Reconstruction and Extension of the Family–Vicsek Scaling Hypothesis for Growing Rough Interfaces

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Abstract

A scaling hypothesis for the standard deviation σ of the height of growing interfaces is proposed by extending the Family–Vicsek (FV) scaling hypothesis. A data-collapsing method is adopted for estimating values of three exponents α, β, and z, which characterize, respectively, the roughness, growth, and dynamic properties of growing interfaces. The estimation is carried out through σ, which is a function of both the time and the width of the interfaces. The advantages of the present extended scaling hypothesis are as follows: (A) The value of β can be obtained even if the data for σ in terms of t are few so that its value is not determined precisely from the slope of the lnσ vs lnt plot. (B) Different scaling relations can be obtained during the time evolution of interface growth. (C) By introducing a new exponent, which represents the time dependence of σ for a short width, a scaling argument is possible even for growing interfaces that do not satisfy the FV scaling relation. Successful applications are carried out to a few numerical models and a paper-wetting experiment.

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