A study of the Phase Diagram of the Micellar Binary Solution Models I: Mean Field Approach

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A study of the Phase Diagram of the Micellar Binary Solution Models I: Mean Field Approach

A. Benyoussef, L. Laanait, N. Masaif, N. Moussa

To cite this version:

A. Benyoussef, L. Laanait, N. Masaif, N. Moussa. A study of the Phase Diagram of the Micellar

Binary Solution Models I: Mean Field Approach. Journal de Physique I, EDP Sciences, 1996, 6 (8),

pp.1043-1058. �10.1051/jp1:1996115�. �jpa-00247230�

A Study of the Phase Diagram of the Micellar Binary Solution Models

I: Mean Field

~

Approach

À. Benyoussef (~),

^L.

Laanait (~),

N.

Masaif

_(~>~)

and

N. Moussa

(~)

(~)

L.M.P.H.E.,

Faculté des

Sciences,

B-P- 1014,

Rabat,

Morocco (~)

École

_Normale

Supérieure,

B-P. 5118, Rabat, Morocco

(~)

Département

de

Physique LPT,

Faculté des Sciences et

Techniques,

B-P. 577,

Setta,

Morocco

(Received

22

September

1995, revised 8

February1996, accepted

19 March

1996)

PACS.75.10.-b General

theory

and models of

magnetic ordering

Abstract. Trie _mean field

approximation

is used to

investigate

trie

phase diagrams

in _a three-dimensional lattice of micellar binary solutions

m the _presence of _a chemical

potential

of

trie

amphiphiles

for different values of

coupling

interactions. New

phases

_appear between trie water-ricin and

amphiphile-nch

_ones. Second and first order transitions,

tricritical,

multicritical and critical

end-points

_are obtained.

By

_a

simple

low temperature _expansion argument, _we bave

confirmed the existence of states of lowest free energy ma mean field approximation.

Résumé.

L'approximation

du

champ

_moyen est utilisée pour examiner les

diagrammes

de

phases

d'un modèle _sur réseau à trois dimensions _pour les solutions binaires micellaires _en

présence

d'un

potentiel chimique

des

amphiphiles

_pour différentes valeurs de

couplage

d'interac- tions.

Cependant,

des nouvelles

phases

apparaissent entre l'eau et

l'amphiphiie.

Des transitions du second et du

premier ordre,

les

points multicritiques, tricritiques

et

critiques

ont été obtenus.

Avec _un argument

simple

de

développement

à basse

température

_{nous avons} confirmé les états de basse

énergie

libre

en

champ

_moyen.

1. Introduction

In recent years

great

^effort has been made in

experimental

and theoretical

physics

to understand the behavior of

phase diagrams

^of

amphiphilic systems. Recently,

several models of the

binary

mixtures of _an

amphiphile (it

is _a molecule of both

hydrophilic

and

hydrophobic parts)

and

1A<ater have been

proposed

in order to

reproduce

the

phases

observed

experimentally [1-7].

In

general,

the

amphiphile

molecules in

water-amphiphile

mixtures

try

to arrange themselves

as to

only

_expose their

polar

head _groups to the water molecules. In

particular,

when the

amphiphile

is

wholly

immersed in water, trie molecules

again try

^{to reduce the} area of contact

«<ith water in order to form _various kinds of

aggregates (rods, spheres, lamellar, ...). Precisely,

1A<hen _a

hydrocarbon

is in contact with water, the network of

hydrogen

bonds between water molecules reconstructs itself to avoid the region

occupied by

the

hydrocarbon.

This _constraint

on the local structure of _water decreases the entropy near the

hydrocarbon,

and it results in

an increase in the free _energy of the

system

[8].

©

Les

Éditions

_de

_Physique

₁₉₉₆

For

amphiphilic systems,

^the

aggregation

of molecules of the

binary

solution induces

equilib-

rium processes controlled

by

intermolecular and

interaggregate

forces

[9].

^Micellar

aggregates

appear in various

shapes

and

sizes, depending

_on the

amphiphiles,

the concentration and other

thermodynamic parameters.

^This

suggests

a classification of the

aggregates

^{into three}

general categories

_as follows:

1) globular aggregates

where _a

spherical

micelle is _a

prototype

of this

Mass; 2)

rod-like

aggregates

where _a

cylindrical

micelle is the

typical example

of this

Mass;

3) bilayers

where _a disk-like

aggregate

is an

example

of this Mass. The choice among all the

possible shapes

is determined

by

interactions between the

hydrophilic headgroups

and

geomet-

ric

packing

constraints _on the

hydrophobic tails,

but the transition from _one

shape

to another

may be obtained

by changing

either the

temperature

_or the concentration _or

by adding

_a third

component.

With

increasing amphiphile

concentration _a

variety

^of

lyotropic phases

_are found

(see [9-14]).

The

phase diagram

^{for the} water and

C12E5 system (1.e. C12H25(DCH2CH2)SDH)

that

represents

^such

phenomena

is

reproduced by Strey

et ai.

ils].

At low

temperatures, Gompper-Schick

_{[4] and} Matsen-Sullivan [3] have shown the coexistence of

homogeneous

water-rich and

amphiphile-rich phases using

mean field

theory.

However at

higher temperatures,

there _{is a}

single

^disordered

phase.

The

phase diagram

established in [4]

resembles that of the

C12E5

and water

system

_{[9] for} which

scattering experiments

have carried out

[14].

Conceming

the

microscopic approach,

Shnidman and Zia [16]

recently proposed

_a

lattice-gas model,

based _on

constructing

_a

coarse-grained representation

of different

types

of

aggregates occurring

m micellar

binary

solutions

(MBS'S)

in _terms of

Ising

variables.

Namely,

_one intro-

duces _{on a} lattice site

(1, j, k)

the

Ising

variable

satisfying

_s~,j,k

" +1

(spin-up)

for _a micellar

section and s~,j,k " -1

(spin-down)

for _a

region

of

comparable

size but

predommantly occupied by

_a solvent.

A

single up-spin completely

surrounded

by down-spms

is identified with _a

globular micelle,

and _a linear chain of up-spms surrounded

by down-spins corresponds

to _a rod-like micelle.

Finally,

_an

up-spin

at the end of _a chain of

up-spins

surrounded

by down-spins

is called _an end cap

(see

Ref.

[16]).

The twc-dimensional version of the model _was

recently

studied

Ii?i,

at low

temperatures, using

^the

Pirogov-Sinai theory

and the low

temperature phase diagrams

_were

investigated

in

the

parameter

_space of T

(temperatures)

and h

(chemical potential

of the

amphiphiles)

for certain values of the interactions

parameters

of the model.

Dur motivation _m this work is to

provide,

within the mean-field

approximation,

a

descrip-

tion of the states of lowest free _energy for the three-dimensional version of the model and to describe the

phase diagrarns

in the space of the parameters ^T

(temperature)

and h

(chemical potential

of

amphiphiles)

for _certain values of the parameters

Ko, Ki, K2

and J

characterizing

the interactions of the model under

study (see

Sect. 2 for

definitions).

Then _we check the

conjectured phase diagrams proposed by

Shnidman and Zia. This

study

led to the conclusion that the chemical

potential

of the

amphiphiles, h,

for fixed values of the

coupling interactions,

may

change

the nature of the

phase

transitions _m fundamental _way,

inducing

the appearance of multicritical

points.

We note that the model

describing

the

physical

_case

corresponds

to

positive

values of J.

The _paper is

organized

_as follows: in Section

2,

_we describe the Shnidman and Zia model

in terms of

Ising spin

variables. In Section

3,

_we mtroduce the

ground

states of the model. In

Section

4,

_we establish the

expression

of the free _energy and the _mean field

equations

for the

system

considered. Results and discussions _are

presented

in Section 5.

Finally,

in Section

6,

we

present

our conclusions.

-K

i

K i+ Ko )/2 "§o

~

_'

+

G~ ~~- -~~ ~

+

-)+( _-j+j- +j-

++- ~ ~

~)

^C

b

^~

8

5

~

l'il _~'~2 ~3 ~4

dÉ~ m~ tll~ m~ m~

/

b)

^~

1?ig. ^1.

a)

The

configuration energies

for Ri > Ko > 0.

(a)

A

sphencal

micelles.

(b) End-cap

sections _m rodlike micelles.

(c) Cylindrical

sections in rodlike micelles.

b) (mi,

_m2,

..,

m8)

_are trie values of trie _{spms on} trie

eight

sublattices.

2. Shnidman and Zia Model

The Hamiltonian of the

proposed

model is _{a sum} of three terms:

H=Hk+Hj+Hh,

Where

Hk

is effective at the intramicellar

length scale, representing many-body

interactions

responsible

for self-association and

controlling

the size and

shape

distribution of

aggregates,

^and

Hj

describes _an effective

short-range coupling

between the

aggregates

at

larger

intermicellar

length

scales.

Finally, Hh

is the

part controlling

the concentration of

aggregates.

We _assign

negative

_energies

-Ko, -Ki

and

-K2, respectively,

for the formation of _a

spherical micelle,

_a rod-like micelle and _a

layer-like

micelle

(planes

of +1

spins

surrounded

by

-1

spms).

l?or

simplicity,

_we show the

configurations

of these

energies

for _a two-dimensional

system [16]

in

Figure

la. For _an

end-caps "spherical-rodlike", "rodlike-layer",

the _averages

-(Ko

+

Ki)/2

éLnd

-(Ki

+

K2)/2

_are chosen

respectively.

~ro have _an idea of the

dependence

of the

energies -Ko, -Ki

and

-K2

and the characteristics of the

amphiphile molecule,

let and _u

denote, respectively,

the

length

and the volume of the

molecule. Let also _a be the _area of the

head-group

of the molecule. Then if _a >

3(u/1)

the

éimphiphiles prefers

to

organize

in

spherical

micelles [6] and then the _energy

-Ko

_is less than t.he other _energies

(-Ki

and

-K2).

If _now

2(u/1)

< a <

3(u/1)

the

spherical

micelles _are unstables and the rod-like micelles _are the

preferred

micellar

geometry

and then

-Ki

is less than the others _energies

(-Ko

and

-K2). Finally,

if

(u/1)

< _a <

2(u/1)

the

layer-like

is the

only

stable

aggregation geometry (among

the three basic

shapes considered);

_so

-K2

is less

than the others energies

(-Ko

and

-Ki).

To write H

explicitly,

it is convement to _use the

lattice-gas

variables

t~ _j,k "

(1

+ a~

j,k)/2

and i~_{j,k =}

(1

_a~

j,k)/2.

For further

convenience,

_we define the bilinear

products

U~,j,k "

t~-1,j,kt~+1,j,k

~i>J>k

Î~-1,j,kt~+1,j,k

Wj,j,k "

t~-1,j,kt~+1,j,k

+

i~-1,j,kt~+1,j,k,

and similar _one obtained

by replacing1

_~

j

and _~ k. In terms of these

operators,

^{H is}

given by

H ₌

-Ko ~j _{t$ûjûk (Ko}

₊

_Ki) ~j _t(w~ûjûk

₊

_$wjûk

₊

_$

₊

_ûj

₊

_wk)

2

(Ki

+

K2) ~j _t(u~wjûk

₊

_w~ujûk

₊

_$wjuk

₊

_$ujwk

₊

_w~ûjuk

₊

_u~ûjwk)

2

-Ki ~j _t(u~ûjûk

₊

_$ujûk

₊

_{$ûjuk) K2} ~j _t(u~ujûk

₊

_û~ujuk

₊

_ufijuk)

-J

~j1(u~

+ u~ ⁺ uk h

£(t _1),

where _we have

suppressed

all

except

the underlined indices and the summations _{are over} ail site indices

[16].

The Hamiltonian H is transformed in terms of

Ising variables,

to the form

-H =

Ji ~j

a~ +

2J2 ~j

a~aj ⁺

2J3 ~j

_{a~aj +}

_J4 ~j

a~aj

(1) (2) (3) (4j

+J5 ~j

Sa _j ak +

J6 ~j

Sa _j ak +

J7 ~j

Sa _j ak

(5) (6) (7)

+J8 ~j

S°j °k +

Jg ~j

S°j °k°1 +

J10 ~j

S°j °k°1

(S) (9) (10)

+

J11 ~j

$ajakal ⁺

J12 ~j

$ajakai ⁺

J13 ~j

_$ajakalam

(il) (12) (13)

+J14 ~j

~°j°k°1°m ⁺

J15 ~j

S°j°k°1°m ⁺

J16 ~j

S°j°k°1°m°n

(14) (là) (16)

+Ji7 ~j

a~aj ak ai aman +

Ji8 ~j

a~ajak ai amanap,

(17) (18)

where

Ji

" h

((14Ko

₊

6Ki -15K2 48J) /128),

J2

_"

-(6Ko

₊

6Ki 2K2

+

32J) /128,

J3

_"

-(-4Ko+4Ki+6K2)/128,

J4

=

(2Ko

₊

6Ki

₊

5K2

₊

16J) /128,

J5

"

(2Ko

+

6Ki

+

5K2 16J)/128,

J6

₌

(2Ko 2Ki 3K2)/128,

J7

"

-(Ko+Ki-K2)/128,

J8

₌

-(Ko 3Ki

+

3K2 )/128,

Jg

=

-(Ko

₊

Ki K2)/128,

Jio

=

K2/128,

Jii

=

-(Ko-3Ki+3K2)/128,

J12

"

J13

_"

Ji4

"

K2/128,

J15

"

J16

"

(Ko

+

Ki

+

K2 )/128

and

J17

"

J18

_"

-(2Ko

₊

6Ki

₊

3K2)/128.

'The summation

(1), (2),

,

(18) corresponding

to the

configurations

mdicated in Table I _are the _sums which _{run over} the different

sites,

bonds and

loops

of the

following

cell:

~i

~m q

i~

'Table

,q

q.

q

~

i q~

~ ~___~ ^,

°~~~*~i ~l~'q~

,

?~l

_~i

; q

. q~

J'~

_q~

( ~Îj~ ₍

;

,

~

*,, ^---à~--«

q~ ~Î

OEj

( "Q

^~l

i~ q~.

q

~.

~ qk.

q ~.

_q~

_{qç. q}

qÎ

_{~l » .} » . »

i,' i,'

qi,'

^i,'

qi,'

~ ^{~ ~}

~jj,W~

_~,

_~ ~jj,f~~~

~

ÎÎn ^~'

^Q1

~qn q~ Îq~

3. Ground States of trie Model

To discover the

ground

states of the

model,

_{we propose} the

following:

First,

_we divide the lattice

Z~

into

eight

sublattices and _we denote

by

_{mi, m2,}

, ms the

parameters characterizing

the

magnetizations

_on the

eight

sublattices

(Fig. lb).

Dur _atm here is to determine all the

periodic configurations

of the lattice

Z~ (which

in tum _are

translationally

invariant in each

sublattice)

that mmimize the energy U

(see Appendix).

To do

this,

_we must search

numerically for

the values of

magnetizations (mi,

_m2,

,

m8)

which minimize the _energy,

U,

for

given

values of the

parameters

^of the

system.

We daim that _our

procedure

must be

self-consistent;

hence it is necessary ^to use

correctly

the definition of _a

ground

state

[18]: by

a

ground

state we mean a

periodic configuration having

a minimal energy "Eo" such that any local

perturbation

increases its energy. Therefore it is useful _to look for the excitations of all the

preceding penodic configurations (Tab. Il)

and to indicate the

region

of their

stability by verifying

whether

they

_are

positive

_or not

(see Fig. 2b).

We

point

out that _we

only consider,

in this paper, the different _cases

corresponding

to _com-

peting

interactions of the model

(e.g. Ko

>

Ki

>

K2, Ko

>

K2

>

Ki,

and to show what

happens

to

phase diagrams

at low

temperature

if _we vary the chemical

potential

of the

amphiphiles.

We denote

by Gi

the "water"

ground

state

(g.s.), G2

the set of

"hexagonal" (infinite

"rod"

micelles), G3

the set of

"spherical" micelles, G4

the set of mixed

"rod-spherical" micelles, G5

the set of mixed

"rod-layer" micelles, G6

the set of "reversed"

micelles, G7

the set of rarefied

"spherical" micelles, G8

^the set of rarefied infinite "rod"

micelles, Gg

the _set of "lamellar"

(infinite "bilayer" micelles), Gio

the set of rarefied "reversed" micelles and

Gii

the

amphiphile (see Fig. 2a).

Theirs energies, Eo, are

Eo(Gi)

"

h, Eo(G2)

_"

-)ÎKI

+

2J], Eo(G3)

_"

-jÎKo

⁺

^3J],

Eo(G4

₌

Î2h

+

3Ki

₊

Ko

+

9J], Eo(G5

_"

Î2h

+

Ki

₊

K2

₊

3J],

Eo(G6)

"

Î2h

+

3K2

₊

3J], Eo(G7)

"

Î-6h

+

Ko

+

3J],

Eo(G8)

"

-jÎ-2h+Ki+2J], Eo(Gg)=-j[K2+J],

Eo(Gio)

"

-(Î2h

+ J +

K2), Eo(Gii)

" -h.

4. Mean Field

Approach

Using

the _mean field

approximation

the free _energy,

F,

of the micellar

binary

solutions _at three dimensions _may be

expressed

_as function of the inverse

temperature fl,

variational

parameters h~,

₌

1,

,

8,

the

magnetizations

_m~,

=

1,

,

8,

and

J~,1

=

1,

,

18,

which _are mentioned

previously:

8 8

~

~~ Î~ ~~~~~~~~~~~~~

~

Î~ ^~~~~

^~

^~

where trie _average

magnetization

_{per 8pin}

m~,1

=

1,. ,8,

is

given by

ill~ =

tanin(flà~).

'The _mean field

equations

for this situation _are obtained

by minimizing

the _expression for the free _energy,

F,

with

respect

to the variational parameters

h~,

1=

1,

,

8 combined with the llast

expression

for the

magnetization.

Then the first

equation

^for this

system corresponding

to

'fable II.

Energies of elementary

e~citations

of

trie

ground

states

Gi, G2,

...,

Gii.

Ground

state Excitation

energy Spin flip fn(@ ) J

j----j _j+---1

GI E(ti~)=-2H-Ko

~ l

G~

à=2H+3J+2Kj-Ko

~

-++-

-++-

(Gzfi=-2H+2J+2Ki-3K~ ~~

1-++-1 j-++-j

^1-++-1

j--+-j

G3

3~1=2H+Ko+6J

~

+--+

+--+

(G3fi=-2H+3J+3Ko-3Kj ~~

1+--+~ j+++-j

^Î+--+~

j+-+-j

G4 _E(G4

+--+

+--+

j+++-j j++++j

E(G4 fi

+--+

--+

j+++-j j+-+-j

E(G4 h

+--+

--+

j+++-j j+++-j

4)4=2H+Ko+6J

~

+--+

---+ 8

G5 E(G5~1=-2H+3J+2Ki

+--+1

1++-+1

fi=2H-3J+2K~-Ki-Ko

(~~~~j

+--+ ---+

j++++j j+-++j

E(G5 h

+--+

^--+

j++++j j++++j

5)4=2H+3J+Kj

~

+--+

^---+

j+++-j j-++-j

G6 _E(G6

_~1

-+++

-+

j+++-j j++++j

~~6~2~

~~~~~~~~

Table II.

(continued)

----1

G7

E(G7J1=-2H-Ko ___~j~~~__~

^l/2

1---+1

~-+-+~

E(G7 h

~++--j^+--j

Gg _E(Gg ---- +---

++--j ~++--j

. ----

fi

~++--j-+--j

Gy

~l=-2H+J+2K~-Kil/4

~----Î

~+---Î

----~

~----~ t~o

~o~l=-2H+3J+6K~

+++-

t~ofi=2H+K~⁺⁺⁺⁺ ~~~j~~~~~~j

+++-

+-+-

E(t~oh=2H-3J+2K~-2Kj

~~~~

~+++-Î ~+++-Î

E(t~o )4

+++- ++++j

^+++-

j++++j

Gij

_~ l

++++ +++-

++++ ~++++j j++++j

~m

Gto~

G1i~ ~~

~) Î----~_+++- +

Kj>

~

'

^{' ~}

Kj> K2> Ko Kj=^6J; Ko" 2J; K2" 4J

~

~

_~~

h/J W

G ~ G~ ⁱⁿ G~ ^lsn G~~

K2> Kj> Ko K~ 6J; Kj = 4J; Km 2J

~ ~~

h/J

, , ~

G~ ^{-7/2 G}_~ ^21/2

G,

q> Kj> K~ Koz 6J; Kj = 3J; K2=J G~

h/J

, , ~

G~ ^-9/2 G~ ^+9/2 G~_~

q> K2> Kj Ko~ 6J; K2* 4J; K _j= 2J

~

~

_~~

h/J

~

G -9/2 G~ 3n G~ ^lsn G~~

Kj>K2>

Ko

Kj= 10J;K~=4J; Xj = J G~

h/J

' ~

~) ^~i

^{~~ ~2}

~~

~5

1712

~ii

Fig.

2.

a)

Ground states of the model: Gi trie "water"

ground

state

(g.s.),

G2 the set of

"hexagonal"

(infinite

"rod"

micelles),

G3 trie set of

"sphencal" micelles,

G4 trie set of mixed

"rod-sphencal"

micelles, G5 trie set of mixed

"rod-layer"

micelles, G6 trie set of "reversed" micelles, G7 trie set of rarefied

"sphencal"

micelles, G8 trie set of rarefied infinite "rod" micelles, Gg trie _set of "lamellar"

(infinite

"bilayer" micelles),

Gio trie set of rarefied "reversed" micelles and Gii trie

amphiphile. b)

Phase

diagrams

at zero temperature for different values of the parameters

Ko,

Ri and K2. Trie above _arrows mdicate the points ^{where the three}

phases

coexist.

c

L(~.5,C) ^L(C,4.5)

Gi ^{G3 ~11}

~.5 4.5 h

Fig.

3. Phase

diagram

of trie model for Ko _> Ri > K2

(for example,

Ko

= 6J Ri

" 3J, K2 ₌ J

on trie T h

plane.

Water

(Gi ), "spherical"

micelles

(G3)

and

arnphiphile (Gii

are trie pure

phases.

The tricritical point C separates trie second order

phase

transition fine

L(-4.5, C)

from trie first order fine

L(C, 4.5)

of trie first order transition.

which is trie intersection of _m lines of second order and _n lines of first

order;

-B~ multicritical

point (which

is the intersection of _m lines of second

order);

-A" the

n-phase _point (where

_n

which is the intersection of _n lines of the first

order).

In

particular

_we call C the tricritical

point

which is the intersection of _a line of second order and _a line of first order.

In the _case

Ko

>

Ki

>

K2 (here

_we

choose,

for

example, Ko

=

6J, Ki

=

3J, K2

=

J)

(see Fig. 3)

the

phase diagram

shows the existence of the

phases corresponding

to the

ground

states

Gi, G3

and

Gii

which _are defined

previously.

The tricritical

point

C separates the

second-order

phase

transition lines

L(-4.5, C)

from the first order transition

(Fig. 3)

line L

(C,4.5).

In the _case

Ko

>

K2

_>

Ki (here

_we

choose,

for

example, Ko

"

6J, Ki

_"

2J, K2

"

4J)

(see Fig. 4)

^the

phase diagram

shows the existence of the

phases corresponding

to the

ground

states

Gi, G3, G6

and

Gii. So,

the

triple

_point

^A~

_separates the first order

phase

transition line

L(C,A~), L(1.5,A~)

and

L(7.5,A~).

The tricritical

point

C

separates

the second~order

phase

transition lines

L(-4.5, C)

from the first order transition lme

L(C, A~) (Fig. 4).

In the _case

K2

_>

Ki

_>

Ko (here

_we

choose,

for

example, Ko

₌

2J, Ki

₌

4J, K2

=

6J)

(see Fig. 5)

the

phase diagram

shows the existence of the

phases corresponding

to the

ground

states

GI G6

an~l

Gii

The coexistence Îine

L(-3.5, 10.5), separating

the

phases correspon~ling

to

(Gi, G6)

and

(G6, Gii),

is _a first order transition line

(Fig. 5).

We note that the _case where

K2

_>

Ko

>

Ki

is identical to the _one where

K2

_>

Ki

_>

Ko.

In the _case

Ki

>

K2

>

Ko

_we find _two different situations: The first _one

corresponds

to the

case

Ki

_>

2K2 (here

_we

choose,

for

example, Ko

"

J, Ki

=

10J, K2

=

4J) (see Fig. 6):

the

phase diagram

shows the _existence of the

phases corresponding

to the

ground

states

Gi, G8, G2, G6, G5

and

Gii

which _are defined

previously.

The

triple points (A~)i, (A~)2, (A~)3,

and

(A~)4

separate the first-order

phase

transition lines

L(-6., (A~)i), L((A~)i, (A~)2), L((A~)2,

(A~)3), L((A~)2, (A~)4), L((A~)3, (A~)4), L((A~)3,3.5)

and

L((A~)4,8.5) (Fig. 6).

The second

~

L(cA~)

L175A~)

~3

~ ~~~~

~~ Gii

G~ L(1.5,A~)

~.5 1.5 7.5 h

Fig.

4. Phase

diagram

of the model for Ko > K2 > Ri

(for example,

Ko

" 6J, Ri

# 2J, K2

"

4J)

on trie T h

plane.

Water

(Gi), "sphencal"

micelles

(G3),

"reversed" micelles

(G6)

and

amphiphile (Gii

_are the _pure

phases.

Trie

triple

point A~ separates the first order

phase

transition fines

L(C,

^A~

), L(1.5, A~)

and

L(7.5, A~).

Trie tricritical

point

C separates the second order

phase

transition hne

L(-4.5, C)

from trie first order fine

L(C, A~).

L(-3.5,10.5)

5

~i ~6 GU

3.5 la.5 h

Fig.

5. Phase

diagram

of the model for K2 > Ri > Ko

(for example,

Ko _" 2J, Ri _" 4J, K2

"

6J)

on trie T h

plane.

Water

(Gi ),

"reversed" micelles

(G6),

and

amphiphile (Gii

_are trie pure

phases.

L(-3.5,10.5)

_{is a} fine of the first order transition.

one

corresponds

to trie _case

Ki

<

2K2 (here

_we

choose,

^for

example, Ko

_"

2J, Ki

"

6J,

K2

"

4J) (Fig. 7):

the

phase diagram

shows the existence of the

phases corresponding

to trie

ground

states

Gi, G8, G2, G6, Gio

and

Gii. Moreover,

the cntical

end~point ^BA2 separates

the first order

phase

transition lines

L((A~)i,BA~)

and

L(BA~,C2)

from the second-order

L((A3)~(A3)4) (A3)~

(A3)

Ll(A3)1,(A3jij L((A3)2,(A3)~

~6 ^~

(A3j (A3j~ L((A3)~/A3jj

L(-6,(A3)il G~

Gi _G~ ~5 Gii

3.5 8.5 h

Fig.

6. Phase

diagram

of the model for Ri > K2 > Ko

(for example,

Ri >

2K2,

Ko

" J, Ri _"

10J, K2

"

4J)

_on trie T h plane. Water

(Gi),

rarefied infinite "rod" micelles

(G8), "hexagonal"

(m~

finite "rod"

micelles) (G2),

"reversed" micelles

(G6),

mixed

"rod-layer"

micelles

(G5)

and

arnphiphile (Gii)

_are trie _pure

phases.

Trie

triple

points

(A~)i, (A~)2, (A~)3,

and

(A~)4,

separate trie first _or~

der

phase

transition fines

L(-6., (A~)i), L((A~)i, (A~)2), L((A~)2, (A~)3), L((A~)2, (A~)4), L((A~)3, (A~)4), L((A~)3, _3.5)

and

L((A~)4, 8.5).

Lici,(A3)~)

L(Ci,BA~) Ci _(A3~

~~~"~~~

~~BA2,Cz)

~~ ~~

Llcz,5)

~jj~3~z,7s)

G~ G8 G~ G6 ~ll

-4 7.5 h

Fig.

7. Phase

diagram

of trie model for Ri > K2 > Ko

(for

_example, Ri < 2K2, Ko

"

2J,

Ri _#

6J, K2

"

4J)

_on trie T-h

plane.

Water

(Gi ),

rarefied infinite "rod" micelles

(G8), "hexagonal" (infinite

"rod"

micelles) (G2),

"reversed" micelles

(G6),

rarefied "reversed" micelles

(Gio)

and

amphiphile (Gii)

are trie _pure

phases.

Trie critical

end~point BA~

separates trie first order

phase

transition fines

L((A~) BA~)

and

L(BA~, C2)

trie second order _one

L(Ci, BA~)

The triple points

(A~)i,

and

(A~)2

_separateii

trie first order

phase

transition hnes

L(-4., _(A~ )i ), L((A~)i, _{BA~ ),} L(Ci, (A~)2

and

L((A~ )2, 7.5).

Trie tricntical points Ci and C2 separate the first order

phase

transition hnes

L(Ci, (A~)2)

and

L(BA~,

C2)

from trie second order _ones

L(Ci, BA~)

and

L(C2, .5).

T

L((BA211,(BA~)11 (BA2j~

L((BA2)~~.161

G3 ~~,~~j~~3~

A3 L((BA~)iA~)

L(A3,.5)

Gi G2 ~4 ~ll

4 .5 h

Fig.

8. Phase

diagram

of trie model for Ri > Ko > K2

(for example,

Ko

" 4J, Ri _" 6J, K2 _"

J)

on trie T h

plane.

Water

(Gi), "hexagonal" (infinite

"rod"

micelles) (G2), "spherical"

micelles

(G3),

mixed

"rod~spherical"

micelles

(G4)

and

amphiphile (Gii

_are the pure

phases.

The critical

end-point (BA~)i

and

(BA~)2

separates trie first order

phase

transition hnes

L(-4., (BA~)i), L((BA~)i,A~) L((BA~)2A~)

and

L((BA~)2, 5.16)

from trie second order _one

L((BA~)i, (BA~)2).

transition fine

L(Ci, BA~).

The

triple points (A~)i,

and

(A~)2 separate

^{the first order}

phase

transition lines

L(-4., (A~)i), L((A~)i,BA~), L(Ci, (A~)2)

^and

L((A~)2, 7.5).

The tricritical

points Ci

and

C2 separate

^{the first order}

phase

transition lines

L(Ci, (A~)2)

and

L(BA2, C2)

from the second order _ones

L(Ci, BA~)

and

L(C2, .5) (Fig. 7).

For the _case where

Ki

>

Ko

>

K2 (here

_we

choose,

for

example, Ko

_"

4J, Ki

"

6J,

K2

₌ ^J

(see Fig. 8)

the

phase diagram

shows the existence of trie

phases corresponding

to the

ground

states

Gi, G2, G3, G4

and

Gii. Moreover,

the critical

end-points (BA~)i

and

(BA~)2 separate

^{the first order}

phase

transition lines

L(-4., (BA~)i), L((BA~)i, A~), L((BA~)2, A~)

and

L((BA~)2, 5.16)

from the second order _one

L((BA2)1, (BA2)2) (Fig. 8).

6. Conclusions

We have

presented,

_m this paper, some new theoretical and numencal results _concernmg trie

phase diagrams

^{of the micellar}

^binary

^solutions m three dimensions. We find that this model describes

qualitatively

the micellar

phases

^observed

expenmentally [14]. Also,

we check that the

conjectured phase diagram proposed by

Shnidman and Zia _is correct.

We want

to,mention

that

qualitatively,

the three-dimensional model

proposed by

Shnidman and Zia describes the

phase diagram

similar to those for the twc-dimensional model

[16].

Further,

we

point

ont that there is _some resemblance between our

phase diagrams

and those round

experimentally (see _[13]

and

[21]

^{for the}

binary

mixture of

C12E5

and

water).

For

example

in

Figure

6 of _{our paper} and in

Figure

^{6.3.b of reference} [6] ^{we can}

observe,

for certain

temperatures,

as we increase the concentration of

amphiphiles,

^the passage from the micellar

phase

to

"hexagonal" phase

^and to the

"rod-layer" phase.

Appendix

The _energy of the

configuration

_{mi, m2, ...,m8°}

~

Î~'l~~Î

Îl'l~1

+

m4)(m2

+

m3)

+

l'l~5

+

m8)l'l~6

+

m7)

_{+ mlms + m2m6}

~=1

+m3m7 +

m4m81

~3

((ml

+

m4)(m6

+

m7)

₊

(m2

+

m3)(m5

₊

m8)

_{+ mlm4 + m2m3}

~~5~8 ~

~6~71

~

ÎÎ~ ~ÎÎ _Îl'l~l

₊

m4

)(mÎ +'l~Î)l'l~lml

_~

_'l~4mÎ)

~=l

~

~~ ~~~~ ~~~i

+(m2

₊

m3)l'l~Î

~

~~~

~

~~~~~

_~

~~~~~

_~

~~

_~

~~~~~~

~ _~~ ~ ~

jj~jjiij~ii~iiiiÎiTii~ÎmiiÎ~iÎ~il~8ÎÎ~~~~i5~~7~~

+ ^~~

~~ ^{Îl'I~Î'I~Î ~'l~Î'l~Î} ~'l~Î'l~Î ~'l~Î'l~Î)

_~

l'I~Î'I~Î)l'l~Î ~'l~Î)

_~

_l'l~Î ~'l~Î)l'l~Î ~'l~Î)

-Jii [(m2m3

+

m6m7)(mims

₊

m4m8)

₊

(mim4

₊

m5m8)(m2m6

₊

m3m7)]

~~ ^[m2m3m5(m2

⁺ m3 +

m5)

_{+ ml m4m6}

(ml

_{+ m4 +}

m6)

_{+ m1m4m7}

(ml

_{+ m4 + m7}

+m2m3m8

(m2

+ m3 + m8 + m1m6m7

(ml

+ m6 +

m7)

₊

m2m5m8(m2

_{+ m5 + m8}

+m3m5m8

(m3

+ m5 + _m8 +

m4m6m7(m4

_{+ m6 +}

m7)]

~~ l'l~Î'l~Îl'l~1

+ m4 +

mÎ'l~Îl'l~2

₊

m3) +'l~Î'l~Îl'l~1

+

m7)

+

'l~Î'l~Îl'l~1

+ m6

+m(m((m2

_{+ m5 +}

m(m((m2

₊

m8)

₊

m(m((m~

_{+ mà +}

m(m((m~

_{+ m~}

~'l~ÎRlÎl'l~4

_{+ m6 +}

'l~Î'l~Îl'l~4

+

m7)

+

'l~Î'l~Î(m5

+

m8)

₊

mÎmÎ _(m6

₊

_m7)1

~~ Î'l~lm2m3msl'l~2

_{+ m3 +}

m5)

₊

'l~lm2m4m61'1~l

+ _m4 + _m6 +

mlm3m4m7(m1

+m4 +

m7)

₊

m2m3m4m8(m2

_{+ m3 + Rl8 +}

m1m5Rl6Rl7(ml

+ Rl6 +

Rl7)

+Rl2Rl5Rl6Rl8

(m2

_{+ Rl5 + m8 +}

m3m5m7m8(m3

_{+ m5 + m8}

+m4m6m7m8(m4

_{+ m6 +}

_m7)1

~~~

[m(m( (m5

+ m8 +

m(m((m2

₊

m3)

₊

m(m((m2

_{+ m8 +}

m(m((m3

_{+ m5}

8

+m]m( _(m3

₊

m8)

₊

m(m( _(m2

_{+ m5 +}

m(m( _(m6

₊

_m7)

₊

m(m( _(mi

₊

_m4)

+m(m( (mi

₊

m7)

₊

m(m((m4

₊

m6)

₊

m(m((m4

₊

m7)

+

m(m((mi

₊

_m6))

~~ ^{Îl'I~Î'I~Î}

^~

'l~Î'l~Î)l'l~1m5

+

m4m8)

₊

_l'I~Î'I~Î

_~

'l~Î'l~Î)l'l~2m6

+

m3m7) +(m(m(

₊

m(m()(mim2

₊

_m7m8)

₊

(m(m(

₊

m(m()(m3m4

₊

m5m6) +(m]m(

₊

m(m()(mim3

₊

_m6m8)

₊

(m(m(

₊

m(m()(m2m4

+

m5m7)]

~~ Î'l~Î'l~Î'l~Î

~

~llÎ'I~Î'I~Î

~

'l~Î'l~Î'l~Î

~

'l~Î'l~Î'l~Î

~

'l~Î'l~Î'l~Î

~

'l~Î'l~Î'l~Î

~

'l~Î'l~Î'l~Î +m(m(m(] ^~~ [mim(m(m(

₊

m2m(m(m(

₊

m3m(m(m(

₊

m4m(m(m(

8

+msm(m(m(

₊

m6m(m(m(

₊

m7m(m(m(

₊

m8m(m(]

'fhe

expression

of the

magnetization equation

which

corresponds

to _mi is

given by:

mi "

tanh(fini)

where

hi

"

Ji

₊

4J2(m2

+ m3 +

m5)

+

8J3(m6

+ m7 +

m4)

+

6J4mi

+

J5[2mi(m2

_{+ m3 +}

m5)

+(mÎ

₊

mÎ

₊

mÎ)Î

₊

_{4J6 [m2} (m3

+ _m4 + m5 + _m6 +

Rl3(Rl4

+ Rl5 +

m7)

₊

m5(m6

₊

_m7)1 +4J7[2mi(m6

+ _m4 +

m7)

₊

(m(

₊

ml

₊

_m()]

₊

_8J8[m4m6

₊

m4m7 +

m6m7)

~~ j _~

~2

_~

~2

_~

~2

_~

~2)

_~

~

(~2

_~

~2

_~

~2

_~

~2)

_{~ ~}

~2

_~

~2

_~

~2

_~

~2)

9 2 3 4 5 5 3 2 4 5 7 5 2 3 6 7

+2mi(m2m6

_{+ m3m7} + _m2m4 + _m5m7 + _m3m4 + _m5m6 )) +

2Jio[2mi (ml

₊

m(

+

m()]

+8Jii Îm5(m2m3

+

m6m7)

+

m4(m2m6

+

m3m7))

+

4J12 Î2mi (m4m6

+ _m4m7 +

m6m7) +m4m6(m4

+

m6))

+

J1312mi(m((m2

+

m3)

₊

m((m2

₊

m5)

₊

m((m3

₊

m5)

+(m(m(

₊

m(m(

₊

m(m()]

₊

4J1412mi(m2m4m6

+ m3m4m7 +

m5m6m7)

+m2m3m5

(m2

+ _m3 + _m5 + _m2m4m6

(m4

+ _m6 +

m3m4m7(m4

+ _m7

+m5m6m7(m6

_{+ m7))}

+2J1512mi (ml (m6

+

m7)

₊

m((m4

₊

m7)

+

m((m4

₊

m7))

₊

m(m(

₊

m(m(

+m(m(]

₊

2J1612mi(m((m2m6

+

m3m7)

+

m((m3m4

₊

m5m6) +'l~Îl'l~2m4

+

m5m7))

_{+ 'l~5}

l'I~Î'I~Î

~

'l~Î'l~Î)

~ 'l~2

(RlÎ'I~I

~

'l~Î'l~Î) +m3(m(m(

₊

m(m()]

₊

_J1712mi (m(m(

₊

m(m(

₊

m(m()]

+

J18 Îm(m(m(

+ 2mi

(m2m(m(

₊

m3m(m(

₊

msm(m(

_)]

fhe _mean field

equations corresponding

to m2, m3,

,

and _m8 must be obtained

by exchanging, iespectively, (mi

_{- m2, m3 - m4, m5 - m6, m7 -}

m8)1 (mi

_{- m3, m2 - m4, m5 - m7,}

m6 **

m8)1 (ml

_{** m41 m2 ** m5, m3 ** m6, m7 **}

m8)1 (ml

_{** msi m2 ** m6, m3 ** m7,}

m4 **

m8)1 (ml

_{** m61 m2 ** m7, Rl3 ** m8, m4 **}

m5)1 (ml

_{** Rl7, m2 ** m8, m3 ** m5,}

m4 -

m6)

and

(mi

_{- m8, m2 - m3, m4 -}

m7).

References

[ii

Dawson K-A- and Kurtovic

Z.,

J. Chem.

Phys.

92

(1990)

5473.

[2]

Halley

J-W- and Kolan

A.J.,

J. Chem.

Phys.

88

(1988)

3313.

[3] Matsen M.W. _an Sullivan

D.E., Phys.

Reu. A 41

(1990)

2021.

[4]

Gompper

G. and Schick

M.,

Chem.

Phys.

Lett. 163

(1989)

475.

[5]

Gompper

G. and Schick

M., Self-Assembling Amphiphilic Systems

ⁱⁿ Phase Transitions and Critical

Phenomena,

Vol.

16,

C. Domb and J-L- Lebowitz Eds.

(Academic, London, 1994).

[6]

Micelles, Membranes,

Microemulsions and

Monolayers,

W-M-

Gelbart,

D. Roux and A.

Ben-Shaul

(Springer-Verlag,

New

York, 1994).

[7] Jan N. and Stauffer

D.,

J.

Phys.

France 49

(1988)

623.

[8]

Physics

of

Amphiphiles: Micelles,

Vesicles and

Microemulsions,

V.

Degiorgio

and M. Corti Eds.

(North-Holland, Amsterdam, 1985).

[9]

Tiddy G-J-T-, Phys. Rep.

57

(1980)

1.

[loi Gompper

G. and Zschocke

S., Europhys.

Lett. 16

(1991)

731.

[iii

Seddon

I.M.,

Biochim.

Biophys.

Acta. 1031

(1990)

1.

[12]

^Hoffmann

H.,

Adu. Colloid

Interface

Sci. 32

(1990)

123.

[13] Strey R.,

Schomàcker

R.,

Roux

D.,

Nallet F. and Dlsson

U.,

J. Chem. Soc.

Faraday

7Fans.

86

(1990)

2253.

[14] Degiorgio V.,

Corti M. and Cantu

L.,

Chem.

Phys.

Lett. 151

(1988)

349.

[15] Israelachvili

J-N-,

Mitchell D.J. and Ninham

B-W-,

J. Chem. Soc.

Faraday

Trans. 72

(1976)

1525.

[16] Shnidman Y. and Zia

R.K.P.,

J. Stat.

Phys.

50

(1988)

839.

[17]

Benyoussef A.,

Laanait L. and Moussa

N., Phys.

Reu. B 48

(1993)

16310.

[18]

Slawny J.,

in Phase Transitions and Critical

Phenomena,

Vol.

16,

C. Domb and J-L- Lebowitz Eds.

(Academic, London, 1987).

[19] ^Laanait

L.,

Masaif N. and Moussa

N.,

A

Study

of the Phase

Diagram

^of the Micellar

Binary

Solution Models II: The

Low-Temperature Approach.

[20] Grifliths

R.B., Phys.

Reu. 812

(1975)

345.

[21] ^Mitchell

D.J., Tiddy G-T-J-, Warring L.,

Bostock Th. and McDonald

M.P.,

J. Chem. Soc.

Faraday

Trans. 79

(1983)

975.