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

Surface Structures of Laplacian Erosion and Diffusion Limited

Annihilation

Y.Kim and S.Y.Yoon

Kyung-Hee Univ.Kyung-Hee Univ.DSRGDSRG

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

ys

ymax

Random walking(Diffusing)particle

Dielectric Breakdown Model (DBM)

• In conducting phase, the electric potential ( = 0 = 0)

satisfies Laplace equation 02

• The growth velocity is proportional to some power of local electric field.

Pi,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.)

)(,

)(1,

)(,1

)(1,

)(,1

)(,

)1(, 4

1 kji

kji

kji

kji

kji

kji

kji

jijiji CCp ,,,

jiC ,1


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.


1. The boundary condition (x, y, t) = 1 in far from the particle sea(yb1) 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)

)(,

)(1,

)(,1

)(1,

)(,1

)(,

)1(, 4

1 kji

kji

kji

kji

kji

kji

kji

3. The annihilation probabilities are assigned at the each site (x, y) on the substrates.

hx thx

thxthxP

, |),,(|

|),,(|),,(

yb

(x,h)(x,h)

(x,h)

(x,h)

(x, yb, t) = 1

2=0


Pb= 0.25 (Simple Diffusion Limited Annihilation (Krug & Meakin, 1991)) Pb=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


4. A particle is annihilated by comparing a random number to the annihilation probability.

(1-Pb)/3

ymax

Killed particle

Killing line Starting line

ysPb

(1-Pb)/3

(1-Pb)/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.

zL

tfLthhW ~])]([[ 2/12

Surface Width

Scaling Relations

z

z

LtL

LtttLW

~),( /z

Theorical Interpretations of the Surface Structures


3

1)1,(),1(),1(

)10()1,(

b

bb

PyxPyxPyxP

PPyxP

z = 2 [I.M.Kim & H.Kim (1993)]

),(),( 2 th

t

thx

x

)(2

21

L

tfLW 2,

4

1,

2

1 z

Pb=1, Edward-Wilkinson Universality Class

qqq

hqt

h 1

( hq ; Fourier Transformation of surface height h(x) )

)/(2 LtfW

)]}/4exp(1ln[)/{ln(2

2 LtaLD

W


z =1 [Krug & Meakin (1991)]

ht

h m)( 2

)()(),(),(

0),(

ttxxDtxtx

txd

2

11 mz

1z

ResultsLaplacian Erosion ( = 1)

Diffusion Limited Annihilation ( Pb = 0.25)


122 zLtfWW

Pb= 0.2


Diffusion Limited Annihilation

122 zLtfWW

Pb=0.26 (z = 1.6)


Pb=0.73 (z = 2.09)

22 zLtfLW z

Biased Diffusion Limited Annihilation Scaling Property Summary


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 : Pb 0.16 : smooth phase (no roughening) * regime II : 0.17 Pb 0.25 : z=1 * regime III : Pb > 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 ( ) ???


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