A Field-Theoretical Model of a Microemulsion

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A Field-Theoretical Model of a Microemulsion

Uwe Heißner

To cite this version:

Uwe Heißner. A Field-Theoretical Model of a Microemulsion. Journal de Physique II, EDP Sciences,

1995, 5 (10), pp.1433-1440. �10.1051/jp2:1995192�. �jpa-00248244�

Classification Physics Abstracts

68.10Cr 82.70Dd 68.35Rh

A Field-Theoretical Model of

_a

Microemulsion

Uwe

HeifIner(*)

Universitat Basel Institut fur Physik, CH-4056 Base], Switzerland

(Received

19 June 1995, received in final form and accepted 25 August

1995)

Abstract. A field-theoretical model of _a microemulsion is proposed taking both, geomet-

rical and material, degrees of freedom into account.

Using

methods of conformal field theory and string theory, the most relevant part of the partition function is calculated. It is shown that smooth stable surfaces exist for _a physically meaningful _range of control parameters. ^In the topological sector of spheres _{we prove} explicitly the existence of stable surfaces with finite

extension. This agrees with the well-established droplet-like phase of microemulsions. Expecta-

tion values of geometrical quantities such _as the _area of _a surface _as well _as its fluctuations _can be obtained. In the topological sector of tori, _on the other hand, such stable configurations do

not exist. This work may be considered _{as an} application of methods of conformal field theory

and string theory in another branch of physics chemical physics and colloid science.

A mirroemulsion consists of _two

non-mixing incompressible

fluids A and B such _{as water} and oil, to which _a third surface-active substance T, the surfactant, is added. Without _any

surfactant,

the system is _an emulsion characterized

by

_a

simple phase

behavior. In the

phase diagram only

_one

thermodynamically

stable _state exists in which the fluids _are

separated by

a flat interface. This situation will

change drastically

^if we add _more than _a critical amount of surfactant to the emulsion.

Depending

_on the temperature, ^the amount of T and the

proportion ^of

A/B,

_a

large variety

of different

phases

_are known to be

thermodynamically

stable- for

example droplet-like

and bicontinuous _structures

exist,

_as well _as

liquid-crystal-

like _one~

[1-4].

These states may be classified

by topological

and

geometrical

properties of the

interface,

which has _no

boundary

and which is in

general multiply

connected

[5-8].

The

microscopic

structure of the surfactant T _is

responsible

^for the increase of the interface _area and its

bending.

Two

microscopic properties

of the surfactant molecule _are

important

here:

first,

_a T-molecule consists of _a small

hydrophobous

head and _a

long polymer-like hydrophilic

tail. Due to its

amphiphilic

nature the surfactant _can exist

only

in the interface between A and B. Second, for entropic reasons the

polymer-like tailgroup

induces _a cone-like shape of the

T-molecule, leading

to an intrinsic spontaneous curvature of the interface [9].

I;i the present paper, we construct _a

simple

field-theoretical model of _a microemulsion and

/)present

_{address. Institut fur} Theoretische Physik, Freie Umversitit Berlin, Arnimalee 14, 14195 Berlin-Dahlem, Germany

© Les Editions de Physique 1995

1434 JOURNAL DE PHYSIQUE II N°10

compute the

partition

function

using

quantum field-theoretical methods. We consider the surfactant _as

being continuously

distributed _over the

interface,

_a

description

which is valid _at

length

scales

larger

than _an atomic scale. Based _on

experimental

observations _we introduce the

assumptions:

_cx) the surfactant exists

only

at the interfaces M between A and

B,

and is

described

by

_a scalar matter field

#

_{on a} two-dimensional differentiable manifold

M; fl)

M is _a Riemannian manifold without

boundary,

with

arbitrary topology

and curvature;

~)

_we

consider

configurations

where component A is enclosed

by

the

interface,

and _assume A to be

incompressible,

such that the enclosed volume is

constrained; b)

the _energy of the

model, containing material, topological

and

geometrical degrees

of

freedom,

is bounded from

below;

e)

_a

configuration

of the system is

fully

described

by

the

geometrical

and

topological

structure of the A-B-interface and the distribution of surfactant _on this

interface; ()

in the absense of

surfactant,

the

ground

state of the system ^is a flat interface.

In the

following

we propose an

expression

for the energy ^H which

implements

the

require-

ments

(a) ((),

which has the form of _a modified

string

^model [10]. ^Then we evaluate the

grand-canonical partition

function

Z(~J,T,Ap) depending

_on the chemical

potential

_~J of the

surfactant,

the temperature ^{T and the}

"pressure-difference" Ap,

the

multiplier correspond- ing

to the fixed enclosed volume constraint

(~),

_across the interface. We _extract

general

informations about the

stability

of

topologically

and

geometrically

distinct

phases,

the focus

being

_on ^the

investigation

of the

topological

sectors of

spheres

and tori. For the

evaluation,

the

partition

function is

represented

_{as a} Euclidian

path integral.

Due to the formal _resem-

blance to _a

string

model it is

possible

to

apply

mathematical

techniques developed

in

string theory

and conformal field

theory [11-13].

Three theoretical

approaches

have been used _so far _to describe the

properties

^of _a microemulsion:

I)

the

'microscopic' approach

starts from the molecules and their interactions and is formulated

(after

_{a coarse}

graining procedure)

in

terms of lattice models

11,14,15]. 2)

The

'geometrical'

or 'interfacial'

approach

starts from

the observation of _an interface between oil and water, where the

properties

of the surfactant

(matter)

_are

expressed

in terms of

geometrical coupling

constants

(surface

_tension,

bending rigidity, ...)

_[7,

9,16-18]. 3)

The

'mesoscopical' approach

starts from _a

phenomenological

order parameter equation

(Landau-Ginzburg theory)

and

provides

_{a more}

microscopical

derivation of the mentioned

geometrical coupling

constants

[19-22].

Our work _can be considered _{as a}

simple

_attempt of _a consistent and

conceptional

clear unification of the

approaches (2)

and

(3), including

material _as well _as

geometrical degrees

of freedom. It _was

essentially

motivated

by

a paper of Kleinert [23] ^{and Leibler} [22]. ^The

implementation

of methods of conformal field

theory

enables _us to obtain

general

informations _on the conditions under which _a well defined

partition

function

(freedom

from

complex

conformal

weights

of the renormalized field operator

products)

exists. In this _sense, the considered model represents ^the

simplest

tractable and

non-trivial toy

model,

which contains the

possibility

of _a well defined

partition

function in _a certain _range of parameters. The

presentation

of the actual calculations has been restricted to a minimum; for _a detailed _account the reader is referred _to [24].

Assume that the microemulsion has total volume

V,

and that there exist N disconnected compartments or

"particles"

of fluid A covered

by

_an

arbitrary

two-dimensional surface of

T,

which _are immersed into the other fluid B and

neglect

the interactions between the

"particles".

The

"N-particle"

_energy is

N

y~N

£~n _(~)

,

n=1

~

,

~geo

^~

~top

~

~mat

,

where with the n-th

particle

there is associated _an energy H"

consisting

of _a

geometrical,

a

topological

and _a material contribution. In the absense of

surfactant, HQ~

^and _HQ~~ are

sufficient to describe the state of the system ^determined

essentially by

_a surface with minimal

area between the fluids. Therefore it is not necessary ^to include _{curvature terms} such _{as so-} called Helfrich terms [9] in the _energy of the _pure water-oil system.

Explicitly,

it is assumed that the

geometrical

and

topological

contributions have the form

2

~

ro

/ _{d~~ ~jxn(x)1}

~~~~

~ ~"

~ ~

(~jjR[Xn(T)I

~j~

=

~) ~~d

^{~ ~ "}

here _{g is} the determinant of the metric gab

[Xn]

₌

daXn(x)dbXn(x),

_a, b

= 1, 2, induced

by

_a 3- dimensional

embedding Xn(x)

₌

Xfe~,

_{p =} 1, 2,

3,

_n

=

I,...,

N of _a two-dimensional manifold M _:

(xi, x2)

in _a 3-dimensional Euclidian _space

R(

with _a Cartesian frame

(e[,..., e()

_[25].

The first term is

proportional

to the _area of the interface of the system.

ro

> 0 is the bare surface tension between A and B. The second term is _a

topological

invariant of

M, given by

the Euler index

Xn(Mn)

=

) / _d~xfiR[Xn]

=

2(1- In)

where in is the number of

M

handles of Mm and

R[Xn]

is the Gaussian _curvature of the n-th

particle [18].

The

experimental

determination _{of Ko} which describes

only topological changes

_appears to be difficult.

As the material part ^of the model _{energy we} introduce the

following

expression:

~zati~"1

_"

/~~ ~~~WQ( )9~~i~"1°a<i~"i°b<i~"i

⁺

)<~i~ni ⁽³⁾

+

ao<o

+

aR<iXni

₊

@ _{(RiXni Rs) (RiXni Rs)}

the

scalar,

dimensionless field

#

describes deviations of the

density

of surfactant from _{a con-}

stant

density

_go

Q(x)

_{= go} +

#(x)

=

(#o

+

#(x))/Ao

_(AD

being

_a constant with dimension of

an

area).

In agreement with

requirement ((),

_HQ~~ is

equal

to _zero if the

density

of surfactant

vanishes,

and the system is

governed by

the

geometrical

and

topological

contributions

only.

In order to obtain _a

theory

which is invariant under

reparametrisations

_{x -} x'

=

f(x)

where

f(x)

is _a

diffeomorphism

of

M,

it is necessary ^to express the energy in terms of _{an energy}

density

with respect to an area measure

d~x@.

The first two terms of HQ~~

correspond

to _a

quadratic

Gaussian

theory

of

density

fluctuations.

Physically,

these contributions modelise _an

elasticity

of the

amphiphilic

matter. In this sense,

bj~

^{is the static}

compressibility

_{at constant curvature}

and _aj is the coeflicent of the

corresponding quadratic

wavenumber

dependence.

Both material constants are measurable in

principle by scattering light

_{on a} two-dimensional surfactant

plane.

The last part of the material energy reflects the cone-like structure of the surfactant molecules

leading

to the

preference

of curved arrangements. The

quantity (R[Xn] R~)

_measures the

deviation of the Gaussian curvature R from _a constant

positive

curvature

Rs, corresponding

to the spontaneous curvature of _a

densely packed

arrangement ^of

hypothetical rigid

surfactant molecules

(~).

For _reasons of technical

simplicity (after indroducing

an

auxiliary

metric _we get a very

simple expression

for the energy of the

X-field)

_we

prefer

for the

description

of

this material property ^{the Gaussian} curvature instead of the _{mean curvature.} In the _context of _a

description

of fluid

bilayer

membranes _an

explicit coupling

between

density

and _mean

(~) In general, the spontaneous curvature is _a function of the temperature, because of its entropic origin Additionally it is not necessary always positive. But _we

neglect

this temperature dependence

and _assume Rs > 0

1436 JOURNAL DE PHYSIQUE II N°10

curvature _was

proposed by

Seifert and

Langer

[26]. ^{Safran et al. also stressed the}

importance

of the influence of

density

and curvature fluctuations [21].

Formally,

_{HQ~~ can} be considered

as a

power-series

in

density

fluctuations

#

and curvature fluctuations

(R Rs),

where the

corresponding

^real

expansion

coefficent _aR controls the relative

strength

of the

coupling

be-

tween these fluctuations. The

quadratic dependence

on

(R Rs)

_guarantees that the _energy

is bounded from below. As will be shown

later,

the

positive

constant

bR,

with dimension of _an area, scales the size of the smooth

interface(~).

Now _we turn to the evaluation of the

grand-canonical partition function,

which contains

I)

_a summation

(TrN)

over all

configurations

with _a fixed number of

particles

N and

2)

_a

summation _over all

configurations

^{with different N:}

Z(p, T, Ap)

₌

£ _~TrN

_exp

-fl 7i~ £ _(Apvn

₊

_pnj)

,

i~ (4)

N.

~i ^~

with the three-dimensional enclosed volume:

Vn #

d~TE~~EpugXfdaX$dbXf

6 M

and the total amount of surfactant _{on one}

particle:

RI

=

/ d~Tfi©gixni

M

Taking

the _trace

(TrN

in

equation (4)

includes summation _over all

topologically, geometrically

and

materially

different

configurations:

TrN

~4

£t~p ^f (§)j f _D#,

where

Df

is the _measure of the

diffeomorphism

of the manifold Mm. The difference of the pressure ^at the interface of A and B with volume Vn is denoted

by Ap,

and p is the chemical

potential

of the surfactant.

Microcanonically,

all enclosed fluid A is conserved _as well _as the total amount of T. Since

we have assumed that the

particles

do not interact,

7i~

=

£$~~

H" the

partition

function factorizes:

Z ₌ exp

VZ(p, T, Ap)

,

Z(l~, T, AP)

_" Tt eXP

l~flH(P,

_T>AP)1

,

(5j

H(/t, T, &p)

_#

Htop [xrel]

+ Hgeo

[xrel]

+ Hmat

[xrel,

_4]

~APvlxrel] £ d~T4t ₍₄₀

₊

_41xrel])

0

M

The _appearance of the volume V is _an immediate consequence of the

integration

of _a constant

zero-mode Xo X

= Xo + Xr~i

corresponding

to translation invariance in R~.

(~) In this context _{we use} the term 'smooth interfaces' for such kinds ofinterfaces, which _are not 'spiky',

where _{we mean} by 'spiky surfaces' such ones, which have many spikes corresponding to large local

geometrical fluctuations. At the end of spikes large scalar curvature R is localized. This, in the fields of random surfaces, string theory and membranes, well known phenomena, also called the problem of _a

'branched polymer'

[27-30],

which is accompanied by the _appearance of _a complex conformal weight of the physical

(renormalized)

_area, is circumvent by adding

(at least)

_a R~ _{term to} the _energy [31]. ^Such

a term smooth out surfaces, _suppress wild fluctuations of R and affects essential the short distance property ^{of the surface More} qualitatively, _we call

a surface smooth, if the 'mean Gaussian curvature' is small R(t~ «

(p~,

where _we have defined the 'mean Gaussian curvature' by the _average of

R~(x)

over an area of

an extension of the correlation-length

((

=

aj/bj,

R(t~ ₌

f~

~~

d~x/jR~(x)

o

Although

_we consider _a

relatively simple expression

of the _energy of the

surfactant,

the

computation

of the

one-particle partition

function

Z(T,

_p,

Ap)

is nontrivial since it contains

a

complicated path integral

_over

topologically

different classes of closed surfaces with _{a non-}

polynomial

Hamiltonian with respect to X. In order to calculate Z _{we use} methods of string

theory

and conformal field

theory.

In this

approach rigorous

and

general

results _are obtained

which _are inaccessible

by ordinary perturbational

methods. In _a first step, the

non-polynomial

character of the _energy with respect to the variables X is removed.

Following Polyakov [10,32]

this is done in the

following

_way:

replace

the induced metric

gab[X] by

_{a new}

auxiliary

metric

hab(x),

substitute _a delta function b

(gab[X] hab(x)) _[Dhab)

into the

path-integral,

and inte- grate _over hab. The delta function is

represented

_{as a} functional

Laplace

transformation _{over a} field

l~~(x),

which _can be

performed

without restrictive

approximations

_[32]. As _a result _one

obtains _an additional term, the so-called

Polyakov-term Hp~i

~4 ci

f d~xvlh~~daXdbX

where

ci is _a constant

expectation

value of the field l~~. In order to

integrate

_over the metric

hab,

it is necessary ^to

parametrisize

_a

general

metric deformation

bhab(x). Writing

the metric _as

hab(x)

=

e~'(~)jab

_with

an

appropriate background

metric

jab

_and

_following

the

approach

of Alvarez [33] or Friedan

[13],

it is

possible

to

decompose

the

integration

_over hab into _an inte-

gration

over the scalar conformal field _a and the infinitisimal generators of the

diffeomorphism

of

M(~).

The _area

dependence

of the

integrand

_can be found

by representing

the

one-particle partition

function Z _{as an}

integral

_over the _area A of the

particle:

Z(p, T, Ap)

= const.

/~

_{dA b}

^A / d~xvi

(: ^e~'(~) :) Zxj«~(A), (6)

o ten

with

Zxj«~(A) containing

still

integrations

_over

X,#,a

and _{a new} scalar field ~fi(x) ^which

corresponds

to the representation of the

quadratic coupling

^{of surfactant}

density

and curvature:

exp

-fl~~~° / _{d~xvi (R(x) Rs)~}

r~

(7)

~°~ M

[~~fij(h)

~XP

~ / _d~TV~

~~

~

^{~fi~(T) l~fi(T)}

(~(T)

~S

R 0

where _means normal

ordering,

and the index

(h)

at the _measure denotes the

reparametri-

sation invariant definition.

The _measure of the conformal field

[Da(x)]~j,~

^{is not invariant under a shift of} a, because

it

depends implicitly

_{on a}

itself,

due to the

covinant

formulation of the

theory.

The

partition function,

defined with repect to the metric

hab,

^should not

depend

_on ^the

decomposition

into e~' _and

jab.

_{This makes it}

_possible

_{to obtain}

a proper measure, which is invariant under a

shift of _a

[29, 34].

^For

achieving this,

_{we use}

again

_some

implementations

of conformal field

theory

[11]. ^The consideration of the

corresponding

conformal field

theory,

which contains the desired Jacobian for the transformation to _a proper measure in _a

general form,

enables _us in _a first step

by setting

^{the total central}

charge equal

_zero to determine this Jacobian. As _a second consequence of the

implicit

determination of the transformation to _a proper measure we

have to

require

that all renormalized

products

of fields

including

^{the conformal field} a behave

(~) It should be noted, that after the decomposition of the metric all deformations

(not

_only

small)

of the total metric

hab(x)

in the corresponding topological class of the backgroundmetric

jab(x)

are

contained in the conformal field

a(x).

Friedan [13] ^has emphasized that if 6M

= %, then

jab(x)

_may

be chosen to be _a constant curvature metric. Therefore _our decomposition implies _no approximation

at all.

1438 JOURNAL DE PHYSIQUE II N°10

like

(I, I)-primary

fields. Such _a

requirement

leads to _a strong ^limitation on the renormalized

products

^{and determines their} coefficents of wavefunction renormalisation.

Using again

meth- ods of conformal field

theory (operator-product expansion)

enables _us to determine the desired

renormalisized field

products.

Contrary

to usual

string theory,

which for I < D < 25

(D

is the dimension of the

embedding space)

is

accompanied by

_a

complex

renormalisation of the

physical

_area,

/d~xvi _(: ^e~'(4 _:)

=

/

_d~

Vi _e~~'"I

ren

with _~ E C [29], we obtain because of the

bR40(R Rs)~-term

_on scales A <

flbR4o

at _a trivial renormalisation of the

physical

_{area: ~}

= l

(for

D

=

3).

On scales A »

flbR4o,

_on

the other

hand,

the

bR40(R Rs)~-term

losses its

smoothing influence,

and _we get the usual result of

string theory.

In this _sense

flbR4o corresponds

to _an intrinsic version of the well- known

persistence length

_11,8,_9] This is _one essential result of _our

investigation;

it is similar to the result of [31] on R~

gravity.

This is _a very

general

and

rigorous

statement, because it

originates

from

general assumptions

about the associated conformal field

theory

in the context of

determining

the proper a measure and is

independent

of the

topology

of M.

This result enables _us to conclude that _our model contains the

possibility

of the existence of smooth interfaces _on scales A _<

flbR4o,

which _are unstable _on scales A _»

flbR4o.

On this

larger

scales «-fluctuations dominate and

destroy

the

possibility

of smooth interfaces

[28].

Now _{we are} in _a

position

to compute ^the

remaining integrals

in

Zxj«~(A).

Because of the

«-independence

of the

X-part

of the system, we obtain

Zxj«~(A)

=

Zx(A)Zj,~(A).

Without any further

specifications

of the

topology

and the

background

metric it is

possible

to

integrate

out the constant zero-mode of the conformal field _ao

(a

= ao +

&(x)),

the additional field

~fi and the material field

# (all

with respect to the fluctuation of the conformal field

&). ^The determination of the _{mean square} fluctuation of the material field has been done up to O

([aj /(b~flbR4o))~),

^which requires ^that the effective scale of

##-fluctuations

should be smaller than the scale of the

underlying

geometry

(aj /b~

<

flbR4o).

The

remaining integration

over the fluctuations of the conformal field

&(x)

can be

approximated by

_an

A-independent

function.

In order to

perform

the

X-integration

it is necessary because of the volume term to fix the

topology

and the associated

background

metric. Therefore _we restrict ourselves _{to a} consider- ation of the first two

topological

sectors (1 =

0,1),

the

topological sphere

and the torus. As

a

ground

state of the field

X,

the

embedding

in

R~,

_we obtain _a

representation

of _a

sphere

and _a torus,

respectively.

For

getting stability against

XX-fluctuations _we must demand

Ap

_> 0

(internal

_pressure > external

pressure). Estimating

the _constant

expectation

value of the

multiplier

_ci

originating

from the approximation of the

integration

of the

multiplier

l~~

(responsible

for the constraint gab "

hab)

and

looking

for _a minimum of the exponent in

Zxj«~(A)

r~

e~~(~l

_{with respect to}

_A,

we conclude that tori (I =

I)

_are

always unstable(~).

Only spherical particles

are stable in _a certain range of control parameters

(p, T, Ap).

As final result _we

give

the partition function for

topological spheres

and tori:

Z =

exp[VZ(p,T,Ap)] ₍₈₎

(~) ^{Because of the}

(1-1)~

A~^~ dependence of

F(A),

the essential A~^~ term vanishes and _we get instead

of _a minimum _a maximum of F(A). But in _a further _{paper we} will present _a detailed discussion of

this situation

(including

also

a discussion _on the integration _over the moduli

coordinate),

which _seems to be sinfilar to the situation of _a critical droplet _{size m an} overheated fluid, and _a compansion with

the work of Seifert [35] on vesicles of toroidal topology

Ii

= exp V

£ _(e~~"~

^~~~~~~

22(1-ij(P> T, AP)

,

i=o

Kren/4

=

fl Ro

+

ao<o

+

ii ^aRRs

+

ill

j2 j2

~12 )~ ^~~~

^{~~ ~~}

~

~~

ii

'

22,0

_" Zsphere,torus ~

/

_d2i

e ~~~~~~.°)

,

~ (9)

F(A)(2,o)

" ai~'°~

(T, Ap)Ai

₊

_a[~'°~(T)A

In

I

+ a(~'°~

(T, p)A

A

+a(~'°~

In

I

+ a)~'°~ ^In

~~~ fi)

+

a(~'°~(T)A~~

A 8 ,

with the _area cutoff

(),

which _can be identified with the inverse reference

density

_go, the _cor- relation

length

of the _matter field fluctuations

((

=

aj/bj,

the constant _area

I corresponding

to the fixed

backgroundmetric jab

_{and the}

exponential integral

of the first kind Ei

[36].

^The coeflicents of equation

(8),

where the

symbols

_a)~~ and a)°~

correspond

to _a

sphere

and _a torus,

respectively,

_are known functions of the control parameters

(T,

_p,

Ap)

as well _as the model _para- meters

((~, ~f,

bR,

ro,

_{aR, ao,}

Rs).

But

only

in the _case of

spheres

and for a)~~ ^> (a)~~)~~"~ ^with (a(~~)~~~~ ⁼

~a(~~j~~~

(a(~~)

~~, i~~

=

(()~~~

+

) _())~~~

there exists _a minimum of

F(A)(2j

with respect to A. Thus in agreement ^{with the} strong requirement A _<

flbR4o

the relation ai~~ = (ai~~)~~'~

provides

_an

implicit equation

for _a maximal temperature T~, _a chemical _po-

tential _{~t~ or a} pressure difference

Apr

consistent with the existence of _a minimum of

F(A)j2)

for T > T~, _p > p~ or

Ap

>

Ap~

_no finite A-B-interface exist. In the _range of subcritical parameters, expectation ^{,,alues of} A, A~ _can be obtained

by corresponding

derivatives with respect to the bare surface tension.

In conclusion we find that _on scales A _<

flbR4o

closed smooth interfaces exist contrary to a

Polyakov-model.

On scales A >

flbR4o geometrical

fluctuations _are dominant and

destroy

the relative smoothness of the

interface,

and _we arrive at _a very

spiky regime

of surfaces

l'branched polymers').

It is

explicitly

shown that

simple

tori _are

unstable,

whereas

spherical particles

_are stable in _a certain range of control parameters ^of

(p, T, Ap).

Consistent with the

requirement

A _<

flbR4o

_we can

identify

this parameter _range ^with a

generic droplet phase

of

a microemulsion. Outside of this range we arrive at other

phases

of _a

microemulsion,

which

are characterized

by

_a

diverging

_mean interface _area of the system.

Acknowledgments

am very

gratful

to H.

Thomas,

H.-F.

Eicke,

H. Kleinert and S.

Weigert

^for useful discussions and _many detailed hints. This work _was

supported by

the Swiss-National-Foundation.

1440 JOURNAL DE PHYSIQUE II N°10

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