A different approach for the study of the crystallization kinetics in polymers. Key study: poly(ethylene terephthalate)/ SiO2 nanocomposites

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Research ArticleReceived: 24 June 2009 Revised: 11 December 2009 Accepted: 12 February 2010 Published online in Wiley Online Library: 20 July 2010

(wileyonlinelibrary.com) DOI 10.1002/pi.2896

A different approach for the studyof the crystallization kinetics in polymers.Key study: poly(ethylene terephthalate)/SiO2 nanocompositesGeorge Papageorgiou,a Dimitrios N Bikiarisa and Konstantinos Chrissafisb∗

Abstract

Polymer crystallization is complex and difficult to model, especially when it is non-isothermal and even more so whendescribing cold crystallization. In most cases, two different processes occur, so-called primary and secondary crystallization. Inthe literature, two assumptions are generally made. Firstly, the validity of the Avrami model is assumed a priori. Secondly, forcalculations of the kinetic parameters and activation energy, data from a single differential scanning calorimetry scan at a givenheating rate are used. The other popular model, that of Ozawa, is also based on similar assumptions. In the study reported here,a different approach was adopted, which uses multiple scans at various heating rates simultaneously. Here the experimentaldata of the non-isothermal cold crystallization of an in situ-prepared poly(ethylene terephthalate)/1% SiO2 nanocompositewere used. Data were analysed following both the ordinary procedure and the method proposed in this work. Findings showedthat when the Avrami model is a priori supposed to hold and the data of different heating rates are analysed separately,results are not acceptable. The new approach involves calculation of the activation energy through use of the isoconversionalmethods of Ozawa–Flynn–Wall and Friedman over the whole range of the crystallization conversion. The reaction model f(a)was determined after the evaluation of 16 different models. The best fitting was achieved for the Prout–Tompkins model or fora mechanism involving two steps described by respective Avrami equations with different activation energies.c© 2010 Society of Chemical Industry

Keywords: poly(ethylene terephthalate) nanocomposites; crystallization; activation energy; kinetic analysis

INTRODUCTIONModelling of polymer crystallization kinetics is difficult since it isa complex process.1 More difficult is the case of polymer non-isothermal crystallization, especially when this occurs on heatingfrom the glassy/amorphous state, in other words in the case of coldcrystallization.2 Usually, two different processes are supposed tooccur, so-called primary and secondary crystallization.1 Secondarycrystallization is supposed to begin after a certain degree of relativecrystallinity has been achieved. However, not all crystallizablepolymers behave in the same way. For some polymers secondarycrystallization is more pronounced.2

Various attempts have been made to model polymer non-isothermal crystallization. Among the proposed models, themodified Avrami and the Ozawa models are often used.2 – 5

Unfortunately, deviations always occur and poor fitting tocrystallization data is achieved.6 What must be noted is the use ofdata from a single DSC scan in each analysis. As a result, the valuesobtained for the kinetic parameters are strictly dependent on thescanning rate.

In this work, a new attempt is made to describe polymernon-isothermal crystallization kinetics, using data from scansat different scanning rates. Also, the activation energy ofcrystallization is determined. Of special interest is the case ofmechanisms of two crystallization processes each one obeying arespective Avrami model for the multi-rate crystallization kinetics

analysis. Experiments involved non-isothermal cold-crystallizationtests of amorphous samples of in situ-prepared poly(ethyleneterephthalate) (PET)/1% SiO2 at various heating rates.

EXPERIMENTALMaterialsDimethyl terephthalate (DMT; 99%), 1,2-ethanediol (EG; 99%),antimony trioxide (Sb2O3; 98%) and triphenylphosphate (TPP;95%) were obtained from Fluka. Zinc acetate ((CH3CO2)2Zn;99.99%) was purchased from Aldrich. Fumed silica nanoparticles(SiO2), used for nanocomposite preparation, were supplied byDegussa AG (Hanau, Germany) under the trade name AEROSIL

200, having a specific surface area of 200 m2 g−1, SiO2 content>99.8% and average primary particle size of 12 nm.

∗ Correspondence to: Konstantinos Chrissafis, Solid State Physics Section, PhysicsDepartment, Aristotle University of Thessaloniki, GR-541 24 Thessaloniki,Greece. E-mail: [email protected]

a Laboratory of Organic Chemical Technology, Department of Chemistry,Aristotle University of Thessaloniki, GR-541 24 Thessaloniki, Greece

b Solid State Physics Section, Physics Department, Aristotle University ofThessaloniki, GR-541 24 Thessaloniki, Greece

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Different approach to crystallization kinetics in polymers www.soci.org

In situ-prepared PET/SiO2 nanocompositeThe PET/SiO2 nanocomposite was prepared in situ by the two-stage melt polycondensation of DMT and EG in a glass batchreactor. An appropriate amount of SiO2 (1 wt%) was dispersedin EG by ultrasonic vibration (50 W) and vigorous stirring with amagnetic stirrer (300 rpm) for 10 min prior to polymerization. Zincacetate was added (10−3 mol mol−1 DMT) and DMT at a molarratio of DMT/EG = 1/2.2. The mixture was degassed and purgedwith argon three times. Subsequently, it was heated at 190 ◦C for1 h, under constant mechanical stirring (500 rpm). The methanolproduced by the transesterification reaction was removed fromthe mixture by distillation and collected in a scaled cylinder. Thetemperature was increased to 230 ◦C and the reaction continuedfor a further 2 h, during which the complete removal of theanticipated methanol produced took place.

During the second stage, under an argon atmosphere, thepolycondensation catalyst (Sb2O3, 250 ppm) and TPP (0.03 wt%DMT) as thermal stabilizer, dispersed in a small amount of EG,were added to the mixture. TPP prevents side reactions such asetherification and thermal decomposition. Afterwards, vacuumwas applied (ca 5 Pa) slowly, over a period of 15 min, to avoidexcessive foaming and to minimize oligomer sublimation, whichis a potential problem during the melt polycondensation. Thetemperature was increased to 280 ◦C and the reaction continuedfor 2 h. When the polycondensation reaction was completed, thereaction tube was broken to remove the product from the flask.The nanocomposite, after glass particles were removed with agrinder, was ground in a mill, sieved, washed with methanol anddried at 110 ◦C for 12 h.

The prepared sample had an intrinsic viscosity of 0.5 dL g−1

as measured using an Ubbelohde viscometer Ic at 25 ◦C inphenol/tetrachloroethane (60/40 w/w) at a solution concentrationof 1 wt%.

DSC measurementsDSC measurements were carried out with a PerkinElmer PyrisDiamond DSC. A PerkinElmer Intracooler II cooling device wasconnected to the instrument to allow cooling at constanthigh cooling rates. Temperature and energy calibrations of theinstrument were performed using high-purity zinc, tin and indiumstandards. Samples of about 6 mg were crimped in aluminiumcrucibles; an empty aluminium crucible was used as reference. Aconstant flow of nitrogen was maintained to provide a constantthermal blanket within the DSC cell, thus eliminating thermalgradients and ensuring the validity of the applied calibrationstandard from sample to sample. The samples were first melted to300 ◦C, held at that temperature for 5 min to erase any previousthermal history and then rapidly cooled at a nominal rate of500 ◦C min−1 (quenching). After this, the samples were heatedto 300 ◦C at various heating rates (5, 10, 15 and 20 ◦C min−1).From these scans, the glass transition temperature (Tg), cold-crystallization temperature (Tcc), melting temperature (Tm) andheat of fusion (�Hm) of the samples were obtained.

THEORETICAL BACKGROUNDKinetic analysis of solid-state transformations is usually based onthe single-step kinetic equation

dα

dt= k(T)f (α) (1)

where k(T) is the rate constant, t is the time, T is the temperature,α is the extent of conversion from the amorphous (liquid or solid)to the crystalline phase and f (α) is the reaction model relatedto the mechanism. The explicit temperature dependence of therate constant is introduced by replacing k(T) with the Arrheniusequation, which gives

dα

dt= A exp

(−E

RT

)f (α) (2)

where A (the pre-exponential factor) and E (the activation energy)are the Arrhenius parameters and R is the gas constant. Thenon-isothermal method is the most common method for thekinetic analysis of crystallization in polymers. Understandingthe dynamic crystallization behaviour is of great importance,as most processing techniques actually occur under non-isothermal conditions. Additionally, non-isothermal crystallizationcan broaden and supplement knowledge of the crystallizationbehaviour of polymers. For non-isothermal conditions, dα/dt inEqn (2) is replaced with β(dα/dT), where β (= dT/dt) is the heatingrate.7,8 The ratio dαα/dt of the kinetic process is proportional tothe measured specific heat flow ϕ, normalized per sample mass(W g−1) as

dα

dt= ϕ

�Hc(3)

where �Hc is the total enthalpy change associated with thecrystallization process. The fractional extent of conversion α canbe easily obtained by partial integration of the non-isothermalcurve.

RESULTS AND DISCUSSIONThe study of the crystallization kinetics follows two steps. As thefirst step, the usual way of studying the crystallization of polymersis presented. The Avrami and Ozawa methods are used in orderto calculate parameters n and m, respectively. Also, in this firststep, the applicability of the Avrami equation is examined usingthe Malek method. As the second step, a new more complicatedapproach to the determination of the reaction parameters ispresented. The activation energy is calculated independently ofthe reaction model using three different methods: the Kissingermethod, the Friedman method and the Ozawa–Flynn–Wall (OFW)method. The appropriate reaction model along with the activationenergy and the pre-exponential factor are determined using themodel-fitting method.

For the kinetic study, non-isothermal measurements withvarious heating rates are used. In Fig. 1, the glass transition, coldcrystallization and melting peaks of PET/1% SiO2 are shown. In thisstudy we focused only on the crystallization, and Fig. 2 shows thedependence of the shape of the crystallization peak on heatingrate. It is clear that the peak shifts to higher temperatures withincreasing heating rate, while, at the same time, the peak heightincreases.

In the study of non-isothermal crystallization using DSC, theenergy released during the crystallization process appears to bea function of temperature rather than of time, as in the case ofisothermal crystallization. As a result, the degree of conversion orthe degree of crystallinity α as a function of temperature can be

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www.soci.org G Papageorgiou, DN Bikiaris, K Chrissafis

50 100 150 200 250 300

heat

flow

(ar

bitr

ary

units

)

T (°C)

Tg

Tc

Tm

exo

Figure 1. Glass transition, cold crystallization and melting peaks of PET/1%SiO2 for a heating rate of 10 ◦C min−1.

115 120 125 130 135 140 145 150 155-1.0

-0.8

-0.6

-0.4

-0.2

0.0

heat

flow

(W

g–1

)

T (°C)

1

2

3

4

Figure 2. Cold crystallization peaks of PET/1% SiO2 for various heatingrates: 1, 5 ◦C min−1; 2, 10 ◦C min−1; 3, 15 ◦C min−1; 4, 20 ◦C min−1.

formulated as

α(T) =

∫ T

T0

(dHc/dT)dT

∫ T∞

T0

(dHc/dT)dT

(4)

where T0 and T∞ are the crystallization onset and end tempera-tures, respectively. On the right-hand side of Eqn (4), the numeratoris the relative crystallization heat generated up to temperature Tand the denominator is the total heat produced by the completionof the total crystallization process. To use Eqn (4) for the analysisof non-isothermal crystallization data obtained by DSC, it mustbe assumed that the sample experiences the same thermal his-tory as designated by the DSC furnace. This may be realized onlywhen the thermal lag between the sample and the furnace is keptto a minimum. If this assumption is valid, the relation betweenthe crystallization time t and the sample temperature T can beformulated as

t = T0 − T

β(5)

0 2 50.0

0.2

0.4

0.6

0.8

1.0

rela

tive

degr

ee o

f cry

stal

linity

t(min)

1234

1 3 4 6

Figure 3. Degree of crystallinity versus time for cold crystallization ofPET/1% SiO2 for various heating rates: 1, 5 ◦C min−1; 2, 10 ◦C min−1; 3,15 ◦C min−1; 4, 20 ◦C min−1.

where β is the cooling rate. According to Eqn (5), the horizontaltemperature axis of a DSC thermogram for non-isothermalcrystallization data can readily be transformed into a time scale.

Figure 3 shows the degree of crystallinity α as a function oftime for PET/1% SiO2 samples non-isothermally crystallized at fourdifferent heating rates. An important parameter which can bedirectly obtained from the degree of crystallinity as a function oftime is the half-time of crystallization, t0.5, which is the changein time from the onset of crystallization to the time at 50%completion. According to Fig. 3, it is obvious for the t0.5 value thatthe higher the heating rate, the shorter is the time for completingthe crystallization.

The crystallization kinetics is usually interpreted in terms of thestandard nucleation–growth model formulated by Johnson, Mehland Avrami (JMA).9 – 11 This model describes the time dependenceof the fractional extent of conversion α, usually written in the form

α = 1 − exp[−(kt)n] (6)

where k is the Avrami crystallization rate constant, which is afunction of temperature and in general depends on both thenucleation frequency and the crystal growth rate, and the Avramikinetic exponent n is a parameter that reflects the nucleationfrequency and/or the growth morphology. Usually, k is writtenin the form of the composite Avrami rate constant kA (kA = kn).Parameter kA (the dimensions of which are given in (time)−n) isnot only a function of temperature, but also a function of theAvrami exponent n. As a result, the use of k should be preferableto the use of kA due partly to the facts that it is independent ofthe Avrami exponent n and its dimension is given in (time)−1. Itshould be noted that both k and n are constants with specificvalues for a given crystalline morphology and type of nucleationfor a particular crystallization condition12 and that, based on theoriginal assumptions of the theory, the value of n should be aninteger ranging from 1 to 4. It should be mentioned that for non-isothermal crystallization, k and n do not have the same physicalmeaning as for isothermal crystallization, because the temperaturechanges instantly in non-isothermal crystallization. In this case, kand n are two adjustable parameters to be fitted to the data.However, the use of Eqn (6) can still provide further insight to thekinetics of non-isothermal crystallization.

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-0.75 -0.50 -0.25 0.00 0.25 0.50 0.75-2.0

-1.5

-1.0

-0.5

0.0

0.5

lg(-

ln(1

-α))

lgt

4 3 2 1

Figure 4. Avrami plots for cold crystallization of PET/1% SiO2 for variousheating rates. 1, 5 ◦C min−1; 2, 10 ◦C min−1; 3, 15 ◦C min−1; 4, 20 ◦C min−1.

The Avrami rate equation can be obtained from Eqn (6) bydifferentiation with respect to time:

dα

dt= kn(1 − α)[− ln(1 − α)]1−1/n (7)

Equation (7) is usually referred to as the JMA equation, and it isfrequently used for the formal description of thermal crystallizationdata. It should be emphasized, however, that the validity of the JMAequation is based on the following assumptions: (a) isothermalcrystallization conditions, (b) low anisotropy of growing crystals,(c) homogeneous nucleation or heterogeneous nucleation atrandomly dispersed second-phase particles and (d) growth rate ofnew phase controlled by temperature and independent of time.

For the kinetic description of polymer crystallization, as hasbeen mentioned previously, the Avrami model is used. This modelis used under the assumption that for the kinetic descriptionof the crystallization only one mechanism is necessary with thesame activation energy for the entire crystallization. In fact, thisassumption is not valid in most cases of polymer crystallization.At least two different mechanisms can be recognized, making thekinetic description a complex physical and mathematical problem.For this reason, the conclusions using one kinetic mechanism areonly qualitative.

A first method for the examination of the JMA model’sapplicability is the linearity of the Avrami plot and the valuesof the Avrami exponent n. Equation (6) can be rearranged asfollows by taking its double logarithm:

log[− ln(1 − α)] = n log t + log kA (8)

where kA (= kn) is the composite Avrami rate constant. Plotsof log[− ln(1 − α)] versus log t obtained from curves recorded atseveral heating rates should be straight lines whose slope gives thevalue of n. Nevertheless, it is well known that a double logarithmicfunction, in general, is not very sensitive to subtle changes toits argument. Therefore, one could expect to observe substantiallinearity in the plots of log[− ln(1 −α)] versus log t even in the caseof the JMA model not being fulfilled.

Figure 4 shows double logarithmic plots of log[− ln(1−α)] versuslog t for PET/1% SiO2 nanocomposite samples at various heatingrates. Each curve has a linear portion, which is followed by a markeddeviation at longer times. Usually, this deviation is considered to

Table 1. Avrami exponents n as well as Ozawa exponent m for coldcrystallization of PET/1% SiO2

Avrami exponents n Ozawa exponent m

Rate ( ◦C min−1) n1 n2 T (◦C) m

5 4.8 2.2 130 3.7

10 4.5 2.1 133 4.5

15 4.7 2.6 136 3.6

20 3.9 2.7

be due to secondary crystallization, which is caused by spheruliteimpingement at later stages. The linear portions are almost parallelto each other, shifting to shorter time with increasing β , indicatingthat the nucleation mechanism and crystal growth geometriesare similar for primary and secondary crystallization at all heatingrates. Each region corresponds to a different range of degree ofcrystallinity and gives different values for n (n1 and n2; Table 1).

Usually, the first linear part of the plot is attributed toprimary crystallization and the second part to slower secondarycrystallization. The analysis of cold crystallization was performedwith the combination of two Avrami models describing thecorresponding processes. The resulting Avrami exponent valuesare summarized in Table 1. As can be seen, the values of theAvrami exponent associated with secondary crystallization aremuch lower than those associated with the primary process.

The values of Avrami exponent n1 are higher than the theoreticalones (<4) and vary. This is a first indication that for the kineticdescription of PET/1% SiO2 more than one mechanisms must beused. This is more evident through the fact that all the curvespresent two linear portions.

The Ozawa method,13,14 which can be described as based on theAvrami theory, has also been used to describe the non-isothermalcrystallization kinetics of polymers. Ozawa modified the Avramiequation for non-isothermal treatment, assuming that the polymermelt was cooled at a constant rate and the mathematical derivationof Evans was valid.15 According to Ozawa theory,13,14 the degreeof conversion or relative crystallinity α at a temperature T can becalculated as

1 − α = exp

(−k(T)

βm

)(9)

where β is the heating rate, m is the Ozawa exponent, whichdepends on the dimensionality of crystal growth, and k is theheating crystallization function which is related to the overallcrystallization rate and indicates how fast crystallization occurs.Equation (9) can be written as

log[− ln(1 − α)] = log k(T) − m log β (10)

If this equation describes correctly the kinetics of non-isothermalcrystallization, plots of log[− ln(1 − α)] against log β shouldgive straight lines and kinetic parameters k(T) and m should beobtainable from the intercepts and slopes of the lines, respectively.

For PET/1% SiO2 nanocomposites, the Ozawa plots oflog[− ln(1 − α)] against log β are shown in Fig. 5. As can beseen, only for a few temperatures are straight and almost par-allel lines obtained (Table 1) indicating the failure of the Ozawamodel to provide an adequate description of crystallization. Sincenon-isothermal crystallization is a dynamic process, in which thecrystallization rate is no longer constant but a function of time


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0.7 0.8 0.9 1.0 1.1 1.2 1.3

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8lg

(-ln

(1-α

))

lgβ

T = 130°C

T = 133°C

T = 136°C

T = 139°C

T = 142°C

Figure 5. Ozawa plots for non-isothermal cold crystallization of PET/1%SiO2.

and heating rate, the quasi-isothermal treatment of the Ozawamodel might be questionable. In the Ozawa analysis, comparisonis carried out on experimental data representing widely varyingphysical states of the system; however, these differences are nottaken into account in the model. Thus, if the heating rates varyover a large range and if a large amount of crystallization occursas a result of secondary processes, the Ozawa model would not beadequate in describing non-isothermal crystallization behaviour.Also, the literature indicates that a more plausible explanationas to why the Ozawa plots of many polymers fail to describenon-isothermal crystallization kinetics is that k(T) is a binary func-tion of T and β , instead of the explanation deriving the resultsfrom comparing the degrees of conversion at a fixed tempera-ture for various heating rates, though both resulting conclusionsare similar.16 Furthermore, plots of log[− ln(1 − α)] against log β

may be closer to linearity for some polymers when two selectedtemperatures are near each other or the difference in the heatingrates is limited. Moreover, effects such as transcrystallization arenot considered in the Ozawa theory. This particular phenomenonaccelerates the average kinetics of transformation, and tends tomake up for the volume restriction effects which, in contrast, slowdown crystallization kinetics. As a result, the Avrami exponent n(Ozawa exponent m in this case) has no physical significance anymore when strong surface nucleation occurs, because its evolutioninvolves factors with contradictory effects. Therefore, the way itvaries when fibres or nucleating agents are added to the bulkpolymer becomes problematic and not easily interpretable. In thepresent case, SiO2 acts as an additional phase for the PET matrix,making the problem more complicated; thus it is plausible thatthe Ozawa approach fails.17,18 In a more practical way, since theOzawa equation13,14 ignores secondary crystallization, the reasonwhy the non-isothermal crystallization of PET/1% SiO2 for thelower temperature regions does not follow the Ozawa equationcan be explained by that, at a given temperature, the crystallizationprocesses at different heating rates are at different stages; that is,at lower heating rates the crystallization process is towards theend, whereas at higher heating rates the crystallization process isat an early stage.

The Avrami and Ozawa plots give indications that thecold crystallization of PET/1% SiO2 nanocomposites cannot be

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

z(α)

2015105

α

Figure 6. Normalized z(α) function obtained by transformation of DSCdata for various heating rates (in ◦C min−1) for the cold crystallization ofPET/1% SiO2.

described by only one mechanism which obeys the Avramiequation.

In order to increase the certainty regarding the applicability ofthe JMA model that arises from the above discussion, we usedanother test method introduced by Malek, which is based on theproperties of the y(α) and z(α) functions (see below). Taking intoaccount Eqns (1)–(3), the kinetic equation for the JMA model canbe written as

ϕ = �HcA exp

(−E

RT

)f (α) (11)

where the function f (α) is an algebraic expression of the JMAmodel and can be expressed as

f (α) = n(1 − α)[− ln(1 − α)]1−1/n (12)

The function f (α) should be invariant with respect to procedureparameters such as sample mass and heating rate for non-isothermal conditions. Malek has shown19 – 21 that the functionsϕ(t) and ϕ(T) are proportional to the y(α) and z(α) functions thatcan be easily obtained by a simple transformation of DSC data. Fornon-isothermal conditions, these functions are defined as follows:

y(α) = ϕ exp

(Ec

RT

)(13)

z(α) = ϕT2 (14)

For practical reasons, the y(α) and z(α) functions are normalizedwithin the range 0–1. The maxima exhibited by the y(α) and z(α)functions are defined asαM and a∞

p , respectively. The maximum a∞p

is a constant for the JMA model (a∞p = 0.632) and a characteristic

‘fingerprint’ for it.19 – 22

As can be seen in Fig. 6, the peak position of the z(α) functionfor various heating rates is far from the accepted maximum value(0.632) for the z(α) function. This is a strong indication that the coldcrystallization of PET/1% SiO2 cannot be described by one kineticmechanism which obeys the Avrami model, and it is clear that atleast two different crystallization mechanisms are in operation.

For the determination of activation energy it is preferableto use isoconversional methods.23 The Kissinger, OFW andFriedman methods are used for comparison reasons. Since every


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2.40 2.42 2.44 2.46 2.48 2.50 2.52

-10.4

-10.0

-9.6

-9.2

-8.8ln

(β/T

2 )

1000/T (K-1)

E = 127.3 kJ mol–1

Figure 7. Kissinger’s plot and the corresponding linear regression line forcold crystallization of PET/1% SiO2.

isoconversional method has a different error, the use of more thanone method can give a range of values for the activation energyat every particular value of α.

First, we consider the Kissinger formula,24,25 which is mostcommonly used in the analysis of crystallization data. This formulathat holds in very general cases is suggested to be valid forcrystallization and has the form

ln

(β

T2p

)= −Ec

RTp+ const. (15)

where R is the universal gas constant, β (= dT/dt) the heatingrate and Tp the crystallization peak temperature. The value of Ec isobtained from the slope of a ln(β/Tp

2) versus 1/Tp plot (Fig. 7). Thecalculated value of activation energy from the Kissinger method,which corresponds to the peak temperature of the crystallizationcurve, is 127.3 kJ mol−1.

The second method used to calculate the activation energyfor various extent of conversion values is the isoconversionalOFW13,14,26 method. This is in fact a ‘model-free’ method whichinvolves measuring the temperatures corresponding to fixedvalues of α from experiments at various heating rates β . Plottingln β against 1/T according to

ln β = ln

(Af (α)

dα/dT

)− Ec

RT(16)

should give straight lines, the slopes of which are directlyproportional to the activation energy (−Ec/R). If the determinedactivation energy is the same for the various values of α, theexistence of a single-step reaction could be concluded withcertainty. In contrast, a change of Ec with increasing degree ofconversion is an indication of a complex reaction mechanismthat invalidates the separation of variables involved in the OFWanalysis.14 These complications are especially serious if the totalreaction involves competing reaction mechanisms.

Plots of ln β versus 1/T of the OFW method for PET/1% SiO2

nanocomposites are shown in Fig. 8. The straight lines fitting thedata are nearly parallel, which is an indication that the activationenergies at different degrees of conversion are similar.

The third method considered is the differential isoconversionalmethod suggested by Friedman,27,28 based on Eqn (2), which leads

2.40 2.44 2.48 2.52

1.6

2.0

2.4

2.8

3.2

lnβ

1000/T (K-1)

0.050.95

Figure 8. OFW plots for various degrees of conversion α, from 0.05 to 0.95,for PET/1% SiO2.

2.36 2.40 2.44 2.48 2.52 2.56-7.0

-6.5

-6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

ln(d

α/dt

)

1000/T (K-1)

0.95

0.05

Figure 9. Friedman plots for various degrees of conversion α, from 0.05 to0.95, for PET/1% SiO2.

to

ln

(β

dα

dT

)= ln A + ln[f (α)] − Ec

RT(17)

For constant α, a plot of ln[β(dα/dT)] versus 1/T obtained fromcurves recorded at several heating rates, should be a straight linewhose slope gives the value of Ec (Fig. 9). It is obvious from Eqn (17)that if the function f (α) is constant for a particular value of α, thenthe sum ln[f (α)] + ln(A/β) is also constant.

Comparing the results for the dependence of the activationenergy Ec on the relative degree of crystallinity α (Fig. 10), it canbe seen that this dependence can be divided into two mainlydistinct regions: the first for values of α up to 0.7, in which Ec

is almost stable; and the second for values of α > 0.7, in whichEc presents a monotonic increase. This dependence of Ec on α, amonotonic increase or/and in two different stages, for PET/1% SiO2

nanocomposites is an indication of a complex reaction with theparticipation of at least two different crystallization mechanismswith different activation energies.

For the determination of the kinetic triplet (reaction model f (α),activation energy Ec and pre-exponential factor A), according tothe literature, mainly simulated data or a combination of simulatedand experimental data for a single heating rate or multiple


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0.0 0.2 0.4 0.6 0.8 1.0120

130

140

150

160

170A

ctiv

atio

n E

nerg

y E

(kJ

mol

–1)

FriedmanOFW

α

Figure 10. Effective activation energy as a function of relative extent ofcrystallization for cold crystallization of PET/1% SiO2 nanocompositesobtained using the OFW and Friedman methods.

heating rates have been used. For the study of complicatedsystems, mainly model-fitting methods have been used. Thesemethods involve fitting different models to α–temperaturecurves and simultaneously determining the activation energyEc and pre-exponential factor A. Because all kinetic parametersaffect the regression values through differential equations, theyare, in the algebraic sense, nonlinear. With one-step reactions,however, the determination of the kinetic parameters can beturned into a multiple linear regression problem through suitabletransformations and simultaneous conversion of Eqn (1). Nonlinearregression allows a direct fit of the model to the experimental datawithout a transformation, which would distort the error structure.An additional advantage lies in the fact that there are no limitationswith respect to the complexity of the model. However, onlyiterative procedures can be employed for estimation of the kineticparameters. The multivariate nonlinear regression method appliesa sixth-degree Runge–Kutta process in a modified Marquardt29,30

procedure to solve a system of differential equations, which isessentially based on the differential equations relevant to thereaction types and their combinations. Fundamentally, multi-step processes can only be analysed with nonlinear regression.But nonlinear regression proves to be advantageous for one-stepprocesses as well, because it provides a considerably better qualityof fit as compared to multiple nonlinear regressions.

It has been demonstrated recently that the complementary useof the model-free method with the isoconversional methods forone-step reactions is very useful in order to understand solid-statereaction kinetics.31

In order to determine the nature of the mechanisms throughcomparison of experimental and theoretical data, it is consideredinitially that the crystallization of the polymer can be described only

by a single mechanism without presuming the exact mechanism.If the result of the fitting cannot be considered as acceptable, thenwe must proceed to fit the experimental data with a combination oftwo mechanisms. The multivariate nonlinear regression method isused for the determination of the kinetic triplet. For this calculationvarious kinetic models are used. The calculated values of Ec and Aafter fitting for three kinetic models, for which the quality of themathematical fitting depending on the regression coefficient R isat an acceptable level, are presented at Table 2. In our case, thebest kinetic model is the expanded Prout–Tompkins (Bna). Thevalues of the activation energy for all these models are in the samerange of values that have been calculated with the isoconversionalmethods. Figures 11 and 12 show plots of the fitting with the bestmodel (Bna) and the Avrami model (An) for comparison. Thequality of the fitting with the Bna model is very good.

The value of 3.3 obtained for the exponent n using onemechanism obeying the Avrami model seems to be reasonable.In fact, rather it is a mean value of those obtained from the usualprocedure using separate Avrami models for each heating rateand for primary and secondary crystallization also separately. It isalso closer to the values of the Avrami exponent always observedfor isothermal crystallization.

Although the two methods, isoconversional and model fitting,are used in a complementary fashion, the three different kineticmodels give almost the same values for Ec and A, which are inthe same range as the values calculated with the isoconversionalmethods, and the fitting with these models is good. It is verydifficult to choose the real kinetic model from the acceptablemodels using only the criterion of the largest regression factor.This difficulty seems to be greater if it is taken into account thatthe crystallization mechanism of polymers is very complex. For thisreason, the choice of the appropriate kinetic model, considering aone-step reaction, only denotes a possible mathematical form forthe conversion function.31 Under this assumption, it can be statedthat in order to fit the experimental data of the crystallizationreaction of a polymer, models more complicated than the Avramimodel must be used. For this reason, although the quality of thisfitting is at an acceptable level and the fitting can be stoppedhere, and knowing that the crystallization reaction of polymers is avery complex reaction, we must consider more than one reactionmechanism. So, we used for the fitting two reaction mechanismsusing the Avrami model.

As can be seen in Fig. 13, the fitting using two reactionmechanisms is very good, and of the same quality as that predictedusing one reaction mechanism with the Bna model. The resultsfor the activation energy obey the dependence of Ec versus α

which shows that at first the activation energy is almost stableand then for α > 0.6 there is an increase. The calculated values ofthe activation energy and the pre-exponential factor are given inTable 3.

The value of the Avrami exponent for the second mechanismis almost 1, which means that the second stage could strictly be

Table 2. Calculated values of Ec and A for the three best kinetic models of PET/1% SiO2 nanocomposites, and regression coefficients R (conversionrange 0 < α < 1)

Symbol Kinetic model f (α) log A (s−1) Ec (kJ mol−1) Reaction order n R

Bna Expanded Prout–Tompkins (1 − α)nαm 14.35 120.2 1.1 0.9965

Cn nth order with autocatalysis (1 − α)n(1+KcatX) 12.05 119.6 1.2 0.9955

An Avrami 13.52 119.2 3.3 0.9852


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115 120 125 130 135 140 145 150 155-1.0

-0.8

-0.6

-0.4

-0.2

0.0he

at fl

ow (

W g

–1)

T (°C)

2015105

Exo

Figure 11. Fitting of the experimental results for four different heatingrates (in ◦C min−1) using one reaction mechanism and the Avrami model(An).

115 120 125 130 135 140 145 150 155-1.0

-0.8

-0.6

-0.4

-0.2

0.0

heat

flow

(W

g–1

)

T (°C)

2015105

Exo

Figure 12. Fitting of the experimental results for four different heatingrates (in ◦C min−1) using one reaction mechanism and the expandedProut–Tompkins equation (Bna).

Table 3. Calculated values of Ec and A for the combination of kineticmodels (An-An) of PET/1% SiO2 nanocomposites, and regressioncoefficient R (conversion range 0 < α < 1)

MechanismKineticmodel

log A(s−1)

Ec(kJ mol−1)

Reactionorder n R

First Avrami 14.04 122.9 4.0 0.9962

Second Avrami 17.07 143.5 1.1

correlated with secondary crystallization, which is a slow processresulting in morphologies much different from those resultingduring the primary stage. The higher value of activation energy forthe second stage is also suitable for secondary crystallization. Infact, secondary crystallization occurs after spherulite impingementand most probably inside the spherulites. Such a process occurringin a constrained environment is expected to have high activationenergy.

From the obtained results it can be concluded that in the caseof a complex mechanism of double Avrami models, a satisfactory

115 120 125 130 135 140 145 150 155-1.0

-0.8

-0.6

-0.4

-0.2

0.0

heat

flow

(W

g–1

)

T (°C)

2015105

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Figure 13. Fitting of the experimental results for four different heatingrates (in ◦C min−1) using two reaction mechanisms and the Avrami model(An-An).

fitting is achieved and also the Avrami exponent value (first stagen = 4 and second stage n = 1) seems to be close to the expectedvalue, even for isothermal crystallization (n = 1 up to 4). In fact, inmost cases, even for isothermal crystallization, the phenomenoncannot be described by a single Avrami model. Such an analysis andsimultaneous fitting of data from multiple heating rates give resultsthat may be better correlated with isothermal crystallization data,as the parameters of the phenomenon are better calculated, sincethe results describe crystallization independently of the heatingrate. However, there is much work to be done before arriving atfirm conclusions and before revealing the real correlation withisothermal process results. In fact, the n value is also close tothat obtained from the conventional Ozawa treatment, which infact is an attempt to approximate isothermal crystallization usingnon-isothermal crystallization data. And finally, the Avrami modelwas introduced from the beginning to describe the crystallizationprocess and the related parameters as having some physicalmeaning at least for the isothermal case, in contrast to the othermodels discussed above used to achieve a satisfactory fitting (Bnaand Cn).

From the analysis using the usual procedure, it is clear thatthe application of Avrami or other crystallization model to fitdata from a single heating rate results in values for the kineticparameters which are closely related with the given heatingrate. Data from analysis of a single heating rate are also relatedwith the temperature region where crystallization occurs, whenthat heating rate is selected. It is true that quasi-isothermalconditions cannot be approximated in practice. However, whenusing analysis methods based on simultaneous use of multipleheating rate data, one is expected to arrive at values describingthe process irrespective of the heating rate. This also expands thetemperature region for which crystallization is described. This isa true impact of the method. This is also a foundation for theintroduction of the isoconversional methods for the estimationof the activation energy, meaning that they use data for a givendegree of crystallinity obtained at different heating rates.

CONCLUSIONSData for non-isothermal cold crystallization of an in situ-preparedPET/1% SiO2 nanocomposite obtained at four different heating


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rates were analysed to study the crystallization kinetics. The usualprocedure using a single heating rate separately was used for theAvrami model. Also the well-known Ozawa model was elaborated.Results of the usual methods were found to be far from acceptable.A new approach that involves the use of data from multiple scansat various heating rates for kinetic parameter calculation was alsoapplied with much better results. Among 16 models that weretested, three achieved satisfactory approximation. Assumption ofa mechanism of two sequential processes that can be described bycorresponding Avrami models gave satisfactory fitting, in contrastto what was found when using a single Avrami model. Thissequential process is in line with the usual assumption of primaryand secondary crystallization. In the case of the double Avramimechanism, the results may also have physical meaning, in contrastto the other two models with good fitting. The proposed methodfor the study of non-isothermal cold crystallization is suitable forall polymers. In some cases, perhaps a more complex combinationof different kinetic models ought to be used.

REFERENCES1 Di Lorenzo ML and Silvestre C, Prog Polym Sci 24:917 (1999).2 Papageorgiou GZ, Bikiaris DN and Achilias DS, Thermochim Acta

41:457 (2007).3 Suraphol P, Dangseeyun N, Srimoaon P and Nithitanakul M,

Thermochim Acta 406:207 (2003).4 Xu G, Du L, Wang H, Xia R, Meng X and Zhu Q, Polym Int 57:1052

(2008).5 Chae DW and Kim BC, J Mater Sci 42:1238 (2007).

6 Papageorgiou GZ, Achilias DS and Bikiaris DN, Macromol Chem Phys208:1250 (2007).

7 Chrissafis K, Antoniadis G, Paraskevopoulos KM, Vassiliou A andBikiaris DN, Compos Sci Technol 67:2165 (2007).

8 Chrissafis K, Maragakis MI, Efthimiadis KG and Polychroniadis EK,J Alloys Compd 386:165 (2005).

9 Avrami M, J Chem Phys 7:1103 (1939).10 Avrami M, J Chem Phys 8:212 (1940).11 Avrami M, J Chem Phys 9:177 (1941).12 Wunderlich B, Macromolecular Physics, vol. 2. Academic Press, New

York (1976).13 Ozawa T, J Therm Anal 2:301 (1970).14 Ozawa T, Polymer 12:150 (1971).15 Evans UR, Trans Faraday Soc 11:365 (1945).16 Shangguan YG, Song YH and Zheng Q, J Polym Sci B: Polym Phys 44:795

(2006).17 Caze C, Devaux E, Crespy A and Cavror JP, Polymer 38:497 (1997).18 Weng W, Chen G and Wu D, Polymer 44:8119 (2003).19 Malek J, Thermochim Acta 138:337 (1989).20 Malek J, Thermochim Acta 267:61 (1995).21 Malek J, Thermochim Acta 355:239 (2000).22 Chrissafis K, Kyratsi Th, Paraskevopoulos KM and Kanatzidis MG, Chem

Mater 16:1932 (2004).23 Chrissafis K, J Therm Anal Calorim 95:273 (2009).24 Kissinger HE, Anal Chem 29:1702 (1957).25 Kissinger HE, J Res Natl Bur Stand 57:217 (1956).26 Ozawa T, Bull Chem Soc Japan 38:1881 (1965).27 Friedman HL, J Polym Sci C 6:183 (1964).28 Friedman HL, J Polym Lett 4:323 (1966).29 Marquardt D, SIAM J Appl Math 11:431 (1963).30 Opfermann J, Rechentechnik/Datenverarbeitung 22:26 (1985).31 Khawam A and Flanagan DR, J Phys Chem B 109:10073 (2005).


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A different approach for the study of the crystallization kinetics in polymers. Key study: poly(ethylene terephthalate)/ SiO2 nanocomposites