In order to explain the curved outline of some polymer crystals, we start from a slightly modified Frank's system of equations describing nucleation-controlled crystal growth. But we give an alternative and completely new interpretation of these equations with a view to applying them to the description of normal growth of crystals with rough surfaces. We show first that Mansfield's approximate solution of such equations is not exact, does not satisfy absorbing boundary conditions and does not depend on the rate of motion of these boundaries. Because that solution describes the facets of a crystal in terms of arcs of the same ellipse, it cannot be used to describe the curvature of 2 0 0 faces of those crystals of large elongation ratio which clearly are not parts of a single ellipse. The most puzzling feature of Mansfield's proposal is, however, that pronounced curvature is obtained solely if the mean distance between steps is assumed to be of the same order of magnitude as the width of a single stem. In such an eventuality, the use of the concept of nucleation-controlled growth becomes meaningless. We propose, therefore, a completely new interpretation of Frank's equations, which allows us to use these equations to describe nucleation-controlled growth. Then these equations appear as the mathematical formulation of a model that mediates between the approach of Seto and Frank, and that of Gilmer and Sadler. Detailed analysis of the experimental data, mostly those of Labaig and Bassett, shows that the outline of crystals with curved habit is neither elliptic nor made of elliptical arcs. We provide new and exact solutions of our generalized equations with generalized boundary conditions. These solutions account well for the experimental data. In some cases they may be approximated by Mansfield's solution (despite the fact that the change of curvature along the facets differs from that actually observed) or by the solution found independently by Toda and us (33th IUPAC Polymer Symposium at Montreal), despite the fact that it is unusual that near the tips of the crystals the outline of a crystal is well approximated by straight segments. The most important claim is, however, that our model appears as a bridge between the model of nucleation-controlled growth and that of normal growth of rough surfaces.
Посилання на статтю:
Crystals with curved edges: a unified model that mediates between the theories of nucleation-controlled and rough surface growth / J.-J. Point* and D. Villers // Polymer. – 1992. – Vol 33. – P. 2263-2272.