Mean field theory of n-layer unbinding

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Mean field theory of n-layer unbinding

W. Helfrich

To cite this version:

W. Helfrich. Mean field theory of n-layer unbinding. Journal de Physique II, EDP Sciences, 1993, 3

(3), pp.385-393. �10.1051/jp2:1993100�. �jpa-00247840�

Classification

Physics

Abstracts

64.60 68.10

Mean field theory ^of n-layer unbinding

W. Helfrich

Institut fur Theoretische

Physik,

Freie Universit&t Berlin, Amimallee14, 1000 Berlin 33,

Germany

(Received 3J August J992,

accepted

in

final form

J6November J992)

Abstract. -A two term

Ginzburg-Landau

Hamiltonian for the free _energy

density

of

n-layer

systems ^is in~roduced. _{It is used to derive}

a distribution function for such systems ^and to obtain

scaling

laws

involving

the number of components. ^{We discuss the}

validity

of this mean-field

theory

for stretched

polymers

in two-dimensional space and for fluid membranes.

1. Introduction.

There _are several theoretical treatments of the

unbinding

transition of _a

pair

of fluid membranes.

Fornlally,

this

problem

is

equivalent

to that of _a

single

membrane

unbinding

from

a flat Wall. The earliest

theory,

_a renornlalization group

calculation,

_was

presented by Lipowsky

and Leibler

ill. Utilizing

the scale invariance of thernlal membrane

undulations, they

renornlalized the

potential

which a membrane feels in front of _a wall

by integrating

out

step by

step ^the

short-wavelength

undulations.

Assuming

_a short range

repulsion

and the Van der Waals attraction of thin

plates, they

calculated the critical value of the Halnaker constant, I.e. the

strength

of Van der Waals interaction, at which

unbinding

takes

place.

Later _on,

Lipowsky [2] pointed

out that the renornlalization group flow

equations

for _a fluid membrane in three dimensions _are identical _to those for _a stretched

polymer

in two dimensions

exposed

to the _same one-dimensional interaction

potential.

There is _a difference in the

derivations,

the

scaling being anisotropic

in the _case of the 2d

polymer (or

linear

interface).

The

propagation

of the stretched

polymer

in _a one-dimensional

potential

is known to

obey

a

Schr6dinger

type

equation [3, 4]. Accordingly,

the method of

finding

the

ground

state wave function of the

Schrodinger equation

and

squaring

it _to obtain the

polymer

distribution function _seems transferable from the

polymer

to the membrane. In _a third

approach Lipowsky

and Zielinska

[5]

did _a Monte Carlo

study

of membrane

pair unbinding, expressing

the interaction of the flat membranes

by

_{a square} well

potential.

All these methods

predicted

the _same critical behavior for the _mean membrane

spacing

s as a function of the

strength

u

(~ 0, arbitrary units)

of _an attractive

potential responsible

^for

386 JOURNAL DE PHYSIQUE II N° 3

adhesion. The _common result is

s -v

(u~ u)~~ _(l)

with #f ₌ I, where u~ is the critical

strength.

The critical exponent #f does not

depend

_on the details of the

potential. However, equation (I )

was shown

[1, 4]

to be valid

only

if the

potential

energy of interaction between the flat membranes

drops

off faster at

large spacings

than the

squared

inverse

spacing.

This condition is

generally

satisfied

by electrically

neutral fluid membranes.

Subsequently,

_we

proposed

a « reflection _» model _{as a}

quasi microscopic description

of

pair unbinding [6].

Central _to it is the

assumption

that the membranes fornl

sharp

bends where

they

hit each other

(or

the

single

membrane hits the

wall).

These bends _are

thought

to be confined _to

the _narrow

region

of

strongly negative potential

energy of membrane interaction. At the critical

point,

the

bending

_energy ^of reflection is _on average

compensated by

the

potential

_energy of the well. Adhesion sets in _as the well

deepens further, obeying again

the

scaling

law

(I)

with

#f =

I. The reflection model suggests that the

unbinding

transition of membranes is modified

by logarithmic

corrections of the scale invariance of membrane

shape

fluctuations. In

particular,

the

unbinding

transition appears in it _to be of

weakly

first order

[6].

Very recently,

Cook-Roder and

Lipowsky[7] published

a Monte Carlo

study

of the

unbinding

of three

parallel undulating

membranes.

Finding

#f ₌ 0.8 for three

equal

membra- nes,

they

invoked the necklace model for _a bunch of three linear interfaces

[8]

to

explain

#f _~ l. In this

model,

the internlediate interface acts like _a

repulsive potential

between the _two outer _ones.

Varying

with the

squared

inverse

spacing

of the latter, the

potential

is

marginal

in the

language

of renornlalization group

theory.

^The exponent #f as obtained

by

the

Schr6dinger equation

method

depends

_on the

strength

of the

potential

in this

special

_case.

However,

the necklace model is not

quite appropriate

of the situation considered since attraction acts in it

only

where all three and not

just

two of the interfaces _come

together.

In the

following,

_{we propose} a

Ginzburg-Landau type theory

for the

unbinding

transition of

n-layer

systems,

designed

for both two-dimensional

polymers

and fluid membranes. The

density

defined _as the inverse _mean membrane

(or polymer) spacing

_represents the order parameter. ^In a

multilayer

system

composed

of _a finite number _n of

layers,

_a

gradient

ternl

enters even

though

fluctuations of the order parameter are

disregarded.

In the

expansion

of the

free _energy

density

_we omit all powers of the order

parameter beyond

the

second,

thus

limiting

ourselves _to situations where the _mean

layer spacing

is much

larger

than the width of the attractive internlembrane

potential.

From the

Ginzburg-Landau

Halmiltonian _we calculate _a

layer density

distribution function. Based _on

it,

_we find

equation (I)

with #f ₌ I to

apply

also to

n-layer

_systems and derive additional

scaling

laws that involve _n. The

validity

of the _mean

field

theory

is then

briefly

^{discussed in} tennis of the

Schr6dinger equation

^method.

Finally, arguments

are

presented

Which

suggest

^{that the} mean field

theory

cannot be

expected

to hold

exactly

_even for _n

= 2 in the _case of membranes.

After

finishing

the present

theory

we became _aware of MiIner and Roux's mean-field

theory

of membrane

unbinding

in infinite

multilayer

systems ^{which has}

just

been

published [9].

Although quite

different from _ours, it also

predicts

_#f

= I for the critical

exponent

^of

unbinding.

We will _come back to it at the

beginning

of the discussion.

2. Mean field

theory

and

density

distribution function.

In order to describe the

unbinding

of _n

polymers

or membranes _we introduce the

following

free energy of

multilayer

fornlation _per unit area or

volume, respectively,

f=AP~+Dl Ill

l~ ⁽²⁾

The

mutually impenetrable

components are taken to be

infinitely

extended and

basically

nornlal to the _z axis. Their _mean number

density

is

expressed by

a continuous function p = p

(z) despite

their discreteness. It is

advantageous

to _use the modified

density

P

(Z)

= I Is

(z )

,

(3)

where

s(z)

is the _mean

spacing

of the

components

as a function of _{z, so} that the membrane

thickness,

_an irrelevant

quantity,

does _{not enter} the

equations.

The first ternl of

(2)

contains the control

parameter

^{A of the}

unbinding

transition which is

proportional

to u u~

occurring

ⁱⁿ

equation (I)

for

pair unbinding.

It is

negative

for bound _states and _{zero at} the critical

point.

Vanishing

A _means that there is neither attraction _nor

repulsion

of the components ⁱⁿ a

homogeneous multilayer

_system. The factor _p^~ _comes from the linear ternl in _{a mean} field like

expansion

in powers of _p of the total interaction energy ^between

adjacent polymers

or

membranes per unit

length

_{or area,}

respectively.

There is _{no zero} order contribution to this energy if the

potential

_energy of interaction goes to _zero for

large spacings.

Contributions of

higher

than linear order in _{p are}

neglected

because _{we are} interested in low

densities,

I-e- in

mean

spacings

much

larger

than the

region

where the interaction

potential

is

significant.

The

validity

of the _one-ternl Landau

expansion

will be examined below in tennis of

microscopic

models for

polymers

and membranes. We will also estimate A in the _case of

polymers

for _a

particular potential.

The

p~

_{ternl of}

_{(2) alone,}

_{without the} intervention of _a

gradient

ternl,

would result in _an

abrupt unbinding transition,

I.e. _a

jump

from s=0 to s= oo as

A

changes

from

negative

to

positive.

The

gradient

ternl in

(2)

is unusual in that

(dp/dz)~

_{is divided}

_by

p. This _can be understood for membranes in the

following

_way. The

bending

_energy of _a

single

membrane

and, correspondingly,

the free energy of

purely undulatory

membrane interaction _are known to be scale invariant. The main effect of _a

gradient dp/dz

is _a distortion of the membrane

configurations

_as

compared

to _a uniform distribution. It

seems therefore reasonable _to expect ^the

gradient

term to be also scale

invariant,

which

requires

the form

(I/p (dp/dz)2.

As _a tentative value of the coefficient D for membranes _we may take

from

a for _the

_{energy nsity of purely undulatory}

_in

_a

nifornl ultilayer

system,

f =

D/s~

[10].

Here k

is

Boltzmann's

onstant, T and _K the

rigidity

of

the

single ^embrane.

scaling

arguments

can be

used

in

the case

of

polymers

to

_justify

the (I/p

) _(dp/d2)~

term.

owever, scaling

is

isotropic in the ^olymer

case,

requiring the engths parallel

to

the polymers to be multiplied

gnifying

of s.

~

with _an unknown numerical

factor,

where _« is the line tension of the

polymers.

^A

gradient

ternl of the fornl

(I/p ) (Vp

)~ in the free _energy bulk

density

of unstretched

polymers

has been introduced _some time ago

by

de Gennes

[11].

The distribution function _p

(z)

is obtained

by minimizing

the

integral

F

=

I[Ap~+D (~~ )~+Mpj

^dz.

⁽⁶⁾

P dz

Without the last ternl in the

integrand,

F would be the free _energy of the whole

n-layer system,

JOURNAL DE PHYSIQUE11 T 3, N'3, MARCH iW3 is

388 JOURNAL DE PHYSIQUE II N° 3

per unit

length

for

polymers

or per unit _area for membranes. The last ternl with _a

Lagrange multiplier

M has _to be added _to take _account of the constraint

p

(z) dp

= n

(7)

where _n is the fixed number of

polymers

_or membranes

composing

the system. ^{The Euler}

equation resulting

from 8F

= 0 is

easily

found to be

k~D~~~2p _{~dz ~2D~'}

_~~~

With the reduced

quantities

fit i12

~ 2 D ^~ ~~~

and,

for A _~

0,

~ =

~~

p

(lo)

this transfornls into

~~'l=-~2+ (~'l _)~+1~. ^(ll)

df~ 21~ df

The solution of the second-order differential

equation

is

if the two

boundary

conditions _are chosen such _{as to} obtain _a

symmetric

function

~ = ~

(c )

that tends to _zero for

c

_- ± oo. The universal distribution function

(12)

^is

plotted

in

figure

I.

We are now in _a

position

to derive

scaling

laws for

weakly

bound

n-layer systems.

^The

number of

components

^should

obey

n~p(0)Az (13)

where Az is _{a measure} for the width of the distribution. Because of

(9)

and

(lo)

the factors _on the

fight satisfy

p(0)~ ~ ₍₁₄₎

and

D l12

AZ~

,

(15)

M

so that

n

(MD

)~'~

(16)

Ti

i

o 2 3 ,

Fig. I. Plot of the function ~(c) _as

given by

equation (12).

The last

equation

fumishes the

dependence

of the

Lagrange multiplier

M _{on n.}

Eliminating

M between

(15)

and

(16) yields

A2

~

~

(17)

Accordingly,

the total width of the

multilayer

system ^is

inversely proportional

to both the number of

components

^{and the control}

para1neterA.

^A

corollary

of

(17)

is

p(0)~

~2-D.

(18)

The

scaling

law

(17)

leads back to

(I)

with #f ₌ I ifs is

replaced by

Az/n. This

implies

^{that the}

critical

exponent

^of

n-layer unbinding equals

the

exponent

^{obtained earlier}

by

various other methods for the

special

case n =

2. Of _{course, near} the critical

point

of

unbinding

at

A

= 0 the control

parameter

^will vary

linearly

with any

changes

not

only

of the interaction

potential,

but also of the line tension of the

polymers

or the

bending rigidity

of the membranes, and of temperature.

3. Discussion.

Let _us first compare our mean field

theory

to that of MiIner and Roux

[9].

These authors started like _us from _a two-ternl

expression

for the free energy of

binding

_per unit

volume,

but

;hey

considered _a unifornl

multilayer

_system. One ternl is

positive

and represents

purely undulatory interaction,

thus

varying

_as

p~.

The other stands for the

potential

_energy and varies _as

pi _applying

_to

a uniform distribution of each membrane between its nearest

neighbors

and interaction

potentials

much _{more narrow} than the _mean

spacing.

This ternl, whose

prefactor

is the control parameter ^{of the}

phase transition,

is similar to the first term in _our

equation (2).

However,

in _our

theory

the

p~

tern1combines

undulatory

and

potential

interaction.

(Purely

undulatory

interaction would be associated with _a nonuniforn1distribution and the

potential

390 JOURNAL DE PHYSIQUE II N° 3

energy would

always

be

negative

for _a square well

potential

in front of _a

wall.)

As _a

result, undulatory

interaction alone does not occur but

only

the

gradient

ternl related to it which varies in effect with _p instead of _p~. This

change

of powers, ^while not

changing

the critical

exponent

#f, results in the

collapse

of infinite

multilayer

_systems which _our model

predicts

for all bound states

regardless

of the

strength

of

binding.

Next,

_we check the

validity

of

equation (2)

which underlies _{our mean} field

theory

for the critical behavior of

n-layer

systems. ^{The fornlula}

Ap~

for the free energy

density

of _a

homogeneous multilayer _system

_can be made

plausible,

in the _case of

polymers, by considering

the

simple

_case of _a

single polymer

between

parallel (linear)

walls. Jn order _to

keep things simple

_{we assume as}

usually (

_q~

~)

«

l,

where _q~ is the

angle

_{a monomer} makes with the

preferred

direction. This _can

always

be achieved

by choosing

_a sufficient line tension

«. A

single polymer

between walls does _not

undergo

the

abrupt collapse

which _occurs in _a

homogeneous multilayer

_system when the control

parameter

^becomes

negative.

Its

Schrbdinger type equation

takes the form

[3, 4]

-~~~~~~~~~'+U(z)9'=EP ₍₁₉₎

2 "

d2~

where

U(z)

is the

potential

_energy of interaction with the walls per unit

length

and

E the

corresponding ground

state

eigenenergy. (For (19)

to be

applicable

the interaction energy

per monomer has to less than

kn.

The interaction is

conveniently represented by

_square well

potentials

of

depth

U~0 and width

f~

ⁱⁿ front of the walls. There is _a critical

depth

U~ ^{for which}

P(z)

is _constant outside the wells. In _a

vicinity

of the critical

point

one has

to

E

"

z ^{(U Uc (20)}

where

I

_{» 2}

f~

is the

spacing

of the walls.

(The

result is

easily

obtained

by

a

perturbation

calculation _or

by

a

matching procedure

for _a P that is still

nearly

unifornl outside the

wells).

Since E is _a free energy, we may infer from

(20)

^that the free energy of interaction per unit

length

of _one

polymer

in _a

multilayer

system ^of

(homogeneous) density

_p ^{should indeed} be

Ap

₌

fo(u uc)

_p

(21)

in accordance with

(2).

Trying

to go a step

further,

we write down _a

Schr6dinger

_type

equation

for the entire

n-layer

system.

Considering only

the

(n-I)-dimensional subspace

of

configurations

with

zi + + z~ =

0,

I.e.

disregarding displacements

of the center of

gravity,

the

equation

_may be written _as

~~~~~

A~_1

^4l +

U~yi,

,

y~_1)

4l

= E4l.

(22)

Here yi,...,y~_i are some Cartesian coordinates

(of

the _same scale _as

zi,...,z~),

A~ _i is the

Laplacian,

and

W~yi,

..., y~ _i

)

is the wave

function,

all of them in the

subspace.

U is the

potential

_energy of the mutual interaction of the

polymers

and the

eigenenergy

E is the

ground

state energy, ^both

refemrg

to _a unit

length

of the

system comprising

all

n elements.

Equation (22)

describes the

propagation

of _n

mutually impenetrable polymers

with E

being

the free energy of their interaction. ln the

Schr6dinger equation analogy

the

polymers

are

represented by

n

equal

but

distinguishable particles

in one-dimensional space which

obey

zi w z~ « w z~. The

potential

_energy of

interaction, U,

_may

again

be

expressed by

_narrow

square well

potentials acting

between nearest

neighbors.

The wells _are in front of the contact

(hyper-) planes

_z;

= z;~i with I

=

I,..,

_n I which behave _as

rigid

walls. The effective width of the

potential

well is

only Io/2~~,

so that U~ ^is now

larger

than in the _case of _a

single polymer interacting

with _an actual wall. For two

polymers

and the obvious choice y =

2~ ~'~(x~ xi

)

_one has

W

~

e~"~~ (23)

with

~

2

E~

«

«~ =

~

(24) (kT)

and

~r2 (U U~ f

~~

16 U~ ^~

~ ~~~~

These

simple relationships

lead to the well-known result Az

~

(U~

^U

)~ (26)

Unfortunately,

the situation becomes very

complex

for _n

~ 2. The _n I

degrees

of freedom

are

necessarily coupled

since

subsequent

contact

planes

_are not

orthogonal

but make

angles

^of

60°. At the _same

time,

_a

separation

of 4l into _a radial and _an

angular

function is

impossible

for

two reasons.

First,

the

potential wells,

their width

being independent

of

position, overlap

when

three _{or more} elements _{are near} each other.

Second,

_at

large spacings

the

potential

wells,

acting

like

&functions,

deternline

(a 4lla~ )/W

on the contact

planes,

where _~ is the coordinate

(same

scale _as zi,

.,

z~) along

the nornlal to the

respective

contact

plane.

The constancy ^of

(awla~ )/W

at

large spacings

is

incompatible

with

asymptotic separability.

If

separation

were

possible,

the radial function R

(r)

or, more

precisely,

the modified radial function

fi X

(r)

=

R(ryr

^~

(27)

could be treated in the _{same manner as} the wave function of _a

single polymer feeling

_a

I/r~ potential. (Both

the

separation

and the substitution of the modified function _can

give

rise to an effective

potential

of this

type.)

The

I/r~ potential

in combination with _a square well

potential

has been

shown,

for the

single polymer

in front of _a

wall,

_to result in _a critical

exponent

#f that varies with its

strength [4]. Although

this

theory

does not

apply,

first

numerical results for three

polymers by

Netz and

Lipowsky

indicate critical behavior with

#f ₌ 0.9 for _n

= 3

[12].

As

yet

^{there is} no

analytical proof

that _mean field

theory

is incorrect.

How similar _are the

unbinding

transitions of two-dimensional

polymers

under _a line tension and fluid membranes with _a

bending rigidity

? The

identity

of the renornlalization group flow

equations

for the _two

systems,

which holds for any number of components, presupposes that the thernlal undulations of the membranes _are scale invariant. This

assumption

is not

entirely

correct since the thermal undulations

give rise,

for

instance,

to an

absorption

of membrane _area and _a decrease of the effective

bending rigidity,

the correction terms

depending logarithmically

on the base _area of the membrane

[13].

Both effects do _not exist in the

polymer analog.

However, ⁱⁿ the _case of

fairly

stiff membranes

(K

= lo

kT)

such _as

lipid bilayers they

amount to no more than _a few

percent

^{of the} area and of the

bending rigidity, respectively,

in the

absence of undulations. At their critical

points

^of

unbinding

both

polymers

and membranes

392 JOURNAL DE PHYSIQUE II N° 3

appear ^to be reflected from each other at _no cost of free energy, ^{while the mutual contacts}

give

rise to _a

negative

free energy in the _case of adhesion. The concept ^{of reflection is in accordance} with the

Ap~

_{term of the}

Ginzburg-Landau expansion (2).

It is at this

point

that _a

possibly

crucial difference emerges between

polymers

and membranes. The _mean square

angle

which _a

polymer

makes with its

preferred

direction does not

depend

_on its

length.

In contrast, the _mean square

angle

_(q~

_~)

which _a membrane makes with its basal

plane

increases

logarithmically

^with the size of the

piece.

With

periodic boundary

conditions _one calculates for _a square of size L~

[13]

(

_q~

_~)

=

~~

ln

~

8 "K

(28)

a

where _a is _a molecular

length.

In the

special

case of

purely undulatory interaction,

(q~~)

can also be estimated for

multilayer

_systems. In those systems ^{L in}

(28)

has to be

replaced by

_an effective

length

K ⁱ¹²

L

~ s

(29)

which is the lateral correlation

length

of membrane

spacing.

The numerical factor in

(29)

is

near

unity

but difficult _to calculate

exactly.

Substitution of

I

_{for L in}

₍₃₀₎

_{results in}

(q~~)

₌ ^{~~ ln}

~~~

(30)

8 ""

kTa~

It follows from

equation (28)

that (q~ ~) ^{is much smaller than}

unity

for

lipid

membranes unless

their size is astronomical.

Highly

flexible

amphiphilic

membranes

(K= kT)

_may

crumple

when

single

but,

according

to

(30), (q~~)

ml is

easily preserved

in

multilayer

_systems.

Nevertheless, (q~

~)

can increase

by

a factor of two and _{more as} the size _or the

spacing

is made

larger

and

larger.

Let _us recall that _near the critical

point

of

unbinding

_we

expect

the membranes to be reflected or,

rather,

deflected from each other. The deflection is

thought

to

consist,

at most

places,

in

sharp

bends of the

colliding

membranes in the _narrow

region

where the interaction

potential

is

negative.

A small deflection

angle

_{may seem} more favorable _as it lowers the

bending

_energy

per unit

length along

^{the lines} of deflection and at the _same time increases the width of the

strip

of membrane immersed in the well. On the other

hand,

_more undulations have _to be

suppressed

as the deflection

angle

_gets smaller and the

strip

is widened. There is

possibly

_an

optimal

deflection

angle

_so ^that the free energy of membrane deflection

depends

_on how well bulk and surface

angles

_can be matched. It _seems obvious that _a small

spread

of _q~^~ is _a

prerequisite

for

good matching.

If these ideas _are correct, the

unbinding

transition of _{two or more} membranes need _not be

govemed by (I )

with #f = I. It could _even end up as a first-order transition _{at some} finite

spacing,

^thus

avoiding

the

high energies

^of

matching

_a wide range ^{of bulk}

angles

to the

optimal

deflection

angle.

4. Conclusion.

Starting

from _a

Ginzburg-Landau Hamiltonian,

we derived _a distribution function for

weakly

bound

n-layer

systems, ^{either stretched}

polymers

in two dimensions _or fluid membranes with _a

bending rigidity.

The distribution function is of universal

shape

and _was used _to obtain

scaling

laws of the

unbinding

transition. The critical

exponent

#f of

unbinding

_came out to be

unity,

regardless

of _n, and the width of the distribution function _was found _to be

inversely

proportional

to the number of components.

It _{seems to} be _an open

question

whether _{or not} this _mean field

theory

is _correct for

n _~ 2 in the _case of

polymers.

For membranes its

validity

_may besides be

impaired

_even for

n = 2

by

_a

logarithmic

correction.

However,

the _mean field

theory

is

hoped

to be useful _as a

first

guide

in

interpreting experimental

results

conceming

the

unbinding

of systems ^of more

than _two components. ^For

instance,

it

predicts

the

unbinding

transition _{to occur at} the _same critical

potential

_or temperature etc., for any number of components. This _appears to agree with the

only experimental

observation of membrane

unbinding

known to date

[14].

If there

was a increase of the transition temperature ^{with the number of}

components

which varied from 2 to more than

20,

it _{was no more} than _a few °C

[15].

Acknowledgement.

a1n

grateful

to M. M. Kozlov for

solving equation (I I)

and to R.

Lipowsky

and M. M. Kozlov for discussions. This work _was first

presented

at the

Aspen

Center for

Physics (Workshop

_on

Self-Assembling Systems

and

Membranes,

^Summer

1992).

Jn this

stimulating

environment I

was introduced to reference

[9] by

S. Milner and made _aware of reference

[11] by

A. Liu.

References

[II LIPOWSKY R, and LEIBLER S,,

Phys.

Rev. Lett. 56

(1986)

2541.

[2] LIPOWSKY R.,

Europhys.

Lett. 7 (1988) 255.

[3] KROLL D, M, and LIPOWSKY R.,

Phys.

Rev, B 28 (1983) 5273.

[4] LiPowsKY R, and NIEUWENHUIzEN T, M,, J.

Phys,

A _: Math, Gen, 21

(1988)

L89, [5] LIPOWSKY R, and ZIELINSKA B,,

Phys.

Rev, Lett. 62

(1989)

1572.

[6] HELFRICH W,, Phase Transitions in Soft Condensed Matter, T. Riste and D,

Sherrington

Eds.

(Plenum

Publishing Corporation,

1989).

[7] CooK-RODER J, and LIPOWSKY R,, Europhys. Lett. 18 (1992) 433, [8] FISHER M. E. and GELFAND M, P,, J. Stat. Phys. 53 (1988) 175.

[9] MILNER S. T. and Roux D., J. Phys. I France 2 (1992) 1741.

[10] HELFRICU W., Z. Natu~fiorsch. 33a (1978) 305.

[I II _DE GENNES P. G., J. Chem.

Phys.

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see also HAI TANG and FREED K. F., J. Chem.

Phys.

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1572.

[12] NETz R. R. and LIPOWSKY R,, poster

presented

at STATPHYS 18, Berlin, August 2-8, 1992, and preprint,

[13] See, _{e, g,,} HELFRICU W.,

Liquids

at Interfaces, Les Houches XLVIII (1988) J. Charvolin _et al, Eds.

(Elsevier Science Publisher, 1990).

[14] See also GROTEUANS S. and LIPOWSKY R., Phys, Rev. A 41 (1990) 4574.

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(1989)

2883.