Relaxation Of Steps Research Articles

Algorithms for optimizing systems with multiple extremum functionals

The problem of minimizing (maximizing) multiple extremum functionals (infinite-dimensional optimization) is considered. This problem cannot be solved by conventional gradient methods. New gradient methods with adaptive relaxation of steps in the vicinity of local extrema are proposed. The efficiency of the proposed methods is demonstrated by the example of optimizing the shape of a hydraulic gun nozzle with respect to the objective functional, which is the average force of the hydraulic pulse jet momentum acting on an obstacle. Two local maxima are found, the second of which is global; in the second maximum, the average force of the jet momentum is three times higher than in the first maximum. The corresponding nozzle shape is optimal. Conventional gradient methods have not found any maximum; i.e., they were unable to solve the problem.

Role of step stiffness and kinks in the relaxation of vicinal (001) with zigzag [110] steps

We present a kinetic Monte Carlo study of the relaxation dynamics and steady state configurations of 〈110〉 steps on a vicinal (001) simple cubic surface. This system is interesting because 〈110〉 (fully kinked) steps have different elementary excitation energetics and favor step diffusion more than 〈100〉 (nominally straight) steps. In this study we show how this leads to different relaxation dynamics as well as to different steady state configurations, including that 2-bond breaking processes are rate determining for 〈110〉 steps in contrast to 3-bond breaking processes for 〈100〉−steps found in previous work [Surface Sci. 602, 3569 (2008)]. The analysis of the terrace-width distribution (TWD) shows a significant role of kink-generation-annihilation processes during the relaxation of steps: the kinetic of relaxation, toward the steady state, is much faster in the case of 〈110〉−zigzag steps, with a higher standard deviation of the TWD, in agreement with a decrease of step stiffness due to orientation. We conclude that smaller step stiffness leads inexorably to faster step dynamics towards the steady state. The step-edge anisotropy slows the relaxation of steps and increases the strength of step-step effective interactions.

Relaxation of surface steps after thermal quenches: A numerical study within the terrace-step-kink model

We study the out-of-equilibrium relaxation of surface steps after thermal quenches using numerical simulations of the terrace-step-kink model for a vicinal surface. We analyze both single and interacting steps in a situation where the temperature is suddenly changed at a given quench time. We focus on a physically relevant range of temperatures and show that the relaxation of the roughness is compatible with a power-law behavior with an effective relaxation exponent close to γ = 1/2 in all cases. This value is consistent with a one-dimensional Edwards-Wilkinson equation. In particular, this means that, although the case of interacting steps is effectively a two-dimensional system, its relaxation is dominated by short length-scale fluctuations, where steps are not interacting.

Elastic displacements and step interactions on metallic surfaces: Grazing-incidence x-ray diffraction andab initiostudy of Au(332)

We have studied the energetics, relaxation, and interactions of steps on the Au(332) vicinal surface using a combination of grazing incidence x-ray diffraction, anisotropic linear elasticity theory, and ab initio density-functional theory. We find that the initial force distribution on a bulk-truncated surface, as well as the resulting pattern of atomic relaxations, can be reproduced excellently by a buried dipole elastic model. The close agreement obtained between experimental and calculated x-ray diffraction profiles allows us to precisely determine the value of the elastic dipole density at the steps. We also use these results to obtain an experimental estimate of the surface stress on an unreconstructed Au(111) facet, ${\ensuremath{\tau}}_{\text{Au}(111)}=2.3\ifmmode\pm\else\textpm\fi{}0.4\text{ }{\text{Nm}}^{\ensuremath{-}1}$, and the value of the step-step elastic interaction energy $A=950\ifmmode\pm\else\textpm\fi{}150\text{ }\text{meV}\text{ }\text{\AA{}}$.

Understanding surface limiting processes occurring during the relaxation of steps on vicinalsurfaces

In our recently constructed Fokker–Planck formalism for describing the equilibration of theterrace-width distribution of a vicinal surface from an arbitrary initial configuration,the meaning of the relaxation time, related to the strength of the random noisein the underlying Langevin equation, was rather unclear. Using kinetic MonteCarlo simulations we show that the time constant exhibits activated behaviorwith a barrier that has a physically plausible dependence on the energies of thegoverning atomistic model. Thus, the Fokker–Planck time has some physical meaning.

Continuum Relaxation of Interacting Steps on Crystal Surfaces in $2+1$ Dimensions

We study the relaxation of crystal surfaces in $2+1$ dimensions via the motion of interacting atomic steps. The goal is the rigorous derivation of the continuum limit. The starting point is a discrete scheme from the Burton, Cabrera, and Frank model, which accounts for diffusion of point defects (“adatoms”) on terraces and attachment‐detachment of atoms at step edges. It is shown that the macroscopic adatom current involves a tensor mobility, which describes different fluxes in directions parallel and transverse to step edges, although the physics of each terrace is assumed isotropic. When the steps are everywhere parallel (straight or circular) the tensor character of the mobility is unimportant; in the general $(2+1)$‐dimensional setting, however, it is crucial. Our methods consist of (i) the solution of the diffusion equation for adatoms via the separation of local space variables into “fast” and “slow,” and (ii) the treatment of the step chemical potential for a wide class of step energies and repulsi...

Continuum model for low temperature relaxation of crystal steps.

High and low temperature relaxation of crystal steps are described in a unified picture, using a continuum model based on a modified expression of the step-free energy. Results are in agreement with experiments and Monte Carlo simulations of step fluctuations and monolayer cluster diffusion and relaxation. In an extended model where mass exchange with neighboring terraces is allowed, step transparency and a low temperature regime for unstable step meandering are found.