Benchmark Computation of Morphological Complexity in the Functionalized Cahn-Hilliard Gradient Flow

TL;DR

This work addresses the numerical simulation of morphological complexity in amphiphilic diblock suspensions by studying the gradient flow of a regularized functionalized Cahn-Hilliard energy. It develops a benchmark suite (sub-critical, critical, super-critical, Foot 1, Foot 2) that tunes absorption rates and left-well stiffness to elicit curvature-driven growth, pearling, and curve-splitting, and evaluates four second-order schemes (IMEX, PSD, SAV, ETDRK2) under adaptive time stepping with Fourier pseudo-spectral spatial discretization. The authors prove energy decay for the SAV scheme and provide a comprehensive comparison of accuracy and efficiency, highlighting that linear-implicit methods (IMEX, SAV) often outperform nonlinear-implicit PSD in stiff regimes, while PSD remains the most accurate at a fixed local error. The results offer practical guidance for selecting robust, efficient solvers for stiff, nonlinear gradient flows and emphasize that energy decay alone is not a sufficient measure of numerical accuracy.

Abstract

Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the "morphological complexity" observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order BDF2-type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a PSD nonlinear solver, or upon IMEX, SAV, ETD approaches. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper "preconditioning" to achieve robust performance that can enhance efficiency by several orders of magnitude.

Figures (18)

• Figure 1: (left) Experimentally observed bifurcation diagram for the morphology of blends of Polyethylene oxide (PEO) - Polybutadiene (PB) amphiphilic diblock in water. The horizontal axis, $w_{\rm PEO}$, is the weight fraction of PEO as a percent of the total diblock weight, and the vertical axis denotes the molecular weights of the PB component of the diblock, fixed at $N_{\rm PB}=45$ or $170$ (vertical axis). Morphological Complexity is observed for $N_{\rm PB}=170$ but not for the shorter $N_{\rm PB}=45$ chains. (right) Experimental images from the morphological complexity regime showing (top) network structures and (bottom) a mixture of end caps and $Y$-junction morphology corresponding to regions marked $N$ and $C_Y$ in the bifurcation diagram. From Figures 1 and 2AC of Bates-BD, Reprinted with permission from AAAS.
• Figure 2: (left) Graph of scaled singular well $W_{\textrm{S}}$ as recovered by reduction of SCMF for $N_P=45$ (red) and $N_P=170$ (blue-dotted). (right) Graph of the regularized well, $W_{\textrm{q}}$ for $\textrm{q}=0, ~0.2,~0.5$.
• Figure 3: (left) A $1D$ cross-section of the grid function $u^0_{\rm 256}$, along with finer mesh realizations $u^0_{\rm 512}$ and $u^0_{\rm 1024}$. (right) The initial data $u^0_{512}$ constructed from \ref{['dcoef']} with width $R=0.14725$ and $\textrm{d} = 0$. The red number on the colorbar indicates $\max\limits_{i,j} \{ u^0_{512,i,j} \}$.
• Figure 4: (left/center) Total FFT calls on log scale at time $T=10$ verses stabilization parameters $\beta_1$ with $\beta_2=3-\beta_1$ for IMEX and SAV for each of the 5 benchmark simulations. (right) Total FFT calls on linear scale at time $T=50$ verses stabilization parameter $\kappa_2$ for ETD for supercritical benchmark and three choices of $\kappa_0.$ The peaks in FFT calls correspond to onset of a shape bifurcation that generates an extra endcap in the ETD simulation. The black arrow indicates choice of parameters in simulations of Section 4.
• Figure 5: Simulation of the sub-critical benchmark with $\textrm{q}=0$, $\sigma_{\rm tol}=10^{-5}$ and $N=256$ at times $T=10$(left) and $T=250$(right). All schemes agree to within $L^2$ relative error $7\times10^{-3}$ as reported in Table \ref{['t:Comp-error']}. The red number on colorbar indicates $\max \{u\}$.

Theorems & Definitions (3)

• Remark 1.1
• Remark 3.1
• Theorem 3.1