Modified Avrami Equation for the Bulk Crystallization
Kinetics of Spherulitic Polymers
I. H. HILLIER,* Chemistry Department,
Synopsis
An explanation of the anomalous fractional values of the Avrami exponent found for the crystallization of a number of polymers is
presented. The interpretation is based on a model which postulates the constant radial growth of spherulites, followed by an increase in crystallinity within them by a first order process. The model is supported by direct microscopic observations of other workers. Crystallization isotherms for polymethylene, poly(ethylene oxide), and poly(decamethy1ene terephthalate) are fitted to this model. Apart from the removal of the fractional values of the Avrami exponent, which have no physical meaning, this model gives a considerably better fit than the Avrami equation to most isotherms analyzed. The temperature dependence of the rate constants found for the two rate processes of this model is also discussed. An interpretation of the results of seeded experiments is presented in terms of this model.
INTRODUCTION
The crystallization kinetic of bulk polymers is usually interpreted [1], although with varying degrees of success, in terms of the Avrami equation:
where χ is the crystallinity, t the time, z a constant depending upon the nucleation and growth rates, and n an integer depending on the shape of the
growing crystalline body. As spherulites are observed in most crystalline polymers and in thin films are seen to grow with a constant radial rate, a value n = 3 (instantaneous nucleation) or n = 4 (sporadic nucleation) is expected if the density of the
spherulites is constant. Recent work on a number of polymers [2-7] has revealed that values of n of 3 or 4 are rarely fo
und. The results of detailed analysis of crystallization isotherms which appear in the literature are summarized in Table I on the following page. The Avrami exponent n may vary with temperature and be a noninteger less than 4 but rarely exceeds this value. Deviations of n from the integral values required by eq. (1) may be explained by postulating either (a) a nonconstant growth rate or (b) a nonconstant density of the growing spherulites.
A model in which the crystalline bodies are considered to be single crystal lamellae growing by a chain-folding mechanism has been shown to be a possible explanation of the observed kinetics of polymethylene [6]. This model leads to a nonconstant growth rate, the single crystal lamellae having a constant density.
A number of authors have independently proposed [8-10] that the “primary” (Avrami) and “secondary” (post-Avrami) crystallization procesees occurring in polyethylene may be described by the growth and subsequent relaxation of the crystalline regions, leading to a variable density within the sample. Gordon and Hilliel.8 have shown that such a scheme, in which the secondary relaxation is described by a general rate equation due to Hirai and Eyring [11] successfully describes the overall crystallization rate in bulk polymethylene. In this paper, it is shown that the fractional values of the Avrami exponent may be explained by the constant radial growth of spherulites (termed the primary crystallization in this work) followed by further crystallization within the spherulite which obeys a first-order law. This model is fitted to crystallization isotherms of polymethylene, poly(ethy1ene oxide) and poly(decamethylene terephthalate). It must be stressed that this postulated subsequent crystallization is distinct from the secondary crystallization which varies linearly with Ln(time), observed in polyethylene and other polymers. The mechanism of this secondary or post-Avrami crystallization has been fully discussed elsewhere [8], and is interpreted in terms of an increase in lamellar thickness. The firsborder process proposed in this paper differs from this post-Avrami crystallization both kinetically and in the morphological interpretation of the kinetics. The slow secondary crystallization is absent in the samples analyzed here.
References
1. Mandelkem, L., Growth and Perfection of Crystals, Chapman and Hall,
2. Sharples, A., and F. L. Swinton, Polymer, 4, 119 (1963).
3. Banks, W., M. Gordon, R. J.,Roe, and A. Sharples, Polymer, 4, 61 (1963).
4. Hatano, M., and S. Kambara, Polymer, 2, 1 (1961).
5. Gordon, M., and
6. Banks, W., and A. Sharplea, Makromol. Chem., 59, 33 (1963).
7. Parrini, P., and G. Conieri, Maknrmol. Chem., 62, 83 (1963).
8. Gordon, M., and
9. Peterlin, A., J. Appl. Phys., 35, 75 (1964)
10. Price, F. P., private communication.
11. Rirai, N., and H. Eyrihg, J. Appl. Phys., 29, 810 (1958)
