Vanishing tension of fluctuating membranes

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Vanishing tension of fluctuating membranes

F. David, S. Leibler

To cite this version:

F. David, S. Leibler. Vanishing tension of fluctuating membranes. Journal de Physique II, EDP

Sciences, 1991, 1 (8), pp.959-976. �10.1051/jp2:1991120�. �jpa-00247568�

classification

Physics

Abstracts

05 20 82 65 68 10

Vanishing tension of fluctuating membranes

F David

(*)

and S Leibler

Service de

Physique Thkonque (**),

CE

Saday,

F-91191 Gif-sur-Yvette

(Received

3 January J99J, rev~ed 5

April

J99J,

accepted

J9

April

J99J)

Abstract. The statistical behavior of

fluctuating,

flexible membranes _or films _can be

governed

by

their

(surface)

tension or by ^their curvature energy,

depending

_on

imposed boundary

conditions We present ^here

simple thermodynamic

arguments which enable _us to

classify

the different

physical

situations. In

particular,

we

danfy

the notion of

vanishing

_tension, and the (or

some of the) conditions under which it can occur Then, m view of this classification, _we reconsider the effects of thermal fluctuations _on tension and

rigidity,

and

bnefly

discuss the

expenmental

systems in which renormalization effects should be

expected

and could be observed

1. lnwoducfion.

When

amphiphihc

molecules _are

brought

into contact with _water

they

_can assemble to form

bilayers

_{SO as} to Onent their

polar

parts ^towards water and their

Oily hydrophobic

tails _away from it These

amphiphihc

membranes have many

interesting properties

^{from the} point Of

view Of Statistical mechanics

they

_are

(quasi-)

two-dimensional

Objects,

assembled

through weak, nonspecific

interactions

[I]

Morover, recent expenments, ^Such aS quantitative Studies Of the

Shapes

and undulations Of

bilayer

vesicles

[2, 3],

and Of the entropic interactions

between membranes

[4, 5],

^have demonstrated that these flexible sheets _are Often

governed by

their

rigidity

[6] ^This means that membranes _are dominated

by

their curvature energy rather than

by

their surface _{tension, as is} the _case for usual

fluid/fluid

interfaces This _is often

summanzed

by

the claim that the membranes have

vanishing surface

tension

There exists,

however,

_{quite a} lot of confusion _as for the precise meaning of such _a statement What does _One

really

call the surface _tension Of _a membrane ? Is _it

strictly

_{zero Or} just very small ^? If the _tension indeed

vanishes,

then _in which situations does this _Occur ? And

finally,

how _{can One measure} membrane _tensions and show that

they

_can vanish? In

particular,

do the measured tensions

depend

_on the

lengthscales

involved _in

experiments

~ It

seems tO _us that

although

these questions are

really

essential for _many recent theoretical and

expenmental developments

it is difficult _tO find _in literature clear _answers tO them. The _aim Of this paper is therefore _to try to

danfy

_some of these _issues

Amphiphilic

membranes _are not the

only

_systems

govemed by

curvature effects ThJn

plates,

_or

shells,

_are also

objects

for which the

dominating

_{energy is}

bending _[7]. By

(*) Physique Thkonque

CNRS

(**) Laboratoire de la Direction des Sciences de la Matidre du Cornrnissanat I l'Energ1e Atomique

960 JOURNAL DE

PHYSIQUE

II N 8

introducing

constraints

(e.g by appropriate boundary conditions)

_{on a} free shell _{one can} introduce _an internal tension

lpressure).

For small _strains such _a tension will be

equilibrated by

cohesive

forces,

_or

compressibility

There is _an

important difference, however,

between such _a mechanical

plate

and _a membrane : the

ngidity

modulus for the later _is of order

k~

T.

Therefore,

_a membrane fluctuates due to thernlal excitations and _one has _to consider

the _tension from _a statistical mechanical rather than a mechanical point of _mew

For _a

fluctuating

membrane there _{are two}

independent thernlodynamic

variables the total

area,

A,

which _can be modified

through

for

example compressibility

effects _or

exchange

of

molecules with _a

reservoir,

and the

projected

_area,

_A~,

which _can be modified

through

the constraints

imposed

_on the system or can vary due to thermal undulations (~) ^We discuss th~s

important

_{point m}

chapter 2,

where _we also _argue that the

(surface)

_{tension, r,} of the

membrane _{is an} intensive

thermodynamlcal

variable

corresponding

to

A~,

^{whereas the}

vanable _r

corresponding

to A _is related to the chemical

potential

of the

amphiph1llc

molecules

Although

there _{is no reason m}

general

for _r to be

equal

_zero

(contrary

to what _is often claimed _in

literature)

in some

physical

situations _{r can} indeed vanish.

Having

defined

thermodynamic

ensembles

corresponding

to different

physical

situations we reconsider _in

chapter

3 _an

example

of _a theoretical calculation for which the _issue of

vanishing

or

nonvamshing

surface _{tension is} crucial

Namely,

we discuss the

problem

of the effect of thermal fluctuations _on the

ngldity

constant of fluid membranes

[10, 11]

We try to point out which

lenght

scales _are important in different situations, and _in which situations _one

can expect to observe the subtle

logarithmic

renormalization effects _to be present. ^This seems to be _a relevant _{issue m view} of recent

expenments claiming

to observe such effects

[12, 13].

The

expenmental problems

_are also discussed _in the last

chapter

4 We try there to

clanfy

what is indeed measured in reflection type expenments

[14, 15]

and what _are the relevant

lengthscales

for such experiments.

2.

Elementary thermodynamics

of

fluctuating

membranes.

Consider _a

fluctuating

membrane made from

amphiphilic

molecules Let _us suppose, for

simplicity,

that it spans a

planar

frame of _a total _area

A~ (Fig. I).

^{If the number of molecules}

forming

the membrane _is 2 N and the

equihbnum

area per molecule _a~ then the total _area of the membrane _is A

= a~ N We shall suppose for the moment that the

compressib1llty

of the

monolayers

is very

low,

and thus the total _area A _can vary

only through changes

_m the

number of molecules N

(exchange

with _a

reservoir).

The

quantities

A and

A~

can be

Fig,

I A

fluctuating

membrane spanning a

rectangular planar

frame of _area A~.

ii)

The distinction between A and A~ is

already

present m reference [6] but we disagree on their conclusions The need for these _two

independent

vanables _was

previously

introduced for

lyotropic

systems m reference [8] and reference [9]

considered _as

independent thermodynamical

variables

[6] Indeed,

_in real systems

they

_can be made to vary

independently

For instance, for black

lipid

films _or

monolayers

films _{on air-}

water interface the

projected

_area

_A~

_is fixed

by

the geometry of the

expenment,

^{whereas the} total area A

(directly proportional

to

N)

_can fluctuate due to the

exchange

of the

amphiphiles

with _{a reservoir} On the contrary, in

lipid vesicles,

for which the

exchange

times are

long,

the

total _area does not vary, whereas the

projected

_one does fluctuates.

From the

thernlodynamical

_point of _view, the _area A and the

projected

_area

A~

are extensive variables. It _is important ^{to realize that the}

corresponding

intensive variables represent ^distinct

physical

_quantities The _area

coefficient

_{conjugate to} the total _area

A,

which _we denote _{r, is} for

incompressible

films

directly proportional

to the chemical

potential

_~1, of

amphiphihc

molecules The

film

tension conjugate to the

projected

area

A~,

^which we denote _r,

corresponds

to the

physical

« surface tension _». As _we shall show

below,

the _situation of

vanish~ng

surface tension _» considered

by

_many authors

[6, 16, 11]

does _not, in

fact, correspond

to

vanishing

_r but rather to

vanishing

_r

With two

independent

extensive variables _{one can} define four different

thernlodynamical

ensembles

ii) IA,

A

~)-ensemble.

^{We call the membranes}

^belonging

to this ensemble

isolated, framed

systems

iii) IA,

_r

)-ensemble.

We call the membranes

belonging

to this ensemble isolated,

unframed

systems.

iii) jr, A~)-ensemble

^{We call the membranes}

belonging

to this ensemble open,

framed

systems.

IN) jr,

_r

)-ensemble.

We call the membranes

belonging

to this ensemble open,

unframed

systems.

In

particular

two of these four ensembles _are

important

^for

expenmental

systems ^and we

shall discuss them _{now :}

Open, framed

_systems.

In this ensemble the

projected

_area

_A~

_is

fixed,

for instance

by

_{imposing proper}

boundary

conditions _{on a}

frame,

and the total _area A fluctuates. The fluctuations _are

govemed by

a

Hanultonian of the fornl

H=rA+H~i (1)

where H~i contains the contribution of elastic internal forces

(bending

_energy, shear

modulus, etc.)

The partition function _is written _as the _{sum over} all film

configurations

C with fixed

A~

z~

=

£~-flH(e) p

~

l

j~)

~

'

k~

T

The free energy in this ensemble _is

Go(A~,

r

)

=

k~

T In

Zo(A~,

_r

(3)

and the film tension is

simply

defined _as free energy per ^unit area

r~ = lim

°~j~~'~~

(4)

A~-w p

962 JOURNAL DE

PHYSIQUE

II N 8

Isolated, unframed

_systems.

In this ensemble the total _area A _is fixed while the

projected

_{area may} fluctuate. The

thennodynamical potential

_is obtained from

G~ by

first _{going to} the internlediate

IA, A~)

ensemble where both A and

A~

are

fixed,

and the

thermodynanfical potential

is

obtained

by

_a

Legendre

transform

F

(A~,

^A

⁾

=

Go(Ap,

_r ^{rA A} =

I (5)

' °r

A~

Then _{one goes} to the

isolated,

unframed film ensemble

by

a second

Legendre

transform which defines the associated

thermodynamical potential

G

jr,

A

)

=

F~ iA~,

^A

⁾ rA~ ¹⁶⁾

where the surface tension _r is defined

by

dF~ dGo

~

dAp

A

dA~

~'

~~~

f

(a)

<ap> ap

f

~o /

~

/ II

1' /

-'

~

i

ap

Fig.

2. The free _energy density

f

as a function of the area ratio a~

(a) f

has a minimum for

a~ m 0 and the tension r vanishes (b) f has its minimum at a~ = 0 and the tension _r may be positive

(i)

or zero

(ii)

It _is clear that _m the

thermodynamical limit,

A

- oJ, the surface _tension defined _in th~s ensemble

by (7)

coincides with the surface tension defined for the open, framed systems

by (4)

We _{can now}

analyse

the _meaning of the tension

r for

isolated,

unframed systems ^{with fixed} total _area A. Let _{us assume} that the

projected

_area fluctuates around its _mean value

(A~).

This _mean value _can be obtained

by

_{minimizing F~}

(A~,

^A

(given by (5))

with

respects

to

A~,

^while A is fixed In the

thernlodynamical limit,

A

- oJ, it is better _to consider the free energy

density

f

F~

=

j ¹⁸⁾

as a function of the _area ratio

A~

~P ^~

j (9)

Two situations _are then _m

general possible,

_as

depicted

_m

figure

2

ii) f(a~)

has its minimum for _{a non zero ratio} 0 _<

a~

< I

(Fig 2a)

The membrane _is then said to be

flat,

_since

although

shrunk due to thermal

fluctuations,

it still

keeps

the

global

structure of _{a two} dimensional

object.

In such _{a case, we} conclude from

(7)

that the surface tension vanishes

r =

~~ (a~)

= 0

(10)

da~

_A

iii) f(a~)

has its minimum at

(a~)

= 0

(Fig 2b)

The membrane _is then said _to be

crumpled,

since it is _so shrunk

by

thernlal fluctuations _so that its extension in space does not scale

linearly

with _its intemal extension (2). ^In this _case it is obvious from

figure

2b that the

tension _{r is} positive, ^but has _in

general

_no reason to vanish

One should also include _in this case

iii)

_a

marginal

situation in which

although

the _minimum of

f(a~)

is at

(a~)

₌

^0,

its

slope

at this point ^does

vanish,

and therefore _r

=

0.

Each of these _{cases can in} fact be encountered for

isolated,

unframed systems, at least _in idealized theoretical situations

Indeed,

at low temperatures

polymerized

membranes _{are in a} flat

phase,

where

(a~)

_is non zero

[17]

Therefore their tension _r

vanishes,

_except if

they

_are

submitted to _an extemal stress. An

isolated,

unframed

fluid

membrane with fixed

topology,

on the other

hand,

_is in

pnnciple crumpled Ill _],

with

(a~)

=

0,

and has therefore _{a non zero} tension

[18, 19],

as we shall discuss _in next section We shall also _see that _{r is} in fact

r~iated

to the

persistance length f~ by

the relation _r

~k~ Tfj~ _Finally,

_free hexatic sheets

[17],

characterized

by

_a

quasi-long

_range onentational order _are

crumpled

_(«

crinkled»)

_since

(a~)

=

0,

but

they correspond

to the

marginal

situation where _{r is} also _{zero since} their

persistence

lenght

_is infinite

(the slope

of

flap)

at

(ap)

=

0

vanishes) [20].

(2) One

might

be worried that _m this situation A~ ^{scales with the number of molecules} N _as N~ (v _< I and is therefore not an extensive variable However from (7) the tension is then

defined by first considering the _situation where a~ =

A~/A

> 0 (in ^which case A~ ^{is indeed}

extensive)

and then taking the limit a~ -

(a~)

=

0

964 JOURNAL DE

PHYSIQUE

II M 8

Free fluid sheets which do _not

change

their

topology

_are hard to realize

expenmentally However,

_{our argument is} much _more

general

and _can also be

applied

for

interacting

membranes,

_{e g. in a} stack of lamellae

[4]

In the _case of _a stack of fluid

membranes,

or in the

case of _a membrane

fluctuating

m a

confining potential le.

_g. of _a

adsorbing wall),

_no

topology changes

will take

place

_as

long

as the average distance d between membranes

(or

between the membrane and the

wall)

_is small

compared

with the persistence

length, id

«

f~).

Then the interactions force

(a~)

to be _{non zero,} while

A~

^still can fluctuate

freely

In these situations the tension _r of each membrane vanishes

(as

described _in the _case

ii) above)

It _is

important

to stress that the argument

presented

here _to

explain why

_{r can} vanish for

fluctuating

membranes _{is very} different from usual arguments

leading

to the so-called

« Schulman _» condition

[21].

This term _is used to describe situations _m which the _area per

molecule,

_a~, of _a

non-fluctuating film,

_{can vary} The _area

(a~)

chosen

by

the system

corresponds

then to the minimum of the free energy,

4 (a~, N),

^and at this minimum

(am)

= 0

(12)

However,

this quantity does not coincide with the tension, _r, of the film defined _m

(10).

Indeed,

the above argument ^is

purely mechanical,

_since it considers

only

the internal elastic forces associated to the finite

compressibility

of the

membrane,

and since it does _not take into

account the thernlal fluctuations. On the contrary, ^the

argument presented

here takes into

account the thernlal

fluctuations,

and does not

depend

_on the

compressibility.

3. Effective

rigidity

and tension of

fluctuating

membranes.

In tl~is section _we shall discuss the _case of fluid membranes

subjected

_to thernlal undulations A

phenomenological

Hamiltonian for such membranes which _is often used _is

[2,

_3]

H~i =

ids( ^{f H~+} _kK) (13)

2

where dS _is the element of surface.

H= ₊

),

K=

(14)

Ri R2 R>R2

are

respectively

the _mean curvature and the Gaussian curvature

(Rj

and

R~

are the

pnncipal

radii of

curvature),

and _K and k _are

respectively

the _mean and the Gaussian

bending ngldity.

We _assume that the membrane is symmetnc so that there _{is no} spontaneous curvature ternl,

linear _in H. For flexible fluid

membranes,

with _K not too

large,

thernlal undulations renornlahze the effective

bending ngldity

and the _tension at

large

distances. Our purpose is not to rederive th~s well known result _m

detail,

but rather to discuss the

meaning

of th~s renornlalization for the _tension, and _its domain of

validity,

_in the

light

of the discussion of section 2

Most of the calculations of the effect of the renormalization of the

rigidity

and _tension have been done _in the _« constrained ensemble _» where the

projected

_area,

_A~,

_is

kept fixed,

while the total _area, A, fluctuates.

Indeed,

if _one

neglects

the effect of k, which _{is not} important as

long

_{as no}

topology changes

_{occur, one can} start from the Hamiltoman

(1)

lKo

~

Ho

= dS

ro

+ H

ii 5)

2

and then integrate out thernlal fluctuations with

wavelength larger

than _a microscopic cutoff

a =

«IA (which corresponds typically

to the width of the

membrane,

_or to the

equihbnum

extent of _a

amphiphihc molecule)

In

doing

_{so one}

usually

_assumes that the

amplitude

of the

undulations above the

equilibnum plane

of the membrane vanishes _at

infinity (or

_is

periodic)

and thus

neglects

contnbutions

coming

from the

boundary.

Th~s _{is m} fact

stnctly equivalent

to the choice of the constrained ensemble described above.

When integrating out thernlal fluctuations one can define _two types ^{of effective} quantities

Effective potential

The _so called effective

potential r~~r

is obtained in the standard _way

by adding

_{a source term}

dS XJ to the Hamiltonian

Ho IX

denotes the position ^{of the membrane} in 3 dimensional bulk

space), computing

the free energy

(3),

and

taking

its

Legendre

transform with

respect

to J

la

_precise definition must take _{into account} the

reparametrization

_invariance of

(15) by

some gauge

fixing)

The

resulting

effective

potential r~~(X)

_may be

expanded

_{in powers} of the curvature

K~~ _~

r~~

= ds

r~~

+ H ₊

ii 6)

2

r~~

_is the generating ^{functional for all} static correlations functions for the membrane at thermal

equilibrium.

In

particular

_its minimization gives the _mean

equihbnum configuration

of the membrane. If this

equilibrium configuration

_is flat

(which

_is not

obvious,

since this

depends

_on the sign of _K~~ and _on the _sign of the

neglected h~gher

order

tennis)

all curvature

terms vanish _at the minimum _m

(16)

_so that

refmin

"

G01Ap, ~0)

_"

Ap

~e~

l17)

Thus from

(4)

_we obtain _again, under the assumption that the

equihbnum configuration

_is

flat,

_{an important}

equality

between the effective surface _ternl and the surface tension

;~~ = r

j18)

Renormafized Hamiltonian

An effective Wilson Hamiltonian

[22] H~~

s is obtained after integrating out all fluctuations

with _wave vector k _{in a} shell

As

=

AS~'

< k _< A

(19)

where Sm I. This nonlocal effective Hamiltonian _can

again

be

expanded

as

K~~(S) Heff.S

_" dS

~eff(S~

~

+ ~

H ₊

(2°)

and it governs the fluctuations of the membrane _{at wave vectors}

(k(

_<

As.

One _can then rescale back the distances and momenta

X-SX, k-S~~k (21)

m order to obtain the renornlahzed Hamiltonian

lKs

_~

Hs= ds~rs+-H+... (22)

2

966 JOURNAL DE

PHYSIQUE

II N 8

with the initial cutoff

A,

and the renornlahzed

couplings

_given

by

rs =

S~ r~~is)

, K s ^{= K}

~~is~ 123)

The operation ^which transfornls the initial Hamiltonian

Ho

into

Hs

is the Renornlahzation

Group

transfornlation

its

In practice,

however,

the renormalized Hamiltonian

Hs

is not calculated

directly.

One

performs

rather _an «infinitesimal R-G- transfornlation» for _an infinitesimal

shell,

S

=

I _{+ e,}

jt,

~ ~ ⁼

l ₊ ES

I

its)

^{= I + et}

⁽²⁴⁾

Then _one

integrates

out the

corresponding

differential equation

S ^~

Hs

=

Hs,

H

~s =,~ ⁼

Ho (25)

frim

I to S _m order to obtain

Hs

From the definition of the internlediate effective Hamiltoman

H~rr

_s

(18, 20),

it _is clear that

in the limit S

- oJ it becomes

equal

to the effective

potential (16)~

llm

Heff,

_{S "}

reff

~~~~

s

_ ~

Therefore the relation with the renornlalized

couplings (23)

and the effective

couplings

in

r~rr

_is

K~~ =

hm K s, r~~ =

hm

(rs/S~) (27)

S

- w S _- w

The R-G

analysis performed

for fluid membranes consists now m

studying

the

dependence

in the scale factor S of the renormahzed

ngidity

modulus Ks and the renormahzed surface term rs In order to

apply

these results to

physical

situations in which the total area A of the membrane does not fluctuate and the membrane _is

subjected

to a

physical

tension

r, one has then _{to use}

(18).

Hence when

performing

the R G calculation _one should start from _a Hamiltonian

Ho

with

microscopic ngldity

_Ko

(corresponding

_to the elastic properties ^of the membrane at microscopic

scales)

and with _a microscopic surface coefficient _ro

adjusted

_in

such _a way that the effective surface _ternl r~rr, obtained from the R G calculation

by (27),

coincides with the

physical

tension r

(we

^still assume that the

equilibrium configuration

_is

flat)

The R G. calculation _{at one}

loop

_is recalled in

Appendix.

Its final result

(A being

the

sharp regulator

_in momentum

space)

_is

~

dS ^{~ ~} 4 _ar rs

~~~~

~ ~

l

~

^~

~

Ks A

a

kg ^TA~

_r~

S rS " 2 rs ⁺ In ₊

(29)

°S 4 _"

Ks

A~

In order to understand the meaning of

(28)

and

(29)

_it is important to define the domain of

their

validity.

The calculation is _{a one}

loop

approximation, therefore it _is valid if the terms of order

kg

T _are small. This

implies

K m

k~

T _{or r m} k

B

TA~ (30)

depending

whether at wave vectors

(q(

~

A it is the _curvature energy or the surface _term

which dominates The

phase diagram

in the

ix ~~, r) plane (for

_positive

r)

_is

depicted

_m

figure

3. It _can be

separated

into three distinct domains.

ii

r

A

k~T

~

^{/. ?}

~

~.

- /

K-1

Fig

3 Phase diagram for fluid membrane and Renormalization

Group

flows _{K is} the inverse of

the

bending

ngidity and

r is the surface term A _is the _tension dominated region, ^B the

ngldity

donunated _regJon and C the therrnal fluctuations dominated _region The three difierents kind of R G trajectories (i), (ii) and (m)

corresponding

to systems ^{with the} same microscopic rigidity _Ko but different

microscopic surface term ro are

depicted

as thin lines

A. Tension dominated _region

r m

KA~

_and

r m k

~

TA~ (31)

In this domain fluctuations _are small and tension dominates _over curvature energy The

properties

of the model _can be described

by

a

simple

interface model

(drumhead model)

characterized

by

_a tension _r. The R G. flow

corresponds

to _{a naive}

scaling

~~ ~ ~°~~~~~~

~

rs =

S~ T

j~~~

B.

Rigidity

dominated _region

K m

k~

T and _r

< KA ^~

(33)

In this _region fluctuations _are small and

rigidity dominates

over tension energy The

renormalization group flow

corresponds

to the

perturbative

^result of

[11, 18]

a 3

k~

T

S

j

^{K s =}

j (34)

a

k~

T

S- rs =

2

+ rs.

(35)

as 4 _"K

s

JOURNAL DE PHYSIQUE it T I V 8 AOOT ₁₉₉₁ 44

968 JOURNAL DE PHYSIQUE II N 8

The renormalization of the surface term _can be

simply

attributed _to the _increase of the ratio total

area/projected

_area due _to thermal undulations

Indeed,

at

leading

order in

k~

T

ii-e-

in

the Gaussian

approximation),

the contribution of undulations with _wave vector k _in the shell

(20)

to this ratio is

equal

to

3

(A ) _/A~

=

(~

~~~

~

~~ ~)

=

~~

^~

ln

IS) (36)

2

As

(2 ar)

Kk

~

4 _arK

(35)

follows from the fact that _{r is} the conjugate parameter ^{to the} total _area A.

C Thermal

fluctuations

dominated _region

K <

kg

T and _{r <}

kg ^{TA~ (37)}

In this domain thermal fluctuations _{are so} important ^that

perturbation theory

breaks down.

One

expects

^that wild fluctuations of the membrane will take

place,

and that

changes

of

topology

and _stenc interactions will be very

important

The

properties

^of the membrane _{at a}

length

scale

I

_are

simply

obtained

by considering

the

RG. flows and

deciding

_in which region of

phase diagram

_one finds the values of the

renormahzed

couplings

K s and rs ^at this scale _{i e.} with S

=

I la

₌

(Alar

_For

a membrane with microscopic

bending ngldity

_Ko

(for

obvious _reasons Ko has to be

larger

_or of the order of

k~

T, but cannot be much smaller than

k~ T,

otherwise the membrane will _not have any

internal

stability)

three different kind of R G trajectories may be encountered

depending

_on the value of the microscopic surface _{term ro} These three _{cases are}

depicted

in

figure

3.

ii)

If ro ^is

large enough

_one start from domain A. Tension dominates and _at all scales

«IA

= a <

f

< oJ the membrane _may be described

by

_a

simple

interface model with

physical

tension _r

= ro

land ngldity Ko)

iii)

_{If ro is} small

enough,

_one starts from domain

B,

but _one then flows towards domain A.

This _means that there exists a crossover

length f~, larger

than the microscopic ^cut off

a =

«IA,

which separates two regimes :

la)

At scales

f

m

i~

_{one is in regime} A, the

physics

is described

by

^the Hamiltonian

(15)

with _an effective

rigidity

_K~~

m

kg

T and _an effective surface _{term r~~}

= r, small but _nonzero.

From

(31)

and

(32)

the _crossover

length i~

_{is given} in terms of the

physical

parameters

r and K~~

by

f~

₌

jd

138)

T

16)

^At scales

I

<

i~

_{one is in regime} B. Tension effects _may be

neglected

and the

physics

_is described

by

the elastic Hamiltonian

(13),

but with _an effective

rigidity xii

_{which decreases}

logarithmically

_as

^I

_increases

according

to

(34)

4

ii)

= Ko

j

In

((la) 139)

until

f

=

i~,

where the tension takes _over the

ngidity

Note that _{m case}

ii)

_{one can} also define

i~ by (38)

but it _is then smaller than the cut off

a

(iii)

If ro is still

smaller,

_one start from domain B but _one flows into domain C before

tension becomes

important

^This occurs at a crossover

length i~

^{such that}

xii (as

_given

by

(39))

_is of order

k~

T Therefore this

length

_is

nothing

but the persistence

length [21] (usually

denoted

f~),

and from

(39)

_{we see} that it

depends exponentially

of the microscopic

bending ngidity

4 arKo

i~

₌

f~

= a exp

(40)

3

kg

T

At scales

larger

than

f~

one enters therefore the

nonperturbative

_regime where thernlal fluctuations _{are very}

large

and where both

rigidity

and _{tension are} ineffective The membrane will be

crumpled,

with _a correlation

length

for its nornlals of order

f~

^At this scale

IS

=

f~la)

the effective surface _ternl

r~~(i~)

= S~

~rs

_is of order

[23]

r~~(i~)

_oz

^~~ )

(41) ix

In th~s regime C the

physics

of the membrane _is not

adequately

described

by

the

Hamiltonians

ii 3)

or

ii 5),

^and effects such _as

topology changes (partially govemed by

the Gaussian

bending ngidity k)

or strong stenc

repulsion

between distant parts of the membrane have _to be taken _{into account}

Phenomenological

models which describe this

phase

_are for

instance the lattice models

[24]

of

non-ngid

self

avoiding

surfaces with

f~ playing

the role of the lattice spacing and

fj r~~(i~)

that of the chemical

potential

of

plaquettes

Therefore

r~~(i~)

_may be considered _as the effective surface tension _r at that scale.

Indeed,

if there is

coexistence between this membrane-rich

phase (where

_{space is} filled with

membranes)

and

another

surface-poor phase (see below),

those models

predict

that the tension of the interface between those two

phases

will be of order _r

=

r~~(i~)

This

provides

_a

justification

of the

commonly adopted

relation

r oc

k~ Tfj~ ₍₄₂₎

between the tension _r and the persistence

length f~

in microemulsions

As _we shall discuss _{in more} details _in the _next

section,

it _is

possible

to encounter these three situations in

expenmental

systems ^In

figure

4 _we have

schematically depicted

the idealized

example

of _a temary

(water

+ oil ₊

surfactant)

_system

fornling

_a rnlcroemulsion

(sponge) phase

At scales of lo

I

_{the surfactant film}

can be described

by

_a membrane model with moderate

ndigity

and small tension

(regime B)

At scales

larger

than the average

distance, d,

between the films _in the sponge

phase id being

identified with the persistence

length f~)

_{one is in regime} C and _{one is}

looking

at the properties ^{of the} sponge

phase Finally

at much

larger

scales the interface between the sponge

phase

and the water-or-oil-nch

phase

_can also be described

by

_a model of interface with _a small tension

jr k~

Td~ _~) but _{one is now in}

regime ^A.

Let _us also stress that _we have discussed _in this section the situation where the _transverse undulations of _a membrane are

only

limited at

large

scales

by

the effect of the _tension. Other effects _may limit those undulations and _{act as an} effective _« infrared cutoff _». For instance if

one considers _a

monolayer

at the interface between two fluids with different

densities,

_gravity supresses undulations _at

wave-lengths larger

that the

capillary length i~~ [25]

^For a

membrane between _two

walls,

_{or m a} stack of

membranes,

stenc

repulsion

will also suppress undulations with

amplitude larger

than the average distance between the membrane and the wall

(or

between

neighbounng membranes) [4].

In those situations the R-G- equations

(28),

(29)

_are valid up to th~s I R cut off scale

iiR [23, 9]

^In

particular,

m the _case of _a stack of fluid membranes

already

discussed _at the end of section

2,

_{we are in} the situation where the tension

970 JOURNAL DE

PHYSIQUE

II N 8

'~~~~~/~y@I?

p~

c~

oil

oil water

oil rich-

Fig.

4 Schematic

description

of _a microemulsion at three different scales The description of the surfactant film at short scales

corresponds

to regime B. The

descnption

of the sponge

phase

at

intermediate scales

corresponds

to regime C. The

descnption

of the interface between the di~erent

thermodynarnical phases corresponds

to regime ^A.

r vanish

(as long

_as the average distance between

membranes, d,

is small

compared

to the

persistence length f~ (as

given

by (40))

^It follows that at all scales _a

<

I

<

ii~

d _one is in

regime

B and that the renormahsed tension rs vanishes

4. Possible

experimental

consequences.

It _is

interesting

to

reconsider,

from the point of _view described _m the last

Chapter,

different

measurements done in the

past

m order to

study

the role of thermal fluctuations _on the

ngldity

of films and membranes.

The existence of three distinct regimes for the

thernlodynamical

behavior of the

membranes _can be

explored directly by

inspecting ^the mesoscopic ^structure of different

phases

^The progress made _m freeze fracture microscopy

[26]

allows for instance the direct visualization of _vanous structures made of membranes, such _as lamellar

crystals,

bicontmuous

fluid

phases (e.g

_a

fascinating _L~

_{sponge »}

phase [27])

_or ensembles of closed vesicles. With the

help

of careful studies of different _« cuts »

through

_a bicontinuous

phase

_{one can} for instance establish the _existence of

highly

curved

regions

of

complicated topology (correspond-

ing to regime

C)

at

large length

scales. One could in

pnnciple

obtain _m th~s way quantitative infornlation about curvature

distnbutions, dependence

of the _{curvature on} the

length scales,

fluctuations _m

topology

etc.

In the

meantime, however,

_one relies

mainly

_on diffraction

experiments

_to obtain information about the structure and fluctuations of membrane

phases.

This infornlation _is

mainly

contained _in the structure factor

S(q),

the Fourier transfornl of the

pos1tlon

correlation function Two types of

expenments

give access to

S(q)

_:

ii) Specular reflexion experiments [28, 14]

In these

expenments

one measures

Iiq=)

=

d2x d2y expiiq=iuix) uiy)))) ¹⁴³⁾

where q~ is the _{wave vector}

perpendicular

to the reflexion

plane

^of the membrane

u(x)

_is the

perpendicular displacement

of the membrane above the

plane

of the membrane

ix

and _{y are} coordinates of the

plane)

^{In harnlonic}

approximation

^this can be

replaced by

Ijq~)

=

d2x d2y

_exp

(- ( _{jju ix)}

u

jy))2j j44)

m wh~ch

1lu lx)

_u

lY))~l

= 2

fi _Ii

e~ ~~~~ ~~)

Slq) 145)

iii) Ellipsometry [14, 15]

^This

powerful technique

allows _{one to measure}

£ qsjq j46)

and therefore check whether

S(q)

has _an assumed fornl

These

methods,

in which _one

integrates

over a whole range of _wave vectors, can be useful

in practice

only

if _one puts

together

the results obtained _in the different regimes descnbed above. For instance, ^if we make _an

e1llpsometric

measurements for _a membrane which at small q ^{is in}

regime

A wh~le for

large

_q in regime B _it would be useful to have _a consistent fornl for

S(q)

which _we could then substitute _m

(46)

We _can do this in the

following

_way for q < q~ =

I

one takes

K

k~

T

S(

_q

= ~ ~,

(47)

Tq ⁺ Kq

while for q~ < q <

"

= A _one takes

a

k~

T

S(q)

= ~,

(48)

K

(q)

_q

where

K(q)

_{is given}

by

x

iq)

= xo +

£ _kg

_Tin

_iq/A)

149)

972 JOURNAL DE

PHYSIQUE

II N 8

One _can then make

S(q

continuous at q~

by adjusting

the value of _Ko, which

corresponds

to

the _«

microscopic

_»

bending ngidity

at momentum q =

A

A third type ^of expenment, which has

already

started to

explore

the interesting

renormalization effects of the regime ^B are the measurements of _area fluctuations. In these

experiments

_{one measures} how the ratio

projected area/number of

molecules _vanes with external parameters. ^For instance, in reference

[13] by

_varying the pressure acting on a

fluctuating

vesicle _{one measures} the functional

dependence

of the

projected

area on the tension _r. In this _way the

logarithmic

corrections due _to thernlal fluctuations _in regime B _were indeed observed.

Similarly by

_varying the distance between the membranes _{m a} lamellar

phase,

and therefore

modifying

the effective IR cutoff i~R

(as

discussed at the end of last

section),

the authors of reference

[12]

_were able _to show the _existence of such corrections for multimembrane systems.

Note, however,

that these

logarithmic

corrections _to the _ratio

A/A~

~

= l ₊

~~ ~ln _(AijR)

(50)

AP

^~ " ~

are

already _present

_m the Gaussian

approximation,

^{while the}

loganthmlc

corrections to the

ngldity

described above _are

h~gher

order

perturbation

effects. It would be therefore interesting to try to _measure

directly

the scale

dependence

of the effective

ngidity

m the

regime

B. In

principle

th~s could be also done

through

_a careful

analysis

of fluctuations of _a

single

free vesicle

[29]

This

important

case is not discussed here

In real systems ^the

simple

_picture

presented

in section 2 _{can m}

pnnciple

be altered

by

_a finite

compressibility

of the membrane

However,

_{as we} shall show _now, the argument

explaining why

the tension

r can vanish stays valid

Indeed, although

the total _area A _{is no}

longer exactly proportional

to the total number of molecules

N,

_{one can}

replace

A

by

N _in the definition of the different ensembles without

modifying

the arguments ^of section 2. In

particular (7),

which defines the surface _{tension r, is}

simply replaced by

G~

_is now the

thernlodynamical potential

_in the constrained ensemble where

A~

^{is fixed while} the fluctuations of the total number of molecules N _is controlled

by

the chemical

potential

of the molecules _r. F _is the

potential

_in the ensemble where both

A~

^{and N are fixed It follows}

that the surface tension vanishes _{as soon as} the

equilibnum

_area ratio

a~,

^defined as

(A~)

a~

=

(52)

(A )

is _nonzero. The effects of _a finite

compressibility

will be felt

only

in fluctuations around the

equilibnum

value

a~

^if

a~ approaches1.

S. Conclusions.

We have tned _in this _{paper to}

classify possible

_regimes ^of the behavior of

fluctuating

membranes and films.

First, by defining

_various

thermodynamic

ensembles _we have

pointed

out the differences between several

physical

situations. In

particular,

_we have _{given a} few

examples

of systems for which the effective surface tension vanishes These include free

polymerized

sheets _or fluid membranes

fluctuating freely

in _a lamellar

crystal.

Our arguments

leading

to the conclusion of

vanishing

surface tension for _a

fluctuating

membrane _are different