Directed self-avoiding walks and statistics of rigid-chain polymer molecules

The statistics of rigid-chain polymer conformations is described on the basis of a model of directed self-avoiding walks. The generating functions for the distribution function of a chain in one-, two-, and three-dimensional spaces are constructed. It is shown that the statistics of the conformational states of chains with finite interunit flexural stiffness can differ strongly from Gaussian statistics. If the chain length is comparable to the Kuhn segment length, then the molecule is strongly anisotropic (almost rectilinear), but as the chain length increases, the molecule starts to bend and ultimately coils up. However, since a coil contains extended, almost rectilinear, chain sections, the coil is not truly Gaussian, even though the squared average size of the coil is directly proportional to the chain length. It is shown that under certain conditions the existence of almost rectilinear chain sections results in the appearance of orientational order in the system.