Geoffrey B. McFadden

A phase-field model for highly anisotropic interfacial energy

Abstract A computationally efficient phase-field model is developed for two-phase systems with anisotropic interfacial energy. The approach allows for anisotropies sufficiently high that the interface has corners or missing crystallographic orientations. The method employs a regularization that enforces local equilibrium at the corners and allows corners to be added or removed without explicitly tracking their location. Numerical simulations for various degrees of anisotropy were performed and they show excellent agreement with analytical equilibrium shapes and yield accurate time dependent solutions for a wide variety of initial conditions.

A phase-field model of solidification with convection

We develop a phase-field model for the solidification of a pure material that includes convection in the liquid phase. The model permits the interface to have an anisotropic surface energy, and allows a quasi-incompressible thermodynamic description in which the densities in the solid and liquid phases may each be uniform. The solid phase is modeled as an extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to derive the governing equations. We investigate the behavior of our model in two important simple situations corresponding to the solidification of a planar interface at constant velocity: density change flow and a shear flow. In the former case we obtain a non-equilibrium form of the Clausius–Clapeyron equation and investigate its behavior by both a direct numerical integration of the governing equations, and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parallel shear flow we are able to obtain an exact solution which allows us to investigate its behavior in the sharp interface limit, and for large values of the viscosity ratio.

The Contribution of Lipid Layer Movement to Tear Film Thinning and Breakup

PURPOSE To investigate whether the tear film thinning between blinks is caused by evaporation or by tangential flow of the tear film along the surface of the cornea. Tangential flow was studied by measuring the movement of the lipid layer. METHODS Four video recordings of the lipid layer of the tear film were made from 16 normal subjects, with the subjects keeping their eyes open for up to 30 seconds after a blink. To assess vertical and horizontal stretching of the lipid layer and underlying aqueous layer, lipid movement was analyzed at five positions, a middle position 1 mm below the corneal center, and four positions respectively 1 mm above, below, nasal, and temporal to this middle position. In addition, in 13 subjects, the thinning of the tear film after a blink was measured. RESULTS The total upward movement could be fitted by the sum of an exponential decay plus a slow steady drift; this drift was upward in 14 of 16 subjects (P = 0.002). Areas of thick lipid were seen to expand causing upward or downward drift or horizontal movement. The velocity of the initial rapid upward movement and the time constant of upward movement were found to correlate significantly with tear film thickness but not with tear-thinning rate. CONCLUSIONS Analysis indicated that the observed movement of the lipid layer was too slow to explain the observed thinning rate of the tear film. In the Appendix, it is shown that flow under a stationary lipid layer cannot explain the observed thinning rate. It is concluded that most of the observed tear thinning between blinks is due to evaporation.

Thermosolutal convection during directional solidification

During solidification of a binary alloy at constant velocity vertically upward, thermosolutal convection can occur if the solute rejected at the crystal-melt interface decreases the density of the melt. We assume that the crystal-melt interface remains planar and that the flow field is periodic in the horizontal direction. The time-dependent nonlinear differential equations for fluid flow, concentration, and temperature are solved numerically in two spatial dimensions for small Prandtl numbers and moderately large Schmidt numbers. For slow solidification velocities, the thermal field has an important stabilizing influence: near the onset of instability the flow is confined to the vicinity of the crystal-melt interface. Further, for slow velocities, as the concentration increases, the horizontal wavelength of the flow decreases rapidly — a phenomenon also indicated by linear stability analysis. The lateral in-homogeneity in solute concentration due to convection is obtained from the calculations. For a narrow range of solutal Rayleigh numbers and wavelengths, the flow is periodic in time.

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.

The effect of anisotropic crystal-melt surface tension on grain boundary groove morphology

Abstract The shape of a stationary solid-liquid interface in a temperature gradient near a grain boundary in a pure material is calculated for anisotropic crystal-melt surface tension and equal thermal conductivities of crystal and melt. Results are compared with those for the well-known problem of the two-dimensional equilibrium shape of a crystal. For small anisotropy, the resulting interface shapes have continuously turning tangents but differ in detail from the grain boundary groove shapes that have been calculated for isotropic surface tension. For larger anisotropy, the interface shapes have discontinuities in slope as a result of missing orientations; these missing orientations are the same as those that would be missing on the corresponding equilibrium interface shape. In cases where a normal to the grain boundary or to the macroscopic interface is in the range of missing orientations on the corresponding equilibrium shape, the groove shape may contain some of these orientations as well as having varifold surfaces. Detailed numerical results are presented for a surface tension with fourfold symmetry.

Numerical simulation of morphological development during ostwald ripening

Abstract A boundary integral technique is employed to determine the morphological evolution of a small number of particles during Ostwald ripening in two dimensions. The approach allows the bodies to change shape consistent with interparticle diffusional interactions and the interfacial concentrations as given by the Gibbs-Thomson equation. It is shown that the strong interparticle diffusional interactions which occur at small interparticle separations can induce significant motions of the centers of mass of the particles. Such motion is shown to be a strong function of the spatial distribution of particles. The generality of the mechanism responsible for the particle migration suggests that particle motion is a generic aspect of the ripening process at high volume fractions of coarsening phase. It was found that significant shape distortions of particles during ripening requires particle arrangements which induce significant diffusional screening of regions of interface. Through particle arrangements similar to those found in solid-liquid systems during liquid phase sintering, it is shown that the formation of regions of flat interface between particles is completely consistent with an Ostwald ripening mechanism.

Step bunching on a vicinal face of a crystal growing in a flowing solution

Abstract The effect of a parallel shear flow and anisotropic interface kinetics on the onset of (linear) instability during growth from a supersaturated solution is analyzed including perturbations in the flow velocity. 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 (negative shear) decreases interface stability. For large wavenumbers kx, the perturbed flow field can be neglected and a simple analytic approximation for the stability-instability demarcation is found. A shear flow counter to the step motion (positive shear) enhances stability and for sufficiently large shear rates (on the order of 1 s−1) the interface is morphologically stable. Alternatively, the approximate analysis predicts that the system is unstable if the solution flow velocity in the direction of the step motion at a distance(2kx)−1 from the interface exceeds the propagation rate vx of step bunches induced by the interface perturbations. The approximate results are applied to the growth of ADP and lysozyme. For sufficiently low supersaturations, the interface is stable for positive shear and unstable for negative shear. More generally, there is a critical negative shear rate for which the interface becomes unstable as the magnitude of the shear rate increases. For a range of growth conditions for ADP, the magnitude of this critical shear rate is2kxvx. Even shear rates due to natural convection may be sufficient to affect stability for typical growth conditions.

Effect of a forced Couette flow on coupled convective and morphological instabilities during unidirectional solidification

Abstract The effect of a forced Couette flow, parallel to a horizontal crystal-melt interface during directional solidification of an alloy of lead containing tin, on the onset of convective and morphological instabilities, is calculated numerically via a linear stability analysis. Such a flow does not affect perturbations with wave vectors perpendicular to the flow. For perturbations with wave vectors parallel to the flow, the onset of morphological instability is somewhat suppressed and thermosolutal convection is greatly suppressed. When instabilities occur, they are oscillatory and correspond to travelling waves. For values of the crystal growth velocity for which mixed morphological and convective modes occur, the presence of a forced flow produces sufficient decoupling to allow otherwise degenerate branches to be identified.

Thin interface asymptotics for an energy/entropy approach to phase-field models with unequal conductivities

Karma and Rappel [Phys. Rev. E 57 (1998) 4342] 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 [SIAM J. Appl. Math. 59 (1999) 2086]. 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 phase-field equations can be used to eliminate the anomalous terms that arise for unequal conductivities.