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Interaction of vacancies with grain boundaries (GBs) is involved in many processes occurring in materials, including radiation damage healing, diffusional creep, and solid-state sintering. We analyze a model describing a set of processes occurring at a GB in the presence of a non-equilibrium, non-homogeneous vacancy concentration. Such processes include vacancy diffusion toward, away from, and across the GB, vacancy generation and absorption at the GB, and GB migration. Numerical calculations within this model reveal that the coupling among the different processes gives rise to interesting phenomena, such as vacancy-driven GB motion and accelerated vacancy generation/absorption due to GB motion. The key combinations of the model parameters that control the kinetic regimes of the vacancy-GB interactions are identified via a linear stability analysis. Possible applications and extensions of the model are discussed.e's comments, Comment: 35 pages, 9 figures. Minor revisions per referee's comments. Published in Acta Materialia. Supplementary file containing additional figures is available from first author (mcfadden@nist.gov)

The kinetic behavior of a phase field model of electrochemistry is explored for advancing (electrodeposition) and receding (electrodissolution) conditions in one dimension. We described the equilibrium behavior of this model in [J. E. Guyer, W. J. Boettinger, J.A. Warren, and G. B. McFadden, ``Phase field modeling of electrochemistry I: Equilibrium'', cond-mat/0308173]. We examine the relationship between the parameters of the phase field method and the more typical parameters of electrochemistry. We demonstrate ohmic conduction in the electrode and ionic conduction in the electrolyte. We find that, despite making simple, linear dynamic postulates, we obtain the nonlinear relationship between current and overpotential predicted by the classical ``Butler-Volmer'' equation and observed in electrochemical experiments. The charge distribution in the interfacial double layer changes with the passage of current and, at sufficiently high currents, we find that the diffusion limited deposition of a more noble cation leads to alloy deposition with less noble species., Comment: v3: To be published in Phys. Rev. E v2: Attempt to work around turnpage bug. Replaced color Fig. 4a with grayscale 13 pages, 7 figures in 10 files, REVTeX 4, SIunits.sty, follows cond-mat/0308173

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, Poisson's 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-H\"uckel theories. We calculate the surface 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., Comment: v3: To be published in Phys. Rev. E v2: Added link to cond-mat/0308179 in References 13 pages, 6 figures in 15 files, REVTeX 4, SIUnits.sty. Precedes cond-mat/0308179

We present the first application of phase field modeling to electrochemistry. A free energy functional that includes the electrostatic effect of charged particles leads to rich interactions between concentration, electrostatic potential, and phase stability. The present model, explored for the equilibrium planar interface, properly predicts the charge separation associated with the equilibrium double layer at the electrochemical interface and its extent in the electrolyte as a function of electrolyte concentration, as well as the form expected of surface energy vs. potential ("electrocapillary") curves., Comment: 14 pages, 10 figures

We have assembled an experimental setup based on Michelson interferometry with the growing crystal surface as one of the reflective surfaces. The crystallization part of the device allows optical monitoring of a face of a crystal growing at temperature stable within 0.05 C in a flow of solution of controlled direction and speed. The reference arm of the interferometer contains a liquid crystal element that allows controlled shifts of the phase of the interferograms. We employ an image-processing algorithm, which combines five images with a pi/2 phase difference between each pair of images. The images are transferred to a computer by a camera capable of capturing 60 frames per second. The device allows data collection on surface morphology and kinetics during the face layers growth over a relatively large area (approximately 4 sq. mm) in situ and in real time during growth. The estimated depth resolution of the phase shifting interferometry is approximately 50 Angstroms. The data will be analyzed in order to reveal and monitor step bunching during the growth process. The crystal chosen as a model for study in this work is KH2PO4 (KDP). This optically non-linear material is widely used in frequency doubling applications. There have been a number of studies of the kinetics of KDP crystallization that can serve as a benchmark for our investigations. However, so far, systematic quantitative characteristics of step interaction and bunching are missing. We intend to present our first quantitative results on the onset, initial stages and development of instabilities in moving step trains on vicinal crystal surfaces at varying supersaturation, flow rate, and flow direction. Behavior of a vicinal face growing from solution flowing normal to the steps and periodically changing its direction in time was considered theoretically. It was found that this oscillating flow reduces both stabilization and destabilization effects resulted from the unidirectional solution flow directed up the step stream and down the step stream. This reduction of stabilization and destabilization comes from effective mixing which entangles the phase shifts between the spatially periodic interface perturbation and the concentration wave induced by this perturbation. Numerical results and simplified mixing criterion will be discussed.

Karma and Rapped recently developed a new sharp interface asymptotic analysis of the phase-field equations that is especially appropriate for modeling dendritic growth at low undercoolings. Their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis, and has the added advantage that interfacial kinetic effects can also be eliminated. However, their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases; when applied to a standard phase-field model with unequal conductivities, anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp-interface solidification models, as discussed further by Almgren. In this paper we apply their asymptotic methodology to a generalized phase-field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities. The additional degrees of freedom associated with the generalized phased-field equations can be chosen to eliminate the anomalous terms that arise for unequal conductivities.

In previous work, we found approximate solutions for paraboloids having perturbations with four-fold axial symmetry in order to model dendritic growth in cubic materials. These solutions provide self-consistent corrections through second order in a shape parameter e to the Peclet number-supercooling relation of the Ivantsov solution. The parameter e is proportional to the amplitude of the four-fold correction to the dendrite shape, as measured from the Ivantsov paraboloid of revolution. We calculate e by comparing the dendrite tip shape to the portion of the equilibrium shape near the growth direction, (001), for anisotropic surface free energy, where the ni are components of the unit normal of the crystal surface. This comparison results in epsilon = -2(epsilon 4), independent of the Peclet number. From the experimental value of epsilon 4, we find epsilon approximately 0.011, in good agreement with the measured value epsilon approximately 0.008 of LaCombe et al.

Step bunching results in striations even at relatively early stages of its development and in inclusions of mother liquor at the later stages. Therefore, eliminating step bunching is crucial for high crystal perfection. At least 5 major effects causing and influencing step bunching are known: (1) Basic morphological instability of stepped interfaces. It is caused by concentration gradient in the solution normal to the face and by the redistribution of solute tangentially to the interface which redistribution enhances occasional perturbations in step density due to various types of noise; (2) Aggravation of the above basic instability by solution flowing tangentially to the face in the same directions as the steps or stabilization of equidistant step train if these flows are antiparallel; (3) Enhanced bunching at supersaturation where step velocity v increases with relative supersaturation s much faster than linear. This v(s) dependence is believed to be associated with impurities. The impurities of which adsorption time is comparable with the time needed to deposit one lattice layer may also be responsible for bunching; (4) Very intensive solution flow stabilizes growing interface even at parallel solution and step flows; (5) Macrosteps were observed to nucleate at crystal corners and edges. Numerical simulation, assuming step-step interactions via surface diffusion also show that step bunching may be induced by random step nucleation at the facet edge and by discontinuity in the step density (a ridge) somewhere in the middle of a face. The corresponding bunching patterns produce the ones observed in experiment. The nature of step bunching generated at the corners and edges and by dislocation step sources, as well as the also relative importance and interrelations between mechanisms 1-5 is not clear, both from experimental and theoretical standpoints. Furthermore, several laws controlling the evolution of existing step bunches have been suggested, though unambiguous conclusions are still missing. Addressing these issues is the major goal of the present project. The theory addressing the above problem, experimental methods, several figures which include: (1) the spatial wave numbers at which the system is neutrally stable as a function of growth velocity for linear kinetics and supersaturation for nonlinear kinetics; (2) a schematic of the experiment of lysozyme crystal growing under conditions of natural convection; (3) fluctuations in time, t, of the normal growth rate, R(t), vicinal slope, p(t) and Fourier Spectra of R(t), discussions and conclusions are presented.

The authors review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. The authors discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, the authors address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, related computational approaches to describe interfacial phenomena, and related approaches describing fully-miscible fluids.

The effect of a parallel shear flow and anisotropic interface kinetics on the onset of instability during growth from a supersaturated solution is analyzed. The model used for anisotropy is based on the microscopic picture of step motion. A shear flow (linear Couette flow or asymptotic suction profile) parallel to the crystal solution interface in the same direction as the step motion decreases interface stability. A shear flow counter to the step motion enhances stability and for sufficiently large shear rates the interface is absolutely morphologically stable. For large wave numbers, the perturbed flow field can be neglected and a simple analytic approximation for the stability-instability demarcation is found.

For growth of a vicinal face at constant velocity, the effect of anisotropic interface kinetics on morphological stability is calculated for a binary alloy. The dependence of the interface kinetic coefficient on crystallographic orientation is based on the motion and density of steps. Anisotropic kinetics give rise to traveling waves along the crystal-melt interface, and can lead to a significant enhancement of morphological stability. The stability enhancement increases as the orientation approaches a singular orientation and as the solidification velocity increases. Shear flows interact with the traveling waves and, depending on the direction of the flow, may either stabilize or destabilize the interface. Specific calculations are carried out for germanium-silicon alloys.

The linear stability of a modulated Taylor-Couette system when the inner cylindrical boundary consists of a crystalline solid-liquid interface is considered. Both experimentally and in numerical calculations it is found that the two-phase system is significantly less stable than the analogous rigid-walled system for materials with moderately large Prandtl numbers. A numerical treatment based on Floquet theory is described, which gives results that are in good agreement with preliminary experimental findings. In addition, this instability is further examined by carrying out a formal asymptotic expansion of the solution in the limit of large Prandtl number. In this limit the Floquet analysis is considerably simplified, and the linear stability of the modulated system can be determined to leading order through a conventional stability analysis, without recourse to Floquet theory. The resulting simplified problem is then studied for both the narrow gap geometry and for the case of a finite gap. It is surprising that the determination of the linear stability of the two-phase system is considerably simpler than that of the rigid-walled system, despite the complications introduced by the presence of the crystal-melt interface.

During vertical directional solidification of a binary alloy at constant velocity, buoyancy-driven solutal convection may occur due to the solute gradient associated with the solidification process. This problem is further complicated if time-periodic forcing is considered, which is relevant to materials processing in a microgravity environment or as a means of dynamic control of flow instabilities. The effect of time-periodic modulation is studied by introducing a gravitational acceleration which is a sinusoidal function of time. The onset of solutal convection is treated by a stability analysis of the linearized governing equations and boundary conditions. Solutions are obtained numerically employing two distinct computational implementations of Floquet theory. Results are presented for materials with large Schmidt number, and an analysis for large frequency and large Schmidt number yields a more complete description of the behavior in this relevant limit.

A variety of problems in material science and fluid dynamics require the solution of Laplace's equation in multiply connected domains. Integral equation methods are natural candidates for such problems, since they discretize the boundary alone, require no special effort for free boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains in two space dimensions, regardless of shape. Current integral equation methods for the Dirichlet problem, however, require the solution of M independent problems of dimension N, where M is the number of boundary components and N is the total number of points in the discretization. In this paper, we present a new boundary integral equation approach, valid for both interior and exterior problems, which requires the solution of a single linear system of dimension N + M. We solve this system by making use of an iterative method (GMRES) combined with the last multipole method for the rapid calculation of the necessary matrix vector products. For a two-dimensional system with 200 components and 100 points on each boundary, we gain a speedup of a factor of 100 from the new analytic formulation and a factor of 50 from the fast multipole method. The resulting scheme brings large scale calculations in extremely complex domains within practical reach.

We examine the equilibrium shape and the dynamics of the morphological evolution of a coherent misfitting particle in an elastically anisotropic medium. No a priori assumptions are made on the possible particle morphologies; the particle evolves in a manner consistent with both the diffusion and elastic fields surrounding the particle and the thermodynamics of interfaces in stressed solids. Through these calculations we find the thermodynamically stable equilibrium shape of a misfitting particle. By thermodynamically stable we mean that all the equilibrium conditions, elastic, chemical and interfacial are satisfied simultaneously, in contrast to previous treatments which determined the equilibrium shape by minimizing the sum of the elastic and interfacial energies for certain classes of particle shapes. We find that the equilibrium particle morphologies are not simple geometric shapes, have fourfold symmetry and a continuously varying interfacial curvature with position along the interface. Furthermore, the results show that the times required for a particle to evolve to its equilibrium morphology are likely to be within those which are accessible experimentally. In addition, we develop a general approach for determining the equilibrium shape of a particle in an elastically stressed solid.

During vertical directional solidification of a binary alloy at constant velocity, buoyancy driven solutal convection may occur due to the solute gradient associated with the solidification process. This problem is further complicated if time periodic forcing is considered, which is relevant to materials processing in a microgravity environment or as a means of dynamic control of flow instabilities. The effect of time periodic modulation is studied by introducing a gravitational acceleration which is a sinusoidal function of time. The onset of solutal convection is treated by a stability analysis of the linearized governing equations and boundary conditions. Solutions are obtained numerically by employing two distinct computational implementations of Floquet theory. Results for materials with large Schmidt number are presented, and an analysis for large frequency and large Schmidt number yields a more complete description of the behavior in this relevant limit.

In a gravitational field, the opposing effects of components of different diffusivities, for example, temperature and solute, in the density profile in a fluid may produce convective instabilities that exhibit a broad range of dynamical behavior. The effect of time periodic vertical gravity modulation on the onset of these instabilities in an infinite horizontal layer with stress free boundaries is examined. This work is viewed as a first step in expanding previous results in solidification to the full problem of characterizing the effects of gravity modulation in thermosolutal convection during the directional solidification of binary alloys. Calculations carried out both with and without steady background acceleration are presented, the latter results being relevant to microgravity conditions.

The effect of compositionally-generated elastic stresses on the conditions for morphological instability during directional solidification of a dilute, ideal binary alloy is investigated using the Gibbs-Thomson equations for an elastically-stressed solid with zero surface stress in equilibrium with a liquid. It is found that these stresses lead to a small stabilization of the Mullins and Sekerka cellular mode of instability. The steady mode is stabilized by a stress-induced modification to the interfacial concentration of the solid, which in turn alters the amount of solute rejected to inhibit the growth of perturbations. The presence of elastic stresses could generate a new oscillatory instability which is most likely to be found in experiments near absolute stability for materials with segregation coefficients near unity and large solute expansion coefficients.

The effect of time-periodic vertical gravity modulation on the onset of thermosolutal convection in an infinite horizontal layer with stress-free boundaries is investigated using Floquet theory for the linear stability analysis. Situations for which the fluid layer is stably stratified in either the fingering or diffusive regimes of double-diffusive convection are considered. Results are presented both with and without steady background acceleration. Modulation may stabilize an unstable base solution or destabilize a stable base solution. In addition to synchronous and subharmonic response to the modulation frequency, instability in the double diffusive system can occur via a complex conjugate mode. In the diffusive regime, where oscillatory onset occurs in the unmodulated system, regions of resonant instability occur and exhibit strong coupling with the unmodulated oscillatory frequency. The response to modulation of the fundamental instability of the unmodulated system is described both analytically and numerically; in the double-diffusive system this mode persists under subcritical conditions as a high-frequency lobe.

A new phase field model is described which models isothermal phase transitions between ideal binary alloy solution phases. Equations are developed for the temporal and spatial variation of the phase field, which describes the identity of the phase, and of the composition. An asymptotic analysis, as the gradient energy coefficient of the phase field becomes small, was conducted. From the analysis, it is shown that the model recovers classical sharp interface models of this situation when the interfacial layers are thin, and they show how to relate the parameters appearing in the phase field model to material and growth parameters in real systems. Further, three stages of temporal evolution are identified: the first corresponding to interfacial genesis which occurs very rapidly; the second to interfacial motion controlled by the local energy difference across the interface and diffusion; the last taking place on a long time scale in which curvature effects are important and which correspond to Ostwald ripening. The results of the numerical calculations are presented.

During directional solidification of a binary alloy at constant velocity in the vertical direction, morphological and convective instabilities may occur due to the temperature and solute gradients associated with the solidification process. The effect of time-periodic modulation (vibration) is studied by considering a vertical gravitational acceleration which is sinusoidal in time. The conditions for the onset of solutal convection are calculated numerically, employing two distinct computational procedures based on Floquet theory. In general, a stable state can be destabilized by modulation and an unstable state can be stabilized. In the limit of high frequency modulation, the method of averaging and multiple-scale asymptotic analysis can be used to simplify the calculations.