Compact or fractal patterns in diffusion limited growth

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Compact or fractal patterns in diffusion limited growth

Martine Ben Amar

To cite this version:

Martine Ben Amar. Compact or fractal patterns in diffusion limited growth. Journal de Physique I,

EDP Sciences, 1993, 3 (2), pp.353-363. �10.1051/jp1:1993136�. �jpa-00246728�

Classification

Physics

Abstracts

61.50C 68.70 47.20

Compact

_or

fractal patterns ^{in diffusion limited} growth

Martine Ben Amar

Laboratoire de

Physique Statistique,

Ecole Normale

Supdrieure.

24 _rue Lhomond, 75231 Paris Cedex 5, France

(Received

23 June 1992,

accepted

22

July 1992)

Rdsumd. Des _structures fractales ont 4t4 observ4es _en

digitations visqueuses

_en cellule de Hele-Shaw et _aux temps

longs

lorsque ^{les forces} capillaires deviennent

n4gligeables.

Par opposi- tion, la croissance de monocristaux h

partir

d'un _germe ponctuel fait apparaitre des dendrites

croissant dans 4 _ou 6 directions selon la

sym4trie

cristalline. Une

comparaison

entre des r4sultats

num4riques

et des donn4es

exp4rimentales

explique

l'origine

de l'instabilit4 de d4doublement des

pointes

_en croissance radiale et montre

qu'elle

peut Atre supprim4e _par

l'anisotropie

cristalline.

Abstract. Fractal viscous

fingering

patterns are observed in _an infinite Hele-Shaw cell at

long

times when the

capillary

forces become

negligible.

On the contrary,

growth

of

monocrystals

from _a

punctual

seed shows dendrites

growing independently

in 4 _or 6

directions, according

to the

crystal

symmetry. A close

comparison

of numerical and

experimental

data

explains

first the

origin

of the

tip-splitting instability

in radial

growth

and shows that it _can be inhibited

by

the

anisotropy

of surface tension.

Among

all his

subjects

of

interest,

Rammal

studied,

around

1984,

the statistical

properties

of fractal structures and has shown that

they

_can be described

by

_very few

parameters

such _as the fractal and the

spectral

dimensions

[1, 2].

^Our

point

of view here will be rather different

mostly geometrical,

since _we

plan

to understand

why,

in nature, ^fractal structures _{occur or}

do not _occur. We consider here

growing systems

which _are

generally

^believed to

satisfy

the

approximation

relevant to the

dynamics

of continuous media but which

reveal,

at

long times,

a fractal

pattern:

_a ^{well known}

example

is the

instability

of Saffman and

Taylor (S-T)

in the radial

geometry [3].

We will restrict _our

study

to this situation that _we would like to compare to

an

analogous

_one: solidification from _a

tiny

nucleus in infinite medium. What is rather

amazing

in the viscous

fingering instability

in _a Hele-Shaw

cell,

is that the observed

structure,

at

long times, crucially depends

_on the

geometry

^{of the}

experimental

cell: it has been known for _a

long time,

since the

pioneering

work of Saffman and

Taylor

_[4], that the

instability

in _an infinite

linear cell

gives

_{a very}

steady finger

whose relative width

(compared

to the width of the

cell)

is _very close to

1/2,

_a result which has

intrigued

_many

physicists working

in this field

[5].

^This very stable

finger

_proves~ if necessary, that this

growth, governed by diffusion,

_can

give compact

354 JOURNAL DE

PHYSIQUE

i N°2

structures in the _presence of

walls, contrary

to what

happens

in infinite

geometry.

Another

strange experimental

fact is the difference between

patterns

^obtained in viscous

fingering

[3]

and solidification at low

growth

rate

[6].

The

equations

which govern both interfaces _are

quite

the _same but not the

shape

observed at

long

times

(see Fig. i).

Solidification from _a

unique

nucleus

gives

^{several dendrites}

growing along

the main axis of

symmetry

of the

crystal.

Each dendrite _grows

steadily

with

a

tip roughly parabolic

whose size is

given

in terms of

macrocospic properties

^{like the}

anisotropy

of surface tension

[7].

^This

opposition

between observed

patterns

in two different contexts _can be

explained by

deterministic

equations

^without noise. This is

the aim of this _paper that _we would like to dedicate to Rammal's _memory.

1. Position of the

problem:

the

tip-splitting instability.

Fractal

growth

_occurs when the

system experiences repeated tip-splitting

events: that is _a

splitting

in two _or three

parts

^of a

protuberance

which becomes unstable

during

the

growth

and _{recovers some}

stability by

_a

systematic

division of its characteristic

lengthscale.

This time

dependent

_process is _very hard to

handle,

_even

numerically,

for diffusive instabilities due to its non-linear and non-local character. This is

why

most of the work _on

crystal growth

and viscous

fingering

has been restricted to the

steady growth regime

of isolated structure

[7].

^This

corresponds

to

investigate

the stable

tip

of _a

long pattern growing

^with a constant

velocity,

it

also _means that the

growth

is unaffected

by

the other

growing

structures

always _present

in

a

real

experiment.

But this

asymptotic regime

is reached

only

after _some time: the dendrites _are

usually

initiated from _a

punctual seed,

then grow

radially

from this seed with _a

"petal" shape.

Here _we want to

study

this intermediate

growth regime

where all

fingers

initiated from the

same seed interact. We will mimic this

complicated

interaction

by

_a suitable

geometry

deduced from the

experimental

observation: _we will

imagine

that each dendrite _grows in _a fictitious but

well-defined

sector-shaped

cell. This situation is also met in radial viscous

fingering

which has been studied in detail _very

recently _{[8, 9].}

In _a Hele-Shaw

cell,

the air-oil interface is circular and stable _up to _a critical

radius,

then

develops fingers

which _grow in

independent

fictitious sector. The number of

fingers

varies from five to seven, it

depends

on the initial conditions and

on defects _on the

plates.

This

growth

is _{more or} less constrained

by

the

neighbours.

It remains self-similar in time _as

long

_as the

tip

radius is less than _a critical value. At this

point,

the

tip

becomes too

large

to be stabilized

by

the

capillary

effects and

tip-splitting

_occurs. After the

tip-splitting,

two _or three _new

fingers

_{reappear, grow,} then

tip-split again.

The cascade of

successive

tip-splitting

events leads to a very ramified structure and at very

long

times to the well known fractal _structure characteristic of this

geometry (see

Viscek's book for _a detailed

discussion and

photographs, _[10] ). Recently,

^{much work has been}

performed

in order _to

explain

the existence of self-similar

fingers

and to

quantify

at least the first

tip-splitting

event [9] ^which takes

place

when the self-similar solutions

disappear.

Crystal growth

is

generally

believed to be

qualitatively

similar to viscous

fingering,

when the

experiment

is realized at low

supersaturation

li and in _a bidimensional

geometry. But,

such _a cascade of

tip-splitting

events has not been

reported

for

crystals

to _our

knowledge.

The

growth

^scenario _seems to be the _same

except

^{that the} number of

petals

is

perfectly

well defined

by

the

anisotropy

of surface tension: 90 _or 60° for _a four-fold _or six-fold

anisotropy.

In earlier

experiments [10],

^the

importance

^{of the}

anisotropy

_on viscous

patterns

^has been revealed

by engraving

the Hele-Shaw.

plates.

At fixed

anisotropy,

_a transition between ramified and dendritic

patterns

was observed when the

applied

_pressure is increased and _a

phase diagram

of the

patterns

^{has been sketched. The lower the}

anisotropy,

the

higher

the _{pressure necessary} to

a)

Fig. _i. - a)

Long

_time behaviour of a _gitalized viscous fingering ^{pattern by njecting} air

into a ilicon oil. _Courtesy of

356 JOURNAL DE

PHYSIQUE

I N°2

modify

the

dynamics.

These features motivate _our numerical

investigation

for the existence of

growing

self-similar

fingers,

in the

Laplacian regime,

^with an

anisotropic

surface tension

Tic).

2. The

free-boundary problem

of

Laplacian growth

in _a

sector-shaped

cell.

Dimensionless

quantities

make _more

transparent

the

similarity

between solidification at low li and

Laplacian growth.

In _{one case, we} will

mostly speak

of _a dimensionless

velocity potential

# while,

in

crystal growth,

_we will deal with _a dimensionless

impurity

amount defined

by:

u =

~

~°,

,

with

Co

the

impurity

concentration in the

liquid

_on the

interface,

_as

given [Co (I

Ii

)]

by

the

binary phase-diagram

and It the

partition

coefficient

(for details,

_see

[I I]).

The

impurity

concentration in the

liquid obeys

the diffusion

equation

which _can be reduced to the

Laplace equation (Au

=

0)

^{for small}

growth

velocities. The

similarity

of the

patterns

^{has been noticed}

experimentally

in

[6],

at least for low Peclet numbers.

Moreover,

for li _<

0.2,

the _area of the germ increases

linearly

in

time, indicating

_a self-similar

growth regime

in

t~'~

_for

any

typical length.

^{These results} are characteristic of viscous

fingering

in _a

wedge

realized with _a constant flux

lo.

A self-similar

regime

^of

growth

exists if _one

neglects

the

capillary

effects.

Assuming

the

stretching

factor:

R(°, t)

=

Ru f(t) r(°)

does not

modify

the

Laplace equation

in the bulk:

hi

=0

(1)

but the

continuity equation

at the interface:

R~ fit) @r

n =

nV# j2)

Here,

n is the normal at the interface while

Ro,

the

tip position

at time t ₌

0,

is _our

length

^{unit. A}

time-independent

free

boundary problem

_can be handled if _{one assumes} that

the

quantity f(t)~ ^~~~

_is

a

constant,

^which means a

growth regime

in

t~'~ _[8].

_For

a

crystal,

t

the

continuity equation (named

the Stefan-Lamd relation

[7])

^{is identical} to

(I)

if _one

formally replaces # by

_-u. This

free-boundary problem

has _a continuous

family

of

petal-shaped

solu-

tions,

labelled

by

_a

parameter I,

the relative value of the

petal angle

at the center of the germ

compared

to the fictitious sector

angle

go ^For a

physical

value of this sector

angle

of 90°

or

60°,

the whole range of I values 0 _< 1 < is accessible

[9].

Let _us recall the

analytical expression

of these

finger shapes

in absence of

capillarity:

if _xi and _vi denote the coordinates of _a

point

of the

interface, they

read:

xi " c

(90) s~°~~~~)l'F ^~°~~ ~~, ~~°, j,

I

s)

T T

vi "

As~°~~~~)l'(

I

s)~/~F ()

+

~°~~ ~~, ~~°, ~,

^l

s)

T '

with-

Y

(1

^{~ Y}

(1+ _~(

~ ~ ~~~ ~~~~~~~ ~~~

()

^~

⁽⁾

+

i~)

^~~~

ii

-0

(a)

0 0.2 Off O.6 o-B I-o 1.2

Fig.

2. Exact self-similar shapes for _a

divergent

flow

corresponding

to _a sector of So _" 60°. The chosen parameters ^{increase from} the inner to the outer profile and _are

equal successively

to o.25, o-s and o.75.

r is the Euler

function,

F _a

hypergeometric function,

_c

(go)

^the

sign

of go and _{s a}

parameter

between 0 and I. As _an

example, figure

2

displays

several

shapes

for _a

sector-angle

of 60°.

As

usual,

the selection of the I

parameter by

the surface tension is

expected

but the surface tension effects

destroy

the

self-similarity.

For _a constant flux

experiment,

it _means that the

capillary parameter

a

(defined below)

decreases

during

^the

growth

as shown

by

the

Laplace (for

the viscous

flow)

or the Gibbs-Thomson relation

(for

the

crystal).

_a is

given by:

'~~~ l

2p

~)~UoRj@]

_~~~~~

'~

Uof~~~]9]

^~~~

In

(2)

_we have introduced the

velocity

of the

tip

Uo at time t

= 0 and the

capillary length:

do

₌

7Tm /mLCo(I It),

_m is the

slope

of the

liquidus,

L the latent heat and

Tm

the

melting temperature. Nevertheless,

_we will

adopt

_an adiabatic

approximation, assuming

that at

long times,

the

dynamics

is slow

enough

_so that the

finger

has time to

adjust

itself _to the

slowly varying

surface tension

a(t).

This

approximation,

which consists in

examining

the selection

by

_a time

dependent capillary parameter ail)

is rather

good

for viscous

fingering

^{since I}

varies rather

slowly

when _a decreases

(for

_a

comparison

between theoretical

predictions

and

experimental results,

_see

[9]).

^{In this} _case, ^{the selection}

by

_a fixed _{a can} be

investigated numerically.

These

approximations

have transformed _a rather

complicated time-dependent

free

boundary problem

into _a

"steady

one" _more suitable to handle at least

numerically, using

methods which _are standard

by

now. Let _us recall that _one way ^to handle this

question

is to transform the free

boundary problem, taking

into account the three

equations _{11, 2,} 4)

^into a

unique

set of

equations

for the

interface,

either

by

Green's function

techniques,

_or

by

conformal map and

hodograph

method

[4]. Here,

_we

present only

the last method with _our numerical

results,

both in viscous

fingering

and in

solidification,

for two sectors

angles

60 and 90°

(for

details,

_see

[9]).

358 JOURNAL DE

PHYSIQUE

i N°2

3 Numerical result _s.

3. I THE METHOD. Let _us

briefly

recall the details of the

hodograph

method and the _cor-

responding algorithm

which have been

published

elsewhere

[9]. First,

the sector

geometry

is transformed into the _more familiar infinite linear cell

by

_a conformal _map:

z =

~

ln

(zi), (z

go

describes the

complex plane). Then,

_we have

adapted

the

hodograph

method of McLean and Saffman

[12] since,

for

go

_"

0,

it appears

remarkably

efficient. Somewhat

unfortunately,

this method introduces rather

unphysical quantities

such _as the derivative _q e~~~ of _a

complex

_pc-

tential which transforms the interface into _a streamline

[9]. Originally,

for the S T

instability,

this

quantity

_was the

complex velocity

of the fluid in the

finger

frame

but,

in _{our case,} it has lost this

physical meaning. Nevertheless,

let _us

give

the

equation:

« q s

tag

q =

~i

~>~

,.

i~/>

^{P P.}

_£~

^dt

~~l'°il~- l~~~

15)

with

Ho

_{= exp}

(- ^~°~ _)[l

_{c cos}

(m(T

+

x/2

+

90v/2)] q

s

~~

+

~° (l I) in(T)j

(6)

2 as 2'

and

~

'~

(l -~2 ^~jo

IA

^{~~~ ~ ~}

and the

following boundary

conditions:

r(0)

=

0, r(I)

₌

-x/2, q(0)

= 1,

q(1)

= 0. As

usual,

P P _means the

principal

value of the

integral.

A second conformal transformation _maps the

finger

_over the

segment

s E

[0,

1], ^{which is} very convenient for the numerical calculations. Note that the

position

of

M,

the

point

of the interface enters

equation (5)

via both the curvature and the

integral.

This is due to the

modification of the Stefan-Lamd and Gibbs-Thomson law via the first conformal

mapping

transformation.

Equation (5)

which relates three unknowns functions _{q, r} and _z

requires

two other

equations

in order to be solved.

They

_are deduced from

analyticity

of _q and _z

[12].

Here,

in order _to also _treat the solidification at low Piclet

number,

_we have introduced _a

key ingredient,

that

is,

the

anisotropy

of surface tension. It

requires

to

modify

_a in

equation (5)

_as

follows: _a

ii

_c

cos(m9)] (with

_m

= 4 _or

6)

with _c the

anisotropy

of the surface free energy:

Tiff)

+

7"1°).

The numerical

procedure

consists in

writing

_a discretized version of these three

equations

at every

point

_so of _an

adequate

mesh. The

resulting

non-linear

algebraic

system for the three unknowns _{q, r, z} is solved

by

Newton's method. The iterative

algorithm requires

_a

starting

function and _we

always

_use the S -T solution at 0° because its

expression

is much

simpler

than the zero-surface tension

profile

at

arbitrary

sector value. in

fact,

to obtain

quick

convergence,

a

good

_guess of the initial

eigenvalue

is

required

_once the surface tension _a is fixed. Before

going further,

_we will

briefly

recall the main results

concerning

the linear

geometry

^{which is} well understood _now. Note that in this _case, the

finger

is

steady

and

especially

stable

against

perturbations.

3. 2 SELECTION _{BY AN ISOTROPIC SURFACE} TENSION.

Perhaps,

it is useful to recall

briefly

the main results

concerning

the viscous

fingering instability

in _a linear cell. It _concerns the selection of the relative width

ii)

of the

finger by

the

capillary

effects

(a),

in both the

isotropic

Although

_{years many}

we will refer to them for _our discussion.

isotropic

^surface tension selects _an infinite discrete set of

profiles

_among the zero-surface-tension set of Saffman and

Taylor.

At fixed surface tension

a or «, there exists _a discrete set of

eigenvalues

labelled

by

_n,

l~,

all of them

merging

^from the 0.5 1-limit at

vanishing

_{a or «}

[12].

^{The 0.5 1-limit} for all the

eigenvalues

_was known from the

pioneering

work of Saffman and

Taylor

_[4] and since has

suggested

_a lot of numerical

[12]

and

analytical

works

[5].

^This

puzzle

has been solved

only recently by singular perturbation

theories

(W.K.B, _[5]).

When _a

increases,

_so does

l~,

another

amazing

result is that the discrete _set

l~ collapses

to I for _a finite value of the

capillary

_{parameter amax m} 0.35.

Any eigenvalue

_goes from 0.5 up to I when _a increases from 0 _up to 0.35

[13].

^A

stability criterion,

shown

numerically

and

analytically _[14],

_proves that

only

the first

eigenvalue ii

is

experimentally relevant, although,

at

extremely

low surface tension _or

large pulling speed,

the

finger

becomes unstable. This is

mainly

due to the _common limit 0.5 for all the

eigenvalues

I

they

are very close _to each other and the

finger

cannot fix itself

on one of them.

Adding

four-fold

anisotropy

in the _same way as the

crystalline

_one,

destroys

^the 0.5 _common limit which _{now goes} to _zero and

gives

_narrow

fingers _[15].

This fact shows off the close

similarity

between solidification at low Pdclet

number,

and viscous

fingering

since the

vanishing

I limit

(which

_means that the walls of the Hele-Shaw cell _are

rejected

at

infinity) corresponds

to the

needle-crystal growth.

What is

probably

less known is the range of this

anisotropic

effect when the surface tension increases.

Figure

3

displays

a

comparison

between the

(c

₌

0) )-spectrum

and

(c

₌

0.2)

_{one versus a.} We have chosen this

anisotropy

value which _{can appear} rather

large compared

to the

crystalline

_one, because if fits rather well the

experimental

data

(Couder, private communication).

o

,o

$

Fig.

3.

Eigenvalue

spectrum of the viscous

fingering

pattern ⁱⁿ the linear geometry: the relative width _{versus a:}

(-)

with

isotropic

surface tension,

(o o)

with

anisotropic

surface tension

(e

=

o.2),

(+ +) experimental

data of Couder and collaborators.

360 JOURNAL DE

PHYSIQUE

I N°2

As

predicted by

the W.K.B

approach _[15],

the first

anisotropic eigenvalue ii

is lowered but

only

_{on a} rather restricted _range of _a values

(a

is less than

5~

of its extremum

amax).

As for the

highest levels,

the

anisotropy

does not affect

them,

at least in the numerical accessible

range. This is the _reason

why anisotropic fingers

_appear more stable: at low surface tension the characteristic

length

scale

ii

is

smaller,

and the stabilization

by

the surface tension is increased.

Moreover,

the first and second

eigenvalues

_{are more} distant than in the

isotropic

_case and noise cannot induce _a transition between two levels.

Nevertheless,

from

figure 3,

_we conclude that the

efficiency

of the

anisotropy

in the Hele-Shaw cell is limited to very low surface tension _or

high finger speed.

This is not _an intuitive result. The

anisotropy

coefficient is _a dimensionless

parameter, independent

of the surface tension. It is difficult to

explain why

it does not affect the

eigenvalues

_{as soon as a} increases. We well

keep

in mind that the

anisotropy

of surface

tension stabilizes the viscous

fingering pattern

at low

capillary

number when it _appears less stable

against perturbations.

3.3 SELECTION _{IN THE SECTOR SHAPED GEOMETRY.} The

isotropic capillary parameter

a(t),

which is

time-dependent

in this _case, selects

a

unique

relative

angular

width

I(t),

at least when it is not too small.

However,

a

surprising phenomenon

_occurs at low values.

By opposition

to the linear

geometry

where all the

eigenvalues

_merge at 0.5 at

vanishing

_a

[12],

^the levels

l~

and

l~+i (n

is

odd)

_merge

by pair

above _a

non-vanishing

surface tension threshold

~~, _~+i ^and

disappear

below

(this labelling

of the

eigenvalues

is deduced from their order at

large a).

The threshold a~, _n+i decreases when _n

increases,

and the first and second levels

disappear

first

(see Figs.

4 and

5).

^This is

important because,

when it

exists,

the

computed

value

ii

_agrees

quite

well with the

experimental data,

measured at different times

[9].

^{When it}

disappears

_a close

comparison

with

experimental

data shows that

tip-splitting

_occurs. This is

the reason

why

_we have claimed that

tip-splitting

_occurs in radial

growth

because _a self-similar

growing pattern

cannot exist at

long time,

when _a is less than ai, ~.

~.

+.

+m Woo

+

+

o,8 ~l

~ ^~

~ m

~+

~~$ +,

~

(

m '

0.COO ^.O02 0006 DOOB

O

0010

Fig.

5. -

The

_relative

angular ^A

the

_apillary

arameter a for a _{ixfoldisotropy}

Figure

4

(resp. 5) displays

_our results

concerning

the 90

degree _geometry (resp.

60° for two different values of the

anisotropy

coefficient _{c =}

0.1, 0.2, compared

to the

isotropic

_case which is

always

_our reference.

Qualitatively,

the

eigenvalue spectrum

^{is the} same for the two

angles

but differs

depending

_upon the values of _c. For _a small _c

value,

_{say c =}

0.I,

one

only

notices

a small shift

compared

to the reference

(c

₌

0)

^{for the levels}

ⁱⁱ

^and

12.

Let _us recall that

only ii

is believed to be relevant for the

experiments.

Since these two levels

only

exist for

a =

0.004, they

_are

poorly

affected

by

the

anisotropy.

On the

contrary,

^it affects the upper

levels,

^which _merge at lower _a. As _an

example, 13

_seems to fall at zero when _a vanishes and the

loop _13,4

has

disappeared.

For this range of _c

values,

_our

prediction

is that the

dynamics might

be dominated

by tip-splitting

events. This is in

agreement

^{with the}

experimental

results

on

anisotropic

radial

laplacian growth [10].

^{At low}

anisotropy,

when the

applied

_pressure is not too

large,

several

tip-splitting

events are observed

giving

_a ^ramified

pattern. But,

for any

anisotropy value,

when the

applied

_pressure is

large enough,

six needles _can be

observed,

without

tip-splitting

events.

According

to

equation (4),

_a

large applied

_{pressure means a} small initial value for _a, which

nevertheless,

will decrease

during

the

experiment.

So, by comparing

the numerical and the

experimental data,

_{we can}

imagine

the

following

scenario: _at low

applied

_pressure, the

experiment

starts _on the level

ii,

and

during

the

growth,

since _a

decreases, ii

describes the lower level up ^to the

loop

and at this

point

the

finger tip- splits

_as in the

isotropic

case. When the

applied

_pressure is

larger,

the

experiment

starts _on level three _{so as} time _{runs, a} decreases and also

I, continuously.

Narrow

fingers

_are observed which in the limit of

vanishing

I look like needles. Due to the lack of

quantitative data,

this

interpretation

is audacious since this third level is

generally

believed to be unstable. This is true for the linear cell but

here,

the

spectrum

is rather different and theoretical

predictions concerning stability

_are

missing.

For

greater

^{values of} _{c, say c =}

0.2,

the

eigenvalue spectrum

^is rather different

as shown

by

figures

4 and 5. The

ii eigenvalue

decreases

monotonously

from the reference value down to zero as a

decreases,

_{so as} time increases. Our

prediction

here is that the

crystal shape

modifies

362 JOURNAL DE

PHYSIQUE

I N°2

itself

continuously

from the

petal shape

for short times to the dendritic

shape

for

long

times.

There _{are never}

tip-splitting

events.

Although

measurement of _c is

impossible

for

engraved plates,

the absence of

tip-splitting

events at

high anisotropy

is

suggested by

the

phase diagram

of

[10]. Ordinary crystal anisotropy

values _c4 varies from 0.075 for

succinonitrile,

_up to 0.016 for

pivalic acid,

and it reaches 0.2 for helium.

Concerning thermotropic

columnar

mesophase,

c6 has been measured to be around 0.I.

Ordinary transparent

materials _are

expected

to have

a different

dynamics

in radial

growth.

Oswald et al. do not mention any

tip-splitting

events but _we think that their

petal-shaped regime (A

<

0.2) corresponds probably

to

ii

while the

dendritic

regime (observed

when 0.2 _< A _< 0.6 that is smaller initial values for

a) corresponds

to

l~.

in the dendritic

regime,

due to the

sharp

variation of I with _a, when both _go to zero, the selected solution labelled

by

^I

(as

well _as the

finger shape)

varies very

quickly

and the

self-similarity

is _no

longer

verified. This _can

explain

the

change

in the

dynamics

from _a

t~/2 regime

to _a constant

velocity

_{one, as} indicated

by

the

experiment _[6].

In _summary,

anisotropic laplacian growth

in the circular

geometry

^{has been examined. We}

showed that the

anisotropy

of surface tension is _a

key ingredient

to describe the

growth dynam-

ics from _one seed. It

explains why crystal growth patterns

^{do not}

display

the fractal structure characteristic of viscous

fingering

at

long

times.

But,

to be

operative,

this stabilization

by anisotropy requires

_a sufficient

supersaturation.

Acknowledgements.

would like to -thank the

Organizers:

J.C.

Angles d'Auriac,.B. Dougot

and J.M.

Maillard,

for this

moving,

sober and sincere

meeting

_on condensed matter

physics

dedicated to _our friend Rammal. I would like to thank them for

participating

to this

special

issue dedicated to

Rammal.

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