Ordered spans of unrestricted and self-avoiding random-walk models of polymer chains. I. Space-fixed axes

Abstract

An N-step random walk on a cubic lattice is adopted as a model of a random polymer chain. The spans, or extents, of each random walk configuration in the principal lattice directions are arranged in order of magnitude, ξ3?ξ2⩾ξ1. In the case of the unrestricted random walk, the average values of the ordered spans 〈ξi,u〉 and 〈ξ2i,u〉, i=1, 2, and 3, are calculated analytically in the limit of large N. The limiting relative values of the first moments, 〈ξi,u〉 are 1.637 : 1.267 : 1; and the limiting values of the second moments 〈ξ2i,u〉 are 2.710 : 1,600 : 1. In the case of the restricted or self-avoiding walk, the corresponding average spans 〈ξi,r〉 and 〈ξ2i,r〉 are estimated for N?150 by using a Monte Carlo procedure. The same Monte Carlo procedure is used to estimate the values of 〈ξi,u〉 and 〈ξ2i,u〉 for N?1000. On the assumption that the rate of approach of the average ordered spans of the self-avoiding walks to their asymptotic forms is similar to the rate of approach of the average ordered spans of the unrestricted walks to their asymptotic forms, the following estimates are obtained for the ordered spans of the self-avoiding walks: 〈ξi,r(N) 〉=?i,rN0.61 for N≫1 and ?3,r : ?2,r : ?1,r =1.75 : 1.31 : 1. The relative values of the estimates of the second moments of these ordered spans for N≫1 are 〈ξ23,r〉 : 〈ξ22,r〉 : 〈ξ21,r〉 =3.08 : 1.71 : 1. A simple ellipsoidal model is analyzed in order to obtain estimates of the intrinsic spans of the two kinds of random walks. It is assumed that all random walks have the same intrinsic dimensions and form (ellipsoidal), but that their principal axes are oriented at random. The average ordered spans with respect to a set of orthogonal space-fixed axes of this randomly oriented ensemble of ellipsoids depend uniquely on the principal diameters of the ellipsoid (intrinsic spans). A procedure is devised to solve the inversion problem of determining the relative lengths of the principal diameters which correspond to a given set of relative ordered moments with respect to space-fixed axes. The relative values obtained for the principal diameters of the ellipsoid are 2.59 : 1.61 : 1 in the case of the unrestricted random walk and 3.05 : 1.77 : 1 in the case of the self-avoiding walk.