Analysis of crystal habits bounded by asymmetrically curved faces: Polyethylene oligomers and poly(vinylidene fluoride)

Curved lateral faces of lamellar polymer crystals have previously been described mathematically using a model of initiation and spreading of molecular steps on the growth surface. Previously it has been assumed that the steps spread with equal velocity v in both directions, and hence the model only applied to faces that have a two-fold axis or are bisected by a mirror plane normal to the lamella. Many lateral faces in polymer crystals do not have such symmetry. We recently solved the growth equation and reconstructed the growth profile for the case where the velocities in the left and right directions (vl and vr) are different, using a square lattice model. Here we adapt the model to oblique lattices suitable for {110} growth faces of polyethylene oligomers (ultralong alkanes) and PVDF. Very good fits are obtained with the observed unusual habits of crystals with curved {110} faces. It is shown that the shape of an asymmetric lateral crystal face is defined by two kinetic parameters, vr/vl and il02/(vr + vl), where i is the nucleation rate and l0 the interchain distance in the growth plane. The value of vr/vl is found to be furthest from 1 at the smallest supercooling, which is consistent with the general principles of crystallization kinetics. The observed ten-fold difference between right- and left-moving steps on α-crystals of PVDF is attributed to the difference between edge-on and flat-on chain attachment at these steps, respectively. The high value of il02/(vr + vl), the cause of high curvature, gives a picture of a crystallization process that is only marginally nucleation controlled. Specific combinations of the above two parameters are predicted to produce exotic single crystals with re-entrant corners, so far unobserved.