Theory of Directed Polymers

Abstract

We develop a theory of polymers in a nematic solvent by exploiting an analogy with two-dimensional quantum bosons at zero temperature. We argue that the theory should also describe polymers in an {\sl isotropic} solvent. The dense phase is analyzed in a Bogoliubov-like approximation, which assumes a broken symmetry in the phase of the boson order parameter. We find a stiffening of the longitudinal fluctuations of the nematic field, calculate the density-density correlation function, and extend the analysis to the case of ferro- and electrorheological fluids. The boson formalism is used to derive a simple hydrodynamic theory which is indistinguishable from the corresponding theory of polymer nematics in an isotropic solvent at long wavelengths. We also use hydrodynamics to discuss the physical meaning of the boson order parameter. A renormalization group treatment in the dilute limit shows that logarithmic corrections to polymer wandering, predicted by de Gennes, are unaffected by interpolymer interactions. A continuously variable Flory exponent appears for polymers embedded in a {\sl two}-dimensional nematic solvent. We include free polymer ends and hairpin configurations in the theory and show that hairpins are described by an Ising-like symmetry-breaking term in the boson field theory.