Kinetics of Nonisothermal Polymer Crystallization

Jiao Yang and Benjamin J. McCoy†

Department of Chemical Engineering, Louisiana State UniVersity, Baton Rouge, Louisiana 70803

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

ReceiVed: April 29, 2005; In Final Form: July 8, 2005

We adopt a cluster size distribution model to investigate the kinetics of nonisothermal polymer crystallization.The time dependencies of polymer concentration, number and size of crystals, and crystallinity (in Avramiplots) are presented for different cooling rates. The incubation period is also investigated at different coolingrates and initial temperatures. The relationship between cooling rates and incubation time is presentedgraphically and compared with experimental measurements. The initial temperature (relative to melting point)has a significant effect on nonisothermal crystallization. A comparison of moment and numerical solutions ofthe population balance equations shows the influence of Ostwald ripening. Agreement between modelingresults and experimental measurements at different cooling rates supports the application of the distributionkinetics model for nonisothermal crystallization.

Introduction

Experimental and modeling studies of polymer crystallizationhave often been carried out under idealized conditions ofconstant temperature. A polymer sample is melted at or abovethe equilibrium melting point, and is rapidly quenched to thetemperature chosen for crystallization. In practice, however,industrial processes and some experimental studies generallyproceed under dynamic and nonisothermal conditions, in whichthe crystallization rate depends not only on the instantaneoustemperature but also on the rate of temperature change. Becausethe thermal history experienced by the polymer sample is criticalin the determination of the final product properties, an under-standing of nonisothermal crystallization is necessary. This isthe motivation for extending isothermal crystallization kineticsto nonisothermal conditions.

The overlapping of primary nucleation and crystal growthcomplicates the kinetics of polymer crystallization,1-3 ensuringthat a fairly sophisticated model will be required. The firstattempt to describe the kinetics of polymer crystallization wasmade by Evans4 via statistical considerations, and by Avrami5-7

via the concept of so-called extended volume. Avrami derivedthe well-known expression for crystallinity as a function of time

whereV(t) is the extended volume, denoting the total volumeof all domains growing from all nucleation attempts occurringat time t. For isothermal crystallization, the Avrami equationsimplifies to

whereK is a rate coefficient depending on both nucleation and

crystal growth rate, andm assumes the values of 1, 2, 3, and 4depending on the nucleation type and the crystal dimensionality.8

For years the kinetic analysis for polymer crystallization hasinvolved plots based on the Avrami equation, even though itwas often criticized as being oversimplified.8-10 Although theempirical basis of eq II allows it to describe some fundamentalfeatures of crystallization, the exponent valuem, determinedexperimentally for various polymers (mainly on the basis ofDSC measurements of latent heat during crystallization), israrely in agreement with the predicted integer numbers.11-14 Anumber of authors15-17 also pointed out that the Avrami plotof experimental measurements, log[-ln(1 - X)] vs ln(t), isactually a curved rather than straight line proposed by theAvrami theory. These observations suggest that the assumptionsleading to the Avrami equation are simplistic, and that itsfoundation is inexact.

Neither eq I nor II can be used directly in nonisothermalcrystallization unless the temperature dependence ofV(t) or Kis known. Nonisothermal crystallization has been modeled byapplying a linear temperature change to the Avrami equation.18-21

To evaluate kinetic parameters, the Tobin22 and Ozawa19 modelsallowedK to depend on the cooling rate. Taking into accountthe transient and nonisothermal effects, Ziabicki23,24 proposeda new model for polymer crystallization in which externalvariables, such as temperature, pressure, and stress, change intime. Generalized equations for polymer crystallization undervariable external conditions were developed by introducing threetemperature-dependent material functions: steady-state crystal-lization rate, relaxation time, and isothermal nucleation. Ex-perimental studies of nonisothermal crystallization at coolingrates between 2 and 40°C/min with the DSC method20 showedan increase in the Avrami exponent as the cooling rate increased.The effect of the cooling rate on the crystallinity9 indicated thatthe Avrami equation could not quantitatively describe noniso-thermal crystallization, and required an additional compensatingparameter. The mathematical description of the spheruliticpattern formation in nonisothermal conditions was elaborated

* Corresponding author. Phone: 91-080-2293-2321. Fax: 91-080-2360-0683. Email: giridhar@chemeng.iisc.ernet.in.

† Current address: Department of Chemical Engineering & MaterialsScience, University of California, Davis, CA 95616.

X(t) ) 1 - exp[-V(t)] (I)

X(t) ) 1 - exp(-Ktm) (II)

18550 J. Phys. Chem. B2005,109,18550-18557

10.1021/jp052219b CCC: $30.25 © 2005 American Chemical SocietyPublished on Web 09/13/2005

on the basis of the assumption of the momentary randomnessof primary nucleation.25-27 This probabilistic approach makesit possible to describe the formation of spherulitic structuralelements in the course of nonisothermal crystallization. Thetemperature dependencies of the material functions are stilluncertain, however, and the Avrami exponent for nonisothermalcrystallization requires further investigation.

The Avrami equation, with its basis in heuristic ideas, is stillapplied to polymer crystallization, even though the controversyabout the underlying mechanism has continued since thederivation of this equation. This leads us to propose a newcluster distribution model for polymer crystallization kineticsby considering the similarity between crystal growth andmonomer deposition on clusters.2,3 Based on the fundamentalconcepts of nucleation and crystal growth, the model accountsfor homogeneous or heterogeneous nucleation and crystalgrowth for isothermal crystallization. The model proved capableof quantitatively describing experimental data for isothermalpolymer crystallization. In the current paper, we will investigatenonisothermal crystallization kinetics with linear cooling rates.The temperature effect is incorporated into the model by thetemperature dependencies of the interfacial energy, equilibriumsolubility, and growth rate coefficient.

Theory

We hypothesize that polymer crystal growth is similar to thedeposition of molecules on clusters. This deposition process bywhich polymer molecules of massx′ ) xm are reversiblyattached or dissociated from an established crystal of massxcan be represented as a reaction-like process28

where C(x) represents the crystal of massx and M(x′ ) xm) isthe macromolecule with a uniform molecular weight distributionwritten in terms of monomer concentrationm(0)(t) and monomermolecular weightxm, asm(x, t) ) m(0)(t) δ(x - xm). The mass-dependent rate coefficientskg(x) andkd(x) are for cluster growthand dissociation, respectively. An activation energy,E, isassumed to account for the temperature dependence of thegrowth rate coefficient,kg(x) ) γxλ exp(-E/RT) whereγ is atemperature-independent constant. Exponentλ accounts for themass dependence of the growth rate coefficient, representingsurface-independent, diffusion-controlled, and surface-dependentgrowth, respectively, whenλ equals 0, 1/3, and 2/3.

Temperature influences crystal growth through kinetics andthermodynamics. The activation energy for the growth ratecoefficient accounts for the temperature effects through kinetics,whereas the temperature dependence of the equilibrium solubil-ity for a flat surface

accounts for the temperature influence through thermodynamics.Here,∆H is the molar enthalpy of the phase transition andµ∞represents the flat-surface equilibrium solubility at high tem-perature. For a lamellar crystal with a flat growth front, theGibbs-Thomson effect vanishes, so that the difference betweenthe local equilibrium interfacial concentrationmeq

(0) and thesolubility m∞

(0) for a flat surface vanishes, thusmeq(0) ) m∞

(0).The rate coefficient for growth and the dissociation ratecoefficient are related,kd(x) ) meq

(0)kg(x), by microscopicreversibility.29 Nucleation, the initiation of crystal nuclei, occurssimultaneously with crystal growth. By including the nucleation

term, the kinetics of polymer crystallization can be describedby the cluster size distribution equations30

and

where x* represents the critical crystal mass andI denotesnucleation rate. For a supersaturated (metastable) system, thecluster energy of a crystal of radiusr reaches a maximum valueat the critical crystal radius. According to classical nucleationtheory,31,32the nucleation rate (moles of nuclei per vol per time)is the flux over the maximum energy barrier at the critical radius

whereW* represents the energy barrier for the nucleation ofcrystal with critical massx*, and the nucleation prefactor writtenin terms of crystal density,F, and crystal interfacial energy,σ,is

The classical nucleation theory utilizes the sum of the surfaceenergy and the formation free energy; for an equilateral lamellarcrystal of characteristic lengtha

whereb is the thickness of the lamellar crystal andσ is thecrystal interfacial energy with its temperature dependenceapproximated as3,33

wheren and σ0 are constants dependent on the polymer. Theequation indicates that the interfacial energyσ vanishes atreference temperatureTm. The chemical potential difference(-RT ln Ssat) between the two phases is expressed in terms ofsupersaturationSsat, which is the ratio of bulk monomerconcentrationm(0) over the equilibrium monomer concentrationmeq

(0). The maximum energy occurs at the critical crystal size

which can be converted into critical crystal mass. Thus thenucleation energy barrier, according to eq 7, is

With the definitions of dimensionless quantities in ourprevious work3 we may write the cluster size distribution

C(x) + M(xm) y\zkg(x)

kd(x)C(x + xm) (1)

m∞(0) ) µ∞ exp(-∆H/RT) (2)

∂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)(t)∫0

∞δ(x′ - xm) dx′ +

m(0) ∫0

xkg(x′) c(x′, t) δ(x - xm) dx′ + Iδ(x - x*) (3)

∂m(x, t)/∂t ) -m(0)(t)∫0

∞kg(x′) c(x′, t) dx′+

∫x

∞kd(x′) c(x′, t) δ(x - xm) dx′ - Iδ(x - x*) x*/xm (4)

I ) kn exp(-W*/RT) (5)

kn ) (m(0))2(2σxm/π)1/2F-1 (6)

W(a) ) 4abσ - a2b(F/xm)RT ln Ssat (7)

σ ) σ0(1 - T/Tm)n (8)

a* ) 2σxm/(FRT ln Ssat) (9)

W* ) 4xmbσ2/[FRT ln(m(0)/meq(0))] (10)

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 (11)

Kinetics of Nonisothermal Polymer Crystallization J. Phys. Chem. B, Vol. 109, No. 39, 200518551

equations in dimensionless form

and

The initial conditions areS(θ ) 0) ) S0 andC(ê, θ ) 0) )0, whereê is defined as the number of macromolecules in thecrystal. The source termJδ(ê - ê*) represents the scalednucleation rate of crystals havingê* macromolecules. Thenumber of macromolecules in the critical crystal nucleus is

which varies with time, because of the time dependence of thescaled number of crystallizing polymersS, and is stronglydependent on temperature. When ripening occurs, a sink termappears representing the denucleation of crystals smaller thanthe critical size. By defining the temperature-independentprefactor,J0 ) (2σ0/π)1/2/Fγxm

λ-1/2, the nucleation rate can bewritten as

With the definition of thejth moment,C(j) ) ∫0∞ êjC(ê,θ) dê,

the dimensionless cluster size distribution equations, for thespecial caseλ ) 0, can be rewritten as moment equations

and

The zeroth moment,C(0), and the first moment,C(1), representthe time-dependent molar (number) and mass concentrationsof crystal, respectively. The ratio of the two moments is theaverage size of the crystals,Cavg ) C(1)/C(0). In terms of theinitial number of polymer moleculesS0, the scaled massconservation for a closed system follows from the populationbalance equations, eqs 12 and 17

whereC0(1) is the initial cluster mass and is zero for homoge-

neous nucleation andS0 is the initial number of polymermolecules.

To use eqs 13-15 in the analysis of nonisothermal crystal-lization, it is assumed that the polymer sample experiences aconstant cooling rate. Though we adopt a linear decrease oftemperature with time, one can easily incorporate any variationof temperature with time in the model. The relationship betweencrystallization timet and the crystallization temperatureT isthus

where â is the cooling rate (K/min) andT0 is the initialtemperature. With the dimensionless quantitiesθ ) tγµ∞xm

λ and

Θ ) T/Tm, where temperatures are represented in Kelvins, therelationship can be rewritten as

whereΘi ) T0/Tm is the dimensionless initial temperature andφ ) â/[γµ∞Tmxm

λ] is the dimensionless cooling rate. Thus eqs12, 16, and 17, for the crystallization in which a polymer sampleis cooled from its melting point,Tm, can be rewritten in termsof dimensionless timeθ

and

Based on eq 20, eqs 21-23 can be rewritten in terms oftemperatureΘ

and

The degree of crystallinityX is defined as the ratio of polymercrystallized at timeθ over the total crystallized polymer mass

Substitution of the mass conservation equation, eq 18,simplifies eq 27 to

Thus the crystallinity time or temperature dependence ofcrystallinity can be obtained by solving the above ordinarydifferential moment equations with various cooling rates,φ.

Results and Discussion

For the non-steady-state nonisothermal polymer crystalliza-tion, our general concern is the effect of the cooling rate on thecluster size distribution. On the basis of the relationship betweentemperature and time, eq 19, we investigated the cluster sizedistribution as a function of temperature and time, respectively,as shown in eqs 21-26. Our choice of parameters is based onearlier papers,2,3 for example, eq 8, proposed in our previouswork with suggested values ofn,3 and is used to present thetemperature dependence of interfacial energy. The interfacialenergy parameter,w, written in terms of density, is chosen tobe w ) 2 in our computation. The molar enthalpy of phasetransition,h, is usually around 100. The activation energy forgrowth rate,ε, is typically smaller thanh, and a valueε ) 1.0is chosen. The homogeneous nucleation rate is very sensitiveto supersaturationSsatand temperature, and a constant nucleationrate prefactor,J0 ) 106, is proposed. We assume the polymermelt is cooled from the melting temperature,Θi ) 1, where

dS(θ)/dθ ) exp(-ε/Θ)[-S(θ) + exp(-h/Θ)]C(λ) + Jê*(12)

∂C(ê, θ)/∂θ ) 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*) (13)

ê* ) [w (1 - Θ)n/Θ × (ln S+ h/Θ)]2 (14)

J ) J0(1 - Θ)n/2 S2exp[-(w(1 - Θ)n/Θ)2/(ln S+ h/Θ)](15)

dC(0)(θ)/dθ ) J (16)

dC(1)(θ)/dθ ) -exp(-ε/Θ)[-S(θ) +exp(-h/Θ)]C(0) - Jê* (17)

C(1)(θ) + S(θ) ) C0(1) + S0 (18)

T ) T0 - ât (19)

Θ ) Θi - φθ (20)

dS(θ)/dθ ) exp(-ε/(1 - φθ))[-S(θ) +exp(-h/(1 - φθ))]C(0) + Jê* (21)

dC(0)(θ)/dθ ) J (22)

dC(1)(θ)/dθ ) -exp(-ε/(1 - φθ))[-S(θ) +exp(-h/(1 - φθ))]C(0) - Jê* (23)

dS(Θ)/dΘ ) -(1/φ)[exp(-ε/Θ)[-S(Θ) +exp(-h/Θ)]C(0) + Jê*] (24)

dC(0)(Θ)/dΘ ) -(1/φ)J (25)

dC(1)(Θ)/dΘ ) -(1/φ)[-exp(-ε/Θ)[-S(θ) +exp(-h/Θ)]C(0) - Jê*] (26)

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

(1) - C0(1)) (27)

X ) (S0 - S(θ))/(S0 - e-h/Θ) (28)

18552 J. Phys. Chem. B, Vol. 109, No. 39, 2005 Yang et al.

the subscript i denotes the initial condition. There are no particlespresent initially for homogeneous nucleation; thus the initialconditions areS(Θ ) Θi) ) S0, C(0)(Θ ) Θi) ) 0, andC(1)(Θ) Θi) ) 0.

The temperature dependence of crystal size distribution ispresented in Figure 1 at different cooling rates. The dimension-less cooling rateφ varies from 10-4 to 6 × 10-4. Figure 1apresents the effect of cooling rate on the evolution of the numberof macromolecules,S, which decreases sharply as temperaturedrops. The effect of the cooling rate on the crystal size ispresented in Figure 1b, in which an increased cooling ratediminishes the average crystal size. This can be explained bythe formation of more nuclei as the temperature decreases at alarge cooling rate. The extent of crystallization2,3 is denoted byrelative crystallinity,X ) (S0 - S(Θ))/(S0 - e-h/Θ), presentedin Figure 1c as a function of temperature at different coolingrates. A sharp increase of crystallinity occurs after a shortincubation time. According to the definition of crystallinity,S(Θ)andX(Θ) are mirror images.

According to eqs 21-23, the crystal size distribution can alsobe presented as a function of time. Figure 2 presents the timeevolution of nonisothermal crystallization of PTT (poly(trim-ethylene terephthalate))34 starting from the melting temperature,Θi ) 1. Figure 2a presents the time evolution ofSat differentcooling rates. We notice that the decrease ofS is delayed bythe increasing induction times at smaller cooling rates. Theinduction time prediction, not present for isothermal crystal-lization,3 but occasionally reported in nonisothermal measure-ments,12,13supports the current model. The average crystal size,pictured in Figure 2b for different cooling rates, increasessharply and finally reaches a constant limit. The final crystal

size is also controlled by the cooling rate: a large average crystalsize is caused by a small cooling rate. This can be explainedby the temperature dependence of nucleation; when the coolingrate is small, nucleation and growth occur within a temperatureregion closer to the reference temperature where the nucleationrate is small. Time evolution of crystallinity is presented inFigure 2c. The conventional Avrami plot of log[-ln(1 - X)]vs ln(t) is presented in Figure 2d. A constant asymptote at theend of crystallization is observed as in isothermal crystallization.The straight midportion of the Avrami plot represents nucleationand growth in which denucleation and ripening are negligible.The Avrami exponent, the slope of this straight portion of theAvrami plot, has little dependence on the cooling rates as thepolymer sample is cooled from the melting temperature.

All the crystallinity curves (Figure 2c) include the incubationtime ∆θinc, defined as a time period from initial temperature,Θi, to the onset temperature,Θonset, where the polymer meltactually begins to crystallize. Thus the incubation period34 isformulated as∆θinc ) (Θi - Θonset)/φ. We define the crystal-lization onset temperature,Θonset, as the temperature when thecrystallinity reaches 1% in the numerical modeling results.

The incubation period,∆tinc, is calculated by quantifying thephysical properties of the polymer35 (∆H ) 28.8 kJ/mol). Themelting temperature,Tm, for PTT is reported34 as 548 K (275°C). The dimensionless cooling rates,φ, are converted into realcooling rates,â, according toφ ) â/(γµ∞Tmxm

λ). Thus withλ) 0, γµ∞ ) â/(Tmφ) and is 27.36 min-1. The dimensionlessincubation time,∆θinc, determined from the moment evaluationof the kinetics model, is converted into a real incubation period,∆tinc, according to eq 11. Withλ ) 0, γµ∞ ) 27.36 min-1,∆tinc (min) ) ∆θinc/(γµ∞) ) 0.03655∆θinc. The calculated

Figure 1. Distribution properties. (a) Supersaturation, (b) average size and (c) relative crystallinity as a function of scaled temperature at differentcooling rates, withC0

(0) ) 0, C0(1) ) 0, S0 ) 10, ε ) 1.0, h ) 100,n ) 20, w ) 2.0, J0 ) 106, λ ) 0, andΘi ) 1.

Kinetics of Nonisothermal Polymer Crystallization J. Phys. Chem. B, Vol. 109, No. 39, 200518553

incubation times at different cooling rates are listed in Table 1for the crystallization of PTT.

The equation relatingφ and â provides an adjustableparameter based on material properties. The relationship betweenincreasing cooling rate and decreasing incubation time is pre-sented quantitatively in Figure 3. Experimental measurementsof incubation for PTT are also provided (Figure 3) for com-parison. The points represent the experimental measure-ments34 of incubation time for PTT, and the solid line standsfor the incubation period calculated by the distribution kineticsmodel at different cooling rates. Good agreement is seenbetween model simulation and experimental results.

Usually nonisothermal crystallization experiments are con-ducted by decreasing the temperature from the melting point.However, as the onset temperature is below the melting tem-perature,34 our model can also represent nonisothermal crystal-

Figure 2. Time evolution ofS, Cavg, crystallinity, and the conventional Avrami plot for nonisothermal crystallization of PTT at different coolingrates, withC0

(0) ) 0, C0(1) ) 0, S0 ) 10, ε ) 1.0, h ) 100,n ) 20, w ) 2.0, J0 ) 106, λ ) 0, andΘi ) 1.

Figure 3. Effect of the cooling rate on the incubation time fornonisothermal crystallization of PTT: (9) experimental measurements,34

(s) simulation results, withC0(0) ) 0, C0

(1) ) 0, S0 ) 10, ε ) 1.0, h) 100,n ) 20, w ) 2.04,J0 ) 106, λ ) 0, andΘi ) 1.

TABLE 1: Calculated Incubation Period at DifferentCooling Rates with Θi ) 1

scaled coolingrateφ(10-4)

real coolingrateâ (°C/min)

scaled incubationtime ∆θinc

real incubationtime ∆tinc (min)

1 1.5 633.5 23.22 3 356 13.03 4.5 236.5 8.65 7.5 166 6.17 10.5 103 3.8

10 15 93.5 3.420 30 52.5 1.9

Figure 4. Avrami plots showing the effect of the cooling rate fornonisothermal crystallization, withC0

(0) ) 0, C0(1) ) 0, S0 ) 10, ε )

1.0, h ) 100,n ) 20, w ) 2.0, J0 ) 106, λ ) 0, andΘi ) 0.97.

18554 J. Phys. Chem. B, Vol. 109, No. 39, 2005 Yang et al.

lization starting from a temperature below melting. Figure 4shows the effect of cooling rate when the initial temperature ofcrystallization is 0.97. Although there is no apparent Avramiexponent change in Figure 2d as the crystallization temperaturedrops from the melting point, the slope differences for theAvrami plots in Figure 4 indicate a considerable influence ofcooling rates as the polymer sample is cooled from a temperaturebelow the melting point. This confirms that the nonisothermalcrystallization is sensitively influenced by nucleation, anddepends not only on the instantaneous temperature but also onthe change rate of temperature, consistent with experimentalmeasurements.21

The initial temperature is another influential parameter fornonisothermal crystallization, as shown in Figure 5, which pre-sents time evolution of crystallinity for nonisothermal crystal-lization starting from different initial temperatures.

The incubation time, increasing as initial temperature in-creases, reaches a maximum when the polymer melt is cooledfrom the melting temperature. The incubation periods fordifferent initial temperatures for PTT (Tm ) 275.0°C) are listedin Table 2.

Figure 6 presents the dependence of a scaled incubation periodon the dimensionless initial temperature, revealing the increasein incubation time with increasing initial temperature. The initialtemperature significantly influences the Avrami exponent(Figure 5b), which increases with initial temperature, as shownby the slope increase of the Avrami plots when the initialtemperature increases.

The moment method provides an easy and quick solution inmodeling polymer crystallization with mass-independent crystalgrowth (λ ) 0). When the crystal growth rate is mass-dependent(λ > 0), a more general form,kg(x) ) γxλ exp(-E/RT), is usedto express the crystal growth coefficient. The effect ofλ on thecrystallization kinetics needs to be explored because it is themost important parameter influencing the Avrami exponent forisothermal crystallization.3 The moment equations, however, do

not have a closed form forλ > 0; thus eqs 12 and 13 weresolved at different values ofλ by a numerical proceduredescribed in detail in our previous publications.29,30 A com-parison is made between moment and numerical solution forλ> 0 (Figure 7a). Solid lines and dotted lines represent thenumerical solution and the moment solution, respectively.Except for the discrepancy at the end of crystallization, causedby the increasing effect of denucleation and ripening, goodagreement is confirmed at various cooling rates. Crystal resizingand ripening can be considered only with the numerical method,and thus we observed a constant asymptote after the straightAvrami plot by numerical solution. This modeling result agreeswith experimental observations that the Avrami plot is a curvedrather than straight line.16,17,19Panels b and c of Figure 7 presentthe effect of cooling rates on diffusion-controlled (λ ) 1/3) andsurface-dependent (λ ) 2/3) crystal growth, respectively. A largeoverall crystallization rate is caused when the cooling rateincreases, as confirmed in the moment solution. A slope increaseof 2.7 in the Avrami plots is observed as the cooling rateincreases from 10-4 to 6× 10-4, in agreement with experimentalstudies.21 The effect of λ, the exponent of the growth ratecoefficient, is presented in Figure 7d. No apparent slope increasefor the Avrami plots occurs whenλ increases, although theoverall crystallization rate rises because of the increasingdependence of the growth rate coefficient on crystal mass. It isimportant to notice that, different from isothermal crystalliza-tion,3 the effect of changing temperature dominates over theeffect of λ because of the sensitivity of the nucleation rate on

Figure 5. Effect of the initial temperature for nonisothermal crystallization, withC0(0) ) 0, C0

(1) ) 0, S0 ) 10, ε ) 1.0,h ) 100,n ) 20,w ) 2.0,J0 ) 106, λ ) 0, andφ ) 1 × 10-4. (a) Time evolution of crystallinity X. (b) The conventional Avrami plots at different initial temperatures.

TABLE 2: Calculated Incubation Period at Different InitialTemperatures with O ) 4 × 10-4

scaled startingtemperatureΘi

real startingtemperatureT (°C)

scaled incubationtime ∆θinc

real incubationtime ∆tinc (min)

1 275.0 520 19.00.99 269.5 420 15.40.98 264.0 220 8.00.97 258.6 130 4.80.96 253.1 104 3.80.95 247.6 65 2.40.94 242.1 30 1.1

Figure 6. Effect of the initial temperature on the computed incubationtime for nonisothermal crystallization. The line segments connect thecomputed values.

Kinetics of Nonisothermal Polymer Crystallization J. Phys. Chem. B, Vol. 109, No. 39, 200518555

temperature. Thus an apparent slope change in the Avrami plotsis not observed for differentλ values for nonisothermalcrystallization.

To verify the application of the cluster size distribution modelto nonisothermal crystallization, we fitted experimental mea-surements with the model. Figure 8 presents the comparisonbetween simulation results of the model and experimental data21

for PPT [poly(propylene terephthalate)]. The solid lines representthe model fit, and the points denote the measurements at variouscooling rates. A unit transposition is made on the data becauseour model calculation is based on dimensionless time whereasthe experimental data is based on real time. Here a horizontaltransposition of log(γµ∞xm

λ) units (γµ∞ ) 125 min-1) is appliedto convert the experimental data into plots based on dimension-less time θ.3 The experimental measurements present theconventional Avrami plots at a cooling rate,â, equal to 1.0,2.5, 5, 10, 20, and 40°C/min. Good agreement betweenexperimental and simulation results is observed, indicating arealistic description of nonisothermal crystallization by thecrystal size distribution approach.

Conclusion

The crystal size distribution kinetics approach2,3 applied inthe present paper permits an investigation of nonisothermalcrystallization. Following our earlier studies on isothermalpolymer crystallization,3 we have investigated the effect ofcooling rates in detail for nonisothermal crystallization withlinearly decreasing crystallization temperature. The temperaturedependencies of crystallinity, supersaturation, and averagecrystal size are presented for nonisothermal crystallization atdifferent cooling rates. Time evolutions of crystal size distribu-tions are also revealed at different cooling rates. The effect ofinitial temperature is investigated in detail. An apparent Avramiexponent increase occurs as the initial temperature increases.

Figure 7. Conventional Avrami plots, withC0(0) ) 0, C0

(1) ) 0, S0 ) 10, ε ) 1.0, h ) 100,n ) 20, w ) 2.0, J0 ) 106, andΘi ) 1. The effectof the cooling rate when (a)λ ) 0, (b) λ ) 1/3, and (c)λ ) 2/3. (d) The effect ofλ for nonisothermal crystallization with a cooling rate ofφ )10-3. Solid and dotted lines represent the numerical and moment solutions, respectively.

Figure 8. Fits of the cluster size distribution model, withC0(0) ) 0,

C0(1) ) 0, S0 ) 10, ε ) 1.0,h ) 100,n ) 20, w ) 2.0,J0 ) 106, λ )

0, andΘi ) 0.96, to the nonisothermal crystallization measurements21

for PPT at the following cooling rates (â, °C/min): (9) 40, (×) 20,(2) 10, (0) 5, (1) 2.5, (b) 1.

18556 J. Phys. Chem. B, Vol. 109, No. 39, 2005 Yang et al.

The effect of the cooling rate, however, is counteracted by anincreasing initial temperature; in other words, increasing thecooling rate has little influence if the crystallization temperaturebegins from the melting point. The incubation period was alsoinvestigated in detail at different cooling rates and differentinitial temperatures. An increasing incubation period is causedby an increase in the initial temperature or a decrease in thecooling rate. We also investigated the effect of power depen-dence for growth rate coefficientλ, and did not observe theapparent Avrami exponent increase when the polymer melt wascooled from the melting point, as indicated in isothermalcrystallization. The validity of the crystal size distribution modelwas also examined by comparison with experimental measure-ments. The model accurately describes the effect of the coolingrate on incubation time and on Avrami plots of crystallinity vstime. Although an analytical, closed expression cannot bederived in our approach, the model and simulation results forthe degree of crystallization provide a theoretical foundationfor the Avrami plots. That the characteristic Avrami plot forexperimental data is curved rather than straight indicates thatour approach is capable of providing a more realistic predictionthan the oversimplified Avrami theory.

References and Notes

(1) Robson, J. D.Acta Mater.2004, 52, 4669.(2) Yang, J.; McCoy, B. J.; Madras, G.J. Chem. Phys. 2005, 122 (6),

64901.(3) Yang, J.; McCoy, B. J.; Madras, G.J. Chem. Phys. 2005, 122(24),

244905.(4) Evans, U. R. J . Chem. Soc., Faraday Trans.1945, 41, 365.(5) Avrami, M. J. Chem. Phys. 1939, 7, 1103.(6) Avrami, M. J. Chem. Phys. 1939, 7, 212.

(7) Avrami, M. J. Chem. Phys. 1939, 7, 177.(8) Hay, J. N.Polym. Int. 1971, 3, 74.(9) Wasiak, P.; Sajkiewicz, P.; Wozniak, A.J. Polym. Sci., Part B:

Polym. Phys.1999, 42 (37), 2821.(10) Piorkowska, E.J. Phys. Chem.1995, 99, 14007.(11) Vilanova, P. C.; Ribas, S. M.; Guzman, G. M.Polymer1985, 26,

423.(12) Xu, J.; Fariclough, J. P. A.; Mai, S. M.; Ryan, A. J.; Chaibundit,

C. Macromolecules2002, 35, 6937.(13) Xiao, J.; Zhang, H.; Wan, X.; Zhang, D.; Zhou, Q.; Woo, E. M.;

Turner, S. R.Polymer2002, 43, 7377.(14) Ciora, R. J.; Magill, J. H.Macromolecules1990, 23, 2350.(15) Hay, J. N.; Booth, A.Polym. Int.1972, 4, 19.(16) Price, F. P.J. Appl. Phys. 1969, 36, 3014.(17) Grenier, D.; Prodhomme, R. E.J. Polym. Sci., Part B: Polym. Phys.

1980, 18, 1655.(18) Di Lorenzo, M. L.; Silvestre, C.Prog. Polym. Sci.1999, 24, 917.(19) Ozawa, T.Polymer1971, 12, 150.(20) Kratochvil, J.; Sikora, A.J. Appl. Polym. Sci.2005, 95, 564.(21) Achilias, D. S.; Parageorgiou, G. Z.; Karayannidis, G. P.J. Polym.

Sci., Part B: Polym. Phys.2004, 42, 3775.(22) Tobin, M. C.J. Polym. Sci., Part B: Polym. Phys.1976, 12, 399.(23) Ziabicki, A. Colloid Polym. Sci.1996, 274, 209.(24) Ziabicki, A. Colloid Polym. Sci.1996, 274, 705.(25) Piorkowska, E.J. Phys. Chem.1995, 99, 14007.(26) Piorkowska, E.J. Phys. Chem.1995, 99, 14016.(27) Piorkowska, E.J. Phys. Chem.1995, 99, 14024.(28) McCoy, B. J.J. Phys. Chem. Solids2002, 63, 1967.(29) Madras, G.; McCoy, B. J.Chem. Eng. Sci.2004, 59, 2753.(30) Madras, G.; McCoy, B. J.J. Chem. Phys.2002, 17, 8042.(31) McClurg, R. B.; Flagan, R. C.J. Colloid Interface Sci.1998, 194,

201.(32) Oxtoby, D. W.J. Phys.: Condens. Matter1992, 4, 7627.(33) Rowlinson, J. S.; Widom, B.Molecular Theory of Capillarity;

Oxford University Press: New York, 1982; p 248.(34) Apiwanthanakorn, N.; Supaphol, P.; Nithitanakul, M.Polym. Test.

2004, 23, 817.(35) Chung, W. T.; Yeh, W. J.; Hong, P. D.J. Appl. Polym. Sci.2002,

83, 2426.

Kinetics of Nonisothermal Polymer Crystallization J. Phys. Chem. B, Vol. 109, No. 39, 200518557