Rashmi C. Desai

Statistical mechanics of a nonlinear stochastic model

A multivariable Fokker-Planck equation (FPE) is used to investigate the equilibrium and dynamical properties of a nonlinear stochastic model. The model displays a phase transition. The equilibrium distributions are found to be non-Gaussian; the deviation from Gaussian is especially significant near the transition point. To study the nonequilibrium behavior of the model, a self-consistent dynamic mean field (SCDMF) theory is derived and used to transform the FPE to a systematic hierarchy of equations for the cumulant moments of the time-dependent distribution function. These equations are numerically solved for a variety of initial conditions. During the time evolution of the system from an initial unstable equilibrium state to the final equilibrium state, three distinct time stages are found.

Steady-states and kinetics of ordering in bus-route models : connection with the Nagel-Schreckenberg model

Abstract:A Bus Route Model (BRM) can be defined on a one-dimensional lattice, where buses are represented by “particles” that are driven forward from one site to the next with each site representing a bus stop. We replace the random sequential updating rules in an earlier BRM by parallel updating rules. In order to elucidate the connection between the BRM with parallel updating (BRMPU) and the Nagel-Schreckenberg (NaSch) model, we propose two alternative extensions of the NaSch model with space-/time-dependent hopping rates. Approximating the BRMPU as a generalization of the NaSch model, we calculate analytically the steady-state distribution of the time headways (TH) which are defined as the time intervals between the departures (or arrivals) of two successive particles (i.e., buses) recorded by a detector placed at a fixed site (i.e., bus stop) on the model route. We compare these TH distributions with the corresponding results of our computer simulations of the BRMPU, as well as with the data from the simulation of the two extended NaSch models. We also investigate interesting kinetic properties exhibited by the BRMPU during its time evolution from random initial states towards its steady-states.

Simulations of cubic-tetragonal ferroelastics

We study domain patterns in cubic-tetragonal ferroelastics by solving numerically equations of motion derived from a Landau model of the phase transition, including dissipative stresses. Our system sizes, of up to 256 3 points, are large enough to reveal many structures observed experimentally. Most patterns found at late stages in the relaxation are multiply banded; all three tetragonal variants appear, but inequivalently. Two of the variants form broad primary bands; the third intrudes into the others to form narrow secondary bands with the hosts. On colliding with walls between the primary variants, the third either terminates or forms a chevron. The multiply banded patterns, with the two domain sizes, the chevrons and the terminations, are seen in the microscopy of zirconia and other cubic-tetragonal ferroelastics. We examine also transient structures obtained much earlier in the relaxation; these show the above features and others also observed in experiment.

Kinetic theory of rough sphere fluids: Linear and angular velocity correlation functions

We present a renormalized kinetic theory for tagged particle motion in a rough sphere fluid which includes the effects of correlated collisions. The kinetic theory is a generalization of a recent theory for monatomic fluids by Mazenko. The correlated collisions create slowly decaying memory in the linear (ψv) and angular (ψω) velocity correlation functions on account of the coupling of the test particle density fluctuations to various moments of the full phase space density correlation function. (These moments are connected with the conserved variables in the system). We use the theory to study ψv and ψω and compare the results with recent molecular dynamics experiments of O’Dell and Berne. The kinetic theory yields the correct long time power law decay for both ψv and ψω without imposing any wave vector cutoff characteristic of the hydrodynamic theories. It also yields useful new insights into various mode couplings responsible for the behavior of ψv and ψω at small and intermediate times. The qualitativ...

Epitaxial growth in dislocation-free strained alloy films: Morphological and compositional instabilities

The mechanisms of stability or instability in the strained alloy film growth are of intense current interest to both theorists and experimentalists. We consider dislocation-free, coherent, growing alloy films which could exhibit a morphological instability without nucleation. We investigate such strained films by developing a nonequilibrium continuum model and by performing a linear stability analysis. The couplings of film-substrate misfit strain, compositional stress, deposition rate, and growth temperature determine the stability of film morphology as well as the surface spinodal decomposition. We consider some realistic factors of epitaxial growth in particular the composition dependence of elastic moduli and the coupling between the top surface and underlying bulk of the film. The interplay of these factors leads to new stability results. In addition to the stability diagrams both above and below the coherent spinodal temperature, we also calculate the kinetic critical thickness for the onset of instability as well as its scaling behavior with respect to misfit strain and deposition rate. We apply our results to some real growth systems and discuss the implications related to some recent experimental observations.

Stress-driven instability in growing multilayer films

We investigate the stress-driven morphological instability of epitaxially growing multilayer films, which are coherent and dislocation-free. We construct a direct elastic analysis, from which we determine the elastic state of the system recursively in terms of that of the old states of the buried layers. In turn, we use the result for the elastic state to derive the morphological evolution equation of surface profile to first order of perturbations, with the solution explicitly expressed by the growth conditions and material parameters of all the deposited layers. We apply these results to two kinds of multilayer structures. One is the alternating tensile/compressive multilayer structure, for which we determine the effective stability properties, including the effect of varying surface mobility in different layers, its interplay with the global misfit of the multilayer film, and the influence of asymmetric structure of compressive and tensile layers on the system stability. The nature of the asymmetry properties found in stability diagrams is in agreement with experimental observations. The other multilayer structure that we study is one composed of stacked strained/spacer layers. We also calculate the kinetic critical thickness for the onset of morphological instability and obtain its reduction and saturation as the number of deposited layers increases, which is consistent with recent experimental results. Compared to the single-layer film growth, the behavior of kinetic critical thickness shows deviations for upper strained layers.

ELASTIC PROPERTIES OF HOMOPOLYMER-HOMOPOLYMER INTERFACES CONTAINING DIBLOCK COPOLYMERS

We study the elastic properties of homopolymer/homopolymer interfaces containing diblock copolymers by means of a theory of Gaussian fluctuations. The interfacial tension and the bending rigidity of the interface in the two-phase coexistence region are calculated from the power spectrum of capillary modes. Our theory shows that in the limiting case of a pure binary homopolymer mixture, while the interfacial tension increases monotonically with increasing χN (where χ is the Flory-Huggins parameter and N is the homopolymer molecular weight) the bending rigidity does not. The bending rigidity increases rapidly at first for small values of χN, but then decreases with further increase of χN. In the presence of diblock copolymers, the interfacial tension always decreases with increasing diblock copolymer volume fraction at a given χN. However, the bending rigidity can show either a decrease or an increase depending on χN and the ratio γ between the molecular weights of a diblock copolymer and that of a homopoly...

Rotational correlations in rough sphere fluids

The coupling of the angular velocity of a test rough sphere to the collective modes of the surrounding bath of rough spheres is studied as a function of the mass and diameter of the test sphere using renormalized kinetic theory. As the diameter of the tagged particle increases, the collective contribution to the relaxation changes from a situation where coupling to the fluid transverse angular velocity field dominates to one in which coupling to the transverse linear velocity field dominates. As the mass of the test particle increases, the collective contribution becomes increasingly sensitive to the nonhydrodynamic states which appear in the kinetic theory. The long time behavior of the angular velocity correlation function and the associated memory kernels are found to have interesting dependence on the test sphere diameter. The results are used to elucidate some features of the hydrodynamic character of microscopic relaxation processes.

Sum rules and their application to surface tension

Abstract Sum rules for inhomogeneous fluids are derived and then related to the surface tension of a liquid—vapor interface. It is found that this surface tension can be expressed in terms of the static correlation for the component of force normal to the interface. This formula is shown to be equivalent to two previous expressions for surface tension that generally are regarded as being exact. It is demonstrated that in the high-frequency or short-time limit this definition for surface tension leads directly to the standard hydrodynamic equation of motion for the normal component of the interfacial velocity.

IMPLICATIONS OF A MODEL FOR INSTABILITY DURING FILM GROWTH FOR STRAINED INGAAS AND SIGE LAYERS

We analyze experiments on the morphology of strained InGaAs and SiGe layers using a nonequilibrium stability analysis. Stability diagrams for growing films as a function of the deposition rate, the temperature and the misfit are calculated and compared to experimental reports. We show that for InGaAs layers, the onset of surface roughening is due to an instability against simultaneous modulations of the surface profile and the composition. For SiGe, the onset of surface roughening cannot be described by an instability, but rather, is due to a nucleation mechanism.