Variational theory of undulating multilayer systems

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Variational theory of undulating multilayer systems

P. Pieruschka, S. Marčelja, M. Teubner

To cite this version:

P. Pieruschka, S. Marčelja, M. Teubner. Variational theory of undulating multilayer systems. Journal

de Physique II, EDP Sciences, 1994, 4 (5), pp.763-772. �10.1051/jp2:1994163�. �jpa-00247999�

J.

Phys.

II IYance 4

(1994)

763-772 _MAY 1994, PAGE 763

Classification

Physics

Abstracts

05.40 68.10 61.30

Variational theory of undulating multilayer _systems

P.

Pieruschka,

S.

Martelja

and M. Teubner

(*)

Department

of

Applied

Mathematics, Australian National

University,

Canberra 0200, Australia

(Received

5 November 1993,

accepted

in final form 15

February1994)

Abstract. We _{use a} variational

approach

to determine the

equilibrium properties

of lamellar surfactant

phases.

The variational

theory

yields a

general expression

for the renormalization of the

bending

constant of

undulating

sheet-like membranes. The method is then

applied

to lamel- lar ensembles characterized

by

conserved surfactant film _area and the

full,

non-linear

bending

Hamiltonian. In the limit of

large bending

modulus the

theory

_converges towards Helfrich's model. For realistic values of the

bending

constant _we find _an increase in the

equilibrium

_crum-

pling

and

layer separation

and characteristic

changes

in the _structure factor and

swelling

law

due to film _area conservation and non-linear terms in the Hamiltonian. The

scaling

of the free energy

density,

however, _appears to be

largely

unaffected

by

first order

crumpling

corrections.

Introduction.

Amphiphilic

molecules when dissolved in _water

(and oil)

often self-assemble in two-dimensional

bilayers (monolayers)

which extend _over scales much

larger

than molecular size

[Ii.

A

commonly

observed film

geometry

is

lamellar,

where the bulk material is

separated by regularly

stacked sheets of surfactant film with _a characteristic _average

layer spacing

d which _can be measured in

scattering experiments.

In effective interface models

[2,

3] the

physical parameters

which

specify

the

equilibrium

state of _a lamellar

system

_are

essentially

reduced to the

amphiphile

concentration

is

and the

bending

stiffness _~ of the elastic surfactant film. These model

systems

have _so far been described

by

a harmonic

approximation

^of the

bending

Hamiltonian [4] which allows for

application

of the

equipartition

theorem to determine the mode distributions of

thermally undulating layers

and to estimate the

repulsive entropic

force which is due to the excluded volume

occupied by neighboring layers

_[5]. The

theory operated

in _an ensemble

characterized

by fluctuating

film _area and fixed _area of the associated

projected

surface

(open-

framed ensemble in the classification of David and Leibler

[2]).

^The

crumpling

of the

system

was assumed

negligible

at all concentrations. This _can be shown to be self-consistent at

large

_~.

Therefore,

to zeroth

order,

the

layer spacing

d and surfactant concentration

is

were

essentially

(*)

Permanent address: Max-Planck-Institut fur

Biophysikalische

Chemie,

G6ttingen, Germany.

equivalent quantities;

the steric force law for the free _energy

density

which _was

expressed

_as

f

_cc

d~~

in [5] ^is

equivalent

to the

scaling

relation

f

_{~ ~j3}

(~)

S

which _can be

directly

derived from the scale invariance of the

bending

Hamiltonian and stan- dard

scaling _{arguments [6, 7].}

We

investigate

_a different ensemble in which the film _area is

fixed,

but the

projected

^film

area can fluctuate

(closed-unframed

ensemble

[2]).

^{This ensemble} appears suitable to model the _most

commonly

studied

experimental

situation where _a

given

amount of surfactant

is

is

confined in _a sealed container. The

remaining

^free

parameter

in _our

theory

is the

bending

stiffness _~ of the film. Other

physical quantities

_can be derived from the minimization of the free _energy

density.

In

particular,

_we

consistently

evaluate the

scattering

structure

factor,

the

layer density

and

crumpling,

renormalization of the

bending

modulus

(which

finds _a

simple

formulation in the variational

theory),

and the steric

repulsion

force for _an ensemble charac- terized

by

the

full,

non-linear

bending

Hamiltonian. For

large

values of _~ the

theory

is shown to be

equivalent

to Helfrich's

theory. Away

from this

regime, however,

_we find characteristic

corrections to the _structure

factor, swelling law,

and renormalization of the

bending

constant which become

significant

for small _{~ <} 5kT. In

contrast,

the

scaling

of the steric force law is found _to hold _even for low values of the

bending

constant.

Theory.

The

thermodynamics

of larnellar surfactant

phases

_can be

conveniently

studied

using

_a model ensemble of

essentially parallel,

but

thermally undulating

interfaces _as

long

_{as a}

single layer

has

essentially

two-dimensional character I.e. does not

crumple

too mucl~

[8].

^The mean

positions

of the

undulating

interfaces with surface _area S _are

given by

a set of

flat, parallel

surfaces with

projected

_or base _area A. The surface

position

of _an individual

undulating layer

can be

described

by

the

displacement

variable

t~(r)

^normal to the

projected

surface

i~(r)

_"

~ji~(k) ^e'~~ (2)

Although commonly used,

the

Monge representation

of states,

equation (2),

is

only

an ap-

proximation

_as it is

single-valued

and does not describe surfaces with

overhanging parts

_or

topological defects,

such _as saddle structures which could connect

neighbouring layers.

How- ever, even in the most swollen

experimental samples _[9],

the ratio of real to

projected

surface

area or

crumpling

ratio is C

=

S/A

m 1.2.

Usually

the

crumpling

ratio is close _to

unity [10],

and hence the

single-valuedness

should be _a minor

deficiency

of the _states

equation (2).

On theoretical

grounds

it has been

suggested

that

topological changes

can be

neglected

_as

long

as

the

interlayer spacing

is much smaller than the

persistence length

d _<

fk

_{" Tm}

exp(47r~/3)

_[2]

I-e- for membranes with not too small

bending

modulus _~. We restrict _our

study

therefore to membranes with _~ > lkT. Below this value _{a more}

complete

state

representation

has to be chosen.

The undulations

t~(r)

_are of thermal _nature and _{we assume} their Fourier

amplitudes

to be uncorrelated and Gaussian with structure factor

v(k) ill].

The structure factor is related to the fluctuation of _a mode

by v(k)

= A

((t~(k)(~)o

^where

()o

^denotes statistical

averaging

_over the Gaussian ensemble.

The

closed,

unframed lamellar ensemble is

essentially

determined

by

the

bending

^Hamilto-

njar~

and

non-local interactions caused

by

the

topological

and surface _area constraints. The

N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 765

local steric constraint due _to

adjacent layers

is

approximated by

the usual

global

constraint [5]

d =

1 ₁₃₎

where _p is _a numerical factor. We follow _[5] and will set p =

1/24 (in

_our definition _we

operate

^{with walls} at

+d/2)

later for numerical calculations. Due to the

incompressibility

of the surfactant film

[12],

^{closed surfactant}

systems

with surfactant volume fraction

is

have _an

approximately

constant surface to volume ratio

is

_cc ^~

=

~

d~~

= Cte.

(4)

I-e- in the closed-unframed ensemble the total surface _{area is}

kept

constant

(whereas

the in- dividual

layer

_area and

crumpling parameter

_may

vary).

The

projected

_area to volume ratio

A/V

is not _a conserved

quantity. Symmetric systems

are characterized

by

_a

simple bending

Hamiltonian without

spontaneous

^and

saddle-splay

curvatures

7i = 2~

/

_dS

_{H~ (5)}

s

where _~ is the bare

bending

^modulus

(in

units of

kT)

and H is the _{mean curvature.}

Minimization of the free _energy

density comprising

the relevant interactions determines

equi-

librium structure

factor, layer spacing

and

crumpling,

^{and free} _energy at

given

^{values of} ~ and

is. Using

Gaussian model

states,

^{the free} _energy ^F

(in

units of

kT)

associated with the

bending

Hamiltonian 7i can be

approximated [13-15]

F

s

F

=

Fo

+

17i 7io)o

=

-TSO

₊

(7i)o

₌

~

in

AT

₊

_{(7i)o (6)}

where the

subscript

0 refers _to Gaussian states characterized

by

the Hamiltonian

7io

₌

~j v(k)~~ t~(k)t~(-k).

The

entropic

term

TSO

"

-Fo

has been derived from the

partition

2

~

function of the Gaussian ensemble

Zo

"

e~'°

=

f Dqe~~°~~)

and the _average of the

bending

energy

(7i)o

_can be calculated

using

the

joint probability

distribution

p(t~~,

_{t~y, t~~~, t~~y,}

t~yy)

^of the first and second derivatives of the

height

field

t~(r)

which is

given by

_a Gaussian distribution

[16]

^{with the} non-zero correlation matrix elements

l~l)0

"

l~()0

"

)(k~l' ^l~lz)0

_"

l~]Y)0

"

)(k~l,

_(t~lV)0

"

(UzzUyy)0

"

~(k~). (7)

The moments of the structure factor _are defined

by

(kn)

=

~~ kn+i v(k)

dk

(8)

and the cut-off k~ is of the order of _an inverse molecular

size,

k~ ₌

~'~

= l. We will _use

r~

(k°)

₌

(I)

as a convenient notation for the 0th moment which is

proportional

to the _mean square average

(t~~)o.

Thus _{we can} write

(7i)o

_"

(2~ /

_dA ^~~

_H~)o

" 2~A

(~~ H~)o (9)

A dA dA

with

~~ ~~

~ 2

~~~

fi

^' dA

~

~~~~

and _we find for the ensemble _{average over} the

weighted

_{mean square} curvature

i$H~)

₌

_(ik~) ⁱⁱⁱ

+

(vl~)~)~~/~

+

(ivl~)~ (i

₊

(vl~)~)~~/~)o

₌

)ik~) ^Giik~)) iii)

where

G(x)

- T

II i

₊

₁₁

13 ^{4T +}

4T~) §(1 _erffi)1 (12)

with _x

=

27r/(k~). G((k~))

is bounded and

monotonically decreasing

0

£ G((k~)) £ I, G'(ik~)) _§

0 for

(k~) /

^0.

(13)

For small

(k~)

it _can be

expanded

into

G((k~))

= i

)(k~)

+

$lk~)~ ^{O((k~)~). (14)}

The function

G((k~))

contains the non-linear

coupling

between modes. As the harmonic _ap-

proximation

for the

bending

_energy is

proportional

to ~

(k~), comparison

with

equation (II) yields

the

effective, thermally

softened

bending

constant

[iii

'~e? "

G((k~))'~. (15)

Therefore _our variational method is

equivalent

to a Hartree

approximation

which

replaces

the non-linear Hamiltonian

by

_a ^Gaussian with effective

parameters

that _are determined self-

consistently.

For _a free membrane the _structure factor is known _to be

v(k)

=

(~k~)~~ Applying

_equa-

tions

(12), (14)

to this _{case we} retrieve the well-known first order renormalization correction of the

bending

constant:

G((k~))

_m 1-

3/(47r~) In(k~/km;n)

in

agreement

with the results of

[17, 18] (km;n

^is a lower

frequency cut-off).

For closed

systems

the

exict

form of the _renor-

malization is _{as we} will _see below different. It is in

general

not

possible

to _use the renormalization derived for _a free membrane in _a system characterized

by

other

physical

_con-

straints.

A second useful _average which will be used below

gives

_an

explicit expression

^for the _crum-

pling

factor

C

=

($)o

=

iii

₊

(vu)~)~/~)o (16)

which _can be

expanded

for

(k~)

« 1

C(lk~))

= i +

)lk~) £(k~)~

+

Ollk~)~). (17)

Results.

With the averages

equation (II)

and

equation (16)

the free _energy

density

_can be written _{as a} functional of the _structure factor

v(k) (using

relation

Eq. (3)

^for

d)

jjv(k)j

=

(

=

£ ^tt _{(k~) G((k~))} ^~~

_kin

_v(k) ^kl. ₍₁₈₎

~~ °

N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 767

This

expression

has to be

functionally

minimized with

respect

to

v(k)

under _one

constraint, i~

_cc

S/V

₌ Cte. which _can be

coupled

to

equation (18) by

_a

Lagrange multiplier.

The result will be _a minimal

v(k)

from which the

equilibrium layer spacing

and

crumpling

and the steric force law _can be calculated _as functions of

i~

and _~. We note that the free energy per area

usually

contains

self-energy

terms

proportional

to

k)

I-e- the number of

degrees

of freedom associated with the base surface. These terms

represent

an

insignificant

additive _constant in the discussion of the free _{energy per area,} but have to be omitted when

going

to the free energy per

volume;

_we will continue this discussion later when this

point

becomes relevant for the calculation.

The minimization _can be

conveniently

carried out

by variationally minimizing

jjv(k)j

₌

ttjk~)G((k~)) ~~

kin

v(k)dk

₊

~i C((k~))

₊

_~2 ((1))~)

~~~(i~)) (19)

°

under two constraints

~

~~~~~~ l~~~" ^~(~)~~

_~~~'

~~~~

The additional constraint _on d will be removed later

by df/dD

= 0. We

prefer

the notation

D((1))

to stress that the

layer density

is in this context not

preset

as in _zero order

theories,

but _a functional.

The result of

df/dv(k)

= 0 is

~~~~ ~4

/~2

+

(4

^~~~~

with

a =

~~~ G((k~))~~

=

~j/ ₍₂₂₎

~~~

~~

(k~) G'((k2))

~

j c'((k2))j ^j~

^~

^~~ D((i))D'((1))

°

Gjjk2))

_~

Gjjk2))

^'

~G(jk2))

~~~~

where _a >

0, i~

> 0 while

k(

_{can assume any}

sign.

Whereas for

given

surfactant concentration

is

and bare

bending

stiffness _~ the coefficient _a is

readily given by equation (22) [19],

^the coefficients

k(, ^i~

have to be evaluated from the non-linear

equation system

~

arctan

~~

+ arctan

~~~

~~

l= _{27rpd~ (24)}

@~ @~ fi~

~ ~ ~

(4

~~

~°

^~ _~

4

~~

kf k)k( ⁺¹⁴

^{~ ~~~~~ ~~~~}

~~

= 0

(26)

where

equations (24, 25) correspond

to

equation (20)

^and

^C~~

^denotes an inverse function.

The function

f(ko, I, _d;

~,

i~)

_can be obtained

by inserting

the relations

(k~)

=

j(k( ^2i~)(1) (k(

^In

^hi

⁺

)k) ⁽²⁷⁾

/~~

^kln

^v(k)

^{dk =}

)(k( 4i~)(1) (k(

^In

^hi

⁺

jk)

^In

^h2

+

k) (28)

o a

with the abbreviations hi

=

i~/(k) k(k]

₊

i~)

and h2

=

a/(k) k(k]

₊

i~),

and

equation (22)

into

equation (19)

f(ko, I, _d;

~,

is

₌

d~~

(»(2a)~~ ^{l~ d~ (87r)~~} kl (i

+ in _v

(kc ))j ⁽²⁹⁾

Thus _we have reduced the

problem

to the solution of three

relatively simple equations.

In

partic- ular,

solution of

equation (24)

and

equation (25)

is

straightforward

and reduces

f(ko, I, _d;

~,

#~)

to

f(d;

_~,

#~)

so that _{we are} left with the

single equation df(d;

_~,

_#~) lad

₌ 0. At this

point

we have to consider the

self-energy.

It is _a harmless _energy offset in

problems

where the total

projected

_area is constant. In the

present problem, however,

the number of lamellar

layers

is allowed to vary and the offset would _{cause a}

spurious ^d~~

term in the free energy. In order to subtract the

self-energy

_we fix the

quantities

_a,

ko, I,

_{and d at their}

_{physical values,}

_consider

the limit k~ - cc and discard all

diverging

terms. After

subtracting

the

divergences

the free energy

density

reads

f(ko, I, _d;

~,

is)

=

d~~ (p(2a)~~ ^i~

^{d~ +}

(87r)~~k) In(1 k(kj~

₊

i~kj~)j ⁽³⁰⁾

The

equation system equations (24-26)

with

f given by equation (30)

defines the solution of the

problem

to all orders in

~~~.

We start

by solving analytically

to first and second order in

~~~

_Since

ko/k~

and

k/k~

are very small

quantities

_{we may}

expand

the

logarithmic

term in

equation (30)

which then becomes

independent

of k~ and

equal

to

-(87r)~~ k(. Solving

the

simplified equation system yields

the well-known first order results for the structure factor

[5],

k(

_m

o, i~

m

(8p)~~ ~~~ #) (31)

the

crumpling

factor C

=

#sd

m 1 + ci

~~~ c2~~~

with

[3],

the renormalization factor G _m I

gi~~~

₊

g~~~~

with

[3],

gi =

-] ini(8»)-1/2 ~-i/21si (33)

and the free _energy

density

_[3]

f(~, Is)

m

(128»)~~ ~~~4i (34)

In second

order,

_a k~ term with

positive

coefficient

k(

m

~

~~~#) (35)

128p

emerges in the _structure factor which should be observable in

systems

^with low

bending

stiffness

(~

m kT

[10, 20]

as a

pronounced rounding

_or

slight bump

in the

scattering

structure factor at low k.

The second order corrections to the

swelling

law and renormalization _are

~~

~2 ~2

~~~~~~~~ ~~~'~ _{~~~~~~ ~~~~}

N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS ₇₆₉

~~~

~~

~2

~

/7r2

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

Both contain

non-logarithmic

terms whicl~ _are

directly

related to film _area conservation.

The free energy

density

_up to second order in

~~~

_reads

f(~Q, ~is) * b1 _~G ~~i~ ~2 ~G ~~i~ + ~3~G ~~i~

(~~)

with

bi

"

(128p)~~,

b2 _"

3(10247rp)~~,

b3 "

(5127rp~)~~

^{The first} term on the rhs does

not contain

logarithmic

renormalization terms which could

originate

from the non-linear

part

of the

bending

_energy G and the

crumpling

factor C because these two contributions cancel each other in second order. This indicates that Helfrich's results remain

largely

unaffected

even

by

second order _terms and it

might explain why

Helfrich's

high

_~ model _can in fact be used for the

interpretation

of data taken in

semi-rigid systems [20]

^this

point

has caused _some

controversy

^{in the literature}

[21].

^{The second} term _on the rhs of

equation (38)

is identical to _a term found

by

Golubovid and

Lubensky

_[3] in their

perturbation analysis

and _can be

rationalized _{as a} non-local interaction term due to the surface _area constraint. The last term in

equation (38)

is

proportional

to

if;

Wennerstr6m and Olsson

[22]

^have

recently

discussed

such terms

although

derived in _a different

theory

from

higher

order

elasticity

terms in the

context of the lamellar to sponge transition. This term becomes

significant

at

high

surfactant

concentration.

However,

the above

approximations

turn out to be unreliable at low

bending rigidity.

We have therefore solved the

equation system yquations (24-26) numerically.

In _a series of

figures (1-3)

_we show numerical results for

k(

and k~

(Fig. I),

the

crumpling C,

the renormalization

G,

and the free _energy

density f

for realistic values of _~ and

is.

The

swelling

factor

i~d

in

figure

2a shows the

typical logarithmic dependence

_on

is

which has been verified in

experiment [10]

for stiff film _~

=

5kT,

but _a

systematic upward

deviation for

high

dilution in the _case of soft

membranes,

_{~ =} ^{1kT. This deviation should be measurable and} characteristic for soft lamellar

phases.

When

comparing

numerical and first order results _we note

significant

differences in

the _{case ~ =}

1kT;

this _{casts some} doubt _on the first order

fitting procedure

used in

[10]

^to

estimate the value of the

bending

modulus in lamellar

phases

and _we believe that the values for the

bending

^moduli

(of

the soft

systems) reported

there _are underestimated

by

factors of

m 2-3.

Indeed,

this correction factor _seems to reconcile the results of the measurements of _~

given

in

[10]

^{with the results of} alternative measurement

techniques [23].

^In

figure

2b _we show the concentration

dependence

of the renormalization correction to the

bending

modulus. As

expected, higher

anharmonic terms lead in the _case of soft membranes to

strong

^deviations from the first order

approximation. Finally,

in

figure

3 the free _energy

density

_{as a} function of the

bending

modulus and the surfactant concentration is shown. At

given is

the steric

repulsion

is

always

lower than

predicted by

first order

approximation.

For _a realistic

regime, is

=

0.I,

lkT < _~ <

10kT, (Fig. 3a)

_we find that the

approximation

is valid down to _{some ~ m} 5kT.

For softer

systems

^the

complex interplay

of anharmonic corrections to the Hamiltonian and the

swelling

corrections due to surface _area conservation lead to deviations from the

1/~

force law.

However,

as

argued above,

due to cancellation of renormalization and

swelling

terms up to second order in

~~~

the

scaling f

_cc

i(

is

practically unchanged

_even for small _~

= 1kT

(Fig. 3b).

Conclusions.

Finally,

_we want to discuss the

shortcomings

and merits of the

presented approach. Monge

gauge cannot, as mentioned

above, represent

states with

complex shape

and

topology

1-o 2.5

2- ~

,,' ,"

,"

i 5

o

o-z o.5

o-o o-o

0.0 o-Z 0.4 0.6 o-B I-o 0.00 0.05 o-lo 0.15 0.20 0.25

#~

a) b)

Fig.

I. The coefficients

k(

and

i~

_{in the}

scattering

structure factor equation

(21)

_as functions of the

bending

constant _K and the surfactant concentration is.

a) kl'10~

us. K~~

(solid line)

and

k~.10~

us. K~~

(dotted line)

for

is

₌ 0.I.

b) k(

2.5

x10~

us.

#( (solid line)

and

i~

_10~

us.

#( (dotted line)

for _{K =} lkT

(upper curves)

and _K

= 5kT

(lower curves).

1.6 1.0

o-B

_,,,

---'"''"

.4

0.6

C

' ",

', _, ,

"', ~

"

, ,

", _{0 4} ,'

', '

1-Z

",,

_,"

', ,

', ,'

0.2 ,' ,

,'

"'- _--- '

''--,,, '

,'

1-O 0.0 "

-3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5

In(#~) ln(#~)

a) b)

Fig.

2.

a)

The

crumpling

ratio G

= <ad of ensembles of

undulating

membranes for _K

= lkT

(upper curves)

and

K = 5kT

(lower curves)

_{as a} function of the surfactant concentration. Solid lines denote accurate numerical solutions, and broken lines the respective first order

approximations (Eq.

(32)).

The solid lines show _a small deviation from the

logarithmic

law.

b)

The renormalization of the

bending

constant G _{as a} function of surfactant concentration: numerical solutions

(solid)

and first

order

appro~imations (broken, Eq. (33)),

for the _{K =} lkT

(lower curves)

and _K

= 5kT

(upper curves).

N°5 VARIATIONAL THEORY OF UNDULATING MULTILAYER SYSTEMS 771

f _ii

o.7j f _ii

o.~j

o-B i-o

, ,

, ,

, ,

, ,

,' _0.8 ,'

0.6 ,' ,'

,, ,

, ,

, ,

, ,

,' 0.6 ,'

, ,

0.4 ,"

,

, ,

, ,

, ,

, , , ,

o-Z

o-o o-o

o-o o-Z 0.4 0.6 o-B I-o 0.00 0.05 o-lo 0.15

K~~ ^~~~

a) b)

Fig.

3.

a)

The free _energy

density f

_{as a} function of the

bending

_constant, at is ₌ 0.I

(solid

line);

it deviates at low _K

visibly

from Helfrich's law

equation (34) (broken line). b)

The free energy

density f

_{as a} function of the surfactant concentration for

K = lkT

(upper curves)

and

K = 5kT

(lower curves),

where solid lines denote numerical

solutions,

and broken lines the

corresponding

Helfrich

approximation.

fluctuations. Therefore the Gaussian curvature

f

K dS which is

coupled

to the

bending

_energy

by

the

saddle-splay

modulus k, does _{not enter} the calculation. Inclusion of this term leads to k

dependent

contributions to the structure factor and free energy

density, f

_cc

b(~, k) #(,

and is

likely

to be crucial for the still

poorly

understood lamellar to sponge transition

[24].

^This

requires

_a

sophisticated

state

representation

which includes

topological

defects. Work _on this

important non-perturbative generalization

will be

presented

elsewhere.

Nevertheless,

the

approach presented

here

is,

within the

validity

of its

assumptions,

able _to

provide

_a

simple

and consistent

description

of multilamellar

phases

in terms of _structure

factor, swelling law,

renormalization of the

bending

constant and the steric force law _as functions of the surfactant concentration and the

bending

modulus. Its range of

validity

_goes well

beyond

that of low temperature ^theories

[3, 5]. However,

the

emphasis

of the

presented

treatment of closed multilamellar

systems

is _on

principles

rather than numbers.

Changes

in the constants p ^or k~ affect

although

not

strongly

the numerical results without

changing qualitative

features,

thus somewhat

restricting

the

predictive

_power of the model.

The results _are in

agreement

^{with known}

observations,

and reveal _new features which _are related to the _more accurate inclusion of

layer crumpling,

the constant _area constraint and the

usually neglected coupling

terms in the

bending

Hamiltonian. These should be observable in the structure factor and

swelling

law of soft and dilute lamellar

phases

_[9]. Our results also show that Helfrich's first order steric force law is in fact also _a

good

second order

approximation,

indicating

that

simple predictions

of the Helfrich

theory might

be

applicable

_even in

semi-rigid

regimes.

Acknowledgments.

It is _a

pleasure

to

acknowledge

^useful discussions with S. A.

Safran,

D.

Roux,

J. S.

Huang,

W.

Helfrich,

B. Ninham and R. Menes. P. P.

acknowledges partial

financial support from the

von Hoesslinschen Foundation of the

City

of

Augsburg, Germany.

References

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For reviews: Statistical Mechanics of Membranes and

Surfaces;

D.

Nelson,

S.

Weinberg

Eds.

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[2] David

F.,

Leibler

S.,

J.

Phys.

II IYancel

(1991)

959; the authors discuss in detail the extensive variables

(real

and

projected

film

area)

and their

conjugate

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[3] Golubovid L.,

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T.

C., Phys.

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(1989)

12110.

[4] ^The

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gives

_a

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W.,

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305.

[6] ^Porte

G.,

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M.,

Billard I., et

al.,

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R. P., Statistical Mechanics

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A.,

Statistical

Thermodynamics

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Wes-

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L.,

Leibler S.,

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1690.

[19] ^{Note that} is determines

C((k~))

which in turn determines

G((k~)).

[20] Safinya C. R., Roux D., Smith G. S., et al.

,

Phys. Rev. Lett. 57

(1986)

2718.

[21]

Gompper G.,

Schick

M.,

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