Self-expanding and self-flattening membranes

S. Y. Yoon and Yup Kim

Department of Physics, Kyung Hee UniversityAsia Pacific Center for Theoretical Physics, Seoul, Korea

2

Abstract

We study the dynamical scaling properties of self-expanding surfaces with the comparison to those of self-flattening surfaces [1,2]. Evolution of self-expanding surface is described by monomer deposition-evaporation model in which the deposition or evaporation process that reduces the globally maximal height or increases the globally minimal height is suppressed. We find that equilibrium surface fluctuation of the expanding surface shows anomalous behavior with roughness exponent = 1 in one-dimension (1D) and = 0 (log), zW 5/2 , where the 2D result is nearly the same as that of self-flattening surface. In case of nonequilibrium growing (eroding) surfaces, self-expanding surface dynamics shows the scaling behavior depending on the suppression probability u. [1] Yup Kim, S. Y. Yoon, and Hyunggyu Park, Phys. Rev. E 66, 040602(R) (2002). [2] Yup Kim, T. S. Kim, and Hyunggyu Park, Phys. Rev. E 66, 046123 (2002)

3

Backgrounds

In equilibrium state,

1. Comparison of static walk and surface roughening

a. Self-attracting walk Self-flattening surface

b. Self-repelling (avoiding) walk Self-expanding surface

In 1D, we can map a configuration of static random walk to a surface morphology of the restricted solid-on-solid (body-centered solid-on-solid) model. But in 2D, the random walk cannot be mapped to the surface morphology.

x

t

0

1

h = 2

-1

Random walk

Surface roughening1

23

4

5

1

2 3

4

5

4

2. General partition function of surface growth with global constraint

Two models are decided by sign of K in this partition function.

a. Self-flattening membrane ( K <0 ) 1d : anomalous behavior with exponents 1/3, 2/9, zW=/1.5 2d : The surface width grows logarithmically with time and its saturated value also increases logarithmically with size.

b. Self-expanding membrane ( K >0 ) 1d : ? 2d : ?

)}({

)}({21

)}({

1

)}({

)2ln(exp

lnexp

2

1)1(

2

1 minmaxmax

min

rh

rh

S

rh

hhn

rh

h

hh

S

zZ h

1,0 minmax hhSz

)2ln(][exp)}({

KSKZrh

Where, nh= the number of columns with height h ,

5

Step 1. Select randomly a site .r

Simulation algorithm of self-expanding membrane

* All the process should be satisfy the restricted solid-on-solid (RSOS) constraint,

( is one of nearest-neighbor bond vectors of a site in d-dimensional hypercubic lattice.)

1)()( arhrh

a

Step 2. Find the globally maximal (minimal) height hmax (hmin) among the height distribution on a d–dimensional substrate.)}({ rh

))(()( maxmin hrhhxh r

b. Deposition (evaporation) is attempted with the new probability u , if deposition (evaporation) process occurs at column with the minimal (maximal) height , if the minimal (maximal) height exists only one on the given configuration.

Step 3. a. Deposition (evaporation) is attempted with the probability p (1- p ) with the exception of b-case.

1)()( rhrh

1)()( rhrh

with p , with 1- p

u

u 0)( rh

6

Simulation Results

1. 1D results of self-expanding(SE) and self-flattening(SF) membranes in equilibrium state

a. Raw data ( u = 0.6 , p = 1/2 )

SE SF

7

b. Effective exponent

)1024~32,2(

ln

)()(ln)(

Lm

m

LWmLWL ss

eff

SE SF

* We expect normal RSOS behavior(Random walk) at u=0 in both of SE and SF membranes.

8

c. Scaling law of SF and SE membranes in 1D

TABLE 1 (Effective exponent, eff , t » LzW )

u 0.1 0.3 0.6 0.8

SE 0.33(1) 0.33(1) 0.32(1) 0.344*

SF 1.00(1) 1.00(1) 1.005* 0.969*

From above simulation results, we get

5.1,9222.0,3

133.0 WzSF : ( SF follows normal scaling relation.)

1SE : ( SE does not follow normal scaling relation. (X) )

( * simple extrapolation data )

)(~ WzLttW

WzL

tfLW

W

W

z

z

LtL

Ltt

,

,

Surface width, W, satisfies the dynamic scaling relation

-> Self-repelling (avoiding) walks

9

2. 2D results of self-expanding(SE) and self-flattening(SF) membranes in equilibrium state

a. Raw data ( u = 0.5 , p = 1/2 )

SE SF

10

b. In saturated regime (t » LzW)

174.02

1a

KG

Slope of Normal RSOS

TABLE 2 (Slope a, t » LzW )

u 0.1 (SE) 0.5 (SE) 1.0 (O. RSOS) 0.1 (SF) 0.5 (SF)

Slope a () 0.175 0.175 0.174 0.174 0.174

* M. Den Nijs, J. Phys. A 18, L549 (1985)

( KG : the effective Gaussian coupling constant (KG 0.9 ))

11

WzG L

tgL

KtLW ln

2

1),(2

W

W

z

G

z

WG

LtLK

LttzK

,ln2

1

,ln2

1

WW zz xxgLttW1

2 )()(ln

From W 2 (t«LzW), we can estimate zW as zW 2.5.

c. Scaling law of SE and SF membrane in 2D

Surface width, W, satisfies the dynamic scaling relation

* D.-S. Lee, and M. den Nijs, Phys. Rev. E 65, 026104 (2002)

Compared to the conventional logarithmic divergence (the above scaling law), the lattice size L is replaced by a logarithmically modified effective length, L/(ln L)1/4.

)()(ln

ln2

1),(

41

2

L

L

LL

KtLW

G

12

3. 1D& 2D results of self-expanding(SE) and self-flattening(SF) membrane in growing phase

SE (1D)

SF

SE (2D)

* In growing phase, SF follows normal RSOS behavior in 1, 2D. But SE membrane is more rough in 1D and 2D, membrane becomes more rough depending on substrate size L in 2D.

13

Conclusion

1. Self-expanding(SE) and self-flattening(SF) membranes in equilibrium state

a. 1D :

b. 2D : The surface width grows logarithmically with time and its saturated value also increases logarithmically with size.

5.1,9222.0,3

133.0 Wz

1SE ->

SF ->

( SE does not follow normal scaling relation.)

( SF follows normal scaling relation.)

From this scaling law, two models have the slope of RSOS 1/(2KG) 0.174 in saturated regime(t » LzW ) and we can estimate zW as zW 2.5.

2. Self-expanding(SE) and self-flattening(SF) membranes in growing phase

a. SF follows normal RSOS behavior in 1, 2D.

b. SE membrane is more rough than its SF in 1D, 2D, and it becomes more rough depending on substrate size L in 2D.

WzG L

tLg

KtLW ln

2

1),(2