A disordered lamellar structure in the isotropic phase of a ternary double-chain surfactant system

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A disordered lamellar structure in the isotropic phase of

a ternary double-chain surfactant system

Ian S. Barnes, Paul-Joël Derian, Stephen T. Hyde, Barry W. Ninham,

Thomas Zemb

To cite this version:

A

disordered lamellar

structure

in the

_{isotropic phase}

of

a

ternary

double-chain surfactant

_system

Ian S. Barnes

_{(1, 2,}

_*),

Paul-Joël Derian

_(2),

_Stephen

T.

_Hyde

_(1),

Barry

W. Ninham

₍₁₎

and Thomas N. Zemb

_{(1, 2)}

(1)

Department

of

Applied

Mathematics, Australian National

University,

Canberra ACT 2601,

Australia

(2)

Département

des Lasers et de

_{Physico-Chimie,}

C.E.A.

Saclay,

F-91191

Gif-sur-Yvette,

France

(Received

13 December _1989, revised 10

_July

1990,

accepted

13

_{July 1990)}

Résumé. 2014 La

_phase

fluide

_isotrope

« anormale » du

_{système DDAB/tétradécane/eau}

est très différente des autres microémulsions dans des

_systèmes

voisins. A

_partir

des mesures de diffusion

de

_rayons-X

et de neutrons à l’échelle absolue nous montrons _que la structure est celle d’une bicouche froissée aléatoire contenant de l’eau. Les autres structures en lamelles aléatoirement

connectées décrites

_{jusqu’à présent}

sont restreintes à un domaine étroit de

composition

et de

température.

Dans le

système

décrit _ici, la

_{rigidité intrinsèque}

de la bicouche et sa courbure faible

permettent la formation de cette structure dans une

_{grande région}

du

diagramme

de

phases

ternaire à

température

ambiante. Les autres modèles structuraux de microémulsions sont discutés

et leurs

_prédictions

confrontées aux résultats

_{expérimentaux.}

Abstract. 2014 The « anomalous » fluid

isotropic phase

in the

DDAB/tetradecane/water

system

differs in

_important

ways from microemulsion

phases

in related _systems, and is thus a useful test

for models. On the basis of absolute scaled neutron and

_{X-ray scattering}

data we show here that

the microstructure is best characterised as a

_randomly

folded reverse

_bilayer.

All

previously

reported examples

of this structure are restricted to a narrow range of

composition

and

temperature. In this case of a stiff

_bilayer

with low _spontaneous interfacial curvature it extends

over a

_{large region}

of the _ternary

_{phase diagram}

at room _temperature, the boundaries of which are

_explained

in terms of

simple geometric

constraints. Other

possible

microstructures are

critically

reviewed. Classification

Physics

Abstracts

61.10L - 61.12E - 64.70M 66.10E

1. Introduction.

Ternary

systems

containing

the double-chain surfactant

_{didodecyldimethylammonium}

bromide

_(DDAB)

form microemulsions over a

_{large composition}

range. We have

previously

explained

the behaviour of these

fluid,

isotropic

solutions

_{by adopting}

a

_{geometric approach}

[1-5] :

the constraints on microstructure are that the

_polar

volume fraction

4S,

the interfacial

area _{per unit}

volume 1

and the interfacial curvature

_{(set by}

the surfactant

_packing

_parameter

Vlat

[6])

must match the values set

_by

the

_composition.

We have constructed an

_approximate

model which satisfies these constraints

_{using spheres}

and

_{cylinders decorating}

a random

Voronoï lattice

_(the

DOC

_{cylinders model).}

For the

_systems

with

_DDAB,

water and alkanes from

_cyclohexane

to

_dodecane,

this allows

_prediction

of

_{approximate phase}

boundaries

_[7]

and

_conductivity

thresholds

_[4]

and calculation of full

_scattering

_spectra

on absolute scale

[8]

for all these

_systems.

Other microemulsion models either introduce free

_parameters

or are

incompatible

with the

_experimental

data

_[9].

The

_system

_{DDAB/tetradecane/water}

exhibits

_{qualitatively}

different behaviour to

_ternary

systems

containing

the same surfactant but shorter-chain alkanes and alkenes

_[10-13].

The

phase diagram

has been studied in detail

_by

Larché

_[14]

from whose work

_figure

1 is redrawn. At room

_temperature

there is a

_{large isotropic phase}

in the centre of the

_{three-phase triangle,}

which we refer to as

_Lx.

This solution does not show

_{Bragg peaks}

and hence must have a

disordered structure.

Unlike the

_L2

_phases

for the shorter

oils,

the

_{L,, phase}

cannot be diluted with tetradecane without

_separating

into two

_phases.

On dilution with water it

_separates

into two

_phases

with the water-rich lamellar

_phase

_L’

in excess, rather than

_demixing

to

_spill

out _water, which is the usual behaviour of such

_systems

_[15].

Towards the

top

of the

_{phase diagram,}

the

L,, phase

is in

_equilibrium

with excess

_oil,

whereas increase in surfactant concentration

_gives

rise to

_equilibrium

with the surfactant-rich lamellar

_phase

_L2a.

If one dilutes the

_{Lx phase}

with a

_{binary surfactant/water}

mixture at constant surfactant to water

_ratio,

a cubic

_phase

is obtained. This transition is

_{extremely temperature-sensitive [14].}

Fig.

1. - Phase

_Diagram

of the

_{DDAB/tetradecane/water}

_system, redrawn from Larché

[14].

The label

Lx

refers to the microemulsion

_phase

of

interest,

L’

to the water-rich lamellar

_phase,

_{L 2}

to the surfactant-rich lamellar

_phase,

and Cub to the cubic

_phase.

The lines marked a and b refer to the water

dilution lines

_{studied ; along}

each the

_samples

are numbered in order of

increasing

water content.

Compositions

are

_given

in table I.

_{Slight discrepancies}

between the marked

_{sample positions}

and the

phase

boundaries are the result of the different _temperatures used. The tie lines drawn are illustrative

As for the other

oils,

the cubic

_{phase region}

is

_probably

a reversed

_{bilayer (water}

inside the

film

_separating

two bulk oil

_{regions) lying}

on a

periodic

minimal surface with a cubic

underlying

lattice

_[16].

For the other

oils,

this

_region

contains several structures with different

topologies

[17, 18].

In contrast to the

systems

with short chain alkanes or alkenes

_[10-13],

the

_conductivity

of the

_{Lx phase}

is

_{always high,}

which means that the structure has to be water continuous

throughout.

There is no

_{antipercolation.}

If oil

_penetration

determines interfacial curvature

[13],

then the surfactant

_parameter

_v/al

is

_expected

to be close to one since tetradecane does

not

_penetrate

the surfactant tails. It is reasonable to

_expect

a value between 0.95 and 1.05.

These observations

_suggest

a

_{qualitatively}

different

_picture

to that established for the

systems

with shorter-chain oils. As the tetradecane does not

_{interpenetrate}

the surfactant

tails,

the interface is

_consequently

much flatter. This conclusion is

_{supported by}

the

observation _{that upon} addition of

_strongly

_penetrating

hexane,

the tetradecane

system

reverts to the

_previous

behaviour,

with both a

percolation

threshold and the

possibility

of dilution to

the oil corner. Addition of a

_single-chain

surfactant has the same

_effect,

while

_adding

a

long-chain alcohol to the dodecane

system

switches its behaviour to that of tetradecane. All of this

can be

explained

by

a

_qualitative

difference in behaviour between

_systems

with

_{v lal=}

1 and

v/aQ > 1.

Since there is no cosurfactant added and

_essentially

no oil

_penetration,

we cannot be satisfied

_by

a model which

_requires

ad hoc variation of the total interface _per unit

volume 1

or

the

_packing

_parameter

_{v lat}

with

_composition.

Without the fixed area constraint one could fit

all

_scattering

curves, even on absolute

_scale,

to a model of

_ellipsoids

or

_{polydisperse spheres}

[9].

As for the other

_systems,

the relative lack of

_temperature

_dependence

of the

_phase

diagram

and of the

_scattering

- in this case,

repeat

scattering

runs at 80 °C show little difference in the results

_{- suggests}

that the interface is rather stiff.

At first

_sight,

no

_{satisfactory explanation}

for the behaviour of this

_phase

exists. It cannot be

a

_dispersion

of

_{spherical droplets :}

water

_droplets

in oil would not conduct and could be diluted to the oil _{corner ;} oil

_droplets

in water could be diluted with water.

_Further,

neither would

_give

reasonable curvature. It also seems

_unlikely

to be a

_DOC-cylinder

structure like that used to model the microemulsion

_phases

for the shorter-chain

_oils,

because this structure can be diluted

_by

oil,

has a

_high

interfacial curvature and

_displays

a

_{conductivity percolation.}

A DOC-lamellar or random

bilayer

structure,

already proposed

for other

_{ternary systems}

_[2],

is not

_expected

to exist over a

large composition

range ; it is

usually

found in

temary

or

quaternary systems

only

over a _very narrow

_temperature

and

_composition

range close to a

lamellar

_{phase [19, 20].}

The structure of the remainder of this article is as follows. First we review the main microemulsion models found in the

_{literature ;}

then

_present

the results of our

_{experiments ;}

and

_finally

_{compare the}

_predictions

of the models with our data. 2. Microemulsion models.

2.1 INTERACTING SPHERICAL DROPLETS. _{- One supposes that} one of the two volumes

-polar

or

_nonpolar

- is « internal » and is distributed in

a

_dispersion

of

_interacting

monodisperse spherical droplets.

The

interface 1

and the internal volume fraction 0 are set

by

the

_composition,

and hence as

The scattered

_{intensity 1 (q)}

can therefore be calculated on absolute

_scale,

without _any

parameter,

by using

the factorisation

_[21]

_]

where the form factor

_{P (q )}

reflects the

_scattering

of one

_sphere

and the structure factor

,S(q )

the interference between

_{adjacent particles. Hayter}

and Penfold

_[21]

showed that the

structure factor can be calculated

_analytically

for identical

_{spherical particles interacting}

via a

screened electrostatic

_{potential using only}

three

_physical

_{parameters :}

the effective

_charge

_per

micelle,

the

_aggregation

number and the

_hydration.

This has been extended to various

_types

of attractive interactions

_using

the work of Sharma and Sharma

_[22].

This model has been

_extremely

useful in the

_exploration

of micellar structure and interactions _{in many} surfactant

_systems

_[23].

In many

situations, however,

the

_assumption

of

spherical droplets

with

_{predetermined}

radius R fails to

_predict

the observed

_{scattering [1, 24].}

In this

situation,

there is a

_strong

_temptation

to fit the data to another

_droplet

radius or to

allow

_polydisperse

or

_elliptical

_aggregates,

as this

_always

allows reconciliation of the observed

scattering peak

with a

_simple

structural

_{picture [25].}

This will not

_do,

because then 1 is

incompatible

with the

_composition

or the measured Porod limit. In other

systems

the

_droplet

model is also

_incompatible

with the electrical

_properties

_[9,

_26]

and the

isoviscosity

lines of the

phase diagram [27].

A review of inconsistencies obtained from the

_droplet

model is available

[28].

2.2 PARAMETRIC MODELS. - A

_totally

different

_approach

to microemulsions is embodied in the

_parametric

models. On the basis of

_{general thermodynamic}

_arguments,

Teubner and

Strey [29]

deduce that the scattered

_intensity

should have the form

where _{a2, cl} and _c2 are

_parameters

with _cl

_0,

_{a2, C2 :>} 0. This

_expression

has been

_extremely

successful at

_fitting

a

_large

number of

_scattering

curves even on absolute scale. From the

damped periodic

correlation function which

_gives

this

_scattering,

one can then derive values for the «

_{spatial periodicity » d,}

the correlation

distance e

and the internal surface X.

This

_expression

has been recast

_by

Chen et al.

_[30]

in terms of a different set of

_{parameters :}

CI,

the ratio of the

_{peak positions predicted by}

the Cubic Random Cell model

_{(see below)}

and measured in the

_{experiment ;}

C2,

an

analogue

of osmotic

compressibility

extended to more

_general

_networks,

and

C3,

which is related to interfacial behaviour and is seen at the

_high-q

limit.

An

_interesting

_{phenomenological}

observation made

_using

this model is that when the

_peak

width is

_large

and there is

_strong

_scattering

at

_high

_q, then

_C3

exceeds 10. It is asserted that this is a condition for

bicontinuity

which is

independent

of such measurements as

_conductivity

or self-diffusion

_[30].

Another

_general

_expression,

similar to that of Teubner and

_Strey,

has been derived

_by

Vonk,

Billman and Kaler

_[31].

For an ideal lamellar structure with

periodicity

D,

water

fraction ~

and thickness

_{polydispersity}

u,

they

derive the correlation function

y °( r ) .

They

then introduce distortions

_{- arising}

from

_twisting

or

_bending

of the lamellar structure

where d is the « distortion

_length

». This

_gives

a _very similar

_expression

for the

scattering

to

that of Teubner and

Strey.

However there is no

_guarantee

that a

_given

set of

d, e

and Z or of

D, d

and a leads to a

geometrically possible

microstructure,

nor do these models include _{any treatment} of

spontaneous

curvature.

_Furthermore,

both

_{expressions depend}

on three _parameters and

hence will fit

_virtually

any

scattering

curve

_[9].

2.3 THE RANDOM WAVE MODEL. - This model was

_{proposed by}

Berk

_{[32] using}

an

algorithm

of Cahn

_[33].

The idea is to

_produce

a random structure of

_typical

size D *

_by

_superposing

waves

_{of wavelength}

D * with random

_phases

and directions.

_Space

is then divided into

_polar

and

_{nonpolar regions by}

_cutting

the

_resulting

random field at some chosen

threshold value - the

_{polar region}

is that in which the total

_amplitude

exceeds the threshold.

The threshold value is set

_by

volume fraction.

This random structure

_produces

a

_{scattering peak}

in the desired

_position

but it is

_sharp

rather than broad. To

_rectify

_this,

a

_{polydispersity}

in the

_magnitudes

of the wavevectors needs

to be introduced. This has been done in two dimensions

_{by Welberry [34]}

and in three dimensions

_by

Berk and

_by

Chen et al.

_[30].

With a suitable choice for the functional form of the

wavelength polydispersity,

this allows

control over the total interface X and the

_morphology

of the structure, but in a rather

non-intuitive way. A better solution

_[35]

is to remove the

_long-range

order

_{implied by}

the infinite

propagation

of the _waves, and to introduce local correlations which allow the interfacial area

and the curvature to be controlled.

2.4 THE TALMON-PRAGER MODEL. - This was the first model to offer a continuous

transformation from water

_droplets

to oil

_droplets

via a bicontinuous network

[36].

It is constructed as follows :

1. choose a random distribution of

_points

in _space with

density n

and construct a Voronoï tessellation of space

using

the bisector

_planes.

The Voronoï cell

_belonging

to each of these

centres is thus the

_region

_{of space which is closer} to that centre than _any

_{other ;}

2. fill the Voronoï cells with water or oil at random in the

_{proportions 4S}

and 1

- 0 ; e

is thus both the number fraction of water-filled cells and the water volume

fraction ;

3.

_place

a surfactant

monolayer

between

_adjacent

cells with different contents.

This structure is bicontinuous for

_polar

volume fractions between 18 % and 82 % with a

percolation

threshold at 18 %. Observed electrical

_conductivity

behaviour is often much more

complicated

than this. The

_scattering

from the

_{Talmon-Prager}

model has been calculated

analytically [37]

and has no

_{peak, making}

this model

_incompatible

with most microemulsions. At this

_point

it is useful to introduce

_formally

the

_{Talmon-Prager Repulsive (TPR)}

model. This is constructed as for the

_{original Talmon-Prager}

model

_except

that the initial distribution of

_points

is

_subject

to a condition of closest

_approach,

thus

_replacing

the

_original

random

distribution

_by

a « hard

_{sphere »}

distribution. This has the effect of

_introducing

a

characteristic distance in the structure and hence a

_peak

in the

_scattering

at D

n _ 1/3.

This structure is drawn

_{schematically}

in

_figure

_{2 ;}

it is the

_simplest

of the models derived

from a « hard core » Voronoï lattice. It is bicontinuous over some volume fraction range and

is

_{roughly equivalent}

to the CRC model

_explained

below.

The

_{simplest prediction}

of the TPR model is that the correlation

length

fe

should be

_equal

to the

_periodicity

D *. Such behaviour has never

_yet

been found

_{experimentally [38, 39].}

When

_fe

is much smaller than

_{D *,}

a local microstructure within the cells must

_exist,

_reducing

Fig.

2. - Schematic two-dimensional illustration of the different model structures derived from the hard core Voronoï lattice

_by

the addition of microstructure. The distribution of centres of

_imaginary

hard

_spheres

allows the construction of a Voronoï tesselation of space. This random lattice is then used as the base on which are

_{placed spheres,}

a connected network of

spheres

and

_{cylinders (the}

DOC

cylinders model)

or of

_{bilayers (the}

DOC lamellar

model). Filling

the cells at random with water or oil

gives

the

_{Talmon-Prager repulsive}

model. Since the

_underlying

lattice is the same for all four _models,

cells must be

_intrdduced,

_moving

_{thescattering peak}

while

_{keeping Pc}

constant. It is

_possible

using

this latter

_{approach (with}

the CRC

model)

to

_explain

both the

_{phase diagram}

and the

peak position [40, 41].

2.5 THE CUBIC RANDOM CELL MODEL. - This model was first introduced

by

De Gennes as a

modified

_{Talmon-Prager}

model

_[42],

and the two are

_{clearly closely}

related. The construction is

_exactly

that of the

_{Talmon-Prager}

model but with the Voronoï lattice

_{replaced by}

a

_simple

cubic lattice of

_repeat

_{distance e. Again}

the cells are filled with water or oil at random

according

to the volume fraction 0. The

_edge

_{length e}

is fixed in the

_{random-filling}

approximation by

and the

_{peak position}

will be at D * = 2 ir

/qmax

= 03BE.

This model has

only

one distance built

in. It is bicontinuous over

_roughly

the same volume fraction as the TPR model and

produces

identical

_conductivity

variation with water content.

Calculation shows that this model

_gives

no

peak

at all

_in

_{scattering [5, 8] :}

the zero of the

form factor of the cubes

_exactly

cancels the

_sharp

_Bragg

_peak

due to the

_underlying

lattice. This is an artefact of the exact calculation and of the random

_filling

_{approximation.}

The

_peak

will reappear when even

_slight

correlations between

_neighbouring

cells are introduced.

As noted

_{by Auvray et}

al.

[43],

the

_peak

in

_scattering

from microemulsions is

_usually

observed at about D * = 2

e,

twice the distance

predicted by

the model. This is indeed the

prediction

of Milner et al.

_[44]

when local anti-correlations between the

_{filling of neighbouring}

cells are introduced. It is worth

_noting

that this result is not automatic :

_longer-range

correlations could

_{easily produce}

still different

_{peak positions.}

2.6 THE DOC CYLINDERS MODEL. -

_Starting

from the same hard core Voronoï construction

as for the TPR

model,

we add local structure so that the model

_depends

on two distances.

These are

_essentially

the

_spontaneous

curvature of the

_interface,

determined

_by

the nature of the surfactant

film,

and the characteristic distance or average lattice

spacing,

which is

determined

_by

the

_composition.

The three fundamental

_geometric

constraints on structure have to be satisfied

_[4].

(i)

The

_polar

volume fraction 4S is set

_by

the

_composition.

(ii)

The

_water/oil

interfacial area _{per unit}

volume X

is set

_by

the surfactant concentration.

(iii)

The average interfacial curvature must _{agree with the}

_spontaneous

curvature of the surfactant film as set

_by

the surfactant

_parameter

_{v /al}

_{(see Appendix 2).}

Within the hard-core Voronoï

_tesselation,

we construct a random connected surface with

variable

_{connectivity by gluing together spheres}

and

_cylinders.

This

_gives

an

_{approximation,}

at the resolution of the

_{scattering experiment,}

to the actual

_{microstructure,}

which we

_imagine

as a surface of constant

_v/al

_separating

water and oil domains.

_Depending

on the

composition,

the three constraints are satisfied

_by

structures

_ranging

from a

_dispersion

of

spherical droplets

to

_highly

connected networks of

_cylinders.

The actual construction

_together

with the

_predictions

of

_{peak position, approximate phase}

boundaries and

_scattering

curves

have been described elsewhere

[4,

7,

8].

This model

_applies

when the

_spontaneous

radius of curvature of the surfactant film -related to

_{persistence length}

and manifested here as the

_sphere

radius

_Rs

- is small

_compared

to the

_{spatial periodicity}

D *. For

_{persistence lengths}

of the same order of

_magnitude

as the

periodicity,

that is for flatter

_interfaces,

the behaviour _goes over to that of the DOC lamellar

2.7 THE DOC LAMELLAR MODEL. - When

_{v lai}

is close to _one, the DOC

_cylinders

model

gives

way to a random lamellar structure. This is identical to the

_{Talmon-Prager repulsive}

model

_except

that instead of water and oil

_{separated by}

a

_monolayer,

we have two fictitious

different

_types

of water

_{separated by}

a normal

_bilayer

or two

_types

of oil

_{separated by}

a

reversed

_{bilayer [2].}

This structure is

_{clearly closely}

related to the sponge

phase

model of Cates et al.

_[19]

with the cubic lattice

_{replaced by}

the hard

_sphere

Voronoï. The

structure

is

constructed as follows.

1. Take the same « hard core » Voronoï lattice as for the TPR or DOC

_cylinder

models.

2. Label the cells A or B at

_random,

with

_{probabilities (and}

thus volume

_{fractions) cl}

and

1-03C8.

3. When two

_adjacent

_polyhedra

have different

labels,

set a

_bilayer

on the

_polygon

between the two

_polyhedra.

For the direct model this is a normal oil-swollen

_bilayer

of thickness

2f

+ 2 t

_separating

water A from water

_{B ;}

for the reversed model it is a reversed

bilayer

with water thickness 2 t

_separating

oil A from oil B.

Schematic

_drawings

of the different microstructures derived from the Voronoï lattice are

shown in

_figure

2 ;

the local

_geometries

of

_monolayers

and direct and reversed

bilayers

are

shown in

_figure

9.

_{When cl is}

close

to 1

the film is connected

_throughout

_space, as are the two

2

bulk

_regions

A and

_{B ;}

_{when cl}

is close to 0 or 1 the structure is

_essentially

that of

_{single-walled}

vesicles

_(Fig.

_3).

Fig.

3. - Two-dimensional illustration of the evolution of the DOC lamellar structure with

increasing

values of the

pseudo-volume

fraction _03C8.

_{When 4r}

is small the structure consists of isolated _vesicles, while

for 03C81/2

it is a random connected

_bilayer.

Note that while two-dimensional structures can never be 2

bicontinuous, in three dimensions the film and both bulks are continuous over a

_large

_range of values of tb.

Derivation of the

_{area-averaged}

mean and Gaussian curvatures

_(H>

and

_(K>,

and of the

packing

parameter

v /al

as a function of the

pseudo-volume cl

are

_given

in

_Appendices

2 and

3. Since the

_bilayer

thickness can be of the same order as

_{D *,}

_{the average} curvature at the

oil-water interface is

_by

no means

negligible ;

moreover, at

_{given polar}

volume fraction

_4l,

interfacial

area Z

and surfactant

_parameter

_{v laf,}

_only

one combination of

_{half-thickness t,}

density

of Poisson

_{points n}

and

_{pseudo-volume}

fraction cl

will

_satisfy

these three constraints

(Appendix 4). Assuming

a DOC-lamellar structure therefore allows

_prediction

of the

approximate peak position assuming

2.8 « BICONTINUOUS » MODELS. These models derive from a

_{proposition originally}

made

by

Scriven

_[45]

and revived more

_recently

_[46,

_16].

The idea is to « melt » or disorder the structures of cubic

_{liquid crystal phases}

to obtain a structure with the same

_properties

but

which does not

_{give Bragg peaks}

in the

_scattering.

Cubic

_phase

structures have been shown in several cases to consist of a normal or reversed

bilayer

centered on

_(and

_presumably

fluctuating about)

a

_periodic

minimal

_(zero

mean

_curvature)

surface which

_separates

two

distinct external subvolumes

_{[16] ;}

when « melted » this is believed to

_give

rise to the structure

of the

_isotropic

« _{sponge »} or

_L3

_phase

_[20].

This model

_envisages

the microstructure as that of a random

_bilayer,

and is thus rather close

to the DOC lamellar

_model,

_remembering

that the latter is to be

_thought

of as an

approximation

to the true structure at

_experimental

resolution. The difference is that for the molten cubic

_phase

the

_{area-averaged}

mean curvature

_(H)

is

_always

zero,

independently

of

the volume fractions on either

side,

while for the DOC lamellar model

(H)

depends

on the

pseudo-volume fraction 1/1

and is

_only

zero

when 03C8 =1/2

_(see

_{Appendix 3).}

Such considerations

2

must

_depend

on the treatment of the

_bending

energy and in

particular

on the

_importance

or

otherwise of

_higher-order

terms than those in the familiar Helfrich form

_[47].

Unfortunately,

there is no known

_algorithm

for

_generating

a zero or other constant mean

curvature surface on a random network. So this model is a

_qualitative

one which cannot

_yet

be tested

_{by experiment.}

3.

_{Experimental procédure}

and results.

Compositions

of the

_samples

are indicated in table

_I,

and also on the

phase diagram

in

figure

1. Measurements were made at

_{28 °C ;}

at that

temperature

the

samples

were all

monophasic.

Since the molecular volumes are

_known,

the

_polar

volume fraction 0 can be

calculated from the

_{composition by adding}

the volumes of water and ionic

_headgroups.

Since

no cosurfactant is

_used,

a

_good

estimate of the interfacial area _per unit

volume 1

can be

_made,

assuming only

that the area per molecule is the same as in the

_neighbouring

lamellar

_phases.

Small

angle

X-ray

scattering experiments

were done on the

_high

resolution

_small-angle

Table I.

_{- Quantities determined from}

the

_composition

_for

the two series

_{of samples}

examined

in the _system

_{DDAB/tetradecane/water. Cs}

is the

_surfactant

concentration,

_{[W ] / [S]}

the

camera D22 at LURE

_{(Orsay, France),}

and neutron

_scattering

oh the same

_{samples (all}

with

D20)

at the PACE

_facility

at

_{Orphée (Saclay, France).}

Data reduction was made

_using

standard

methods,

including

absolute

_{scaling by comparison}

with

_{scattering of pure D20 [23].}

Spectra

from

_along

a

typical

water dilution line are shown in

_figures

4 and 5. The

peak

position

shifts with water content, D *

_increasing

with water dilution. A well-defined Porod

regime

allows an

_unambiguous

determination of

03A3

The neutron and

_X-ray

_spectra

are _very

similar,

hence the

_assumption

of constant bromide counter-ion concentration is valid within

the

_experimental

resolution

_(qmax

= 0.5

Â-1,

giving

_a resolution of

12 À

in real

space).

This is

contrary

to the result obtained with SDS

_systems,

in which details of the local electronic densities or

scattering length density

distributions are

responsible

for the

peak shape

and

position

[48].

We can calculate several

_quantities

of interest from the

scattering

curves before

_making

reference to _{any model. The}

_simplest

of these is the

_{peak position}

D* = 2

’TT /qmax-

The

experimental

value of the

_specific

surface X

can be deduced from the absolute value in

cm-5

of the

_plateau

in a Porod

_plot

q4 I (q )

against

_q. The invariant

Q *

is

given by

Fig.

4. - Absolute scaled

small-angle

neutron and

_X-ray

_spectra for the

samples

on dilution line a in the

Fig.

5. - Plot

of log (I ) against log (q )

for neutron

_scattering

from the

_{samples along}

dilution line a in

the _system

_{DDAB/tetradecane/water.}

Note the

extremely

clear Porod law behaviour at

_{large angles.}

but can also be calculated from the

_{composition by}

the

_expression

Agreement

of these two

_expressions

can be

checked,

as can the

_agreement

between the values

of X calculated from the

_scattering

and from the

_composition.

This ensures :

(i)

that the two-media

_assumption

- that the

sample

can be treated as made up of

homogeneous polar

and

_{nonpolar regions}

- is

_{justified ;}

(ii)

that a

_negligible

fraction of the surfactant molecules are

_molecularly

_dispersed

instead

of

_{participating}

in

_forming

the surfactant

film ;

(iii)

that the calculated volume fractions and electronic densities are correct ; and

(iv)

that the minimum and maximum values

_{of q}

are

correctly

chosen.

4. Discussion.

The

_parameters

_resulting

from

_fitting

a

_{Teubner-Strey expression}

to the

_spectra

are

_given

in

table

_III,

_along

with values derived from the TPR and

_{(random filling)}

CRC models. The fits

to the

_{Teubner-Strey expression}

are _very

_good

_indeed,

but as usual one can infer little from

these

_results,

in

_{particular nothing}

about the microstructure. Here we note

_{that e}

is of the

same order of

_magnitude

as

D *,

which means that a Voronoï lattice is a

_reasonable,

choice for

structural models. We have also calculated values of the «

_{bicontinuity »}

_parameter

C3

of Chen et al.

_{[30] (see}

Sect.

_2.2).

_{While the values vary from} over 20 down to 8 we know

from

_conductivity

measurements and the fact that the surfactant is insoluble in

_tetradecane,

that the

_phase

is bicontinuous

_throughout.

As a result we are forced to conclude that this criterion is not valid.

The

_predictions

of the other models for the

_{peak position}

D * are summarized in tables IV

Table II.

_{- Quantities}

_{measured from}

_X-ray

and neutron

_scattering

_{experiments for}

the system

DDAB/tetradecane/water.

D * is the

_{real-space peak}

_position,

_fc

is the correlation

_length,03A3

is

the internal

surf47,ce

and o, the calculated

_headgroup

area _per molecule.

Table III. - Results

_{of fitting}

_X-ray

data

_for

the system

DDAB/tetradecane/water

to the

parametric expression

of

Teubner and

_Strey

and to the

Talmon-Prager

and cubic random cell models. For the

_{Teubner-Strey}

_expression,

d is the

spatial

periodicity

and e

the correlation

distance ;

for

the other models D * is the

_{real-space peak}

_position

and

_{v laf}

the

_effective

surfactant packing

parameter.

local

corrélations.

Full

_scattering

_spectra

_{predicted by}

the water-in-oil

_spheres

and DOC

lamellar models are shown

_compared

to the observed

_scattering

in

_figures

6 and 7. Of

these,

only

the DOC lamellar model appears reasonable.

The

_{peak positions}

and

_scattering

_spectra

for the oil-in-water and water-in-oil

_spherical

droplet

models were calculated

using

the

Hayter-Penfold

RMSA

procedure [49],

with the

droplet charge

set to zero. It turns out that the

_{peak positions}

obtained for a

_dispersion

of

water

_spheres

in oil are in reasonable

_agreement

with those

measured,

even with the radii

imposed by

the volume fractions and interfacial areas

_(R

=

3 03A6/03A3).

The

shapes

of the

Table IV. - Results

_of

_attempting

to

_fit

data

_for

the _system

_{DDAB/tetradecane/water}

with a

model

_of

_interacting

_{monodisperse spheres.}

R is the

_sphere

radius,

D * the

_{real-space peak}

position

and

_{v /al}

the

_{effective surfactant packing}

parameter.

Table V. - Result

_of

_attempting

to

_fit

data

_for

the _system

_{DDAB/tetradecane/water}

with the DOC models. For the DOC

_cylinders

model

_{(water-filled cylinders}

in

_oil),

Z is the

co-ordination number and r is the

_cylinder

radius. For the DOC lamellar

model,

t is the oil or

water

_{half-thickness}

_(the

total

_bilayer

thickness is thus 2 t + 2

_f)

_{and fk}

is the

_{pseudo-volume}

fraction.

D *

is the

_real-space

_peak

_position

and

_{v laf}

the

_{effective surfactant packing}

_parameter.

explain

the

_{high conductivity}

of the

_{Lx phase.}

It also

_requires

an ad hoc variation of the

surfactant

_parameter

_vlaf

from 1.13 to 1.5 with

_composition,

which we

_regard

as

unlikely.

Oil-in-water

_{droplets give systematically}

_wrong

_{peak positions}

in the

_scattering.

Dismissal of the

_DOC-cylinder

model is more difficult. This structure would

_explain

the

conductivity

of the

_L,,phase,

but not the nature of the

_phase

transitions and

_equilibria.

In the

adjacent

cubic

_phase

and in both the lamellar

_phases

the surfactant forms a

_bilayer

and not a

monolayer.

Furthermore,

the

_scattering

can

_only

be reconciled with a connected

_cylinder

structure if one assumes an

_unlikely

variation of the surfactant

_parameter.

The

_proposed

Fig.

6.

_{- Comparison}

between measured and calculated _spectra for

_sample

la. The full curve is the

normalised

_{X-ray scattering data,}

the dashed-dotted curve is the

_prediction

of water

_spheres

in oil and the dashed curve that of the DOC reversed lamellar model.

Fig.

7. -

Comparison

between measured and calculated _spectra for

_sample

3a. Curves are labelled as

for

_figure

6.

this would

_{certainly give}

the

_high

conductivities

_observed,

the

_packing

constraint would have

to be relaxed and

_vlaf

allowed to _vary

_by

over 20 % for this to work. The

_DOC-cylinder

structure must therefore be

_rejected,

even

_though

it is

_compatible

with the

_{peak position.}

This leaves the DOC lamellar model. This structure has

_already

been found in other

systems

and is referred to as the

_L3

or _{« sponge »}

_{phase by}

other authors. Since the

Lx phase

is in the middle of the

temary

phase diagram,

the DOC-lamellar structure could be a

priori

a reversed

_(water

inside the

_bilayer)

or a direct

_{bilayer (oil inside).}

The calculations in

Babinet’s

_{principle :}

an

_exchange

of

_polar

and

_{nonpolar regions}

does not

_change

the

scattering.

However,

since the

_neighbouring

cubic

phase

is a reversed

_bilayer

_structure, we conclude

that the

_{L,, phase}

should also

_be,

in order to

_explain

the

_{extremely temperature-sensitive}

phase

transition between them. This structure would also

_{explain why}

the

_phase

cannot be diluted with water or

oil,

since

_unfolding

or excessive

swelling

of this random lamellar

structure would induce forbidden variations of

_vlaf.

While it is

_{always possible}

to calculate the full

_scattering

curve

_using

the cube method

_[8],

setting

up the array of cubes is

extremely

cumbersome for Voronoï models like this. Instead

we obtain a

_{fairly good approximation}

to the full

_scattering

curve

_{I (q ) by multiplying}

the

S(q)

of the

_underlying

lattice

_by

the

_{P (q )}

_{of the average}

_polygonal

face. The excluded

volume is determined

_{by assuming}

that if the distribution of centres is too

_random,

then there will be

_places

at which the

_bilayer

is forced to bend too

_{sharply. Excluding}

such

_{configurations}

1

is achieved

_{by imposing}

a minimum distance of

_approach

of the order

of 1 n _1 3 .

This

_gives

an

2

excluded volume of

_{approximately}

50 %. This causes the distribution of the centres to be

quite regular.

The form factor is obtained

_{by approximating}

the

_polygonal

faces

_by

disks of thickness 2 t for the reversed

_{bilayer (or}

_2(f

+

t )

for the normal

_bilayer)

and radius one

_quarter

the

nearest

_neighbour

distance.

_{(This gives roughly}

the correct surface area for the

_cells.)

This

procedure neglects

correlations between

_adjacent

facets - in

_particular

the fact that the facet-facet correlation function will

_certainly

be

_anisotropic

- but is otherwise

_reasonably

sound. It has been checked

_{against optical}

Fourier transforms of two-dimensional realisations of the

DOC-lamellar model

_[34]

and found to

_give

fair

_agreement.

It is

_possible

to deduce the

_geometric

limits on the random lamellar structure from the

positions

of the

_phase

boundaries

_(see

_{Fig. 8).}

This is somewhat similar to the

_{phase boundary}

calculation made for the

_DOC-cylinder

model

_[7].

The left-hand

_{phase boundary}

lies

essentially along

a line of constant

_{surfactant/water}

ratio. This is

_equivalent

to an _{upper limit} on the water film

_thickness,

which is never more than about 64

À.

_Beyond

this limit it seems

that the structure is no

_longer

stable and the dilute lamellar

_phase,

with water thickness at

least 70

À

is in excess.

The

_{right edge}

is

_{approximately}

an _upper limit on the surfactant concentration. This is

directly

linked to the cell

size,

and thus to the

_{peak position}

D *. Once D * is less than about

80 Â

_{(Fig. 8)}

the

_{phase prefers}

to _go into

_equilibrium

with the concentrated lamellar

_phase,

which has

_hydrocarbon

thickness about 27

_Á,

_slightly

less than two

_{fully-extended}

chain

lengths.

Larché

_[14]

has noted that this

_equilibrium

is

_{extremely temperature-sensitive.}

This could be one of the first

_examples

of an

_unbinding

transition between

_{highly charged}

concentrated

_bilayers

_[50].

The

_top

and bottom boundaries of this

_-phase

appear to be set

_by

limits on the allowed

curvature of the reverse

bilayer.

Dilution with oil has the effect of

_trying

to flatten out the

bilayer,

which has

_negative

Gaussian curvature. _{The limit appears} to be

_roughly

(K> t2 %:_

10- 2 :

in other words the

_principal

curvature radii cannot be

_greater

than about five

times the total water thickness. A _{first effect of this appears} to be a

breaking

of the

_symmetry

between the two sides of the

_bilayer.

While

_{making cl}

different

to 1

at constant cell size 2

changes

the curvature in the _wrong

_direction,

at constant

_composition

it has the

_compensatory

effect of

_reducing

the cell size and in fact

keeps

_(K)

negative

maintaining

the desired

_degree

of « foldedness ». The limit is reached

when cl

approaches

0.18,

at which

point

(K>

must

change sign

and no amount of

_change

in the cell size will

_keep

the surface

_sufficiently

folded

(see Fig. 3). Approximate phase

boundaries deduced from

_geometric

constraints on the DOC

lamellar structure are shown

_compared

to the

_{experimental phase}

boundaries in

_figure

8. Our model calculations

give

values of the

_{pseudo-volume}

_{fraction qi}

_ranging

between 0.25

and 0.5. This

_gives

a bicontinuous _structure, in

_agreement

with the

_conductivity

measure-ments, with a mean curvature which is not too far from zero. The total

_bilayer

thickness

(2 t

+

2 f)

is

_generally

of the order

_{of D * /2.}

The

_{persistence length,}

the

_typical

distance over

which the

_bilayer

normal remains

_{roughly parallel}

to itself is also about

_{D * /2,}

as for all

Voronoï models.

This model also

_gives

some

_{understanding}

of the slow increase of

conductivity

with water content.

_Along

_any water dilution line the

_bilayer

thickness increases. The

connectivity

of the

bilayer

also increases

_{as aJi}

relaxes to its desired value

of 1 .

2

5. Conclusion.

In the

_{DDAB/tetradecane/water}

system

all the standard models run into

_problems

when tested

_against

the

_scattering

data. The cubic random cell model and both variants of the

Talmon-Prager

model

_predict

the

_peak

_{in the wrong}

_position

unless local correlations are

introduced.

_Droplets

or DOC

cylinder

models

_{require unlikely}

ad hoc variation of the

surfactant

_parameter

_vlaf

with water content and cannot be reconciled with the

_phase

diagram.

Of the

_quantitative

models,

only

the DOC-lamellar

model,

describing

a random

bilayer,

is

_compatible

with the observed

_scattering.

We therefore conclude that the observed microemulsion

_region

is a DOC-lamellar

_{(or L3) phase.}

This model is

compatible

not

_only

with the

_scattering

data but also with the

_conductivity

measurements and the

_experimental

phase diagram.

It

_gives

a

_{simple explanation}

for the

_positions

of the measured

phase

At

_high

water contents it may appear

strange

to be

_proposing

a « reversed » structure for this

_phase.

It should be noted however that in this structure the curvature is not towards the

water ; in fact

_{v laf}

is

_slightly

less than one,

indicating

curvature of the surfactant

_monolayers

towards the oil.

_Also,

even for

_sample

5a,

at the water-rich corner of the

_{single-phase region,}

the total water thickness

_proposed

is

only

60 Â

out of a total cell size of 180

À.

While this

certainly

means we are not in the

regime

of

_validity

of

_{thin-plate theory,}

the reversed

_bilayer

description

still makes sense.

For surfactant

systems

with lower

_bending

constant

_kc,

this structure is restricted to a _very narrow channel near the

_edge

of the lamellar

_phase.

In this

_system

it extends over a much

broader range of

composition,

but still has in common with other

_L3

phases

the

ability

to be in

equilibrium

with excess oil and lamellar

phase.

This is not the first

_example

of a more

concentrated

_L3

structure : such a

phase

was

mapped

out

_by

Ekwall in the central

_region

of the

_ternary

sodium

_{octanoate/octanoic acid/water}

system

[51].

In that

_system,

as in this one,

the disordered lamellar structure exhibits a

_peak

in

_scattering.

In diluted

_L3

structures on the

other

hand,

where no

_spontaneous

curvature restricts the

_{swelling [20, 52],}

the

_scattering

can

be divided into a

low-q

_part

where

I(q)

is

approximately

constant, followed

_{at q >}

2 ir ID *,

by

a

region

in which

1 (q) ’" q - 2,

and

finally

a Porod

_region

where

_I(q)

-q - 4.

In the case studied

here,

the

scattering

is dominated

by

the interaction

peak

which

merges into the Porod limit at

_high

_q.

Closely

related to the DOC lamellar model is the molten cubic structure. This is also a

random

_bilayer,

the main difference

_being

that it

_always

has zero mean curvature, even if the

pseudo-volume

fractions on the two sides are

_different,

but does not offer the continuous transition to disconnected vesicles built into the DOC lamellar model. Without more

knowledge

of such structures this remains

_{spéculative ; perhaps}

the

_symmetry

_breaking

predicted

here upon oil dilution means that the « molten cubic » model would have to include similar features in order to work.

Acknowledgments.

We thank Francis Larché for

_letting

us use his

_{phase diagram,}

Loic

_Auvray

at

_Orphée

and Claudine Williams at LURE for their assistance with the

_scattering

measurements, Didier Roux and H. S. Chen for access to

_preprints

before

_publication,

Reinhard

_Strey

for

experimental

hints on

_{making phase diagrams}

and communication of his model calculation programs and

Stjepan Marcelja

for

extremely

valuable discussions.

This work was made

_{possible by}

_{the program of Franco-Australian scientific collaboration} instituted

_by

Professor Michel Ronis of the French

_Embassy

_{in Canberra. This program} enabled three of us to make extended visits to

_foreign

laboratories.

Appendix

1.

Curvature of facetted surfaces.

Consider a surface made up of flat

pieces joined along straight

lines. The mean curvature H

and the Gaussian curvature K are both zero wherever

_they

exist - that is

_everywhere

_except

at the

_edges

and

vertices,

where

_they

are undefined.

_Nonetheless,

the

_integrated

curvatures

are not in

_{general equal}

to zero, and neither are the

_{area-averaged}

curvatures

and

All the curvature is concentrated at the

_edges

and vertices. To

_proceed,

we consider

_edges

and vertices as the zero-radius limits of suitable

_pieces

of

_{cylinders, spheres}

or other

_surfaces,

and thus calculate the correct contributions to JC and X

_[53].

First consider an

_edge

of

_length

L and dihedral

_angle

a. This must

_always

be measured on

the same side of the surface and will therefore be

_greater

than or less than 7T

_according

as the

bend is towards or _{away from} the chosen side. We choose our normal vector to

_point

in the direction of this chosen

side,

so that if the curvature is towards this side it will be counted as

positive.

Thus we

_expect

that if a 7T then the contribution to JC will be

_positive

and if

a > 7T then it will be

_negative.

Let us now remove narrow

_strips

from

_along

each side of this

edge

and

_replace

them

_by

a

piece of cylinder

of radius r which

_{joins smoothly}

to the

_remaining

flat

_pieces.

In order for this to

_happen,

this

_piece

must subtend an

angle

of

17T - a

1

and have

length

L

_(plus

a small correction of order r which takes account of what

_happens

at the

_ends).

At every

point

of this

piece,

the mean curvature is

_{1 /2 r}

if a « 7r

and - 1 /2 r

if a > 7T. As for _any

cylinder,

the Gaussian curvature is zero. So the surface

_integral

of the

Gaussian curvature is zero and that of the mean curvature is

Notice that the

_sign

is

_always

correct here : the absolute value and the

_change

of

_sign

in the

curvature have cancelled.

_Taking

the limit as r - 0

gives

the mean curvature contribution for

a

_{single edge}

The situation at a vertex is similar.

_Suppose

that n faces meet at a

_point

and that the face

angles

are

_Pi.

_{If E}

_Pi

_: 2 7r then this vertex _{may be}

replaced by

a

spherical polygon

with

radius r, n sides and vertex

_angles

ir -

_Bi.

As this has area

and the mean and Gaussian curvatures at _every

_point

are

_{1 /r}

and

11r2

_{respectively,}

the

integrated

mean curvature is

_{(2 ir - ESi ) r}

and the

_integrated

Gaussian curvature is

.

2

_{71’ - L}

_{{3 i. Letting}

r - 0 as

before,

we see that the vertex contributes

_nothing

to

_Je,

and

2 ir -

_03A3Bi

_i to K.

If L {3

> 2 7r then a

piece

of a

_sphere

will not

_{do ;}

_{something saddle-shaped}

is needed. But

clearly

the mean curvature is

_{always going}

to be zero : no matter what the

details,

the area

must be

_proportional

to

r2

while H _goes like

_{1 /r.}

For the Gaussian curvature, the correct

unit

_{sphere corresponding}

to the direction of the normal vector. The area of the

image

of a

region

under this _map is its

_integrated

Gaussian curvature, with the

_proviso

that the area

should be counted as

_negative

if the vertex order is reversed

_by

the

_mapping.

Careful

application

of this method

_gives

the

_required

result :

_regardless

of whether the vertex

_angles

sum

to less or more than 2 7r, the Gaussian curvature contribution of a vertex is

So _{for any surface built} up in this way, the total curvatures je and X are

_simply

obtained

_by

summing

these contributions over all

_edges

and vertices

_{respectively. Dividing by}

the total

surface area then

_gives

the

_{area-averaged}

curvatures.

Appendix

2.

Packing

on curved surfaces.

In order to relate the surfactant

_packing

_parameter

_vlaf

for various models to the

area-averaged

curvatures, we recall that for a small element of

_surface,

where A

₍₀₎

= A is the area of the

_original

surface and A

_(x)

is the area of the

_parallel

surface

displaced

a distance x in the direction of the normal vector

_[54].

We now consider three

different decorations to our facetted surface :

(i)

a surfactant

_monolayer

with the tails of

_{length f}

on the «

_{negative »}

side of the surface

(see Fig. 9a) ;

(ii)

a normal oil-swollen

_bilayer

with oil half-thickness t and surfactant tails of

_length

_f,

so

that the total

_bilayer

thickness is 2

f

+ 2 t

_{(see Fig. 9b) ;}

and

(iii)

a reverse

_bilayer

with water half-thickness t and

_again

tails of

_{length f}

_(see

_Fig.

_9c).

Fig.

9. - The three ways to decorate a surface with a surfactant film :

(a)

a

_{monolayer, (b)}

a normal

(oil-swollen) bilayer,

and

(c)

a reversed

bilayer.

The

_original

surface is labelled 0,

parallel

surfaces are

labelled

_by

the distance

_they

are

displaced

in the direction of the normal vector.

For the

_monolayer

the total volume

_{occupied by}

the tails is