Brownian motion and shape fluctuations of single-layer adatom and vacancy clusters on surfaces: Theory and simulations

Abstract

In recent observations of Brownian motion of islands of adsorbed atoms and of vacancies with mean radius R, the cluster diffusion constant D c varies as R Ϫ1 and R Ϫ2 . From an analytical continuum description of the cluster's steplike boundary, we find a single Langevin equation for the motion of the cluster boundary, rather than three special cases. From this we determine D c and the correlation function G sh for fluctuations of the shape around an assumed equilibrium circular shape. In three limiting cases we find the scaling relations D c ϳR Ϫ␣ and, at early elapsed time t, G sh ϳt 1/(1ϩ␣) , where ␣ϭ1, 2, and 3, corresponding to the three generic surface mass-transport mechanisms of straight steps. We thereby provide a unified treatment of the dynamics of steps and of clusters. To check how well the continuum results describe clusters of the size in experiments, we perform Monte Carlo simulations of simple lattice gas models. Further, we estimate atomic diffusion parameters from the available experimental data on diffusion of large clusters.

Statistics of interfacial fluctuations of radially growing clusters

“…In that sense, this model would not be able to describe growing cell colonies, precisely because it assumes spurious rigidity of bulk cells. On the other hand, it would be better suited to describe the radial growth of crystalline structures [51]. We have also found that reparametrization invariance as defined in [22] implicitly implies dilatation dynamics.…”

Dynamics of curved interfaces

“…Despite the usefulness of the Cartesian representation in many cases, there are some growth profiles that can not be described according to it. Physical settings such as fluid flow in porous media [1], grain-grain displacement in Hele-Shaw cells [2], fracture dynamics [3], adatom and vacancy islands on crystal surfaces [4], and atomic ledges bordering crystalline facets [5,6] present interfaces that violate the hypothesis of the Cartesian representation. Biological systems are also characterized by an approximate spherical symmetry: bacterial colonies [7], fungi [8], epithelial cells [9], and cauliflowers [10] develop rough surfaces which are not describable from a planar reference frame.…”

Determination of Step Free Energies from Island Shape Fluctuations on Metal Surfaces

“…Since the origin of the coordinate system can be chosen freely, we define it at any time in such a way that it coincides with the center of mass of the cluster. If we define the function R͑u͒ as the time-averaged position of the cluster's edge relative to its center of mass, we may describe the deviation from the average shape in terms of a dimensionless variable g͑u, t͒ [15]:…”

“…To quantify the shape fluctuations, we follow the work of Khare and Einstein [15] and define the instantaneous position of the cluster edge in cylindrical coordinates by a function r͑u, t͒, where r and u are the radius and polar angle, respectively, and t is the time. Since the origin of the coordinate system can be chosen freely, we define it at any time in such a way that it coincides with the center of mass of the cluster.…”