Conditions under which a natural iterative method for calculating the orientation distribution of rodlike particles decreases the free energy at each step†

Abstract

It is shown that under certain conditions the iterative method suggested by the calculus of variations for calculating a cylindrically symmetric orientation distribution function (ODF) for rodlike particles strictly decreases the free energy at each full step. This monotonic behavior has strong implications for convergence of the sequence, or a subsequence, of the calculated ODFs. The result is valid not only for the reference system of hard core particles with indistinguishable ends, but also for free energy functions of similar type. The effect of an applied field is also permitted. Since the behavior of the iteration is intrinsic to the general form of the free energy function it applies to rod mixtures and may extend to a wider class of free energy functions. Outside the conditions that guarantee a monotonically decreasing free energy, we find that the iteration can fail to converge.