Solution of the growth equation for asymmetric crystal faces

Most melt-grown and many solution-grown lamellar polymer crystals have curved lateral faces. Mathematical treatments by Mansfield, Point and Villers, and Toda, have provided a satisfactory interpretation of the shape of such crystal faces in terms of nucleation and relatively slow propagation rates of layers of attaching stems. The treatments by these authors, which start with the Frank-Seto growth model, assume that the propagation rates of growth steps to the right (vr) and to the left of the secondary nucleus (vl) are equal. However, for many crystal growth faces this is not the case; faces which lack a mirror plane perpendicular to the lamella have vr≠vl, resulting in asymmetric curvature. Here, we set up and solve the differential equations and reconstruct the shape of the growth front for the case of asymmetric spreading of steps. The solution is presented for the simple square lattice model. The asymmetric growth front is still described as part of an ellipse, as in the symmetric case, except that the centre of the ellipse is translated parallel to the underlying crystallographic plane in the direction of fast v. In forthcoming publications we will adapt the solution to other 2D Bravais lattices, appropriate to the crystal structures of specific polymers. Thus we will analyze complete habits of polymers such as polyethylene, poly(ethylene oxide), and poly(vinylidene fluoride), whose {110}, {120} and {110} growth faces, respectively, are asymmetric. The results of the present work allow a detail kinetic analysis of any well-developed polymer growth face in terms of the step initiation rate i and the propagation rates vr and vl. The present work also quantifies explicitly the deviations from elliptic shape and the substrate edge effects, and discusses when these can be ignored.