Arkady Vilenkin

A traveling wave solution for coupled surface and grain boundary motion

Abstract We study the coupled motion of a grain boundary in a bicrystal which is attached at a “groove root” to an exterior surface which evolves according to surface diffusion in a “quarter loop” geometry (Dunn et al., Trans. Am. Inst. Min. Engrs. 185 (1949) 708), and prove the existence of a unique traveling wave solution for various partially linearized formulations. Our results complement and complete the earlier analysis by Mullins (Acta metall. 6 (1958) 414) where the groove root and the velocity as a function of groove depth were determined. We demonstrate that the net effect of the coupling to the exterior surface is to reduce the overall velocity relative to a freely moving grain boundary by a factor which is small (≈3.5%) for typical parameter values. For extreme values of the parameters, the coupling may cause an increase in the overall grain boundary velocity. No “jerky” or “stop and go” motion is predicted by our solution, and we conjecture why this is so.

Large Deviations of Surface Height in the Kardar-Parisi-Zhang Equation

Using the weak-noise theory, we evaluate the probability distribution P(H,t) of large deviations of height H of the evolving surface height h(x,t) in the Kardar-Parisi-Zhang equation in one dimension when starting from a flat interface. We also determine the optimal history of the interface, conditioned on reaching the height H at time t. We argue that the tails of P behave, at arbitrary time t>0, and in a proper moving frame, as -lnP∼|H|^{5/2} and ∼|H|^{3/2}. The 3/2 tail coincides with the asymptotic of the Gaussian orthogonal ensemble Tracy-Widom distribution, previously observed at long times.

The role of segregation in diffusion-induced grain boundary migration

The problem of grain boundary motion in the diffusion field of a solute is formulated for the case of infinitely fast diffusion along a straight boundary. The steady state solution suggests that (de)alloying occurs by two different modes, namely: the solute diffusion through the stationary boundary to the bulk, or by diffusion-induced grain boundary migration (DIGM). The transition from one mode to another depends on the grain boundary segregation coefficient. The result enables an assessment of the relative importance of different possible driving forces. When the equilibrium concentrations of the bulk solute with the external gas is low, the entropy of mixing is the leading driving force. DIGM does not occur in isotope solution because the solute atom does not segregate to the boundary. Based on this theory, we construct the phase diagram in the plane of the (gas/bulk) equilibrium concentration vs the segregation coefficient, representing the transition from DIGM to alloying via stationary boundaries.

Interaction of Solute Impurity with Grain Boundary: The Impurity Drag Effect

An equation of grain boundary motion in a binary polycrystal is derived. The derivation is based on minimization of free energy of the total systems. The equation takes into account an impurity segregation at the grain boundary, grain boundary curvature and energy.As an example, we apply this equation to the analysis of the impurity drag effect problem. It is shown, that the sign of the impurity effect on grain boundary velocity (delay or acceleration) does not depend on kinetic coefficients. The sign of the effect is determined by a thermodynamic function which combines the grain boundary segregation coefficient, the derivative of grain boundary energy with respect to absorbed impurity concentration, and the derivative of bulk free energy with respect to bulk impurity concentration.

Survival of interacting diffusing particles inside a domain with absorbing boundary.

Suppose that a d-dimensional domain is filled with a gas of (in general, interacting) diffusive particles with density n_{0}. A particle is absorbed whenever it reaches the domain boundary. Employing macroscopic fluctuation theory, we evaluate the probability P that no particles are absorbed during a long time T. We argue that the most likely gas density profile, conditional on this event, is stationary throughout most of the time T. As a result, P decays exponentially with T for a whole class of interacting diffusive gases in any dimension. For d=1 the stationary gas density profile and P can be found analytically. In higher dimensions we focus on the simple symmetric exclusion process (SSEP) and show that -lnP≃D_{0}TL^{d-2}s(n_{0}), where D_{0} is the gas diffusivity, and L is the linear size of the system. We calculate the rescaled action s(n_{0}) for d=1, for rectangular domains in d=2, and for spherical domains. Near close packing of the SSEP s(n_{0}) can be found analytically for domains of any shape and in any dimension.

Grain Boundary Segregation and Grain Boundary Wetting

We give an explanation of the phenomenon of a liquid film penetration along grain boundary (GB) using a kinetic mechanism that does not rely a wet st at of the GB. The mechanism is as follows. Fast diffusion of the liquid’s atoms (LA) along a s tr ight stationary GB induces its instability in concentration field of the LA and results in displace ment of the GB. A high concentration of the LA can then be found in the area has been that swept by the GB. Due to its composition, this area has low melting temperature, and becomes liquid. The necessary thermodynamic condition for such a scenario is * lim ) ( T kc Tm < , where m T is the solid phase melting temperature depends on the LA concentration in the bulk, * T is the temperature in the experiment, k is the GB segregation coefficient, lim c is solubility limit of the LA in the solid matrix at * T T = . We assume fast mass transport in the melt, and formulate a set of kinetic equations describing the process. Our solution describes the linear growth of a thin channel filled with liquid.

Nonequilibrium steady state of a weakly-driven Kardar–Parisi–Zhang equation

We consider an infinite interface in

Scaling and Self-Similarity in an Unforced Flow of Inviscid Fluid Trapped Inside a Viscous Fluid in a Hele-Shaw Cell

d>2

Grain Boundary Migration with Thermal Grooving Effects: A Numerical Approach

dimensions, governed by the Kardar-Parisi-Zhang (KPZ) equation with a weak Gaussian noise which is delta-correlated in time and has short-range spatial correlations. We study the probability distribution of the (properly shifted) interface height

Geometric Interfacial Motion: Coupling Surface Diffusion and Mean Curvature Motion

H