Neutron scattering from solution-grown polymer crystals: a statistical model

Abstract

An earlier statistical model for the spatial arrangement of isotopically labelled ‘stems’(single molecular traverses) within single crystals has been extended, with the probability of deposition of a labelled stem following another labelled stem (P_A), the probability of labelled stem deposition after an unlabelled one (P_∪) and the number of labelled stems per ‘sheet’(the crystallographic plane containing the labelled stems)(σ) as variables. Analytical expressions have been derived for the dilution of one isotopic species with the other and for the size distribution for clusters of labelled stems for a one-dimensional sheet of labelled stems. This opens the possibility of comparison with predictions from different possible crystallisation mechanisms. Calculations for the neutron scattering intensity from this model, with ‘superfolding’(folding from one crystallographic fold plane to its neighbour) occurring after a specified number of labelled stems, show good agreement with those based on the earlier model for a set of variables appropriate to solution-crystallised polyethylene. Experimental data are also well fitted. Conversely, a model with superfolding occurring at random along the molecular length is shown to provide a poor fit to experimental data. A test is demonstrated for the number of computer simulations required to obtain adequate statistics.