Chapter 13 - Davidson Physics

CHAPTER 14. FRACTALS AND KINETIC GROWTH MODELS 547some simple models that generate a class of fractals that are self-similar only for scale changes incertain directions.Suppose that we have a flat surface at time t = 0. How does the surface grow as a result ofvapor deposition and sedimentation? For example, consider a surface that is initially a line of Loccupied sites. Growth is in the vertical direction only (see Figure 14.12).As before, we simply choose a growth site at random and occupy it (the Eden model again).The average height of the surface is given byh = 1 ∑N sh i , (14.11)N swhere h i is the distance of the ith surface site from the substrate, and the sum is over all surfacesites N s . (The precise definition of a surface site is discussed in Problem 14.11.)Each time a particle is deposited, the time t is increased by unity. Our main interest is howthe width of the surface changes with t. We define the width of the surface byi=1w 2 = 1 ∑N s(h i − h) 2 . (14.12)N si=1In general, the width w, which is a measure of the surface roughness, depends on L and t. Forshort times we expect thatw(L, t) ∼ t β . (14.13)The exponent β describes the growth of the correlations with time along the vertical direction.Figure 14.12 illustrates the evolution of the surface generated according to the Eden model.After a characteristic time, the length over which the fluctuations are correlated becomes comparableto L, and the width reaches a steady state value that depends only on L. We writewhere α is known as the roughness exponent.w(L, t ≫ 1) ∼ L α , (14.14)From (14.14) we see that in the steady state, the width of the surface in the direction perpendicularto the substrate grows as L α . This scaling behavior of the width is characteristic ofa self-affine fractal. Such a fractal is invariant (on the average) under anisotropic scale changes,that is, different scaling relations exist along different directions. For example, if we rescale thesurface by a factor b in the horizontal direction, then the surface must be rescaled by a factor ofb α in the direction perpendicular to the surface to preserve the similarity along the original andrescaled surfaces.Note that on short length scales, that is, lengths shorter than the width of the interface, thesurface is rough and its roughness can be characterized by the exponent α. (Imagine an ant walkingon the surface.) For length scales much larger than the width of the surface, the surface appears tobe flat and, in our example, it is a one-dimensional object. The properties of the surface generatedby several growth models are explored in Problem 14.11.Problem 14.11. Growing surfaces