Modeling soft interface dominated systems: A comparison of phase field and Gibbs dividing surface models

The two main continuum frameworks used for modeling the dynamics of soft multiphase systems are the Gibbs dividing surface model, and the diffuse interface model. In the former the interface is modeled as a two dimensional surface, and excess properties such as a surface density, or surface energy are associated with this sharp interface. Its motion and deformation, and the time evolution of the excess fields associated with it are calculated by a set of partial differential equations (the jump balances), which have to be solved together with the mass, momentum, and energy balances of the bulk phases. A wide range of phenomena have been modeled in this framework, such as the deformation of emulsion droplets in a flow field, the effects of interfacial rheology on multiphase flows, wave phenomena in stratified flows, bubbles rising in a quiescent liquid, phase separation in immiscible mixtures, wetting phenomena, and in-plane and cross-plane mass and heat transfer. In the diffuse interface model the interface is modeled as a three dimensional region of finite thickness, in which order parameters change continuously from their value in one bulk phase to their value in the adjoining bulk phase. The model starts from a free energy functional which contains nonlocal contributions, which to leading order are represented by a square gradient term of the order parameter. The spatial distribution of density or concentration within the interfacial region is then determined by free energy minimization. When a simple pairwise inter-particle potential is assumed, a natural extension of the van der Waals model of phase transitions is obtained. Assuming a simple constitutive relation for the diffusive material fluxes in very viscous binary mixtures, the Cahn-Hilliard theory of spinodal decomposition can be obtained. For systems where convective material fluxes cannot be neglected, a reversible body force, called Korteweg force, must be added to the Navier-Stokes equation, which is proportional to the chemical potential gradient. This force is non-zero only for systems not at equilibrium, and is responsible for the strong convection observed in phase separating mixtures, while it is absent in systems where chemical potentials are uniform. Like the Gibbs dividing surface model, the diffuse interface model has been applied to a wide range of phenomena, such as mixing and demixing in binary mixtures, buoyancy driven detachment of wall-bound droplets, droplet break-up and coalescence, Marangoni effects, and flow in nano- and microchannels. Here we discuss the derivation of the governing equations of these two frameworks, and compare their strengths and weaknesses. We also show how these two frameworks can be combined to solve multiphase problems more effectively.