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Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard (FCH) 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 re- sult is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split di- rectly into networks of interfaces. We evaluate a number of schemes that use second or- der backward differentiation formula (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 preconditioned steepest descent (PSD) nonlin- ear solver or upon linearly implicit time discretization based on the standard implicit- explicit (IMEX) and the scalar auxiliary variable (SAV) approaches. We add an ex- ponential time differencing (ETD) scheme for comparison purposes.