This paper reviews methods and applications of the phase field technique, one of the fastest growing areas in computational materials science. The phase field method is used as a theory and computational tool for predictions of the evolution of arbitrarily shaped morphologies and complex microstructures in materials. In this method, the interface between two phases (e.g. solid and liquid) is treated as a region of finite width having a gradual variation of different physical quantities, i.e. it is a diffuse interface model. An auxiliary variable, the phase field or order parameter φ( x), is introduced, which distinguishes one phase from the other. Interfaces are identified by the variation of the phase field. We begin with presenting the physical background of the phase field method and give a detailed thermodynamical derivation of the phase field equations. We demonstrate how equilibrium and non-equilibrium physical phenomena at the phase interface are incorporated into the phase field methods. Then we address in detail dendritic and directional solidification of pure and multicomponent alloys, effects of natural convection and forced flow, grain growth, nucleation, solid-solid phase transformation and highlight other applications of the phase field methods. In particular, we review the novel phase field crystal model, which combines atomistic length scales with diffusive time scales. We also discuss aspects of quantitative phase field modeling such as thin interface asymptotic analysis and coupling to thermodynamic databases. The phase field methods result in a set of partial differential equations, whose solutions require time-consuming large-scale computations and often limit the applicability of the method. Subsequently, we review numerical approaches to solve the phase field equations and present a finite difference discretization of the anisotropic Laplacian operator.

More and more countries are showing a significant slowdown in the number of new COVID-19 infections due to effective governmentally instituted lockdown and social distancing measures. We have analyzed the growth behavior of the top 25 most affected countries by means of a local slope analysis and found three distinct patterns that individual countries follow depending on the strictness of the lockdown protocols: rise and fall, power law, or logistic. For countries showing power law growth we have determined the scaling exponents. For countries that showed a strong slowdown in the rate of infections we have extrapolated the expected saturation of the total number of infections and the expected final date. Three different extrapolation methods (logistic, parabolic, and cutoff power law) were used. All methods agree on the order of magnitude of saturation and end dates. Global infection rates are analyzed with the same methods. The relevance and accuracy of these extrapolations is discussed.

PACS 64.60.A-Specific approaches applied to studies of phase transitions PACS 64.70.Hz-Solid-vapor transitions PACS 92.40.ed-Snow Abstract.-A Monte Carlo algorithm to simulate the isothermal recrystallization process of snow is presented. The snow metamorphism is approximated by two mass redistribution processes, surface diffusion and sublimation-deposition. The algorithm is justified and its parametrization is determined. The simulation results are compared to experimental data, in particular, the temporal evolution of the specific surface area and the ice thickness. We find that the two effects of surface diffusion and sublimation-deposition can accurately model many aspects of the isothermal metamorphism of snow. Furthermore, it is shown that sublimation-deposition is the dominant contribution for temperatures close to the melting point, whereas surface diffusion dominates at temperatures far below the melting point. A simple approximation of gravitational compaction is implemented to simulate density change.

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[1] Recently, Seybold et al. (2007) proposed a reduced complexity model which simulates the process of delta formation on geological time scales. It includes subaerial and subaqueous growth in a three-dimensional framework. In this paper we apply this model to the formation of a river-dominated delta and compare the model dynamics with observations of the formation of the Balize Lobe of the Mississippi River Delta. The model generates both subaerial and subaqueous channels and lateral levee formations as well as a profile morphology with steep drop-offs and a flat delta surface which is similar to natural ones. We show that the dimensionless parameters of the model may be consistently rescaled to match the Balize Lobe. This means that after rescaling the water flows, the subaerial geometry and time, the deposited (subaqueous) lobe volume, the sediment and water flows, the age, as well as the sediment capture ratio match the observed data. Finally, we use detrended fluctuation analysis to show that the modeled long-term dynamics of the delta formation process shows a complex temporal correlation structure. A characteristic time scale separates periods of consistent delta growth by gradual sediment deposition at the mouths of distributary channels from periods during which random large-scale channel avulsions lead to rapid change and the formation of new channels and subaqueous-dominated deposition.