James A. Warren

A continuum model of grain boundaries

Abstract A two-dimensional frame-invariant phase field model of grain boundaries is developed. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. With an appropriate choice of functional dependencies, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275]. In addition, dynamic (one-dimensional) solutions are presented, showing rotation of a small grain between two pinned grains and the shrinkage and rotation of a circular grains embedded in a larger crystal.

Extending phase field models of solidification to polycrystalline materials

Abstract We present a two-dimensional phase field model of grain boundary statics and dynamics. We begin with a brief description and physical motivation of the crystalline phase field model. The description is followed by characterization and analysis of several microstructural implications: the grain boundary energy as a function of misorientation, the liquid–grain–grain triple junction behavior, the wetting condition for a grain boundary and stabilized widths of intercalating phases at these boundaries, and evolution of a polycrystalline microstructure by solidification and impingement, followed by both grain boundary migration and grain rotation. Simulations that demonstrate these implications are presented, with a description of the numerical methods that were used to obtain them.

FiPy: Partial Differential Equations with Python

Many existing partial differential equation solver packages focus on the important, but arcane, task of numerically solving the linearized set of algebraic equations that result from discretizing a set of PDEs. Many researchers, however, need something higher level than that.

Vector-valued phase field model for crystallization and grain boundary formation

Abstract We propose a new model for calculation of the crystalliztation and impingement of many particles with differing orientations. Based on earlier phase field models, a vector order parameter is introduced, and thus orientation of crystal/disordered interfaces can be determined relative to a crystalline frame. This model improves upon previous attempts to describe this phenomenon, as it requires far fewer equations of motion, and is energetically invariant under rotations. In this report a one-dimensional simulation of the model will be presented along with preliminary investigations of two-dimensional simulations.

Grain boundaries exhibit the dynamics of glass-forming liquids

Polycrystalline materials are composites of crystalline particles or “grains” separated by thin “amorphous” grain boundaries (GBs). Although GBs have been exhaustively investigated at low temperatures, at which these regions are relatively ordered, much less is known about them at higher temperatures, where they exhibit significant mobility and structural disorder and characterization methods are limited. The time and spatial scales accessible to molecular dynamics (MD) simulation are appropriate for investigating the dynamical and structural properties of GBs at elevated temperatures, and we exploit MD to explore basic aspects of GB dynamics as a function of temperature. It has long been hypothesized that GBs have features in common with glass-forming liquids based on the processing characteristics of polycrystalline materials. We find remarkable support for this suggestion, as evidenced by string-like collective atomic motion and transient caging of atomic motion, and a non-Arrhenius GB mobility describing the average rate of large-scale GB displacement.

Phase Field Theory of Heterogeneous Crystal Nucleation

The phase field approach is used to model heterogeneous crystal nucleation in an undercooled pure liquid in contact with a foreign wall. We discuss various choices for the boundary condition at the wall and determine the properties of critical nuclei, including their free energy of formation and the contact angle as a function of undercooling. For particular choices of boundary conditions, we may realize either an analog of the classical spherical cap model or decidedly nonclassical behavior, where the contact angle decreases from its value taken at the melting point towards complete wetting at a critical undercooling, an analogue of the surface spinodal of liquid-wall interfaces.

Modeling reactive wetting

Abstract When a liquid alloy spreads on a metal substrate, interdiffusion may result in partial dissolution of the substrate and/or formation of intermetallic phases. We investigate the former case and use a diffusion/fluid flow analysis to describe the change in shape of the liquid–solid interface and the resultant motion of the triple junction along the substrate. Matching of flux boundary conditions for the liquid surface and the liquid–solid interface at the triple junction lead to an expression for the speed of spreading. The governing equations, for a drop whose height is much smaller than its width, are solved numerically for a variety of conditions. The roles of chemical driving force, liquid–solid interface energy, and angles between the various boundaries are established. During rapid spreading, large variations in curvature of the liquid–solid interface near the triple junction can lead to experimentally measured apparent angles much different from those determined by capillarity.

Phase field modeling of electrochemistry. I. Equilibrium

A diffuse interface (phase field) model for an electrochemical system is developed. We describe the minimal set of components needed to model an electrochemical interface and present a variational derivation of the governing equations. With a simple set of assumptions: mass and volume constraints, Poissons equation, ideal solution thermodynamics in the bulk, and a simple description of the competing energies in the interface, the model captures the charge separation associated with the equilibrium double layer at the electrochemical interface. The decay of the electrostatic potential in the electrolyte agrees with the classical Gouy-Chapman and Debye-Hückel theories. We calculate the surface free energy, surface charge, and differential capacitance as functions of potential and find qualitative agreement between the model and existing theories and experiments. In particular, the differential capacitance curves exhibit complex shapes with multiple extrema, as exhibited in many electrochemical systems.

Simulation of the Cell to Plane Front Transition During Directional Solidification at High Velocity

Using the alloy phase-field method with a frozen temperature approximation, interface morphology and solute segregation patterns during directional solidification are examined near the high velocity (absolute stability) condition for planar growth. The dynamics of the breakdown of initially planar interfaces into cellular structures are shown. At sufficiently high solidification speed, a planar interface is reestablished after breakdown during the initial transient. The cell spacings, depths, tip temperatures, tip radii, and concentration patterns are determined as a function of solidification velocity. The presence of solute trapping is manifest in the variation of the degree of solute partitioning across the interfacial region with interface speed.

The phase-field method: simulation of alloy dendritic solidification during recalescence

An overview of the phase-field method for modeling solidification is given and results for nonisothermal alloy dendritic growth are presented. By defining a “phase-field” variable and a corresponding governing equation to describe the state (solid or liquid) in a material as a function of position and time, the diffusion equations for heat and solute can be solved without tracking the liquid-solid interface. The interfacial regions between liquid and solid involve smooth, but highly localized variations of the phase-field variable and the composition. Simple finite-difference techniques on a uniform mesh can be used to treat the evolution of complex growth patterns. However, large-scale computations are required. The method has been applied to a variety of problems, including thermally driven dendritic growth in pure materials, solute-driven isothermal dendritic growth in alloys, eutectic growth (all at high supercoolings or supersaturations), solute trapping at high velocity, and coarsening of liquid-solid mixtures. To include thermal effects in solute-driven dendritic growth in alloys, a simplified approach is presented here that neglects the spatial variation of temperature in the computational domain but provides for changes with time and thus includes recalescence. Growth morphologies and solute patterns in the liquid and solid obtained for several values of an imposed heat flux are compared to results for isothermal growth.