Coarsening Exponent Research Articles

Noise-assisted mound coarsening in epitaxial growth

We propose deposition noise to be an important factor in unstable epitaxial growth of thin films. Our analysis yields a geometrical relation H=(RWL)^2 between the typical mound height W, mound size L, and the film thickness H. Simulations of realistic systems show that the parameter R is a characteristic of the growth conditions, and generally lies in the range 0.2-0.7. The constancy of R in late-stage coarsening yields a scaling relation between the coarsening exponent 1/z and the mound height exponent \beta which, in the case of saturated mound slope, gives \beta = 1/z = 1/4.

Faceting kinetics of stepped Si(113) surfaces: dynamic scaling and nano-scale grooves

Time-resolved synchrotron X-ray scattering studies are presented of the faceting kinetics of stepped silicon surfaces misoriented by 2.1° and 1.3° from the cubic [113] direction towards the [001] direction. Following a quench from the high-temperature one-phase region of the orientational phase diagram into the two-phase coexistence region, originally-uniformly-distributed steps rearrange to form a grooved superstructure. Time-resolved surface X-ray scattering measurements reveal the behavior of the grooved surface morphology as a function of time. For times up to several hundred seconds after a quench, the behaviors of the specular and diffuse scattering intensity support a dynamic scaling description of the surface morphology. Specifically, the surface is found to be self-similar in time, with a characteristic groove side ( L) varying as a power-law versus time (t): L = L ot ф with a coarsening exponent of ф = 0.164 ± 0.021 ⋍ 1 6 . ф = 1 6 is consistent with a theory which focuses on thermally fluctuating step bunches and takes their collisions as the key growth mechanism. At later times, the groove size approaches a limiting value which depends on the stepped phase misorientation angle. This is consistent with the behavior expected for faceted surfaces in the case that elastic effects are important [V.I. Marchenko, Zh. Eksp. Teor. Fiz. 81 (1981) 1141].

Coarsening of B2 Ordered Domains by Monte Carlo Simulations

ABSTRACTWe present results concerning the study of growth and coarsening kinetics of ordered domains in a binary alloy exhibiting a B2 ordering at the stoichiometric composition. This study has been performed within the framework of an atomistic kinetic model based on vacancy migration using Monte Carlo simulations techniques. Contrary to the standard Lifshitz-Allen-Cahn law, it is observed that the coarsening exponent is a function of the annealing temperature, and also of parameters controlling the migration of the vacancy.

Viscosity and structure variations of Al-Si alloy in the semi-solid state

The viscosity and microstructure evolution of A356 Al–Si alloy in the semi-solid state were investigated. During continuous cooling, the viscosity increases slowly at the onset of the semi-solid region, and starts to increase abruptly after reaching a critical temperature. The higher the shear rate, the lower is the critical temperature. During isothermal stirring, the viscosity decreases and the sphericity increases as the shear rate increases. The grain size decreases and the sphericity increases as the shear rate increases for short stirring times. The viscosity increases as the temperature decreases during isothermal stirring. The spheroidizing effect is not significant for short stirring time. During isothermal stirring, the viscosity increases as the rest time increases. Both the grain size and sphericity increase as the rest time, following stirring, increases. The coarsening exponent is 3.2, suggesting that the coarsening is controlled by the volume diffusion of the grains. During isothermal stirring, the viscosity decreases as the stirring time increases. The sphericity increases and the grain size decreases as the stirring time increases. We propose that there is a critical stirring time, above which the grain size increases with stirring time, but below which the grain size decreases with stirring time.

Effects of crystalline microstructure on epitaxial growth.

The results of kinetic Monte Carlo simulations of epitaxial growth on fcc(100) and bcc(100) surfaces in which the correct crystal geometry is taken into account are reported. The existence of downward funneling to fourfold hollow sites leads to a downward current for large angles and to angle selection as observed in a variety of experiments. We have used our model to simulate Fe/Fe(100) deposition at room temperature and have compared our results with recent experiments. Excellent agreement is found for the selected angle, mound coarsening exponent n, and kinetic roughening exponent \ensuremath{\beta} as well as for the mound morphology. A theoretical analysis also leads to an accurate prediction of the observed mound angle for Fe/Fe(100) deposition at room temperature. The general dependence of the surface skewness, mound angle, and coarsening kinetics on temperature, deposition rate, and strength of the step barrier to interlayer diffusion is also studied and compared with recent experiments. While for a moderate step barrier we find an effective coarsening exponent n\ensuremath{\simeq}0.16--0.25, for the case of a very large step barrier we find n\ensuremath{\simeq}1/3, which is significantly larger than found in previous models but in agreement with recent experiments on Rh/Rh(111). \textcopyright{} 1996 The American Physical Society.

Faceting of Stepped Silicon (113) Surfaces: Step Unbinding, Dynamic Scaling, and Nano-Scale Grooves

AbstractX-ray scattering studies are reviewed of the faceting kinetics of stepped silicon (113) surfaces misoriented towards [001]. Following a quench from the one-phase region of the orientational phase diagram into the coexistence region, initially-uniformly-distributed steps rearrange to form a grooved superstructure. Our measurements reveal the behavior of the grooved surface morphology as a function of time. The surface is found to be self-similar in time, with a characteristic groove size (L) varying as a power-law versus time (t): L = L0tøwith a coarsening exponent of ø = 0.164 ± 0.021 ≃ 1/6, consistent with a theory for the kinetics of faceting which focuses on thermally fluctuating step bunches and their collisions. At later times, the groove size approaches a limiting value which depends on the stepped phase misorientation angle, as expected for faceted surfaces in the case that elastic effects are important.

Mound Formation and Coarsening in Homoepitaxial Growth

AbstractThe effects of instabilities which lead to mound formation and coarsening in homoepitaxial growth on metal (100) surfaces are discussed. These include an instability due to the Ehrlich-Schwoebel step barrier to interlayer diffusion as well as an instability due to step-adatom attraction at ascending steps. A unified picture of the effects of attractive and repulsive interactions at ascending and descending steps on surface morphology and island nucleation is presented. An analytic calculation of the selected mound angle and critical temperature for mound formation as a function of both the Ehrlich-Schwoebel step barrier and the barrier to diffusion towards an ascending step is also presented. Depending on the sign of the step barrier and the magnitude of the prefactor for diffusion over a step various scenarios are possible, including the existence of a critical temperature for mound formation above which (for a positive step barrier) or below which (for a negative step barrier) quasi-layer-by-layer growth will be observed. A theoretical analysis also leads to an accurate prediction of the observed mound angle for Fe/Fe(100) deposition at room temperature. The general dependence of the mound angle, surface skewness, and mound coarsening exponent on temperature, deposition rate, and strength of the step barrier is also studied via kinetic Monte Carlo simulations of bcc(100) growth and compared with recent experiments.

Critical phenomena in morphology transitions of growth models with competition.

We investigate some simple stochastic models for the propagation of a growth front which borders a condensed phase consisting of two species competing with each other. When both species propagate at the same average speed we identify a critical point and define a coarsening exponent. A difference in average growth rates between the two species defines a deviation from criticality leading to exponential coarsening. We propose a scaling relation including the specific form of the scaling function and an additional competition exponent in apparent agreement with the data.

Scaling behavior of two-time correlations in a twisted nematic liquid crystal.

We measured the coarsening exponent \ensuremath{\varphi} and the nontrivial scaling exponent \ensuremath{\lambda}, which characterize how the two-time correlation function C(t,t') scales with the correlation length L(t), C(t,t')\ensuremath{\propto}L(t${)}^{\mathrm{\ensuremath{-}}\ensuremath{\lambda}}$ when t\ensuremath{\gg}t', for a twisted nematic liquid-crystal system quenched from the isotropic phase to the nematic phase. This system is expected to be Ising-like with \ensuremath{\varphi}=0.5 and \ensuremath{\lambda}=1.25. Our values, \ensuremath{\varphi}=0.515\ifmmode\pm\else\textpm\fi{}0.026, measured over two decades in time, and \ensuremath{\lambda}=1.246\ifmmode\pm\else\textpm\fi{}0.079, measured over the late time period, 30 to 150 s, are in good agreement with theory.

Ostwald Ripening and Relaxation in Dendritic Structures

The development of microstructures in cast materials is described as a sequence of processes, among which are dendritic growth and coarsening. Dendritic evolution is analyzed first as the deterministic formation of the tip and its neighborhood, followed by the incubation of stochastic events, including Ostwald ripening and coalescence. Mean-field theories of coarsening are discussed briefly, and the problem of relating classical coarsening of convex spherical particles to that of highly branched dendritic interfaces is shown to be resolved by considering the distribution of chemical potentials over the solid-liquid surface. Experiments relying on stereological measurements yield kinetic coarsening experiments in reasonable agreement with the statistical theories. The influence of volume fraction on the coarsening kinetics is shown to affect the rate constant, which increases about four-fold when the solid fraction rises to 50 pct, but the coarsening exponent remains at the classical value of 1/3. Attempts to measure mean-field intensive thermodynamic properties such as temperature and concentration during dendritic coarsening yield consistent kinetics, although the precise relationships of these mean-field quantities to the stereological parameters are not yet established. The application of morphological scaling laws is stressed throughout, as a fundamental method to predict cast structures in more complex materials.