Cahn-Hilliard vs Singular Cahn-Hilliard Equations in Phase Field Modeling

Abstract

The Cahn-Hilliard equation is often used to describe evolution of phase boundaries in phase field models for multiphase fluids. In this paper, we compare the use of the Cahn-Hilliard equation (of a constant mobility) for the phase variable with that of the singular or modified Cahn-Hilliard equation (of a variable mobility) in the context of physical derivation of the transport equation and numerical simulations of immiscible binary fluids. We show numerically that (i). both equations work fine for interfaces of small to moderate curvature in short to intermediate time scales; (ii) the Cahn-Hilliard equation renders strong dissipation in simulations of small droplets leading to dissolution of small droplets into the surrounding fluid and/or absorption of small droplets by larger droplets nearby, an artifact for immiscible binary fluids; whereas, the singular Cahn-Hilliard equation can significantly reduce the numerical dissipation around small droplets to yield physically acceptable results in intermediate time scales; (iii) the size of droplets that can be simulated by the Cahn-Hilliard equations scale inversely with the strength of the mixing free energy. Since the intermediate timescale is the time scale of interest in most transient fluid simulations, the singular Cahn-Hilliard equation proves to be the more accurate phase transporting equation for immiscible binary fluids.

Ordering kinetics of a conserved binary mixture with a nematic liquid crystal component

“…For simulations, most operate in two dimensions but some expand to three dimensions [2,3,36,15,52]. These models are usually employed in simulating spinodal decomposition [7,2,3,35,36,8] or effects related to drops such as retraction, collision, or coalescence [7,2,56,58,57,60,62,15]. One point of interest is the coarsening rate of the binary mixture [2,3,49].…”

“…The dynamics are driven by energy minimization with conserved / (Model B in the nomenclature of Hohenberg and Halperin [28]). Phase field models like this one have been used extensively for the investigation of phase separation with and without an underlying flow [18,43,30,7,2,3,35,36,8,56,58,57,4,60,16,62,15,49,26,52,55]. CahnHilliard type equations are used to handle the phase separation and are coupled to the Navier-Stokes equations to introduce flow.…”

Splitting Schemes for a Navier-Stokes-Cahn-Hilliard Model for Two Fluids with Different Densities

“…In [27], Zhang and Wang present a study of the influence of the mobility term in a model of two-phase incompressible flows with different densities but no physical background of the derivation of the model is presented or cited. Finally, Abels et al derived in [3] a new thermodynamically consistent model for incompressible two-phase flows with different densities while in [2] the existence of weak solutions for this model is proved.…”

A Class of Conservative Phase Field Models for Multiphase Fluid Flows

“…Multiple phase-field methods can be devised to study multiphase materials [15]. Recently, phase field models have been applied to study liquid crystal drop deformation in another fluid, liquid films, polymer nanocomposites, and biofilms [9][10][11][12][13][14][15][16][17][18][32][33][34].…”