Nonlocal Phase-field Models Research Articles

Morphological evolution and migration of void in bi-piezoelectric interface based on nonlocal phase field method

This paper reports the result of investigation into the morphological evolution and migration of void in bi-piezoelectric material interface by utilizing nonlocal phase field model and finite element method (FEM), where the small scale effect containing the long-range forces among atoms is considered. The nonlocal elastic strain energy and the nonlocal electric energy around the void are firstly calculated by the finite element method. Then based on the finite difference method (FDM), the thermodynamic equilibrium equation containing the surface energy and anisotropic diffusivity is solved to simulate the morphological evolution and migration of elliptical void in bi-piezoelectric films interface. Results show that the way of load condition plays a significant role in the evolution process, and the boundary of void's long axis gradually collapses toward the center of ellipse. In addition, the evolutionary speed of left boundary gradually decreases with scale effect coefficient growth. This work can provide references for the safety evaluation of piezoelectric materials in micro electro mechanical system.

The evolution of void defects in metallic films based on a nonlocal phase field model

In this paper, a nonlocal phase field method is presented to solve anisotropic diffusion-driven morphological evolution and migration of void defects in finite metallic film interconnects by utilizing a nonlocal phase field model considering a small scale effect. The nonlocal elastic theory is used to describe a small scale effect on the morphological evolution and migration of the void. In example calculations, the effects of the stress field, the electric field, and the anisotropic diffusion characteristic on the evolution of void defects in finite metallic film interconnects are described and discussed. The result in comparison with literature shows that the small scale effect based on nonlocal elastic model induces the migration diffusion of the crack tip to decrease.

Influences on Pattern Formation During Non-Isothermal Phase Separation in Local and Nonlocal Phase-Field Models

The classical phase-field model represents a coupling of an Allen-Cahn type nonlinear equation with a standard diffusion equation and has been proposed to describe uniform phase separation in a pure substance. This research considers an extension of that model which incorporates a more accurate approximation to the diffuse interface between states of matter through the use of a nonlocal operator. Results of simulations under this model are compared with those of the classical model in order to understand the effects of the nonlocal contribution. Attention has been given to the behavior of the underlying temperature field during early phase separation. We use the tools of computational homology to quantitatively compare patterns in the phase field under both models. Simulations show that the complicated patterns in the phase field persist longer during the solidification process in the nonlocal extension of the classical model.

A semi-implicit spectral method for stochastic nonlocal phase-field models

The classical phase-field model has been introduced as a model fornon-isothermal phase separation processes in materials. In order to overcomesome of the model's shortcomings, a variety of extensions have recently beenproposed which include nonlocal interactions as well as stochastic noise terms.In this paper, we study these extensions from a functional-analytic and numericalpoint of view. More precisely, we present a functional-analytic framework forestablishing existence, uniqueness, and qualitative dynamical results,and then propose a spectral method for solving stochastic nonlocal phase-fieldmodels. In particular, we establish convergence properties of our method and studythe effects of noise regularity and of the nonlocal interaction term on theseconvergence properties. Finally, numerical studies relating to the associated phaseseparation process are presented.

A nonlocal phase-field model with nonconstant specific heat

We prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.

A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries

Ferroelectric perovskites are used in various transducer, memory and optical applications due to their attractive properties. In these applications, the ferroelectric materials often have complex geometries and electrode arrangements, and are subjected to domain switching. Therefore, it is important to understand the domain pattern that forms in these geometries. However, models of domain evolution often make assumptions like periodicity and complete shielding which make them unrealistic for these applications. In this paper, we develop a phase-field approach that can study domain patterns and domain evolution in complex geometries with no a priori assumption on geometry, electrode arrangement, and dielectric properties. The key idea is a boundary element method to resolve the electrostatic fields. We illustrate the method by examining the closure domains that form at a free surface and domain switching under cyclic electric field in a device with interdigitated electrodes.

Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions

In this paper a nonlocal phase-field model for non-isothermal phase transitions with a non-conserved order parameter is studied. The paper extends recent investigations to the non-isothermal situation, complementing results obtained by H. Gajewski for the non-isothermal case for conserved order parameters in phase separation phenomena. The resulting field equations studied in this paper form a system of integro-partial differential equations which are highly nonlinearly coupled. For this system, results concerning global existence, uniqueness and large-time asymptotic behaviour are derived. The main results are proved using techniques that have been recently developed by P. Krejčí and the authors for phase-field systems involving hysteresis operators.

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.