Coupling between Phase Separation and Geometry on a Closed Elastic Curve: Free Energy Minimization and Dynamics

Abstract

We study the free energy and dynamics of a closed elastic filament (a one-dimensional curve in two dimensions) whose local internal state is specified by curvature, stretch, and a scalar density field representing, for example, the concentration of an absorbed species. The density variable has a tendency to phase-separate whereas the local spontaneous curvature is concentration-dependent. There is also a coupling between concentration and the stretching of the filament, although our main interest is in the nearly inextensible regime. We formulate and simulate the dynamics, comprising a coupled Willmore flow and Cahn-Hilliard gradient flow on the full differential geometry of a closed filament, addressing issues that previous work typically sidestepped by restricting to the Monge gauge. We use a numerical strategy for global free energy minimization, presenting the equilibrium shapes and density profiles across a wide range of model parameters. The phase diagram is dominated by a relatively small number of simple shapes that exhibit, as expected, strong coupling between local curvature and concentration. We also find regimes where curvature and/or stretching energies suppress phase separation altogether. For selected parameter values we present fully dynamical results, tracking the time evolution of the various contributions to the free energy. The dynamics often arrive at metastable minima rather than the equilibrium state - for example, at states with more than the minimum number of interfaces between coexisting phases. The metastability of these states is absent for phase separation on a rigid circular domain and thus a direct result of the coupling between geometry and density.

Figures (8)

• Figure 1: Macroscopic shape evolution driven by coupled local dynamics.a,Adsorption: Adsorbed particles that induce a spontaneous curvature locally bend the filament. Because of the dynamical coupling, such particles collectively drive the global evolution of the closed filament away from the circular shape with the lowest bending energy to different morphologies in which symmetry is broken as phase separation and loop deformation set in concurrently. The right panel shows representative final morphologies featuring different numbers of domains and distinct shapes. b--d, Local dynamical mechanisms governing the evolution, illustrated between arc-length positions $X_1$ and $X_2$. b,Advection: particles are advected together with the tangential motion of filament at velocity $v_t$. c,Deformation: local particle concentration is changed by the bending and stretching of the curve. d,Phase separation: particles spontaneously partition into dense and dilute phases along the curve. Feedback among these local processes drives the macroscopic morphological changes illustrated schematically in figure a.
• Figure 2: Illustration of a closed reference curve parameterised by the material coordinate $\sigma \in S^1$ and its embedding $\Gamma \subset \mathbb{R}^2$, with arclength coordinate $s$.
• Figure 3: Free energy minimisers for the case $\alpha=1024$, $\beta=20$, $\epsilon=0.05$, $\kappa_0=3$, $C=0.43$. From left to right: circle ($N=0$), acorn ($N=2$), peanut ($N=4$), polygon ($N \ge 6$). Dashed line represents the shape of the curve at rest without the coupled concentration field. The $N=0, 4, 6$ cases satisfy the optimal relation \ref{['eq: optimal_kappa']}. The $N=2$ case does not, because of competition between energies induced by the closure constraint \ref{['eq:closure']}.
• Figure 4: Effect of the closure constraint. Upper panels: plots of concentration, curvature and metric. Lower panels: real-space shapes with concentration field in colour scale. For $\alpha=64$ and $C=0.5$ (left), imposing the closure constraint yields a closed global minimum (a 'peanut' curve) with $N=4$, whereas with only the turning number constraint \ref{['eq: turning']} and periodic boundary conditions the optimum has $N=2$. The global concentration is $C=0.5$ so that the two phases have the same total arc length. For $\alpha=1024$ and $C=0.24$ (right), the global optimum remains in the $N=2$ (acorn) category; removing the closure constraint eliminates the need to trade bending energy for closure and therefore lowers the optimal energy. Other parameters are $\kappa_0=3, \varepsilon=0.05, \beta=20$ as in Fig. \ref{['fig: metastables']}; for more details see Appendix \ref{['secapp: numerical']}.
• Figure 5: Phase diagrams in the ($\alpha, C$) plane of minimizer morphologies, for two values of the interfacial parameter $\varepsilon=0.05, 0.15$. Within each morphology, the red-blue color scale depicts the local composition $c$. The solid red and green lines delineate phase boundaries of the $N=4$ (peanut) and $N=2$ (acorn) phases; outside these (yellow shading) the minimizer is a compositionally homogeneous circle ($N=0$). Increasing $\varepsilon$ raises the interfacial cost and shifts the optimum toward states with fewer interfaces. Further parameters: $\kappa_0 = 3$, $\beta=20$; numerical details are in Appendix \ref{['secapp: numerical']}.

Theorems & Definitions (10)

• Proposition C.1
• proof
• Proposition C.2: absence of kinks
• proof
• Proposition C.3
• proof
• Proposition C.4
• proof
• Proposition C.5: degeneracy or uniqueness of closed minimizer solutions
• proof