A Distribution Kinetics Approach for Crystallization of Polymer Blends

Jiao YangDepartment of Chemical Engineering, Louisiana State UniVersity, Baton Rouge, Louisiana 70803

Benjamin J. McCoyDepartment of Chemical Engineering and Materials Science, UniVersity of California, DaVis, California 95616

Giridhar Madras*Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India

ReceiVed: April 21, 2006; In Final Form: June 7, 2006

The cluster distribution approach is extended to investigate the crystallization kinetics of miscible polymerblends. Mixture effects of polymer-polymer interactions are incorporated into the diffusion coefficient. Themelting temperature, activation energy of diffusion, and phase transition enthalpy also depend on the blendingfraction and lead to characteristic kinetic behavior of crystallization. The influence of different blending fractionsis presented through the time dependence of polymer concentration, number and size of crystals, and crystallinity(in Avrami plots). Computational results indicate how overall crystallization kinetics can be expressedapproximately by the Avrami equation. The nucleation rate decreases as the blending fraction of the secondpolymer component increases. The investigation suggests that blending influences crystal growth rate mainlythrough the deposition-rate driving force and growth-rate coefficient. The model is further validated bysimulating the experimental data for the crystallization of a blend of poly(vinylidenefluoride)[PVDF] andpoly(vinyl acetate)[PVAc] at various blending fractions.

Introduction

Polymer blending is a useful and economical way to producenew materials with a variety of properties. Many high-performance thermoplastics are prepared by the crystallizationof polymer blends. The polymer-polymer interactions duringcrystallization can alter crystal structure, thermal stability, andmechanical properties such as rigidity and toughness. Under-standing how adding a polymer component affects the morphol-ogy, crystallization, and mechanical and thermal properties ofthe polymer blend is a significant scientific challenge. To explorethe crystallization kinetics of neat polymer melt, a cluster sizedistribution approach for nucleation accompanied by crystalgrowth and Ostwald ripening has been developed.1 Polymernucleation and crystal growth involve the dynamics of polymerchains, including the folded chain structures. Population balancesequations based on crystal size distribution, and the concentra-tion of amorphous polymer segments were established. Therelated moment and governing differential equations were alsosolved numerically under isothermal and nonisothermal condi-tions.2,3 The approach presented a fundamental, quantitativeexplanation for the mechanism of nucleation and crystal growth.In the current work, we extend this approach to the applicationof polymer blend crystallization. The miscibility of polymerblends complicates the kinetics and dynamics of crystallization.Understanding how the addition of one component affects thecrystallization behavior of the other can potentially be used todesign and develop new types of high-performance polymermaterials and to control product properties, e.g., microscopic

structure, average size of polymer crystals, and crystal-sizepolydispersity.

Several studies have investigated the crystallization of neatpolymer,4-7 but binary polymer blend crystallization hasreceived less attention. Experimental investigations on themiscibility and crystallization of polymer blends have beenreported occasionally,8-11 but only a few studies have beendirected toward fundamental theoretical investigations. Theunderstanding of how the presence of one component affectsthe morphology and crystallization of the other is still not clearfor polymer blends. Polymer crystallization involves twophenomena: nucleation and crystal growth. The addition of asecond polymer alters the crystallization kinetics of polymersolute mainly through these two steps. The temperature depen-dence of crystallization rates for neat (unmixed) polymers isapproximately parabolic; crystallization rate is zero atTg (glasstemperature) andTm (melting temperature), and the maximumrate appears between these two points. The addition of a secondpolymer to a semicrystalline polymer can act as a diluent,causing either a decrease in crystallinity by decreasing crystalnumber concentration or an increase in crystallinity by increasingchain mobility. The existence of polymer-polymer interactionsalso influences the deposition mechanism, causing either anincrease or decrease of the crystal growth rate.

For example, the spherulite radial growth rate of poly(ethylenemethacrylate) was found to decrease as poly(ethylene oxide) isadded as the binary component.12 By contrast, the presence ofpoly(vinyl methyl ether) was found to enhance the spherulitegrowth rate of polystyrene.13 Increases in the spherulitic growthrate and crystallinity were also found for poly(vinyl methylether) added as a second ingredient in isotactic polystyrene.14

* Corresponding author. E-mail: giridhar@chemeng.iisc.ernet.in. Tele-phone: 91-080-2293-2321. Fax: 91-080-2360-0683.

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10.1021/jp062451x CCC: $33.50 © 2006 American Chemical SocietyPublished on Web 07/14/2006

The miscibility and crystallization of PES poly(ethylene suc-cinate) (PES)and poly(vinyl phenol) (PVPh) blends15 wasinvestigated, and it was found that the growth rate of neat PESwas higher than that of blended PES crystallized at the sametemperature, which indicates that the addition of PVPh reducesthe spherulite growth of PES in blends. The Avrami exponentwas almost the same despite the blend composition andcrystallization temperature.

A theoretical model was developed by Rostami16 to explainthe slower rate of spherulite growth in thermodynamicallymiscible blends of semicrystalline and amorphous polymers.Harris and Robeson17 proposed a hypothesis to explain theenhancement of the crystallinity of a crystallizable componentdiluted with a miscible polymer. The interlamellar region, whichcontains the amorphous fraction, increased as the amorphouscomponent is added to semicrystalline polymer. The increasedamorphous fraction reduces the glass-transition temperature andlends additional mobility to the crystallizable ingredient, thusallowing a higher fraction of this ingredient in the crystallinephase.

For immiscible polymer blends, crystallization occurs withina single phase domain of a single component. The miscibilityand crystallization of PHBV poly(3-hydroxybutyrate-co-hy-droxyvalerate) (PHBV) and poly(ε-caprolactone) (PCL) blendswere investigated, and these two polymers were not miscible.18

The crystallization of the blends was studied using a two-stepprocess: as the blend was cooled, the component with highermelting temperature crystallized first and the lower meltingcomponent crystallized later. The crystallization rate of PHBVdecreased with increase of PCL, while the crystallization rateof PCL increased with increasing PHBV fraction. The Avramiexponent for crystallization of PCL varied as the componentfraction of PHBV changed and was almost the same forcrystallization of PHBV as fraction of PCL changed.

In this study, we focus on the crystallization kinetics of binarymiscible polymer blends; immiscible blends present additionaldifficulties that will be addressed in future work. Our approachis to represent the dynamics and kinetics of nucleation andgrowth by a cluster size distribution model for binary polymerblends. The influence of the blending fraction on crystallizationis incorporated into the effect of blending on the nucleation andcrystal growth rates.

Theoretical Model

The addition of a second polymer can act as a diluent, eitherdecreasing crystallinity by decreasing concentration of crystal-lizable component and nuclei numbers or increasing crystallinityby enhancing nucleation rate or increasing chain mobility. Fivepatterns of crystallinity development are identified upon additionof a crystallizable polymer additive:19 (1) the diluent does notaffect the crystallization, (2) the diluent retards the crystallizationrate, (3) the diluent prevents crystallization (particularly at highconcentrations), (4) the diluent accelerates crystallization, and(5) the diluent provides enough thermal mobility to causecrystallization of a normally noncrystalline polymer. As men-tioned, the addition of a second polymer ingredient affectscrystallization mainly through nucleation and crystal growthrates. We use classical nucleation theory to account for thenucleation of polymer blends. The nucleation rate, accordingto classical nucleation theory,20 can be represented as the fluxover the energy barrierW*,

with prefactor

and

Here, m(0) is the bulk concentration of amorphous polymersegments in the miscible polymer blend,σ accounts for theinterfacial energy of nucleus,δ represents the thickness of thecrystalline lamellae, andxm is the molecular weight of polymersegments. The thicknessδ of the crystalline lamellae isapproximated by the length of polymer segments, which isconstant for a certain polymer. The equilibrium concentrationmeq

(0), which is constant for a certain polymer blend, exclusivelydepends on crystallization temperature and the miscibility ofthe polymer blend. Thus, blending influences the nucleation ratethrough the dilution effect on the crystallized componentconcentration,m(0). As we increase the blending fraction of thesecond component, the nucleation prefactorkn decreases, andso does the exponential term because of the increase of thenucleation energy barrier. Nucleation vanishes when the polymerconcentration,m(0), is equal to or less than the equilibriumconcentration,meq

(0), because of the high energy barrier.The addition of a second component affects crystal growth

by altering the growth rate coefficient and deposition rate drivingforce. As proposed in our previous studies,1-3 crystal growthis represented by a reversible mechanism of addition to andseparation from nuclei. The polymer-polymer interactionintroduced by blending may affect the deposition and, conse-quently, alter the growth rate coefficientkg. Thus, the establishedpopulation balance equation in our previous work1-3 can beapplied to polymer blends. Reasoned thusly, the governingequations for crystallization of neat polymer and polymer blendsare the same, and the mixing effects are incorporated into theevaluation ofm(0), meq

(0), and growth rate coefficientkg(x). Thegoverning equations are

and

whereI accounts for nucleation rate given in eq 1. A generalexpression for growth rate coefficientkg(x) is a power lawdependence on crystal massx, kg(x) ) κgxλ, where the value ofpower exponentλ depends on the deposition mechanism. Apolymer segment that deposits on a crystal surface diffusesthrough the solution to react on the crystal surface; suchdiffusion-controlled reactions have a rate coefficient representedby

whereD is the diffusion coefficient. The crystal size is relatedto its massx by rc ) (3x/4πFc)1/3, in terms of crystal densityFc, which is constant at a given temperature. Thus, the powerλ

kn ) (m(0))2 (2σxm/π)1/2F-1 (2)

W* ) 4xm δ σ2/[FRT ln(m(0)/meq(0))] (3)

∂c(x, t)/∂t ) -kd(x) c(x, t) + ∫x

∞kd(x′) c(x′, t) δ(x -

(x′ - xm)) dx′ - kg(x) c(x, t) m(0) ∫0

∞δ(x′ - xm) dx′ +

m(0)∫0

xkg(x′) c(x′, t) δ(x - xm) dx′ + I δ(x - x*) (4)

∂m(x, t)/∂t ) -m(0)(t) ∫0

∞kg(x′) c(x′, t) dx′ +

∫x

∞kd(x′) c(x′, t) δ(x - xm) dx′ - I δ(x - x*) x*/xm (5)

kg ) 4πDrc (6)

I ) kn exp(-W*/RT) (1)

Crystallization of Polymer Blends J. Phys. Chem. B, Vol. 110, No. 31, 200615199

) 1/3 represents diffusion-controlled crystal growth. In dilutesolutions,D is linearly dependent on the blending fraction.21,22

However, the polymer segment-segment interactions compli-cate the determination of the diffusion coefficient in polymer-polymer mixtures. We may express the diffusion coefficient interms of tracer diffusion of each componentD*, degree ofpolymerizationN, blending fractionφ, and the Flory-Hugginsinteraction parameterø,23-26

with subscripts A and B representing the two polymer compo-nents. As the activation energy for the growth coefficientaccounts for activated diffusion, eq 7 can be expanded into apolynomial in the blending fractionφA ) 1 - φB,

wherea1, a2, anda3 are constants for a certain polymer blenddepending onD*, N, andø. The diffusion activation energyEalso varies as the blended polymer or its fraction changes,leading to the variation of growth rate coefficient. Substitutingrc ) (3x/4πFc)1/3 and eq 7 into eq 6 gives a detailed expressionfor the growth rate coefficient

with γ ) 4πD0(3/4πFc)1/3. The volume fractionφA can also berelated tom(0), the molar concentration of component A, by

whereνm is the molar volume of crystallized component A.Thus, the growth rate coefficient can be represented in termsof m(0) as

with b1 ) a1/νm, b2 ) a2/νm2, andb3 ) a3/νm

3. When the growthis limited by the crystal surface, the rate coefficient isproportional to the crystal surface area,rc

2, so that the powerexponent of growth isλ ) 2/3, which represents the surface-controlled deposition rate coefficient. If the deposition rate isindependent of the crystal surface area, then the power exponentof growth is λ ) 0, indicating that crystal growth rate isindependent of mass. With the definition of dimensionlessquantities as follows

Equations 4 and 5 are rewritten as,

and

with d1 ) b1µ∞, d2 ) b2µ∞2, and d3 ) b3µ∞

3. The scalednucleation rateJ can be written in terms of supersaturationS,interfacial energyw, temperatureΘ, and phase transitionenthalpyh

with

The parametern is the exponent of temperature in the expressionfor interfacial energy.1,2 For the simple case of surface inde-pendent deposition,λ ) 0, the moment technique can be usedto rewrite the population balance equations as

For any nonzeroλ, the moment equations lack closure, thus anumerical technique is required to solve eqs 13 and 14.

The number of macromolecules in the critical crystal nucleusis

which varies with time because of the time dependence of thescaled number of crystallizing polymersS and is stronglydependent on temperature. In terms of the initial number ofpolymer moleculesS0, the scaled mass conservation for a closedsystem follows from the population balance equations, eqs 17and 19

whereC0(1) is the initial cluster mass, representing heterogeneous

nuclei and seeds, and is zero for homogeneous nucleation, andS0 is the initial number of polymer molecules. The degree ofcrystallinity, X, is defined as the ratio of the mass crystallizedat time t divided by the total mass crystallized,

Substitution of the mass conservation equation, eq 21, simplifiesthe above equation to

The growth rate coefficient is not the only parameter subjectto change under different blend fractions, although only itsdependence on blend fraction is mathematically approximatedin the population balance equations. Melting temperature, phasetransition enthalpy, and activation energy may also change with

D ) (DA*NAφB + DB*NBφA)(φA/NB + φB/NA - 2φAφBø)(7)

D ) D0 exp(-E/RT) (1 + a1φA + a2φA2 + a3φA

3) (8)

kg(x) ) γx1/3 exp(-E/RT)(1 + a1φA + a2φA2 + a3φA

3) (9)

m(0) ) φA/νm (10)

kg(x) ) γx1/3 exp(-E/RT)(1 + b1m(0) + b2m

(0)2 + b3m(0)3)(11)

C ) cxm/µ∞, C(n) ) c(n)/µ∞ xmn, ê ) x/xm, θ ) tγµ∞xm

λ,

S) m(0)/µ∞, Θ ) T/Tm, w ) 2σ0 (xmb/F)1/2/RTm,

J ) I/γµ∞2xm

λ, ε ) E/RTm, h ) ∆H/RTm (12)

dS(θ)/dθ ) (1 + d1S+ d2S2 + d3S

3) exp(-ε/Θ) [-S(θ) +exp(-h/Θ)]C(λ) + Jê* (13)

∂C(ê, θ)/∂θ ) (1 + d1S+ d2S2 + d3S

3) S(θ)

exp(-ε/Θ) (-êλC(ê, θ) + (ê - 1)λC(ê - 1, θ)) -êλ exp(-(ε + h)/Θ)C(ê, θ) + (ê + 1)λ exp(-(ε +

h)/Θ)C(ê + 1, θ) - Jδ(ê - ê*) (14)

J ) J0(1 - Θ)n/2 S2 exp[-(w(1 - Θ)n/Θ)2/(ln S+ h/Θ)](15)

J0 ) (2σ0/π)1/2/Fγxmλ-1/2 (16)

dS(θ)/dθ ) (1 + d1S+ d2S2 + d3S

3) exp(-ε/Θ) [-S(θ) +

exp(-h/Θ)]C(0) + J ê* (17)

dC(0)(θ)/dθ ) J (18)

dC(1)(θ)/dθ )-(1 + d1S+ d2S

2 + d3S3) exp(-ε/Θ)[-S(θ) +

exp(-h/Θ)]C(0) - J ê* (19)

ê* ) [w(1 - Θ)n/Θ (ln S+ h/Θ)]2 (20)

C(1)(θ) + S(θ) ) C0(1) + S0 (21)

X ) (C(1) - C0(1))/(Ceq

(1) - C0(1)) (22)

X ) (S0 - S(θ))/(S0 - e-h/Θ) ≈ 1 - S/S0 (23)

15200 J. Phys. Chem. B, Vol. 110, No. 31, 2006 Yang et al.

blending fraction, e.g., the melting temperature of poly-(vinylidenefluoride)[PVDF]/poly(vinyl acetate)[PVAc] blendsvaries from 170.7°C to 152.4°C as the fraction of PVAcincreases from 0 to 30%.27 To keep the governing equationssimple, we consider the effects of the blending fraction on thoseparameters in computation rather than approximate themmathematically in the population balance equations.

Results and Discussion

For the crystallization of miscible polymer blends, we areconcerned with the effect of blending on the kinetics ofcrystallization for the easily crystallizing component, i.e., thecomponent crystallizing at the relatively higher temperature. Theincrease of the second ingredient fraction, according to eqs 1,2, and 3, leads to decreasing nucleation rate because of thedilution of concentrationm(0). The dilution also slows the crystalgrowth rate by decreasing the deposition rate and the drivingforce,-S(θ) + exp(-h/Θ). Thus, by incorporating the blendingeffects for melting temperatureTm, enthalpy of fusion∆H, andactivated diffusion energyε, introducing a second polymercomponent generally delays the crystallization by decreasingnucleation and crystal growth rates.

For the simple case of surface-independent deposition (λ )0), the governing moment eqs 17-19 are solved by NDSolvein Mathematica to assess the effect of blending on nucleationand crystal growth rate. The moment method provides an easyand quick model for polymer crystallization with surface-independent crystal growth (λ ) 0). However, a generalapproach is to solve the equations numerically based on methodsdescribed previously.1,3 The differential eq 19 is solved by theRunge-Kutta technique with an adaptive time step andC(ê,θ) is evaluated at each time step sequentially. The mass variable(ê) is divided into 5000 intervals, and the adaptive time (θ)step varied from 10-5 to 10-2, ensuring stability and accuracyat all values of the parameters. At every time step, the massconservation (eq 21) was verified. The numerical results arefurther validated by comparison with the analytical momentsolutions obtained forλ ) 0.

Our choice of parameters is based on earlier papers1-3 andexperimental measurements.28 Previous work suggested a value1

n ) 20 in eqs 15 and 20 to represent the temperature dependenceof interfacial energy. The interfacial energy parameterw, writtenin terms of density, is chosen to bew ) 2. The molar enthalpyof phase transitionh is usually about 10. The activation energy,ε, for growth rate is typically smaller thanh, and a constantnucleation rate prefactor,J0 ) 106, is proposed. The values ofvarious parameters used in the model are summarized in Table1. There are no particles present initially for homogeneousnucleation, thus the initial conditions areS(θ ) 0) ) S0, C(0)(θ) 0) ) 0, andC(1)(θ ) 0) ) 0.

Figure 1 presents the computational results of crystallizationtime evolution as the fraction of the second polymer component

TABLE 1: Parameter Values Used in the Model for Figure1

φA S0 ε h Θ C0(0) C0

(1) J0 λ w n d1 d2 d3

1.00 10 2.0 10 0.950 0 0 106 0 2.0 20 1 0 00.90 9 2.5 9 0.955 0 0 106 0 2.0 20 1 0 00.80 8 3 8 0.96 0 0 106 0 2.0 20 1 0 00.70 7 3.5 7 0.965 0 0 106 0 2.0 20 1 0 0

Figure 1. Time evolution of (a) supersaturation, (b) number of crystals, (c) average crystal size, and (d) nucleation rate for various blendingfractions (parameters are listed in Table 1).

Crystallization of Polymer Blends J. Phys. Chem. B, Vol. 110, No. 31, 200615201

changes. The initial values of scaled solute concentrations,S0

) 10, 9, 8, and 7, correspond to the second component fractionof 0% (neat polymer), 10, 20, and 30%, respectively. Theparametersd1, d2, andd3, which depend on polymer physicalproperties, are chosen to equal 1.0, 0, and 0, respectively. Sucha linear approximation for the diffusion coefficientD mayconstrain applicability to a certain range of blend fraction. Fora miscible blending system, crystallization is initiated only abovethe critical blending fraction where the two components are inequilibrium. Most of crystallization processes are initiated bycooling; and polymer-polymer solutions become supersaturatedas the temperature decreases. Consequently, the supersaturatedcomponent begins to nucleate only after the fraction ofcrystallized component in the liquid phase decreases to theequilibrium blending fraction corresponding to the crystallizationtemperature. Thus crystallization of miscible polymer blendsoccurs only within a certain range of blending fraction.

Figure 1a shows the time evolution of supersaturationS. Theconsumption rate of free solute (in the amorphous state)decreases as the fraction of the second polymer increasesbecause the neat polymer has the largest solute consuming rate.Compared with neat polymer melt, the blended system requiresmore time to complete the crystallization. The time evolutionof crystal number and crystal size is represented in Figure 1band c. The influence of the second polymer in the blend (Figure1b) shows that the fraction increase of the second polymerdilutes the solute concentration. Meanwhile, the initial criticalsize ê0* (eq 20) is also subject to increase because thesupersaturation decreases as the blending fraction increases.Figure 1c presents the time evolution of the average crystal size,

indicating the influence of blending fraction on crystal growthrate. The rise of the second polymer component fraction slowsthe crystal growth rate by decreasing the growth rate coefficientand deposition rate driving force. As a consequence, the smallestaverage crystal size at a blending fraction of the secondcomponent under 30% is observed. The influence of the secondpolymer in the blend shows that the fraction increase of thesecond polymer decreases nucleation rate, as shown in Figure1d.

The model is also used to represent the Avrami plots of thecrystallinity time evolution, as shown in Figure 2. The evolutionof crystallinity with time integrates the blending effects on bothnucleation and crystal growth and indicates the negativeinfluence on crystallization time for the second polymercomponent. The variation of the blending fraction does not affectthe Avrami exponent significantly. Denucleation, the reverseof nucleation, occurs when the crystal size is smaller than thecritical nucleus size. Denucleation has considerable influenceon the characteristic Avrami plots, especially at the end ofcrystallization when denucleation dominates over nucleation.At the beginning of crystallization, most polymer crystals arelarger than the critical crystal size, thus denucleation, comparedwith nucleation, is negligible. The critical crystal size, however,increases as the supersaturation declines when polymer depositson the crystal, thus more crystals become smaller than the criticalsize. These unstable crystals will dissolve rapidly. At the endof crystallization, denucleation and ripening become dominantbecause the supersaturation is close to unity, and consequently,nucleation vanishes. Figure 2 presents the effect of denucleationby comparison of moment and numerical solutions. The solidlines represent the characteristic Avrami plots of momentsolutions for which denucleation is not considered. The dottedlines represent the numerical solutions that account for denucle-ation. At the beginning, the denucleation rate is small and themoment solution matches the numerical solution very wellbecause the small denucleation rate delays the dominatinginfluence of nucleation.29 As supersaturation decreases, thediscrepancy between moment solution and numerical solutioncaused by denucleation becomes apparent at long times. Thenumerically computed crystallinity reaches its asymptote,whereas the crystallinity computed by the moment methodcontinues to increase.

Figure 1 presents model predictions when the diffusioncoefficient D is approximated by a linear dependence onblending fractions,φA, and Figure 3 shows the time evolutionof crystallinity and crystal size based on the linear approximationof diffusion coefficient by varyingd1 in the range of 0.1-10

Figure 2. Avrami plot showing the time evolution of crystallinitycomparing the moment solution (solid line) and the numerical solution(dotted line) with the parameters listed in Table 1.

Figure 3. Effect of d1 on the time evolution of (a) crystallinity and (b) crystal size withφA ) 0.80 and the parameters listed in Table 1.

15202 J. Phys. Chem. B, Vol. 110, No. 31, 2006 Yang et al.

(see eqs 13 and 14). It is observed that a stronger dependenceof the diffusion coefficient on the blending fraction leads tofaster crystallization rates. The slope of the Avrami plot (Avramiexponent) does not depend ond1, as shown in Figure 3a. Theeffect of d1 on average crystal size is presented in Figure 3b.According to the linear expression of the diffusion coefficient,a large value ofd1 leads to an increase in the diffusioncoefficient, resulting in a larger average crystal size, as shownin Figure 3b.

We next determine the effect of a polynomial dependence ofD on blending fractions. The time evolution of crystallizationbased on linear and polynomial expressions for diffusioncoefficientD is shown in Figure 4. The polynomial expressionfor diffusion coefficient does not lead to a significant variationin Avrami exponent, suggesting that the linear expression is avalid approximation.

The presence of the second polymer in the blend19 may eitherdecrease or increase the absolute value ofTg and Tm of thesemicrystalline ingredient, depending upon the differencebetween the isothermal crystallization temperature,Ti, and (Tg

+ Tm)/2. Depending on whether this difference increases ordecreases with the addition of the second ingredient, thecrystallization rate decreases or increases, respectively. In theproposed model, eq 19 indicates that the crystal growth andcrystallization rate would increase or decrease depending onwhetherΘ increases or decreases with the blending fraction,respectively, consistent with the observed experimental data.19

The validity of the distribution kinetics model is alsoexamined by comparison with the experimental data27 of thePVDF/PVAc blend (Figure 5), whereΘ decreases with increas-ing blending fraction, and therefore, the crystallization ratedecreases with the blending fraction. The simulation results arebased on dimensionless timeθ ) tγµ∞xm

λ, so a transpositionof the original experimental data, based on real time, is appliedto compare with simulations. According to the definition ofdimensionless time, a horizontal transposition of log(γµ∞xm

λ)units is applied to the experimental measurements to convertthe experimental data into plots based on dimensionless timeθ. The points are experimental data,27 and the model parametersare listed in Table 2. A moment solution with surface-independent crystal growth (λ ) 0) does not fit the experimentaldata (Figure 5a), soλ ) 1/3 is chosen3 and the population balanceequations are solved numerically. The measured horizontaltransposition is unity, independent of the blend fraction ortemperature. Good agreement between experimental data andsimulation results (Figure 5b) withλ ) 1/3 is observed, indicatingthat the crystal growth for the PVDF/PVAc blending system isdiffusion dominated.

Conclusions

On the basis of a previous distribution kinetics approach, wehave investigated the crystallization kinetics of miscible polymerblends. This model accounts for the effects of polymer blendingon nucleation and crystal growth rates. Adding a second polymerinfluences the crystallization mainly through thermodynamicsand kinetics (equilibrium solubility and deposition mechanism).These effects are incorporated by varying the initial values ofmonomer concentration and growth rate coefficient. Assumingthat the deposition mechanism does not change during crystal-lization, we have investigated the crystallization kinetics forvarious blending fractions. Computation results indicate thatincreasing the second polymer fraction leads to the decrease ofboth nucleation and crystal growth rates, while the Avrami

Figure 4. Comparison between linear and nonlinear expression fordiffusion coefficient withC0

(0) ) 0, C0(1) ) 0, φA ) 0.80,J0 ) 106, λ

) 0, andn ) 20. The dotted line is the linear approximation forD (d1

) 1, d2 ) 0, andd3 ) 0) and the solid line represents the polynomialexpression forD with d1 ) 1, d2 ) -0.02, andd3 ) 0.02.

Figure 5. Model prediction of the experimental data of the PVDF/PVAc blend with different blending fractions with (a)λ ) 0 and (b)λ ) 1/3;(1): PVDF/PVAc ) 100/0, (2): PVDF/PVAc ) 90/10 (b): PVDF/PVAc ) 80/20, (9): PVDF/PVAc ) 70/30.

TABLE 2: Melting Temperature Tm, Heat of Fusion ∆Hm,and Diffusion Activation Energy E of PVDF and PVDF/PVAc Blends under Different Blending Fractions27 and theParameter Values Used in the Model to Fit the ExperimentalMeasurements in Figure 5a

PVDF/PVAc Tm/°C ∆Hm/J/g E/J/g φA S0 ε h Θ

100/0 170.7 64.54 0.96 1.00 10 1.3 10 0.94990/10 168.4 55.27 1.03 0.90 9 1.4 8.56 0.95480/20 167.3 42.69 2.05 0.80 8 2.8 6.61 0.95670/30 152.4 40.06 5.30 0.70 7 7.5 6.21 0.989

a The other parameters are the same as listed in Table 1.

Crystallization of Polymer Blends J. Phys. Chem. B, Vol. 110, No. 31, 200615203

exponent does not change. The diffusion coefficient plays animportant role in determining the time evolution of polymerblends because of the strong polymer-polymer interactions. Apolynomial expression and a linear approximation for thediffusion coefficient on the blending fraction are compared, andit is found that the linear expression is an adequate approxima-tion. The model is used to simulate the experimental data ofthe time evolution of crystallinity for the blend of poly-(vinylidenefluoride) [PVDF] and poly(vinyl acetate) [PVAc] atvarious blending fractions.

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