Elastic properties of self-assembled bilayer membranes: Analytic expressions via asymptotic expansion

Abstract

Bilayer membranes self-assembled from amphiphilic molecules are ubiquitous in biological and soft matter systems. The elastic properties of bilayer membranes are essential in determining the shape and structure of bilayers. A novel method to calculate the elastic moduli of the self-assembled bilayers within the framework of the self-consistent field theory is developed based on an asymptotic expansion of the order parameters in terms of the bilayer curvature. In particular, the asymptotic expansion method is used to derive analytic expressions of the elastic moduli, which allows us to design more efficient numerical schemes. The efficiency of the proposed method is illustrated by a model system composed of flexible amphiphilic chains dissolved in hydrophilic polymeric solvents.

Analytical Calculation of the Elastic Moduli of Self-Assembled Liquid-Crystalline Bilayer Membranes

“…Then, the elastic moduli can be calculated by eqs 25−27. Note that for flexible bilayers, our previous study 13 shows that…”

“…Unlike the PFM that should solve MDEs in eqs 17 and 18, the MDEs in the AEM are eqs 30 and 32 where the main difference is the source terms g i . Our previous work 13 developed the numerical method for solving MDE of coils; therefore, here, we only give the numerical method for MDE of rods, i.e., eq 32. Furthermore, we reformulate eq 32 with initial value q 0 and a reflection bound condition as…”

“…11 The AEM regards the curvature of bilayers as a small parameter and carries out an asymptotic expansion of the free energy in terms of the curvature to derive analytical expressions for the elastic constants. 13 Both the PFM and the AEM require accurate computation of the free energy of bilayer membranes. Among the different theoretical frameworks developed for amphiphilic molecules, the self-consistent field theory (SCFT) provides a versatile platform for studying self-assembled bilayer membranes.…”

“…Cai et al 18 extended the PFM to a semiflexible chain model described by the wormlike chain to study the elastic properties of liquidcrystalline bilayers and found that the liquid-crystalline order of the bilayers could have significant effects on the elastic properties of the membranes. Since the AEM could provide the analytical expression of elastic moduli and allows one to design more efficient numerical schemes, 13 it is valuable to extend the AEM to semiflexible bilayers and verify its numerical efficacy compared with the PFM.…”

“…Compared with the PFM, the AEM analytically gives the elastic moduli corresponding to a series of SCF equations. Our previous study 13 derived these MDE and SCF equations for flexible bilayers. In the next sections, we extend the AEM to the liquid-crystalline bilayers and design numerical methods to solve the obtained equations.…”