 1.0 pm was used, based on the result of the morphological study in the previous section. Com- positions described in the Experimental section were used here as well. Specifically, an initial concentration of 7%/3O/6/90% was used for PMAA/MAA/water, respectively. The phase dngram, shown in Rg. 10, was obtained by Shi (16) for the PMAA/MAA/water system using cloudpoint experiments was used for this investigation. From Fig. 11 one can obtain approximate values for the equilibrium compositions in the polymer-rich phase (cy = 6%, cs = 82%, c; = 12%) and polymer-lean phase [cf = 0.5%, cl = 97%, c$ = 2.5%).

A computer program was written in FO- in order to cany out the numerical calculation of Eq 14. A series of computer experiments were performed using the following values for a&:

all = Di1/ Dii = Di/ Dii = 1 a12 = D12/ D,i = 0

a22 = 4 2 / Dll = Q/ Dll = 0.96, based on the in- clination of the tie line in Fig. 10.

The value of Dll = -5.84 X lo-" cm2/sec was obtained from the matching of the slopes in Fig. I I for both simulation and light scattering results. The dif- ference in the position of the lines in the time axis can be explained by the level of prevailing random composition fluctuations between the simulation and experimental systems. The experimental system experi- ences a much larger random composition fluctuation than the simulation system. Thus, induction period of spinodal decomposition is longer for the simulation system. The extent in the reduction of this induction period has been verified through simulations of spin- odd decomposition in binary systems (34). Note that Eq 10 provides the values of 6i from the above-mentioned values of ai. As shown in Fig. 12, parameters obtained from experimental and computer simulation results predict reasonable composition profiles for all the components in the system.

MAA

40 0 00

PMAA Water m. 10. Phase diagram of the temary PMAA/MAA/water system showing the ouerQll composition cp at 7% PMAA/3% MAA/90% water. Points d show extent of evolution of a composition pro@ of the polymer-rich and polymer-lean phases. %se points fd on tie line. Equilibrium compositions in the polymer-rich phase (cia) and polymer-lean phase (cIP). l k phase diagram was obtainedfrom Wang et d (16).



0 10 20

Time, sec 30 40 50 60 70

5

3 1.5

z 1 W

C 0.5 -I

0

I I

Light Scattering - Data

110 120 130 140 150

Time, sec 160 170 180

5

4.5

4

3.5-

3 -*

2 Y

2 v -!

2.5 2 0 -I

1.5

1

0.5

0

Fig. 1 1 . Plot of the natural logarithm of the integrated intensity us. time from the time-resolued light scattering ewperiment of the ternary system 7% PMAA/3% MAA/90% water system at 80°C. Also shown is the plot of the natuml logarithm of the extent of completion uI for p o l y ( m e t ~ l i c acidl (based on Eq 20) us. timefrom the simulation of the spinodal decomposition in ternary polymers systems (Eq 14). Linear behnuwr in both data sets con$nn the occurrence of spinodal decomposition and coincidence of parameters in both experimental and simulation results.

13

12

11

10

9

8

7

6

5

4

3

3

1

- Time = 136.78 sec.; PC = 62.0% - 1 1 -1 ,

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Distance, microns Fig. 12. Composition projiles of PMAA. during the early stage of spinodal decomposition.



CONCLUSIONS

We have therefore been able to show that through a combination of data from phase equilibria, time- resolved light scattering, and morphological experiments, we can determine approximate parameters needed to analyze early stages of spinodal decomposition in a ternary polymer/solvent/nonsolvent system. We have done this by extending the Cahn-Hilliard expression to ternary systems, and solving numerically the resulting set of partial differential equations. Spe- cifically, the following assumptions have been shown to be a good model for fitting three experimentally determined ternary parameters in the system:

1. Constant density in the system 2. Single interdomain distance for all the compo-

3. Independent pseudobinaries between the minor

All the above-mentioned assumptions are specific to our system, but they are nevertheless valid for ternary systems that comprise a polymer, its solubilizing monomer, and a major nonsolvent component.

nents

components and the major component.

ACKNOWLEDGMENTS

This project was partially supported by the National Science Foundation (CTS-9404 156). We also acknowledge partial fellawship support for the student (R.S.) in this project from the Michigan Polymer Consor- tium. Finally, we wish to acknowledge Amway Corpo- ration for allowing the student (R.S.) to cany out some of the simulation work within its R&D facilities.

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Documents

Studies of spinodal decomposition in a ternary polymer-solvent-nonsolvent system