Dynamics of Growing Interfaces in Disordered Medium: The Effect of Lateral Growth

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Dynamics of Growing Interfaces in Disordered Medium:

The Effect of Lateral Growth

Semjon Stepanow

To cite this version:

Semjon Stepanow. Dynamics of Growing Interfaces in Disordered Medium: The Effect of Lateral

Growth. Journal de Physique II, EDP Sciences, 1995, 5 (1), pp.11-17. �10.1051/jp2:1995110�. �jpa-

00248134�

Classification Physics Abstracts

05.70L 05.40 47.55M 68 70

Dynamics of Growing Interfaces in

_a

Disordered Medium:

The Efllect of Lateral Growth

Semjon Stepanow

Martin-Luther-Universitht Halle-Wittenberg, Fachbereich Physik, Friedemann-Bach-Platz 6, 06108

Halle/Saale,

Germany

(Received

I October1994, received _in final form it October 1994, accepted 18

October1994)

Abstract. We study the model of _a driven interface _{m a} disordered medium including the

KPZ-nonlinearity The coupled functional renormalization group flow equations for the disorder correlator and the coupling constant associated with the KPZ-nonlinearity _{possess a} strong coupling fixed-point for the interface dimensions d

= 1, 2. In d

= 1 _we get the roughness exponent and the dynamical exponent ⁱⁿ one-loop approximation respectively _as ( ₌ 0.8615 and

z = 1

The interface

growth

_appears in many different situations such _as fluid

displacement

in porous media

II,

_2], domain

growth

in random magnets [3-6], ^ballistic

deposition

[7, 8],

growth

of Eden clusters [9],

crystallization

of solid from _vapor

[lo-14].

Driven interfaces between two fluids in disordered media and driven domain walls in random magnets _possess a threshold in

dependence

_on the

driving

force. Below the threshold the interface becomes

pinned.

The

critical behavior of such interfaces above the threshold has been studied

recently [15,

16] on

the basis of the

equation proposed by Koplik

and Levine [2]

(fluid displacement)

and Bruinsma and

Aeppli

_[3]

(random-field Ising model)

~t~~ dz

fat

₌

7V~z

₊ F +

g(x, z), II

where _a

single-valued

function

z(x, t)

describes the

height profile

above _a basal

plan

_{x, ~t is} the bare

mobility,

F is _a

driving

force, _{i is} the surface tension.

g(x,z)

_is assumed to be

a Gaussian variable with the _{zero mean}

(g(x,z))

₌ o and the correlator

(g(x, z)g(x', z'))

₌

6~(x x')li(z z'), ~l(z)

has _a finite width and is assumed to

decay exponentially

for

large

z. The Eden

model,

ballistic

deposition,

etc _are believed to be described

by Kardar,

Parisi,

Zhang (KPZ) equation ii?]

~t~~dh fat

₌

iv~h

₊

'(Vh)~

₊

_{~(x, t), (2)}

where

h(x, t) (z

= h ₊

~tFt)

_is the interface

height

in the

co-moving

coordinate system, ^{I is} responsible for the lateral

growth

and

q(x, t)

is _a Gaussian white noise

satisfying

the condition

jnjx, t)njx', t')j

=

2D6dix x')61t t').

₁₃₎

© Les Editions de Physique 1995

12 JOURNAL DE PHYSIQUE II N°1

In contrast to the

quenched

noise g, the thermal noise ~ is delta-correlated with respect to t.

In connection with the

equations (I)

and

(2)

there _appear the

questions

_I) how the interface

dynamics changes

if in

(2)

instead of white noise _{one uses} the

quenched

randomness

g(x,z)?

and

ii)

how the lateral

growth

influences the

dynamics

of

(I)?

The

quenched

randomness _can be introduced into

(2) by considering

the

dynamics

with _a

quenched

lateral

growth. Repeating

the derivation of

(2) according

to

ii?] by assuming

_an

inhomogeneous

lateral

growth

instead of

(2)

we get

~-iazlat

=

iv2z

₊

_{vjx, z)ji}

₊

jvz)2 _)1/2

=

iv~z

₊

F(I

+

(Vz)~)~/~

₊

g(x, z)(I

+

(Vz)~)~/~, (4)

where

V(x,

_z

= F+

g(x, z)

is _a

quenched

random variable

being responsible

for

inhomogeneous

lateral

growth (compare

to

ii?]).

In contrast to

ii?]

_we write

~t~l

in front of

dz/dt

_on the left-hand side of

(4),

_so F _on the

right-hand

side is _a force.

Expanding (4)

in powers of

(Vz)~

we

get ii

with

KPZ-nonlinearity. Comparing (4)

^after the

expansion

mentioned with

(2)

_we

see that the bare value of I coincides with the

driving

force F. The

practical

motivation for

considering

_{a more}

general description

for the interface

dynamics

is the anomalous

roughness

found _m recent

experiments by

Rubio et al. [18]

((

₌ o

73),

Horvith et al. [19]

((

₌

o.81),

He et al. [20]

((

= o.65 o

91),

and in simulations

by Martys

et al. [21]

((

=

o.81),

and Nolle et al.

[22]

((

=

o.8).

The apparent

disagreement

between these

experiments

and KPZ'S

predictions

have stimulated much theoretical interest

[23-26].

The aim of this paper is to present the results of the renormalization group

analysis

of the equation

(4)

in the vicinity ^{of the threshold.} The main result of this

analysis

is I)

equation (4) belongs

to a new

universality

class

differing

both from that of

(2)

and

ii ),

and

ii)

equation

(4)

possesses a strong

coupling fixed-point

in space dimensions d ₌ 1, 2.

Just from the dimensional

analysis

_we find that the present

problem

has two relevant

coupling

constants: gi "

(li"(o)/i~)l~ ii

is

length)

and _g2

"

(li(o)/i~)l~

_l~ with _e

= 4 d. gi

is the

expansion

parameter of the

equation

of motion of the interface _{in disordered} medium without lateral

growth (1).

_g2 is

analogous

to the expansion parameter of the

KPZ-equation (2).

We find also two combinations of the parameter which have the dimension of the force

F~

t

Jl(o)(1/7~)

l~~~ _and

Fa

m

(li'(o) Ii)

l~~~.

F~

and

Fa give

rise to _an additive shift of the external bare force

F,

I.e. these _terms generate a threshold. Fa differs from _zero if the

correlator

li(z)

_{possesses an}

appropriate singularity

at the

origin,

which if _not present in the bare propagator _may ^be

generated by

the renormalization [15].

In order to carry ^out a

quantitative analysis

of

(4)

_{we use} the

path integral

method [27].

To do this _we rewrite the nonlinear

Langevin

equation

(4)

as a Fokker-Planck

equation

for the

probability density P(z(x),t).

The conditional

probability density P(z(x),t;z°(x),t°) averaged

_over disorder to have the

profile z(x)

at time t

having

the

profile

^z°

ix

at time t°

can

be

represented by path integrals

_as

P(z(x),

_t;

z°(x), t°)

= Dp

Dze~~, ₍₅₎

S=So+Sg+S~, (6)

So _" D

/

_dt'

/

_dx

_{p~(x, t')}

-1

/~

^dt'

/

_dx

_{p(x, t')(~t~~ I(x, t')}

_F

7V~z(x, t') _{), (7)}

to

s~ ₌

j~

^dt'

j~

^dt"

j

dx

pjx, t')

x

~)

+

ii)ix, t'))2/~izix, t') zix, t")) »ix, t")~/i

₊

_ivzix, t"))a,

₁₈₎

s~

=

j _j~

_dt'

j

_dx

_pjx, t')jvzjx, t'))2. _jg)

To

study

the shift of F and I _we consider the _average <

g(x,z(x,t))

>, where _{< >} is the average over disorder and the overline

implies

the average over the _zero mode

(center

of

mass)

of

z(x).

These _{averages can} be carried out

by using

the conditional

probability density P(z(x),t; =°(x),t°).

The result of

computing

the latter to the first order in the disorder

strength

and in the limit F _- o is

<

g(x,z(x,t))

> =

-Fffi, _(lo)

where

Ff

= -~t

fj

_dt

_f~ G(~j(t)li'(o+),

^and

_G(~j(t)

₌

exp(-i~tk~t)

_is the Fourier transform of the Green function of the diffusion equation. The

one-loop

computation of

F/ _yields F/

=

-(1/2)li(o)I _f~

^k~~. Thus

f~

_< o when I _> o which _means that the

KPZ-nonlinearity

acts

as a

driving

force.

Combining (lo)

and

(4)

_{we see} that F and I _are shifted in _a different _way.

The shift of I is two times

larger

than that of F. This difference _appears because in order to carry ^out the average over

g(x, z(x, t))

_we have to take the term g ⁺

(Vz)2 (see Eq.(4))

in the distribution function

P(z(x)~t; z°(x), t°),

with which the averages should be carried out.

Because of the different shift F and I _even

having

the _same bare values will become different under the renormalization. From this _we conclude that at the

threshold,

F Fc

- o, ^I

could differ from _zero. There is _a

widespread opinion

in the literature that the >-term at the threshold should be _zero [16,

23].

The argument ^{used in} [16] ^is based _on the

following. By dividing

the both sides of

(4) (without

the _term

iV2z) by (1+ (Vz)2)1/2

and

expanding

in

powers of

(Vz)2

_we get the

KPZ-nonlinearity

in the form

~-iazlatjvz)2. _iii)

By comparing

II with the

KPZ-nonlinearity I(Vz)2

_one concludes that I should be identified with the

velocity,

with the consequence that the latter

disappears

at the threshold. This

argument ^does not

apply

to

(4),

since the factor

ii

₊

(Vz)2)1/2

is missed in front of _i in

equation (4).

Let _us consider the

quenched

_average of

ill)

_over disorder and the thermal average over the center of _mass of the interface

analogous

_{as we} have done it above for the

quenched

force _{g. To do} this _we eliminate

~t~ldz fat

in II

according

to the

equation

of motion.

It is easy ^to see that the result of the average of this

quantity yields

the shift of I which is two times

larger

than that of F in accordance with the

analysis

carried out above.

Therefore,

at the threshold _m contrast to the conclusion made in [16] ^{1 is not}

expected

to be _zero. The quantities u and I behave

differently

also under the

multiplicative

renormalization. Whereas the

velocity given by

u ci

~t(F Fc)

renormalizes like

mobility

in accordance with

equation (12),

1 renormalizes in

according

to

(14).

Already

the existence of the

coupling

constant g2 is _a

signal

that the

universality dais

of

equation (4)

^{could differ from that of}

(1). Despite

this the

coupling

constant g2 could be irrelevant if I) the

fixed-point

value of the latter,

g(,

is _zero, and

it)

if the bare value g2,bare is

proportional

to F

Fc

and will

disappear

at the threshold. This would be the _case if the shift of F and I would be the _same.

Intuitively

_we do not see reasons for I to be _{zero at} the threshold. The bare value of I is the

driving

force, which in contrast to equation

(1)

acts

normally

to the surface.

14 JOURNAL DE PHYSIQUE II N°1

From the above discussion _we conclude that

equation (4)

represents a new

universality

class.

The _same conclusion _was drawn in [28], ^{where the authors studied} a model similar _to

(4) numerically.

We _now outline the

procedure

of

evaluating

the

path integral (5) by

Wilson's shell

integration

method. First _we separate

z(x, t)

= z<

(x, t)

+ z>

(x, t)

and

p(x, t)

= p<

(x, t)

+ p>

(x, t),

where

z>(x,t)

and

p>(x,t) only

contain Fourier components in the momentum shell A _< k < Ao.

Then _we

expand (5)

_up to the second order _{in S~} and _{up to} the third order in

S~,

_carry out the _{average over}

z>(x, t)

and

p>(x,t)

and represent ^the result of the _{average as}

e~~(~<,P<J

with renormalized

quantities

_{~t, 7,}

1, li(z).

The flow equations for the quantities under the renormalization _are obtained in the

one-loop approximation

_as follows

l~In~t=

_gi

(12)

1~ ^{ln1= ~~~g2, (13)}

l~

In1

=

-jg2, ⁽¹⁴⁾

l~li(z)

=

)li~(z)l~l~/7~ 7~~l~$ _()li~(z) li(z)li(o))

,

(Is)

z

where the factor

(d/20r)~/~

is absorbed into gi and _g2 and

=

I/A.

For I ₌ o these

equations reproduce

the flow

equations

^derived in [15]. ^In case of the uncorrelated

disorder, li(z)

_e

lio, equations (13-15)

reduce to the equations derived

by

Nattermann and Renz [29]. ^{It is} easy ^to

give

the connection of the flow

equations (13-15)

and that of the

KPZ-equation.

The number of time

integrations

associated with each

loop being

two for

quenched

disorder reduces to _one for the delta-correlated thermal disorder given

by (3).

Each time

integration

in the

loop bring

the factor k~~

resulting

in that the critical dimension for the

quenched

disorder is four and reduces to two in _case of thermal disorder

(KPZ).

The

diagram

representation of the

perturbation

series

is the _same in both _cases. To get ^the

right-hand

side of the flow equation ^of _i ^{in the}

KPZ-case,

besides the redefinition of the

coupling

constant g2 (g2 _"

li(o)l~/i~l~~~)

_one should

multiply

the

right-hand

side of

(13) by

an factor

1/2.

The factor

I/d

in

(14) changes

to zero in the KPZ-

case due to

compensation

of

diagrams.

The

right-hand

side of

(Is)

in KPZ-case is obtained

by

the

change li(z)

_-

~li(o).

2

The flow equation for the function

f(z)

defined

by li(z)

=

i~ l~~~~f(z/l~)

is obtained from

(IS)

_as

1(flz)

=

e

2( +

jg2) _fix)

+

(zf'lz)

+)g2f~lz)/f1°) $

()f~lz))

⁺

f"lz)no). j16)

The

fixed-point

solution of

(16)

for small _z is found in the form

with

~~~

ii

₌

e

2(

+

~~~ ~gl) ^l18)

12d

²

⁾ ₍₁₉₎

j*

₌

(1/3)

_e

(

+

~ ^§2

We note that

f(

is the

fixed-point

value of the

coupling

constant gi The

nonanalyticity

of

f(z)

at the

origin

contributes _to the flow

equation

of _g2 which is obtained

by using (12-19)

_as

1(§2

^"

^2(§2 _)§~

~~~~

with the

fixed-point g(

=

2d( le.

The

roughness

exponent

(

is obtained from

demanding

that

f* (z) decays exponentially

for

large

_{z [30].} We solved the

equation, rhs(15)

₌ o, and found for

(

₌ o.8615

(d

= 1) ^and

(

₌ o.815

(d

₌

2) positive exponentially decaying

^solutions. In fact in

solving rhs(15)

₌ o

numerically

_we

captured

the values of

(

at which

f(z) decays

in most fast way. The _same

procedure

in the _case of random

Ising

model (g2 _"

o) yields (

₌

e/3 (random fields)

and

(

₌ o.2083e

(random bonds)

_as it _was

previously

obtained

by

Fisher [30]. ^{At d} = 3 there is _no

positive exponentially decaying

solution. We expect that for random

fields,

which

we consider here, the bare disorder correlator

being

positive for all distances remains

positive

under renormalization [15]. ^There are also other solutions where

f*(z)

has

negative

values.

The

roughness

exponents associated with solutions

having

_one node _are

(

=

o.548,

o.412, o.3 at d ₌ 1, 2, 3,

respectively.

The critical exponents which _we will compute ^below can be

expressed through

the

roughness

exponent

(.

The

fixed-point

value of _gi is obtained from

(19)

_as

g(

₌ (e

5(

₊

12(le)/3. (21)

The

dynamical

exponent z is derived from the relation iilLit _r~ ^{l~ and} t

r~ l~ _as follows

z = 2

e/3

₊

2(/3 2(le. (22)

The renormalized

quantities

_~ti and _{ii are} solutions of the

equations (12, 13)

at fixed

points g(

and

g(.

The correlation

length (

can be introduced _as the characteristic

length

above which

the term ut in the random force

g(x,

ut +

h(x, t))

dominates the term

h(x, t)

_[15].

Using

the relations t _r~ l~ and h

r~ l~ _we get

(

_r~

u~l/~~~~J. Combining

the

power-law dependence

of the correlation

length

_{on u} and that of the

velocity

_on

f

= F

Fc,

_u

r~

f°,

_{we get for} the correlation

length (

r~

f~",

with _u

=

@/(z ().

The

velocity

exponent _@, u r~

f°,

_is derived from the relation _u

= ~ti

f

with ~ti r~

(~l

_as

= 1

ug(. Combining

this expression with the

expression

for the correlation

length

exponent u we get the

velocity

exponent _as

g =

(-6e

₊ e2 ₊ 6< + e<)

/(6(-e

+

e<)). (23)

Substituting

the values for and _z into the above derived expression ^for the correlation

length

exponent u we get ^{the latter} as follows

u =

e/(21e

+

e<)). (24)

The

fixed-point

^{value for}

f(

is

f(

₌

-(e 5(

+

8(le)1/~. _{By using}

the numerical values for the

roughness

exponent

(

obtained from the numerical solution of the

fixed-point equation

for

fix)

_we give in Table I the values of the exponents ^{and the}

fixed-point

values of the

coupling

constants computed for d

= 1, 2.

16 JOURNAL DE PHYSIQUE II N°1

Table I.- Numerical values

of

the crttical exponents ^{and the}

fixed-point

values

of

the

coupling

constants

for different

_space dimensions.

d

(

_z fl _"

§I §I

I o.8615 1 o.163 1.174 o.713 o.574

2 o.815 1.062 o.208 o.844 o 938 1.63

The renormalization group

analysis presented

here

gives

a

fixed-point

for dimensions d < 3, which _is

probably

_{a strong}

coupling fixed-point.

In _case of _a

weak-coupling fixed-point

the

fixed-point

values of the

coupling

constants

being proportional

to _{e =} dc d

(dc

is the critical

dimension)

tend to zero in

approaching

the critical dimension. In the present case

(see

the two last columns of Tab.

I)

the

coupling

constants increase

by increasing

the

dimensionality.

The situation resembles that _one in the

KPZ-equation

where the

fixed-point

_was found below d ₌

3/2.

The

roughness

exponent,

(

₌ o.8615 _at d ₌ I, lies in the _range of

experimental findings [18-20] ((

= o.63

o.91).

Because _we have to do with _a strong

coupling fixed-point

_we

should not expect to get a

good

agreement with the

experiment.

We note that the

fixed-point

values of _the

coupling

constants achieve their minimal values

(<

I _at d

= I, _so the

comparison

to the experiment is expected to be the best at d

= I.

In conclusion _we have considered the

dynamics

of _an interface in _a disordered medium

taking

into _account the

KPZ-nonlinearity.

This model

belongs

to _{a new}

universality

class

differing

from both that of interface in _a disordered medium without lateral

growth

and that of KPZ

equation.

From the

(strong) fixed-point

of the

coupled

^functional renormalization group flow

equation

for the disorder correlator and the flow equation for the

coupling

constant associated with the >-term _we have derived the critical exponents for the dimensions d

= 1, 2.

Acknowledgments

I wish like to thank T. Nattermann and L.-H.

Tang

for _very useful discussions.

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