Mesostructured Fluids: Supra-Aggregate Made of Interdigitated Reverse Micelles

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 174 (2000) 221–232

Mesostructured fluids: supra aggregates made ofinterdigitated reverse micelles

A. Filankembo a, P. Andre a, I. Lisiecki a, C. Petit a,T. Gulik-Krzywicki b, B.W. Ninham c, M.P. Pileni a,*

a Laboratoire SRSI, URA CNRS 1662, Uni6ersite P. et M. Curie (Paris VI), B.P. 52, 4 Place Jussieu,F-752 31 Paris, Cedex 5, France

b Centre de Genetique Moleculaire, CNRS F-91190 Gif-sur-Y6ette, Cedex, Francec Department of Applied Mathematics, PO Box ACT, Canberra 0200, Australia

Abstract

In this paper we describe in detail the phase diagram Cu(AOT)2–water–isooctane in the oil rich region. The phasediagram is very rich. All the phases described are thermodynamically stable. In this paper we concentrate in theisotropic region obtained at rather large water content (29Bw= [H2O]/[AOT]B40). A transition from intercon-nected cylinder to reverse micelles is observed. At high surfactant concentration the system does not have enough oilto solvate the surfactant alkyl chains and large aggregates made of interdigitated reverse micelles are surrounded byisolated water in oil droplets. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Mesostructured fluids; Oil rich region; Micelles

www.elsevier.nl/locate/colsurfa

1. Introduction

The microstructure of mixtures containing sur-factant, oil and water has been the subject ofmuch debate [1,2]. As recently as a decade agosome of the most active workers in the field ofcomplex fluids held that microstructure did notexist. Some considered that ionic microemulsionscould not exist without co-surfactant, others madea distinction between reverse micelles and microe-

mulsions. Furthermore emulsions have alwaysbeen regarded as thermodynamically unstable sys-tems, as a matter of definition.

Most of the ideas on self-assembly that connectstatistical mechanics to thermodynamics draw onnotions of local curvature of a surfactant interfaceset by a balance of molecular forces, together withglobal packing constraints.

Several years ago we described a phase diagramof various divalent double chained surfactantX(II) bis(2-ethylhexyl)sulfosuccinate, X(AOT)2

[3]. These last few years several groups describedsome parts of this phase diagram [4–7].

In the present paper we present a completedescription of the phase diagram of Cu(II) bis(2-

* Corresponding author. Tel.: +33-1-44272516; fax: +33-1-44272515.

E-mail address: [email protected] (M.P. Pileni).

0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.

PII: S0927 -7757 (00 )00513 -6

A. Filankembo et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 174 (2000) 221–232222

Fig. 1. Phase diagram of Cu(AOT)2 in isooctane solution.

ethylhexyl)sulfosuccinate, Cu(AOT)2–water–iso-octane in the oil rich region. A quantitative studyin the isotropic region obtained at rather highwater content is described. The component ratiosand microstructures that form can be predictedquantitatively from elementary geometrical argu-ments.

2. Phase diagram

The phase diagram of copper II bis(2-ethyl-hexyl)sulfosuccinate, Cu(AOT)2, isooctane, watersolution is described (Table 1).

We first give an outline of the phase diagram. Itwas prepared as follows: Cu(AOT)2 was mixedwith isooctane at a given concentration and water

added progressively to the solution. (Sampleswere sealed, vigorously shaken and centrifuged.They were allowed to stand for several days. Infact the equilibrium phases form in a few min.Each sample was checked for reversibility to tem-perature cycles). Phase boundaries appear alonglines essentially of constant water to surfactantratios, corresponding to discrete jumps in headgroup area as the surfactant is progressively hy-drated. The resulting ternary diagram is shown onFig. 1. All phases are separated by very flatmenisci, of ultra low interfacial tension. The sin-gle phase regions are shown in one solid color orwithout color. Hatched regions correspond to twophases. Triphasic regions are represented bysquares made of points. Subscripts b and t indi-cate biphasic and triphasic regions. The notation

A. Filankembo et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 174 (2000) 221–232 223

Tab

le1

Sum

mar

yof

the

phas

edi

agra

mof

Cu(

AO

T) 2

/wat

er/i

sooc

tane

inth

eoi

lri

chre

gion

Num

ber

ofSt

ruct

ure

[7]

inC

u(A

OT

) 2%

wit

h2%

inm

ass

wP

hase

regi

onSt

ruct

ure

inC

u(A

OT

) 2an

dm

etho

dph

ases

ofN

a+de

riva

tive

san

dm

etho

dC

u(A

OT

) 2C

u(A

OT

) 2/N

a+

Isot

ropi

cph

ase:

reve

rse

mic

elle

s(c

ondu

ctiv

ity)

Isot

ropi

cph

ase:

reve

rse

mic

elle

s(c

ondu

ctiv

ity,

10–

50–

5I

pack

ing

theo

ry)

5–9

5–10

Isot

ropi

cph

ase:

inte

rcon

nect

edcy

linde

rs2

IIb

Isot

ropi

cph

ase:

inte

rcon

nect

edcy

linde

rs(F

FE

M,

cond

ucti

vity

)(F

FE

M,

cond

ucti

vity

,pa

ckin

gth

eory

)B

iref

ring

ent

phas

e:es

sent

ially

lam

ella

ean

dII

I tB

iref

ring

ent

phas

e:es

sent

ially

lam

ella

ean

dfe

w9–

1010

–13

3sp

heru

lites

(FF

EM

)fe

wsp

heru

lites

(FF

EM

)Is

otro

pic

phas

e:in

terc

onne

cted

cylin

ders

Isot

ropi

cph

ase:

inte

rcon

nect

edcy

linde

rs(F

FE

M)

(FF

EM

)Sp

heru

lites

+pl

anar

lam

ella

e(F

FE

M)

Sphe

rulit

es+

diso

rder

edla

mel

lae

(FF

EM

)2

IVb

13–1

810

–15.

515

.5–2

618

–25

Bir

efri

ngen

tph

ase:

sphe

rulit

es(s

upra

2V

bB

iref

ring

ent

phas

e:sp

heru

lites

(sup

raag

greg

ates

);F

FE

M,

SAX

S)ag

greg

ates

);F

FE

M,

SAX

S,pa

ckin

gth

eory

)Is

otro

pic

phas

e:in

terc

onne

cted

cylin

ders

wit

hIs

otro

pic

phas

e:in

terc

onne

cted

cylin

ders

(FF

EM

,SA

XS)

smal

lob

ject

s(F

FE

M,

SAX

S,pa

ckin

gth

eory

)Is

otro

pic

phas

e:in

terc

onne

cted

cylin

ders

wit

h25

–31

Isot

ropi

cph

ase:

inte

rcon

nect

edcy

linde

rsV

I b2

26–2

85sm

all

obje

cts

(FF

EM

)an

d/or

cubi

cph

ase

VII

\28

.5\

31Is

otro

pic

phas

e:in

terc

onne

cted

cylin

ders

Isot

ropi

cph

ase:

28.5B

wB

321

and/

orcu

bic

phas

e(F

FE

M)

inte

rcon

nect

edcy

linde

rsan

dw\

32sp

here

sat

low

conc

entr

atio

nan

din

terd

igit

ated

mic

elle

sat

high

conc

entr

atio

n(F

FE

M,

SAX

S,pa

ckin

gth

eory

)


‘phase’ I, II, III… is used to indicate visuallydifferent phase regions. The ternary phase dia-gram exhibits the following features.

Let w= [H2O]/[AOT] be the ratio of the num-ber of water to surfactant molecules. Whateverthe Cu(AOT)2 concentration, at very low watercontent, a single isotropic phase is observed (re-gion I). At low water content (wB5), isolatedwater in oil droplets form.

Above w=5, region IIb is an isotropic phase inequilibrium with excess isooctane. Within this re-gion, an increase of water content results in areduction in the isotropic phase proportion. Anincrease of Cu(AOT)2 concentration causes theisooctane to be taken up to give the singleisotropic phase II. The structure of the isotropicphase has been seen by small angle X-ray scatter-ing, SAXS1, and conductivity2 is that of an inter-connected cylindrical network of constant meancurvature.

At w=9, a middle birefringent phase appearsin equilibrium with isooctane and the sameisotropic phase (region IIIt). Such behavior israther surprising. Usually the birefringent phase ismore densed than the isotropic one and appearsin the bottom of the vessel. In the present case itis less densed. This is attributed to a mixture totwo phases. The FFEM patterns of the birefrin-gent phase show the presence of spherulites(oignon phases) and planar lamellar phases. As amatter of fact, the FFEM patterns of the isotropicphase show presence of small objects which couldbe attributed to interconnected cylinders.

On increasing Cu(AOT)2 concentration, the ex-cess isooctane disappears and two phases (IIIb)coexist. There is a clear upper phase with a cloudylower phase. It is not birefringent. Regions IIIt

and IIIb exist over a very small water contentrange (9BwB10).

At w=10, a new birefringent phase emerges, inequilibrium with isooctane (region IVb). PhaseIVb exists over a wide range of w (10BwB15.5).On increasing Cu(AOT)2 concentration, moreisooctane is taken up until only a single birefrin-gent phase is obtained (region IV). The FFEMpattern3 shows a multilamellar phase usuallycalled spherulites. On heating the solution, anisotropic phase is formed, under cooling samespherulites, in same proportions, are obtained.

At w=15.5, a middle isotropic phase emergesin equilibrium with a new lower turbid birefrin-gent phase and excess isooctane. This phase re-gion Vt exists over a wide water content rangefrom 15.5 to 26. On increasing the water contentin this region from 15.5 to 26, the isotropic phaseincreases at the expense of the birefringent phasewhich disappears at w=26. The birefringentphase is made up of spherulites, spontaneous‘emulsion’ droplets. These are large closed multil-amellar objects surrounding an interior mi-crophase. On increasing Cu(AOT)2 concentrationwithin this region, the isooctane phase progres-sively disappears to become at first a turbid andbirefringent phase (region Vb). As for the othersamples, with time and/or under centrifugation,two well-defined phases emerge: an isotropicphase (phase Vb1) and a birefringent one (regionVb2). The FFEM pattern of phase Vb1 is similar tothat observed with interconnected cylinders withsmall spherulites (Fig. 2(A)). Fig. 2(B) shows thatphase Vb2 is made of multilamellar objects.

On increasing the water content from w=15.5to 26, the birefringent phase progressively de-creases whereas the isotropic one increases toreach a single isotropic phase at w=26. Phase Vb2

appears to be a single phase and is a spontaneousthermodynamically stable emulsions. The mi-

3 A thin layer of a sample (20–30 mm thick) was placed ona thin copper holder and then rapidly quenched in liquidpropane. The frozen sample was fractured at liquid nitrogentemperature, in a vacuum close to 10−7 Torr, with the liquidnitrogen cooled knife in a Balzers 301 freeze-etching unit. Thereplication was done using unidirectional shadowing, at anangle of 35°, with platinum–carbon 1–1.5 nm size of meanmetal deposit. The replicas were washed with organic solventsand distilled water and were observed in a Phillips 301 electronmicroscope.

1 SAXS experiments were performed at the D22 difractome-ters of the LURE, (CNRS-CEA-Paris XI) in Orsay, France,with high spatial resolution, the wavelength is 1.46 A, and thescattering vectors (q) range from 0.007 to 0.4 A, −1.

2 The measurements were made with a Tacussel CD 810instrument using a TD 100 (platinum) electrode from the samemanufacturer.


crostructure of the supra self-assembled aggre-gates, large spherulites, contained in the systemhas new features. The interior collapses to aninterconnected microemulsion. The exterior of thespherulites is comprised of the same isotropicmicroemulsion phase. In the interior and exteriorphase the interconnected cylinders has the samewater volume fraction. The whole system is inequilibrium with this isotropic microemulsion.The lamellae, which separate an interior and exte-rior, then adjust their spacing and componentfractions to allow equilibrium. Microphase segre-gation and compartmentalization can occur inany system in which geometry imposes differentphysical conditions inside and outside.

Above w=26, region VI is formed. At lowCu(AOT)2 concentration an isotropic phase is inequilibrium with isooctane (region VIb). On in-creasing Cu(AOT)2 concentration, one isotropicphase is observed. We did not do any experimentsin the domain because the domain phase is tooshort (26BwB28.5).

At w=28.5, one phase region is reached (re-gion VII). This is observed at any Cu(AOT)2

concentration. Structural study of the phase isdeveloped below.

As it has been already mentioned, the wholephase diagram is thermodynamically stable. Onheating each region, the system turns into a singleisotropic phase. On cooling the phase region re-mains unchanged than that observed beforeheating.

Very recently we observed that the phase dia-gram described above change with a low percent-age of impurities. Figs. 1 and 3 compare the phasediagram obtained without any impurities and inthe presence of 2% sodium derivatives mass, re-spectively, the phase diagram behavior is thesame. By comparison with Figs. 1 and 3, it isclearly observed slight changes in the water con-tent to which the phase transitions take place.Furthermore the existing domains of the variousregions changes. However the structure inside thevarious phases remains unchanged and the wholesystem remains thermodynamically stable.

3. Transition from interconnected cylinders toreverse micelles

Structural study of the region VII is presented.Whatever the Cu(AOT)2 concentration, phase VIIis isotropic.

Conductivity measured at various water con-tents is especially deserving of note (Fig. 4). Atlow Cu(AOT)2 concentration ([Cu(AOT)2]=5×10−2 M), the conductivity is rather large at w=28.5 in the range of milli Siemens. On increasingwater content, the conductivity dropped markedlyto level off at water content up to 33. The conduc-tivity is then in the range of nano Siemens. Thisindicates a decrease in the number of connectionswith increasing the water content. At higher

Fig. 2. (A) The FFEM pattern of phase Vb1 at [Cu(AOT)]0=4.2·10−1 M, w=21. (B) The FFEM patterns of phase Vb2 at[Cu(AOT)]0=4.2·10−1 M, w=21.


Fig. 3. Phase diagram of Cu(AOT)2 in isooctane solution containing 2% in mass of sodium derivative.

Fig. 4. Variation of conductivity of 5×10−2 M Cu(AOT)2 in isooctane solution forming phase with increasing water content.


Cu(AOT)2 concentration ([Cu(AOT)2]=1.2×10−1 M and [Cu(AOT)2]=4.2×10−1 M), the con-ductivity is also large but still decreases withincreasing water content.

From SAXS (Tables 2 and 3), at 29BwB32, thescattering is characteristic of cylinders. At w=32,a transition has taken place. It is difficult todistinguish between scattering of spheres and cylin-ders. Probably a mixture of both exists. Abovew=32, scattering characteristic of spheres is ob-tained. The FFEM patterns clearly show presenceof small objects.

The FFEM patterns at w=40 show, at lowsurfactant concentration (1.2×10−1 M), reversemicelles in the 20–40 nm diameter range (Fig. 5(A)and Fig. 5(B)).

Fig. 6(A) shows, at w=30, a monotonous in-crease in the conductivity with polar volume frac-tion, fp. This indicates an increase in the numberof connections of the ‘cylinders’. In apposite, Fig.6(B) shows, at w=36, a low conductivity at lowpolar volume fraction. Below, fp=0.17 isolateddroplets are dispersed in the solution. At fp=0.17,a percolation onset is observed and the conductivityincreases drastically. Comparison on a logarithmicscale of the conductivity variation with the polarvolume fraction at w=30 and 36 indicates a strongchange in microstructure (Fig. 6(C)). On increasingthe polar volume fraction, at w=30, the conductiv-ity increases by four orders of magnitude and ten

orders of magnitude at w=36. From these data, itcan be concluded that in phase VII, a transitiontakes place in the water content range. On increas-ing water content, an anti percolation process takesplace whereas on increasing surfactant a percola-tion is observed. Such complex conductivity behav-ior can be explained in term of connected cylindersand spheres (Figs. 4 and 6). It is concluded that atransition takes place around w=32: as observedby SAXS experiments, below this water contentconnected cylinders exist whereas above w=32spheres are formed. This is confirmed by the changeof the conductivity with polar volume fraction,measured at w=30 and 36. The head group is fullyhydrated and curvature at the interface remainsconstant. With increasing water content, the onlysolution to the problem of maintaining constantaverage curvature consistent with global packing isachieved by reducing connections in the networkwhich ultimately becomes disconnected spheres.

Elementary considerations that require only no-tions of local curvature and global packing con-straints are sufficient to determine microstructure(see Appendix). The surfactant and water volumefractions determined experimentally at low (Table2) and high (Table 3) surfactant concentration arealways lower than those calculated (Table 4).However, as mentioned above, this packing doesnot take into account aggregate interactions.

Table 2Water content, w ; head polar group area s ; shape of the objects cylinders (c); spheres (s); length of the cylinders, Lc; width of thecylinders, lc; polar radius of spherical droplet, R s·k the conductivity; surfactant, fS; water, fW, and oil, fO, volume fractionsmeasureda

29 30 31 32 33 34 35 36 37 38w– ––55s (A, 2) –50 50–55––Shape –s– –s–c/s–c

150 – – – – – – – –Lc (A, ) –110 – –lc (A, ) –– – – –

– – 60 – 65 – 70R s (A, ) –fS·100 7.98.08.08.08.18.18.18.18.28.2

13.813.513.112.812.5 13.912.111.811.511.1fW·10081.3fO·100 8080.7 79.7 79.4 79.1 78.8 78.5 78.3 78.1

425 414 – 320 –k (mS) 232 – 97 – 929.839.83 9.839.839.839.839.839.83 9.83fO/fS (exp) 9.83

fO/fS (cal. s) 2.28 2.31 2.34 2.37 2.41 2.44 2.472.25 2.51 2.54

a fO/fS (cal. s) calculated ratio of oil versus surfactant volume fractions for interdigitated spheres packed in a simple cubicstructure ([Cu(AOT)2]0=1.2×10−1 M).


Fig. 5. The FFEM patterns of phase VII obtained at w=40 at [Cu(AOT)2]0=1.2×10−1 M (A, B) and [Cu(AOT)2]0=4.2×10−1

M (C, D).

4. Interdigitated reverse micelles

In the region in which reverse micelles areformed (w]32), the SAXS measurements indi-cate scatter of spheres (Tables 2 and 3). TheFFEM patterns drastically change withCu(AOT)2 concentration. Fig. 5(A) and (B) showsmall objects. On increasing Cu(AOT)2 concentra-tion to 4.2×10−1M the FFEM pattern clearlyshows appearance of large objects surrounded bysmaller one (Fig. 5(C)). This is confirmed byFFEM high resolution. Fig. 5(D) indicates thatthese large objects are made of the smaller one.These differences in the FFEM patterns cannot beattributed to artifact. They are highly repro-

ducible. This could be explained in term of lack ofoil in the reverse micellar solutions. On increasingCu(AOT)2 the amount of oil needed to solvate thealkyl chains is not enough. So to reach stability,the hydrocarbon chains tend to interact betweeneach others to form large aggregates made ofinterdigitated reverse micelles. To prove this, wecompare the ratio of oil versus surfactant volumefraction measured and calculated by assuming asimple cubic packing of the reverse micelles (seeAppendix). At low Cu(AOT)2 concentration, theratio of oil to surfactant volume fraction mea-sured is larger than that calculated (Table 2). Thisclearly indicates that the alkyl chains have suffi-cient oil to be solvated and reverse micelles are


randomly dispersed in isooctane. At highCu(AOT)2 concentration, the ratio of oil to sur-factant volume fractions measured is smaller thanthat calculated (Table 3). This indicates that theoil is insufficient to solvate the chains. Hence atw=40:

(i) Whatever the Cu(AOT)2 concentration is,scatter of spheres is observed (Tables 2 and 3).The water and surfactant volume fraction mea-sured (Tables 2 and 3) are always smaller to thecalculation values (Table 4) indicating thatenough space is valuable for spheres to pack.

Fig. 6. Variation of the conductivity with polar volume fraction at w=30 (A), w=36 (B). Comparison in the log scale ofconductivity at w=30 and w=36 (C).

Table 3Water content, w, head polar area, s, shape of the objects cylinders (c), spheres (s), width of the cylinders, lc, polar radius ofspherical droplet, R s. Surfactant, fS, water, fW, and oil, fO, volume fractions measureda

31 32 33 34 35 36 37w 3829 30––55–55–s (A, 2) 55–53–

–Shape c/s– – s – – s –clc (A, ) ––––––––104–

60 – 65 – 70– –– ––R s (A, )22 21.8 21.6 21.3fS·100 21.122.5 20.9 20.7 20.522.2

30.5 31.2 31.9 32.6 33.3 34 34.6 35.3 35.9fW·100 36.545.6 45.1 44.7 44.3 43.846.1 43.4fO·100 4347 46.6

2.09 – – 1.64 –k (mS) 1.522.29 1.42 1.322.122.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1fO/fS (exp) 2.1 2.12.312.28 2.542.512.472.442.412.22 2.37fO/fS (cal. s) 2.34

a k the conductivity, fO/fS (cal. s) calculated ratio of oil versus surfactant volume fractions for interdigitated spheres in a simplecubic structure ([Cu(AOT)2]0=4.2×10−1 M).


Table 4Calculated surfactant and water volume fractions for a set of water content, w, in various structuresa

29w 30 31 32 33 34 35 36 37 3831XS

s · fcc·100 30 30 29 29 28 28 27 27 2728 27 27 26 2628 26XS

s · bcc·100 25 25 2422XS

s · sc·100 21 21 21 20 20 20 19 19 1938 37 37 36 3539 35XS

s · hcp·100 34 34 3333 32 32 31 31XS

c · s·100 3033 30 29 2939 39 38 37 3740 36XS

c · i · c·100 35 35 3434 34 33 33 32 32XS

c · i · fcc·100 3135 31 3042 43 44 44 4542 46XS

s · fcc·100 46 47 4738XS

s · bcc·100 39 40 40 41 41 42 42 43 4329XS

s · sc·100 30 30 31 31 32 32 33 33 3353 54 55 55 5652 57XS

c · hcp·100 58 58 59XS

c · s·100 4645 47 47 48 49 49 50 50 5155 56 57 58 5854 59XS

c · i · c·100 60 60 6148 49 50 50 51 52XS

c · i · fcc·100 5248 53 53

a (i) for spheres: XSs · fcc, XW

s · fccfor FCC, XSs · bcc, XW

s · bcc for BCC, XSs · sc, XW

s · sc for simple cubic; (ii) for cylinders: XSc · hcp, XW

c · hcp,for hexagonal, XS

c · s,XWc · s for square, XS

c · i · c,XWc · i · c, for interconnected cubic, and for FCC interconnected cylinders, XS

c · i · fcc,XW

c · i · c.

Fig. 7. Cartoon of interdigitated reverse micelles.


(ii) At low Cu(AOT)2 concentration, the FFEMpatterns at low (Fig. 5(A)) and high (Fig. 5(B))magnifications indicate presence of smallobjects with an average size in a same order ofmagnitude as that determined from scatteringmeasurement (Table 2). The ratio of oil versussurfactant volume fraction is larger thanthat calculated for packing of spheres (Tables 2and 4).

(iii) At high Cu(AOT)2 concentration, theFFEM pattern at low (Fig. 5(C)) magnificationshows large objects surrounded by smaller one. Athigh magnification (Fig. 5(D)) the larger aggre-gates seem to be formed by smaller ones. Theaverage size of the smaller aggregates is in thesame order of magnitude as the size determinedby SAXS experiments. In apposition to what isobserved at low Cu(AOT)2 concentration the ra-tio of oil over surfactant measured is smaller thanthat calculated indicating that not enough oil ispresent to solvate the alkyl chains of thesurfactant.

From these data it is concluded that large ag-gregates made of interdigitated reverse micellesare surrounded by fully solvated water in oildroplets. Hence supra aggregates made of inter-digitated micelles are in equilibrium with isolatedreverse micelles as shown in Fig. 7.

5. Conclusions

In this paper we describe the diagram ofCu(AOT)2-isooctane–water. It is shown that thisphase diagram is highly sensitive to presence ofvery low amount of impurities. However the gen-eral behavior remains the same.

Structural study of isotropic phase in the watercontent range from 29 to 40 is presented.

A phase transition between interconnectedcylinder to isolated sphere is demonstrated. In thesphere region, at high surfactant concentration,interdigitated reverse micelles surrounded by fullysolvated water droplets and formed.

Hence supra aggregates made of interdigitatedmicelles are observed. From our knowledge, it isthe first example of supra aggregates made ofreverse micelles.

Appendix A. Packing constraints and structures

We process to region of the phase diagramwhere the polar head group is totally hydratedand remains constant as determined by SAXSmeasured and shown on Tables 2 and 3.

Before proceeding to the necessary analysis weneed to list here some theoretical preliminaries.

The following parameters and results will beneeded:

the water content, w :

w=NW

NS

(1)

where NW and NS are the number of water andsurfactant molecules respectively.

Similarly w is defined as:

w=fW

fS

=NW · 6WNS · 6S

=w6W6S

(2)

where 6W, 6S, fW, fS, are the volume of watermolecules (30 A, 3) and that of the surfactant (639A, 3), the water and the surfactant volume frac-tions, respectively.

The surfactant parameter [8], s :

s=6S

aS · 1S

(3)

where aS is the head group area (55 A, 2) and lS thealkyl chain length of the surfactant (10 A, ). 6S isthe surfactant volume which has been determinedelsewhere [9]. In the packing model 6S is thevolume of the chains. In the calculation this willnot change drastically the data. It is assumed thatCu(AOT)2 behaves as NaAOT with head polargroup area following a logarithmic law with watercontent [10].

Several types of packing are assumed:(i) Packing of spheresFor a simple cubic packing [11] we have:

fOb +fS+fWB

p

6(4)

where fOb is the ‘bound’ oil volume fraction in a

‘dressed’ micelles. Bound oil is the fraction of oiltaken up by surfactant tails. Then:


fSBp

6(ws)2s�

ws+13�3=XW

s · sc (5)

fWBp

6(ws)3�

ws+13�3=XW

s · sc (6)

fO=1− (fS+fW) (7)

The parameter p/6:0.52 can be replaced by3p/8:0.68 or p/32:0.74 for a BCC orFCC packing, respectively.

(ii) Packing of cylindersThe cylinders can pack into square structure

[11] if:

fOb +fS+fWB

p

4(8)

This gives:

fSBp

4w · s2�

w · s+12�2=XS

C · S (9)

fWBp

4(w · s)2�

w · s+12�2=XW

C · S (10)

For cylinders in hexagonal, interconnected FCCand interconnected simple cubic structure, theparameter p/4:0.78 has to be replaced by p/23:0.91, 0.82 and 0.94, respectively.

(iii) Interdigitation of spherical reverse micellesIn first approximation it is assumed that

spheres are packed in a simple cubic structure,which provides the equivalence of a single dropletand the elementary cell of the array. The equa-tions are much more complicated for BCC, FCCrandom packing. By assuming RW the water coreminimal conditions for interdigitation for spheresare:

Minimal conditions for interdigitation ofspheres are:

4p

3RW

3 =NW6W

4pRW2 =NSaS

which ratio gives: RW=3wslS and NSaS=4p(3wslS)

(2RW+2lS)3=NW · 6W+NS · 6S+NO · 6O (11)

Eq. (11) can be written as:

lS3

NS6S

�2

RW

lS+2

�=

NW6WNS6S

+1+NO6ONS6S

Hence, using (Eqs. (2) and (3)) and RW relationand for interdigitations of spheres, the volumefraction ratio of oil and surfactant is defined as:

fO

fS

=(6ws+2)3

ps(6ws)2 − (w+1) (12)

The system has two options. It can interdigitateor go to a denser (BCC, FCC) packing as fO

decreases.

References

[1] M.P. Pileni, Reactivity in Reverse Micelles, Elsevier, Am-sterdam, 1989.

[2] S. Hyde, S. Andersoon, K. Larsson, et al., The Languageof Shape, Elsevier, Amsterdam, 1997.

[3] C. Petit, P. Lixon, M.P. Pileni, Langmuir 7 (1991) 2620.[4] J. Eastoe, G. Fragneto, B.H. Robinson, et al., J. Chem.

Soc. Faraday 88 (1992) 461.[5] J. Eastoe, B.H. Robinson, R.K. Heenan, Langmuir 9

(1993) 2820.[6] E. Bardez, V. Nguyen Cao, T. Zemb, Langmuir 11 (1995)

1081.[7] J. Tanori, T. Gulik-Krzywicki, M.P. Pileni, Langmuir 13

(1997) 632.[8] J.N. Israelachvili, D.J. Mitchell, B.W. Ninham, J. Chem.

Soc. Faraday Trans. 2 (72) (1976) 1525.[9] M. Magalhaes, D. Pusiol, E. Ramia, et al., J. Chem.

Phys. 108 (9) (1998) 3835.[10] L. Motte, I. Lisiecki, M.P. Pileni, In hydrogen bond

networks, in: J.C. Dore, M.C. Bellissent-Funel (Eds.),NATO ASI Series C, Kluwer, Dordrecht, 1994, p. 447.

[11] D.F. Evans, D.J. Mitchell, B.W. Ninham, J. Phys. Chem.90 (1986) 2817.

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Mesostructured Fluids: Supra-Aggregate Made of Interdigitated Reverse Micelles

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