Exact result for the effective conductivity of a continuum percolation model

Journal ArticleA random two-dimensional checkerboard of squares of conductivities 1 and 8 in proportions p and 1 - p is considered. Classical duality implies that the effective conductivity obeys o* = V8 at p = 1/2. It is rigorously found here that to leading order as 8--0, this exact result holds for all p in the interval (1- pc,pc), where pc=0.59 is the site percolation probability, not just at p = 1/2. In particular, o*(p,8)=78+O (8), as 8 -- 0. which is argued to hold for complex 8 as well. The analysis is based on the identification of a "symmetric" backbone, which is statistically invariant under interchange of the components for any pE(1--pc,pc), like the entire checkerboard at p =1/2. This backbone is defined in terms of "choke points" for the current, which have been observed in an experiment

Lattice Boltzmann simulations of 3D crystal growth: Numerical schemes for a phase-field model with anti-trapping current

A lattice-Boltzmann (LB) scheme, based on the Bhatnagar-Gross-Krook (BGK) collision rules is developed for a phase-field model of alloy solidification in order to simulate the growth of dendrites. The solidification of a binary alloy is considered, taking into account diffusive transport of heat and solute, as well as the anisotropy of the solid-liquid interfacial free energy. The anisotropic terms in the phase-field evolution equation, the phenomenological anti-trapping current (introduced in the solute evolution equation to avoid spurious solute trapping), and the variation of the solute diffusion coefficient between phases, make it necessary to modify the equilibrium distribution functions of the LB scheme with respect to the one used in the standard method for the solution of advection-diffusion equations. The effects of grid anisotropy are removed by using the lattices D3Q15 and D3Q19 instead of D3Q7. The method is validated by direct comparison of the simulation results with a numerical code that uses the finite-difference method. Simulations are also carried out for two different anisotropy functions in order to demonstrate the capability of the method to generate various crystal shapes

Quantitative phase-field model for phase transformations in multi-component alloys

Phase-field modeling has spread to a variety of applications involving phase transformations. While the method has wide applicability, derivation of quantitative predictions requires deeper understanding of the coupling between the system and model parameters. The book highlights a novel phase-field model based on a grand-potential formalism allowing for an elegant and efficient solution to problems in phase transformations

TWO-DIMENSIONAL SIMULATION OF SOLIDIFICATION IN FLOW FIELD USING PHASE-FIELD MODEL|MULTISCALE METHOD IMPLEMENTATION

Numerous efforts have contributed to the study of phase-change problems for over a century|both analytical and numerical. Among those numerical approximations applied to solve phase-transition problems, phase-field models attract more and more attention because they not only capture two important effects, surface tension and supercooling, but also enable explicitly labeling the solid and liquid phases and the position of the interface. In the research of this dissertation, a phase-field model has been employed to simulate 2-D dendrite growth of pure nickel without a flow, and 2-D ice crystal growth in a high-Reynolds-number lid-driven-cavity flow. In order to obtain the details of ice crystal structures as well as the flow field behavior during freezing for the latter simulation, it is necessary to solve the phase-field model without convection and the equations of motion on two different scales. To accomplish this, a heterogeneous multiscale method is implemented for the phase-field model with convection such that the phase-field model is simulated on a microscopic scale and the equations of motion are solved on a macroscopic scale. Simulations of 2-D dendrite growth of pure nickel provide the validation of the phase-field model and the study of dendrite growth under different conditions, e.g., degree of supercooling, interface thickness, kinetic coefficient, and shape of the initial seed. In addition, simulations of freezing in a lid-driven-cavity flow indicate that the flow field has great effect on the small-scale dendrite structure and the flow eld behavior on the large scale is altered by freezing inside it

Modeling of drift-diffusion systems

We derive drift-diffusion systems describing transport processes starting from free energy and equilibrium solutions by a unique method. We include several statistics, heterostructures and cross diffusion. The resulting systems of nonlinear partial differential equations conserve mass and positivity, and have a Lyapunov function (free energy). Using the inverse Hessian as mobility, non-degenerate diffusivity matrices turn out to be diagonal, o