Dynamics of growing surfaces by linear equations in 2+1 dimensions (original) (raw)
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Interfacial Super-Roughening by Linear Growth Equations
International Journal of Modern Physics B, 2001
We give an extensive analytical study of a class of linear growth equations in 1+1 dimensions which describe certain interfacial super-roughening processes. With our calculation, we give a first rigorous analytical affirmation on the applicability of the anomalous dynamic scaling ansatz, which has been proposed to describe the dynamics of super-rough interfaces in finite systems. In addition, we explicitly evaluate not only the leading order but also all the sub-leading orders which dominate over the ordinary dynamic scaling term. Finally, we briefly discuss the influence of the macroscopic background formation on the interfacial anomalous roughening in super-rough growth processes.
Scaling of Local Slopes, Conservation Laws, and Anomalous Roughening in Surface Growth
Physical Review Letters, 2005
We argue that symmetries and conservation laws greatly restrict the form of the terms entering the long wavelength description of growth models exhibiting anomalous roughening. This is exploited to show by dynamic renormalization group arguments that intrinsic anomalous roughening cannot occur in local growth models. However some conserved dynamics may display super-roughening if a given type of terms are present.
Physical Review E, 2004
A study on the ͑1+1͒-dimensional superrough growth processes is undertaken. We first work out the exact relations among the local interfacial width w, the correlation function G, and the pth degree residual local interfacial width w p with p =1,2,3,. . .. The relations obtained are exact and thus can be applied to any ͑1 +1͒-dimensional growth processes in the continuum limit, no matter whether the interface is superrough or not. Then we investigate the influence of the macroscopic structure formation on the scaling behavior of the superrough growth processes. Moreover, we show analytically that the residual local interfacial width w p excludes only the influence of the macroscopic structure on the scaling behavior of the system and retains the true scaling behavior originating from the stochastic nature of the system. Finally, we analyze and simulate some superrough growth models for demonstration.
Anomalous roughening of curvature-driven growth with a variable interface window
Physical Review E, 2010
We studied the curvature-driven roughening of a disk domain pattern with a variable interface window. The relaxation of interface is driven by negative "surface tension". When a domain boundary propagates radially at a constant rate, we found that evolution of interface roughness follows scaling dynamic behavior. The local growth exponents are substantially different from the global exponents. Curvature-driven roughening belongs to a new class of anomalous roughening dynamics. However, a different "surface tension" leads to different global exponents. This is different from that of interface evolution with a fixed-size window, which has universal exponent. The variable growth window leads to a new class of anomalous roughening dynamics.
Kinetic roughening of surfaces: Derivation, solution, and application of linear growth equations
Physical review. B, Condensed matter, 1996
We present a comprehensive analysis of a linear growth model, which combines the characteristic features of the Edwards-Wilkinson and noisy Mullins equations. This model can be derived from microscopics and it describes the relaxation and growth of surfaces under conditions where the nonlinearities can be neglected. We calculate in detail the surface width and various correlation functions characterizing the model. In particular, we study the crossover scaling of these functions between the two limits described by the combined equation. Also, we study the effect of colored and conserved noise on the growth exponents, and the effect of different initial conditions. The contribution of a rough substrate to the surface width is shown to decay universally as w i (0)(ξ s /ξ(t)) d/2 , where ξ(t) ∼ t 1/z is the time-dependent correlation length associated with the growth process, w i (0) is the initial roughness and ξ s the correlation length of the substrate roughness, and d is the surface dimensionality. As a second application, we compute the large distance asymptotics of the height correlation function and show that it differs qualitatively from the functional forms commonly used in the intepretation of scattering experiments.
Roughening phase transition in surface growth
Physical Review Letters, 1990
We present systematic numerical simulation results which show a strong indication of phase transitions, in both 2+1 and 3+1 dimensions, between weak-coupling and strong-coupling regimes in a surface-growth model. A modified ballistic-deposition model is used to demonstrate the transitions and the roughness scaling exponent is measured. While the transition in 3+ I dimensions confirms the prediction of the renormalization-group analysis, the one in 2+ 1 dimensions had not been previously anticipated and exhibits a complex critical behavior.
Universal macroscopic background formation in surface super-roughening
1997
We study a class of super-rough growth models whose structure factor satisfies the Family-Vicsek scaling. We demonstrate that a macroscopic background spontaneously develops in the local surface profile, which dominates the scaling of the local surface width and the height-difference. The shape of the macroscopic background takes a form of a finite-order polynomial whose order is decided from the value of the global roughness exponent. Once the macroscopic background is subtracted, the width of the resulting local surface profile satisfies the Family-Vicsek scaling. We show that this feature is universal to all super-rough growth models, and we also discuss the difference between the macroscopic background formation and the pattern formation in other models.
Generic Dynamic Scaling in Kinetic Roughening
Physical Review Letters, 2000
We study the dynamic scaling hypothesis in invariant surface growth. We show that the existence of power-law scaling of the correlation functions (scale invariance) does not determine a unique dynamic scaling form of the correlation functions, which leads to the different anomalous forms of scaling recently observed in growth models. We derive all the existing forms of anomalous dynamic scaling from a new generic scaling ansatz. The different scaling forms are subclasses of this generic scaling ansatz associated with bounds on the roughness exponent values. The existence of a new class of anomalous dynamic scaling is predicted and compared with simulations.
Surface Growth with Power-Law Noise in 2+ 1 Dimensions
1991
Abstract Large scale simulations of stochastic growth of 2+ 1 dimensional surfaces were carried out on a 16K processor Connection Machine. We introduce a family of models for which we could reproduce the known scaling behavior of kinetic roughening in the presence of bounded noise. For noise amplitudes eta\ eta eta distributed to $ P (\ eta)\ sim\ eta^{-\ mu-1} ,thegrowthexponentsdependon, the growth exponents depend on ,thegrowthexponentsdependon\ mu $ and they are well described by the recently formula based on the scaling of rare events.
Anomalous Roughness in Dimer-Type Surface Growth
Physical Review Letters, 2000
We point out how geometric features affect the scaling properties of non-equilibrium dynamic processes, by a model for surface growth where particles can deposit and evaporate only in dimer form, but dissociate on the surface. Pinning valleys (hill tops) develop spontaneously and the surface facets for all growth (evaporation) biases. More intriguingly, the scaling properties of the rough one dimensional equilibrium surface are anomalous. Its width, W ∼ L α , diverges with system size L, as α = 1 3 instead of the conventional universal value α = 1 2. This originates from a topological non-local evenness constraint on the surface configurations.