On the growth of two dimensional crystals. I. Fluctuations and the relation of step free energies to morphology

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Abstract
Some fundamental aspects of crystallization of two dimensional (2D) crystals are examined, including an analysis of the fluctuations in step positions, computer simulation results for the step density and growth rates, and analysis for predicting crystal shapes. A discussion is given of whether nucleation can occur on the 1D surface of a 2D crystal. During the course of the paper it is shown that a more appropriate description, for any infinite 1D surface, is of a step density related to the equilibrium value and a growth rate which is proportional to the density and speed of these steps. The speed depends essentially linearly on the net thermodynamic driving force for crystallization. (This description is not appropriate for a 2D surface since the equilibrium step density can then be zero.) Nevertheless, one can define a criterion, depending both on the density of steps and on the extent of the fluctuations in step positions, for a regime of growth with some similarities to nucleation (1D nucleation). Kinetic (nucleation) theories for polymer crystallization, using a rate equation approach, have implicitly assumed the conditions for 1D nucleation to hold. Even when the conditions do not hold, the results of this rate equation approach are in reasonable agreement with the simulation results (though a rather curious discrepancy of a factor of 21/2 occurs since the rate equation approach omits a growth mediated approach to equilibrium). The agreement in general trend is shown to be the fortuitous consequence, in the rate equation approach, of neglecting both the enhancement of step annihilations caused by fluctuations and the creation of pairs of steps at cavities. If the latter effect is neglected but not the former, anomalous results can be obtained which violate thermodynamic principles. Growth rate results from simulation are obtained as a function of the obliquity of the growth face with respect to the principal directions in the lattice. These growth rate anisotropy data are used to predict the shapes of crystal outlines. Comparisons between these results and experimental observations on shapes of polymer crystals are made, and a preliminary account is given of how the step free energy per length of step decreases with increasing temperature. In general, the total step free energies tend to decrease as the crystal thickness increases for polyethylene.