Dynamical Eigenmodes of a Polymerized Membrane

TL;DR

This paper derives exact eigenmodes for the dynamics of a polymerized membrane model, enabling precise calculations of physical properties and understanding membrane behavior under tension.

Contribution

It provides the first exact analytical derivation of Rouse modes for a two-dimensional polymerized membrane, extending polymer dynamics theory.

Findings

Exact eigenmodes are derived for the membrane model.

Analytical expressions for radius of gyration and displacement are obtained.

Rouse modes remain valid even under tensile forces.

Abstract

We study the bead-spring model for a polymerized phantom membrane in the overdamped limit, which is the two-dimensional generalization of the well-known Rouse model for polymers. We derive the {\it exact} eigenmodes of the membrane dynamics (the "Rouse modes"). This allows us to obtain exact analytical expressions for virtually any equilibrium or dynamical quantity for the membrane. As examples we determine the radius of gyration, the mean square displacement of a tagged bead, and the autocorrelation function of the difference vector between two tagged beads. Interestingly, even in the presence of tensile forces of any magnitude the Rouse modes remain the exact eigenmodes for the membrane. With stronger forces the membrane becomes essentially flat, and does not get the opportunity to intersect itself; in such a situation our analysis provides a useful and exactly soluble approach to the dynamics for a realistic model flat membrane under tension.