Exact results for a meniscus in a three-phase system within an SOS-type approximation

The shape of a meniscus of one phase between two others is studied in two dimensions using random walk models. An interface with a meniscus is approximated by two random walks forming microscopic droplets of the intruding phase before and after a macroscopic lens. Within this class of models, we establish a Wulff construction and prove the Herring relations between contact angles. We give explicit formulas for the contact angles as functions of temperature, both at low temperatures and near the wetting transition.     

The equilibrium shape of a two-dimensional crystal between parallel planes

By applying rather standard techniques for equilibrium crystal shapes (Wulff construction), we derive a construction for the equilibrium shape of a 2D crystal grown between two parallel plane substrates. The critical distance of the substrates at which this crystal splits into two parts is computed as a function of the wall free energy of the substrates. This may open new perspectives for the measurement of wall free energies.     

Brownian motion with inert drift, but without flux: A model

We study the motion of a Brownian particle which interacts with a stationary obstacle in two dimensions. The Brownian particle acquires drift proportionally to the time spent on the boundary of the obstacle. The system approaches equilibrium, and the equilibrium distribution for the location and drift magnitude has the product form. The distribution for the location is uniform, while the drift distribution depends on the shape of the obstacle, resembling a gamma function for the circular or elliptic obstacle.     

Surface energy anisotropy of iron surfaces by carbon adsorption

To investigate the surface energy anisotropy of carbon-adsorbed iron surfaces related to carbon nanotube growth we have performed self-consistent pseudopotential density-functional calculations. The iron particle’s equilibrium shape is obtained from the Wulff construction using the calculated surface energies. We investigate the adsorption and diffusion of carbon atoms on the iron surfaces. It is found that the desorption energy of the carbon atoms and the activation energy for carbon diffusion are very different on different facets. Using the energetics of carbon-adsorbed iron surfaces, we evaluate the formation energies of the surfaces as a function of carbon chemical potential. Since the surface energies of the low-index iron facets are affected differently by the presence of carbon, the crystal shape is changed correspondingly.     

Cube–Root Boundary Fluctuations¶for Droplets in Random Cluster Models

For a family of bond percolation models on ℤ² that includes the Fortuin–Kasteleyn random cluster model, we consider properties of the “droplet” that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Kotecky and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the “local roughness”) is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l ^1/3(log l)^2/3) in probability, where is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of “bottlenecks”, which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present. Received: 20 January 2000 / Accepted: 7 August 2001     

Exact equilibrium shapes of Ising crystals on triangular/honeycomb lattices

The anisotropic surface tension for an Ising system below the critical point on a triangular or a honeycomb lattice can be computed through duality. Using the Wulff construction, the equilibrium shape of a crystal (droplet of one phase inside a sea of the other) is found. An exact and simple equation for this shape is derived.     

Structure and morphology of nanosized lead inclusions in aluminum grain boundaries

Grain boundary lead inclusions formed by ion implantation of mazed bicrystal aluminum films have been investigated by transmission electron microscopy. The vapor-grown bicrystal films contained mainly 90°(110) tilt boundaries with fixed misorientation but variable inclination, as well as some growth twins with 70.5°(110) symmetrical tilt boundaries and a few small-angle boundaries. It was found that the shape, size and orientation of the inclusions in the grain boundaries depend on the orientation of the aluminum grain boundary plane. Inclusions at 90°(110) tilt boundaries were invariably sharply faceted toward one aluminum grain and more rounded toward the other grain. The faceted side was a section of the cuboctahedral equilibrium shape of bulk lead inclusions in parallel topotaxy with the aluminum matrix. The rounded side, where the aluminum grain was rotated by 90° with respect to the lead lattice, approximated a spherical cap. At specific low-energy segments of the grain boundary where a (100) plane in grain 1 meets an (011) plane in grain 2, only two of several possible shapes were observed. One of these was preferred in as-implanted samples while both types were found after melting and re-solidification of the lend inclusions. The observations are discussed in terms of a modified Wulff construction.     

The summertop construction: Crystals in a corner

Equilibrium shapes of crystals in contact with more than one substrate, e.g., droplets at a corner or the edge of two planes, are described. This generalizes a construction due to Wulff and Winterbottom and, unlike them, allows non-convex equilibrium shapes. Since this construction may require a central inversion of the Wulff plot, it is dubbed the summertop construction.     

Ground States of the 2D Sticky Disc Model: Fine Properties and N 3/4 Law for the Deviation from the Asymptotic Wulff Shape

We investigate ground state configurations for a general finite number N of particles of the Heitmann-Radin sticky disc pair potential model in two dimensions. Exact energy minimizers are shown to exhibit large microscopic fluctuations about the asymptotic Wulff shape which is a regular hexagon: There are arbitrarily large N with ground state configurations deviating from the nearest regular hexagon by a number of ～N ^3/4 particles. We also prove that for any N and any ground state configuration this deviation is bounded above by ～N ^3/4. As a consequence we obtain an exact scaling law for the fluctuations about the asymptotic Wulff shape. In particular, our results give a sharp rate of convergence to the limiting Wulff shape.     

Surfaces and roughening

Some recent results in the application of statistical mechanics to surfaces are discussed. Only exactly soluable models are described. First, we consider phase separation below the critical temperature in uniaxial ferromagnets and their analogs. We then consider the determination of the equilibrium shape of a crystal having fixed volume, given the orientation-dependent surface tension, using the Wulff construction.     

Numerical investigation of electrostatic interactions in nanoscale substances based on finite-size particle simulation

We use the optimized finite-size particle techniques derived from plasma simulations to investigate the electrostatic interactions in nanoscale substances. In conjunction with electron tunneling, the substance surface is modeled as a potential well that confines simulated electrons for reaching equilibrium in an electrostatic system governed by Poisson's equation. This scheme avoids the mathematical difficulty of handling sophisticated boundary conditions at the interface and easily treats complicated shapes. We demonstrate the performance of the proposed method by simulating millions of electrons propagating in isolated substances at nanoscale. Numerical results are consistent with theoretical predictions of electrostatic properties in equilibrium.     

Droplet dynamics for asymmetric Ising model

Nucleation from a metastable state is studied for an anisotropic Ising model at very low temperatures. It turns out that the critical nucleus as well as configurations on a typical path to it differ from the Wulff shape of an equilibrium droplet.     

The rigid molecule approximation in lattice dynamics

The theory of the lattice dynamics of molecular crystals with and without the rigid molecule approximation is developed and the two compared. To make the two compatible, the equilibrium conditions for the internal molecular dimensions are replaced in the rigid molecule approximation by equations representing the constrained constant values of the molecular dimensions expanded in vibrational coordinates.     

Slow Dynamics for the Dilute Ising Model in the Phase Coexistence Region

In this paper we consider the Glauber dynamics for a disordered ferromagnetic Ising model, in the region of phase coexistence. It was conjectured several decades ago that the spin autocorrelation decays as a negative power of time [Huse and Fisher, in Phys Rev B 35(13):6841–6846, 1987]. We confirm this behavior by establishing a corresponding lower bound in any dimensions d ≥ 2, together with an upper bound when d = 2. Our approach is deeply connected to the Wulff construction for the dilute Ising model. We consider initial phase profiles with a reduced surface tension on their boundary and prove that, under mild conditions, those profiles are separated from the (equilibrium) pure plus phase by an energy barrier.     

A microscopic justification of the Wulff construction

We report results about a rigorous microscopic justification of the Wulff construction for the two-dimensional Ising model at low temperatures and under periodic boundary conditions. The idea of the proof is sketched.     

Two families of analytic γ-plots and their influence upon homogeneous nucleation kinetics

The equilibrium shapes corresponding to two different families of γ-plots are constructed. One γ-plot family comprises continuous variations from a sphere to an oblate ellipsoid. This set of γ-plots yields a sharp edge in the corresponding equilibrium shape when its aspect ratio is less than $\text{1}\text{√2}$. The other family consists of nephroids of revolution, varied in cross-sectional form from two slightly overlapping near-circles to an ellipse-like morphology. This family exhibits a facet at one boundary orientation in the equilibrium shape. For the analytical expression of the equilibrium shape, the ξ-vector formalism of Cahn and Hoffman is used and found to give results identical to those from the Euler-Lagrange method. The effects of the variations in equilibrium shape within the two families treated upon the principal parameters in the general equation for the time-dependent rate of nucleation are assessed in order to ascertain their relative influence on nucleation kinetics.     

Capillarity-Induced Surface Morphologies

Surface energetics is reviewed including expressions for the chemical potential of a curved surface element and the Legendre transform relation between the projected surface free energy as a function of orientation and the Wulff equilibrium shape. A well known equation is derived describing surface evolution by surface diffusion, assuming local equilibrium. Solutions are reviewed including a decaying sinusoid and a developing thermal groove. Breakdown of local equilibrium is considered. The structure, energetics and dynamics of steps on a vicinal surface are discussed. Facet sizes on the Wulff shape and the surface profile at the edge of a facet are related to the step self and interaction free energies respectively. Fourier analysis of step fluctuations is described, revealing the underlying transport processes. Analysis of the decay of a sinusoidal profile on a vicinal surface in terms of step behavior is given. Finally, examples are reviewed of surface evolution below the roughening temperature T _R in which case facets move by the lateral spreading of steps. Results differ greatly from those of the continuum theory applicable above T _R.