Effect of Chain Transport in the Secondary Surface Nucleation Based Flux Theory and in the Lauritzen−Hoffman Crystal Growth Rate Formalism

Abstract

A previous study (Snyder, C. R.; Marand, H.; Mansfield, M. L. Macromolecules 1996, 29, 7508) by the authors suggested that, within the context of the Lauritzen−Hoffman secondary surface nucleation theory, the apportionment factor for the free energy of crystallization (ψLH) and the retardation factor due to transport of chain segments (βLH) may differ for placement of the first vs all subsequent stems. In this manuscript, we investigate the consequences of this treatment on the crystal growth rate of polyethylene in regimes I, II, and III. It has been impossible so far to find an analytical solution for the secondary nucleation rate, i, and the average lamellar thickness, 〈l〉, when the apportionment and the retardation factors for the first stem (ψ‘ and β‘) are treated as being different from those used for all subsequent stems (ψ and β). It is shown in this work that approximate solutions, which cover most conditions, can be obtained by assuming two extreme cases. In the first case, it is assumed that the rate constant for initial stem removal (B1) is larger than the difference between the rate constants for subsequent stem placement and removal (A − B); i.e. A − B ≪ B1. Such an approximation leads to the conclusion that the crystal growth rate in regimes I and II is only dependent upon the retardation factor for subsequent stem placement. In the second case, it is demonstrated that if (A − B) ≫ B1, the resulting flux equation is valid over most other circumstances where the first assumption (B1 ≫ A − B) does not hold. This leads to the conclusion that the crystal growth rate in regime I is only dependent upon the path followed for initial stem placement whereas in regime II it is dependent upon both the initial and subsequent stem placement paths. Conversely, it is shown, without any assumption regarding the relative magnitudes of β‘ and β, that in regime III the crystal growth rate is dependent upon the path of initial stem placement and independent of the path of subsequent stem placement. Under these conditions, the molecular weight dependency of the crystal growth rate in regimes I and II vs III may differ substantially. Finally, some remarks are given on the consequences of this approach for the crystallization of other polymers at larger undercoolings than encountered for poly(ethylene).