Hydrodynamic modes and topology in microemulsions and L3 phases

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Submitted on 1 Jan 1990

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Hydrodynamic modes and topology in microemulsions

and L3 phases

S.T. Milner, M. E. Cates, D. Roux

To cite this version:

Hydrodynamic

modes and

_topology

in

microemulsions

and

L3

phases

S. T. Milner

_(1),

M. E. Cates

₍₂₎

and D. Roux

₍₃₎

(1)

Exxon Research and

_Engineering,

_Annandale, NJ 08801, U.S.A.

(2)

Cavendish

_{Laboratory, Madingley}

Road,

Cambridge

CB3 0HE, G.B.

(3)

Centre de Recherche Paul Pascal, Domaine Universitaire, F-33405 Talence _Cedex, France

(Received

20

_April

_1990,

_accepted

in

_{final form}

2

_{July 1990)}

Résumé. 2014 Le _temps

_{caractéristique}

de déclin des fluctuations de concentration à

petit

vecteur

d’onde est estimé pour des

phases

de membranes aléatoires telles

_qu’on

en rencontre dans les microémulsions ou les

phases éponges.

Nous considérons les effets de deux temps

microscopiques

(supposés longs) :

l’un

_{correspondant}

à la relaxation de la

topologie

du film, et l’autre à celle des volumes

respectifs

des deux

régions

continues. Ces effets peuvent être observables par des

expériences

de diffusion

_dynamique

de la lumière, saut de

_température

et

_{biréfringence}

magnétique.

L’ordre de

_grandeur

du _temps de relaxation

_topologique

pour des

phases éponges

de

SDS/dodecane/eau/pentanol

est estimé être de l’ordre de 0,01 s.

Abstract. 2014 We estimate the

decay

rates for surfactant and

_oil/water

_{long-wavelength}

concentration fluctuations in

_phases

of random fluid _films, such as microemulsions and « _{sponge »}

L3

phases.

We consider the effects of two

_{(possibly long) microscopic}

time scales : one for local relaxation of the film

_topology,

and one for relaxation of the relative volumes of the two

_regions

partitioned by

the film

(when

this is not

_conserved).

These effects may be observable in

dynamic

light scattering,

temperature

jump

and

_{magnetic birefringence}

studies. For dilute

SDS/dodecane/pentanol/H2O

sponge

phases

we use

_experimental

data to estimate bounds for the

topological

relaxation _time, and find this to be of order 0.01 s.

Classification

Physics

Abstracts 82.70 - 68.10 - 47.55

1. Introduction.

In certain

_systems,

_{self-assembly}

of surfactant molecules leads to the formation of extended random surface structures. In balanced

_{microemulsions,}

for

_example,

a surfactant

_monolayer

divides

_regions

of oil and water. In _{the « sponge »}

_{(L3) phase}

_[1-3]

of

_{bilayer-forming}

surfactants,

it appears that an extensive

_bilayer

interface divides _space into two

_regions

of identical

solvent,

which may be

designated

« inside »

(I)

and « outside »

_{(0) (Fig. la).}

In either of these two

_systems,

we can

_expect

two additional conserved

_quantities

not

_present

in

ordinary single-component

fluids. In a

_{microemulsion,}

these may

obviously

be taken to be the volume

_{fraction 0}

of

_surfactant,

and the fraction

_cl

of the

_remaining

volume

_{occupied by}

oil

(we loosely

refer

_{to 03C8}

as the volume fraction of oil in this

_paper).

The surfactant volume

fraction is

_obviously

conserved in an

_{L3 phase}

as well. It is less obvious but still true that if

there is no flow

of solvent

through

the

bilayers,

then the

_partitioning

of fluid into 1 and 0

regions

is maintained

_{dynamically [4].}

Hence,

the volume

_{fraction cl}

of « inside » fluid is a

conserved

_quantity

if we assume

_dynamics

in which

_bilayers

fuse and reconnect without

tearing,

as shown in

_figure

1 b. More

_physically,

if fluid may be

slowly transported

across the

bilayers,

the volume

_{fraction cl}

of « inside » fluid is not

_quite

a conserved

variable,

but has some slow non-conservative

_dynamics

associated with the

_leakage

of solvent

_through

the

bilayer.

Neither of the currents

_{corresponding}

to the two conserved

_{quantities e}

and .0

described above is itself

conserved,

so the related

_hydrodynamic

modes are both

diffusive,

i.e.,

the

_dispersion

relation must be of the form oi =

Dq 2.

Fig.

1.

_{- (a)}

The

_L3

_phase

or « _sponge »

_{(Ref. [2]).}

The surfactant forms a

_bilayer

surface without tears,

edges

or seams. Such a

_surface,

even if it has disconnected _components, divides space into two

_disjoint

regions

which may be called « inside » and « outside ».

_(Here,

the inside volume

(say)

has been shaded ; however, the fluids in the inside and outside volumes are identical in an

_{L3 phase.)}

The free energy is

unaltered if the

labelling

of these

regions

is

_{interchanged ;}

this symmetry is not

_usually

_present in microemulsions for which e.g., « inside » _may be identified with oil and « outside » with water.

_(b)

A

change

in the

_topology

of a surfactant interface may occur when a narrow « neck » between two fluid

regions

is created or

_destroyed.

In this paper, we estimate the values of D and describe the

_{corresponding hydrodynamic}

modes

_{(eigenvectors}

of the

_{hydrodynamic matrix),}

and how

_they

each

_depend

on

quantities

characterizing

thé microemulsion or

_L3

_{phase, namely}

the mean volume fraction of

(respectively)

oil or

_{«inside» 03C8>}

and

_{surfactant ~> .}

We shall show that the

symmetric

state 03C8> =

1/2

has

_{particularly simple dynamics, with 03C8 decoupled}

from

_~.

As the

_symmetry

is

_broken,

the

_decays

of oil and surfactant autocorrelations are mixed.

Also,

we consider the effect of a slow

_microscopic

timescale for relaxation of the

_topology

(as depicted

in

_{Fig. 1 b)}

of the surfactant

_(bi)layer

on the oil and surfactant autocorrelation

functions. This

_gives

rise to a crossover from

_decay

rates

_{corresponding}

to «

_{quenched »}

Euler

characteristic at

_high

_wavenumbers,

to

_decay

rates for « annealed »

_topology

at lower wavenumbers. We may

expect

the

_microscopic

timescale to be slow when the activation energy for the fusion process is

large.

In the case of a balanced

microemulsion,

the

_symmetry

of oil and water is

_approximate,

but

can be made almost exact

_{by tuning}

the salt content etc. so a to obtain a

_{symmetric phase}

diagram.

For the

_L3

_phase

the

_symmetry

is exact in the

Hamiltonian ;

it is maintained in the

symmetric

sponge

(t/J = 1/2)

state, but can break

_{spontaneously}

to make an

_asymmetric

sponge or vesicle state. From now on we use the

_language

of balanced microemulsions

(«

oil » and « water

_»)

as

_opposed

to « inside » and «

_{outside » ; however,}

_many of our results are

2. Relaxation processes and conserved

quantities.

2.1 LOCAL SOLVENT FLOW. -

We may make an estimate of the diffusion constant for

transport

of

_(say)

oil in the microemulsion

_by

_equating

the work done

_by

viscous

_dissipation

and the

_lowering

of the free _{energy upon} the relaxation of a

fluctuation,

as follows. We

imagine

that

_transport

of oil occurs

_by

_hydrodynamic

flow on the characteristic « cell size » or

length scale e (corresponding

to the

_peak

in

_{S(q )),}

which must involve

_velocity

_gradients

Vv of the order

_{of e -}

v .

Now we consider an initial state in

_{which 03C8}

has _{the average value}

o,

and the

_{value 4> + 6 e}

in some

hydrodynamically

large (» region

of size

q - 1.

The

_{corresponding}

_{free energy}

_change

_{per unit volume}

_{is &FI& e.}

The

_microscopic

flow

_velocity

is of some

characteristic

magnitude

v, and the

corresponding

current is

Viscous

_dissipation

does work _per unit volume at a rate

The

_{corresponding change}

_{in the free energy}

_density

occurs at a rate

Equating

these two rates of work done

_gives

the scale of

_{microscopic velocity}

as

(The

factor

_{of q}

arises from the term V. v, since it is

_{only gradients}

of the

_macroscopic

velocity (coarse-grained

on a scale

_{> ))}

which have the

_{hydrodynamic length}

scale

q- 1,

and are effective in

_relaxing

_ôj

and hence

_F.)

Now we _may substitute the value of v into the

_{continuity equation}

to obtain the

_{hydrodynamic equation}

for relaxation of

03C8,

where

_F03C803C8=

_{8 2F 18,p 2,}

with similar notation to be used

_throughout

_{the paper.}

_(For

the

_L3

phase,

this

_{equation neglects}

fluid flow across

_bilayers,

which would lead to nonconservative relaxation of

_03C8,

discussed

below.)

We may make an

_analogy

with so-called

_permeation

flow in porous media

_[5].

There it is

argued

that a _pressure

_gradient

_Vp

across a porous rock induces a

_macroscopic

flow

_velocity

v.

On dimensional

_grounds,

if we

_require

that

_op/v

is

_proportional

to

_viscosity,

we find

immediately

Darcy’s law,

If we now

identify p

with osmotic pressure variation

f/Jo

_{F 03C803C8}

8

03C8

and use the

expressions

for

the current and conservation

_{of e,}

we arrive at the same

_{hydrodynamic equation,}

2.2 SURFACTANT MOTION. - Now consider the

_transport

of surfactant. In microemulsions and

_L3

_phases,

we _may consider two

_possible

modes of

_transport

for surfactant :

₁₎

diffusion

across

_intervening

fluid

regions,

where the surfactant

solubility

is small but

_{nonvanishing ;}

and

2)

flow which « smooths out wrinkles » in the surfactant

_{(bi)layers separating}

fluid

regions,

and sweeps surfactant

along

the fluid-fluid interface

_[6].

We examine these in turn.

_First,

the

_solubility

of surfactant in the bulk fluids is

_quite

low,

and the

_{resulting macroscopic}

diffusion constant is

_proportional

to the volume fraction of surfactant in the bulk fluid. We will

_neglect

this _{process in the}

_present

work.

Next,

the

_smoothing

out of wrinkles of

_{all length}

scales f

up

to f = e

involves

_microscopic

velocity gradients

on the scale

f.

If some excess area is to be added or

_removed,

_perhaps

forced

_by

some local

_adjustment

of the surfactant chemical

_potential,

the

_energies

of the lowest modes are most

_affected,

and so their

_amplitudes

will be most

_changed.

In other

words,

the

_largest

wrinkles

_give

the dominant contribution to the

_change

of surfactant

_density

among all undulations. Hence we _{may say that the} viscous

_dissipation

associated with

smoothing

out undulations occurs

_primarily

on the

scale f - e.

Then the

_argument

_given

above for the rate

_{of decay}

of fluctuations of the oil volume

_{fraction e}

_{may be}

_repeated

for the surfactant volume fraction

4>,

and an

_Onsager

coefficient

_{corresponding}

to a _pore

_{size C}

is

again

obtained

_[6],

_{D 2 1 o 0 2F,00.}

This result for

_{Do ,}

and

_{equation (6)}

for

_D,,

involve the

_{thermodynamic}

derivatives

F t/Jt/J

and

_F..,

whose

scaling

will be discussed in more detail in section 3.2. As shown

_there,

the basic

_{scaling expected}

for both

_Dk

and

_{D 03C8}

is

_(to

within

_{logarithmic corrections)}

D - kTI (~e).

This behavior is similar to that in semidilute

_polymer

solutions and other

systems

with thermal

_energies

and a well-defined correlation

_length.

With a

_viscosity

of 1 cP and a cell

size e

in _{the range 100-1 000}

Á,

we have D -

10 - 6_ 10- 7.

2.3 « LEAKAGE ». In the

_L3

_phase,

we

_expect

the inside volume

_{fraction «/J}

to be

nonconserved,

on a timescale

_governed

_by

an activated

_{« leakage »}

_{process ;}

_namely,

the « inside » and « outside » fluid in the

_L3

_phase

_may

_{exchange by opening}

small holes in the

bilayer.

These holes are

_expected

to _{be very} rare so that their effects are

_negligible

for static

properties ;

however,

their

_dynamical

_{effects may be}

_significant.

If these holes are molecular

in

size,

we are

_{really describing}

diffusion of solvent molecules

_{directly through}

the

bilayer,

a

process which is known in

thermotropic

smectics as

permeation [7].

_Larger

holes

(of

size

03BE)

would lead to actual

_hydrodynamic

flow of fluid

_through

the

_bilayer.

In _{any case,} the effect of «

_{leakage »}

across the

_bilayer

is to add a term

T4 1 03C8

to the

_right-hand

side of the relaxation

_{equation (6)}

for

cl.

Note that

_{7- V,}

= oo for

microemulsions ;

such a term is absent for

1

microemulsions since oil and water are not

_{locally interchangeable.}

2.4 TOPOLOGICAL RELAXATION. In addition to this

_{long microscopic}

timescale,

there is another

_important

activated local _process to consider in the

dynamics

of microemulsions and

L3

phases.

This process is fission or coalescence of fluid

_regions

which

_changes

the

_topology

of the surfactant

_{layer (Fig. 1 b).}

We may

imagine

a _{process in} which a « neck » in the surfactant

layer

is

₁₎

created

_by

the collision of two

_{nearby layers}

and the

_opening

of a neck between

them ;

or

_{2) destroyed, by pinching}

off a neck and

_retracting

the

_separated

_pieces.

_(Physically,

we

_expect

such

topological

changes

to be activated because the intermediate state of the interface

_undergoing

such a

_change

is

_{energetically}

_unfavorable,

_{containing sharp}

bends

and/or bilayers

in close

_contact).

If we forbid such

_changes

in the

_topology

of the surfactant

_layer,

then it is clear that a

different state is reached if some control

_parameter

is

_changed.

For

example,

consider the

Fig.

2. - In the

L3

phase,

in which the fluids on either side of the

_bilayer

are

_identical,

_transport across

the

_bilayer

may occur either

through

the

opening

of

large

holes in the

_{bilayer through}

which fluid can

flow

_(2a),

or

_by

diffusion of fluid molecules across the

_{bilayer (2b).}

different 0

to be related

_by

a dilation

_{[1, 8] (this}

is true _up to

_logarithmic

_corrections,

at least in the

_{symmetric state),}

and the number of handles _{per unit}

_volume,

for

_example,

would scale

with - 3.

Without

_topological

_relaxation,

the Euler characteristic for the surfactant film is

fixed ;

we

_might

_expect

this state to contain more surfactant

_(to

form the extra

_handles)

than the

_totally

relaxed state at the same chemical

_potential.

If no

_{topological changes}

could occur in the surfactant

_layer,

we would have a new

conserved variable in the

_system,

which

_might

be taken to be the Gaussian curvature

_[9],

or

equivalently

the Euler number per unit

volume ;

we shall define h to be the

_negative

of this

quantity

and refer to it

_loosely

as the « handle

_density

». Since we

_expect

the kinetics of local

processes

involving

the formation and removal of handles in the surfactant

_layer

to be

activated,

the relaxation of the handle

_density

h should be

_{non-conserved,}

but may be rather

slow if the activation energy is

large.

We characterize this

_by

a time scale _Th. In

_general

this timescale will be rather different from

_{r ,}

as different activation barriers are involved for

changing topology

and

_leaking

solvent across

_bilayers.

It is

_possible

in an

_L3

_phase

to have

Th «

_{T 03C8}

if

_leakage

of solvent across the

_bilayer

is slow.

3.

_{Thermodynamic couplings.}

We now turn to the issue of the

_{thermodynamic}

and kinetic

_couplings

between the variables

03C8

0,

and

h,

and how these

_{couplings change}

when the oil-water or « inside-outside »

symmetry

« f/J)

=

1/2)

is broken.

3.1 HYDRODYNAMICS IN THE SYMMETRIC STATE. - First we consider the

_{thermodynamic}

couplings,

i.e.,

the entries in a matrix

where

_{u, f3}

are one

_{of {f/J, l/J, h }.}

These entries _may be estimated from some «

_{microscopic »}

model of

_{microemulsions,}

for which we shall use the model of Safran et al.

_{[10, 2].}

The model Hamiltonian has

_(in

the absence of _spontaneous

_curvature)

the

_{symmetry 03C8}

-->

(1 - 03C8),

which means that all

_off-diagonal

terms

in X -1 1 involving f/J vanish

in the

_symmetric

state.

_(This

may say that in the

symmetric

state, a small

_change

in

03BC~ or _{1£ h} cannot

_produce

a linear

change

in t/J )

because the

_sign

of the

_{supposed change}

reverses under the

symmetry

operation.

In contrast, there should be linear

_{couplings between}

and h even in the

_symmetric

_state, as we

_argued

above that a state with more handles must have more

_surfactant,

all other

_things

being equal.

We _{may guess the size of this}

_coupling

to be such that a

_doubling

of the number

of handles

_gives

rise to a similar increase in the amount of surfactant

_present.

In other

words,

Here 0 0

and

_ho

are the mean volume fraction of surfactant and handle

_density

_{respectively.}

From the model of Safran et al. it can be shown that

_{ho -}

const.

x ç - 3 in

the

_symmetric

state.

We may make similar

arguments

for the

_vanishing

of the

_{off-diagonal couplings}

_{involving f/J}

in the matrix of

_Onsager

coefficients.

_Namely,

the direction of a current of oil volume fraction

j,,

in response to a

_gradient

in

_{either »,,}

_{or gh} reverses under the

_symmetry

operation

of

relabeling

oil as water

_{(e --+ (1 - e», so}

such a current cannot exist in the

_symmetric

state.

We could in

_principle

have

_{off-diagonal couplings}

in the

_Onsager

matrix

between 4>

and

h,

which would be

_proportional

to

q 2

_{(since 0}

is

conserved,

and the

_Onsager

matrix is

symmetric).

Since these variables are

_{already coupled thermodynamically,}

we will

_neglect

these crossterms for

_simplicity

in the

_present

work.

_Finally,

h itself can be relaxed

by

conservative

transport

(diffusion

of

_handles)

as well as activated local processes ; this

_suggests

that

_Ahh

should have a

_{permeation-like}

term similar to

_A",,,,

and

_A-00.

Hence we

_have,

for the

_symmetric

state,

where a,

b,

and c are coefficients of order

_unity,

and

_{03C4 03C8 = oo}

for microemulsions. The inverse

susceptibility

matrix is

The linearized

_{hydrodynamic equations}

for our microemulsion

_degrees

of freedom are then

(in

matrix

_form)

3.2 DYNAMIC LIGHT SCATTERING PREDICTIONS. In a

_{dynamic light-scattering experiment,}

we _may observe most

_easily

in microemulsions the autocorrelations

_{of f/J,}

and in the

_L3

_phase

(where

there is no contrast between inside

_{( 03C8 )}

and outside

_{(1 - e) regions,}

which contain identical

_solvent)

the autocorrelations

_{of çb [11].}

Hence we are interested in the

_{decomposition}

of the oil or surfactant volume fractions in terms of the

_{hydrodynamic eigenmodes}

of the

The autocorrelation of the surfactant volume

_{fraction 0}

is more

complicated ;

its

coupling

to a

_{nearly-conserved}

handle

_{density gives}

a

_q-dependence

to the

_apparent

diffusion constant

for

.0.

The effect _{may be summarized} as follows.

_{For q}

such that the relaxation time

Th is short

compared

to the timescale

(D. q 2)-l

1 of conservative relaxation of surfactant

density,

the handle

_density

may be

regarded

as annealed. This would be the conventional

hydrodynamic

assumption,

and

_gives

In contrast,

_{for q large enough}

that

_{D t/J q2 Th 1,}

the relevant

_{susceptibility}

is that found at

fixed

h,

and we have

The ratio of the two results for

_{D .~}

is

_{given by}

the

_{thermodynamic}

relation

Hence there should be a crossover from the

_{quenched prediction}

_{D03C8/h to}

the annealed

expression

_{D0 1 Mh’}

which is smaller

_by

a factor of order

unity [12],

on the timescale

7"h.

Note

_finally

that when the

_dependence

_{of Fko}

and

_F,,,

on cell

_{size e is}

taken into account, both

_Do

and

_{D .}

scale as

Here a stands for

_{either 0}

or

_03C8

_eK

is the

_{persistence length,}

and the

_{xa}

are constants. This

scaling

holds because the dimensionless inverse

_{susceptibilities}

_{a2 Fa a}

both scale

_{[8, 13]}

as

kTle3 K

x In

(x/xa)/x3,

with x -

elCK

and

{x,,,l

as above.

_Except

for the

_logarithmic

corrections

_{[8, 13],}

this is evident from dimensional

_analysis,

as the

_{thermodynamic}

derivatives are both energy

densities,

and the scales of energy and

length

in these

systems

are

kT

and e

respectively.

3.3 BROKEN SYMMETRY. - The oil-water or inside-outside

_symmetry

_{may be}

broken,

either

spontaneously

in the

_{L3 phase,}

or

by

the

imposition

of a nonzero

_spontaneous

curvature of the surfactant

_monolayers

in a microemulsion. There is no

_{longer a cl - ( 1 - cl )}

_symmetry,

hence there are

_{off diagonal couplings involving cl}

in the

_{thermodynamic}

and

_Onsager

matrices. One may guess that for the

L3

phase,

these

_couplings

will be of order

s -

(2 f/J) -

1 )

times the

appropriate coupling

scale. That

is,

we

_expect

For a « minimal »

_{thermodynamic}

model,

we _may take such an

off-diagonal coupling

between

f/¡

and the _{surfactant volume fraction in the free energy. For}

_simplicity

we assume an annealed handle

_density

in the broken

_symmetry

case. This is

_equivalent

to

_taking

a linear

_coupling,

of the form

_given

above,

between e

and an annealed

_{variable . 4> - hF h4> / (2 F 4>4»’}

and

omitting

a

_separate

_{coupling between e}

and h.

where d is a

_positive

coefficient of order

_{unity. (Here}

the

sign

is determined

_by

the

supposition

that the surfactant current will tend to follow the current of the

_minority

fluid

phase ;

this is motivated

_by

a

picture

of

moving droplets.)

With these

_assumptions,

we _{may summarize} the effect of the

_{symmetry-breaking}

on the

eigenmodes (in

the case of annealed

_topology)

as follows. The

_{O(E) couplings}

shift the two

eigenvalues

of the

_hydrodynamic

matrix

_by

an amount of order 0

_(82),

and

_split

the modes

by

a similar amount. The

_eigenvectors

are also

_changed,

such that the mode which was for e = 0

purely

surfactant

_transport

at fixed oil volume fraction _now involves oil flow _at

O (E).

This

_mixing

may be difficult to see

_{experimentally,}

for the

_following

reason. As remarked

above,

the dimensionless inverse

_{susceptibilities}

_{u- 2 Fa a}

for a

_{= cP, ’" scale [8, 13]}

in the same

way

with e; similarly,

the

permeabilities

for oil and surfactant

transport

were

_argued

to be of the same

order,

(qç)2 TI -1.

Hence the two

_eigenvalues

which are

_being

_{mixed may} not be

widely separated,

and have the same

scaling dependence

_{on 03BE}

etc. The

_symmetry

breaking

introduces an

_{0 (e)}

admixture of a relaxational mode of a

comparable decay

rate to the mode

already

observed for the

_symmetric

state. Because

_separation

of the sum of two

_exponentials

of similar

_decay

rate can be

experimentally

difficult,

this effect could be

misinterpreted

as an

9(c)

shift in the

_decay

rate, with a decrease in the

_goodness

of fit.

In

_addition,

if we examine an

_{asymmetric L3 phase}

in which _T,

_(presumed

here

>

_{T h)}

falls in the

_experimental

_timescale,

we should

_expect

to see a

_q-dependent

crossover in

D,o

as we _pass from wavenumbers with

_{r 4, «,,:}

_(D,#

q 2)-1,

at which

03C8

is

_{quickly relaxing}

(annealed) compared

to 0 fluctuations,

to wavenumbers at

_{which cl}

is

_{slowly relaxing}

(quenched).

This fractional shift in

_D~

_{for q crossing}

_D~

_{q 2 ~}

_{T 03C8}

1 will

be

_{0 ( e 2).}

4.

_{Estimating T h.}

As a final

_topic,

we turn to the

_question

of how we can determine the timescale

Th for relaxation of the microemulsion

topology.

We have

already argued

that the kinetics of

this relaxation is

_activated,

_according

to _{the process shown in}

_figure

1 b. The

_attempt

frequency

for the activated process should be set

_by

the so-called « Zimm time »

TO ’" ç 2 D (ç)-

required

for a fluid

region

of

size e

to diffuse its own

length.

The relaxation

time _{T h} then has the form

_{(/3 == (kB}

_{T )-1 )}

where the diffusion constant for a

region

of

size e

is

_simply

Rather than

_trying

to determine AE from some

microscopic

model of the surfactant

layer,

we will

_try

to bound this T

by considering 1)

the

viscosity

of the microemulsion or

_{L3 phase,}

and

₂₎

the

_sensitivity

of the

_system

to shear.

_(Our

numerical estimates are

pertinent

to the very dilute

SDS-dodecane-pentanol-water L3 phase

of Ref.

[3].)

We

_expect

a contribution to the

_viscosity

from the

_long

relaxation time T to _go as

8 TJ -- G (0) T,

where the instantaneous modulus

_{G (o)}

of the

_mesophase

is of order G

_{(0) -}

_{kT / ç 3}

_(by

a

_scaling

_argument,

which maintains that the characteristic energy of a cell

of

_{size e}

is of order

_k1).

This

_gives

rise to a contribution to the

_viscosity

which is

_simply

8 TJ --

71

exp[/3 DE ] ,

where q

is the solvent

viscosity.

Because microemulsions and

L3 phases

dilute

_L3

_phases

of reference

_[3],

we _{may take}

_{8 Tl $}

_{100 Tl 0’}

_{and ) -}

1

000 À ;

then we find

Th

10-1

s.

On the other

_hand,

_preliminary

_{experiments [14]}

in most of these dilute

_L3

_phases

indicate that characteristic shear rates for shear-induced

_{birefringence (and possibly}

a transition to

lamellar

_order)

are on the order of

_{y - 100 s - 1.}

This

_suggests

some relaxation time

Th which is no shorter than about

10- 2

s. The two estimated bounds taken

_together

_suggest

that the

_topological

relaxation time T in the dilute

L3

phase

is in the

neighborhood

of

10- 2

s

_{(though perhaps}

much shorter in

microemulsions,

which are

_usually

less viscous and

less

_dramatically

affected

_{by shear).}

This estimate

_implies

an

_energetic

barrier to creation of handles of around 5 kT for the most dilute

_L3

_phases

of reference

_[3].

Relaxation times of order

10- 2

s are

readily

observable,

for

_example

in

_{temperature-jump}

experiments [15]. (Shorter

time

scales,

as

_might

arise in microemulsions or less dilute _sponge

phases,

can also be

_easily

resolved

_by

this

_technique.)

_Furthermore,

the crossover in the

q-dependence

of the

_decay

rate for surfactant fluctuations _may

_give

evidence of the size of

Th. We may compare a relaxation time of order

10- 2

s with the conserved relaxation rate to

determine the

_{magnitude of q}

for which the diffusion constant and

_amplitude

of fluctuations

might

be

_expected

to

_{change ;}

we obtain

which for the

_typical

_L3

_parameters

used above

_{gives q - 104}

cm - 1,

which should be accessible

by dynamic light scattering.

5. Conclusions.

We have described the

_hydrodynamic

modes associated with fluctuations of oil and surfactant volume fractions

_{("’, cf> )}

in microemulsions. For the

_analogous

« _{sponge »} or

_L3

phase, e

is not « oil » but the volume fraction of « inside » solvent

_[2].

Two relevant timescales are :

₁₎

_{r ,}

the

_{« leakage »}

time of solvent across the

_bilayer

in the

_L3

_phase

(T f/I

= oo in

_{microemulsions,}

where solvents on

_opposite

sides of a surfactant interface are

distinct) ;

and

₂₎

_’rh, the relaxation time for the

_topology

of the surfactant

interface,

which may be very

long

in

phases

with

sharply-defined

interfaces and

high

activation barriers.

In the

_simplest

case, in which the

phase

respects

the

_oil/water

or

_{inside/outside}

_symmetry

( 03C8 >)

=

1/2),

the

hydrodynamic decay

_rates

for 03C8

and .0

fluctuations _are

displayed

in

figure

3.

To summarize : fluctuations

_{of cl decay}

as

D03C8q2

for

_large

_q, and as

T-1

1 for small q.

Fluctuations

_{of 0 decay}

as

Db (q) q 21

with

_{Db (q)}

_crossing

over from a value at

_{small q}

appropriate

to an annealed

_topology,

to a

_larger

value at

_{large q corresponding}

to a

quenched

topology.

The crossovers occur when the relaxation rates

_{of cl}

_{and cf>}

fluctuations

_equal

T;

1 and

_{T h 1 respectively.}

In the

_{broken-symmetry phase}

_(03C8>

#=

1 /2 ), 03C8

and 0

fluctuations

are

_coupled

at

_{0 ( 03C8>) -}

1 /2 )2,

and the

_resulting

fluctuation

_decays

are not

single-exponen-tial.

By

considering

the observed

sensitivity

to shear and moderate

_viscosity

of

L3

phases,

we

conclude that _Th for the dilute

_{L3 phases}

of reference

_[3]

should be of order

10- 2

s’. This relaxation time _may be observed either in

_small-q

crossover of relaxation rates

_off

fluctuations in

_L3

_phases,

or in

temperature-jump experiments [15].

A

_{magnetic birefringence}

measurement, in which a _square wave

_pulse

of

_magnetic

field is

applied

to the

_system,

would

_{give striking}

behavior for the

_{optical signal}

as the duration T of

the

_pulse

is increased

_beyond

_Th. For short

_pulses

with T « _{T h,} the

_topology

would be unable

Fig.

3. - Shown are the different

wavenumber-dependences

at small

_{q 2(qe .-r.}

1 )

of the characteristic

decay rates to .

and w ,

for the volume fractions of oil and surfactant in a

_symmetric

microemulsion. The

dashed curves are

_asymptotic

diffusive behaviors ; the dotted line indicates the crossover

_frequency

Th 1,

near which the _apparent diffusion constants of the two modes

_changes

from the «

_quenched

topology »

(high-frequency)

value to the « annealed

_{topology »}

(low-frequency)

value.

_(We

have assumed

_arbitrarily

for this

_figure

that

_w,,(q)

lies above

_w,(q).)

In the

_symmetric

sponge

phase,

the timescale r,, is finite, and the « oil » mode

_{(unobservable by}

_{scattering) decay}

rate

_approaches

T;

1

as q

approaches

zero. In the

broken-symmetry phase,

the modes are mixed as described in section 3.3.

optical signal

should be

_govemed

_by

local solvent and surfactant flow with a relaxation time of order

_{7-0 _ e2 IDO(e-’) -}

6

_{_ffq e 3/ T.}

The

_topology

could

_adjust

to a new

_equilibrium

value if

the duration of the

_pulse

exceeded _{7-h ; then the}

_decay

of the

_{optical signal}

after the removal of the field would show the slow relaxation

_{time r h - 7- 0 exp (,B AE)}

in addition to a fast

_part

with time _To. As discussed

above,

these times may be well

separated

if the activation energy

for

_{topological change}

is

_{large enough.}

Acknowledgements.

This research was

supported

in

_part

by

the National Science Foundation under Grant No.

PHY82-17853,

supplemented by

funds from the National Aeronautics and

Space

Administra-tion,

and in

part

by

EEC Grant No. SC 10228-C. The authors are

_grateful

to the Institute for Theoretical

_Physics,

UCSB,

for

_hospitality

and to F.

Nallet,

G.

_Porte,

and S. J. Candau for

helpful

discussions.

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