Freeze-fracture observations in the Lα phase of a swollen surfactant in the vicinity of the L3 and the L1 phase transitions

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Freeze-fracture observations in the Lα phase of a swollen surfactant in the vicinity of the L3 and the L1 phase

transitions

P. Boltenhagen, M. Kleman, O. Lavrentovich

To cite this version:

P. Boltenhagen, M. Kleman, O. Lavrentovich. Freeze-fracture observations in the Lα phase of a

swollen surfactant in the vicinity of the L3 and the L1 phase transitions. Journal de Physique II, EDP

Sciences, 1994, 4 (8), pp.1439-1448. �10.1051/jp2:1994209�. �jpa-00248052�

J.

Phys.

If Franc-e 4 (1994) 1439-1448 _AuGusT 1994, _PAGE 1439

Classification Phj,sics Ab.itiacts

61.30 64.70 62.20D

Freeze-fracture observations in the L~ phase of

_a

swollen surfactant in the vicinity of the L~ ^and the Li phase transitions

P.

Boltenhagen (', *),

M. Kleman

(2)

and O. D. Lavrentovich

(3)

('_j Laboratoire de Physique des Solides, Bit, 510, Universitd de Paris-Sud, 91405 Orsay Cedex, France

(2j Laboratoire de

Mindralogie-Cristallographie,

Universitds Pierre et Marie Curie (Paris Vii et Paris VII, 4

place

Jussieu, _case I15, 75252 Paris Cedex 5, France

(~) Kent State University, Liquid Crystal Institute, Kent, Ohio, 44242, U-S-A-

(Reieii.ed 2 July J993, revised /3 April /994, aiiepted /8 Apri/ /994)

Abstract. Freeze-fractured specimens ^of the L,, pha~e of _a swollen quasi-ternary surfactant

(cetylpyridinium

chloride/hexanol/btine), have been observed by electron

microscopy.

The

textures show the _same features _as those oberved by optical microscopy, viz. defects with negative

Gaussian curvature (focal conic domains of the usual type) ^close to the phase boundary with the L~ _sponge phase, and defects with positive Gaussian curvature (focal conic domains of second

species,

mostly

spherolites)

close to the

phase

boundary with the Lj micellar

phase.

We have studied _{in more} detail this latter region the density of spherolites increases _{as one} approaches the

Lj-L,,

transition, _up to the

point

^where they

seemingly

fill the whole space their size distribution

obeys

_an exponential law _we

explain

by _a

simple

thermodynamic ^{model which}

predicts

_a transition when the elastic coefficient A 2 K + k _vanishes. Our experimental data

yield

indeed, with _a

reasonable precision, _{a very} ~mall value of Ad/L~ T for _a composition ^of the surfactant clo~e _to the phase boundary.

1. Inroduction.

Swollen lamellar

phases (L,,)

constituted

by equidistant bilayers

of

amphiphilic

molecules

separated by

water have been

extensively

studied for their structural

properties [I],

their

stability

with respect to their

neighbouring phases

in the

phase diagram [2],

their defects

[3, 4].

Regarding

this last

point,

^{it should be noted} that these defects have been first observed _at

large

scales

by optical microscopy [3, 4],

^but very little is known _at

microscopic

^scale. This paper reports on electron

microscopy

observations of freeze-fractured

specimens

which have been

prepared by

the standard method _a small

drop

of the

L~ phase

is inserted between two copper

plates

and

quenched

into

liquid

_propane, then fractured in _{vacuum a} thin film of carbon-

(+) Peinianent addle.I.I In~titut de Physique. 3 _rue de l'Universitd~ 67084

Stra~bourg

Cedex, France.

platinum

is

evaporated

_on the fracture, detached from the

sample,

and

finally

observed

(for

details _on this

technique,

see for

example

Ref.

[5]).

These observations confirm the results

obtained _at

larger

scales,

implying

that the defects _are of the _same nature at all-scales, and

bring

_{some new} results which _are of _a

particular

interest _near

phase

transitions.

Amongst those,

the lamellar-micellar

(L~-L~)

transition is studied in _more detail _; the defects appear ^to be

thermodynamically

stable, and the

quantitative study

of their distribution in size allows _one to determine the values of the

rigidity

moduli of the

layers.

We have

investigated

the

L~ phase

of the

quasi-ternary

_system

cetylpyridinium

chloride (noted

Cpcl)/hexanol/brine

(I fl

by weight

of

Nacl).

This lamellar

phase

extends in the

phase diagram (see

Ref.

[I])

between _a micellar

(L~) phase

for low hexanol _i>eisus

Cpcl

ratio (r

=

h/c 0.5 and a sponge

(L~ phase

for _r l, Let _us

briefly

recall the results obtained

by optical microscopy

observations,

In the

vicinity

of the

L~-L~ phase transition,

the

typical

observed defects

layers

are the well-

known focal conic domains of the first

species (FCD-I),

characterized

by

constant

layer

thickness and

negative

Gaussian _curvature of the

layers

I,e.

« «~ ~ 0, On the contrary, ^{in the}

vicinity

of the

L~-L~ phase transition,

the defects have

positive

Gaussian _curvature

«j «~ ~ 0 _{; most} of them _are

spherolites,

that _are

degenerate

focal conic domains of the second

species (noted

hereunder

FCD-II),

with _a few FCD-II'S of _{a more}

generic

nature

(for

details see

Ref.

[4]).

The differences between the defects

representative

of the _two

regions

of the

phase diagram

close _to the

phase

limits _can be traced to a difference in the

sign

of the

saddle-splay

elastic constant K which govems the Gaussian _curvature

(Fig,

I _{; a}

quantitative analysis

of the results has

yielded

_an estimation of the values of

thil

_{material parameter}

_{[3, 4].}

FCD-1, _aia2 <0 FCD- II, aia2 ^>0

Fig. I. Focal _conic domains, degenerate _{cases :} a) FCD-In _{« «,}

< 0 FCD-lI, _{« «~}

> 0. The circle is virtual in the

case of FCD-II.

2.

Energy density

of _a lamellar

phase.

This section summarizes _a few well-known aspects of the

phenomenological description

of

lamellar

phases

and of their defects which _are of

importance

in the

following

discussion.

The energy

density

of _a Iamellar

phase

reads, to

quadratic

order

f

=

1/2

K(«~

+

«~)2

₊

_K«j

_{«~ +} l/2

BF~ (1)

N° 8 PHASE TRANSITIONS STUDIES IN A L,, PHASE 1441

In this

expression

_«~ and «~ are the

principal

curvatures of the

bilayers,

_e is the relative variation of the

layers thickness,

K and K _are the

splay

and

saddle-splay

^elastic constants, and B is the

compressibility

modulus

[6].

Equation (I)

contains three _terms.

The first _{two are} related to the _mean curvature «j ⁺ «~ and the Gaussian _curvature

«j «~ of the

bilayers, respectively.

The second _term is

generally

omitted because it does not

enter in the minimization of the energy. This is _a direct consequence of the Gauss-Bonnet

theorem which _states that the

integral

of the Gaussian _{curvature over a} closed surface

depends only

_on its

topology

_:

1«~ ^{«~dI=4 ar(I -N) (2)}

(N is the number of handles _on the

surface).

For

example

the

integral

of

equation (2)

is

equal

to 0 for _a torus and _to 4 _ar for _a

sphere.

Thus this _term is _{a constant} and _can be

neglected

in any

smooth continuous deformation of the system ^{however it}

plays

an

important

role in any

change

in the

topology

of the

layers.

A

complete description

of the energy of the system

requires

to take into _account the energy of interaction between the

bilayers.

This is the third _term of

equation (I).

For _an

electrically

neutral system, as it is the _case here, this term is of

entropic

nature and has been studied

by

Helfrich

[7].

Focal conic domains

(FCD)

_are the

typical

defects of lamellar

phases.

^Their characteristics

can be

simply

inferred from

equation (I).

^The

energies

^associated with _a pure ^curvature deformation _or with _a pure

compression

deformation at

large

scales

(larger

than _a

typical layer periodicity)

_{are very} ^different in

magnitude, telling

_us that the system ^{will choose} to deform in

a set of _a

quasi-equidistant

and

parallel surfaces,

I.e. with

vanishingly

small

compression

energy. It _can be shown

that,

if _so, the system ^forms FCD'S, ⁱⁿ which the

layers

take the

shape

of

Dupin cyclides [8].

Let _us recall the basic features of _a FCD. In each domain the

layers

are folded around two

conjugate

focal

lines,

viz. _an

ellipse

^and a

hyperbola,

in such _a way that

everywhere they

are

perpendicular

to the

straight

lines

joining

_any

point

E _on the

ellipse

to any

point

H _on the

hyperbola.

If the actual part ^{of the}

layers

^is between E and H, _{«~ «~}

~ 0 in the

opposite

situation «j «~ > 0, These _are the defects observed at

optical

scales.

Figure

I illustrates the

cases where the

ellipse

is

degenerate

to a circle and the

hyperbola

to a

straight

line, and

illustrates the differences between _a FCD-1 (« _{«~ ~} 0) and _a FCD-II

(«

_«~

> 0

).

^{The reader} will

easily recognize

that the _{two cases} do _not

correspond

to two different realizations of _a FCD built _on the _same focal lines.

3. Results.

Starting

from the

vicinity

of the

Lj-L~ phase transition,

_a first remarkable result is that the lamellar

phase

is filled with

spherolites

whose

density depends critically

on the h/c ratio

(Fig. 2).

In each

spherolite

the lamellae form concentric

spheres.

^The

spherolites

are

polydisperse

in size and their size distribution follows _some characteristic

exponential

law which repeats

remarkably

well from _one part ^{of the}

sample

to another

(Fig. 3).

At first

sight,

the whole space seems to be filled

by spherolites

of different

sizes,

but _a closer look at the

pictures

reveals the presence of the

layered

^matrix in _a reasonable

proportion

indeed. An estimate of this

proportion (made

farther

on)

shows that the

planar

^{matrix fills}

approximately

half the volume of the

sample,

The presence of this

large

matrix of well-oriented

L~ phase

suggests also that the fracture which is

performed

at low temperature in the freeze-

/

^--

~ i

14t^. 'Z

~

~ ,

'~

~

'"~' ,

~' ' '~

' ' "

'

'

'~"i y

"^'.-,[

~$ ~ _" ~

Fig.

2.- Freeze-fractured specimen, electron

microscopy

observation

spherolites

_near the Lj-L,~

phase transition. The lamellar phase ^{is filled with} polydisperse spherolites. Note the pre~ence of

elongated

FCD-II (arrow),

hex/Cpcl

o.7 85 % solvent. Bar 5 _~m.

o n exp

suriac;c dens"ty 1P2m _{x n} calculation

0.4 0.35

0.3

0.25

0.2

0.15 j x

O-I i x

o.05

~ ^m)

I 1.5 2 2.5

Fig.

3. Hodograph ^{of the surface} density of

spherolites

_{as a} function of the radius measured in free2e- fractured

specimens.

The horizontal _error bar (AR)is the

experimental

_error of measurement of the radii.

The

density

_error (~n) is _not

represented,

it is equal to ~n

~"

~R.

6R

N° 8 PHASE TRANSITIONS STUDIES IN A L,, PHASE 1443

fracture

experiment,

and which is well-known to

easily

follow the

layers

in such _a type ^of

material,

is

practically planar.

The

typical

size of the

spherolites

which _are observed is about

one micrometer. Note the presence of

generic FCD-II'S,

which _{are not}

perfectly spherical

«

spherolites

_», ^but have the

shape

^of

rugby

balls. It has been shown elsewhere

[3b, 4]

that

spherolites

_are the FCD-ii's of smallest energy, for _a

given

volume.

The increase of the hexanol _iersus

Cpcl

ratio _i'

=

h/c leads _{to a}

progressive disappearance

of the

spherolites.

In the middle of the lamellar

phase,

for h/c between the

Lj-L~

and

L~-L~ phase

transitions, _we observe isolated

spherolites

embedded in the lamellar

phase (Fig. 4).

These observations confirm the

large

scales results.

In the

vicinity

of the

L~-L~ phase

transition the lamellae _are very distorted

(Fig.

4j. A number of them _are in _contact

along

areas whose

typical

size is of the order of _one micrometer _:

we do _not know whether these _{contacts are}

dynamic

or static, but

they clearly

_announce the formation of the _numerous pores which _are present ^{in the} L~

phase.

Furthermore _we observe _a

large

^{number of} screw dislocations, whose

stability

is _most

probably

related _to the

negative

Gaussian

geometry

of their _cores

[6].

Note that the _areas which surround the _zones of _contact between lamellae _are often of

negative

Gaussian _curvature.

4. A

simple

model for

spheroJites.

We restrict here _our discussion _to the most novel aspect ^{of this} work, viz, the formation of _a

thermodynamically

stable _set of

spherolites,

whose

density

decreases when h/c. increases, _near the

L~

-L

phase

transition. This part ^is theoretical. The

comparison

with

density

measurements is made in the _next section.

Let R be the diameter of _a

perfect spherolite according

to

equation

its total _curvature energy writes

Wc(R R

= 4 _w

f(r ^r~

di

=

4 _w

(2

K ₊

K)

R

(3)

To this energy should be added _{two terms.} The first _one,

W~,

relates to the disclination

loop

which circles the

spherolites

at the contact with the outer

layers (Fig.

lb) this energy scales like the

perimeter

and is of the order of KR. The second _one,

W~,

is due _to the

layer

compressibility;

_as shown in reference

[3b],

it scales like

BAR~

_{with A}

=

(K/Bj'~~

for

spherolites

whose _mean distance 3 is

larger

than

R~/A, (R~/A

is the distance _on which the

stresses decrease

exponentially [9].

But, if R _< 3 _<

R~/A,

_a different model should be used

we believe indeed that the stresses are relaxed

by

the presence of dislocation

loops

of size

comparable

to 3, _so that

W~

scales _{as a core} energy ^K3 -

BA~

_{3. We shall check that}

3 -R, _so that

W~

w BA

~R.

Furthermore, the situation _we have

investigated

is close to the L

phase,

where B is small, so that this term should _not be very relevant.

Putting together

these

results, we have

W,~~ =

4 WA'R

(4)

where A' is _a renormalized value of the

quantity

A

=

2 K ₊ K.

We _assume that the free energy

density

F is _{a sum} of

independent

terms

£ F~

+

Fo.

Each

R

F~

refers to the set of

spherolites

of radius R _# 0, and

Fo

refers to the part ^{of the}

sample

which

is not in _a

«spherolitic»

state. We shall also _{use a} continuous version of F, viz.

~

~R ~~

⁺

^~0,

^{Wh~~~ d IS th~ ~~P~~t d'~t~~lc~ Of th~}

^{~a Phase.}

^{The VolUme}

^density

^Of

_, _{- ~_}

' '

m

~- .~

Fig. 4. - specimen, electron bservation :

in

the middle of ^the

L~

Note the presence of _embedded in the ^lamellar phase. The lamellar

phase

_positive

aussianurvature (small _arrow) as _well

,

j

.~

~~

~

~

~~

~'~ ,' /j~

< ~~ R ^~~'

'w,

$~/

~,

,,- ~» .~£-SL d

Fig.

5. Freeze-fractured specimen, ^electron

microscopy

observation _: aspects of the membranes _near the L~-L~ phase transition. Note the _numerous pores in formation.

hex/Cpcl

=

1.0 _; 85 % solvent. Bar 0.5 _~m.

N° 8 PHASE TRANSITIONS STUDIES IN A L~, PHASE 1445

spherolites

of radius R is noted n~ =

#~/v~,

where

#~

^{is the} volume

fraction,

and v~ = 4

wRh3.

_{We have}

:

F~

=

[4 WA'R#~

+ l.~

T~~

In

~~] (5)

v~

where

~

=

z ~~.

The estimation of

F~

is _a real

difficulty

in the present

problem.

In

fact,

while

R

the _structure of the free energy

density Fo

remains

unknown,

its variation _can be reached _more

easily.

It would be w<tong ^to w>rite _:

Fo

=

k~ T(1

~ ) ln

(1

~ where _Do is _{some mean} v~

volume of _{an «}

elementary object

_» relative _to the _{« non}

spheroJitic

_» part ^{of the}

sample.

The molecules

themselves, being

_so small

compared

to the

spherolites,

do not know whether

they belong

_or not to a

spherolite

_; hence

they play

_no r61e in the total balance of the entropy ^when a

variation 3

~~

occurs, except may be

along

the

singularities (the

_equators of the

spherolites

and their

centers),

but this contribution scales like the entropy ^{of the}

spherolites

themselves and does _not

bring anything

_new. The molecules _are not these

elementary objects.

On the other

hand,

the variation of the

density

^of the

spherolites strongly

affects the entropy ^{of the} non-

spherolitic

part. ^Consider a variation 3~

=

3~~,

I-e- restricted _{to some}

particular

value of R. This variation affects

F~, but

_also

_Fo,

_{since ~ varies. The}

_quantity

SR =

~ (l

~ ^In

(I

_~b

(6)

would be the entropy of the

non-spherolitic

part

supposed

filled (in _a

gedanke experimentj

with

non-spherolites

of size R. The

reasoning

follows in fact

closely

the

approach

used _to calculate the entropy ^{of vacancies in} a monoatomic

crystal.

This _means also that the appearance (or

conversely,

the

disappearance)

^of some relative volume

3#~

^of

spherolites

^of radius

R involves _a relative number of _«

elementary objects

_»

3#~/v~.

^Hence the variation of the

entropy ^of the

non-spherolitic

part ^{would be}

35~

₌ ^~

3~~

In e(1 ~ ), and the variation of v~

the total free energy

Fo,

which is the _sum of the entropy ^{variations should therefore read} :

3

#~

3Fo

₌

k~

T

jj

In e

(1

# )

(7)

R

~R

In the continuous version, we

replace z #~ by i~~

^{~~ and}

^{equation (7) by}

~

d

3F~= -k~T j~ ^{~~~Ine(I -#j~~}

o ~R d

Entropy

has been taken into _account in

equations (5)

and

(7)

with the

assumption

that the

spherolites

form _a

regular

solution of

species

labelled

by

the different values of R. This _mean- field

picture

overestimates the total entropy ^{and the total internal} energy, since it

neglects

all

correlations between

spherolites

of different sizes. The latter

assumption might

however be

valid _{to a}

large

extent. As indicated above, the interaction is

mostly

of elastic

origin

(B term).

Consider then _two sets of

spherolites Ro

and

Rj

,

they

form _two distinct disordered ,nets with uncoirelated characteristic distances

30

and &j ; hence the energy of interaction of

the _{two nets} does _not

depend

on the small fluctuations of their individual

spherolites,

_nor on the relative _«

phase

_» of the two nets. Therefore

we expect ^{that the}

long-range

elastic interactions

are

mostly

^intra-net.

Short-range

interactions, I,e, those related to the excluded volume effects between

spherolites belonging

to different nets, will be

neglected.

Note further that the

B _term becomes

probably

less

important

_near the

L~-Lj phase transition,

_a fact which

decreases _{even more} the elastic interaction and

eventually brings

A' closer _to A. Let _{us now} examine the free energy in _more detail _{; we}

distinguish

_{two cases :}

I) ^A~ _~ ^0.

Minimizing

F with respect to

#~ yields

#~

=

(l

# exp

(-

^{~ "~'~}

⁽⁸⁾

kB ^T

When

integrating

this

expression

with respect to R, _we get

k~

^T

~

k~

^T + 4 w-t'd ~~~

The

inequality (d~F/d#

_~)~

~ 0 is

satisfied,

whatever R may be, when

~~

takes the value of

equation (8)

_: the solution is stable. It _can also

easily

be shown that, for the _same value of

#~, (dF~/dR)~~

cannot vanish. Hence it is _not

possible

to minimize the free energy with respect to R and there is _no

unique

solution for R _{: we} expect a distributinn of

spherolites

of

different sizes. One _can

finally

show that the appearance of these defects in the

L~ phase

decreases the free energy of the

perfectly

flat

phase.

According

to

equation j9),

the medium is

entirely

invaded

by

the

spherolites

(I,e.

#

-

),

when A'

- 0. This

implies

that the

L~ phase

with

spherolites

is _no

longer

stable, and another

phase

does appear. The present

approach

does not enable _us to define it any better. It is

probably

the

Lj phase.

ii)

A'

~ 0. The _same

equations

as above

(Eqs. (8, 9))

still hold, but _now

equation (8j yields

#

_~ l, which is inconsistent with the definition of

#.

Therefore

equation (7)

does not describe

properly

the _case A'~ 0

(the Lj phase ?).

5.

Density

measurements.

The

theory developed

for the _case A' _~ 0 _{seems to} describe

reasonably

well

oiJr experimental findings.

^The volume

density

_{n~ is} related to the surface

density n(,

I,e, the

density

measured

experimentally

in _a 2D cut,

by

the relation

n(

= n(~~, ^which is

easily

obtained if _one notices that

nl _3i

~ nR. Here

3i

~ ni ^{~~~ is the} mean distance between _two

spheroiites

of radius R. We _can

safely

use this estimates of n~, because the fracture is

qiJasiplanar,

_as

argued

above.

Comparison

of

equation (8)

with

experimental

data

(Fig. 3) yields

the

following expression

for

~,

R

n(

"

((1-~)

^~

_{j~~~~p(_2~}

~~ ~~

~R

4

wR~

3

j/ ⁽¹⁰⁾

where 3.6 _x

10~~

_{is the}

slope

of the

logarithmic plot

of

n(

^R~ as a function of R/d. The

corresponding

^value of ~ ^{obtained from}

equation (9)

is 0.56, while the

experimental

value obtained from the

plot

^of

figure

^{3 is 0.52.}

Since this

expression yields

_so small _a value for

A',

we assume A'~A and get

K/K _{as :}

~= -2+3.6x10~~ ~~~

K 4

"kc (ll)

where k~ ₌ Kd.

N° 8 PHASE TRANSITIONS STUDIES IN A

L~~ PHASE 1447

The

periodicity

d of the lamellar

phase

^{studied here is}

typically

of the order of

1501;

_the

curvature modulus of the lamellar

phase

is about _a few

t~

T, thus

~ = + 3.6 _x 10~ ^~

k~

^TM

wk~

is small and

positive.

We found that K/K is very close _to the value _at which A

- 0, where _we expect a

divergence

of the volume fraction of

defects,

I-e- _a

phase

transition.

If,4

=

2 K ₊ K is small, the

phase

must be filled with

spherolites,

which is the _case when h/c is small. A contrario when A increases the

phase

should have _a lower

density

of

spherolites.

Experimentally

_we found that this is the _case. Thus the increase of h/c leads _{to an} increase of K/K.

6.

Why

the distribution is not fractal.

We end this discussion

by

_a brief _{comment on} the fact that the distribution is

exponential,

ant

algebraic,

I-e- _not fractal.

Let _us first recall that _a fractal distribution has

already

been invoked in the

problem

of

filling

space with

(conically shaped)

FCD-l's

[10].

There _are similarities, but also differences

between the _two

problems.

In

particular,

while the

filling

with

spheres

is _a

genuine

3D

problem,

^{the FCD-I} _space

filling

is 2D the

length

of the _cone is

equal approximately

to the

characteristic size L of the

sample,

while the base has _a radius _a which ranges from

L to some small value _a *.

According

to reference

[10],

_one first

places

^inside the

sample

the

largest possible domain,

with base radius _a L, then the gaps are filled with smaller

domains,

in such _a way that

neighboring

domains _are in _contact

along

a

generatrix

of the _cones

(Friedel~s

law of

corresponding

cones

[8])

and the bases _are tangent. ^This process continues down to

molecular scales, with _a * A

,ilB.

_{The result}

a ^* A is _an intrinsic feature of the model,

since the space

filling

is the result of _a

competition

between

layer compressibility

and

curvature

elasticity.

However, there is another natural limit for _a ^* in the FCD-I

filling,

which

originates

in the

anisotropy

Am of the surface tension. As _{a matter} of fact, the part of the

sample

which does _not

belong

to a FCD-I _can be either filled with FCD-11

(at

least in certain

geometries [I I]),

_or

by

dislocation

loops

which relax the

layer elasticity.

The

regions

with FCD-I and FCD-II

fillings

have different surface orientations _as a

result,

the balance of the surface and _{curvature terms}

yield

_{a new} characteristic

length [12],

_a ^*

K/(-

Am ) which _can be very different from A.

Let _{us now} consider in turn the other type ^of space

filling,

with FCD-II'S. There is

only

one

characteristic

length

for each

spherical FCD-II,

viz. its radius _a

(we

restrict _our considerations to

spherolites),

^{which varies} from _some

macroscopic

value L

(in general

the

sample

size _at

rest a

length depending

_on the shear rate, if any, has been introduced in Ref.

II 3])

to _some minimal _a * There _{are no}

special geometrical

conditions akin _to Friedel's law of

corresponding

cones when two

spherolites

are

neighbors (there

is no way ^to eliminate the elastic distortion of

the

layersj

_as in the FCD-I _case, and there is _no

anisotropy

of the

global

geometry, ^{which is} 3D,

consequently

_we do _not expect

special

surface tension effect which would

specify

a*.

Having pointed

out these differences with FCD-I's iterative

filling,

let _{us now} put ^the stress on the differences with the former

e.iponential filling.

A fractal of

spheris

of different sizes

would be

inhomogeneous

at all scales, _so that _our

previous

arguments about the irrelevance of the elastic interactions would fail

('). Assuming

that the

spheres pack randomly,

the elastic

(') Reciprocally,

_{we may} wonder whether the

growth

_process of _a set of

objects

into the

shape

of _a fractal is not due _{to a} strongly

inhomogeneous

distribution of _stresses which do not relax

during growth.

energy now

probably

scales like the residual volume

[10].

Let g =

(L/r)Y

be the numbers of

spheres

of radius R

larger

than _r

packed

in _a volume fl

mL~.

_{The residual volume is}

V(r)

= fl

(L/r)Y

^~~ the total free energy F

(r)

scales like _:

F

(r

_m An (L/r _)Y ^' ₊ Bfl

(L/r

_)Y ^~ l

2)

There is _no entropy term, because the

packing

_processes which starts from the

biggest

spheres

(R _m L

),

in _a volume of size L, continues with

spheres

^of

decreasing

sizes, ^is

pratically

deterministic

(but

to

indiscernability

of

equal spheres).

The smaller the residual

volume,

the

more so.

According

to reference

[14],

_a relevant value of the exponent y is _ca. 2.8. This value

yields

indeed ij _a minimum of F

(r)

when A

~ 0

(the

condition for _a minimum to exist

being

I _{< y <}

3)

and

it)

_a minimum value for _r, viz. r~,~ m

fi _{(y 1)/(3}

y

).

^The latter minimum does _not show up in the former

theory

_; furthermore the energy is _now

positive.

In other words, a fractal random

packing

cannot describe _a

phase

with

thermodynamic defects,

but _a

possible phase

of another nature.

Acknowledgments.

O. Lavrentovitch thanks the Laboratoire de

Physique

des Solides

(Orsay)

and the NSF

(grant

DMRB9-20147

ALCOM)

for support.

References

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Appell

J., Bassereau P.,

Marignan

J. and Porte G., J.

Phys.

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P., Lavrentovich O. and Kleman M., al J. Phys. II Fiance 1 (1991) 1233 _; b) Phys.

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