The Dynamic Scaling Theory

The concept of scaling was introduced in the field by Family and Vicsek in 1985  to provide a framework for understanding fractal-like topologies of nonequilibrium surfaces.

The dynamic scaling theory describes the development of a contour (.Drop = 2, where Drop is the topological dimension), on a flat surface of size L on the x-axis at time t = 0 (Fig. 1), and roughness surface growth proceeds in a single direction normal to L (z-axis) increasing in height, h, without overhangings. The instantaneous surface mean height, hs (t), can be described by h(x,t), a function of the x-coordinate and time. Then, co(L,t), the instantaneous surface width, can be taken as a measure of the surface roughness. The value of ca(L,t) is given by the root mean square of the interface height fluctuations.

According to the scaling theory, the discreteness of the depositing material is the main factor in turning the growing surface into a self-affine fractal surface. The standard deviation of the interface height can be expressed as where j[t/V) denotes a function of the t/V ratio, resulting in fit/Lz) = constant for t/V 00, and fit/If) = (t/L)a/z for t/V -» 0. Likewise, z = aJfi, where a and ¡3 are the roughness exponent and the growth exponent, respectively, and z is the coarsening exponent. Then, for small L and t-> oo, eq. (1) becomes which corresponds to a quasi-steady-state regime for the roughness development. It implies the appearance of a saturation roughness in the growth process. On the other hand, for L —» oo and t —» 0, co(L,t) = LaJ[t/Lz)

Equation (3) represents the nonsteady-state roughness growth regime. The transition from the nonsteady- to the steady-state regime implies the existence of a transition time, tx which is given by the proportionality

For t > tx the surface becomes a scale-invariant self-afSne fractal. The value of a is related to jDfrac, the local fractal dimension of the surface , as Z>frac = 3 - a. It should be noted that due to the fact that hs , is proportional to t, Eqs. (3) and (4) can be formulated indistinctly in terms of these variables depending on the available experimental data.

Key parameters a, ¡3, and z can be derived from the analysis of surface profiles resulting from adequate imaging procedures. In fact, those profiles from the STM and AFM images are adequate for the dynamic scaling analysis of solid topographies because STM and AFM provide high lateral resolution 3D images in real space. For this purpose, Eqs. (1) - (4) can be applied to nanomicroscopy imaging data, taking aiL,t) - onM(is,0) where fiJhM(Is>0 represents the root-mean-square average roughness determined from image profiles scanned in either the x- or ^-direction. The determination of a>nM(Ls,t), the fluctuation of h over each profile size L = Ls along both the x- and ^--directions, allowed the calculation of a, /?, and z, starting from the relationship:

and considering equations (1) -(4) for a{L,i) = iOhM(is>0- h Eq. (5) <hs(t)> stands for the instantaneous average height of the surface profile of size Ls measured from either STM or AFM images in the x- and y-directions. A detailed discussion on methods for the dynamic scaling analysis of nanomicroscopy image analysis is given elsewhere k~Lz

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