Bond-length and bond-angle distributions in coarse-grained polymer chains

Abstract

The joint distribution pN(ω,R1,R2) of the lengths of the vectors (R1,R2) and the angle ω between the vectors joining the center of mass of a block or bead in a chain with the centers of mass of the preceding and succeeding beads of a coarse-grained chain is investigated. Somewhat unexpectedly, the a priori and the conditional distributions for the connector length pRN(R1) and for the interconnector angle pωN(ω) can be determined in closed form for the Gaussian random walk for any number of monomers N per bead. Additionally, pωN(ω) is also obtained for an unperturbed polymethylene chain by a Monte Carlo scheme. In general, pωN(ω) is asymmetric and shifted to values greater than π/2 for all bead sizes, implying that the chain made up of the centers of mass of the beads tends to be locally more extended or spread out than a random walk. Convergence to the (analytically known) asymptotic distribution pω∞(ω) for the Gaussian random walk is given analytically and turns out to be very nearly quadratic in 1/N. For the polymethylene chain convergence is linear. The a priori and the conditional distributions are used to formulate a Monte Carlo scheme for the generation of the coarse-grained chain. The bias of pωN(ω) to angles greater than π/2 should be taken into account by any model that attempts to represent a linear chain polymer by lumping its detailed structural information in coarser units at a larger length scale.