Equilibrium properties of polymers from the Langevin equation: Gaussian self-consistent approach

We investigate here the dynamics of polymers at equilibrium by means of a self-consistent approximation that can be applied to arbitrary Hamiltonians. In particular we show that for the case of two-and three-body excluded volume effects, and the Oseen hydrodynamic interaction, the Gaussian self-consistent approach can recapture what we believe to be the essential features across the collapse transition. This method is based on the approximation of the complete Langevin equation by a Gaussian stochastic ensemble obeying a linear equation of motion with some unknown effective potential Δ${\mathit{V}}_{\mathit{q}}$(t) and friction. Self-consistency equations for this potential are derived and studied in a variety of regimes across the collapse transition. Here we have calculated the friction ${\mathrm{\zeta }}_{\mathit{q}}$ scaling behavior. The results of a simple power counting analysis of the equations, applicable for sufficiently large polymers, confirm the expected law ${\mathrm{\zeta }}_{\mathit{q}}$∝${\mathit{N}}^{\nu }$ ${\mathit{q}}^{1\mathrm{-}\nu }$, and give exponent values ν=3/5 for the Flory coil, ν=1/2 for so-called θ point, and ν=1/3 for the collapsed globule phase. Further applications of the method for various experimental observables of interest, e.g., the dynamic structure factor of light scattering, are presented, and again simple applications are discussed.