Fluctuations of compressible droplets: the role of surface excitations

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Fluctuations of compressible droplets: the role of surface excitations

A. Zvelindovsky, A. Zatovsky

To cite this version:

A. Zvelindovsky, A. Zatovsky. Fluctuations of compressible droplets: the role of surface excitations.

Journal de Physique II, EDP Sciences, 1994, 4 (4), pp.613-626. �10.1051/jp2:1994151�. �jpa-00247986�

Classification

Physic-s

Abstracts

68. IO 82.70

Fluctuations of compressible droplets

_:

the role of surface excitations

A. V.

Zvelindovsky

and A. V.

Zatovsky

Department of Theoretical

Physics

and Research Institute for

Physica,

I. I. Mechnikov Odessa State University, Petra Velikogo 2, Odessa 270100, Ukraine

(Received 9 May J993, iei,i.ved 6 Janttaiy J994, ar.cepted J4 Janttaiy 1994)

Abstract. The thermal fluctuations of nearly spherical micelles _are considered. The

viscosity

and compressibility of the surfactant layer and of the bulk fluids _are taken into account as well _as the bending _energy. The dynamical equations for surface and bulk variables _are the linearized

hydrodynamic equations

with random _sources of stresses.

Employing

the

fluctuation-dissipation

theorem the spectral densities of fluctuations related with surface excitations _were derived. The behaviour of the

dynamical

structure factor S(k, _w is discussed for various

droplet

parameters.

1. Introduction.

The existence of so-called

amphiphilic

molecules is the

origin

of local constraints which lead to spontaneous

aggregation

into microemulsions. The microemulsion is _a

dispersion

of

micelles which _are

liquid droplets

stabilised

by

a

monolayer

^of tensioactive molecules in aqueous or

apolar

solvent. The nature and

dynamics

of interface

play

a great role for _a wide

variety

of

physical

and chemical

phenomena

in microemulsions

[1-3].

The surface is _an

important

factor in many industrial

applications

such _as

oil-displacement

mechanisms in enhanced oil recovery, coalescence of fluid

droplets, rheology

^of

emulsions,

etc.

[4, 5].

Fluctuations of micelles _are

intensively

studied

by experimental techniques,

for

instance, by spin-echo

neutron

scattering [6]

_or video

microscopy [7].

Already Rayleigh

and Lamb have studied

theoretically

the behaviour of

fluctuating droplets

in _terms of the

phenomenological

_parameters such _as surface tension and densities of

liquids

[8].

In order _to consider the

dissipation

effects the oscillations of viscous

droplets

immersed in viscous fluid have been

investigated by

a number of authors

[9-12]. Sparling

and Sedlak have

developed

the viscous

droplet

model

taking

also into _account the

viscosity

^and

compressibility

of surface film

[13].

A

description

of microemulsion

droplets requires

additional

physical

concepts ^{because the} coefficient of the micelle surface tension is _{zero or}

practically

_zero, and the

dependence

of the surface energy on the curvature should be taken into consideration. Such

approach

has been put

forward

by

^Helfrich

[14],

then it has been

developed

and used in many works (see, ^for

example, [7, 15-20]).

In Helfrich's

theory

^{the surface} energy

density

is

given by

the

expression

g=«-p()+ _+j ()+

^'

j~+fi, _(i)

~ ~ ~

where _K and k _are the elastic _constants for

cylindrical bending

and saddle

bending,

respectively, (I/Rj

₊

I/Rj)/2

and

I/Rj R~

are the _mean and Gaussian _curvature,

a determines the usual surface tension of the

planar

interface,

p/2

_K is so-called spontaneous curvature. The last _term of (I is in

general

omitted _as the

integral

of Gaussian _{curvature over} the surface is

topological

invariant. (It is 4 ark for

spheres,

^ideal or

deformed.)

The _most

general development

of Helfrich's

approach

has been carried _out in reference

[17]

where the surfactant

redistribution mode _was studied in addition to the ~eformations of micelle form. However, the authors of the papers

[7, 14-19] disregarded

the

viscosity

of surface

layer.

Moreover, in all the above papers the bulk fluids _were assumed to be

incompressible

and

only

the fluctuations of

quantities

that _are

closely

associated with the deviations of the

droplet shape

from

spherical

have been studied. For this _reason the

approximate expression

for the spectrum ^{of the}

density fluctuations,

which is

solely

determined

by

the

shape

fluctuations and the redistribution of surfactant _at the micelle

surface,

has been obtained. However, _{as was}

noticed

long

_ago in reference

[21],

the

neglect

of the bulk

compressibility

is

inappropriate approximation

in _case when the

scattering

with small energy transfers is examined.

Recently

we have

already

studied the

hydrodynamic

fluctuations of _a

compressible droplet

immersed in

a

compressible liquid [22-24].

In these

articles,

however, the interface is described

by only

one

phenomenological

parameter, the constant surface tension

a

[22, 23]

_or effective surface tension

at

[24],

and

only

the radial components ^{of random forces} on the

droplet

surface _are

considered. This _means that surfactant fluctuations _are

negligible

and

only

the

shape

fluctuations _occur.

In this paper ^we present ^the most

general approach

to the

study

of the thermal fluctuations of

nearly spherical

micelles. We

regard

the micelles _{as an} ideal _gas,

assuming

rather dilute

micelle

dispersions.

So, _we can focus _our attention _on the

study

of the

dynamic properties

of _a separate ^{micelle. The}

viscosity

and

compressibility

of the surface film and the bulk fluids inside and outside the

droplet

are taken into account as well _as the curvature energy ) and the redistribution of the surfactant _on the interface.

Low-frequency

collective excitations of the microemulsions _are described in the framework of _two- and three-dimensional

hydrodynamics.

The

spectral

densities of the correlation functions for

hydrodynamic

variables _are found based

on the

fluctuation-dissipation

theorem. The obtained results _can also be

helpful

for the

investigation

of the excitations of vesicles and

biological

cell solutions.

2.

Theory.

2. I THE PHENOMENOLOGICAL MODEL. The micelles _are modelled

by

fluid

drops

immersed

in _a fluid with different

properties

and coated with _an

infinitely

thin film. The

fluctuating

velocities _v and densities of the interior and exterior fluids _are solutions _to the linearized

continuity equation

and the Navier-Stokes

equation containing

random stresses «),~

~

3p +pdivv

=o,

3Uf _~

~ at

~~

~~

~~ _~ ~°~~ ~°~ _{~ ~}

3 ^{~ ~} ~

~~

~~~ _{~ ~}

~""~"

~~~

Here, _~ and ( are the shear and bulk viscosities,

8p

is the deviation from the

equilibrium

_mass

density

_{p, c} is the sound

velocity.

For the sake of

simplicity

we

neglect

the fluctuations of

temperature. ^{In what}

follows, quantities referring

to the interior

(exterior)

fluids will be labelled with

1(2).

Using spherical polar

coordinates jr, o, _~g

),

with

origin

at the _centre of the

droplet,

_{we can}

describe _a

slightly

deformed

spherical

micelle

by

the series of

spherical

^harmonics

Rio,

_~,

t)

~

Ro(

^{i +}

z ui,,,(t) yi~,io, ~)) ⁱ

» 2,

13)

tn,

where

Ro

^{is the}

equilibrium

radius of the

droplet,

^I _{w mm}

I,_and

the maximum mode

number will be of order

i~~~ arRo/d,

where d 5

h

_is

a

typical

molecular diameter

[13].

The redistribution of the surfactant _on the interface _can be described

analogously

to

(3)

n~

lo,

_~g, t) ₌ n~ ₊ 3n~

=

n~ ₊

~

_vt~

it) Yt~

lo, _~g

,

I

m

,

(4)

t~~

where 3n~ is a small deviation of surfactant

density

from the

equilibrium

value n~. The surface tension

depends

_on the surfactant concentration

(4) [25],

and _we define the elastic modulus of

interface film _as B

=

n' _da

(n')/3n~

_~ 0. The

dependence

_on n~ for the other coefficients of

expansion

can be

neglected [17].

We assume that the

infinitely

thin surface

layer

behaves _as

a

compressible

Newtonian surface fluid described

by

two viscosities _~~ and

(~.

Besides, the surface

layer

is chosen _to be insoluble in the

surrounding liquids.

Then the

boundary

conditions

derived from the surface _momentum balance _are

P~

l~

+

Ii

_~~~~

+

~~

_"in

~in ~P' ~P2

+

~r/~~ ~«~'+

+

f~

_{~, rot~} rot v~ ₊ ~~ ⁺

(,

_V~ div v~

,

(5)

3u(

_~~

Psi

=

I"& ni)l~' I«& ni)l~

+ m V< 3n~ ₊

f<

an

~, rot, rot v~ +

~

~~ +

(,

V, ^div v',

(6)

3

providing

that there is _no flow of _mass

through

the surface film. Here the indices _r and t denote the radial and

tangential

_components of the _vectors in the

spherical

coordinate system, n is the

normal to the interface (here and

below,

_n is directed outside the

sphere),

_{p~ v~ is} the surface momentum

density,

_pi

=

3pj

+ po + 2

a/Ro,

_pi

=

3pj

_{+ po, po is} the pressure at r = oJ, and

3pj

₌

c) 3p,,

f is the random surface forces. The components ^{of the viscous} stress tensor

«,',

_are

[26]

~,(> ^v,~

~~

~

'~'~Jr

⁺

l'j

71~

div

v~,

~~'~ ~()~

^{~ l}

~~~

' ' '~J

J; _; v« ⁺ 71,

v,v~~,

since the bulk fluids _are

compressible.

The second term of

(5)

has the form of the

fluctuating

part ^of

Laplace

_pressure. Parameter _at is determined

by

the coefficient~ of

(1)

2

p Kf(f

₊ i

"~ " "

j

^~

_Rj

^'

~~j

it must be

positive

lest the micelle is instable with respect to small

perturbations.

The surface

density

can be

expressed

in _terms of the velocities

by using

the

surface-continuity equation

~~~

+ ii'div v'

=

o.

(9)

3t

'

In addition to (5),

(6), (9),

the

velocity

must be continuous _across the interface and

equal

to the

velocity

of surfactant motion

v, = v~ =

v', v)

₌ ^I..

(lo)

In the linear

approach

all the

boundary

^conditions

(5), (6), (9), (10)

_may be taken _on the surface of the undeformed

sphere

_(>.

=

R~

).

Is is convenient to rewrite all

equations

in _a Fourier transform

retaining

the _same notation for

the Fourier components ^{of the}

quantities

and

replacing

the differentiation in time

by

iw.

Owing

to the

linearity

of the

hydrodynamic

and

boundary equations

the total solution of the

boundary-value problem

_can be

represented

as a sum of _two contributions. The first _one is the solution of Dirichlet

problem

for the

velocity

which satisfies the

inhomogeneous equations (2).

The second _one is determined

by

the

inhomogeneous boundary

^conditions

(51, (6), (9), (10)

and

by equations (2)

without random bulk forces

V,,«

),,, Thus, the total

spectral

densities of the

hydrodynamic

correlation functions

split

into _two

independent

contributions

corresponding

to the random volume stresses

«),,

in

(2)

and _to the random surface forces t. The first _{term was}

earlier found in

[27]

where _we also took into _account fluctuations of the temperature. ^It turns out that the first contribution to the

dynamic

structure factor is small

compared

to the second

one when the energy transfer is small and the sizes of micelles _are of the order of

10" -10~1 _[?3].

_{For this}

reason we shall

investigate only

the

hydrodynamic

fluctuations

excited

by

the surface random _sources.

2.2 THE _{SPECTRAL DENSITIES OF THE} FLucTuATioNs. The solutions of the Navier-Stokes

equation

in the

spherical

coordinate system can be found in two different ways

using

the

following expansions

vj(r, w)= jj [P~(9,

_~

)Fj~(>., w)+B~(9, ~)Gj~(>.,

wl+

+

C~ (9,

_~

Hi

~(>., w

II

=

I,

m,

II)

or

vi jr, ml

=

~j [C(~

_{(w )}

_{L~ (kj}

i <.) +

C(~

_{(w )}

_{N~ (k,}

~ >.) +

C'f~

(w

M~ (kj

~

>.)]

,

(12)

where P, B and C _are

spherical

vector functions, L, N, M _are basic functions for the vector Helmholtz

equations. They

_are

[28]

~fn< _- ~fn<~^~~ ~ ~

in<

,,fi

_~~fn<

~fn %( i~~l

'~i

Here,

jj(kj r)

_are the

spherical

Bessel functions. From

equations (2)

without

inhomogeneous

terms one can

easily

find the relations of the coefficients of

expansions F~, G~,

H~

to

Cl _(below,

_{the indices of the}

_spherical

_harmonics,

=

I,

_{ni, will be}

_omitted)

:

G

,i

lcLJfl[,.>~

⁺

^{cN Ji(L}

i

;i

Jfl(~£,>~

II ^its)

H' ~'"ill

₊ I

Clji(k'£

_"J'

where

j)

(xl

=

dji(,r)/3,<,

^and we have used the notation k

i

'ii

^"~

k>i

i _Ii ~°~l

^~

vii < + 7/ /P '

6)

Substituting

the

spherical

^{Hankel functions} of the first kind,

h)"

for the

spherical

Bessel functions

ji

and the indices I for ? we obtain the exterior solutions.

They correspond

to

diverging longitudinal

and _transverse

damped

waves. Note that the

quantities k,~

and k,i have _two

complex conjugate

values. To exclude the

exponential divergence

of the solutions

as _{>- -} m we use the values which have

positive imaginary

parts.

From the

boundary

conditions

(5), (6), (9), (10)

and the solutions

(15)

_we obtain the

inhomogeneous algebraic equations

for the

integration

_constants

C(.).~

The

equations determining

the coefficients

C(~

_are the

simplest, they

are

independent

from the other _ones.

For

example,

Cl

=

At

h~ (17)

Al

=

~~ (-iwp,Ro+ 7~j(dlJ',J-

II

7~~(,K(j'~)-1)+ 7~,~~~/

^~~

.

(18)

jf("i ),/f(f

₊ I1 °

The coefficients

C(.)

_are determined

by

the

following

set of six

equations

©j(I Ii

₊

₁₎

2

3C(,i~)i

+

©j iii

₊

_(.K~y~)

=

iwRo

_vi~,,

©jii(I+1)-2tl(.ij))+©ji(I+ _I)(d(yj)-

_I)=

-iwRovi~,,

/~(Q(>.,)+/~(iji+

_I)=

_-iwRoui~,, ()JCj>.,)+C(I(I+ _{I)= -iwRoui~,,}

llfn<

i"f(f

₊ 2

ii

_p~

W

~Rli

-Cil~1iY'+4dii12fif +1)1+ 1,~~~~i

_'~

ii ~<>))~

+

+©ji(I+1)

(27~j(tl(v,)- I)+h _ii(I+

I)-dly,)- ljl+

Ro

+©jy~~~yj+4Jc(x~)~2i(I

₊

iij~©j I(I+1)2 y~~(Jc~y~)~ i)+

P, w ~

R(

₄ 2 p, w

~

R(

~~~ ~

iii+1)

~°~

3'~~~~~~~

^~"~~

i(I+1)

^~

+2 _7~>

(©ha(x>)- i)-©i[a(yj)+

i

-f(f

₊

ii+jvljj

-27~2 lcii~c(>,)')-CilJCiY2i+

^'

f(f+')+I>.ill =~~,,fi

where, in addition, we have introduced the notation

()

=

C)

_u,

_ix,

_if,<,

,

C)

=

C)

_{u, (v~}

_ill',

,

u, =

~~' hl'~

I

=

2,

Q(z

= z

I

In

jj

(=

,

3C(z

= =

I

In

hi'

_~(=

,

= =

jr,,

_y,

j

,

(201

Xi. ₂ ~

ki.

2jj

Ro,

_Y>,

2 ~ k>, _?i

Ro,

and

f~, g',

h~ _are the

amplitudes

of the random force f

expanded

in the series of the

spherical

vector functions P, B and C

(cf. (I

)). As above, the index _s denotes the

quantities

related to

the surface. The _roots of the determinant of the set of

ii?)

and

(19) correspond

to all kinds of the collective excitations of micelles in our model.

Equations

ii 9) _can be transformed into the

form that is

analogous

to

equation

(17)

/~)=A)Jf'+A)~g~, /~)=A)'f~+A)Yg~, j=1,2. (21)

The

explicit expressions

for

A§

(A

=

(I, m)

K

=

(Lf, Lg,

N

f, Ng

_can be

easily

found

by using

Kramer's rule. As

they

have _a very

bulky

form, we do _not write them here. Note, that

we shall set

A(fj~f

m 0 because u,

~ m 0.

To

employ

the

fluctuation-dissipation

theorem _{we must} know the

expression

for energy

dissipation

^{in the} system ^{under the action of the random} sources

[29].

In _{our case,} the

dissipation

is caused

by

^{both the bulk} viscous forces and the surface _ones. Therefore the random surface _{sources can} be

represented

_{as a sum} of _two terms, f

=

f~" ₊

f'~~''.

The first _one is determined

by

the difference of the bulk spontaneous stresses of

(2)

_at the

interface, f)~'~

_~

l«ll~~ "ll'~)

_n,

~. The second _one is the random surface _sources proper

f)~~'~

₌

V~«,'~.

Applying

Gauss theorem to the well-known

hydrodynamic

formula for the

dissipation

of energy

[26],

_we obtain the

expression

for the _power absorbed in the bulk fluid due _to random surface _sources,

Q~"~

=

ds

[vl'~ «,'j'

_{n~ v)~~«,?j~~}

_n~]

=

ds

v) f)~'~ (22)

where

ids

^{denotes the}

integration

_over the

droplet

surface S.

Analogously

_we find the energy

dissipated

in the two-dimensional surface fluid

~j~~

_

l~~~~~ ^v~v~

^-

j

ds(v vi«~i

~~

~~~~"~

~~~~

Averaging

the _sum of

(22)

^and

(23)

over time and

inserting ill) yield

Q=ReR[~(Fj fj~+Gjgj*+Hjhj*),

₍₂₄₎

and, i<ia

(10)

and

(15),

Q

₌ Re

RI I i©lA fl*

₊

_©lA fl~ +Ji~>),~ _{cf~ hj*), (25)}

~

where

f~.~

are the _new

conjugate generalized

forces for the

generalized

coordinates

/jL,N

~L

~

a*(<, fs

₊

,~ _g',

fN

~~~~

~

iii

₊ i

f~

₊

~ _(a*(y,)

+

i)g"

Application

of the

fluctuation-dissipation

theorem _to

equations (17),

(21) and

(25)

leads _to the

following spectral

densities of the

fluctuating velocity amplitudes

C

(

~

~)

=

~~

j

Re

~~~

~ ~ ^~

~~'~ '~ _~~~

,

(27)

W

grRo (I+Q*(y,))tl*(.<,)-I(I+I)

jj~~ _j~j ^k~T At~a*(.<i)/v~-Ajf

'~

"

~j~~(i

^+a

jj,))a*(.<,)-t(t+ ^il'

~~~~

~~~~~~~~" ~R(~~ji*b'i)h~'

~~~~

where

k~

is the Boltzmann constant, T is the absolute temperature ^{and the asterisk denotes}

complex conjugation.

Note that the

spectral

densities do not

depend

on m. The

spectral

densities for the

amplitudes

of the exterior

velocity

have _an

analogous

form, for

example,

The formulae obtained,

equations (27-30), essentially

differ from what _was found in _our

preceding

articles

[22-24],

where _we

neglected

the

tangential

components ^of spontaneous

surface _sources of stresses, g~ and h~. Those results

correspond

to the fluctuations of _a

compressible droplet

in the absence of surface concentration modes at the fluid-fluid interface is described

by

the surface tension alone. In this _case, the

spectral density

for the

amplitude C(,

_for

_example,

_is

_{(cf. (27))}

(/jL

^{£jL* ~B}

~~~ ^~A

^~AA'

_(3j)

' IA'

w

~j~2

*(~,

j

^'

0

where

Al

is determined

by

the

equations

derived from the proper

boundary

conditions

[2?, 24],

Finally,

the

spectral

densities (27-30) allow _{us to} write the Euler

velocity

correlation

function,

(v, (r, t)

_v,

(r', t'))

=

~j [C/~ _[~)

W~

(r) Wf(r'),

~fi

=

(L,

_N,

M) (32)

~

> w

~

and the

density

correlation function,

by

means of

continuity equation,

(3pi(r, t)3p>(r', t'))~

=

z ()Clim)~)~~'~

^~'

ji(k>i >.)J?(kn r')

_x

i~

~

^~

X

Yfn<(9.

~

Yfi(9', ~') (33)

To obtain such correlation functions for the outer fluid _{one must use} the suitable

spectral

densities of

amplitudes (e.g. (30))

and the Hankel functions instead of the Bessel _ones.

3. Results and discussion.

3.I SURFACE FLucTuATioNs OF MICELLES. In

spite

of cumbersome form of the final results

they

_can be

simplified considerably

in

limiting

cases when _some parameters are taken _to be

small. Let _us demonstrate this for two main surface variables. The

spectral density

of

fluctuations of the

displacement

of _a

droplet

^{surface is}

easily

recovered from

equation (19)

and

fluctuation-dissipation

theorem

Ii

_Mimi

_~l

~

#

~)~~~

[f~(f

⁺

1)~(1°iA l~l~

⁺

la(><)l~l1°iA _{l~l ~j} 134)

The character of the collective excitations

depends

on the

relationships

_among the dimensions of the

droplet,

the

penetration depth

of the viscous _wave and the acoustic

wavelength.

^Since a

typical

micelle size is of the order

10~-10~1,

one has

wRo/c

« I for

low-frequency

acoustic excitations. If the

penetration depth

of the shear _wave is

large compared

to the micelle

radius,

then

wR(/v

_« I, and the oscillations will be hindered _one is

dealing only

with

shape

distortions.

Suppose

likewise that the effects of

compressibility

_are

insignificant,

_so that the conditions 11-,, ~ «

ii,

~ « l hold.

Neglecting

all surface parameters except at and

keeping only

the

leading

terms in powers of

)y),

_we find from

(34)

and

(27-28)

"fm

~)

~

~

j j~(t~+~~~it

~'i _((~

'

~~~~

where

rj'~ai(I+2)ji+i)iii-1)(2i+ij'~'+'~2,

Roqipi

qj~2(12~i)y~j+(212+1)y~~, pj~(212+4i+3)y~,+2i(I+2)y~~, ~~~j

~2

r, =

m (pi iii ₊₁₎₍₂

I ₊

1)(7~,

+ 7~~) +

2(7~j

_{7~2) x}

p,f(I+2)qi p~i12-1)pi

jj

~

~Ji(2i+3)(2i+5)~~J~(2i-3)(2i-11

'

~~"~'~ ~~~~~

~ ~'

This result agrees with that obtained in reference

[19],

where _a

dispersion

relation _was studied for micelles and vesicles formed in

incompressible liquids.

Since

w r « I, the spectrum

(35)

is close _to Lorentzian, with _a width determined

by (36),

and

corresponds

to _an

overdamped

mode of surface oscillations. The

single-time

correlation function of the surface

displacements

is determined

by integrating (35)

over all

frequencies

kB T Tj ^>'2

(Uj~ (t Uj(

It

))

=

~

l

(37)

~0 "f(I

I

)(I

₊

₂₎

_To

As the square-root ^{factor in}

(37)

is close to

unity,

the main term of

(37)

coincides with the

expression

obtained

previously by

^{MiIner and Safran}

[16].

Then

compression

effects _are taken into account, the relaxation time r,

acquires

contributions linear in 7~/c" _p, so that _r,

- r, ⁺ y,

7~,/c(

_pi + y~

7~j/cj

_p~ with dimensionless coefficients yj, yj.

Analogously

to

(34)

we can write the correlator for surfactant concentration

(4)

:

ii

_yin

_i?i~

-

)i[j~j~~~

ii1

~)[~i~~ ~(

icL~ i~i~

⁺

ia~i) -112 ii ©i~ ~i~j

(38)

Let the

penetration depth

of the shear _wave be small,

ii'j ii

_w I, and the vibration

frequencies

at small viscosities _are

large compared

to those for the

capillary

waves. If ),<,

ii

_« I as before and

p~«R~pj,

_we obtain the

following simplification

of

(38) neglecting

surface film

viscosities

~~~

21~

m

~~ (jflJ

^kB

)~

~~

~~i~jf

⁺

j~

~

~~j

,

~~~~

7T 0 +

~ 2 0 +

2

~J~~<~l<'

The zero of the denominator of (39)

fully

coincides with solution of

dispersion

relation

obtained earlier in

[17, 19].

The static correlator has the

simple

form

k~

^T

(yin, Iii vi(, (t ))

= j,

(40)

RIB

which is the _{same as} is obtained from the Boltzmann distribution with the surface energy

[17].

3.2 THE DYNAMICAL STRUCTURE FACTOR. In this subsection _we shall

study

the behaviour

of the

dynamical

structure factor

(DSF)

that is of _concern for the fluctuation

theory.

^The

structure factor is determined

by

the

density-density

correlation function

S(k,

_w

=

idr dr'e'~~~~ ~'~(3p

(r,

t)

3p

jr', t')) (41)

~

where dr denotes

integration

_over volume. The structure

factor, S(k,

_w

),

_can be calculated

by

the way of reference

[30],

where we have examined the

density

fluctuations of

compressible liquid

within the

rigid spherical cavity.

Substitution of

(33

) ^reduces

equation (41)

to the

expression

that contains square of volume

integral. Using

the

decomposition

then leads, after summation over v, p, m and

integration

over 9, _~ ^and _>.

(0

_< r < R

~),

to the final result for the DSF of _a micelle

S,(k,

WI _cc ^{~' ° ~°}

~ ~ x

~~ ~

l ₊

W~l

i/~i)

x

~

(2 I + I

) j

_I

(kRo

^~

^~~~°

~

~

~'

^~

C

[

j

~)

~

(43)

kR )- ~-

i o

The _structure factor for the micelle environment

(Ro

_{w r <} m ) is found

analogously

51(k,

_{W OC} ^{~~ '~ ~}

, ,

X

~~

^~

l ₊

Wv~jj/Ci)~

x

~j ^{(2 I}

^{+ I}

^j

I

(kRo)

^~

^~~~°

^{~ ~~~}

~

Cl

I

~

(44)

I

(kRo)~ xi

W

Numerical

analysis

of the

density

fluctuations excited

by

^surface sources in the

surrounding liquid

shows that

they decay rapidly

when

moving

_away from the micelle.

Already

when the

distance is

comparable

with micelle radius

they

_are

practically

zero.

Consequently,

the

environment

overlap

contributions for different

droplets

_are

negligible.

For this _reason the _sum of the contributions

(43)

and

(44)

is _a

good approximation

for the _structure factor of dilute

dispersions

of micelles which do _not interact with each other. In the

hydrodynamic approach

the

expressions

derived for DSF,

(43)

and

(44),

are strict in contrast to the

simplified

results found

by

the other authors

[6,

13,

16]

who

approximated

the

density-density

correlator

by

the

correlation function of surface

displacements.

The

spectral

densities of fluctuations have _a

complicated dependence

_on

frequency owing

to the presence of the Bessel function combinations,

Figure

I shows the

frequency dependency

of the DSF of fluid

droplet

when

only

the normal components ^{of surface random forces} are taken

f

~ ~

i

~

i

d

_fi

~- ~-

~

3 ,'

/"

, ^/

~ ~ '

~§ ~§

~'i ~'i ^'~

to' to'

_to ° to ^" to ^"

to'to'to'to

^°

to'to'to'to

"to

"to

CJ, S~' _CJ, W

Fig.

I.

Fig.

2.

Fig. I. Dynam<cal structure factor of emulsion droplet in the absence of surface concentration modes for LRO 0.2 (Ro

4001)

L~ =

400 _m s~ ^' p~

i0'kg

_{m~ ~, (,}

=

0.75 ~, and

various solvent

viscosities _~~ [kg m~'

s~']

0.00i (I). 0.1(2), 0,2 (3), 0.3 (4), 0.4(5), The inner

liquid

parameters corre~pond to castor oil at standard conditions.

Fig,

2,

Dynamical

structure factor of microemulsion for _two values of surface elastic modulus B 5 _x 10~^~ i),

10~'(2)

_{[J m~ ~] and LRjj 0.3 jRo}

4001)

pi 900, _p~ 10~

kg

^m~' _{; ~} 1,

~~ 0.2

kg

m~ ^' s~ ^' (, 0.75 ~, c-j I 400, _c~

=

500 _m s~ ^' «I 10~'J _m~ ~

p, 4.5 _x 10~? kg m~~

into account

(see Eq. (31)).

The

viscosity

is found to have _a

profound

influence _on the DSF. As

the solvent

viscosity

increases both

frequency

of DSF maximum and DSF

height

decrease,

while the increase of

kRo

^leads to the increase of maximum of the spectrum ^without

frequency

shift

[23].

The

tangential

components ^{of the random surface forces}

engender

the surface concentration modes, which

modify considerably

the structure factor. As shown in

figure

2 the _new

step-like

low-frequency

contribution appears. It

depends essentially

on the

elasticity

^{of surface film, the}

low-frequency wing

of DSF increases with

decreasing elasticity

modulus. The _{curves are} the

result of numerical calculation of

equations (43), (44), (27), (30).

The

low-frequency

contribution of DSF increases with the solvent

viscosity (see Fig,

3 in contrast to the

peak

that decreases

similarly

to the _case in the absence of the concentration modes

(Fig.

I

),

whereas the

dependence

of the maximum

position

_on the

viscosity

of

liquid

differs

appreciably

from that shown in

figure

I. It should be observed that the spectrum ^{is sensitive also} to the bulk

viscosity

(

(Fig. 3,

dashed

line).

The spectrum parameters are

easily

measured _{as a} function of _wave

number k. Such functions for the

peak

and

low-frequency plateau heights

are

presented

in

figure

4, in which, besides, the great ^{effect of the}

liquid viscosity

_on the _wave number

dependencies

is shown.

Figure

5 illustrates

changing

DSF when

viscosity

of surface fluid increases. It is remarkable that the influence of the surface

viscosity

_on DSF differs

utterly

from the influence of the viscosities of the bulk fluids. In this _case all the contributions increase

or decrease

together.

^{As is}

expected,

the surface fluctuations _are

larger

when the

elasticity

and

io~ I

~-

i

z ~~3

3-

-44 Ul lo

,

,

0

,

~ ~ ^-i

kR~

Fig.

3.

Fig.

4.

Fig, 3. Dynamical structure factor of microemulsion for LR,, ^o.2 (Ro 400 h_{), B o.05 J m~} ~ and various solvent viscosities _~~ o.05 (1), o.2 (2)

[kg

m~ ^' s~ ^']. The rest parameters are the _{same as} those d~scr'bed f°r

f'gure

2. The dashed line differs from the solid line 2 only by ^{the value of the bulk} viscosity of inner liquid (j ₌ 1.5 _~

j.

Fig,

4. The heights ^{of the} peaks ^and plateaux ^of the

dynamical

structure factor of emulsion _{as a} function of _wave number k for two viscosities of droplet fluid. Lines and 3 correspond to the peak and plateau

heights respectively

when _~j 0.I kg m~ ^' s~

',

and lines 2 and 4 when _~ kg m~ ^' s~ ^' The _rest parameters are the _{same as} those for line in

figure

3.

~

~ i

~' f

~

~

l

~

£ ~

G~

S

~

/ '0'

'0~

GJ, s~' Q, s ~'

Fig. 5. Fig. 6.

Fig.

5.

-

p, ⁼ x lo~~ kg m~ ~ ~

j

₌ ^1, ~~ ^o.

I

_{kg '}

s~

'

lo~ ( _l), 2.5 x lo~ ^~

(2), (3) [kg '

].

Fig.

6.

Dynamical

structure factor of microemulsion for various values of _a~

ma -2

p/Ro

+

Ki(I

_{+ I}

)/R(

[J

m~~]

5 _x lo~~ _{(sol<d line), 5}

x lo~~ _{(dotted line),} 5 _x lo~~ (dashed line). The _rest parameters are the _{same as} those described for line in

figure

2.

viscosity

of the

droplet

film _are smaller. As it is evident from

equations (35-37)

the

height

and

width

_{of spectrum of the surface}

displacement

correlation function

depend

on the parameter

at and therefore _on the coefficients of Helfrich's

expansion (I).

For micelles the _term

a in _at is

anomalously

low

(of

the order

10~~

_J

m~~

or

smaller). Taking

into _account that

a

p

~/K

[17],

where

p/2

_K

=

Rj

^' is the spontaneous curvature

(for

vesicles

p

=

0),

_{one can}

easily

see that the _term Ki

ii

+ I

Rj

^~ in _at becomes

predominant

for small

droplet

radii. In

figure

6 _we show the influence of combined parameter _at, that contains Helfrich's coefficients a,

p,

_K, on the

shape

of the

dynamical

structure factor. Unlike correlator

(35)

the

dynamical

structure factor includes all excitation

modes,

_so this influence is

significant

in the low-

frequency

_range.

Our main attention has been

paid

to the construction of a correlation

theory

of thermal fluctuation of

nearly spherical

micelles _or vesicles modelled

by

the

compressible

viscous

droplets.

We have studied

consistently

the collective

hydrodynamical

excitations of interface

taking

into _account the thermal bulk flows inside and outside the

droplet

as well _as the

dynamical properties

^{of insoluble} surfactant. The

spectral

^{densities of}

fluctuating hydrodynam-

ical fields have _a

parametric dependence

on the

thermodynamical

and kinetic

properties

of all

three mediums,

and,

in this _sense,

they

are universal, The

poles

of the fluctuation spectra contain both bulk and surface modes, The

analysis

carried _out shows that in

limiting

_cases the

obtained correlators coincide with the known results of other authors

[16,

17,

19],

The main contribution to the

spectral

densities of

shape

fluctuations has been found when the excitations

decay strongly

due to

viscosity

effect

(Eq. (35)),

The surfactant concentration mode has _a

specific

fractional power

dependence

on the

frequency

and

viscosity

when the vibration

frequencies

are

larger

than those for the

capillary

_waves

(Eq. (39)).

In _our

approach

the spectra of _mass

density

and

velocity

fluctuations _are found

by using

a

common scheme. On this _reason the

dynamic

structure factor, determined

by

twofold Fourier

transform of

density-density

correlator, is

specific

and differs from the fluctuation spectra ^of other scalar densities of

thermodynamic

variables.

The numerical

analysis

of

presented expressions

showed that the surfactant

dynamics

leads

to the considerable contribution _to the structure factor in

low-frequency

_range. The time

correlation functions have

essentially nonexponential

form. These results _can be verified

by experiments

on elastic

scattering

of thermal neutrons and

dynamical

Brillouin

light scattering.

The small

angle

neutron

scattering

and the

dynamic light scattering experiments

on micellar solutions, microemulsions, small

proteins

and

large

colloid

particles

_are

already

known

[31- 33],

where the main attention _was focused _{on structure} aspects ^and concentration

dependence

of spectra. ^We propose the details of spectrum

shape

to be examined. The spectra _are

expected

to be sensitive _to the surface collective excitations _as it takes

place

for the flat

liquid-vapour

interface

[34].

Neutron

spin

echo

experiments [6]

and dielectric relaxation

[35, 36]

also show that

dynamic

structure factor

strongly depends

on interface fluctuation in microemulsions. In

article

[37]

the internal

viscosity

^{of micelle} was studied

by using

the measurement of

fluorescence

intensity.

^{Our results}

provide

the

possibility

of

extracting

the

complete

information about

viscosity dependence.

Note that in _our

approach

the solution of

problem

^of

diffusion motion inside confined

region

can be

easily expressed

in terms of

velocity

correlation functions obtained. This should be

important

in model studies of solute diffusion in cells _or vesicles

[38],

4,

Concluding

remarks.

So,

we have

investigated

the

hydrodynamical

fluctuations of fluid

droplets (micelles)

coated

with _a thin film and immersed in _a fluid medium. In the present theoretical

study

the

compressibility

and

viscosity

of the surface and bulk fluids have been

explicitly

taken into

account. The surface energy is considered _to

depend

_on curvature. The thermal fluctuations

excited

by

the surface random sources are

primary

for the micelles with diameter of order

10"-10~ 1.

_The

frequency

spectra of the fluctuations _are found _to

depend strongly

on the values of the

phenomenological

parameters.

The effect of

compressibility

is of considerable interest in various

applications.

For instance, the fluctuations of

density

determine the spectra ^{of inelastic}

scattering

from microemulsions.

The collective excitations of micelles have _an effect _on the cross-section of

scattering

with small energy transfer

(e.g.

slow _neutron

scattering).

We

hope

that the results

obtained,

in

particular

those

concerning

the _structure of the DSF line

shape,

^{will stimulate additional}

experimental

studies of the

dynamical

behaviour of micelle solutions.

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