Equilibrium dynamics of microemulsion and sponge phases

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phases

G. Gompper, M. Hennes

To cite this version:

G. Gompper, M. Hennes. Equilibrium dynamics of microemulsion and sponge phases. Journal de

Physique II, EDP Sciences, 1994, 4 (8), pp.1375-1391. �10.1051/jp2:1994205�. �jpa-00248048�

Classification Physics Abstracts

64.20G 64.60H 82.70

Equilibrium dynamics of microemulsion and _sponge phases

G.

Gompper

and M. Hennes

Sektion Physik der

Ludwig-Maximilians-Universitit

Miinchenj Theresienstr. 37, 80333 Miinchen, Germany

(Received 13 April 1994j received in final form 19 April 1994, accepted 29 April 1994)

Abstract. The dynamic structure factor

G(k,

w) is studied in _a time-dependent

Ginzburg-

Landau model for microemulsion and sponge phases in thermal equilibrium by field-theoretic

perturbation methods. In bulk contrast, we find that for sufficiently small viscosity _~, the

structure factor develops _a peak at _non-zero frequency _w, for fixed wavenumber k with ko <

k £ q. Here,

2n/q

is the typical domain size of oil- and water-regions in _a microemulsion, and ko _~ ~q~. This implies that the intermediate scattering function,

G(k, t),

oscillates in time. We

give _a simple explanation, based

on the Navier-Stokes equationj for these temporal oscillations by considering the flow through _a tube of radius R _ci

n/qj

with _a radius-dependent tension.

1. Introduction.

The behavior of

self-assembling amphiphilic

systems ^{has attracted much attention}

recently

_[1, 2]. The

unique properties

of these systems can

easily

traced back to the

ability

of the

amphiphile

to reduce the interfacial tension between two

normally

inmiscible fluids like oil and _water

by

several orders of

magnitude.

One of the most

interesting phases

of ternary

amphiphilic

systems is the microemulsionj a

homogeneous, isotropic

mixture of all three components. It is _now well established that microemulsions with medium- _or

long-chain amphiphiles

consist of coherent oil- and

water-regions,

which _are

separated by arnphiphilic monolayers. Similarly,

the sponge

phase

in aqueous

amphiphilic

solutions consists of distinct

water-regionsj

which _are

separated by

_an

amphiphilic bilayer

_{[3j 4].} Thus, the static behavior of these systems ^is now

relatively

well understood. Much less effort has been made _so

far, howeverj

to

study

the

dynamical

behavior of these systems. ^Most of these studies have considered

non-equilibrium

situations,

typically spinodal phase separation

[5, 6]j spontaneous emulsification [7], ^oil solubilization in

aqueous surfactant solutions [8], ^and

amphiphile aggregation

at _an

oil/water

interface [9].

We want to

investigate

^{here the}

dynamic

structure factor

G(k,

_uJ), both with bulk and film contrastj of microemulsions in thermal

equilibrium.

It is well-known from many studies _near critical

points

that systems

belonging

to the _same static

universality

class _may show

completely

different

dynamical

behavior

[10,

11]. ^{It is thus essential to}

clarify

which variables have to be

included _{in a}

dynamic

^{model of} microemulsion and _sponge

phases.

A

microemulsion,

for

example,

has two conserved

densities,

the

amphiphile

concentration

~(r, t),

and the difference of oil- and water-concentration

4l(r, t).

A sponge

phase,

_on the other

hand,

_may also have two

conserved

densities,

if the

arnphiphilic bilayers

_are almost

inpenetrable

to water, or

only

_one,

the

amphiphile

concentration, ^if

they

_are not [12].

Nevertheless,

the

arnphiphile

concentration

~

_may not have to

be,

^considered

explicitly,

if its relaxation is fast

compared

to that of the order parameter 4l. This is the _{case we want to}

study

in this _paper. The

opposite limit,

where the

amphiphile dynamics

is slow

compared

to the order parameter

dynamics,

is _more

complicated

and will be considered elsewhere. We will show below that it is essential to include the

hydrodynamic

variables in _a model for microemulsion and _sponge

phases.

In

general,

these

are the _pressure

p(r,t),

and the

lon#tudinal

and transverse components ^{of the} momentum

density, jL(r,t)

and

jT(r,t).

It has been shown that in the

vicinity

of _a critical

point,

_p and

jL

_are irrelevant

variables,

and thus _can be omitted in the model [10]. ^{We will} assume that the _same is the _case here.

However,

_p and

jL

_are essential for studies of sound

propagation

in

these systems

[13-15].

2.

Ginzburg-Landau

model.

The

time-dependent Ginzburg-Landau

model _we consider is _a

generalization

of the

Siggia, Halperin,

and

Hohenberg

model H

[16, 10],

consistent with linear

hydrodynamics.

^Our

hy- drodynamic

variables _are the local order parameter

4l(r, t),

which is

proportional

to the local concentration difference between oil and water, ^{and the} transverse component ^{of the} momen- tum

density jT(r, t).

In this case, the stochastic

equations

of motion _can be written _as

~

=

r~V~~(

+

gov 4l $1+ (~ (1)

JT

~~ ~~~~~)

^{~ ~°}

~~~$)~

^~

^~~

' ~~~

where the

Langevin

forces (, have the usual correlations [16],

<

(~(r, t)(~ (r', t')

_>

=

-2T~i7~b(r r')b(t t') ₍₃₎

<

(T,n (r, t)(T,p(r', t')

_>

=

-2TTi7~b(r r')b(t t')hap (4)

The

equilibrium

free _energy functional has the form

F =

/ _d~r C(V~~)~

+

g(~)(V~)~

₊

_f(~)

₊

)jij ⁽⁵⁾

It has been shown

[17-22]

that the static behavior of ternary

amphiphilic

_systems is very well described

by

this free _energy

functional,

with

jT(rj t) integrated

out, when the functions

f

and g ^are chosen to be

g(4l)

= bo + b24l~

(6)

f(4l)

=

r24l~

₊

r44l~

+

r64l~

,

(7)

with _c >

0,

b2 >

0,

and _r6 > 0.

Three-phase

coexistence between

oil-rich,

water-rich and microemulsion

phases

is obtained for _{r4 <} 0, r2 > 0 and bo < 0. As discussed

above,

_{we assume} that the transport ^{of the}

amphiphile

is much faster than all the other transport _processes. Thus the deviation of the

amphiphile

concentration

~(r, t)

from its _mean value

#

= <

~(r~ t)

_> does

not appear

explicitly

in _our model.

However,

the average concentration

#

_{and the}

_properties

of the

amphiphile

enter the model via the parameters ^{of the functions}

f

and _g. The value of

g(4l)

in the

microemulsion,

bo, for

example,

_can be made

negative by increasing

the

amphiphile

concentration _or the

strength

of the

amphiphile

_[23]. Since _{we are} interested here

only

in the behavior of the microemulsion

phase,

it is sufficient _to retain

only quadratic

terms [17] ^{in the} free _energy functional

(5),

_so that

g(~)

" bo

(8)

f(~)

=

r2~~ ₍₉₎

The _same model _can also be used to describe _sponge

phases

_{[24, 4,}

25].

In this _case, the order parameter

4l(r, t)

is identified with the local concentration difference between _{water on}

one side

("inside")

and the other side

("outside")

of the

amphiphilic bilayer.

Since _{we use}

a conserved order parameter in

(1)

and

(2),

_our results

apply

to sponge

phases

in which the

leakage

time of water

through

the

bilayer

is very

large [12].

3, van-Hove

theory.

When non-linear contributions in the

Langevin

equations

(1, 2)

_are

neglected,

all structure factors _can be calculated

easily

_[11]. In this

approximation,

the Fourier transform of equation

(1) reads,

for

example,

iuJ4l(k,

_uJ)

=

-2r~k~ _(ck~

_{+ bok~ +}

r2) ^4l(k,

uJ) +

(~(k,

_uJ)

(10)

Since

(10)

is _a linear relation between the random variable

4l(k,

_uJ) and the noise

(~(k,

_uJ), the noise correlation function

(3) immediately implies

that the

dynamic

water-water correlation

function has the form

G©((k,

_uJ) ~

=

~

~~~~

~ ,

(11)

uJ +

[r~k2xo(k)-~]

where

xo(k)~~

₌ 2

(ck~

+ bok~ +

r2) (12)

is the static structure factor. For

b(

<

4cr2,

the static correlation function in real _space is

given by

[17]

G~~(r)

=

~e~~/~ _{sin(qr) (13)}

r

~vhere

(,

with

~~ lbo

i~~ (14)

"

§ j

^~

ii

is the correlation

length,

and

27r/q,

with

q~ =

~fi-

~°

(IS)

2 _C 4 _c

is the characteristic domain size of coherent oil- and

water-regions.

It follows from

equations (12)

and

(13)

that there _are three

important

lines in the

phase diagram [17, 18],

^With de-

creasing

bo, ^{the first} is the "disorder line" _bo

=

@,

where the correlation function

changes

its behavior from _a monotonic

decay

to

damped

oscillations. The second line is the "Lifshitz line" bo _{" 0~} where _a

peak

in the structure factor

begins

to _move to _non-zero wavenumber.

Table I.

Asymptotic

behavior of the characteristic

frequency

_uJc in the van~Hove

approxi-

mation.

regime

_uJ~

k »

(~

,q

2cr~k

k _<

(~

_,q

2cr~((~

_+q k q < k <

(~ 2cr~(~

^k

(~

< k _{< q}

2cr~q

k

(~

= 0~ k _{m q}

8cT~q (k q)

~~

(Q~ /cr~)~

i o

oo

0.0

~

2.0

k~

k/q

^{3.0 ~.°}

4.0

Fig. 1. The inverse scaling function

Qo(kf, k/q)~~

=

k~/wc

in the van-Hove approximation.

Finally,

the line bo "

-@

is the

spinodal

to the lamellar

phase.

Note that the

shape

of the static structure factor is characterized

by

a

single

dimensionless parameter

qt.

The intermediate

scattering function, G~~(k, t),

is obtained from

equation (11) by

Fourier

transform,

and reads [11]

G©((k, t)

=

xo(k)

_exp

[-r~k~xo(k)~~t] (16)

Both

equations (ii

and

(16)

can be used to define _a characteristic

frequency~

wc +

r~k2xo(k)-i (ii)

This characteristic

frequency

_can be written in the

scaling

form uJ~ =

k~oo(k(, k/q)

,

(18)

where the

dynamic

exponent z = 6 and the

scaling

function

no (~, y)

₌ 2cT~ _[1 +

2(~~~ y~~)

₊

(~~~

+

y~~)~j (19)

are obtained

easily

from equation

(17).

The

asymptotic

behavior of _uJc in various limits is listed in table I. The inverse

scaling function,

I.e. k~/uJ~, ^{is shown in}

figure

1. Two slow modes

are present in a structured

microemulsion,

_one for k _m 0, the other for k _m q. In the first case, the conservation of 4l _causes the

divergence

of the relaxation time in the limit k _- 0,

as in any other

binary

fluid. In the latter _case, critical

slowing

down _{occurs as} the transition to the lamellar

phase

is

approachedj

this slow mode is thus due _to the

increasing stability

of

fluctuations with the wavevector q, which characterizes the structure of the microemulsion.

4. Perturbation

theory.

In order to treat the non-linear interactions, we

employ

_a field-theoretic

approach [26,

27]

using

response fields

4

_and

jT.

The

twc-point

correlation and response functions

G~,,~~(1~2)

=

<

~,(1)~lj(2)

_{> are}

given exactly by Dyson's

equation,

[G~,,~~(1, 2)]

^~~ =

(G~~~ (1, 2)j ^Z~,,~~ (1, 2) (20)

in terms of the

self-energies Z~,,~~,

^with

^~,

E

(4l~4~jT,jT).

Here, G~°~~

(1,2)

denotes the correlation and _response functions of the linearized

theory.

The inversion

it (he Dyson equation yields

in

particular

the correlation function

G~~(k ~)

=

2T~k~

₊

Z++

iw +

T~k2xo(k)-1 g~~j2 ⁽²¹⁾

In the

one~loop approximation,

the

self-energy Z~+(k~

uJ) ^is

given by

_[16]

~~~~~~~~ ~~~ ~~~ _~~~~

~1°J+~T(~~~~~~~P~X0(P)

~ ~~~~

with the

projection

operator

~~

"

~°fl (~~

(~~)

The

Feynman diagrams

_are shown in

figure

2a. The second

self-energy~ Z+j,

ⁱⁿ equation

(21)

can be obtained from

equation (22) by

the fluctuation

dissipation theorem~

which leads _to

Z&j(k~uJ)

₌

-2xo(k)

Re

(Z~+(k, uJ))

,

(24)

or

directly

from the

loop expansion, just

_as

Z~j.

The calculation of the other correlation and response functions is described in the

appendix.

To calculate the characteristic

frequency

_uJ~ in

perturbation theory,

_we define [28]

In the _case of the van-Hove

theory,

this definition _agrees with

equation (17).

The results of van-Hove and

perturbation theory

for _{uJ~ are}

compared

in

figure

3. The

dependence

of _{uJ~ on} the wavevector k shown in

figure

3a demonstrates that _uJ~ becomes

larger (I.e.

the

typical

relaxation times becomes

smaller)

when the non-linear interactions _are taken

into account. This shows that the

hydrodynamic

flow

speeds

_up the relaxation _process of the

order-parameter

correlations. This

hydrodynamic

relaxation mode must be _a flow of oil and water

through

their

multiply

connected networks which _are characteristic for _a microemulsion.

p, UJ~

k, a k, a

-igoP -igok

p-k, m-a' P' °~~

k, a k, a

p-k, a-w' ~~~OP

~fv

p-k, a-a'

k, _a k, _a

fl0u -igok

~

p, ^@'

b)

a)

Fig. 2. Feynman diagrams which contribute to one400p order to the self-energies

(a)

Z~a and

(b) Z~j.

Here, the straight lines represent the physical fields, the _wavy lines the response fields. The solid lines denote order parameter propagators, the lines with bars momentum propagators. The diagrams

contain the vertex contributions _u

=

pxo(p)~~ kxo(k)~~

and _v

= p

[xo(p k)~~ xo(p)~~j.

.o

t0~

OS

,' --, ,

o-o

°.° °.~ ~.°

k/q

a)

t0c to~

i-o

oio

,,,

o.5 o.05

""~,,,,

.,

,

~ ~ ooo

""'~

"

~'~ ~'°

°'~iq~)-1°'°

^{°'~ ° °°~}

lq~)~

^{° °°}

b)

C)

Fig. 3. The results of van-Hove

(dashed lines)

and perturbation theory

(full lines)

for _wc,

(a)

as a

function of

k/q

for fixed qf

= 8.888,

(b)(c)

_{as a} function of qf for fixed k

= q. (c) The behavior of _wc for large qf in detail. Here, the dashed-dotted line is calculated with the one-loop approximation for

Gjj and G. The parameters in equations

(1), (2), (5),

(8) and (9) _are r~ ₌ 1, rT ₌ 1, _{go =} 2, _{c =} 1 and _r2

= 1~~

Figure

3b shows _uJ~ for k

= q. This demonstrates

again

the

point

_we have made above for the van-Hove

theory

that there is _a slow

mode,

with _a relaxation _time which

diverges

_as the

spinodal

of the transition to the lamellar

phase

is

approached,

I-e- for

1/(q()

_- 0.

However,

it _can also be _seen from

figure 3b,

_c that the relaxation time

obtained

_{from the}

perturbation theory

remains finite in the limit

q(

_{- cc.} This is _a

disturbing result,

since it _can

easily

be

seen from

equations (5), (8)

and

(9)

that the static

spinodal

remains

unchanged.

It is _not difficult to find the

origin

^{of this}

problem.

It is the _use of the van-Hove response function G(°~ of the momentum

density

in equation

(49)

for the

self-energy,

_compare

figure

2a. In the

~an-Hove approximation,

the _response and correlation functions of the momentum

density

contain _no information about the structure of the

microemulsion,

_{as can} be _seen from

equation (46).

To avoid this

problem,

all

one-loop

correlation and _reponse functions have to be evaluated

self-consistently.

Since this

requires

_a considerable numerical

effort,

_we

only

_use the

one-loop

result for

G~

^and

Gjj

in

equation (49),

which is the dominant contribution. With this

modification,

the characteristic

freqency

for k

= q now

correctly

vanishes at the

spinodal,

as can be _{seen in}

figure

3c.

The

dynamic

correlation function

G~~(k,uJ)

_{as a} function of _uJ for three different values of

the wavevector k is shown in

figure

4. Note that for k _m

q/2

_a

peak develops

for _{uJ >} 0 in

the limit of

large q(,

I-e- for

strongly

structured microemulsions. This

peak

indicates that the correlation function oscillates in time.

Indeed,

the Fourier transform of the data shown in

.o

G4~4~

", _k/q

= 1.3

,

'

0.5

',

' '

' '

,

o-o

O-O 5.0 ~/~- ^10.0

a)

~©©

,

Gaa

,

k/q

= 0.5

k/q

=

0,1

~'~ 40.0

i '

',

i '

2.5

',

20.0 ',

,~

',,

0.0

'''

0.0

''

O-O i-O 2.0 ~/~+ ^{3.0 O-O O-1} ~/r ^0.2

© ©

b)

^c)

Fig. 4. The dynamic correlation function Gww

(k,

_uJ)

as a function of the scaled frequency

uJ/rw

for three values of the wavevector,

(a)

k

= 1.3q,

(b)

k

=

q/2

and (c) k

= q

/10.

The dashed line is the van-Hove result, the full line the result of the perturbation theory. The parameters ⁱⁿ equations

(1), (2),

(5), (8) and (9) _are ro #1, rT ₌ 0.3, _{go "} 5, _c =1, _{r2 =} and qf

= 8.888.

1.o

~©© _k/q

= 1.3

5.o

o-o

O-O 2.0

~_~- ^4.0

a)

~~~5

~ ~

~4~4~

~ '~

o.4

k/q

= 0.1

0.25 0.2

0.0 0.25

0.0 5.0 lO.0 ~_~- ^{15.0 00 20.0} ~_~- ^40.0

© 4~

b)

_c)

Fig. 5. The intermediate scattering function

Gww(k,t)

_{as a} function of the scaled time trw for

three values of the wavevector,

(a)

k

= 1.3q,

(b)

k

=

q/2

and (c) k

=

q/10.

The parameters _are the

same as in figure 4.

60.0

~©© to/ro

₌ 0.3 40.0

20.0

,

o-o

°.° °.~ ~.°

k/q

^~.~

Fig. 6. The dynamic correlation function Gww(k,uJ) as a function of the scaled _wavevector

k/q

for

uJ/rw

= 0.3. The dashed line is the van-Hove result, the full line the result of the perturbation theory.

The parameters are the _{same as} in figure 4.

figure

4b has

temporal oscillations,

_see

figure

5b. For k _{> q,} the

peak

is much

broader,

_so that there _are still oscillations, but with _a

rapid decay,

_see

figure

5a.

Finally,

for k _{< q,} the

peak

at finite _uJ is

absent,

and the correlations function for

large

times

decays monotonically,

as can be _seen in

figure

5c. The

peak

at finite

uJ and the

temporal

oscillations of the

dynamic

correlation function _are reminiscent of the

peak

at finite k and the

spatial

oscillations of the

static correlation function.

However,

while the latter has been

interpreted convincingly

ⁱⁿ

terms of the microemulsion structure

[17, 29],

^{the mechanism of the former is unclear.}

Finally,

_we ^show the

dynamic

correlation function

G~~(k,

uJ) as a function of the wavevector k for fixed dimensionless

freqency uJ/Tj

₌ 0.3 in

figure

6. This function vanishes in the limit k _- 0 due to the conservation of the order parameter.

5. Channel flow.

To understand the

temp_oral

oscillations of the correlation function in

microemulsions,

_we follow _an idea which _was used to

explain

the

dynamics

of

spinodal phase separation

^{in the}

bicontinuous

regime

[30]. ^{It is}

argued

in that _case that the relaxation is dominated

by

the flow

through

^the

channels;

this leads to _a

growth

law for the

coarsening

of the bicontinuous

structure, which is linear in time t, ^rather than the

t~/~

behavior observed for

droplets

_[31].

The force which drives the flow

through

the channels is the interfacial tension.

To derive _an

equation

which describes the time

dependence

observed in

microemulsions,

we

start from the linearized Navier-Stokes equation [32] ^{for the}

velocity

field

v(r, t),

~v

=

Vp

₊

~V~v ₍₂₆₎

t P P

where _p is the _mass

density,

and _~ is the

viscosity.

Consider _a

spherical droplet

of radius L within the microemulsion

phase,

in which the concentrations of oil and water deviate

slightly

from their average

values,

_as shown

schematically

in

figure

7. This

implies

that the _cross- section of the oil- and water channels in this

region

^{is somewhat}

^larger (or smaller)

than _on

average. The easiest way for the system to get rid of the extra oil and

/or

water is to let it flow

through

the channels to the outside of the

droplet.

We _{now assume} that the flow

through

this

multiply

connected network of _a microemulsion is similar to the flow

through

_a

cylindrical

tube of

length

L. For _a

cylinder

in the

z-direction,

the flow field is approximated

by

Poiseuille flow,

v(r, t)

=

o(t)[R(t)~ r~]ez (27)

Fig. 7. - A spherical ^roplet of radius _L

(marked

by the dashed line) within the

phase, in which

the

where

R(t)

is the radius of the tube and

o(t)

describes the

magnitude

of the

velocity. Further,

we

employ

the

Laplace equation

to relate the _pressure

gradient

to the tube geometry and to the interfacial tension _«,

~~~ ~/~~~ _iL~~

~~~~

for _a tube of

length

L. The Navier-Stokes

equation (26)

^then

implies

that the average

velocity (averaged

_over the cross-section of the

cylinder),

@(t) =

jj ^~~~~ _{u(r, t)dr (29)}

is determined

by

~ +

~

=

"

~~

_~

,

(30)

at R

pRL

_p R

where l~

=

dR/dt.

Note that it follows

immediately

from

equations (27)

and

(29)

that

=

~aR~.

_{Since for}

an

incompressible

fluid 3

@(t) =

-(R(t) (31)

we

finally

arrive at

~ ~ ~

~

2L2p p12

^{~ ~}

R~~

^~~~~

The

analysis presented

so far is _not

specific

to bicontinuous

microemulsions,

but

equally

well

applies

to the _case of bicontinuous

phase separation.

^The

specifics

of

amphiphilic

systems

come in when _we consider the interfacial tension _«. It is well-known that the interracial tension between the oil-rich and the water-rich

phase

in these systems ^is very small. What _we

really

need in

equations (28)

and

(32), however,

is not the tension of the

oil/water interface,

but the

driving

force for _a

change

in the tube radius. It has been shown in references [33] ^and [21] ^{that in the lamellar}

phase

the tension vanishes

identically

^for the

equilibrium separation

of the interfaces.

However,

if the

spacing

between the lamellae is increased

(decreased),

a

negative (positive)

tension

develops

_[21] which drives the interfaces back to their

equilibrium

distance. We _assume here that the _same is true in the microemulsion. For small deviations from the

equilibrium

radius

Ro,

_{we can} thus write

a(R)

=

>(R Ro

+

o((R Ro

₎₂

133)

with _a

positive

constant

ji,

and

R(t)

₌

Ro

+

e(t) (34)

By inserting

these results into equation

(32),

_we find that

e(t)

is determined

by

I + 6 ~

~

i _{+ e}

= 0

(35)

PRO ^2L P

to

leading

order in

e. Thus,

e(t)

satisfies the differential equation ^of a

damped

harmonic oscillator. The radius oscillates if the

damping

is small

enough,

I,e. if

]

>

(3§) ₍₃₆₎

P P _o

~

Thus,

there should be oscillations if the

viscosity

_~ of oil _or water _are small

enough.

We want to

emphasize

that the

R-dependence

of the tension

a in

equation (33)

is unrelated to _a

stretching

of the

amphiphilic monolayer (I.e.

_a

change

in the _{area per}

headgroup),

since _{we are}

assuming

in _our model

(5)

that the

amphiphile

concentration

samples

its full equilibrium distribution _on

a time scale

rapid compared

to

changes

in the oil- and water-concentrations.

With these

results,

_{we can now}

predict

the behavior of the

dynamic

water-water correlation function. We

identify

the tube radius Ro with

7r/q,

and the tube

length

L with

7r/k.

The

viscosity

in _our

Ginzburg-Landau

model is

given by

_~

=

pTT

_(34]. In terms of the variables _q, k and

TT,

the condition

(36)

then reads

k~ _>

aoT(q~

,

(37)

where _{ao =}

18p/(7r~jJ). Here,

both the parameter ji and the _mass

density

_p

depend

_on the coefficients _{c, bo} and _r2 in the free _energy functional

(5),

but not _on any

dynamic

coefficients in

equations (1, 2).

While the variation of p is

probably small,

ji can vary

considerably.

^Its

functional

dependence

_can be obtained from _a calculation of the interfacial tension [21]. ^{In the}

following discussion,

_we will consider _a fixed set of parameters ^{of the free} energy functional

(5), (8), (9),

_so that _ao and _{q are} constant.

In _case the

inequality (37)

is

satisfied,

the solution of

equation (35) implies

that the _asymp- totic

decay

of the correlation function should have the form

G~~(k,t)

= B

exp(-t/r) cos(v(k)t

₊

q) (38)

where

r =

~~

~

rTq (39) v(k)~

=

~~k~

a3T(q~ (40)

ao

with constants

B,

_q, and _{al, a2, a3.} We want to

emphasize

that the relaxation of

regions

with

increased oil- _or water-concentrations is facilitated

by

the flow mechanism _on

length

scales

larger

than the diameter of _a

tube,

I-e- for k « q. This is the

regime

where _we expect

our result

(38)

with

(37)

to describe the

dynamics correctly.

On the other

hand,

_on

length

scales smaller than the tube

radius,

I.e. for k _{> q,} the

dynamics

should be determinded

by

undulations of the

microscopic

oil

/water interfaces,

_or

by

the fluctuations within the

channels,

which _are

ignored

in _our derivation.

Our

predictions (37)

and

(38)

with

(39)

and

(40)

for the correlation function _{can now} be

compared

with the result obtained from the

time-dependent Ginzburg-Landau theory.

First,

note that the

inequality (37) implies

that there should be _no oscillations for

sufficiently

small

k,

in agreement ^{with the result shown in}

figure

5c. To extract the relaxation time and the oscillation

frequency

from the

perturbation theory,

_we fit the inverse correlation function for small _uJ

(up

to the

peak position)

to a fourth-order

polynomial,

~~~(ki~J)

^~

"

/~0(~)

+ /~2(k)~J~ + /~4(~)~J~ +

°(~'~) (~~)

and then _use the

equivalent

of equations

(14)

and

(15)

to calculate _r and _v. We have

compared

the results in _a few _cases with the

explicit

Fourier

transform, G~~(k, t),

and have found

good

agreement. ^{The relaxation time} r and the oscillation

frequency

_{v as} functions of k and

TT

_are

shown in

figures

8 and 9. It _can be _seen that in the

regime

^k < q there is

semi-quantitative

agreement with

predictions (39)

and

(40).

The linear

dependence

of I

IT

_on

TT

and of v2 _on

0A 10.0

a) b)

Fig. 8. The relaxation time _T of the asymptotic form

(38)

of the intermediate scattering function

Gww(k,

t), _as obtained from pertubation theory.

(a) 1IT

_as

a function of rT for k ₌

q/2 (full line)

and k =

q/3 (dashed line). (b)

_{T as a} function of k for rT

= 0.2. The parameters in equations

(1), (2), (5),

(8) and (9) _are rw =1, rT ₌ 0.3, _{go =} 5, _c

= 1, _{r2 =} 1 and qf =19.97.

~2 ~2

~.°

~ l.5

",,

_1.5

1',,

1.0

0.9

",,

,, lA

',,~~

^0.5

O-B

""

1.3 O-O

O-O 0.5

~+~ ^{l 0 0.0 0.5 1.0}

~2

a) b)

Fig. 9. The oscillation frequency _v of the asymptotic form

(38)

of the intermediate scattering

function

Gww(k,

t), _as obtained from pertubation theory.

(a)

v~

as a function of

r(

_{for k}

=

q/2

(furl

line)

and for k

=

q/3 (dashed line). (b)

^v~ _{as a} function of k~ for rT

# 0.2. The other parameters _are the _{same as} in figure 8.

r[

is found _to be satisfied _very

nicely,

_see

figures

8a and 9a.

However,

there is _a considerable

k-dependence

of _{r, see}

figure 8b, although

from

(39)

it is

expected

to be _constant. In the

regime

k j~ q, where _we do not expect our arguments

leading

to equations

(39)

and

(40)

to

apply,

the behavior of _r and _{u as a} function k is indeed

qualitatively

different. The relaxation time is found _to have _a maximum at k ₌ q, while the oscillation

frequency

almost

vanishes,

see

figures

8b and 9b.

Finally,

for k > q, the relaxation time decreases

rapidly.

6.

Sponge phases.

It has been

pointed

out in references

[24,

4] ^{that in} sponge

phases

the order parameter ^{4l and} its fluctuations cannot be observed

directly,

since water _{on one} side of the membrane cannot be

distinguished

from water on the other side. The order parameter fluctuations _can been _seen

indirectly,

however, via the fluctuations of the

amphiphile density

~fi(r,

t).

Although

our model

(5)

does not contain _any

explicit amphiphile degrees

of

freedom,

_we

Gq~q~

,

I.°

k/q

₌ 0.5

' '

0.5 '_'

', ',

', '-,,_

~.~

~.~ ~.~ QJ/J~ ^~.~

~

Fig. lo. The dynamic amphiphile correlation function

G~~(k,w)

_{as a} function of the scaled fre- quency

w/rw,

for wavevector k

=

q/2.

The dashed line is the van-Hove result, the full line the result of the perturbation theory. The parameters _are the _{same as} in figure 4.

can nevertheless calculate _an

amphiphile

correlation

function,

if _{we assume} that most of the

amphiphile

is located at the

4l(r, t)

₌ 0 surfaces. Here, _{we assume} again that the

amphiphile

concentration

equilibrates instantaneously

with

relatively

slow

changes

of the

order-parameter

field

4l(r,t).

In this _case, the

amphiphile

correlation function is

given by Gfijm(r,t)

= No <

6(4l(r,t)) 6(4l(0, 0))

>

,

(42)

with _a normalization factor No, which is chosen such that limr-n~

limt-n~ Gfijm(r, t)

₌ I. The average can be calculated

easily

for Gaussian

fluctuations,

where _one finds

[35,

36]

Gfiim(r, t)

=

~° (< 4l(0,0)~ ^>~

<

4l(r, t)4l(0, 0)) >~]

^~~~~

(43)

For not too small _{r or} t, this result _can be

expanded

to

give

Gfiim(r, t) Gfiim(r

_{= cc, t =}

cc)

+~ <

4l(r, t)4l(0, 0)) ^>~ (44) Interestingly,

in the static _case, equation

(44)

is just ^{the result obtained from} the two-order-

parameter

model,

where the

amphiphile density

is included

explicitly

_[24]. We thus

identify

the

expression (44)

with the

amphiphile

correlation

function, G~~(r, t).

The results for the

arnphiphile

correlation

function,

calculated with the van-Hove and the

one-loop approximation

for the

order-parameter correlations, equations (21)

and

(22),

_are

shown in

figure

10 _{as a} function of the

frequency

_w for k

=

q/2.

It _can be _seen that the

scattering intensity

in film contrast does not have _a

peak

at finite _w, but

develops

_a shoulder at the _same value of _w, where the

order-parameter

correlation function has its

peak.

We want to

point

out that _{a very} similar behavior is found when the expression

(44)

is used to calculate the static film correlation function [2].

Only

when additional

coupling

terms in _a free _energy functional for the two order parameters 4l and

~fi are taken into account, a

peak

at finite wavevector k appears [25]. Thus, we expect ^that a

time-dependent Ginzburg-Landau

model for 4l and

~fi will show _a

peak

of

G~~(k, w)

at _a finite

frequency

_w.

By

_a numerical Fourier transformation of the data of

figure

10, we obtain the behavior of the intermediate

scattering

function in film contrast,

G~~(k, t),

_{as a} function of time t. The

result is shown in

figure

11 for the _same wavevector k

=

q/2

for which the

order-parameter

correlation function oscillates in time, _compare

figure

5b. We find

again

_an

oscillatory behavior,

G~+ ~~

°'~

k/q

= 0.5

o oo4

k/q

= 0.5

,

0.2

",,~~

~

0.002

~~

''''''''''''~

0.000

~~

o-o 5.0

t,r~

^{lo-O °.° 5.° ~°.°}

t.rq~

~~

b)

Fig. ii. The intermediate scattering function in film contrast, G~~

(k, t),

_{as a} function of the scaled

time trw, for _wavevector k

=

q/2. (a)

van-Hove result

(dashed line)

and the result of the perturbation theory

(full

line).

(b)

A magnification of

G~~(k, t) (perturbation theory)

on intermediate time scales.

The parameters _are the _{same as} in figure 4.

with _a characteristic

frequency

which is

roughly

^the _{same as} ^the characteristic

frequency

_u of

G~~(k, t),

but with _a

considerably

smaller relaxation time.

Note that

formally

there is _a strong

similarity

with the

spatial

behavior of the static _corre-

lation,

where _one finds with equations

(13)

and

(44) G~~(r)

~2

+~

~e~~~/~sin(qr)~

,

(45)

r

a function which shows the _same characteristics _{as a} function of _{r as}

G~~(k,t)

does _{as a} function oft- In

particular, G~~(k,t)

>

G~~(k,t

=

cc).

We want to

emphasize

that these results of the

time-dependent Ginzburg-Landau theory

_agree with the arguments

presented

in section 5. From these arguments, ^{it follows}

immediately

that

G~~(k, t)

should oscillate in the _same intervall of wavevectors _as the

order-parameter

correlation function itself.

Also,

the

oscillation

period

of the two correlation functions should be the _same.

7.

Summary

and conclusions.

We have studied the

dynamic

behavior of microemulsion and sponge

phases

in thermal

equilib-

rium. For _a

time-dependent Ginzburg-Landau model,

_we have

analysed

the

dynamic

structure factor

G(k,w),

both in bulk and in film contrast, ^which _can be

measured,

for

example, by

inelastic _neutron

scattering

experiments in microemulsions. Our calculation shows that for systems with _a

sufficiently

small

viscosity

_{J~, or} with _a

sufficiently large

domain size

2~/q

of coherent oil- and

water-domains, G~~(k,w)

for fixed wavenumbers k of order

q/2 develops

a

peak

_or ^shoulder at finite _w. This

peak implies

that the intermediate

scattering function, G~~(k,t),

oscillates in time. To understand this

surprising result,

_we have studied the flow

through

tubes in _a very

simple, cylindrical

geometry,

employing

the linearized Navier-Stokes

equation.

Under the

assumption

that the interfacial tension is

radius-dependent,

and vanishes

identically

for the

equilibrium

radius of the tube, _{we were} indeed able to

reproduce

the oscilla- tory behavior of

G~~(k, t).

Furthermore, _we found that the dependence of both the relaxation time and the oscillation

frequency

_on the

viscosity

in the two

approaches

_{agrees very}

well,

while the

dependence

_on the wavenumber k is

qualitatively

correct. Thus, _we believe that the

existence of

temporal

oscillations in microemulsions is well-established.

The behavior of the

arnphiphile

correlation

function, G~~(k, w),

is found _to be _very similar.

In the range of wavevectors, ^where

G~~(k,w)

has _a

peak

at finite

frequency

_Lao,

G~~(k,w) develops

_a

shoulder,

which appears also at

frequency

_Lao. The intermediate

scattering intensity

is found to oscillate in time. The characteristic time scale of these oscillations is the _same in film and in bulk contrast.

We want to

emphasize

^{that the} two

approaches

we have used should

apply

in two different limits. The

Ginzburg-Landau approach

is based _on the

assumption

that the order parameter fields _vary

slowly

in space and

time,

I.e. it

applies

to the weak

segregation

^{limit. The}

hydro- dynamic

flow

through

_a tube with hard

walls,

_on the other

hand,

^should describe the behavior of the strong

segregation

limit. Since the _same behavior is found in both

limits,

it should also be found in intermediate _cases.

The mechanism, which leads to the oscillations of the intermediate

scattering

function in bulk and film contrast, _can be summarized _as follows. In _a microemulsion _{or sponge}

phase,

there is _some

preferred

_average distance between the

arnphiphilic

membranes. If

by

thermal fluctuations this distance increases _or decreases in _some

region

of these

phases,

there is _a

force,

the interfacial

tension,

which drives the system back to the thermal _average. It is

negative (positive)

for

large (small)

distances between

membranes,

and thus tends to increase

(decrease)

the interfacial _area. A

restoring

force itself is not sufficient to

produce

oscillations. We also need _a moment of

inertia,

^{which drives} the system

through

the

point

where the force

vanishes;

in _our

model,

this term is

provided by hydrodynamic

momentum conservation. And _we need _a

sufficiently

small friction

coefficient;

in the microemulsion _{or sponge}

phase,

this _can be achieved

by

_a

sufficiently

small viscosity, or

by

_a

sufficiently large

diameter of the tubes,

through

which

the flow _moves back and forth.

Thus,

oscillations _can

only

be observed for

strongly

structured

and swollen microemulsion _or sponge

phases.

We _want to conclude with _a short dicussion of the

dynamic

behavior of the

amphiphile.

We have

already pointed

out that the

radius-dependence

of the interfacial tension is not due to _a

stretching

of the

amphiphile film,

since in _our model the

amphiphile always samples

its

equilibrium configurations.

Let _{us now} consider the

opposite

limit where the

dynamics

of the

amphiphile

is very slow. When the oil- _or water-concentration in _a

drop

of radius L »

~/q decreases,

the interfacial _area must increase. Since the

amphiphile dynamics

is

slow,

its concentration in the

drop

can be assumed to be

approximately

constant. This

implies

that the

amphiphile density

within the

monolayer decreases, leading

to an increase of the _{area per}

headgroup

and thus _{to a}

positive

interracial tension.

However,

this is the situation where _we

already

have _a

positive

tension in _our model

(1), (2), (5)

with _an

equilibrated amphiphile.

Thus,

if the

amphiphile dynamics

is slow, the

stretching

of the

monolayer

enhances the radius

dependence

of the interfacial tension, and

thereby

the oscillations.

Acknowledgments.

Helpful

discussions with R. Hausmann, D. M. Kroll and H.

Wagner

_are

gratefully

^acknowl-

edged.

This work _was

supported

in part

by

the Deutsche

Forschungsgemeinschaft through

Sonderforschungsbereich

266.