Quantitative Phase Field Model of Alloy Solidification

We present a detailed derivation and thin interface analysis of a phase-field model that can accurately simulate microstructural pattern formation for low-speed directional solidification of a dilute binary alloy. This advance with respect to previous phase-field models is achieved by the addition of a phenomenological "antitrapping" solute current in the mass conservation relation [A. Karma, Phys. Rev. Lett 87, 115701 (2001)]. This antitrapping current counterbalances the physical, albeit artificially large, solute trapping effect generated when a mesoscopic interface thickness is used to simulate the interface evolution on experimental length and time scales. Furthermore, it provides additional freedom in the model to suppress other spurious effects that scale with this thickness when the diffusivity is unequal in solid and liquid [R. F. Almgren, SIAM J. Appl. Math 59, 2086 (1999)], which include surface diffusion and a curvature correction to the Stefan condition. This freedom can also be exploited to make the kinetic undercooling of the interface arbitrarily small even for mesoscopic values of both the interface thickness and the phase-field relaxation time, as for the solidification of pure melts [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The performance of the model is demonstrated by calculating accurately for the first time within a phase-field approach the Mullins-Sekerka stability spectrum of a planar interface and nonlinear cellular shapes for realistic alloy parameters and growth conditions.Comment: 51 pages RevTeX, 5 figures; expanded introduction and discussion; one table and one reference added; various small correction

A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification

A fully-implicit numerical method based upon adaptively refined meshes for the simulation of binary alloy solidification in 2D is presented. In addition we combine a second-order fully-implicit time discretisation scheme with variable steps size control to obtain an adaptive time and space discretisation method. The superiority of this method, compared to widely used fully-explicit methods, with respect to CPU time and accuracy, is shown. Due to the high non-linearity of the governing equations a robust and fast solver for systems of nonlinear algebraic equations is needed to solve the intermediate approximations per time step. We use a nonlinear multigrid solver which shows almost h-independent convergence behaviour

Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Non-classical shocks to model oil-trapping

We consider a one-dimensional problem modeling two-phase flow in heterogeneous porous media made of two homogeneous subdomains, with discontinuous capillarity at the interface between them. We suppose that the capillary forces vanish inside the domains, but not on the interface. Under the assumption that the gravity forces and the capillary forces are oriented in opposite directions, we show that the limit, for vanishing diffusion, is not in general the optimal entropy solution of the hyperbolic scalar conservation law as in the first paper of the series \cite{NPCX}. A non-classical shock can occur at the interface, modeling oil-trapping

Boundary effects on non-equilibrium localized structures in spatially extended systems

A study of the effects of system boundaries on bistable front propagation in nonequilibrium reaction-diffusion systems is presented. Two model partial differential equations displaying bistable fronts, with distinct experimental motivations and mathematical structure, are examined in detail utilizing simulations and perturbation techniques. We see that propagating fronts in both models bounce, trap, pin, or oscillate at the boundary, contingent on the imposed boundary condition, initial front speed and distance from the boundary. The similarities in front boundary interactions in these two models is traced to the fact that they display the same front instability (Ising-Bloch bifurcation) that controls the speed of propagation. A simplified dynamical picture based on ordinary differential equations that captures the essential features of front motion described by the original partial differential equations, is derived and analyzed for both models. In addition to addressing experimentally important boundary effects, we establish the universality of the Ising-Bloch bifurcation. Useful analytical insights into perturbative analysis of reaction diffusion systems are also presented

General model for the kinetics of solute diffusion at solid-solid interfaces

Solute diffusion through solid-solid interfaces is paramount to many physical processes. From a modeling point of view, the discontinuities in the energy landscape at a sharp interface represent difficulties in predicting solute diffusion that, to date, have not been solved in a consistent manner across length scales. Using an explicit finite volume method, this work is the first to derive numerical solutions to the diffusion equations at a continuum level while including discrete variations in the energy landscape at a bicrystal interface. An atomic jump equation consistent with atomistic descriptions is derived and scaled up into a compendium of model interfaces: monolayer energy barriers, monolayer interfacial traps, multilayered traps, and heterogeneous interfaces. These can track solute segregation behavior and long-range diffusion effects. We perform simulations with data for hydrogen diffusion in structural metals, of relevance to the assessment of the hydrogen embrittlement phenomenon, and point defects in electronic devices. The approach developed represents an advancement in the mathematical treatment of solute diffusion through solid-solid interfaces and an important bridge between the atomistic and macroscopic modeling of diffusion, with potential applications in a variety of fields in materials science and physics

Lagrangian, Game Theoretic and PDE Methods for Averaging G-equations in Turbulent Combustion: Existence and Beyond

G-equations are popular level set Hamilton-Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly non-convex and nonlinear. In the absence of coercivity and convexity, PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging non-coercive, non-convex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.Comment: 69 page

Iterative construction of conserved quantities in dissipative nearly integrable systems

Integrable systems offer rare examples of solvable many-body problems in the quantum world. Due to the fine-tuned structure, their realization in nature and experiment is never completely accurate, therefore effects of integrability are observed only transiently. One way to surpass that is to couple nearly integrable systems to baths and driving: these will stabilize integrable effects up to arbitrary time, as encoded in the time dependent, and eventually, the stationary state of form of a generalized Gibbs ensemble. However, the description of such driven dissipative nearly integrable models is challenging and no exact analytical methods have been proposed so far. Here we develop an iterative scheme in which integrability breaking perturbations (baths) determine the most necessary conserved quantities to be added into a truncated generalized Gibbs ensemble description. Our scheme significantly reduces the complexity of the problem, paving the way for thermodynamic results.Comment: 10 pages, 5 figures (including Supplemental Material

Diffusion, Nucleation and Recombination in Confined Geometries

In this work we address several problems of surface physics which are all based on the transport mechanism of diffusion. Essential parts consider systems in which particles interact upon meeting, either by reaction or by nucleation. For the astrophysical problem of hydrogen recombination on interstellar dust, the previous analytical treatment is substantially improved upon: The parameter responsible for reaction is defined consistently, and in its calculation, we account for the nature of two-dimensional diffusion. Within several models that are also compared with each other, one obtains explicit results that excellently agree with Monte Carlo simulations. Special attention is given to the influence of the confined geometry, which is analytically examined and explained for a minimal model. We also deal with the role of disorder in the surface structure. A further section presents the closely related problem of nucleation at island edges, which, however, proves to be no longer tractable analytically even under moderate assumptions. Step growth under the effect of codeposited impurities is a technologically relevant process, which is examined in a one-dimensional random walk model. Using microscopic models for the resulting disorder on the surface, one finds interesting results that also bear implications for simulations. Those results concern the general influence on the speed and stability of step growth as well as the nature of boundary conditions in the corresponding continuum model. In the last part we thoroughly analyze a model to describe the diffusion field in desorption experiments, which can be made directly visible on a large scale and with real time resolution. The model agrees well with experimental data and hence constitutes an important step in understanding the fundamental processes involved. Based on this understanding, one can draw direct conclusions from the observation of a surface of complex morphology on its microscopic properties

Coupled Pore-to-Continuum Multiscale Modeling of Dynamic Particle Filtration Processes in Porous Media

Modeling particle transport and retention in porous media is important in fields such as hydrocarbon extraction, groundwater filtration, and membrane separation. While the continuum-scale (\u3e1 m) is usually of practical interest, pore-scale (1-100 μm) dynamics govern the transport and retention of particles. Therefore, accurate modeling of continuum-scale behavior requires an effective incorporation of pore-scale dynamics. Due to current computational limits however, the large spatial and temporal discrepancies of these scales prohibit modeling an entire continuum-scale system as a single pore-scale model. Even if a pore-scale model could incorporate every pore contained in a continuum-scale system, an upscaling scheme that coupled pore- and continuum-scale models should in principle be more efficient and achieve acceptable accuracy. In this work, a continuum-scale model for particle transport and retention has been developed using the concurrent coupling method. In the model, pore network models (PNMs) were embedded within continuum-scale finite difference grid blocks. As simulations progressed the embedded PNMs periodically provided their continuum-scale grid blocks with updated petrophysical properties. The PNMs used a Lagrangian particle tracking method to identify particle dispersion and retention coefficients. Any changes in permeability and porosity due to particle trapping were also determined. Boundary conditions for the PNM simulations were prescribed by fluid velocity and influent particle concentration information from the continuum-scale grid blocks. Coupling in this manner allowed for a dynamic understanding of how particle induced changes at the pore-scale impact continuum-scale behavior