Diffuse Interface Approach Research Articles

Modelling the evolution of phase boundaries in solids at the meso- and nano-scales

Phase boundaries play an important role in setting the properties of multicomponent materials at both the meso- and nano-scales. In this review, we provide an overview of the modelling methods utilized in state-of-the-art research and engineering applications. We review the current physical understanding of how phase boundaries evolve, focusing on multicomponent systems. The recent advances in numerical modelling, fueled by powerful computers, have provided accurate and robust results that allow problems that are beyond the reach of analytic methods to be addressed. While the approaches used in engineering-oriented applications employ simplified microstructures, it is found that such models are quite useful in many problems. The ability to simulate realistic microstructures will further increase the power of materials modelling. We also highlight the differences between the sharp interface and diffuse interface approaches for modelling microstructural evolution. In addition, we identify future research topics in this area.

A diffuse interface approach to Hele–Shaw flow

A diffuse interface model for the one-phase Hele–Shaw problem is derived from a gradient flow characterization due to Otto (1998 Arch. Rat. Mech. Anal. 141 63). The resulting dynamical model yields a generalized form of Darcy's law, and reduces to a degenerate version of the well-known Cahn–Hilliard equation. Formal asymptotics illustrate the connection to the classical Hele–Shaw free boundary problem. Some example computations are carried out to demonstrate the flexibility of the modelling framework.

Coherency Strain-Assisted Equilibrium Segregation at Heterophase Interfaces

Theoretical work in the field of interfacial segregation is focused mainly on the sharp interface approach beginning with the Gibbs' introduction of the dividing surface. In this paper a continuum approach to the problem of interfacial segregation, which deals with continuous distributions of quantities at interfaces, is developed. The current study concentrates on heterophase interfaces in alloys as opposed to grain boundaries or antiphase domain boundaries. The important effect of coherency strains is taken into account. The Gibbs adsorption theorem is discussed in the framework of a continuum approach. A comparison is made between the sharp (Gibbsian) and diffuse interface approaches.

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

A diffuse interface model for microstructural evolution in elastically stressed solids

We present a diffuse interface (DI) model for capturing microstructure formed during the coarsening of a two dimensional, elastically stressed binary alloy. The DI model is based on a generalized Cahn–Hilliard free energy; evolution occurs to lower the free energy. Using a matched asymptotic expansion, we show that the DI model converges to a well-studied sharp interface system as the thickness of the diffuse interface approaches zero. Numerical simulations confirm this equivalence. We develop pseudo-spectral numerical methods to solve the DI system and we carefully investigate the dependence of results on numerical parameters. The DI model is used to follow microstructural evolution through topological transitions such as particle merging and vanishing. We show that in isotropic media, elastic inhomogeneity may lead to interesting topology changes such as a reversal of the roles of the precipitate and matrix phases.

Diffuse interface approach to crystal nucleation in glasses

A diffuse interface theory of nucleation is applied to the crystallization of various glasses. This approach eliminates the problem of ‘anomalous’ nucleation prefactors and offers a method of distinguishing between homogeneous and bulk heterogeneous processes.

A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface

A method is presented to solve two-phase problems involving a material quantity on an interface. The interface can be advected, stretched, and change topology, and material can be adsorbed to or desorbed from it. The method is based on the use of a diffuse interface framework, which allows a simple implementation using standard finite-difference or finite-element techniques. Here, finite-difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Interfacial flow with soluble surfactants is used as an example of the application of the method, and several test cases are presented demonstrating its accuracy and convergence.