Meander and Pearling of Single-Curvature Bilayer Interfaces in the Functionalized Cahn--Hilliard Equation

Abstract

The functionalized Cahn-Hilliard (FCH) free energy models interfacial energy in amphiphilic phase separated mixtures. Its minimizers encompass a rich class of morphologies with detailed inner structure, including bilayers, pore networks, pearled pores and micelles. We address the existence and linear stability of α-single curvature bilayer structures in d ≥ 2 spacedimensions for a family of gradient flows associated to the strong functionalization scaling. The existence problem requires the construction of homoclinic solutions in a perturbation of a 4th-order integrable Hamiltonian system while a negative index argument reduces the linear stability analysis to the characterization of the meander and pearling modes of the second variation of the FCH energy on a family of invariant subspaces, independent of the choice of mass-preserving gradient flow.

Pearling Bifurcations in the Strong Functionalized Cahn--Hilliard Free Energy

“…In particular, we establish explicit pearling stability conditions for both bilayers in R d and filaments in R 3 that apply uniformly to all admissible morphologies. This extends the results of [Doelman et al, 2014], which considered the case of constant single-curvature bilayer morphologies that are exact equilibria of the FCH system. The key obstacle to this extension is to retain estimates that are uniformly valid over the of pearling eigenvalues that are both asymptotically large in number and asymptotically close together as they interact through the nonconstant interfacial curvatures.…”

“…The spectrum of both L and ∆L are purely real, and small eigenvalues of ∆L have an explicit mapping onto those of L, see [Doelman et al, 2014], so that the bifurcation of unstable modes in ∆L is controlled by the zero crossing of eigenvalues in L. The central result of this paper is a rigorous analysis of the eigenvalue problem LΨ = ΛΨ, (1.8) about bilayer and filament morphologies in the context of the strong FCH free energy. In particular, we establish explicit pearling stability conditions for both bilayers in R d and filaments in R 3 that apply uniformly to all admissible morphologies.…”

“…In [Doelman et al, 2014] bilayer equilibria of the strong FCH where explicitly constructed as dressings of single-curvature codimension-one interfaces in R d , corresponding to cylinders and spheres in R 3 . The onset of both the pearling and the meander instability where characterized in terms of the functionalization parameters, with the analysis of the meander instability hinging on a very degenerate, higher-order expansion.…”

“…In the analysis of [Doelman et al, 2014], a family of double-wells was considered for which the constant S b was uniformly positive; numerical evaluation of S b shows that it can be of either sign, and the switching of the sign of S b has a significant impact on association of pearling stability regimes with the far-field chemical potential, and is addressed in detail in the companion paper, [Christlieb et al, tted].…”

A second‐order energy stable backward differentiation formula method for the epitaxial thin film equation with slope selection

“…This model is an H −1 gradient flow of an energy functional containing a four-Laplacian energy density. As another example, for the functionalized Cahn-Hilliard model [32][33][34][35][36][37][38] a convex-concave decomposition for the corresponding energy functional has been revealed in a recent work [24], in which a four-Laplacian term appears in the convex part. For these related models, the numerical approach of this article could be extended to obtain second order accurate, energy stable numerical schemes, and the preconditioned steepest descent and preconditioned nonlinear conjugate gradient solvers, which will be outlined in Sections 4.1 and 4.2, could be efficiently applied.…”