Three-dimensional coarsening dynamics of a conserved, nematic liquid crystal-isotropic fluid mixture

We present a numerical investigation of the three-dimensional coarsening dynamics of a nematic liquid crystal-isotropic fluid mixture using a conserved phase field model. The model is a coupled system for a generalized Cahn–Hilliard equation for the order parameter ϕ, related to the volume fraction of the nematic component, and a simplified de Gennes–Prost evolution equation for the director field n, which describes the mean orientation of the rigid rod-like, liquid crystal molecules. We find that, as in the two-dimensional system, the orientational distortion induced by interfacial anchoring has profound effects both on the morphology and the coarsening rate. However, we identify significant differences in the three-dimensional and two-dimensional coarsening processes. In particular, we find a remarkable, new 3-stage late coarsening process with markedly different coarsening rates in the three-dimensional bicontinuous phase separation with homeotropic anchoring, unseen in the two-dimensional system

Computational phase-field modeling

Phase-field modeling is emerging as a promising tool for the treatment of problems with interfaces. The classical description of interface problems requires the numerical solution of partial differential equations on moving domains in which the domain motions are also unknowns. The computational treatment of these problems requires moving meshes and is very difficult when the moving domains undergo topological changes. Phase-field modeling may be understood as a methodology to reformulate interface problems as equations posed on fixed domains. In some cases, the phase-field model may be shown to converge to the moving-boundary problem as a regularization parameter tends to zero, which shows the mathematical soundness of the approach. However, this is only part of the story because phase-field models do not need to have a moving-boundary problem associated and can be rigorously derived from classical thermomechanics. In this context, the distinguishing feature is that constitutive models depend on the variational derivative of the free energy. In all, phase-field models open the opportunity for the efficient treatment of outstanding problems in computational mechanics, such as, the interaction of a large number of cracks in three dimensions, cavitation, film and nucleate boiling, tumor growth or fully three-dimensional air-water flows with surface tension. In addition, phase-field models bring a new set of challenges for numerical discretization that will excite the computational mechanics community

Simulação tridimensional adaptativa da separação das fases de uma mistura bifásica usando a equação de Cahn-Hilliard

Simulamos a separação dos componentes de uma mistura bifásica com a equação de Cahn-Hilliard. Esta equação contém intrincados termos não lineares e derivadas de alta ordem. Além disso, a delgada região de transição entre os componentes da mistura requer muita resolução. Assim, determinar a solução numérica da equação de Cahn-Hilliard não é uma tarefa fácil, principalmente em três dimensões. Conseguimos a resolução exigida no tempo usando uma discretização semi-implícita de segunda ordem. No espaço, obtemos a precisão requerida utilizando malhas refinadas localmente com a estratégia AMR. Essas malhas se adaptam dinamicamente para recobrir a região de transição. O sistema linear proveniente da discretização é solucionado por intermédio de técnicas multinível-multigrid

An energy-stable time-integrator for phase-field models

We introduce a provably energy-stable time-integration method for general classes of phase-field models with polynomial potentials. We demonstrate how Taylor series expansions of the nonlinear terms present in the partial differential equations of these models can lead to expressions that guarantee energy-stability implicitly, which are second-order accurate in time. The spatial discretization relies on a mixed finite element formulation and isogeometric analysis. We also propose an adaptive time-stepping discretization that relies on a first-order backward approximation to give an error-estimator. This error estimator is accurate, robust, and does not require the computation of extra solutions to estimate the error. This methodology can be applied to any second-order accurate time-integration scheme. We present numerical examples in two and three spatial dimensions, which confirm the stability and robustness of the method. The implementation of the numerical schemes is done in PetIGA, a high-performance isogeometric analysis framework