Deviations from the spontaneous curvature in surfactant films : effects on the second virial coefficient and interfacial properties of the microemulsions

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Deviations from the spontaneous curvature in surfactant

films : effects on the second virial coefficient and

interfacial properties of the microemulsions

L. Auvray

To cite this version:

L-163

Deviations from

the

_spontaneous

curvature

in

surfactant films :

effects

on

the second virial

coefficient

and

interfacial

properties

of

the

microemulsions

L.

_Auvray

Laboratoire de

_Physique

de la Matière Condensée

(*,+),

Collège

de _France,

11, place Marcelin Berthelot, 75231 Paris Cedex _05, France and

Laboratoire Léon Brillouin

_{(+ +),}

CEA CEN _Saclay, 91191 Gif sur Yvette Cedex, France

(Re!ru le 10

_septembre

1984, revise le 5 deeembre, accepte le 20 decembre 1984)

Résumé. 2014 On évalue les contributions de

_l’énergie

de courbure du film interfacial de tensioactifs

aux interactions attractives entre

_gouttelettes

dans les microémulsions diluées d’une _part, et aux

tensions interfaciales des _systèmes

triphasiques

de Winsor loin des

points critiques

d’autre part.

(1) Dans une microémulsion diluée d’eau dans l’huile où la courbure des

_gouttelettes

d’eau _(définie

comme

_positive)

est _légèrement

_{plus grande}

que la courbure spontanée

C0

du film interfacial, une

fluctuation

_possible

à

_{l’équilibre thermodynamique}

est l’union de deux _gouttelettes en un dimère de faible

_énergie

de courbure. Ce processus donne une contribution attractive au second coefficient du viriel entre

_gouttelettes

en bon accord avec certaines observations

_{expérimentales.}

₍₂₎ L’inter-face entre une microémulsion et une

_phase

en excès, de l’huile par

exemple,

est soit

_fermée

_par un film

plan

de tensioactifs de _type

_Langmuir,

dans ce cas un film mouillant d’eau _sépare le film et la

micro-émulsion, soit _{ouverte, i.e. traversée par} des _{pores d’huile. La} courbure du film de tensioactifs de la microémulsion dans la

_région

interfaciale est en _moyenne

_dirigée

vers l’huile dans le

_premier

cas

et vers l’eau dans le second cas. Nous montrons à l’aide du modèle de microémulsions aléatoires

proposé par de Gennes, que, dans les systèmes

triphasiques

de Winsor, l’interface

eau/microémulsion

passe du cas « ouvert » au cas « fermé » et l’interface

_{huile/microémulsion}

du cas « fermé » au cas

« _{ouvert »,}

_quand

la salinité de _l’eau, i.e.

_C0,

_augmente. Cette évolution est en accord avec des

expé-riences

_{d’ellipsométrie.}

_Quand

_{| C0 |}

_{est petit,} les deux interfaces sont « ouvertes » et la différence entre les tensions de surface des deux interfaces est

_{proportionnelle}

à

_C0.

On retrouve _qu’à la salinité

optimale (C0

= 0), cette différence est nulle.

Abstract. 2014 One evaluates the contributions of the surfactant interfacial film

bending

energy first to the attractive interactions between

_droplets

in dilute microemulsions and second to the inter-facial tensions in

_triphasic

Winsor _systems far from the critical

_points.

₍₁₎ In a dilute water in oil microemulsion where the water

_droplet

curvature _(defined as

_positive)

is

slightly larger

than the

interfacial film’s _spontaneous curvature

_C0,

a

_possible

fluctuation at

_{thermodynamic equilibrium}

is the fusion of two _droplets in a dimer of low curvature _{energy. This process gives} an attractive

J. _Physique Lett. 46 ₍₁₉₈₅₎ L-163 - L-172 15 FÉVRIER _1985, ]

Classification

Physics

Abstracts 61.25 - 68.10 - 82.70

(*)

Equipe

de Recherche Associée au _C.N.R.S., no. 542.

(+)

GRECO « Microemulsions » du C.N.R.S.

( + + )

Laboratoire commun CEA-C.N.R.S.

contribution to the

_droplet

second virial coefficient which is in _{good agreement} with certain

experi-mental observations. ₍₂₎ The interface between a microemulsion and an excess _phase, oil for _example, is either closed

_by

a flat

_Langmuir

type surfactant film, in which case a _wetting water

_layer

_separates the film and the microemulsion, or _open, i.e. traversed

_by

_{oil pores. The} microemulsion surfactant film curvature is on _{average directed towards oil in the first} case and towards water in the second

case. We show

_{by using}

the microemulsion random structure model

_{proposed by}

de Gennes that,

in the Winsor

_triphasic

systems, the

_{water/microemulsion}

interface evolves from the case « _{open »}

to the case « closed » and the _{oil/microemulsion} from the case « closed » to the case « _{open »,} when the water _salinity, i.e.

_C0,

increases. When |

C0 |

is small, the two interfaces are « _{open » and the}

diffe-rence between the surface tensions of the two interfaces is _proportional to

_C0.

One recovers the result that this difference vanishes at the _{optimal salinity (C0} = _0).

1. Introduction.

Recently

many features of microemulsion structure and

_phase

behaviour have been

_{interpreted by}

considering

the curvature _energy of the surfactant interfacial film between oil and water, which becomes

_important

because the film surface tension is almost zero

_[1-3].

The curvature _{energy per} unit area is written :

C is the local interface mean _curvature, C =

1/R1

₊

I/R2,

where

R1

and

R2

_are the

principal

radii of curvature.

_{By convention,}

a flat interface

(C

=

0)

has _{a zero curvature}

energy and the curvature is

_positive

for water in oil

_droplets.

The first term describes the

_flexibility

of the interfacial

film;

K is a

rigidity

constant

_(in

micro-emulsions,

K ~

10-14

_{erg) [4] ; if K}

is

_larger

than

_kT,

the

_oil,

water and surfactant are ordered in

birefringent

lamellar

_{phases [1, 4],}

if K is

_smaller,

disordered microemulsions may be formed.

Co

is the

_spontaneous

film curvature.

_{Co depends}

on the

_geometry

of the surfactant

and-cosur-factant molecules

_[5]

and on their interactions with oil and _water; it is related to the Winsor

~ R-ratio »

_[6]

and to the surfactant

_{Hydrophile-Lipophile-Balance [7],}

but has never been

mea-sured

_{quantitatively}

because,

in

_general,

the surfactant film structure in

_{lyotropic liquid crystals}

and microemulsions is not

_only

_imposed

_by

the curvature _energy, but also

_{by long}

_range van der Waals and electrostatic

_forces,

steric interactions and

_dispersion

_entropy;

thus the local film

curvature differs in

_general

from

_Co.

In the two situations discussed

below,

this deviation leads to

remarkable effects :

i)

in dilute

_droplet

_{microemulsions,}

structural

_{rearrangements}

which lower the film

_bending

energy may be the

origin

of

apparent

attractive interactions between

_droplets;

ii)

in Winsor III microemulsions

[6-9],

the curvature _{energy contributions} to the interfacial tensions between the oil

_phase

and the microemulsion middle

_phase

and between the micro-emulsion and the water

_phase

_may

_explain

the interface structure

_changes

and the interfacial tension variations observed as the water

_salinity

varies.

2.

_Spontaneous

curvature and attractive interactions in dilute microemulsions.

Attractive interactions between dilute

_spherical

water

_droplets

have been observed in many

water in oil microemulsions

_{[10-13]. They}

are too

_large

to be

_only

due to _pure van der Waals

interactions between the

_droplet

water cores

separated by

the interfacial film. To describe

them,

Lemaire,

Bothorel and Roux

_[14]

assume that the

_spherical

surfactant

_{(and cosurfactant) layers}

of two

_interacting

_{droplets interpenetrate;}

the

_resulting

increase in van der Waals interactions

L-165 TWO CURVATURE EFFECTS IN MICROEMULSIONS

particularly

in salt free microemulsions with ionic surfactant and

alcohol,

the

_{interpenetration}

effects could be in concurrence with the curvature effects for two reasons :

i)

ionic surfactants like S.D.S.

_{(sodium dodecylsulfate)}

are _very

_hydrophilic.

S.D.S. forms oil in

water

_micelles;

the

_spontaneous

curvature

_Co

of a _{pure S.D.S. film should be very}

_negative;

even if

alcohol addition in the film increases

_Co,

it _may

_happen

that

_Co

remains

_{slightly negative,}

while the actual

_droplet

curvature is

_positive;

ii)

the addition of salt to water diminishes the attractive interactions between neutral water in oil

_{droplets [ 11, 12].}

This effect is not

_explained

_by

the

_{interpenetration}

mechanism.

The idea that curvature effects can lead to

_apparent

attractive interactions arises from the work of Safran

_[16],

who has shown that a

_{spherical droplet}

microemulsion is unstable with _respect to

shape

fluctuations which lower the surfactant

_bending

_energy at small

_{Co R}

values

_{(R, droplet}

radius).

Here,

to evaluate the

_strength

of these

_effects,

we

_specify

one _way

_by

which the Safran

instability begins

and use a

_{perturbation approach.}

We do not take into account all the

_possible

water and surfactant

_{arrangements,}

but consider

_only

the formation of

_droplet

dimers which is the main process

contributing

to the _apparent second virial coefficient of the

_droplets.

In the

_{non-perturbed}

state, the water in oil microemulsion

_(Øw,

water volume

_fraction,

_Cg, surfactant

_{concentration)}

is made of

_monodisperse

hard

_spheres,

each

_sphere

has a radius R

_(R

= 3

~w/c$

E, E,

_{area per} surfactant

polar

head),

volume

V1,

_area

81

and _an internal

curva-ture _energy

_El

_{(E1 =}

8

_{rcK(1 -}

_Co

_R)).

If the surfactant film is

_{incompressible}

and is curvature

independent,

the total film area and surface energy are _constant, and the initial state is

_perturbed

by

the dimer formation in two main _{ways :}

i)

because the total water volume and film area are

_conserved,

a dimer made of two

_spherical

droplets

cannot be

_spherical,

but has a surface with

_regions

of

_high

and low

_{(possibly negative)}

curvature

_{(Fig. 1);}

in the favourable cases

_{(Co R C 1),}

the curvature _{energy will be}

_lowered;

ii)

the translational

entropy

is decreased.

To

_study

these

_effects,

we

_proceed

_follows.

In order to _{separate the microemulsion free energy}

~.~)’ ~tB I

into two terms, the translation

entrt~y

and the curvature _energy, we

_neglect

the small

_entropy

associated to the dimer

_shape

fluctuations.

This amounts to

_{approximating}

_{the dimer free energy}

by

its curvature _energy, which has to be minimized at constant volume and area. As this

minimi-zation is not

_{possible analytically,}

we reduce the variational

_{problem by}

_considering

a _very small

class of dimer

_shape.

_Then,

_assuming

that all the dimers have the same minimal

_shape

and curva-ture _energy, we derive the microemulsion

_equation

of state.

Fig.

1. - Two identical

2.1 DIMER SHAPE AND CURVATURE ENERGY. - The

_problem

of

minimizing

the curvature _energy

of a deformable

_globule

at constant volume and area has been studied

_{numerically by}

Helfrich

_[17].

Applied

to our _case, his results show

_that,

for

_{Co R}

0,

a dimer

adopts

an oblate biconcave

discoid

_shape

rather than a

prolate

dumbbell like

shape,

whereas for

Co ~ ~

0,

it

probably

prefers

an

_{elongated prolate shape.}

We have used these indications to obtain two

_{simple shapes}

(biconcave

or

convex),

which allows an estimation to be made of an _upper bound of the dimer curvature _{energy and the free energy} in both cases :

_{Co ~ ~}

0 and

_{Co R -}

0.

_(In

these

_estimates,

we

_neglect

_possible

cosurfactant

_segregation

as a function of film curvature,

Co

and K are assumed to be curvature

_{independent).}

2 .1.1 Biconcave

_shape.

- We describe the dimer

_{shape by}

a

_development

in

_spherical

harmonics limited to the

_quadrupolar

terms and calculate its curvature _energy

_{by neglecting}

anharmonic

terms

_(cf.

Ref.

_{[ 16]}

and

_note).

In

_spherical

_coordinates,

the dimer surface

_equation

is :

m=2

To the first order in

A,

defined

_by

4 7~4

_{= ¿}

_a2,m

_~’

the dimer volume

_V2,

surface

₈₂

and

m=-2

curvature _energy

_{E2 ~S-}

are :

Equations (2)

and

(3) determine,

A and

_{E 2 b.,.}

uniquely :

2.1.2 Convex

_{prolate shape.}

- The

_{simple shape}

of this kind allows an exact calculation of the

curvature _energy in a

_{spherocylinder (cylinder length}

_L,

radius

_p~)

to be made. The constraints of volume and area conservation

_{impose :}

By

comparing

E2 ~S~

and

_E~

with

_{2 E1,}

the curvature _{energy of} two

_separated

_droplets,

one

deduces that the dimer formation is

energetically

favourable as soon as

_{Co R}

1.2.

_Assuming

that

one _{may compare the}

_approximate

calculation of

E2b-s-

with the exact calculation of

_E’21,

one recovers Helfrich’s

results;

for - 2

_{Co R}

_1.2,

a dimer has a

_prolate

spherocylindrical

shape,

the energy

gain

is

_mainly

due to the

_flexibility

term; for

_{Co R}

-

_2,

the dimer has a biconcave

shape,

the energy

gain

is

_mainly

due to the

_spontaneous

curvature term; note

that,

in the harmonic

approximation

used in this case, a

_prolate

_shape

has the same _energy as an oblate

shape.

L-167 TWO CURVATURE EFFECTS IN MICROEMULSIONS

2.2 EFFECTIVE DROPLET SECOND VIRIAL COEFFICIENT. - With

our

_{simplifying assumptions,}

we can use Tanford’s

_procedure

_[18]

to evaluate the

_apparent

_droplet

virial coefficient. If one

describes the dimer formation

_by

a

single

chemical

equilibrium

(X

+ X ~

_X2),

the

_equilibrium

constant is :

4>1’ 4>2

are the monomer and dimer volume

_fractions,

to second order in

_{4>w :}

The dimer formation reduces the osmotic _{pressure 77}

_by

the

_{quantity :}

The dimer contribution

_Ba

to the second virial coefficient B defined

_{by :}

is :

By including

the

_hard-sphere

contribution

_{BH.s. =}

8

_{(one neglects}

the difference between the hard

sphere

radius and the water core

radius),

the total second virial coefficient is :

This

_equation

has two _{consequences for} water in oil microemulsions :

i)

when the

_droplet

radius is

_kept

constant, B

increases as

_Co

_increases,

i.e. the

_droplets

harden as the interfacial film becomes more

lipophilic (e.g.

if one uses surfactant or alcohol molecules with

a

_longer

aliphatic

tail)

or less

hydrophilic (e.g.

salt is added to

_{water) ;}

ii)

at constant K and

_{Co, B}

decreases as the

_droplet

radius increases if

_Co

is

negative.

2. 3 COMPARISON WITH EXPERIMENT. - The

_theory

_presented

here

_predicts

that dimers associated to the attractive interactions in _very dilute

_droplet

microemulsions

_always

_exist,

this is indeed verified in recent electric

_{birefringence experiments [19].}

Equation (5)

may

interpret

the

yet

unexplained droplet hardening

caused

_by

salt addition to water in oil

microemulsions;

it also

_predicts

the observed evolution when one varies the alcohol

chain

_length

or

_droplet

radius. In this last

_respect,

it _may however be difficult to

_separate

the

present

model from the Lemaire

_model,

because both

_predict

the same observed trends. As an

example,

we have

_analysed

the data of reference

_[13].

In the S.D.S.-hexanol-dodecane-water

system

studied,

the alcohol to surfactant ratio in the film remains

_{approximately}

constant as the

droplet

radius R is

_increased,

thus we _suppose that K and

_Co

are constant.

_Equation

₍₅₎

fits

cor-rectly

the data with a and

#

constant and

_{gives :}

Such

_hypothetical

estimates should be correlated with microemulsion

_phase

behaviour. In the

S. D. S.-pentanol-cyclohexane-water

system

of reference

_[11],

where dimers have been evidenced

_by

neutron

_scattering,

one observes that B does not

_depend

on the

droplet

radius and that these

microemulsions are

_adjacent

to lamellar

_phases

in the

_{phase diagram [4].}

Both observations

suggest

that

_Co

is

_practically

zero.

3.

_Spontaneous

curvature and Winsor III

_phases

interfacial tensions.

structure, which is now

_reasonably

well described

_{theoretically}

_[21-23]

and

_{experimentally}

_{[24, 25],}

are

_yet

unclear. In

_particular,

the local structure of the

_o/m

and

_w/m

interfacial

_regions

is little known. Surface tensions

_[26]

and

_ellipsometry

measurements

_[27]

have evidenced two extreme cases :

i)

the case of a thick diffuse critical interface associated with a _very small interfacial tension

(-

10 - 5

_{dyne/cm) ;}

ii)

the case of a narrow interface associated with a

_{relatively large}

interfacial tension

( ~

10- 2

_dyne/cm).

In this _case, an

_{ellipsometry experiment [27]}

has shown that a

Langmuir

film

probably

separates

the microemulsion from either the oil

_phase

or the water

_phase

and that a thin

wetting layer

of the third

_phase

exists between the

_Langmuir

film and the microemulsion.

As

_salinity

increases from the lower

_salinity

level where the water

_phase

_appears to the _upper

salinity

level where the oil

_{phase disappears,}

the

_m/w

interface moves from case

_(i)

to case

_(ii)

and

the

_o/m

interface from case

(ii)

to case

_(i).

In the medium

_salinity

_{range, the interface} structure is

unknown.

Up

to now, no unified

_theory

of the microemulsion interfacial tensions and interfacial structure

exists. The recent structure models

_[21-23],

which describe the

_middle-phase

as a random

disper-sion of oil and water in cells of size

_~,

the surfactant

_being

at the oil-water

_interface,

have been

_only

applied

to the case of a

diffuse

interface,

this characteristic

_arising

either from the randomness of

the microemulsion or from the

_vicinity

of a critical

_point.

Talmon and

_Prager

have shown that their structure

_model,

which

_neglects

curvature

_effects,

naturally

leads to a diffuse microemulsion interface of thickness

~.

The interfacial

tension,

calcu-lated in a van der Waals

_{theory [28],}

has an

_{entropic origin}

and is small

because ~

is

large

(y-T7~,~100 A).

More

_{recently, by using}

a model which includes the curvature _energy of the surfactant film and

by treating

the

_dispersion

_{scale ~}

as a variational

_parameter,

Widom

_[23]

has

_interpreted

the

Winsor microemulsion

_phase

behaviour and has

_pointed

out the

_special

nature of the Winsor critical

_points,

which would be

tricritical;

he then has discussed the microemulsion interfacial tensions within the frame of critical

_{phenomena theory [29].}

Here,

by using

also the random structure models and

_neglecting

van der Waals

interactions,

whose contributions to the microemulsion interfacial tensions have been calculated

_by

Chun Huh

_[30],

we

_study

the narrow non-critical

_middle-phase

interfaces. We focus the

_(mainly

quali-tative)

discussion on curvature effects and demonstrate a mechanism which

interprets

the

appea-rance of a

Langmuir

film at a microemulsion interface.

3.1 « OPEN » versus « CLOSED » INTERFACES

_(Fig.

_2a-c).

- When a classical microemulsion

made of water in oil

droplets

in a continuous oil

phase

coexists with a water

_phase

in excess, there

is a

_Langmuir

flat surfactant film at the

_{water/microemulsion}

interface,

the interface is « closed ».

Fig.

2. - Three

possible

structures of a microemulsion

middle-phase/oil phase

interface.

Progression

from

and open to a closed interface as the _spontaneous curvature decreases and

_changes

its

_sign.

_(a) and

_{(b) :}

the

L-169 TWO CURVATURE EFFECTS IN MICROEMULSIONS

However,

if the same microemulsion coexists with a _pure oil

phase,

the

_{microemulsion/oil}

inter-face is the

_analog

of a

_liquid

_vapour interface and there is no

_{Langmuir film;}

the interface is then

« _open ».

Inside the middle

_phase,

the interfacial film curvature fluctuates so

_strongly

that the oil and the

water are

_{randomly dispersed}

in

_microscopic

domains. At first

_sight,

it seems that the

micro-scopic

interface between oil and water defined

_by

the surfactant film in the

_w/m

or

_o/m

interfacial

region

cannot be flat. At the

_m/o

interface for

_example,

there should be continuous

_{(but fluctuating)}

path

going through

the oil from the _upper

_phase

into the middle

_phase.

This

_implies

that the

interfacial film should be on _average bent towards water. In this case, we _say that the

_m/o

interface

is « _{open »} to oil

_(Fig.

_2a).

_Similarly

the

_w/m

interface

_might

be « _{open »} to _water, with the inter-facial film bent on _average towards oil.

If the two

_w/m

and

_o/m

interfaces are « _{open » and if the film’s}

_spontaneous

curvature is not zero, the curvature _energy

_Ec

of the _soap film in each interface is different. If

_Co

is

_positive,

_E~(w/m)

is

_larger

than

_E~(o/m)

and one can then

_{imagine geometrical}

_{rearrangements}

of the

_w/m

interface which lower its

_spontaneous

curvature _energy

_(Fig.

_{2b, c).}

The

_experiments

_suggest an extreme case : one can transform the

_high

_energy

_w/m

interface

into a low _energy

_o/m

interface

_{by introducing}

a

wetting

oil

_layer

between the microemulsion and

water

_{(Fig. 2c).}

Then one needs a

Langmuir

flat surfactant film on the new oil-water interface. In

this _case, we

_speak

of a « closed » interface.

Finally, starting

from the « closed »

_w/m

_interface,

one can

_generate

all the

_possible

different

structures of a

_w/m

interface

_by

_creating

water pores

through

the oil

_{wetting layer.}

Here we will

study only

the two extreme cases :

i)

the

_randomly

_open interface

ii)

the

_completely

closed interface.

3.2 SURFACE ENERGY OF A RANDOMLY OPEN INTERFACE. - In the de Gennes

or Widom

_model,

the oil and the water

_(in

volume fraction

₀₀

and

_l/Jw)

are distributed

_randomly

with

_probability

0.

and

_ow

in cubes of size

~,

the surfactant

_being

at the

_oil/water

_{interface. ~}

is the

_{persistence length}

of the surfactant

_film;

as the

_probability

to find a cube facet

_{occupied by}

surfactant is

_{0. l/Jw’}

one has :

Far from a critical

_point,

the

_o/m

and

_w/m

interfacial

_regions

have a thickness of order

_~,

so we

restrict each of them to

_only

one cube

layer.

Our main

assumptions

are the

following : ~

is not a

variational

_parameter

_[23]

and does not

_change

in the interfacial

_regions;

the interfaces are

randomly

open, this means that the oil and the water remain

_randomly

distributed in the inter-facial cubes with the same

probability

as in the bulk

(Fig.

3).

Fig.

3. - The

There are two contributions to the surface tensions. One comes from the

_entropy,

the other from

the curvature _energy.

The

_entropic

contribution has been evaluated

_by

Talmon and

_Prager

_[28],

its order of

magni-tude is :

Note that this contribution is the same for both

_w/m

and

_o/m

interfaces. The surface tension

difference between the two interfaces will

_only

be due to the curvature _energy.

In the cube

model,

the curvature is concentrated on the cube

_edges.

For a

_{given edge,}

the

curvature _{energy 8~ may} take three values :

i)

Bc

= 0,

if there is no _soap on the

_edge

or if the film is flat.

ii)

Bc =

~K(l 2013 ,uCo (A

and ~ positive

constants of order of

_1),

if the film is bent towards

water at the

_edge.

iii)

Bc =

~,K ( 1

+

_{~Co ~),}

if the film is bent towards oil.

At the

_m/o

_interface,

_only

_edges

located on the upper

_plane

of the interfacial

_region

contribute to the curvature _energy excess with

_respect

to the

_bulk,

their curvature energy can

_only

take the

values 0 or

_{~,K ( 1 -}

_{/~Co ~).}

As the number of such

edges

where the film is bent towards water is on

average 4

Po ~W

per square of

area ç

2,

the curvature _energy contribution to the

_o/m

interfacial tension is :

8~

is the mean curvature _{energy per} cube

_edge

in the bulk and is a function of

_K,

_Co,

_~,

_4>0

and

4>w.

It has been discussed

_by

de Gennes from

_{symmetry arguments}

_[22],

it could also be evaluated

by counting

and

_weighting

the _ways of

_filling

with oil and water a cube of size

_{2 ~}

_containing

8 subunits of

_{size ~ ;}

then one describes all the

possible

film

_{configurations}

around a cube comer; one has :

where a 1, a2 and a3 are

positive

constant of order 1.

Finally,

the total

_o/m

interfacial tension is within this model :

Similarly,

the

_w/m

interfacial tension is :

We turn now to the case of a closed interface to establish the

_{corresponding equations.}

3.3 SURFACE ENERGY OF A CLOSED INTERFACE. - We use a _very crude model

where,

_{say, the}

m/o macroscopic

closed interface is a

_{superposition}

of a

_randomly

_open

_{w/m interface,}

a water

wetting layer

and a

_Langmuir

flat surfactant film. We

_forget

the

_possible

_{existence of oil pores}

through

the water

_{wetting layer,}

we

_only

remember that this

_{possibility requires}

the

_{wetting layer}

thickness to be of order

~.

Then,

the oil-microemulsion surface tension

_ym~o

is the sum of :

i)

_ym~W,

the surface tension of the

_randomly

_open interface calculated in section

_{3.2 ;}

ii)

the free _energy of the

_{wetting layer,}

_{yw. 1.} If one

_neglects

van der Waals

_{interactions,}

one

_only

has to consider the

_entropy

loss due to the localization of the water at the microemulsion interface.

As the

_{wetting layer}

thickness is of order

_~,

_{yW.l ~}

_~17~.

L-171 TWO CURVATURE EFFECTS IN MICROEMULSIONS

We _regroup the two last contributions in a

phenomenological

_parameter

y * (of

order

kTlç2),

finally :

- - -- ~

Similarly,

one would have for a

_m/w

closed interface :

3.4 DiscussioN. - We first note from

_{equations (8)}

to

₍₁₁₎

that the

_o/m

and

_w/m

interfaces

cannot be closed at the same time. If one limits the discussion to our _{two extreme cases,} there are three different situations

depending

on the value of the

_spontaneous

curvature with

_respect

to the _parameter

_{Co ,}

which is a characteristic curvature of

_{order ~ -1 :}

~

-~-i)

if

_Co

-

_Cd,

the middle

_phase

will minimize its surface free energy with an _open

_w/m

interface and a closed

_m/o

_interface;

ii)

if -

C*

Co

+

_C*,

the two interfaces are _open;

iii)

if

_Co

>

_CoB

the

_w/m

interface is closed while the

_m/o

interface is _open.

Experimentally,

one increases the

_spontaneous

curvature of an ionic surfactant film

_by

increasing

the water

_salinity,

one then sees that the _sequence

_(i),

_(ii),

(iii) corresponds

to the sequence observed in the

ellipsometry experiments [27].

The case

_(ii),

which

_corresponds

to a

small _spontaneous curvature and a middle _range

_salinity

is

_{particularly interesting.}

_Then,

the

difference

_Ay

between _ym~o and _ym~W is

_directly

related to the

_spontaneous

curvature and

_might

be used to estimate its order of

_{magnitude :}

Taking

K-10’~

_erg

_[4],

_{I/Co-100 A}

and

_{1 /cS ~ ~}

_{100 A,}

one estimates

_{0 y ~}

10- 2

_dyne/cm,

which is the

_right

order of

_magnitude.

Note that if

_{Co =}

_0,

one

_expects

that the middle

_phase

contains as much oil as water

_[23]

and that

_{Ay =}

0. This is indeed what is observed in _many Winsor

_systems

at the

_optimal

sali-nity [31].

4. General conclusions.

We have shown on

_simple

models that the recent ideas

_describing

microemulsion structure and

phase

behaviour as

resulting

from a

competition

between the oil and water

_dispersion

_entropy

and the surfactant film surface and curvature

_energies

could in certain cases

_explain

the observed

values of :

i)

the virial coefficients of

_{interacting droplets}

in dilute

_{microemulsions;}

ii)

the surface tensions of non-critical Winsor III _systems.

By

measuring

these two

_quantities,

one can obtain direct information on the curvature

elasti-city

parameters

of the surfactant interfacial film.

Hopefully,

it will be

_possible

to correlate both

measurements.

We have also

_interpreted

the existence of open or closed microemulsions interfaces. It would

be

_interesting

to

_get

information on the transition between the two

_types

of interfaces as the film’s

spontaneous curvature

_changes,

and td extend the discussion to the case of critical interfaces.

Acknowledgments.

I thank Christiane

_Taupin

and Jean-Marc di

_Meglio

for

_helpful

discussions and criticisms and

one of the referees for a remark about the dimer

_shape.

This work has received

partial

financial

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