The Cahn-Hilliard equation is very important in materials science. It is a conservation law which describes important qualitative behaviors of two-phase systems, namely, the phase separation process in a binary alloy. This phenomenon can be observed, for instance, when the alloy is cooled down sufficiently. One then observes a partial nucleation (i.e., the apparition of nucleides in the materia...

A description of the short time behavior of solutions of the Allen-Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki [9] in spatial dimension n = 2 to arbitrary dimensions. Resumé On étudie le comportem...

Computational scientific simulations have long used parallel computers to increase their performance. Recently graphics cards have been utilised to provide this functionality. GPGPU APIs such as NVidia’s CUDA can be used to harness the power of GPUs for purposes other than computer graphics. GPUs are designed for processing two-dimensional data. In previous work we have presented several two-di...

The Cahn-Hilliard equation is a fourth-order parabolic partial differential equation that is one of the leading models for the study of phase separation in isothermal, isotropic, binary mixtures, such as molten alloys. When a spatially homogeneous alloy is rapidly quenched in a physical experiment, a fine-grained decomposition into two distinct phases is frequently observed; this phenomenon is ...

In this article we formulate a numerical method for the simulation of dealloying of a binary alloy by the selective removal of one component via electrochemical dissolution such that there is phase separation of the other component at the solid/electrolyte interface. The evolution of the interface is modelled by a forced mean curvature flow strongly coupled to the solution of a Cahn-Hilliard eq...

Eutectic freeze crystallisation is a promising way of purifying water for it may require less energy than other methods. In order to simulate such process, phase field models as Allen-Cahn and Cahn-Hilliard can be used. this paper, port-Hamiltonian formulation the equations used coupled heat conduction, which allows thermodynamically consistent system written with help entropy functional. secon...

We prove a weak rate of convergence fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and backward Euler time. Compared Allen--Cahn type partial differential equation, error analysis here much more sophisticated due to presence unbounded operator front nonlinear term. To address such issues, novel direct approac...

The dynamics of the Cahn-Hilliard equations for a ternary alloy are studied during the phase separation process. The results show a primary spinodal decomposition followed by coarsening and then possibly a secondary spinodal decomposition.

Nonirradiated B-lymphocyte-deficient CBA/N mice given T6T6 chromosome-marked normal CBA/CaHN spleen cells became lymphoid chimeras exhibiting donor-type mitoses. Normal CBA/CaHN recipients did not exhibit significant numbers of donor-type mitoses. The lymphoid cell chimerism in the CBA/N host appeared in spleen, lymph nodes, and Peyer's patches, but not in marrow or thymus. Stimulation of CBA/N...

In this second part of the series, we focus on approximating the Hele-Shaw problem via the Cahn-Hilliard equation ut + ∆(ε∆u − εf(u)) = 0 as ε ↘ 0. The primary goal of this paper is to establish the convergence of the solution of the fully discrete mixed finite element scheme proposed in [21] to the solution of the Hele-Shaw (Mullins-Sekerka) problem, provided that the HeleShaw (Mullins-Sekerka...