Interface roughening and pinning

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Stéphane Roux, Alex Hansen

To cite this version:

Stéphane Roux, Alex Hansen. Interface roughening and pinning. Journal de Physique I, EDP Sciences,

1994, 4 (4), pp.515-538. �10.1051/jp1:1994157�. �jpa-00246927�

Classification Pllysics Abstracts

61A3H 64.60A 68.45G 61.72L

Interface roughening and pinning

Stéphane

Roux

(1)

and Alex Hansen (~>*)

(1) Laboratoire de Physique et Mécanique des Milieux Hétérogènes

(**),

Ecole Supérieure de

Physique et Chimie industrielles, 10 _rue Vauquelin, 75231 Paris Cedex 05, ^France

(~) Groupe Matière Condensée et Matériaux, URA CNRS 804, Université de Rennes 1, Campus

de Beaulieu, 35042 Rennes Cedex, France

(Received

27 September1993, accepted in final form 24 December

1993)

Résumé Nous étudions _un modèle simple _pour analyser l'accrochage d'une interface

sur des

impuretés et le décrochage _sous l'effet d'une pression appliquée, dans _une limite quasi-statique.

Ce modèle est très voisin du modèle '~Robin Hood" introduit _par Zaitsev. Il s'applique _en parti-

culier à l'invasion d'un fluide mouillant

(imbibition)

dans _un milieu _poreux hétérogène contenant

un fluide immiscible. Nous discutons les relations entre _ce modèle et d'autres approches _propo- sées _pour décrire

ce phénomène. Le front d'invasion acquiert _une structure auto-affine, _avec

un développement de la rugosité selon _une loi de puissance du volume injecté. La valeur de l'exposant de

rugosité

apparent se compare bien _avec des _mesures expérimentales, mais _nous

montrons que l'exposant réel est hors de portée des méthodes d'analyse habituelles. La distribu-

tion,

f(d, Al),

des distances, d, entre pores envahis à _un intervalle de temps ^Al peut être décrit par une loi d'échelle

f(d, Ai)

₌

d~~~(dllù).

_{Cette loi peut être obtenue}

en identifiant _une structure hiérarchique de "bouffées" dans le signal de pression. Ces "bouffées" sont qualitative-

ment similaires à celles observées lors du drainage

(invasion

d'un fluide _non

mouillant),

dans _un

régime de percolation d'invasion.

Abstract. We study _a simple model for the pmning of _an interface by impunties with

random strengths, and the depinning due to the applied _{pressure, m a} quasi-static propagation limit. The model is very close to the _so called "Robin Hood" model introduced by Zaitsev.

It is designed to describe _e-g- the invasion of _a wetting fluid

(imbibition)

in _a heterogeneous

porous medium

containmg

_a second immiscible fiuid. The relation between this model and

other previously proposed approaches is discussed. The front of the mvaded domain is shown to

develop _a self-affine structure with _an increase of the roughness _{as a} power-law of the injected

volume. The value of the apparent roughness _{exponent can} be favorably compared to _some

expenmental measurements although _{we argue} that the true roughness exponent is out of reach of commonly used methods. We show that the distribution

f(d,

Al) of distances d between

discrete local invasions at _a time interval Al _can be described by _a scaling law

f(d, Al)

₌

d~~~(dllù).

_{This form}

can be obtained from the identification of _a hierarchical _structure of ~'bursts"

m the pressure signal. Those ~'bursts" _are qualitatively similar _to those observed

in quasistatic drainage, (1.e. invasion percolation), although charactenzed by dilferent scaling indices.

(*) Also at Institutt for Fysikk, NTH, N-7034 Trondheim, Norway.

(**) URA CNRS 857.

1 Introduction.

The geometry ^{and trie}

dynarnics

of trie

propagation

of interfaces in _a

heterogeneous and/or noisy

environment is _a

problem

which _occurs in _a

variety

of fields. Much progress bas

recently

been achieved

through

trie

quantitative analysis

of trie fluctuation of trie interface both in _a number of

experimental examples,

and

through

trie

development

of various models which have revealed _very

general scaling

^laws

iii.

in various

growth models,

it has been realized that the

roughness

of the front _can be described

by scaling laws,

^which accounts both for _a self-

afline geometry below _a characteristic

length

scale the correlation

length

and the time

development

of this correlation

length.

A few selected works

focusing

_on this property ^have been collected in _a recent book [2].

However, despite

this _{progress, some} situations still

display

features which cannot be _con-

vincingly

described

by existing

models without

resorting

to some additional

hypotheses (e.g.

long

_range correlation [3],

singular

distribution of _noise [4], or even

inhomogeneous spatial

distribution of

pinning

centers with _a

power-law dependence

_on the distance from _an

injection

side _[Si

...)

whose

physical

^foundation remains unclear. More

recently,

the rote of

pinning

has been outlined in _a series of model [5-10] we will discuss further.

One such

example

of interface

propagation

which is

lacking

a proper

understanding

_is trie structure of imbibition fronts _{even in} trie

simplifying

limit of

quasistatic

fiow. Imbibition is trie invasion of _a

wetting

fiuid in _a porous media filled with another immiscible less

wetting

fiuid. We will in trie

following

_use this

problem

to derive trie rules of trie model _we

study,

and to

highlight

its relation with _a model called trie "Robin Hood" model RH model in trie

following

introduced

recently by

Zaitsev

[loi

and later studied

by Sneppen

_[9]. Albeit _we will

essentially

refer to the imbibition

problem,

the RH model and results _may be

applied

to other _cases.

Among

other fields of

application,

_{one may} quote ^trie

growth

of bacteria

colony [11],

trie motion of vortex lines in

superconductors

_[12], of dislocations in

crystals [loi,

_or

even trie

roughening

of crack fronts in fracture [13]. ^{In addition trie RH model shares} some

similarities with other models

proposed

in trie context of solid friction [14] or

earthquakes [15],

of avalanches in

granular

media

[16j.

It should also be

emphasized

that trie motivation for

studying

those

problems

_is not

simply

to bave _a proper statistical

description

of trie geometry ^{of trie}

interface,

but also to

gain

_some

insight

into trie

global pinning

and

propagation

of _an interface. In trie domain of

wetting,

heterogeneities

are

responsible

for trie _occurrence of _a

hysteretic

behavior of trie

(macroscopic)

contact

angle

between two fiuids [17].

Understanding

and

quantifying

this

hysteresis

is still _a

major challenge.

It bas been

recognized

[18j

long

_ago

that,

_m trie _case of _a

non-wetting invading fiuid,

trie sc-called

~'drainage"

_case, trie

development

^{of trie interface between} trie two fluids in _a porous medium could be

accurately

^described

by

_an "invasion

percolation"

model.

Experiments

[19j bave confirmed trie

validity

of this limit. Trie _case of

~'imbibition",

1-e- when trie

invading

fiuid is

wetting,

is _a

question

which bas been addressed much _more

recently

in terms of statistical

analysis.

Trie

development

of imbibition fronts bas been studied

expenmentally recently

in two or quasi-two dimensional

geometries by

dilferent groups

[20-23]

^{and trie results} they bave

obtained _are

quite comparable (although

trie _very precise value of trie

roughness

exponent

they

measured bas been

slightly

controvers1al

[22]).

However, these values _are much

larger

than

initially expected

from theoretical models.

One of trie most celebrated models in trie field of interface

growth,

trie

Kardar,

Parisi and

Zhang equation

[24], ^which

appeared

_a

priori

to be _a reasonable

description

for such _a

problem

bas led to

quantitative

dilferences with respect to observed

expenmental

data.

Attempts

to

modify

this model in order to match trie measured

properties, through

trie introduction of

a

power-law

noise distribution [4], or a

long

_range time _or space correlated noise [3] ^{or non}

homogeneous spatial

distribution of

pinning

centers [Si seem to be difficult to

justify

for trie

experimental

systems which bave been studied

although

_some

analysis

bas been carried in this direction [25]. ^Other

proposais

bave been made such _as trie

suggestion

that directed

percolation

or vanants [26, 27] may

provide

_a suitable frarnework.

However,

at present, trie

justification

of such models is

mostly

based _on trie numerical agreement ^{between measured and}

proposed

exponents.

A much _more

promising approach

bas been

independently proposed by

Kessler et ai. _[7] and Parisi _[8] who bave both noted that

taking

into account a

quenched

_noise leads _{to a}

change

of trie

scaling

exponents ^{of trie}

original

KPZ

equation. However,

both studies bave been carried out in

regimes

such that trie _new

scaling

which could be

favorably compared

to trie

experiments

was

only

of _a limited

validity. Sneppen

_{[9] bas} shown that _a

simple

modification of trie above

proposed

models with

quenched

noise reveals

naturally

trie

required scaling properties by keeping

trie

driving

force _on trie interface

exactly

at its

depinning threshold,

in _a

spirit

similar to invasion

percolation.

In fact trie model studied

by Sneppen

had

originally

been introduced

a little earlier in _a dilferent context,

namely

trie

propagation

of dislocation

lines, by

Zaitsev

[loi.

Trie "Robin Hood" model _was trie _name chosen in trie latter reference and thus _we will

use it in trie

folllowing.

Cieplak

and Robbins

[28-30]

bave introduced _{a two} dimensional model hereafter referred to _as trie CR model for

finding

static

equilibrium configurations

of _an interface between two

wetting

immiscible fiuids in _a

simple

geometry. Their

analysis

leads to two very distinct

regimes depending

_on trie value of trie contact

angle

between trie

invading

and trie

defending

fluid with respect to _a threshold value. On trie

non-wetting

side

(for

trie

invading fiuid),

_one encounters an invasion

percolation regime

characterized

by

_a fractal geometry ^{of trie invaded} _pores, above

a lower cut-off scale which

depends

_on trie contact

angle.

On trie

wetting

side of trie transition, trie invaded domain is compact, ^{but trie interface}

displays

_a self-affine

roughness.

Trie latter

regime

thus _appears to be _a natural candidate for

modelling

trie above mentioned imbibition

experiments

and _a

quantitative

comparison ^{bas revealed}

good

agreement ^with

experimental

data

[29].

However,

trie CR model is both _{very precise m} its

definition,

and very ricin in trie diflerent types of basic _processes at

play

in trie local

displacement

of trie interface. A very careful

analysis

of trie dilferent types ^of

possible

restructuration of trie interface bas to be done _in order to find

a new

position

^of

equilibrium.

It is thus of _uttermost

importance

to try to

single

out in trie

complex interplay

of trie basic local mechanisms trie essential features which determine trie

"universality

Mass" and thus trie

scaling

properties of trie

problem.

In trie

following

we will discuss _{a very} crude

simplification

of trie CR model aimed _{ai pre-}

serving

trie few

ingredients

that _we believe _are essent1al to

reproduce

trie imbibition

regime.

We will show that _one encounters

naturally

trie RH

model,

and indeed trie

scaling

properties of

trie invasion _are

correctly reproduced.

Trie

simphcity

of trie RH model allows _us to

investigate

in detail _many of trie statistical features which _can be accessed

experimentally.

We focus _our attention _on two aspects: ₁₎ trie

geometrical

features of trie

front,

and ii

)

trie pressure

signal.

Trie

analyses

_we

perform

_are based _on trie usual

description

of front

growth

[31] as well _{as on}

geometrical analyses

of trie interface. We also

study

trie fluctuations of trie pressure

signal

in terms of

"bursts",

_{a concept}

brought

^{forward for} trie

non-wetting

invasion

percolation regime [32-34].

2. Trie CR mortel.

Before

presenting

trie RH

model,

_we recall

briefiy

trie model introduced

by Cieplak

and Robbins

[28-30].

^Trie

starting point

is trie

description

of trie geometry ^{of trie} _porous medium in two

dimensions. It is

represented by

_{an array} of disks whose centers _{are on} trie sites of _a

regular triangular lattice,

and whose radii _are

randomly

distributed. Trie latter rule

provides

trie

only

randomness of trie model which is otherwise deterministic. Trie model is static in trie _sense that trie _pressure in each fiuid is assumed _to be

homogeneous.

Across trie interface there exists _a pressure

discontinuity

due to

capillary

forces which is related _to trie _curvature of trie

menisci,

which is thus constant for all _pores. As _a

result,

trie interface between trie two fluids thus consists of _a series of circular menisci of

equal

radii

connecting

trie disks. An additional

parameter ^{is needed for}

determming

trie front geometry ^for a

given

_pressure

drop, namely

trie contact

angle

which

gives

trie direction of trie tangent to trie minuscus with respect to trie tangent to trie disk at trie contact

point. Being given

trie _pressure

drop

and trie contact

angle,

it is thus _a

simple

exercice to compute trie location of _a meniscus between two

disks,

and to check that trie obtained meniscus does not

overlap

with _a

neighbouring

disk _or another meniscus.

Let _us also note that for _a

given pair

of disks

(a pore),

there exists _a maximum pressure above which _no stable solution _can be found. This allows _one to construct _an admissible

position

of trie interface step

by

_step.

Trie

interesting

aspect of trie CR model is however not to generate ail _{or any} admissible front

locations,

but rather to follow how _a

given

interface will _move to _{a new}

configuration

of

equilibrium

after _an increase in pressure or in

injected

volume. In order _to find this transition,

a detailed

analysis

of trie local

restructuring

_is needed. Three diflerent mechanisms _can be identified:

a

)

Bitrst: when trie pressure exceeds trie maximum value that _can be

supported by

a

given

_pore,

no stable

configuration

_can be found and thus its radius tends to _increase until it encounters a

neighbouring

disk _or

interface,

two cases which _are considered below.

b)

^{Toitch: trie} arc

connecting

two disks

(1- j)

intersects another disk

(k).

In this _case, trie meniscus _is broken into two _arcs

(1- k)

and

(k j) (see Fig. la).

c) Ouerlap:

two arcs

(1- j)

and

(j k) overlap

_on one disk

j.

In this _case, trie two menisci will merge ^to form _one

single

₍₁

k)

meniscus

(see Fig. lb).

Obviously,

_once such _a

restructuring

bas taken

place,

additional _{ones may again} be necessary, until _a stable

configuration

is found.

> w

Touch

Overlap

la) ¹⁶⁾

Fig. l. Schematic _view of the two major mechamsms through which trie front evolves _m the original model introduced by Cieplack and Robbins. The location of the front formed by the _{menisci is}

determmed step by step by searching for the ~'touchs" (a) and "overlaps" (b) which

are the most

limitative _m terms of pressure.

Clearly,

trie "burst" _case is trie _one which is dominant in trie invasion

percolation regime.

For trie

wetting

invasion

regime,

trie

interplay

of ail three mechanisms and

particularly

^of "touches"

and

~'overlaps"

makes it diilicult to

single

out

only

_one mode _as discussed in reference [29].

It is important to note _a

major

diflerence between trie first two _cases and trie third _one.

Bursts and touches _are local

properties

^of one

single

_pore.

Although

that _may not be _an

efficient _way of

handling

trie

problem numencally,

_one could think of

assigning

to each pore

(e.g. pair

of

neighbouring disks)

_an ^{interval of} pressure that _con be

supported. Ignoring

trie

"overlap" mode,

trie

problem

would then reduce to _an invasion

percolation problem

in trie _pore space, (1.e. on a lattice dual to that of trie

disks).

On trie contrary, ^trie

overlap

mode cannot be

expressed

_{as a}

single

_pore property, as it involves trie interaction of two

neighbouring

_pores. It thus _seems

likely

that this mechanism

plays

_a

determining

^role in trie

regime

where compact ^clusters are

generated.

Trie

spirit

of trie

simple

model _we will present below is to retain

only

this

mechanism,

and to

investigate

its

consequences in detail.

We will furthermore take disorder

explicitly

into account since it is

obviously

_an essential

mgredient.

As _a final observation

relying

_on trie

reported experimental results,

_{we note} that trie fronts

generated

in trie

wetting regime

_are

rougir

but

asymptotically

flat

(trie roughness

divided

by

^trie system _size ^tends to _zero for _an infinite system

size).

We will _use this property

by assuming

that trie interface _can be considered _as

being

defined

by

_a

single

valued function of _a coordinate axis

along

trie _mean direction of trie front. Thus _we will

neglect

trie _occurrence of

overhangs.

As far _{as one} is

only

interested in trie

scaling

features of trie geometry of trie front and related

properties,

trie details of trie CR model should not be relevant. Dur aim is to

simplify

but preserve trie essent1al features which determine trie

universality

Mass of trie model. In

some sense we are

looking

^for a model for imbibition

playing

_an

analogous

^role _as ^trie invasion

percolation

model for

quasistatic drainage.

3. Mortel definition and relation witl~ otl~er mortels.

We propose ^to model trie front _{as a}

single

^{valued function of} an abscissa

along

trie front. We

moreover discretize

it,

and represent it _{as a} series of

heights

_% where1 _{is a} discrete index. With

this restriction, _we forbid trie _occurrence of

overhangs,

which _are not

expected

_anyway to

play

a

significant

role in trie

wetting regime.

Such would not be trie _case for trie _non

wetting regime.

We will _see below that in fact trie conclusions _we reach _are in confiict with this

hypothesis.

Hence,

trie domain of

validity

of trie model _we will

study

_is restricted _to scales small

enough

so that trie

slope

of trie front is small

compared

to 1. For

comparison

with trie CR

model,

trie

% could be

thought

of _as trie distance of trie disk which is part ^{of trie interface} at _an abscissa

(see Fig. 2).

We mentioned earlier that _we

only

wished to take into _account trie

"overlap"

mode. This is done

by assigmng

_a threshold _t~ to trie admissible citruatitre

(%-1 2%

+

%+1)

^{of trie interface}

at each point. ^{This threshold is} meant to take into account trie diflerence of orientation in two consecutive menisci, as well _as trie distance between trie disks. Trie directedness _we impose

on trie front allows _one

single pair

^of menisci to be considered _at each

site,

whereas _in trie

original

CR

model,

ail

pairs

of bonds

onginating

from _a site had to be taken into _{account to} describe ail trie

possible overlaps properly.

Trie thresholds _are chosen at random from _a uniform

distribution between o and tmax. For _a

given configuration

of trie

front,

_%, and of trie

pinning

strength

t~, we bave to decide where trie front will _move. When trie _pressure increases, ^trie

angle

between trie two menisci

touching

_a

given

site decreases in _a non-linear fashion.

However,

o-i) o) o+1)

Fig. 2. Illustration of the discrete model studied in this paper. The front shown

as a bold line

consists in _a series of discrete heights _y~. Pinning centers _are located at fixed abscissa along the

thin vertical bues but at random heights and with randomly varying strength. The force exerted by

the front _on the pinning center _is approximated by the dilference of slopes between the two segments (1

1)-j)

and

j)-(1+1).

assuming

that ₁₎ the distance between

pinning

centers _varies

only

in _a small

interval,

and that ii

)

trie deformation of each meniscus is limited to _a small _{range, we con} linearize finis relation.

Moreover,

_we will consider that trie rate of

change

of trie

angle

at any site with trie _pressure

is a constant, ^A.

Or, altematively,

_{we assume} that trie

fiuctuating

part ^{of this}

quantity

is absorbed in trie randomness of trie thresholds. A site1 will be able _to support trie pressure p

only

if trie

following

condition is met

(vi+1 2vi

+ vi-i +

AP

_{< t~}

(1)

Since A _can be _{seen as}

defining

trie scale for pressure, we will in trie

following

set A

= 1.

Therefore,

trie _{pressure necessary} to move trie interface will read

p =

minlti

₊

(2vi

_vi-i

vi+i)1 (2)

We do not include trie

description

of trie fiow

(we neglect

trie viscous

forces),

and _as in trie invasion

percolation

model and in trie CR

model,

_we atm _at

modelling only

trie

quasistatic

limit.

Therefore,

_{we assume}

that,

at each time step, ^trie pressure is

adjusted

to trie threshold value of equation

(2).

This

corresponds experimentally

to _a control of trie fiow _rate rather than trie pressure. Trie evolution of trie interface _is considered _as

irreversible,

and thus trie interface

is not allowed to recede from _an invaded _area.

Once _a condition for

overlap

is

satisfied,

_say at

site1,

trie interface is

moved, by increasing

%

by

_an increment d~, and a new threshold

t[

^is

picked.

~~~ÎÎ~~~

(3)

t~-t

~

The

displacement

d~ is chosen at random from _a uniform distribution _over trie interval [dm,n,

dmax].

^{To check trie elfect of}

having

a range of

possible jumps,

we will consider in trie

following

two cases: either dm,n ₌ o, _or dm;n ₌ dmax _m ^{trie latter} _case, ^{trie interface}

always

lie _{on a}

lattice, y~/dmax being

integer ^{valued. Trie} ratio ₌

tmax/dmax

is trie

only

free parameter ^of trie model. It determines how fast trie

"memory"

of _a

given

conformation vanishes. We will show below that this parameter

only

affects _some aspects ^{of trie model} when

àmin _" àmax.

Trie

important

features _are 1) ^that

pinning

is taken into account, ^with a

qitenched

character

(trie

threshold t at _a

given

site does not

change

with

time), 2)

trie

propagation

is qitasistatic

(trie

interface is moved at one and

only

_one site

during

^each

elementary step).

Trie model _we

bave defined here shares _some similarities with other

approaches

such _as those

developed by

Kessler et ai. [7] and Parisi [8] in

particular. However,

trie second condition

diflers,

in trie

sense that _a constant

driving

force _was

imposed

in both references rather than _a controlled

flow. This diiference _{appears to} be

significant

since at

large

flow rate, ^trie interface

propagation

is

expected

to be described

by

trie

Kardar-Pansi-Zhang equation,

1e. it should

belong

to a

diflerent

universality

Mass.

Both of trie above mentioned features bave been included in _a model introduced

by Sneppen ("model

A" of Ref.

[9]).

Trie

only

_very minor dilference between this mortel and the

one we bave defined above _{concems an} additional rule

imposed by Sneppen

that trie

slope

of trie interface should

only

_assume two

possible

values _-1 and 1. This constraint is enforced _as

an additional rule. This is in contrast with trie hmit _we bave mentioned earlier _- for trie lattice version of _our model dm;n

=

dmax.

In trie latter _case, trie constraint _{y~ y~-1}

naturally

results from trie

updating

rule.

This model is also

extremely

close to _a mortel which bas been introduced earlier

by

Zaitsev

[loi,

in order to model trie

propagation

^of dislocation lines in _a

crystalwith impurities

which act as

pinmng sites,

in _a low temperature

regime.

^{Trie mortel} _was ^then named trie "Robin Hood"

model, RH,

and thus _we will trie _saine

terminology.

In fact trie RH model is very close in its definition to another model which bas received _a lot of attention

recently.

It is _a cellular automaton model introduced

by

Bak et ai. [16] ^{to illustrate} trie concept of

"self-organized criticality". Many

a variant of this model bave been

considered,

and dilferent

applications

bave been discussed

ranging

from

sandpile

avalanches to trie

large

scale _structure of trie universe. Trie basic features however of this model _are trie

following:

on each site1 of _a

regular

network

(for comparison

with trie RH

model,

_{a one} dimensional line is to be

considered)

_a variable _{z~ can assume a} few discrete values which increase very

slowly

in time

land

in _an

irregular

fashion _to introduce _a

noise).

Above _a threshold _{z~ > z~,} often considered to be

independent

of trie site and constant with time, trie site1 relaxes to _a lower value _{z~ - z~} A and

simultaneously,

trie

neighbormg

sites

j

of1 experience _an increase

z~ - z~ ⁺ B where B

=

A/n

if _n is trie number of

neighbors(1).

If _{we now} interpret _{z as} trie local _{curvature m} RH model then trie local front evolution

equation (3)

_appears to be identical _as in trie SOC model.

Hence,

trie dilference with usual SOC is trie

heterogeneous (and quenched)

nature of trie threshold.

Similarly,

it is

possible

to draw _a

correspondence

with friction models such _as trie famous

"Burridge-Knopolf"

model introduced in trie 6o's and which bas

recently

been trie

subject

of _an intense

activity

_[15]. In this

model,

_a linear chain of massive

blocks,

connected

by

elastic spnngs, is

subjected

to a

driving

force

along

^{trie chain direction.}

Moreover,

friction of trie blocks with their support

plane

is described

by

_a

velocity weakening

law of

friction,

and

a maximum

(homogeneous)

static force

they

_cari sustain. Trie

specific

form of trie

velocity dependent

friction law is not believed to be fundamental _as

long

_as it is

velocity-weakemng.

Trie most fundamental dilference between this mortel and the RH model is that inertial eifects

are descnbed. Trie motion in finis _case bas been shown to be

chaotic,

with _{a very}

large

(~ ^{Violation of} trie latter relation has also been investigated in "non-conservative" models. Critical exponents _are observed to be dependent _on the parameter (A

In -B)

but qualitatively, the phenomenon

is unchanged.

JOURNAL DE PHYSIQUE T 4 " 4 APR'L 1994 0

distribution of

slip

events, ^which involve _a number of blocks

ranging

^from _one to trie entire chain. Trie distribution of trie size of

slipping

blocks in _one event shows _a

power-law

behavior for small sizes which _can be

compared

with trie burst size distribution discussed below and _an additional contribution from events very close to trie system ^size. Trie latter _seems to be

specific

of trie inertia of trie blocks.

In trie

following,

_we will

analyse

trie results of extensive numerical simulations.

System

size varies in trie range ^L = 64 to L ₌ 1024. A "time" parameter is introduced _so that _an interval of 1

corresponds

to L individual _moves in _a system ^of size L

(on

_{average one move per} unit

length during

_one time

step).

This unit of time is sucl~ tl~at _{on average a} volume

(dmax

+

dm;n)L/2

is

injected.

Tl~e time range varies

according

to what property is examined. In order to

study

trie

early development

of trie

roughness,

trie time

laps

is chosen to be at most

equal

to trie

lengtl~.

For

analysing

tl~e front structure at _a fate stage time, ^{tl~e front is first} grown for _a time wl~icl~ ensured tl~at most

early

stage ^{features bave died}

off,

and then tl~e furtl~er evolution is

followed _{over a}

long period

^of time. Tl~en ail tl~e data collected _are

averaged

_over

independent

realizations

(up

to loo realizations for L

= 1024 and t ₌ 1024 (1.e. a total of 10~ individual

moves).

Most of tl~e simulations _are

performed

witl~

tmax/dmax

₌ 5, and dm;n

= o.

Trie results _are

presented

in trie

following

_way: first _we will discuss tl~e

geometrical

_aspects

of trie invasion front, then _we will report ^trie

analysis

of trie pressure

signal. Finally,

_we will

investigate

trie influence of various parameters used in trie definition of trie model

(ratio

of

maximum threshold _over maximum

displacement tmax/dmax, amplitude

of tl~e

displacement

interval dmax

dm,n)

and discuss in

particular

_a transition towards _a faceted

growth regime.

Conceming

tl~e

geometrical

aspects of trie front

development, figure

3

gives

_a

qualitative picture

of trie front geometry at diiferent times for

= tmax

/dmax

₌ 5, dmax ₌ 1 and dm;n

= o.

Figure

4 shows

qualitatively thjt

trie parameter does non _seem to

play

_any

significant

role when dm;n ₌ o.

However,

when trie interface is constrained to lie _{on a}

lattice,

1-e- dm;n

=

dmax,

trie situation is diiferent. For _> 1, we find

again

a similar pattem, as for

figure

3, but when

approaches

1, trie

underlying

discretisation of _y becomes much _more visible. A little above 1, ^{trie interface} consists of

straight

facets with

slopes

-1, o _or 1, and thus it can be cast in trie

general

framework of Solid-On-Solid models.

Figure

5 shows _an example of such _a situation for

= I.oS. For _< 1, trie interface is

always straight

if _one starts with _a

straight

line. If trie initial

configuration

is _a sawtooth

profile

_y

assuming

trie values o and

1/2

for odd and

even

index1respectively,

then trie value of wl~icl~

corresponds

to _a facetted

regime

is 0. It is

interesting

to note tl~at in tl~e

original

CR model for _a small

disorder,

tl~e lattice _structure of tl~e support ^{of trie disks also becomes} apparent ^and tl~e interface is facetted in _a similar _way

as m trie

example pictured

in

figure

5.

Starting

from _a fiat

front,

tl~e

roughness

first increases witl~ tl~e

injected

volume and tl~en it saturates and becomes

independent

of lime _as

usually

_seen in sucl~ models. Tl~e widtl~ of tl~e front _{w is} written [31] ⁱⁿ a

scaling

form _w

= L"

ifi(t/L°/~)

witl~ _a

scaling

fucntion _ifi wl~ich

assumes trie

followmg asymptotic

behavior:

ifi(x)

c~ x° for _{x <} 1 and

ifi(x)

_c~

x~

for _{x »} 1.

Therefore trie two important

scalings

of trie front _are trie

steady

state

regime

w c~

L",

and trie

early

lime behavior _{w c~}

t~.

In sections 4 and 5 _we will first discuss trie features of trie

fully developed

front _as it is trie easier to

analyze expenmentally.

Then _we will tum to trie

early

stage

regime

ⁱⁿ section 6.

4. Structure of tl~e front in tl~e

steady-state regime.

We consider trie

profile (y~)

for _very

large

lime _{so as} to _ensure that _{we are m a}

steady

state

regime.

In order to

mvestigate

trie

self-affinity

of trie

front,

we used two diiferent methods:

~= 5.oo ~= ios

Fig.

3

Fig.

4

Fig. 3. Illustration of the front geometry at dilferent limes for

= 5, dmax

" 1 and dn~;n = 0.

Periodic boundary conditions _are implemented in the horizontal direction. The initial geometry ^is the bottom fine. The system size is here 128, and the last interface shown _on the figure _was drawn at lime 128.

Fig. 4. Same _as figure 3 for _a dilferent set of parameters: ₌ 1.05, dmax ₌ 1 and dn~;n " 0. No striking dilference with figure 3 is visible.

1) ^window measurement of the

roughness, 2)

power spectrum of trie front. Trie results _are discussed in trie

following

subsections.

4.1 ROUGHNESS ANALYSIS. Let _us first report trie results of trie first method. We _measure trie

roughness

of trie front _over windows of varying

lengths.

^{Let à} be trie window size. For each

starting point, 1, we define trie two estimates of trie

roughness

w2(1, à)~

=

~2 ~j _y)16 £ _A/à

~<~<i+à

<~<i+à

(4)

wm(1, à)

_{= max}

(A)

_mm

(A)

i<j<1+6 1<j<1+6

We then average w2 and _wm for fixed à _over ail available

starting positions

_1,

W(à)

₌ lW(1,

à))1 (5)

For a self-affine

profile,

trie

roughness depends

_on trie window _size à _{as a}

power-law,

wc~ô", (6)

~

= i 05

Fig. 5. Same _as figure 3 for _a dilferent set of parameters: ₌ 1.05 and dmax

= dmin _# 1. The front

now lies _{on a} lattice which is clearly apparent _on the figure. When À approaches 1, the model tends to _a Solid-On-Solid model, with facets of slope -1, 0 _or 1 without having to impose this condition

artificially.

where _a is the

roitghness

e~ponent [35]. ^{Trie data used to} compute ^this

roughness

_are relative to _a system size of L =512, first _grown for _a lime t ₌ 2048

(during

which _no data is

collected).

Then further 4096 lime units _are used to measure trie

roughness

_every 64 lime steps. A total of ₁₀₀ such

samples

_were

generated

and

averaged

_upon. We limited trie maximum value of à to 100 in order to avoid _an artificial saturation which would result from _an

undersampling

^for

large

window sizes

(too

few

independent windows).

Figure

6 shows trie evolution of both estimates of trie

roughness

_{as a} function of trie window

size in a

log-log

scale. Trie second moment of trie

height

distribution _w2 shows _{a nice} power-law behavior with _an exponent _{a =} o.85. A very

slight upward

curvature is

detectable, although l~ardly

visible _on trie

plot.

Tl~e _wm estimate of tl~e

rougl~ness

also shows _a

power-law

behavior

but with _a

langer

downward curvature. Trie best

regression

line

gives

_an estimate of _{a m} o.88 for à _> 20

(whereas

_a best fit _over trie entire range gives a m

o.92).

From trie curvature of

both

estimates,

it _seems

likely

that trie

slopes

of these two sets of data bound trie value of _a.

Tl~us,

tl~e _use of tl~is metl~od leads to tl~e final estimate

a m o.86 + 0.03

(7)

We note tl~at albeit tl~e

quality

of tl~e lits is quite

satisfactory,

there exists _a marked size- eflect. For _a smaller system size, _very

good

lits witl~

power-laws

_are obtained _over tl~e entire range but with _a smaller exponent ⁱⁿ

particular

wl~en _{w2 is} used. For L

= 128, trie

roughness

exponent is measured to be _{o m} 0.75.

This result is consistent with trie estimates of _{a as}

given by Robbins,

1-e- 0.75 in reference [29] or 0.81 _{as a} later estimate _m reference [30], ^which was obtained

using

a similar method

2.5

w~(ôi

2.0 W~(ôi

m 1.5

lOÎ j Éi _1-Ù

o.5

ù-ù

ù-ù 0.5 0 1.5 2.0

log~~(à)

Fig. 6. Roughness

w(à)

_us. window size à for the long time regime of trie invasion. Trie two data sets refer to the two estimates defined in equation (6); (o) for _w2

là)

and

(+)

for

woo(à).

The system size _is 512, and the other parameters are = 5, dmax _# 1 and dmin _" 0.

with trie _w2 estimate. In reference [7], a value of o.75 is mentioned

although

it is obvious from trie data shown in

figure

1 that _a

power-law

fit is non quite

satisfactory. Experimentally,

there exists _some scanner in trie determmation of _a, trie values

reported

in trie literature _are o.73 [20], o.91

[22],

^o.81 [21] ^{and o.63} [26]. ^{At low}

capillary numbers,

He _et ai. [23] obtained _a broad

scatter in trie determmation of trie exponent in trie range o.8 to o.9.

in most studies of trie

scaling

of

front,

trie

pair

correlation function _or variants of it such

as trie two quantities _w2 or wm introduced in this subsection is trie

only

feature which is studied in order to _assess trie self-affine geometry ^{of trie front. This} is however _a

dangerous procedure

because of trie

following

remark: it is

always possible

to consider self-affine

profiles

with _a

roughness

exponent

larger

than _or smaller than o. This _can be done e-g-

through integration (resp. derivation)

which increases

(resp. decreases)

trie

roughness

exponent

by

_one.

Nevertheless,

it may be shown

mathematically

that trie method used above cannot reveal _an exponent

larger

than 1 _nor smaller than o. In practice, trie limitation is _{even more severe.}

As _{soon as} trie

roughness

exponent of _a

profile

_is

larger

than o.8, trie measured value

using

_a

w2 or wm method lies

always

in trie _range o.8 to

o.9,

^{for ail values} of trie

original

_exponent [36]. ^Trie

forger

trie system size, trie

forger

^trie measured value

(~). Thus,

finis method cannot

be used to

quantify accurately

trie

roughness

exponent, but rather to establish _a lower bound.

We bave thus used another _measure of trie geometry ^{of trie front: trie} power spectrum.

4. 2 FOURIER ANALYSIS oF THE FRONT. Trie front _{structure con} aise be studied

through

trie _power spectrum of trie

profile P(k).

Trie _square modulus of trie Fourier transform of trie

(~) A simple illustration of this _is given by applying the analysis method _to the deterministic

parabolic profile _y

=

~~. _A back-of-the-envelope calculation gives

wm(à)

= ôL. Thus, the scaling

of _wm with à at fixed L grues an exportent 1, whereas for à

= L, the scahng of _wm with L gives the exponent 2, _1.e. the physically relevant value. A similar elfect _can also be shown for the other roughness estimate _w2.