Kinetics of Thermally Induced Phase Separation in TernaryPolymer Solutions. I. Modeling of Phase SeparationDynamics

B. F. BARTON, A. J. MCHUGH

Department of Chemical Engineering, University of Illinois, 600 South Mathews, Urbana, Illinois 61801

Received 24 June 1998; revised 18 December 1998; accepted 16 February 1999

ABSTRACT: Simulations based on Cahn–Hilliard spinodal decomposition theory forphase separation in thermally quenched polymer/solvent/nonsolvent systems are pre-sented. Two common membrane-forming systems are studied, cellulose acetate [CA]/acetone/water, and poly(ethersulfone) [PES]/dimethylsulfoxide [DMSO]/water. The ef-fects of initial polymer and nonsolvent composition on the structure-formation dynam-ics are elucidated, and growth rates at specific points within the ternary phase diagramare quantified. Predicted pore growth rate curves exhibit a relative maximum withnonsolvent composition. For shallow quenches (lower nonsolvent content) near a phaseboundary, the pore growth rate increases with increasing quench depth, whereas fordeep quenches, where the composition of the polymer-rich phase approaches that of aglass, the pore growth rate decreases with increasing quench depth. With increasinginitial polymer concentration, the overall rate of structure growth is lowered and thegrowth rate maximum shifts to higher nonsolvent compositions. This behavior appearsto be a universal phenomenon in quenched polymer solutions which can undergo a glasstransition, and is a result of an interplay between thermodynamic and kinetic drivingforces. These results suggest a mechanism for the locking-in of the two-phase structurethat occurs during nonsolvent-induced phase inversion. © 1999 John Wiley & Sons, Inc. JPolym Sci B: Polym Phys 37: 1449–1460, 1999Keywords: phase separation; polymers; spinodal decomposition; ternary; solutions

INTRODUCTION

Phase separation in a polymer solution can beinduced by solvent evaporation, addition of a pre-cipitating agent (nonsolvent), or a change in tem-perature. A technologically important example ismembrane formation by nonsolvent-inducedphase inversion (NIPS).1 In this process, the so-lution is cast or spun in the form of a thin film orthin-walled hollow fiber and quenched in a non-solvent bath to induce liquid demixing and struc-ture formation. Morphologies ranging from highly

asymmetric films exhibiting columnar macro-voids (fingers) underneath a dense, thin skin, to askinless symmetric sponge can be formed. Theunderlying dynamics of the structure formation iscomplicated by compositional changes that occurin both space and time because of mass transferat the film-bath interface. Nevertheless, experi-mental2–4 and theoretical2,5,6 studies have suc-cessfully correlated observed membrane morphol-ogies with the film-bath mass transfer dynamics.Despite the utility of these approaches, quantita-tive prediction of the phase separation dynamicsand resultant structure development has notbeen possible. The focus our studies has been tosystematically explore the dynamics of phase sep-aration within the two-phase region of commonmembrane-forming systems using computational

Correspondence to: A. J. McHugh (E-mail: a-mchugh@uiuc.edu)Journal of Polymer Science: Part B: Polymer Physics, Vol. 37, 1449–1460 (1999)© 1999 John Wiley & Sons, Inc. CCC 0887-6266/99/131449-12

1449

and experimental analyses of thermal quenching.Since the location inside the miscibility gap isknown, insights into the structure formation dy-namics related to NIPS can also be gained.

Modeling of phase separation in binary andternary systems using Cahn–Hilliard theory, par-ticularly for polymer blends, has been an intensearea of research.7–13 More recent work has alsoexplored the mechanism of domain pinning insome polymer blends,14,15 and several studieshave begun to elucidate the qualitative effectsglass transition or gelation might have on thephase separation dynamics.16–18 A few workshave focused on polymer solutions.19–21 Whereasall these studies make qualitative comparisonsbetween predictions from Cahn–Hilliard theoryand experimental observations, our recent workon a binary polymer solution21 has been the firstto make quantitative predictions of the late-stagestructure growth kinetics. Good agreement wasfound between calculated growth rates andgrowth rates determined from light scatteringmeasurements.22 Near a glass transition, pre-dicted pore growth rates were found to decreasewith increasing quench depth (i.e., decreasingtemperature), consistent with experimental ob-servations. Moreover, over the entire tempera-ture range explored by the simulations, the poregrowth rate curve exhibited a relative maximum,suggesting that an interplay between thermody-namics and transport kinetics can determinetrends in the structure-coarsening rates in poly-mer solutions. Whereas a relative maximum inthe pore growth rate was not observed experimen-tally for a binary polymer solution, this phenom-enon has recently been observed in our studies ofa ternary poly(methyl methacrylate) (PMMA)/1-methyl-2-pyrrolidinone (NMP)/glycerin solutionundergoing a thermal quench.23

In the present work, spinodal decompositionmodeling will be extended to structure formationby thermal quenching in two common ternary,membrane-forming systems, cellulose acetate(CA)/acetone/water and poly(ethersulphone)(PES)/dimethylsulfoxide (DMSO)/water. Qualita-tive and quantitative comparisons of model pre-dictions with light scattering observations for thePMMA/NMP/glycerin system will be presented ina companion article.24

MODEL EQUATIONS

The ternary diffusion equations describing thephase separation dynamics can be written as25

­fi

­t 5 Oj51

2

¹ z ~Mij¹Gj! i 5 1, 2 (1)

where wi is the volume fraction of component i,and Gj is the generalized potential for componentj. Subscripts, 1, 2, and 3 refer to nonsolvent, sol-vent, and polymer, respectively. Our concern isprimarily with the late-stage dynamics, hence,thermal fluctuations are neglected. Moreover,since experimental studies of binary22 and ter-nary24 polymer solutions under similar conditionshave shown that structures grow by diffusivecoarsening only, hydrodynamic coarsening effectsare also neglected.

Using a volume fraction basis, the componentpotentials are defined as

G1 5 m1 2n1

n3m3 2 k1¹

2f1 2 k12¹2f2 (2)

G2 5n1

n2m2 2

n1

n3m3 2 k12¹

2f1 2 k2¹2f2 (3)

where vi is the molar volume of component i, andmi is the chemical potential of component i deter-mined from ternary Flory–Huggins theory.26 Thegradient energy parameters, k1, k2, and k12 areestimated from Debye’s approximation27

k1 513RTg13RG

2 (4)

k2 513RTg23RG

2 (5)

k12 5RT~g13 1 g23!

6 RG2 (6)

where the interaction parameters, g13 and g23,are evaluated at the system mean composition.Equations (4)–(6) ignore the entropic contribu-tion28 to the gradient energy parameters11 sincesimulations in a binary polymer solution wherethis assumption has been employed yielded re-sults in very good agreement with experimentalobservations.21

The ternary mobilities, Mij, are given in termsof the binary friction coefficients, zij, by29,25

M11

5n2n3f1

NA2 Sn3f1f3z12 1 n2f1f2z13 1 n1~1 2 f1!

2z23

M D(7)

1450 BARTON AND MCHUGH

M12 5 M21 5 2n2n3f1f2

NA2

3 S2n3f3z12 1 n2~1 2 f2!z13 1 n1~1 2 f1!z23

M D ~8!

M22

5n2n3f2

NA2 Sn3f2f3z12 1 n2~1 2 f2!

2z13 1 n1f1f2z23

M D(9)

where

M 5 ~n2n3f1z12z13 1 n1n3f2z12z23 1 n1n2f3z13z23!

(10)

and NA is Avogadro’s number. The equality, M125 M21, results from the definition of the compo-nent potentials and is not a general feature ofternary mobility coefficients.

For a fixed temperature and overall composi-tion, phase separation will occur along a singletie-line which may be obtained from binodal cal-culations26 for a given polymer/solvent/nonsol-vent and expressed as

f1 5 mf3 1 b (11)

where m and b are functions of the initial compo-sition and quench temperature. A mass balanceat a given point in the system gives

­f3

­t 5 2 S­f1

­t 1­f2

­t D (12)

A combination of eqs. (1) and (12) results in thefollowing single diffusion equation for the polymer:

­f3

­t 5 ¹ z ~M1¹~ 2 G1! 1 M2¹~ 2 G2!! (13)

where M1 5 M11 1 M21 and M2 5 M12 1 M22.The gradient terms in the potential functions mayalso be rewritten as functions of polymer compo-sition, yielding the following expressions:

2 G1 5n1

n3m3 2 m1 2 k11¹

2f3 (14)

2 G2 5n1

n3m3 2

n1

n2m2 2 k22¹

2f3 (15)

where

k11 5 2mk1 1 ~m 1 1!k12 (16)

k22 5 2mk12 1 ~m 1 1!k2 (17)

Equations (13)–(17) describe phase separationby spinodal decomposition in a ternary polymer/solvent/nonsolvent system where all compositionsin the two-phase system are assumed to lie alongthe tie-line given by eq. (11).

NUMERICAL SOLUTION

Scaling

Equation (13) is scaled using the following dimen-sionless variables:

f*i 5 fi 2 fi0 (18)

M*j 5Mj

M0(19)

G*i 5Gi

RT (20)

x* 5xL (21)

t 5M0RT

L2 t (22)

where the scaling length L, is given by

L 5 aRG (23)

The parameter a is used to ensure the appro-priate size scale needed to model the early-stagestructure in each simulation and is given by

a 51

23/2Î 2 Du2G/RT

(24)

where Du2G is the second derivative of the Flory–

Huggins free energy of mixing along the directionof the system tie-line. In the present work, valuesused for a ranged from 0.5 to 3.

PHASE SEPARATION IN TERNARY POLYMERS. I 1451

Discretization

The dimensionless diffusion equation is dis-cretized using a standard explicit scheme30

fi,jn11 2 fi,j

n

Dt

5 Om51

2 Jm,i11/2,j 2 Jm,i21/2,j 1 Jm,i,j11/2 2 Jm,i,j21/2

Dx*

(25)

where wi, jn is the volume fraction of polymer at

point (i, j) after n time steps, and Ji11/ 2, j is thetotal flux of polymer at point (i 1 1

2, j) and is givenby

Jm,i11/2,j 5 M*m,i11/2,jSG*m,i11,j 2 G*m,i,j

Dx* D (26)

where

¹*2fi,j 5fi11,j 1 fi,j11 2 4fi,j 1 fi21,j 1 fi,j21

Dx*(27)

Numerical integration was carried out using a100 3 100 grid with the application of periodicboundary conditions. As noted earlier,21 careneeds to be taken in the choice of the mesh size,Dx*, to avoid possible discretization artifacts

such as mesh-size dependence and artificial pin-ning effects. Since the scaling parameter, a, isuniquely determined by eq. (24), proper modelingof the early-stage length scale is insured for eachset of quench conditions. Thus, a value of 1.0 forDx* is adequate for all computations.

EVALUATION OF MODEL PARAMETERS

Parameters to be specified include quench tem-perature (T), mean compositions (wio’s), compo-nent molecular weights (Mi), and molar volumes(vi), the thermodynamic interaction parameters( gij’s), and the binary friction coefficients (zij’s).The latter two parameters are determined fromdata on the three limiting binary pairs. In whatfollows, parameters are also determined for thePMMA/NMP/glycerin system, since simulationsand experimental observations for this systemare discussed in Part II.24

Expressions for the interaction parameters forthe PMMA/NMP/glycerin system, and, at a fixedtemperature of 25°C for the CA/acetone/waterand PES/DMSO/water systems, are given in Ta-bles I and II, respectively. Figures 1 and 2 showthe resulting phase diagrams for the CA and PESsystems. For the PMMA/NMP/glycerin system,g12 and g23 were estimated from UNIFAC calcu-lations discussed in detail elsewhere.25,31–33 Thetemperature dependence of the PMMA/glycerininteraction parameter, g13, assumed to be compo-

Table I. Thermodynamic Interaction Parameters for the PMMA/NMP/GlycerinSystem

PMMA(3)/NMP(2)/Glycerin(1)Method of

Determination

g12 5 a 1 bf2 1 gf22

a 5 8.411 3 1022 2 53.19/T 2 5.350 3 104/T2 UNIFACb 5 0.7100 2 567.0/T 1 9.873 3 104/T2

g 5 21.155 1 999.9/T 2 2.612 3 105/T2

g23 5 b1 1 b2f3

M3 5 20,000 Modified UNIFACb1 5 1.666 2 442.6/T with free volumeb2 5 20.2582 1 216.0/T correction

M3 5 75,000b1 5 1.628 2 442.7/Tb2 5 20.2105 1 216.0/T

M3 5 20,000 Empirical fit tog13 5 22.760 1 1234/T cloud point data

M3 5 75,000g13 5 22.652 1 1223/T

1452 BARTON AND MCHUGH

sition independent, was determined by fittingbinodal calculations26 to ternary cloud pointdata34 at several temperatures. Since the poly-mer-rich branch of the ternary phase diagram isrelatively insensitive to molecular weight,26 weused cloud point data for a polydisperse PMMA todetermine g13 for different PMMA molecularweights. Interaction parameters at 25°C for theCA/acetone/water and PES/DMSO/water systemswere determined from experiments reported else-where.2,6

Glass transition boundaries on the phase dia-gram are determined from Kelly–Bueche theoryextended to ternary polymer solutions35

Tg 5R1f1Tg1 1 R2f2Tg2 1 f3Tg3

R1f1 1 R2f2 1 f3(28)

where Tg is the estimated glass transition tem-perature of the mixture, Tgi is the glass transitiontemperature of pure component i, and Ri is re-lated to the difference in the thermal expansivitybetween liquid and glass for component i. Glasstransition temperatures for solvents and nonsol-vents are estimated from viscosity data using theapproach of Barlow et al.36 Parameters used tocalculate glass transition curves are listed in Ta-ble III.

The nonsolvent/solvent friction coefficient, z12,is related to the mutual diffusion coefficient, D12,by

z12 5n2f1

NA2D12

S­m1

­f1D (29)

Table II. Thermodynamic Interaction Parameters for the CA/Acetone/H2O andPES/DMSO/H2O Systems at 25°C

CA(3)/Acetone(2)/H2O(1) PES(3)/DMSO(2)/H2O(1)

g12 5 0.661 1 0.417/(1.0 20.755f2) g12 5 20.218 20.816f2 1 0.383f22

g23 5 0.535 1 0.11f3 g23 5 21.189 2 0.621f3

g13 5 1.4 g13 5 2.7

Figure 1. Phase diagram for the cellulose acetate/acetone/water system (M3 5 27,000) at 25°C. Thickline, binodal; thin line, spinodal; dotted line, glass tran-sition boundary; dashed lines, tie-lines. Double-ar-rowed lines represent range of compositions explored insimulations.

Figure 2. Phase diagram for the PES/DMSO/watersystem (M3 5 80,000) at 25°C. Thick line, binodal;thin line, spinodal; medium line, glass transitionboundary; dashed lines, tie-lines. Double-arrowed linesrepresent range of compositions explored in simula-tions.

PHASE SEPARATION IN TERNARY POLYMERS. I 1453

From diffusion data,37,38 v2RT/(NA2 z12), for ac-

etone/H2O and D12 for DMSO/H2O can be as-sumed constant with values of 5.28 3 1025 and1.0 3 1025 cm2/s, respectively. For the NMP/glyc-erin mixture, the composition dependence of D12is calculated from the relation proposed byVignes39

D12 5~D12

0 !x2~D210 !x1

c; c 5 S­ ln~a1!

­ ln~x1!D (30)

where a1 is the nonsolvent activity, xi is the molefraction of component i, and Dij

0 is the diffusioncoefficient of solute i in pure j which is deter-mined from the Scheibel relation40

Dij0 5

KThjVi

1/3 ; K 5 8.2 3 1028S1 1 S3Vj

ViD2/3D (31)

where hj is the viscosity of component j in centi-poise and Vi is the molar volume of component iin cm3/mol at its normal boiling point. From so-lution thermodynamics, c, is given by

c 5 S­ ln~a1!

­ ln~x1!D 5 f1 1 f2

1 f1f2Sn1

n22 1D 2 2f1f2

2g12 2 2f1f22~1 2 2f1!

3 S­g12

­f2D 1 f1

2f23S­2g12

­f22 D (32)

The resulting dependence of D12 for the NMP/glycerin binary mixture is shown in Figure 3.

The polymer/solvent friction coefficient, z23, isrelated to the mutual diffusion coefficient, D23, by

z23 5n3f2

NA2D23

S­m2

­f2D (33)

and D23 is related to the solvent self-diffusioncoefficient, D2, by41

D23 5 DTD2 5

f2S­~m2/RT!

­f2D

bf22 1 ~1 2 f2!~1 1 2f2!

D2 (34)

where DT is a thermodynamic factor. The param-eter, b, is the ratio of D2 to ND3 in the dilutesolution limit where N is the degree of polymer-

Table III. Kelly–Bueche Parameters Used to Determine the Glass TransitionBoundaries for the Ternary Polymer/Solvent/Nonsolvent Systems

System: P(3)/S(2)/NS(1) Kelly–Bueche Parameters

PMMA/NMP/glycerin Tg1 5 187 K R150.8606Tg2 5 45.4 K R251.397Tg3 5 378 K

CA/acetone/water Tg1 5162.7 K R150.3630Tg2 5 44.17 K R253.856Tg3 5 570 K

PES/DMSO/water Tg1 5 162.7 K R150.249Tg2 5 122.4 K R251.684Tg3 5 498 K

Figure 3. Glycerin/NMP mutual diffusion coeffi-cients. (F) T 5 15°C; (E) T 5 25°C; (�) T 5 35°C; (‚)T 5 45°C; (■) T 5 55°C; (h) T 5 65°C.

1454 BARTON AND MCHUGH

ization and D3 is the polymer self-diffusivitywhich is estimated from Kirkwood–Riseman the-ory.42 Solvent self-diffusion coefficients for thepolymer solvent binary pairs in this work areestimated from the free volume theory of Vrentasand Duda43,44

D2 5 D02expS 2 ~v2V*2 1 jv3V*3!VFH/g D (35)

where vi is the weight fraction of component i, V*iis the specific critical hole free volume of compo-nent i, j is the ratio of the molecular weights ofsolvent to polymer jumping units, g is the averageoverlap factor for the mixture which is assumed tobe unity, D02 is a pre-exponential factor assumedto be independent of temperature, and VFH is theaverage specific hole free volume of the mixturegiven by

VFH

g5 v2

K22

g2~K23 1 T 2 Tg2! 1 v3

VFH3

g3(36)

where K22 and K23 are free volume parameters,Tg2 is the glass transition temperature of thepure solvent, and g2 and g3 are overlap factors forthe pure component free volumes, which are as-sumed to be unity. Since temperatures are belowthe glass transition of the pure polymer, the sys-tem is in the rubbery (Tg,mix , T , Tg3) or glassystate (T , Tg,mix). Under these conditions, the

following relations for VFH3, the specific hole freevolume of the equilibrium liquid polymer, apply

VFH3 5 V30~Tg3!@fH3

G 2 ~a3 2 ac3!~Tg3 2 T!# (37)

for the rubbery region and

VFH3 5 V30~Tg3!@fH3

G 2 v2B~a3 2 a3g 1 ac3g 2 ac3!

1 ~a3g 2 ac3g!~T 2 Tg3!# (38)

for the glassy region, where V30 is the specific

volume of the equilibrium liquid polymer, ai arethermal expansion coefficients, B is a parameterthat measures the ability of the solvent to depressthe glass transition of the mixture, and fH3

G is thefractional hole free volume of the polymer at itsglass transition temperature, Tg3, which is re-lated to the polymer free volume parameter, K33.Parameters for the free volume equations arelisted in Table IV. For the CA/acetone and PES/DMSO systems, D02 was fit to experiment diffu-sion data reported elsewhere.6,2 The resulting dif-fusion coefficients are shown in Figures 4 and 5.

The polymer/nonsolvent friction coefficient isestimated using the relationship proposed by Re-uvers et al.45

z13 5 Cn1

n2z23 (39)

Table IV. Free-Volume Parameters for Polymer–Solvent Systems

Parameter PMMA/NMP CA/Acetone PES/DMSO

V*2 (cm3/g) 0.732 0.943 0.766V*3 (cm3/g) 0.788 0.730 0.694V3

0 (cm3/g) 0.870 0.804 0.763K22 (cm3/g z K) 5.76 3 1024 9.83 3 1024 1.03 3 1023

K23 (cm3/g z K) 3.05 3 1024 3.53 3 1024 5.55 3 1024

K32 (cm3/g z K) 33.0 33.02 19.69K33 (cm3/g z K) 80.0 51.6 47.43Tg2 (K) 45.4 44.17 122.4Tg3 (K) 378 570 498B (K) 165 1293 596j 0.491 0.260 0.239D03 (cm2/s) 1.535 3 1023 5.00 3 1024 1.75 3 1025

a3 (K21) 5.80 3 1024 3.90 3 1024 5.45 3 1024

a3g (K21) 2.50 3 1024 1.30 3 1024 1.65 3 1024

ac3 (K21) 2.00 3 1024 1.34 3 1024 1.10 3 1024

ac3g (K21) 8.66 3 1025 4.47 3 1025 3.33 3 1025

fH3G 0.0464 0.0201 0.0258

PHASE SEPARATION IN TERNARY POLYMERS. I 1455

where C is a constant. Owing to difficulties indetermining C,2 it is generally assumed that C5 1, which effectively assumes that the differencebetween z13 and z23 depends only on molecularsize. Although this is a special case, our simula-tions give results consistent with experimentalobservations.24

COMPUTATIONAL RESULTS

The time evolution of the two-phase structure isquantified using the structure factor, S(k, t),which is the square of the magnitude of the Fou-rier transform, A(k, t), of the concentration fluc-tuations in the system,

S~k, t! 5 uA~k, t!u2 5 U Om50

N21 On50

N21

~f3,m,n 2 f3,0!

3 expS2piN ~mkx 1 nky!DU 2

(40)

where w3,m,n is the volume fraction of polymer atnode (m, n), w3,0 is the mean polymer volumefraction, and k is the 2D position vector (kx, ky) inFourier space. The structure factor offers directcomparison with light scattering experiments be-

cause it is proportional to the scattering inten-sity.46 A similarly useful quantity is the pair cor-relation function, Gp(r, t), which is the Fouriertransform of S(k, t)

Gp~r, t! 5 Ok

exp~ik z r!S~k, t! (41)

Since the phase separation is isotropic, analy-sis can be improved by using circular averagedquantities, S(k, t) and Gp(r, t), where k and r arethe distances from the center of the Fourier spaceplane and real space plane, respectively.20 Theposition of the maximum of the structure factor,km, and the first zero of the pair correlation func-tion, r1, are used as measures of the domain size,because both are proportional to the position ofthe light scattering intensity maximum. Owing todiscretization, km is difficult to determine accu-rately, particularly at larger times where the spi-nodal ring is collapsing. As a result, a better mea-sure of domain size in discrete Fourier space isthe first moment of the structure factor, k1, givenby

k1 5

Ok

kS~k, t!

Ok

S~k, t!(42)

Figure 5. Polymer solvent mutual diffusion coeffi-cients at 25°C determined from free-volume theory. (F)Cellulose acetate/acetone; (E) PES/DMSO.

Figure 4. PMMA/NMP mutual diffusion coefficients.(F) T 5 15°C; (E) T 5 25°C; (�) T 5 35°C; (‚) T5 45°C; (■) T 5 55°C; (h) T 5 65°C.

1456 BARTON AND MCHUGH

This quantity can be determined more accu-rately than km since it is calculated from dataover the entire range of wave vectors, and it hasbeen shown to scale in the same manner withtime as km.15

To model the effects of nonsolvent and polymercomposition on the coarsening rates in a ternarypolymer solution, simulations were performed onthe ternary systems CA/acetone/water and PES/DMSO/water. Moreover, since these two systemsexhibit a wide variation in phase properties (Figs.1 and 2), one can explore the effects of the size ofthe miscibility gap on the predicted phase-sepa-ration dynamics. For each simulation, the coars-ening rate is determined by fitting the time evo-lution data for r*1 to the growth law expression47

R3 5 R03 1 K~t 2 t0! (43)

where K is the structure growth rate constant, Ris the size of the droplet phase at time, t, and R0is the size at t0, the time required for the earlystage of spinodal decomposition. The evolutioncurve determined from this analysis for a samplerun is shown in Figure 6. The linearity of thecorrelation demonstrates that the size of thephase separated structure grows as time1/3, whichis consistent with late-stage diffusive coarseningmechanisms. As shown from gray-scale intensity

plots of the two-phase morphology depicted inFigure 7, the structure grows by one of two mech-anisms, (1) coalescence,48 where solvent-richpores impinge on one another to form a largerpore, or (2) evaporation–condensation (Ostwaldripening),49 where larger pores in the polymer-rich matrix grow at the expense of smaller poresthat are eventually reabsorbed into the surround-ing matrix. As also found in our simulations on abinary polymer solution,21 these two coarseningmechanisms may occur simultaneously. Figure 7is an example of an off-critical quench on thepolymer-rich side which results in the formationof a dispersed solvent-rich phase surrounded byan enriched polymer matrix. Over the wholerange of temperature and compositions, a varietyof morphologies are obtained ranging from thestructures shown in Figure 7 to a bicontinuousmorphology or a dispersed polymer-rich phase.For a real system, this range of morphologies mayor may not be observed because of the finite timerequired for thermal equilibration. However, thedriving force for phase separation and structurecoarsening at the final quench temperature aspredicted by the Cahn–Hilliard approach remainsunchanged.

Figure 8 shows the effect of initial water con-tent on the coarsening rate for cellulose acetatesolutions quenched to 25°C. A relative maximumin the growth rate is observed at an intermediatewater content of the starting solution. This be-havior is a result of the interplay between ther-modynamics and transport kinetics and has alsobeen observed from simulations and light scatter-ing measurements performed on the PMMA/NMP/glycerin system23 (see Part II). Near themiscibility gap, thermodynamics dominate thetrends in the phase separation dynamics leadingto an increase in K with increasing nonsolventcomposition. However, further addition of nonsol-vent causes the polymer-rich phase to approachthe glassy regime, shown in the phase diagram inFigure 1, thereby resulting in a dramatic decreasein the system mobility. As a result, K decreaseswith increasing water content in this regime.These results also suggest a mechanism for thelocking-in of two-phase porous structures duringNIPS. In the latter case, additional nonsolventinflux will drive the system to the region of themiscibility gap where the dynamics effectivelyslow to zero because of the presence of the glasstransition region. This important characteristic ofthe phase-separation dynamics is further eluci-dated in Part II for the system PMMA/NMP/glyc-

Figure 6. Calculated late-stage growth curves forseveral 20 vol % cellulose acetate solutions quenched to25°C. Initial water concentration (vol %), (F) 54.0%; (E)50.1%; (Œ) 46.4%.

PHASE SEPARATION IN TERNARY POLYMERS. I 1457

erin. In addition, as shown in Figure 8, for higherinitial polymer concentrations, the maximum inK shifts to smaller values and higher water con-tents. Moreover, a decrease in the growth ratewith increasing polymer content occurred overthe entire range of water compositions investi-gated.

Similar behavior is shown by the PES/DMSO/water system in Figure 9, where one also observes

a maximum in the growth rate with water com-position of the initial solution. Moreover, athigher water concentrations, the growth rate ef-fectively approaches zero, suggesting that underthese conditions, the resulting late-stage growthstructure will be frozen-in. In addition, thegrowth rate curve also shifts to lower coarseningrates for higher PES concentrations; however, theeffect is not as dramatic as the shift observed for

Figure 7. Evolution of morphology for a 20/43/37 v/v/v cellulose acetate/acetone/watersolution quenched to 25°C. Scaled time a2t, (a) 1100; (b) 2750; (c) 5500; (d) 11,000; (e)22,000; (f) 44,000.

1458 BARTON AND MCHUGH

the cellulose acetate system. The trends in thephase separation dynamics observed for thesesystems are a direct result of the complex inter-play between the thermodynamic driving forcesfor phase separation and the system transportkinetics and seem to be a universal phenomenonof quenched polymer solutions that can undergo aglass transition.21,24

CONCLUSIONS

The effect of quench conditions and system ther-modynamic and transport properties on the late-stage phase-separation dynamics in two commonmembrane forming systems has been investi-gated utilizing Cahn–Hilliard theory. With bothsystems, a relative maximum occurs in thegrowth rate as a function of initial solution non-solvent concentration and the position of thismaximum shifts to larger nonsolvent concentra-tions with increasing polymer concentration. Inaddition, the overall structure formation dynam-ics is slowed with increasing polymer content,with a subsequent reduction in the prevalence ofthe maximum. Similar trends are observed insimulations and experimental measurements forthe PMMA/NMP/glycerin, which are reported inthe companion paper of this study.24 The similar-ity of the trends observed in the coarsening rates

for the ternary systems investigated in this workand the binary PMMA/cyclohexanol system stud-ied previously21 suggest that these results are auniversal phenomenon in quenched polymer solu-tions that can undergo a glass transition. More-over, as will be discussed in Part II, these resultssuggest a mechanism for the locking-in of thetwo-phase structure that forms during nonsol-vent-induced phase inversion.

This work has been supported under a grant from theNational Science Foundation, CTS 97-31509.

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