Dirac chains in the presence of hairpins

摘要

We study semiflexible polymers of arbitrary stiffness subject to nematic and non-nematic elongation forces. The presence of nematic forces is found to cause the formation of hairpins. [A hairpin is an immediate return (or a sharp bend) of a chain in the nematic ordering field.] An analysis of the path integrals for semiflexible (Dirac) chains with these elongational forces indicates that the distribution functions describing these induced hairpins satisfy the Whittaker-Hill (WH) equation in two dimensions. The same equation describes hairpins in three dimensions if (and only if) the Dirac monopole term is included in the corresponding path integral. The solutions of the WH equation indicate that the non-nematic stretching force can only have discrete values corresponding to the sequential destruction of hairpins. This discreteness disappears when the nematic force is absent, as demonstrated in previous work [A. Kholodenko and T. Vilgis, Phys. Rev. E 50, 1257 (1994)]. We also indicate how the hairpin problem is related to other statistical mechanical problems of interest: commensurate-incommensurate transitions, quantum spin chains, Landau-Lifshitz equation, rotational Brownian motion, strings with rigidity, etc.