Equilibrium statistics of an infinitely long chain in the severe confined geometry: Rigorous results

TL;DR

This paper provides rigorous analytical results on the equilibrium statistics of a long polymer chain confined between two lines, deriving partition functions and critical exponents for various separations in a two-dimensional model.

Contribution

It introduces exact analytical expressions for the partition function and critical exponents of a confined polymer chain using a directed self-avoiding walk model.

Findings

Exact critical exponents for different separations were obtained.

Analytical expressions for the partition function were derived.

Grand canonical partition function for semiflexible chains was calculated.

Abstract

We analyze the equlibrium statistics of a long linear homo-polymer chain confined in between two flat geometrical constraints under good solvent condition. The chain is ocupying two dimensional space and geometrical constraints are two impenetrable lines for the two dimensional space. A fully directed self avoiding walk lattice model is used to derive analytical expression of the partition function for the given value of separation in between the impenetrable lines. The exact values of the critical exponents ($\nu_{||}, \nu_{\perp}, \nu $ and $ \gamma_1$) were obtained for different value of separations in between the impenetrable lines. An exact expression of the grand canonical partition function of the confined semiflexible chain is also calculated for the given value of the constraints separation using generating function technique.