Spherulite phase induction from positive Gaussian curvature in lyotropic lamellar liquid crystals

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Spherulite phase induction from positive Gaussian curvature in lyotropic lamellar liquid crystals

J. Fournier, Geoffroy Durand

To cite this version:

J. Fournier, Geoffroy Durand. Spherulite phase induction from positive Gaussian curvature in ly- otropic lamellar liquid crystals. Journal de Physique II, EDP Sciences, 1994, 4 (6), pp.975-988.

�10.1051/jp2:1994178�. �jpa-00248026�

Classification Phj'sics Abstracts

61.30J 64.70 82.70

Spherulite phase induction from positive Gaussian _curvature in

lyotropic lamellar liquid crystals

J. B. Foumier and G. Durand

Laboratoire de Physique des Solides, Universitd Paris-Sud, bfit. 5lo, 91405

Orsay,

France

(Receii,ed 23 Noi,ember J993, revised 31 January J994, accepted 3 Maich J994)

Abstract. The equilibrium

properties

of

spherulite

assemblies in

lyotropic

lamellar phases are

studied within both elastic and statistical models. Following previous models, the spherulites are

assumed _to be stabilized by their positive Gaussian _curvature. Our model predicts that the spherulites should adopt _a regular crystalline piling if their _core energy E~ ^{is much} larger than k~T, _{or a} liquid-like piling ^with large ^radius fluctuations ifE~ k~ T. The spherulites develop an

internal pressure

depending

only on temperature. ^{The existence of} a maximum in the distribution of spherulite radii gives _a practical criterion for checking the Gaussian _curvature efficiency.

1. Introduction.

Lyotropic liquid crystals

show a rich

variety

of lamellar

phases

where the lamellae _are similar to

biological

membranes 11,

2].

For

instance,

the temary system CPCI-hexanol-brine

[3]

exhibits the

following

_sequence of lamellar

phases Lj (isotropic phase

of

micelles)

_- lamellar

phase

with

spherulites

_-

L~ (ordinary

lamellar

phase)

_- L~ (« sponge » lamellar

phase).

The

«

spherulite phase

_» is

quite

novel and _not yet

recognized

_as a true

thermodynamical phase.

It consists of _a dense

packing

of

spherulites,

which _are concentric

pilings

of

spherical bilayers

of the

ordinary

lamellar

phase [3].

A few years ago it was

proposed

^{that the transition from}

lamellar

phase

to sponge

phase (L~

-

L~)

was driven

by

a spontaneous

tendency

of the

lamellae _to

adopt

a

negative

Gaussian _curvature,

I-e-,

_a saddle

shape [4].

The transition would

occur at an elastic threshold _at which the Gaussian _curvature

tendency

would _overcome the

ordinary stabilizing elasticity

of the lamellar

phase.

More

recently,

it _was

proposed [5, 6]

that the

spherulites

result from _an

opposite tendency

of the

bilayers

to

adopt

a

positive

^Gaussian

curvature,

I-e-,

a

spheric-al shape,

close _to the

Lj

-

L~

transition. There exists

probably

a

similar elastic threshold _as for the sponge

phase

transition, _at which the

spherulites

would be

individually

stable.

However,

the model of reference

[6]

assumed that the

spherulites,

close _to but still below the elastic

stability

threshold, were rather stabilized

by

the entropy ^associated with their

disorder, following

_a model

already developed

for dilute vesicules

[7].

In this paper, we revive the idea that the

spherulites

of lamellar

phases

_are created

by

_a spontaneous

tendency

of the

bilayers

to

adopt

a

positive

Gaussian curvature. We

study

the

spherulite equilibrium

both above and below the elastic threshold where the

spherulites

are

individually

stable. In part 2, we describe _a

purely

elastic model in which the

equilibrium

size of the

spherulites

is calculated. Then _we estimate the fluctuations within this elastic model to

check the

importance

of entropy. In part

3,

_we derive from

thermodynamics

the

spherulite

pressure.

Then,

in _a one-dimensional

(lD) model,

_we calculate the statistical mechanics of _a dense

spherulite packing, taking

into _account both the size and

position

fluctuations of the

spherulites.

We derive

(in lD)

the

entropic

contribution _to the

spherulite

_pressure, and _we calculate the fluctuations of the

spherulite

radii.

Finally,

in 3D, we discuss _some

general

features of the

spherulite phase, giving

for instance the Gibbs distribution of the

spherulite

radii as a function of their pressure.

2. Elastic model.

The _curvature

free-energy density

of _a lamellar

phase depends

_on the lamella

principal

curvature

radii,

_p, and pi- For

symmetric bilayers.,

the _most

general

form up to second order in the curvatures is

[8]

f~j

=

~K(

⁺

)~+K ^(l)

2 p, p2 Pi P2

K,

always positive,

is the standard elastic constant

corresponding

to the

stabilizing elasticity

of the lamellar

phase

and

k,

of

arbitrary sign,

is the Gaussian curvature constant. K

~ 0 favors p, p~ < 0 which is realized in the

(L~

sponge

phase [4],

whereas K _< 0 favors pi p~ ~ 0 _as in the

spherulite phase [5, 6].

In the

following,

_we shall _assume K

< 0 but, _as we shall _see, this is not sufficient _to stabilize _a

spherulite.

2. I ISOLATED SPHERULITE. Let _us consider _a

single spherulite

of radius R embedded in the uniform lamellar

phase.

It consists in _a concentric

piling

of

spherical bilayers

with a central

core and _an interface with the

surrounding

lamellar

phase (Fig. I).

We choose _as the

origin

of

energies

that of the condensed

planar

lamellar

phase.

The _core energy is _a

positive

_constant

)

Fig.

I. Schematic representation of a

single spherulite.

E~.

^The interfacial energy may be written _as 4

grR~

y. The _curvature elastic energy of the

spherulite

is obtained

by integrating f~j

from the _core radius

i~~ up ^to R. With pi = p~ = I/r,

equation (I) yields f~j

=

(2

K +

K)/r~.

The

spherulite

curvature energy is then 4

gr(2K+K)(R -r~).

^The term

proportional

to r~

simply

renormalizes the _core energy

E~.

^The curvature energy of the

spherulite

reduces then _to 4

grKR,

with

km2K+K. ₍₂₎

The total energy of the

spherulite,

sum of the above described terms, is thus

w(R

=

E~

+ 4

grkR

+ 4

gryR~ ₍₃₎

Let _us discuss in _more detail the

physical assumptions

included in the derivation of this energy. We have

completely ignored

_any lamella dilation _as is usual in confocal textures

[9].

The _core, in _a continuous model, is the

region

where the order of the lamellar

phase

_goes to zero because of the

divergence

of the lamella _curvature. This

«

melting

_{» occurs}

classically

over a coherence

length

where _curvature and

melting energies

_are

comparable.

^In our case of _a first-order transition, this

length

is of order _a

(the

lamella

period)

and the _core energy E~ ^{is of order Ka. At the}

microscopic level,

the _core is

probably

filled

by

solvent alone, I-e-,

completely

melted. If the _core is

larger

than _a, it still

implies

_a net

melting

_energy. If it is smaller than _{a, one} could think that the _core energy vanishes. In

fact,

the

ordering

of the first

layer

will be very

perturbed, yielding

also a non-zero core energy

E~. Finally,

the interfacial tension _{y can} be visualized in the

following

_way if the

spherulite

radius R compares with a, there may exist _a pure elastic

matching

with the

neighbouring

lamellar

phase.

It has then been demonstrated that the total elastic energy scales _{as a} surface energy

[6].

In the

opposite

limit of

large

R, an elastic

matching

is _no

longer

realistic. The

layers

must be _{cut at} the

spherulite

surface, which involves _a

grain boundary

_energy ^also

proportional

to the

spherulite

surface. We _assume that this property ^{remains valid for all} R, and _to

simplify,

we take y

independent

of R.

For K

<

0,

the

spherulite

energy

w(R)

_{presents a} minimum _at R*

=

K/(2 y),

which is

negative

for

(2 ark

)~ _~ 4 _gr

yE~.

The

spherulite

_appears as the result of _an elastic

instability

of the lamellar

phase.

In terms of K, the

spherulite

is stable for K

< 2 K (

yE~/gr )'/~

In order to write the results in dimensionless form, let _us define

iw

_{K/K and}

e~,

i~, i~

such that

~

=

k~

TE~

K

=

k~ Tli~ (4)

y =

k~ Tli(

The

spherulite stability

threshold and the

equilibrium

radius _can be

expressed

_as

E~ ^1/2

i~

~~

Qr i

f2

~ ~~~

j~*

~

£/

~ ~K

The threshold

spherulite

radius is R

*(i~)

=

(e~/gr )'/~ i~.

^For the consistence of the model, 2

this

quantity

must be

larger

than the _core size, itself

larger

than the lamella thickness

a. A reasonable estimate for E~ ^is a few _K, where _K wKa is the bidimensional curvature

constant

[8].

One _can find in the literature

[10]

that for ternary systems K

k~

T and for pure

amphiphilic

systems _K ²⁵

k~

T. One _can therefore expect _E~ ^{3 for the former} system ^and e~ 60 for the latter _one. In

practice,

with _E~

=

10,

R ^*(k~

)

_{» a}

requires i~

» a. This condition

seems

exaggerated

for _a

thermotropic

smectic but may be reasonable for _a ternary

lyotropic

smectic the cosurfactant may heal the

bilayers

cut at the

spherulite boundary

and

largely

reduce the surface energy

defining i~.

2.2 DENSE _ASSEMBLY oF SPHERULITES. At first

sight,

_one could expect an

assembly

of

spherulites

to consist of _a dense

packing

of

spherulites,

all

having

the

equilibrium

size R

*,

_as

previously

defined. We show in the

following

that

densely packed spherulites

tend _to be

much smaller.

Let _us consider _a dense

assembly

of N

spherulites

in _a closed volume V. We _assume that the lowest energy state of the system ^{will consist in}

spherulites having

^all the same radius R. The average

density

of the

filling

_can be described

by

_a

compacity

factor _{c s} I such that

N ~

grR~

=

c-V

(6)

Densely packed spherulites

still present a surface energy. The interstices between

spherulites

can be filled either

by

_a

long-range

oriented lamellar

phase,

_or

by

lamellae

locally parallel

to the

neighbouring spherulites.

The former _case is identical _to that of _an isolated

spherulite.

^For

the latter _{case, a} distribution of

grain

boundaries will appear within the

planes

that _are tangent

to each

pair

of

spherulites

at their _contact

point.

The energy of these

grain

boundaries scales also _as

R~.

To

simplify,

_we

keep writing

the surface energy as 4

grR~

y and _assume that the

spherulites

interact

only

_as hard

spheres.

The total elastic energy

W(R

of the

assembly

is then W

(R )

= Nw

(R ),

which is

proportional

to

w(R)/R~. _Making

use of

equation (6),

_we find

lE~ ^k _y~

W(R)=3cV ~+j+- ⁽⁷⁾

4grR R R

The

comparison

between

W(R)

and

w(R)

is shown in

figure

2.

W(R)

has extrema for

o

w/v~

0 R~ R*

Fig.

2. The

free-energy density

W(R)/V of _a dense

spherulite assembly

(plain line) and the free- energy density wjR )/u* that would result from filling the _space with spherulites having their individual equilibrium ^{radius R*} (dashed line). The volume v* is ~

arR*~ _c~ '

3

yR~

₊ 2

kR

₊ ^~ _gr~ 'E~ =

0. The minimum

gives

the

equilibrium

radius of the

assembly

_as

4

/j2

R~=2R*

1-

1-~2

(8)

~~~

The

assembly stability threshold,

defined

by

the condition

W(R~)

=

0,

is still

I,.

_{At the}

threshold, _we have

exactly

_R~

=

(e~/gr)'/~i~wR*(I,),

but above the

threshold,

_R~ is 2

definitely

smaller than

R*,

_as announced earlier

(Fig. 3).

This decrease of the radius is

energetically

favoured because the

gain

associated with the increase in the

spherulite density largely

_overcomes the individual loss per

spherulite.

In the limit

)I)

»

)k~),

we have R~

~

(e~ i~/gr) ^i~

^'

Assuming

_e~

= 10

(cf.

discussion in

2.I)

and

classically i~

~ a, ^we

8

find that R~ ^{reaches the}

bilayer

thickness _a for

I

_{l. This}

gives

the absolute limit of _our elastic model.

Within _a dense

assembly,

each

spherulite

is constrainted at a size far from its _own

equilibrium

size

by

the hard

sphere

interactions exerted

by

^its

neighbours.

The

resulting

elastic forces

exchanged

between the

spherulites (Fig. 4)

build then _a pressure, the definition of which will be

given

in section 3.I. The

piling optimal compacity

is obtained

by minimizing W(R~)

with respect to c. Since

W(R~)

cc c, the

spherulite equilibrium piling

is the

crystalline

hexagonal piling

that maximizes _c.

Conversely,

below

threshold,

since

W(R~)

~

0,

the system will rather minimize _c the

spherulites,

which _are then metastable, will separate ⁱⁿ a dilute

piling

and

consequently adopt

their individual

equilibrium

size R ^*

(in

the absence of attractive forces between

spherulites).

The

extrapolated

_R~ variation for

ili~

<

(Fig. 4)

is therefore _not

physical.

,''

,'

"

~

"

J~

#' ,'

' "

R~

o o

Fig.

3.

Fig.4.

Fig.

3.

Comparison

between the

equilibrium

radius R~ ^within a dense assembly ^{and the individual}

spherulite equilibrium

radius R*. Both _are displayed versus the relative Gaussian curvature gain. The

plain

_parts of the _curves

correspond

to the

physical

path.

Fig.

4. Schematic

representation

of the pressure

arising

from the

repulsive

contact forces exened between the spherulites in _a dense assembly.

2.3 ELASTIC _{STABILITY DIAGRAM.} We _now extend _our

previous analysis by

also consider-

ing

assemblies of

spherulites

^{within the}

isotropic phase (L, ).

This is

justified by

^{the fact that in} reference

[6]

the

spherulites

_appear ^close to the

L,

-

L~

transition

(although

no observation of

spherulites

in the

L, phase

was

mentioned).

Since the

L,

_-

L~

transition _occurs when the cosurfactant/surfactant ratio

4

increases and reaches

4~,

we shall consider the

spherulite stability

in the

(I, ₄

_space.

The

spherulites

_can survive within the

isotropic phase

as

long

_as their

negative

elastic energy compensates ^their

positive

condensation

free-energy

^~

grR~ hf.

In the

vicinity

of the

3

L,

-

L~

first-order

transition,

the

free-energy

difference between the

planar

lamellar

phase (4

~

40)

and the

isotopic phase (4

_<

40)

can be

approximated by Af ~A(40 4).

In

principle,

the

boundary

between the

spherulites

and the

isotropic phase

involves a surface energy coefficient y, different from _y. To

simplify,

we shall make the crude

approximation

y, = y. The energy of _an

assembly

of

spherulites,

_now within the

isotropic phase,

^is

given by

W,(R)=W(R)+cVAf, _j9)

since _c is also the volume fraction of the

spherulites.

^The

spherulite equilibrium

radius is still

R~,

^{since the minimum} of

W,(R)

is

independent

of

hf.

The

spherulite stability

threshold

corresponds

to

W, (R~)

=

0,

which is

equivalent

to A

(4 40) R(

=

(4

_gr

)~

^'

3

E~ +

KR~

+

yR).

This condition defines _a transition line

Lj

_-

Si

between the

isotropic phase

and the

crystal

of

spherulites

^within the

isotropic phase.

The

Lj

_-

Sj

line

equation,

^{in the limit}

4

Wow

I in order that the

expression

of

Af

remains valid, is

given by

g~

1/2k~

^T

Ii

40

4 12

j

(10)

EC

Ai~ [

The transition line between the

spherulites

within the

isotropic phase (Sj)

and the

spherulites

within the lamellar

phase (S~)

is

given by 4

₌

4~

since _we assumed y, = y.

Finally,

the

L~

-

S~

line is

given by

k

=

i~

as shown in section 2.2. All the

spherulite

transitions _are first- order since metastable

spherulites

_can survive

beyond

the

stability

threshold. The transition lines

Lj

_-

Sj

and

Sj

_-

S~

are

naturally

limited to the

region I

_~

l, beyond

which _our elastic

model _no

longer

^holds since R~ compares then with _{a as} shown in section 2.2. These results _are summarized in the

phase diagram

of

figure

5.

5~ L~

-i

$ ~o

_~

~ ^°

£j

Fig.

5. Elastic

stability

diagram of the spherulite phases ⁱⁿ the

Ii,

_{~ space.}

I

_{controls the Gaussian}

curvature and ~ the lamellar (L~)

-

isotropic

IL,) transition. S~ ^and S, are the spherulite phases within the lamellar and the

isotropic phase, respectively.

If the

spherulite

entropy was taken into account, the

stability

domains of the

spherulite

phases would become

larger.

2.4 ESTIMATE oF THE DISORDER WITHIN THE SPHERULITE ASSEMBLY. The above discussed model

(Sects.

2.2 and

2.3)

_was

purely

elastic and

naturally yielded

_a

crystal

of

spherulites.

To test its

validity,

let _us evaluate the thermal fluctuations of the

spherulite positions

and radii.

We shall first estimate the

position

fluctuations. The energy

required

_{to create a vacancy}

within the

spherulite crystal

is

w(R~).

From

equation (3)

and the

equation yielding (8),

_we have

w(R~)= -2E~-4grkR~.

_It is

easily

shown from

equation (8)

that

4grkR~=

f (kli~ E~,

where

f (x

)

=

4.i~ I 3/

(4

x~ ). ^The _vacancy creation energy is therefore

w(R~)

= g

(kli~)

_E~

,

(I1)

with g

(x)

₌ 2

f(x).

For

ill,

~

[l,

+ co

(in

the

stability

_range ^of the

spherulites),

_we have

g(ilk~)

_~ 0, Thus,

remarkably, w(R~)

the elastic energy

gain

_per

spherulite

within the 2

assembly always

_compares ^{with the} core energy

E~.

^{This is}

clearly

_apparent in

figure

2. To estimate the relative

density

_x of

vacancies/spherulites

in the

regime

x« I, we treat the vacancies _{as a}

perfect

_gaz. This

gives

_{x exp in>(R~}

)/k~ T]

_m exp

[- E~/k~ T].

It follows that the

spherulite positional

disorder will be

large

if e~ l.

We _now estimate the size fluctuations of the

spherulites

to check _our

hypothesis

of _a

common radius within the

assembly.

Let _us consider _a

spatial

variation of the

spherulite

radius

R

(r

₌

R~

+ 3R

(r

close to the elastic minimum. In the

reciprocal

_space, 3R jr ) ^is

represented by

its components

3R~.

^As q

approaches

Am 2

gr/R~,

the

piling

does not remain

locally

hexagonal

^and

gradient

terms must be taken into _account. The

quadratic

variation of the system elastic energy can be written as

~~

~ ^{~ i}

«

where

f

is the coherence

length

associated with the

gradient

terms. For a constant

3R, we have

3R~

=

3R

3~

_o ^{and 3F reduces to V}

a3R~.

Identification with _our elastic

2 model

gives

_a

=

(d~/dR~) _(~~(W/V

_). This

quantity, easily

calculated from

equation (7),

_can be

given

the form

~

/

"

Rs

The value of

f

is difficult to calculate

exactly,

but it _seems reasonable to expect

f R~ ^{which is the}

piling

characteristic

length.

We have

verified,

in _an elastic model with _two

spherulite sizes,

that this

assumption

is reasonable. The thermal fluctuations of the size modes

are

given by

the

equipartition

theorem

applied

to

equation (12), I-e-, ((3R~(~)

₌

kB

T[Va (I

+

q~ f~)]~

' The local fluctuation is then

classically

obtained

by summing

_over all the

modes,

I-e-, here up to the cut-off A _:

(3R~ (r))

^~

~

4

grq~ dq

^~~

~

~ ~

(14)

12 _ar ) _o va I ₊ q~ f

-)

With the values f

=

R~

^and A

= 2

gr/R~ already

discussed,

equation (14) gives (3R~)

(k~ T)/ («RI ₎

with _a

proportionality

coefficient 2

(2

_gr )~^~

(2

_gr arctan

(2

_gr of order

unity.

From

equation(13),

we have

aR(~E~,

with _a coefficient

again

of order

unity

[since

^{I + g}

((l(,

_~

j1,

^~

). ^Finally,

^{the relative size} fluctuation is 2

fi

_{E i<1}

fi

(15)

R~

k~

^T

Thus the radius fluctuations _are also

expected

to be

large

for s~ l.

There should be two kinds of systems

according

_to whether _e~ is smaller _or

larger

than

unity.

Since _we expect e~ 3 for ternary

lyotropic

systems ^and _E~ ^{60 for} pure

amphiphilic

systems

(cf.

Sect.

2.I),

ternary systems ^should

give liquid-like spherulite packing

^with

large

size and

position

fluctuations, whereas pure

lyotropic

systems ^should

give crystalline spherulite piling (although

^{metastable disordered}

glassy

states are also

expected).

^This

prediction

is in

agreement ^{with the} observations of reference

[6].

3. Statistical models.

We showed in the last section that _an

assembly

of

spherulites

^within the lamellar

phase

is

subject

to

large position

and size fluctuations if the

spherulite

_{core energy E~ compares} with

k~

^{T. In this}

regime,

_our elastic model

clearly

lacks _a

significant entropic

contribution.

To determine the free energy associated with the

spherulite

disorder, _one should

study

the statistical mechanics of the system

by taking

into _account all the

possible

_sources of disorder.

This is _a very

dauting

task. Besides the disorder associated with the

spherulite

size and

position distributions,

various other disorder _{sources are}

possible

_to

explorate

^{the various} radiuses, the

spherulites

must aggregate or dissolve additional

bilayers.

The

corresponding

disorder is the distribution of the

spherulite growth

rates (at least each

spherulite

_{may grow or}

recede).

^More

accurately,

_one could consider the entropy ^{associated with the lines that} are the borders of the

aggregating bilayers.

The _momentum of the

spherulites

could also be taken into _account. The latter may be

larger

when the

spherulites

evolve in the

isotropic phase (cf.

Sect.

2.3).

Within the lamellar

phase,

^there may also be _an entropy ^{associated with the} orientations of the

interstical lamellae between the

spherulites.

The

spherulites

_may not be

exactly spherical.

It has been shown

[5]

that _some

spherulites

are in fact

elongated (they correspond

to focal conics of

second-species II II).

A

corresponding elongation

distribution could be taken into _account.

One could also consider the dilatable elastic deformation modes of the

spherulites,

etc.

Due _to this

complexity,

it _seems

illusory

to calculate

exactly

the total entropy ^{of the}

spherulite

_system. It is however

interesting

to estimate the contributions associated with the

position

and size distributions,

specially

because the size distribution is measurable

[6].

In

principle,

the radii _are

coupled

with the other

degrees

^of freedom mentioned above. In the

following

_we shall _assume that _{we can} consider them _as

decoupled.

Before

discussing

statistical models, let _us first define _some

thermodynamic

parameters ^{that will be} relevant in what follows.

3.I SPHERULITE PRESSURE AND CHEMICAL POTENTIAL. Let _a

fluctuating assembly

of

N

spherulites

be

placed

in _a closed volume V at temperature ^{T. To take into account the}

assembly fluctuations,

_we must consider the

free-energy

F (T, V, N of the

assembly

instead of the elastic energy W defined in section 2.2. The

spherulite free-energy,

measured from the

planar

lamellar

free-energy,

differs from W

by

an

entropic

contribution TS associated with

the disorder of the

spherulites.

Since the total number of

spherulites

^is not

fixed,

N

plays

^{the role of} an intemal parameter ^with respect to which F must be minimized. This

equilibrium

condition reads

p = 0,

(16)

where _p

=

dF/dN (~, v is the

spherulite

chemical

potential. Similarly,

the

spherulite

_pressure

is defined _as P=

-dF/dV(~,~.

From the well-known

Legendre

transformation

G

=

F ₊ P V

=

NV,

_{p can} be

expressed

_{as a} function of P and T

only. Therefore,

_we have

p

(P, T)

=

0 ~ P

= P

(T). (17)

The

equilibrium spherulite

_pressure

depends only

_on the temperature. ^{This is} reminescent of the

black-body,

for which the total number of

photons

is also free. An

interesting

consequence is that if _one reduces the volume of the

spherulite

system

by exerting

an additional pressure,

the numbers of

spherulites

will

simultaneously

decrease _to maintain the pressure to its

equilibrium

value

PIT).

In this

paragraph

we show that the results of the elastic model

(Sect. 2.2)

_can be recovered within this

thermodynamical

formalism. In the elastic

approximation, F(V,

N

)

reduces _to F

=

Nw>(R),

with

w>(R) given by equation (3).

The chemical

potential

is then

given by

p =

dF/dN

= w' + N

(dw>/dR

₎ (dR/dN

;

),

where dR/dN

v is

given by equation (6).

We obtain _p

= w

R(dw/dR)

_cc

d(w/R~)/dR. Thus, writing

_p

=

0 is

equivalent

_to minimize 3

w/R~

_{with respect to R. This is}

exactly

what _was done in section 2.2 and gave R

=

R~.

^The

spherulite

elastic pressure is then

P~

=

(dF/dR (~) (dR/dV (v),

I-e-,

p ^c

d(Nw>) k

y

~ N 4

grRj

^JR N

~

R)

^{~ ~} R~ ~~~~

Since _p

=

0 is

equivalent

to F

=

-P~

V, the volumic elastic energy of the

spherulite

assembly

is

given by P~.

We have thus

P~

=

0 at the

stability

threshold and

P~

~ 0 above threshold.

In _a

fluctuating spherulite assembly,

the total pressure has _an additional

entropic

contribution

P~(T)

that must be determined

by

statistical mechanics. We may therefore write

P

IT)

=

P~

+

P~(T). (19)

Since _p

=

0, the

spherulite

_pressure P

(T)

is _now F (T)/V,

opposite

of the

free-energyy.

At

equilibrium,

F

(T)

will

always

be

strictly negative

since entropy ^will

always

stabilize _{a non-}

zero

spherulite density

(even if the

spherulites

are unstable and must be considered _as

defects).

Consequently, P(T)

will

always

be

strictly positive.

Moreover, the

stabilizing spherulite

entropy ^will

enlarge

the

Sj

^and

S~

domains of

figure

5.

3.2 STATISTICAL _MECHANICS OF A ID SYSTEM OF SPHERULITES. To get some

insight

into

the

entropic

fluctuations, let _us

study

a

simplified

ID

problem.

We consider N

spherulites

with

arbitrary

radiuses,

densely packed along

a line of

length

L

(Fig. 6).

To solve this

problem,

_we

impose

that the

spherulites

remain

densely packed.

^This is _a reasonable

approximation

ⁱⁿ the

regions

of the

phase diagram

where the

spherulites

_are stable.

The

microscopic

states of the system are

uniquely

defined

by

the set of the

spherulite

radii

(Rj,Rz,. ,R~),

each

being equal

to an

integer

number n, of the

bilayer spacing

o L

Fig.

6.

Fluctuating

ID

assembly

of

spherulites,

in contact with _a thermostat _at temperature T and

a pressure P

a. The total

length

of the

assembly

is

L=2~jR,. (20)

If _we fix L, we must take into account the fact that the radii _are not

independent.

It is _more

convenient _to let L fluctuate and fix the value of the

conjugated

_pressure P. We shall _use the Gibbs distribution where the system

exchanges

both energy and volume with _a thermostat _at

temperature ^Tm

(pk~)~

^' and pressure P. The

partition

function of this system

Z(T,P,N)= jj

_e

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

jdRj. ^dR~e ~~~~~~'~~~~~

(21)

in,>

~~

is in

principle

a discrete _{sum over} the

spherulite

number of

bilayers.

In the limit where the

spherulites

are

statistically large enough,

we can

replace

the discrete _sum

by

an

integral

in the range

[0,

₊

ml (also setting

the lower limit _{to zero} instead of

r~).

With the definition of L and the

expression

of

w(R, given by equation (3),

we have

~j

w

(R, )

+ PL

=

~j (E~

⁺

⁽⁴

^{grK + 2 P}

R,

+ 4

gr

yR)) (22)

The radiuses

being independent,

the

partition

function

(21)

factorizes _to Z

=

z~,

with

-flE~+fl~~~~~~~~

_+CCdfi

-fl4ny(R+~(~~~)~

z(T,

P

= e ^{~ "Y} e ^"~

(23)

~

The

integral

in

equation (23)

is Gaussian and _can be

expressed

via the

complementary

error

function

(erfc).

Since the

potential classically

associated with Z is the

free-enthalpy

G

=

k~

^{T In Z} ₌

NV,

the

spherulite

chemical

potential

is

simply k~

^{T In} z

Let _us compare this

expression

with the chemical

potential

of the ID elastic model

N compact identical

spherulites

^{with radius} R

=

L/(2N)

have _an energy

W(R)

=

Nw(R) given by equation (3).

It follows that _p~

= dW/dN (~ =

w(R

R dW/dR

= E~ ⁴

gryR~

_and

P

=

dW/dL

(~

=

dW/dR

=

2

ark

₄

_gryR. Combining

these _two relations

gives

2 p~

(P

=

E~ (2

grK ₊ P )~/

(4

_{gr y} which coincides with the first _{two terms} of

equation (24).

Then, from _p~

= 0

(cf.

Sect. 3.I

),

_{we get} the

equilibrium

radius

R('~~

=

E~/

(4 _{gr y}

)

and the

elastic pressure

P)'~J

=

P

(R)'~J)

=

4

gryE~

2

ark.

_{The elastic threshold}

_{(defined by}

P)'~'

=

0)

is still

i~ given by equation (5).

The limit T-0 in

equation (24) (artificially keeping k

_and

y

constants) gives

p

(T,

P

-

p~(P).

As

expected,

when T

- 0 the statistical model coincides with the elastic

one.

The

equilibrium

_pressure P

(T)

is

given by

_p (P, T)

=

0

(cf.

Sect. 3.I

).

This

equation gives 2grk+P

as a function of T

only.

The

quantity

2

ark

_+P

can be written _as

P)'~~(T)

4

gryE~,

from the

expression

of

P('~'

and

equation (19). Defining

the reduced

entropic

_{pressure as}

p

<iDj~~~

Ps(T)

~

~

(25)

4

gryk~

T

and _p,

=

p/k~

T,

equation (24)

can be rewritten

adimensionally

as

p, = 2

Ell~p~ p)

In ~~

erfc ~p~

ll~)j

(26)

4 _a

The

equilibrium equation

_p,

= 0 _must be solved

numerically

and

gives

_p~

=

p~(e~, i~la).

Figure

7 shows the

entropic

_{pressure p~ i>ersus e~} for

i~la

=

20 _at fixed

(room)

temperature.

We chose

i~la

_» I _to make _sure that the

spherulites

be much

larger

than the

bilayer

thickness.

Consistently

with the fluctuations estimated in section

2.4,

the

entropic

_{pressure p~} vanishes for _s~ » 1.

Ps(E~)

i

I lo

Ec

_loo

Fig.

7. Reduced

entropic

_{pressure p~ versus} the reduced spherulite core energy e~ in the lD model.

The dotted _curve represents ^{the ratio between the} entropic _pressure and the elastic pressure

P(~~l/P)~~l

for

I

= 2i~. ^For

large

values of F~, the

entropic

_pressure is

negligible

compared with the elastic pressure.

The

spherulite

_average radius _can be calculated from

(R)

=

3p/d(4 ark)

_{taken at}

P

=

P

(T),

_or

equivalently

from

(R)

=

(16

_gr

l''/~ i~

dp

/dp~

taken for the

equilibrium

value of p~.

Taking

the derivative of

equation (26)

^and

making

use of _p~

=

0,

we obtain

f

~

(27)

)/>

~ ~~ ~~~~~

4

~~

a

~

~

Similarly,

the average value of the fluctuation 3R =R

(R)

_can be calculated from

(3R~)

=

k~

T

d~~/d(4 grk)~

=

(16

gr)~

^'

i( d~p/dp(.

We find

~~~

=

l ~

e~ ^~~

(28)

R

i~>

2 _{e~ a}

Figure

8 shows

(R)/R)'~'

ml" and

,~/R)'~~w

_3r

versus e~, at fixed temperature ^and

again

^for

i~la

= 20.

Consistently again

with section

2.4,

_we find that the radius fluctuations

are very

large

for _e~ l but vanish for _{E~ »} I,

yielding

the lD elastic model described above.

JOURNAL DE PHYS>QLE >I T 4 N' b JUNE lQ94 3?

r+br(e~)

r(Ec) r-br(i~)

0

10

E~

¹⁰⁰

Fig.

8. Reduced average

spherulite

radius and radius fluctuations _versus the reduced

spherulite

_core

energy _F~ in the lD model. For

large

values of F~, the elastic model is valid.

With respect to the elastic model, the

spherulites

_are further stabilized due to the

entropic

pressure.

Indeed,

the volumic

free-energy being equal

to

(P)'°J

_{+ P)~~~), the}

spherulites

_are

already

stable _at the elastic threshold

i~

defined

by P('~~

=

0. However, the exact

stability

threshold cannot be determined within the present ^model because the initial

dense-packing assumption

is _{no more} valide close to the threshold. The entropy due to the

separation

of the

spherulites,

that _we have

neglected

close _to the

threshold,

will stabilize the

spherulites

_even

more.

Consequently, P('~'

₊

P('~~

= 0 defines _an indicati~>e threshold

I,~~,

^{the absolute value}

of which is overestimated.

Since, P)'~~

₊ P

)'~~

₌ 4 _gr

yE~

2

ark

+

P)'~~,

_we obtain I>nd ~

(i

_{El ~~~Ps)}

is ₍₂₉₎

As _an

example,

for e~ =

I and

again i~la

=

20,

_{we see} from

figure

7 that p~

l.6,

thus I,~~ ^0.6 k~ the

spherulites

_{are now} stable for

la

_{0. Therefore if}

e~

l,

the entropy can

stabilize the

spherulites

in the

region

Km 2

K,

_as

proposed

in reference

[6].

3.3 RADIUS DISTRIBUTION IN A 3D _{SYSTEM OF} SPHERULITES. The extension to 3D of the

above described model

yields

_a

liquid

of hard

spheres

with variable radii. Even with the

simplifying compacity hypothesis,

it is

extremely

difficult to count all the

microscopic

states of such _a system. ^We can however make _use of _our

hypothesis

that the

spherulites

interact

only

_as hard

spheres,

to

apply

the canonical distribution to the system

consisting

in _one

spherulite,

«weakly coupled»

to the thermostat made of the lamellar

phase containing

all other

spherulites.

Discarding again

all other intemal modes, _we

only keep

the

spherulite

radius R _{as an} internal variable. Our system,

consisting

in _a

single spherulite

with energy w>(R

given by equation (3)

and volume V

=

~

grR~, exchanges

both energy and volume with the thermostat _at temperature 3

T

=

(pk~)~

^' and pressure P. The

probability density f(R)

of _a

configuration

with radius R is therefore

proportional

to

exp[- p (w>(R)

₊ PV

)],

I.e.,

f(R

cc exp

[-

p

(E~

^{+ 4}

^grkR

^{+ 4 gr}

yR~

₊ P

(T)

^grR~

(30)

Since the thermostat consists in the

assembly

of

surrounding spherulites,

its pressure

p cannot be chosen

independently

of the temperature

(cf.

sect.

3.I).

The function

f(R)

i~>-2K

K<-2K

~m

Fig.

9.

Typical spherulite

radius distributions in 3D.

P

(T)

is well-defined but unknown. It could be determined

only by solving

the whole statistiial 3D

problem.

Nevertheless we can make _an

interesting prediction

from the fact that P is

always positive

(for obvious

stability

_reasons in

Eq. (30)

and _more

generally

for the

thermodynamical

_reasons

given

in Sect.

3.I).

Since

E~,

_y and P _are all

positive,

w>(R

)

+ P V has _a minimum _at

R~

_# 0

only

if

k

< 0

(Fig. 9). Therefore, f (R )

must have _{a non-}

zero maximum if K

< 2 K and _no maximum if

k

~ 2 K.

3.4 SPHERULITE _{PHASES AND SPHERULITE} FLUCTUATIONS. In section 3. I _we showed that _a

small but finite

spherulite density

of

entropic origin

will

indefinitely persist

within the

L~

and

Lj phases.

This will broaden the

spherulite

transitions (cf.

Fig. 5).

Within the present models _{we cannot} determine whether the

drop

in the

spherulite

_pressure when

going

from the

spherulite region

to the

L~

_or

Lj phases

will be

sharp enough

to define _a transition. To

overcome this

difficulty,

we propose ^to call _a

spherulite phase

a

spherulite assembly

presenting

a maximum in the radius distribution

(K

_< 2

K,

cf. Sect.

3.3)

and _to consider _{as a}

spherulite fluctuation

within the mother

phase

a

spherulite assembly

without maximum

(K

_~ 2 K

).

With this

definition,

the

spherulites

_are

elasticall»

stable

(or metastable)

within the

spherulite phase. Experimentally,

the

spherulites

observed in reference

[6]

showed _an

absence of maximum in the size distribution. This would indicate that

k

~ 2 K and would

correspond

in _our

description

to

spherulite

fluctuations close _to the

S~ phase.

The

S~ phase

would then still remain _to be observed _as well _as the

Sj phase.

4. Conclusion.

In this paper, we

investigated

whether

positive

Gaussian _{curvature can} induce stable

phases

of

spherulites.

We first discussed _a pure elastic model. Above _a

stability

threshold, _an isolated

spherulite acquires

_a

negative

_energy due _to its

positive

Gaussian _curvature and has _a

relatively large

size, limited

only by

its surface energy. However, the lamellar

phase

^fills space with much smaller

spherulites.

The total energy is then lowered due _to the rise in

spherulite density,

which

largely

compensates ^{the associated} energy loss per

spherulite.

An elastic pressure

develops

then within the

assembly

_as the

spherulites

_are constrained _{to a} smaller size than the

equilibrium bY

the hard

sphere

interactions with

neighbours.

The

spherulite density

rise is limited

by

the

spherulite positive

_{core energy.} It follows that the elastic energy per

spherulite

within _a dense

assembly

is

always comparable

to the _core energy

E~.

If

E~»kBT,

the energy per

spherulite

of the order of

E~

^is much

larger

than kB T and we expect ^the

spherulite assembly

to build _a compact

hexagonal crystal,

^We _propose

as a

possible phase

_sequence, controlled

by

the Gaussian curvature constant K and the

phase

component concentrations :

isotropic (Lj )-spherulite crystal

within

isotropic-spherulite crystal

within lamellar-lamellar

(L~).

If

E~ k~ T,

^the

spherulite

fluctuations _are dominant and _we expect a

spherulite assembly

to build _a disordered

liquid phase

with

large

radius fluctuations. The

spherulite

_pressure

acquires

an

entropic

contribution

coming

from the radius and

position exchanges

between the

spherulites.

From

thermodynamics,

the total

spherulite

_pressure is then _a function P

(T)

of

temperature only, strictly positive

and

opposite

_to the volumic

free-energy.

This follows from the fact that the

spherulite

number is _an internal parameter ^that

adjusts

to

equilibrium.

In _a lD

model,

_we have calculated the

entropic

contribution to the pressure, The entropy stabilizes the

spherulite

_system

beyond

the elastic threshold. In 3D, _we cannot calculate P

(T)

but the

previous

ideas remain valid. We

predict

the characteristics of the radius distribution function of the

(unknown)

_pressure. The existence of _a maximum in the

spherulite

radius distributions

gives

a

practical

criterion _to determine the

sign

of the _{curvature constant} combination

2 K K.

Finally,

_to

distinguish

a

spherulite phase

from

spherulite

fluctuations within the mother

phase,

we

proposed

^{the criterion of the existence of} a non-zero maximum in the

spherulite

radius distribution. This «elastic _» criterion _overcomes the

difficulty

that the

spherulite

entropy ^{broadens the}

spherulite

transitions when E~

k~

^{T. If} E~ »

k~

T, the

spherulites

should build _a 3D

crystal.

This is reminiscent of other 3D ordered

phases

in

lyotropics,

between

isotropic

and lamellar

phases [2].

^In the _case where the interstices between the

spherulites

show

long-range

smectic order, the system ^{should be considered} as an eutectic of the

spherulite crystal

and the

ordinary

lamellar

phase.

For E~

k~

T, the

fluctuating spherulites

form an

isotropic liquid phase,

^{which is different from the}

Lj phase.

If the interstices between the

spherulites

still show

long-range

smectic order, this

spherulite packing

then forms _a

lamellar

phase

different from the

ordinary L~ phase.

It would be

interesting

to check the existence of these

phases by

diffraction

experiments.

As for the

predicted

R

(P

distribution for

one

spherulite,

it could be checked

by

micromechanical measurements. The

spherulite

assemblies described up ^to now

[6]

seem to

correspond

to

strong S~

fluctuations in the

L~ phase

and not to the

S~ phase

itself.

Finally,

note that _our model

probably

does not

apply

to multivesicular

spherulites [12, 13],

in which the additional entropy ^{associated with the}

shape

fluctuation may be dominant.

References

[ii Ekwall P., Advances in

Liquid Crystals,

G. M. Brown Ed. (Academic, New York, 1975).

[2] J. Meunier, D.

Langevin

and N. Boccara Eds.

(Springer

Verlag, New York. Berlin Heidelberg, 1987).

[3] Gomati R.,

Appell

J., Bassereau P., Marignan J. and Porte G., J. Phys. Chem. 91 j1987) 6205.

[4] Porte G., Appell J., Bassereau P. and Marignan J., J. Phys. France So (1989) 1335.

[5]

Boltenhagen

Ph., Lavrentovich O. D. and Kleman M., Phys. Rev. A 46 j1992) R1743.

[6]

Boltenhagen

Ph., PhD Thesis jUniversitd Paris-Sud, France, 1993).

[7] Simons B. D. and Cates M. E., J. Phys. JI Franc-e 2 (1992) 1439.

[8] Helfrich W., Z. Naturforsch 28c jl 973) 693.

[9] Fournier J. B., Phys. Rev. Lent. 70 (1993) 1445.

[10]

Safinya

C. R., Sirota E. B., Roux D. and Smith G. S.,

Phys.

Rev. Lent. 62 (1989) l134.

ill

Bouligand

Y., J. Phys. France 33 (1972) 525.

[12] Hervd P., Roux D.,

Bellocq

A.-M., Nallet F. and Gulik-Krzywicki T., J. Phys. II France 3 Ii 993) 1255.

[13]

Diat O., Roux D. and Nallet F., J. Phys. II France 3 (1993) 1427.