Surface Structures of Laplacian Erosion and Diffusion Limited Annihilation Y.Kim and S.Y.Yoon

Kyung-Hee Univ. DSRG  Motivation  Diffusion Limited Aggregation (Deposition) (DLA; DLD) (Witten & Sander 1981) and Dielectric Breakdown Model (DBM) (Niemeyer & Pietronero 1986)  Diffusion Limited Deposition y = 0 Killed particle Killing line Starting line ysys y max Random walking (Diffusing)particle  Dielectric Breakdown Model (DBM) In conducting phase, the electric potential (  =  0 = 0) satisfies Laplace equation

The growth velocity is proportional to some power  of local electric field. P i,j : Growth probability : normalization factor  : over-relaxation parameter  The simulations for DLD and DBM shows a same forest of tree-like structures with nearly the same Fractal Dimension D (D= 1.7 in 2d.) Kyung-Hee Univ. DSRG

 Related Phenomena  Electrolytic polishing The diffusion of accepter ions(such as CN - or water moleclues) to the anode. The accepters reach at the anode, recombines with a metal ion which is removed from the sample.  CORROSION  Diffusion limited etching process Specified surface shapes by etching through inert mask  Aggregations, chemical reactions, particle coalescence, trapping by stationary sink, phase separation processes  Mullins-Sekerka instability The non-linearity of the Laplace equation is manifested.  Saffman-Taylor(1958) The experimental result using the fluids with a high viscosity ratio leads to the growth of disordered interface. Kyung-Hee Univ. DSRG

1. The boundary condition  (x, y, t) = 1 in far from the particle sea(y b  1) and  (x, y, t) = 0 in the particle sea is defined. 2. The Laplace equation  2  =0 is solved by the relaxation method.  Our Model  Laplacian Erosion (Anti-DBM) 3. The annihilation probabilities are assigned at the each site (x, y) on the substrates. ybyb (x,h)  (x, y b, t) = 1  2  =0 Kyung-Hee Univ. DSRG

P b = 0.25 (Simple Diffusion Limited Annihilation (Krug & Meakin, 1991)) P b =1 (Ballistic Annihilation (I.M.Kim& H.Kim, 1993)) 1. A particle starts from a random site on a starting line.  Biased Diffusion Limited Annihilation Kyung-Hee Univ. DSRG 4. A particle is annihilated by comparing a random number to the annihilation probability. (1-P b )/3 y max Killed particle Killing line Starting line ysys PbPb (1-P b )/3 Hopping Probability L

2. The probabilities of the hopping of the particle from a given site to nearest neighbor sites is assigned as 3. A particle moves away from the substrate and reaches a killing line, then this particle is abandoned and a new particle is chosen to start. 4. A particle reaches a nearest neighbor site of a particle in particle sea, then the particle in the sea is annihilated.  Surface Width  Scaling Relations  Theorical Interpretations of the Surface Structures Kyung-Hee Univ. DSRG

 z = 2 [I.M.Kim & H.Kim (1993)] P b =1, Edward-Wilkinson Universality Class ( h q ; Fourier Transformation of surface height h(x) ) Kyung-Hee Univ. DSRG  z =1 [Krug & Meakin (1991)]

 Results  Laplacian Erosion (  = 1)  Diffusion Limited Annihilation ( P b = 0.25) Kyung-Hee Univ. DSRG

 P b = 0.2 Kyung-Hee Univ. DSRG  Diffusion Limited Annihilation

 P b =0.26 (z = 1.6) Kyung-Hee Univ. DSRG  P b =0.73 (z = 2.09)

 Biased Diffusion Limited Annihilation Scaling Property Summary Kyung-Hee Univ. DSRG

 Result and Discussion 1. Laplacian Erosion (Anti-DBM) is physically equivalent to Diffusion Limited Annihilations. Surface Structures in Both models are described well by the linear growth equation with z=1. 2. Biased Diffusion Limited Annihilations have three regimes. In surface structure. * regime I : P b  0.16 : smooth phase (no roughening) * regime II : 0.17  P b  0.25 : z=1 * regime III : P b > 0.25 : z  2 (EW) The crossover from regime II to regime III is very sudden. 3. Laplacian Erosion with  > 1 ; z=1 or smooth phase (???) (*  =  : smooth phase) Laplacian Erosion with  < 1 ; z=2 or 3/2 (KPZ) (????) (*  = 0 : random annihilation (inverse of random deposition ;  =0.5) ) 4. What is the universailty class when erosion is governed by the Drift-Laplace Equation ( ) ??? Kyung-Hee Univ. DSRG