Quasisteady patterns in interfaces: Folding and Faceting

TL;DR

This work develops a systematic, intrinsic-variables framework for gradient flows of interfacial energies on evolving membranes, recasting curve evolution in terms of curvature $\kappa$ and arc length $g$ with $L^2(\mathbb{S})$ dynamics under closure constraints. It derives explicit first- and second-order variational formulas, introduces a Lagrange-multiplier formalism, and formulates gradient flows that couple local energies (e.g., Canham–Helfrich) to nonlocal two-point interactions. The authors apply the framework to a faceting (Allen–Cahn-like in curvature) energy and to bounded two-point adhesion–repulsion energies, obtaining quasi-steady patterns, front coarsening, labyrinthine folds, and layered equilibria in simulations. The results offer a unifying, computationally implementable approach for quasi-steady membrane/interface patterns on evolving curves and lay groundwork for spectral analyses and reaction–diffusion problems on moving interfaces.

Abstract

We present a systematic derivation of the gradient flows associated to a broad class of interfacial energies, emphasizing the relation between intrinsic and extrinsic variations of the interface. We show that the intrinsic variables formulation brings the gradient flow into alignment with the traditional analysis of quasi-steady dynamical systems defined on a stationary domain. Gradient flows are derived for model systems which exhibit quasi-steady pattern formation including coarsening among faceted interfaces and nonlocal interactions that model membrane self-adhesion and self-avoidance.

Figures (7)

• Figure 1: An in-situ transmission electron micrograph of "pancake stacks" of thylakoid membranes within a chloroplast of an Anemone leaf. The stacks self assemble from inter-membrane electrostatic forces and act as light absorbing panels for photosynthesis, WC_chloro
• Figure 2: Faceting in ice crystals. (left to right) The impact of increasing density of antifreeze glycoproteins on morphology, Gibson10.
• Figure 3: Coarsening of interface under the faceting gradient flow, \ref{['e:Facet']} with system parameters $\alpha=0.1, \beta=5, \kappa_*=5$ and $N=600$ grid points. (Top) Images of curve in $\mathbb{R}^2$ and (bottom) plot of the corresponding curvature versus position along $\mathbb{S}$ at times $t=2.2\times10^{-4}, 9.8\times10^{-3}, 3.2\times10^{-2}, 1.35\times10^{-1}$, and $2.0$ respectively. The open circle indicates the location $\gamma(0)$ on the curve $\Gamma$.
• Figure 4: A semi-log plot of system energy ${\mathcal{E}}_{\rm Fc}$ in blue and the functional residual ${\mathcal{R}}_{\kappa}$ in red versus computational time for the simulation in Figure \ref{['f:facet']}.
• Figure 5: (Top two rows) $L^2(\mathbb{S})$- gradient flow for the energy \ref{['e:CHA-Energy']} from crenelated initial data (top-left). Parameters are given in Table \ref{['T:T1']} with $\rho=1.3$ and time of simulation is indicated above image. Interiors are shaded for clarity. (bottom row) Semi-log plot of system energy versus time showing monotonic decay to equilibrium.

Theorems & Definitions (14)

• Lemma 1
• proof
• Lemma 2
• proof
• Proposition 3
• proof
• Remark 1
• Lemma 4
• proof
• Lemma 5