Non-universal roughening of kinetic self-affine interfaces

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Non-universal roughening of kinetic self-affine interfaces

Yi-Cheng Zhang

To cite this version:

2129

Short Communication

Non-universal

_roughening

of

kinetic self-affine

interfaces

Yi-Cheng Zhang

INFN and

_Dipartimento

di

_Fisica,

Università di

_Roma,

Piazzale A.

_Moro,

00185

_Roma,

_Italy

(Received

on

_July

_{24, 1990,}

_accepted

on

_July

_2q

₁₉₉₀₎

Abstract. - Recent

experiments of wetting

immiscible

_displacement

_{in porous media} and bacteria

colony growth

are considered as realizations of the Eden interfaces. I _propose an

_explanation

of

the

_discrepancy

in the

_scaling

behavior between the

_experiments

and the

_theory.

Due to an

_instability

inherent in the

_growth

_processes, the _presence of unbounded non-Gaussian noise can violate the naive

scaling universality.

LE

JOURNAL

DE

_PHYSIQUE

1

_Phys.

France 51

_{( 1990)}

2129-2134 1 er OCTOBRE

_1990,

Classification

Physics

Abstracts 05.70Ln -

61.5OCj

- 68.35Fx -

05.40+j

In _many

_physical

situations an interface

_separates

two distinct

_phases.

When one

_phase

ad-vances into the other under a

_driving

force a kinetic interface is created. Kinetic interfaces cannot

generally

be studied in the fashion of

_{equilibrium physics.}

One

_simple

model is the Eden model

[1]

which was

_originally

motivated to describe

_{biological growth.}

On a

_strip

_geometry

an Eden interface can be described

_by

a stochastic differential

_equation

_[2].

The

_system

is in a d + 1 dimensional

_space,

with the direction of

_growth

_singled

out. Denote

_by

h(x, t)

the

_height

of the interface at time t and

_position

x which is a vector in the d-dimensional base

_plane.

We have the

_following

evolution

_equation:

where the coefficient of the surface tension and the

_coupling

constant are assumed to be

_1,

since

they

can be absorbed

_by

a

_rescaling

_{of h,}

x and t.

_{7](x, t)}

is stochastic noise with a

_two-point

correlation:

where z > denotes the ensemble _average over realizations of

_noise,

for _many

_pratical

_purpose

it can be

_{replaced by}

a

_temporal

_average.

2130

The

_{key ingredients}

of

_{equations (1)}

and

₍₂₎

are the surface tension term Ah which tends to

locally

smoothen out

_{fluctuations;}

a non-linear term

_(Vh)2

from

₁

_{+ (-Vh)2}

=

1+

(Oh)2/2+ ...,

which

_represents

the

_uniformly

lateral

_growth;

and stochatic noise that is believed to be

_ubiquitous

in realistic

_systems.

There is considerable theoretical

_[3]

_(analytical

and

_{computational)}

research

_activity

on

prob-lems related to

_{equation (1).}

It is believed

_by

_many

workers in the field that

_{equation (1)}

pro-vides a valid

_{hydrodynamic description}

of a broad class of interface models

_involving

the local

growth

mechanism. The most

_interesting

observable is the _average width

_{[2] w(L, t),}

which is

ex-pected

to be a

_scaling

function of the

_sample

size L and the

_growth

time t.

_{w(L, t)}

ri

LX f

(t / L Z) ,

f (u -> oo)

->

_const,

f ( u -+ 0)

-->

ux/l.

Surfaces

with X

1 are called self-affine

_fractals,

_they

are

anisotropic

and

_globally

flat

_{(w/L ->}

_0).

Theoretical

_{analysis gives}

the

_scaling

_exponents

_exactly

in

1+1

dimensions: X

=

1/2

and z

= 3/2. Higher

dimensional

_exponents

so far are

_only

known

numerically

[3] .

Very recently

there are some

_experimental

realizations of the interface

_growth,

which should

inject

new enthusiasm into the field. Rubio et aL

_[4]

and Horvath et al.

_[5]

have

_performed

experiments

on

_wetting

immiscible

_displacement

in a

_porous

medium. Another

_{experiment by}

Vicsek et al.

_[6]

studies bacteria

_(Escherichia

coli or Bacillus

_{subtilis) colony development}

on a

nutrient

_{(agar) plate.}

This is _very close to the

_{original spirit}

of Eden.

Despite

of their

_apparantly

dissimilar

_{microscopic origins,}

the above

_growth

mechanisms are

local and

_{approximately}

_homogeneous.

1

_attempt

to view that the above interfaces are

_effectively

described

_by

the

_simple

_{equations (1)}

and

_(2).

Rubio et al.

_reported

that their self-affine interface has the

_roughening

_exponent

_X = 0.73 :f: 0.03. Horvàth et aL found that for their

_experiment

x m

0.81,

they

also measured the time

expo-nent

_{x/ z}

0.65. For the bacteria

_{expansion problem,}

Vicsek et al. estimated that the bacteria

colony

has a self-affine border with the

_roughening

_{exponent X}

= 0. 78 ± 0.06. While these

findings

are

_very

_interesting,

the values of the

_roughening

and time

_exponents

are

_certainly

in

contradic-tion with the current theoretical

_{prediction [2, 3] x = 1/2.}

Elsewhere 1 shall show that

_long

_range correlated noise cannot account for the

_{larger roughening}

_exponent.

In this work 1

_propose

that the

_discrepancy

between the

_experiments

and

_theory

is due to the fact that in nature there is no Gaussian

_noise,

as it is

_usually

assumed in theoretical models. In

most of statistical

_{physics microscopic}

details are not

_important

on

_large

_scales,

this _goes under

the name of

_{universality.}

For the

_present

_{growth problem}

I shall show that

_microscopic

details

can indeed influence

_large

scale behavior in a _{substantial way,} thus

_violating

the naive

_universality

concept.

1

_rely

on the

_hypothesis

that

_equations

_{(1), (2)}

are still a valid

_hydrodynamic

_description

of the above

_experiments,

but abondon the

_requirement

_thatq

is a Gaussian or truncated noise.

To be

_precise

1 assume noise is

_{independently}

distributed on each site

_(a

lattice

_{regularization}

is

understood) according

to the

_following

distribution

_density:

This distribution does not have an absolute

_cutoff,

_{singularly large}

values of _ri can

_appear.

This

does not

_imply

_however,

that

_{infinitely large}

value will

_appear.

_{In any finite}

_sample

of N

_degrees

of freedom there is a

_typical

cutoff on the

_{largest q}

_value,

_?7max- We have

Nlq’+m -

_0(1),

which

implies

1 simulate a discrete version of

_{equation (1)}

on a

_square

lattice

where i and t are

_integers,

i runs over

_only

even indices if t is even, odd if t odd.

h (i, t )

and

_{7(i, t )}

are continuous variables. The initial condition is h =

0,

periodic boundary

conditions are used in the transverse direction

_i,

the transverse size is L.

_{Equation (5) implies}

a checkerboard

_updating

rule. It can be considered as a simultaneous ballistic

_deposition

model

[7] .

Equation

₍₅₎

can be

derived in the directed

_{polymer representation}

_[8]

in the zero

_temperature

limit. Elsewhere I will show that the

_particular

choice of

_parameters

of

_{equation (1)}

does not influence the

_scaling

behavior for the same _M. 1

_conjecture,

on the basis of the

following

simulations,

that there is a

(sub-) universality

for each _M. 1 have taken this

_liberty

to choose the much

_{simpler equation}

₍₅₎

to _carry out most of the simulations.

An interface is said to be in a

_stationary

state when its _average width reaches a constant. In

figure

1

_snapshots

of an interface at successive time

_steps

are shown. The tranverse size is L = 1000. The interface has evolved for 50000 time

_steps

after

_starting

from a

_straight

line

_geometry.

It is

_{for y}

= 3 and the interface’s

_roughening

_exponent

is X

= 0.75 ± 1. The

_growth

direction is

upwards.

The

_longer

time interval between the

_snapshopts

is 30 time

_steps,

the shorter interval at the

_top

is 5 time

_steps.

One can notice that

_occasionally

there are some

_{abnormally large}

thrusts

on the

_growth

front.

Fig.

1. - Successive

snapshots

of an interface evolved

_according

to

_equation

_(5).

The

_longer

time intervals are 30 time

_steps,

the shorter ones are 5 time _steps.

Figure

2

_presents

two

_examples

of the

_scaling

behavior of the interface’s width

_against

_system

size on a

_log-log

scale. The data are for two different values

_{of y}

_(2.5

and

_3).

The statistics is

2132

Fig.

2. - The interface width

w(L) (vertical)

against

the size L for two values of li

_{(2.5, 3).}

The data appear

to scale with the _exponent

_{w(L) - LX , x z}

0.82, 0.75,

respectively.

In

_figure

3 1 show the

_systematic

assessment of the simulation results. The estimates of the

roughening

exponent X

is

_{plotted against}

_M.

_{For M ---+}

2 it

_appears

that the interface has the

max-imally

attainable self affine

_roughening

_{exponent X}

= 1. Elsewhere

[9]

1 will address what can

happen

to an interface

_{if M :5}

2

_{(Lévy distribution).}

In the inset 1

_plot

the sum of the two

expo-nents X

+ z, where the

_{temporal scaling}

_exponent

z has been calculated

_{independently}

[10]

in the directed

_{polymer representation.}

We see that the sum is

_{approximately}

2,

thus

_confirming

the

exponent

identity

which can be derived on the basis of a Galilean invariance

[11] .

We may conclude that the

_growth

_{equation (1)}

combined with unbounded noise

_(3),

can

ac-count for the observed

_{larger roughening}

_exponent.

Even

_though

the simulations are made on

rather small

_lattices,

1

_expect

that

_larger

lattices would not

_change

at least

_{qualitatively}

the

_present

conclusion. It is

_worthy

_noting

that the

_experiments

are also

_performed

on limited sizes.

_However,

there are _many

_questions

to be answered. 1 should

_emphasize

that 1 do not have solid

_microscopic

justifications

as to

_why

in the

_{experiments power-law}

noise distributions are

_preferred

over Gaus-sian or truncated ones. 1 want nevertheless to _argue that it is

_equally

hard to ascertain that

_they

are not there. After

all,

this has to do with

_microscopic

details to which

_{traditionally}

not much attention has been

_paid.

It is

_plausible

that unbounded noise in real

_systems

comes from rare combinations of all

_contributing

factors,

the

_larger

the

_system,

the rarer a combination can be.

Is there a _way to check if the

_present

_approach

is

_applicable

to the

_experiments,

besides the fact that

_roughening

_{exponent X}

can be made to

_agree?

1 believe that the effective noise distribution can be revealed: examine the

_snapshots

on the

_top

of

_figure

1,

if the time interval is short

_enough

between successive

_shapshots

the effective

_microscopic

noise could be

_picked

_up

and

_analyzed.

This should be

_possible

for the

_experiments

as well.

It _may be

_surprising

that

_microscopic

details do influence the

_large

scale behavior for the

_growth

Fig.

3. - Estimates

of x

(vertical)

vs. various values of g. Error bars are from a

_subjective

assessment. In the inset the sum x + z is

compared

to the

_expected

value 2.

thrusts with

_respect

to _average

_places.

These rare thrusts have much more influence than their

small statistical

_weight

would

_suggest

to have. This is so

_by

two kinds of mechanism :

₁₎

Amplifi-cation :

_during

the lateral

_growth

a rare

_large

thrust will

_expand

_laterally

to cover its

_neighboring

area as if there were _many simultaneous

thrusts;

2) Memory:

a

_large

thrust

_probably

will result in

a hill on the

interface,

this

_geometry

will be remembered for a

_long

time. If

_during

this

_long

time there

_appears

another rare thurst

_they

can have combined effect to

_roughen

the interface.

Summary.

In this work 1

_point

out an

_instability

inherent

_{(or hidden)}

in the

_growth

model

₍₁₎

and

_(2).

This

instability

can be attributed to the above

_amplication

and _memery mechanisms of rare,

abnormally

large

noise values. 1

_suggest

that the

_{experimentally}

observed

_{larger roughening}

_exponent

is a manifestation of this

_instability.

Acknowledgements.

1 thank J.

Amar,

H.

Herrmann,

V Horvàth and D. Stauffer for

_helpful

discussions. This work took about lhr CPU on a

_Cray-YMR

References

[1]

EDEN

M.,

Symposium

on Information

_Theory

in

_Biology,

Ed. H.P.

_Yockey

_{(NewYork, Pergamon Press)}

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[2]

KARDAR

M.,

PARISI G. and ZHANG _Y.-C.,

_Phys.

Rev. Lett. 56

₍₁₉₈₆₎

889.

[3]

For recent reviews see KRUG J. and SPOHN

_H.,

Solid Far From

_Equilibrium:

_Growth,

_Morphology

and

Defects,

Ed. C. Godrèche

₍₁₉₉₀₎

to be

_published;

VICSEK

_T.,

Fractal Growth Phenomena

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Scientific,

Singapore)

1989;

FAMILY

_{E, Physica A,}

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[4]

RUBIO

_M.A.,

EDWARDS

_C.A.,

DOUGHERTY A. and

_{GOLLUB J.P.,}

_Phys.

Rev. Lett. 63

₍₁₉₈₉₎

1685.

[5]

HORVÁTH

VK.,

FAMILY F and

VICSEK T.,

Emory preprint

1990,

to be

_published.

[6]

VICSEK

T,

CSERZÖ M. and HORVÁTH

V.K.,

Self-Affine Growth of Bacterial

Colonies,

to _appear in

Physica

A

_(1990).

[7]

FAMILY F. and VICSEK

T., J.

Phys.

A18

₍₁₉₈₅₎

L75.

[8]

KARDAR and ZHANG

Y.-C.,

Phys.

Rev. Lett. 58

₍₁₉₈₇₎

2087.

[9]

ZHANG

Y.-C.,

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_{published (1989).}

[10]

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Y.-C.,

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