Non-Isothermal Crystallization Kinetics of Poly(ethylene terephthalate) from the Point of View of Isokinetic Models

Abstract

The kinetic crystallization model of A vrami is the generally accepted starting point for the analysis of isothermal nucleation and crystallization of polymers.1 -3 It will be shown in this communication that, in the case of non-isothermal crystallization experiments performed with constant heating or cooling rates and analyzed by using differential scanning calorimetry (DSC), even an apparent m-order reaction model, being defined for every real, positive reaction-order m, is appropriate to describe the corresponding crystallization curves in the vicinity of the inflexion points. The latter approach will be shown to be approximately equivalent to the isokinetic nucleation and growth model of Nakamura 4 as well as to the non-isothermal crystallization kinetics theory applied to DSC-curves of Ozawa. 5 The apparent m-order reaction model is given by the following equation, 6 if constant heating or cooling rates are maintained during the DSC-experiments:whereby a(T) represents the conversion factor at the absolute temperature T (varying from O to 1), K 0 the frequency factor, R the gas constant, ER the activation energy, q the heating or cooling rate, k,(T) the temperature-dependent rate constant and m the apparent reaction-order. The inflexion point of the function a(T) is denoted by a(Tin). It is found by solving the equa-This has already been done in a previous publication. 6 The following result has been obtained:In deriving eq 5 some approaches have been made assuming that the investigated chemical reactions, curing reactions, phase transformations or crystallization processes occur over a relatively narrow temperature range around Tin in comparison with the absolute 262 temperature T. This assumption is justified in most practical cases. Thus, a( Tin) has been found to be approximately independent of the temperature, the temperature-dependent rate constant and the heating (or cooling) rate q. a(Tin) is only dependent on the apparent reaction-order m. Because m is approximately independent of q, if an inflexion point exists, eq 5 is valid for a wide range of q-values, although not for q = 0. 6 On the other side, the isothermal crystallization model of A vrami, being based on the nucleation and growth characteristics of crystalline phases, also shows an inflexion point for n > 1 as well as for the formal cases with n < 0. The three-parameter Avrami equation function, 7 • 8 being the result of the isothermal nucleation and growth theory, is given bywhereby 8(t) is the relative crystallinity at time t, k the crystallization rate constant containing the nucleation and growth rates, r the induction period and n the Avrami index. Moreover, the value of B(tin) at the inflexion point is given by 7If a(Tin) is formally set equal to 8(tin), the following relation between the Avrami index n and the apparent reaction-order m is found 6 :Equation 8 implies that, with the exception of the singularity at m = 1, every real, positive value of an apparent reaction-order m corresponds exactly to one real value of n, being larger ...

Polymer crystallization kinetics, master quotients, master curves and Nakamura‐Ozawa‐equations of iPP and PTFE

“…Equation 13 is valid for a wide range of q-values, although not for q ϭ 0 (12). If ␣(T in ) is formally set equal to (T in ), the following relation between the Avrami index n and the apparent reaction-order m is found (7,12,13): (15) Owing to Eq 15, data of non-isothermal crystallization experiments can be treated both with m-order reaction models and with isokinetic nucleation and growth theories (12,13,16).…”

“…Finally, the combination of Eqs 14, 18 and 19 leads to the heating (or cooling) function of the Ozawa equation at the inflexion point (T in ). The use of Eq 14 is justified because of the one-to-one correspondence between (t ), (T ) and the non-isothermal m-order reaction function ␣(T ) with respect to the inflexion points (12,13).…”

“…T in denotes the maximum reaction rate temperature, T the mean-temperature and T f (w) the temperature corresponding to a relative crystallinity w. By performing DSC measurements with varying q, the temperatures T in ,T and T f (w) can be changed considerably (5,12,13). Thus, the independence of (T ) from the cooling (or heating) rate q can also be checked.…”

“…According to Eq 14, the following equations are (approximately) valid (12,13) for any cooling or heating rate q 0:…”

“…Since the apparent m-order reaction model is defined for every real, positive apparent reaction-order m, a distinct Avrami index n, which is valid for the characterization of isothermal and non-isothermal crystallization experiments with constant heating or cooling rates, can always be related to an apparent reaction-order m. Therefore, all three types of Avrami master curves, being merely dependent on the Avrami index n, can be obtained by performing isothermal as well as non-isothermal experiments (8Ϫ13). In addition, apparent m-order reaction models are approximately equivalent to the isokinetic, non-isothermal nucleation and growth models of Nakamura (8) and Ozawa (9) in the vicinity of the inflexion points of the crystallization curves, i.e., at crystallization rates close to their maxima (12,13).…”