Direct observation of the effective bending moduli of a fluid membrane: Free-energy cost due to the reference-plane deformations

TL;DR

This study introduces a direct method to measure the effective bending moduli of fluid membranes by analyzing free-energy costs from reference-plane deformations, revealing that the mean-curvature measure predicts membrane stiffening and that the Gaussian-curvature modulus remains nearly scale-invariant.

Contribution

The paper presents an efficient approach to directly observe effective bending moduli of fluid membranes through free-energy analysis under reference-plane deformations, clarifying the impact of statistical measures.

Findings

Effective bending rigidity increases with flexible membranes under mean-curvature measure.

The Gaussian-curvature modulus remains nearly scale-invariant.

Results differ significantly when using normal-displacement measure.

Abstract

Effective bending moduli of a fluid membrane are investigated by means of the transfer-matrix method developed in our preceding paper. This method allows us to survey various statistical measures for the partition sum. The role of the statistical measures is arousing much attention, since Pinnow and Helfrich claimed that under a suitable statistical measure, that is, the local mean curvature, the fluid membranes are stiffened, rather than softened, by thermal undulations. In this paper, we propose an efficient method to observe the effective bending moduli directly: We subjected a fluid membrane to a curved reference plane, and from the free-energy cost due to the reference-plane deformations, we read off the effective bending moduli. Accepting the mean-curvature measure, we found that the effective bending rigidity gains even in the case of very flexible membrane (small bare rigidity); it has been rather controversial that for such non-perturbative regime, the analytical prediction does apply. We also incorporate the Gaussian-curvature modulus, and calculated its effective rigidity. Thereby, we found that the effective Gaussian-curvature modulus stays almost scale-invariant. All these features are contrasted with the results under the normal-displacement measure.