Undulation-enhanced electrostatic forces in lamellar phases of fluid membranes

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Undulation-enhanced electrostatic forces in lamellar phases of fluid membranes

Renko de Vries

To cite this version:

Renko de Vries. Undulation-enhanced electrostatic forces in lamellar phases of fluid membranes. Jour-

nal de Physique II, EDP Sciences, 1994, 4 (9), pp.1541-1555. �10.1051/jp2:1994216�. �jpa-00248060�

Classification Physics Abstracts

5.40 82.70 87.20

Undulation-enhanced electrostatic forces in lamellar phases of fluid membranes

Renko de Vries

Faculty of Clienfical Engineering, Dept. Polymer Teclin., Delft University of Technology, P. O-Box 5045, 2600 GA Delft, The Netherlands

(Received

23 March 1994, received in final form 23 May 1994, accepted 31 May

1994)

Abstract. A formula for the free _energy of _a stack of highly charged, semiflexible _mem- branes, previously derived by Odijk

(Odijk

T. Langmuir,

1992),

is shown _to be the first order

term in _a formal expansion of the free energy in terms of powers of the electrostatic potential.

The formula describes the transition from _a regime where the free energy is dominated by the electrostatic _energy to _a regime where it is dominated by bending _energy and entropy.

Introduction.

The

physics

of _a stack of semiflexible membranes

interacting

via _an electrostatic

potential,

has been addressed

by

_a number of authors [1-6]. ^For

highly charged membranes,

undulations _are

unimportant

_as

long

_as the distance between the membranes is either smaller than _or of the order of the

decay length

of the electrostatic

potential (~). Upon adding salt,

_or

increasing

the distance between the

membranes,

the undulations become

stronger

^and stronger, ^and may

substantially

^enhance the electrostatic

repulsion ill. Finally,

ⁱⁿ the limit of _very

high

salt concentrations _or

large

distances between the

membranes,

the situation is described

by

Helfrich's

theory

[7], except ^for a

boundary layer

_near the

membranes, preventing

them from

touching

each

other,

due to the electrostatic

repulsion

_[2]. The

regime

to which Helfrich's

theory

does _not

apply,

_{covers an}

appreciable

_range of membrane

separations

and salt concentrations.

For membrane

separations

_on the order of the

decay length

of the electrostatic

potential,

undulations will be weak. Then _a harmonic

approximation

of the

Hamiltonian,

in which

only

terms

quadratic

in the

amplitudes

of the undulations _are

retained,

is

legitimate.

In this

regime,

which will be called the harmonic

regime,

there is _a small enhancement of the electrostatic

repulsion, proportional

to the square ^root of the bare electrostatic

potential

_[3,

4].

^{If the}

(~)For

_very weakly charged membranes the situation is somewhat different. Here undulations _are

important _even for distances between the membranes of the order of the decay length of the potential.

amplitude

^of the undulations becomes of the order of the

decay length

of the electrostatic

potential,

^the harmonic

approximation

^{breaks down.}

A number of theories have been

proposed, dealing

with the

regime

where the harmonic

approximation

is _no

longer legitimate

and where it may be

expected

that there is _a substantial enhancement of the electrostatic

repulsion

_{11, 4, 5].} For this

regime, only

the

Odijk theory ill provides simple

closed

expressions(~). Postulating

a Gaussian

single

membrane distribution

function for the

amplitudes

of the

undulations, Odijk

derived

simple

closed

expressions

for the free _energy, the osmotic _pressure and the _average

amplitude

of the undulations in _a stack of

charged

semiflexible membranes. In view of the

highly

anharmonic interaction

potential however,

it is not clear to what extent the distribution of the

amplitudes

of the undulations will be _a Gaussian.

Here it is shown

rigorously

that the

Odijk expression

for the free _energy, from which _expres- sions for the osmotic pressure and the _average

amplitude

of the undulations _can be

derived,

and which is called the Gaussian

approximation,

is in fact the first order term of _a formal

expansion

of the free _energy in terms of _powers of the electrostatic interaction

potential.

The range of

validity

of the first order _or Gaussian approximation is determined from _an estimate of the contribution of the second order _term. The Gaussian

approximation

describes the tran- sition from _a

regime

where the free _energy is dominated

by

the electrostatic _{energy, to a}

regime

where it is dominated

by bending

_energy and entropy.

Beyond

the

point

where the Gaussian

approximation

breaks

down,

the convergence of the

expansion

for the free energy ^is poor.

The electrostatic

potential

must be calculated

using

the non-linear Poisson-

Boltzmann

equation.

It is assumed that the

decay length

of the electrostatic

potential,

the

Debye length,

is _a great ^{deal smaller than both the}

typical

distance between the membranes and the

typical wavelength

of the undulations.

Thus,

it is assumed that the orientational order is

high

and that the average

amplitude

of the undulations is much smaller than the _average

distance between the membranes. For the interaction between two small

pieces

of membrane opposite to each other and

separated by

_a distance

D,

in _a stack of

undulating membranes,

_we

can then _use the

expression

for the electrostatic interaction energy between two flat

surfaces, separated by

_a distance much

larger

than the

Debye length

_{11, 9]:}

V(D)

=

8xZ(~(~Q~~)e~~~ ₍₁₎

~~~

^{~~ ~}

~~~~~~

~~~

Z =

Qa/~

is _a dimensionless surface

charge density

in terms of the two electrostatic

length scales,

the

Bjerrum length Q

"

e~lekBT

and the

Debye length

^~~~ ₌

(8xQn)~~/~

and the

number of

elementary charges

_per unit _{area, a.} kB is Boltzmann's constant, T is the absolute temperature, e the

elementary charge,

e the

permittivity

of the

solvent,

which contains _an

excess of monovalent

electrolyte

of concentration _n.

V(D)

is the

potential

of interaction per

unit area, scaled

by kBT.

Since _{we are}

only considering

_cases in which the

amplitude

of the membrane undulations is

appreciably

smaller than the distance between the

membranes,

short range

repulsive potentials,

to account for

possible hydration

forces _or collisions _etc. will not be included in

V(D).

The deformation behaviour of the membranes is described

by

_a

single

elastic

bending

modulus

K,

scaled

by kBT.

An electrostatic contribution to K

[10-12]

^should be included

explicitly.

^Since it _was assumed that the orientational order is

high,

_we dust have

K »1.

(2) podgornik ^and Parsegian _[4] derive approximations for several limiting _cases, but not the expo- nential renormalization of the electrostatic free energy that is found by Odijk. Unfortunately, Evans and Ipsen _[5] do not present ^closed expressions for the free energy and the osmotic _pressure.

The reference

configuration

is taken _to be _a stack of flat

membranes, separated

from each other

by

_a distance D. The stack of N

fluctuating cliarged

membranes will therefore be

described

by

the Hamiltonian

(scaled by kBT):

7ilzl

₌

L / _d~x ) _(Azn(x))~

+

v(D

₊

zn(x)

_zn-i

(x)) (3)

~i

where

(x, zn(x)

_are the Cartesian coordinates of the

points

_on the n-th membrane and A ₊

Ax

is the two-dimensional

Laplacian.

The undulation

amplitudes zn(x)

will be

decomposed

into Fourier

modes, along

the lines of David's

analysis

_[8]:

zn(x)

₌

~j _{z(kjj, ki} exp(ikjj x) exp(ikin) ⁽⁴⁾

kjj,ki

Here kjj ^is a twc-dimensional

in-plane

wavevector, while ki labels the

perpendicular

modes.

It is _a

phase

difference between successive membranes.

Values,

allowed for the wavevectors k ₊

(kjj,ki)

follow from

periodic boundary

conditions

zn(zi

+

L,z2)

"

zn(zi,z2

+

L)

=

zn(zi, z2)

for _some

macroscopically large length L,

and

zn(zi, z2)

= zn+N

(xi, z2).

The range of ki is restricted _to allow for

exactly

N

independent perpendicular

^modes: -x < ki < x.

In _terms of

z(k),

the

Hamiltonian,

scaled

by kBT

and

by L~N,

_can be

decomposed

into

7i ₌

V(D)

+ 7io + 7iI, where 7io is

quadratic

in

z(k),

~ojz(k)j

=

I £(Kkj

₊

4sin2(ki/2)v"(D) _{)z(k)z(-k) (5)}

2

~

and

7iI

contains the

higher

order terms:

7iI(z(k)]

₌

j~ _~j

^~

_{~~~~)~~ jj(1}

_e'~~~

_{)z(ki) (6)}

m=3ki,k2;..,km ^~' l=1

V(~')(D)

denotes the m-th derivative of

V(D).

The asterisk indicates that there is _a restriction

on the set of k values to be summed _over:

kill

+

k2

II

+ + km

II

" °

kit+k~i+..+kmi

₌

2xl,1=0,+1,+2,.. (7)

For membrane

separations

_on the order of _a

Debye length,

^the undulations _are weak. In this _case the _energy of _a

configuration

_may be

approximated by

7io. This is the harmonic

approximation,

which _was also used

by

Pincus et al. [6] ⁱⁿ their treatment of the _same

problem.

The renormalization _group flow

equation

for the effective

potential

of David [8] ^{is also} derived in this

approximation.

In the harmonic

approximation,

the Fourier transformed correlation

function is

given by:

So(k)

₊

(z(k)z(-k))o

₌

~~~~

~

-~(~~~~~~

~~~

(8)

Here, lo

denotes

thermodynarlic averaging

^with respect to

7io(z(k)].

The correlation

length

in the harmonic

approximation,

lo ₌

(K/V"(D))~H

will be called the bare correlation

length.

In-phase

undulations of the stack of membranes may have _very

large amplitudes,

determined

only by

^their

bending

_energy. ^Therefore one can not

simply integrate

the correlation function

jOURNAL DE PHYSIQUE II -1 4 N' 9 SEPTEMBER 1994 58

to find _{an average}

amplitude

of the undulations. This

integral diverges logarithmically,

_as it should for _a smectic.

Instead,

the _average

amplitude

of the undulations is taken to be the

mean square deviation of the distance between the membranes with respect to their distance in the reference

situation,

D.

Taking

the limit

N,

L _{- cxJ we} have

d(

₌

j(z~(x) z~-i(x))2)o

=

/ so(k)4sin2(ki/2)

=

£ (9)

k

where

~ ~~ ~ ~~~

/ /_~ (2~ /_~ ^2x)~~

~~~~

The

subscript

0

again

indicates that

do,

which will be called the bare

amplitude

of the _un-

dulations,

is the average

amplitude

of the undulations in the harmonic

approximation.

The free energy per membrane _per unit area, with respect to free

membranes,

in the harmonic

approximation,

is

given by:

Fo

=

V(D)

+

/ In(Sp~(k)/Kk()

=

V(D) (I

₊

_{(~do)~) (II)}

k

As will be shown in the _next

section,

the harmonic

approximation

is valid _as

long

_as

~do

< I.

In its range of

validity,

it

predicts

_a small renormalization of the interaction,

proportional

to the _square root of the bare electrostatic

potential,

that is,

proportional

_to

exp(-~D/2),

_{as was}

already recognized by

Evans and

Parsegian

_[3].

However,

_{as was}

clearly

shown

by Odijk ill,

if the

amplitude

of the undulations is

appreciably larger

than the

decay length

of the electrostatic

potential,

there is _a substantial enhancement of the electrostatic

repulsion. Assuming

_a

single

membrane Gaussian

amplitude

distribution for the

undulations, Odijk

found _an electrostatic free energy of:

Fei

_"

V(D) exp((~d)~ /2) (12)

where

d/vi

_{is the root}

mean square

amplitude

of the undulations. The loss of free _energy due

the confinement of the

undulations,

with respect to free membranes _was estimated to be

?cant

₌

] ₍₁₃₎

with _a constant _c

equal

to about

1/64(3 ).

The value of d is found

by minimizing

the total free energy. If

(~d)~

becomes of order

~D,

the electrostatic free _energy

diverges

_as D _{- cxJ} and the

theory

breaks down [2]. ^{Therefore the}

theory

will not be valid for _very strong undulations.

The electrostatic free _energy is calculated

assuming

that

neighbouring

membranes fluctuate

independently. Also,

director fluctuations _are

neglected.

I find in the

following analysis,

that the

Odijk

result _can also be found without

assuming

that

neighbouring

membranes fluctuate

independently

and without

neglecting

the director fluctuations. The

Odijk

result is shown to be the first order _term in _a formal

expansion

of the free _energy. There _is in fact _a close relation between the present

approach

and the previous variational

approaches

_11, 4, _5] based

on the classic

Gibbs-Bogoliubov (or Peierls) inequality.

To stress this relation it will be _ex-

plicitly

shown that the exact first order result _can

always

be found from the

Gibbs-Bogoliubov inequality, using

_a trial Hamiltonian which is

quadratic

in the undulation

amplitudes z(k).

(3 ~~

jij

tb~ c~~~t~nt c was estimated to be

1/32,

but _{a more} precise calculation shows that it should

b~

1/64

~Thich is the value adopted in [4].

The

expansion

^of the free _energy.

Consider _a trial Hamiltonian

7itr[z(k)], quadratic

in the Fourier transformed fluctuations

z(k):

7itr[z(k)]

=

V(D)

₊

~ _~~~~

~ ~ ~~~~~ ~~~~~~~~~~~~~~~~~ ~~ ~~~~

This form is obtained from the

quadratic

part ^of the full Hamiltonian

7io(z(k)] by replacing

the bare correlation

length lo

=

(K/V"(D))~H _by

an as yet

unknown, k-dependent

correlation

length A(k).

The

Gibbs-Bogoliubov inequality

states that

? I

?tr

+

17ilz(k)1 7itrlz(k)I)tr (15)

Here F is the true free _energy of the system, Ftr is the free _energy of _a system with Hamiltonian

7itr(z(k)]. Further, )tr

denotes

thermodynamic averaging

with respect to

7itr(z(k)].

The correlation

length A(k)

is obtained

by minimizing

^the

right-hand

side of

(15)

with respect to

>(k).

It will _now be shown that the trial Hamiltonian 7itr ^{in fact}

yields

the first order _term in _a formal

expansion

of the free _energy in terms of powers of the

potential V(D).

Let _us

expand

the free _energy in the usual way in terms of contributions of connected

diagrams.

A connected

diagram

with 2n-external lines will be called _an

n-diagram,

_{n =} 0,1,... The free energy per membrane _per unit area, with respect to free membranes _can be written _as:

F

=

v(D)

+

( in(Sii(k)/Kki)

~j [contributions

of all

topologically

distinct

0-diagrams] (16)

where the minus

sign

accounts for the fact that the free _energy is minus the

logarithm

of the

partition

^function. The rules to find the contribution

represented by

_a

diagram

of

given topology

are:

I)

associate _a factor

-V(~')(D) £[

~~ ~

fl]~(l

e'~~~

)z(ki)

with every m-th order vertex, where

ki,k2;. km

_are the

wile tic(rs _labeling

_{the vertex;}

it)

associate _a factor

So(ki)b(~

_~~ with _every

line,

internal _or external, where ki and k2 are

the _wave vector labels

on'that _line;

iii)

divide

by

the symmetry ^{number S of the}

diagram.

The symmetry number is the

product

of the number of _ways of

permuting

the vertices and the number of _ways of

permuting

the

lines,

while

keeping

the contractions

represented by

the

diagram

the _same.

The correlation function is the _sum of the contributions of all

topologically

distinct I-

diagrams.

This _{sum can} be written in terms of the sc-called _proper

self-energy, Z(k).

A

diagram

is called _proper if it cannot be

split

into two

pieces by cutting

_one internal line. The contribution of all proper

I-diagrams

to

S(k)

_can be written _as

So(k)Z(k)So(k)

and

S(k)

is

given by:

~~~~ ~~~~~~~ ~~~

Sp~(k/- _Z(k)

^~~~~

Any

insertion of _a

I-diagram

into _an internal line of _some other

diagram

is called _a

self-energy

insertion. Let _us call _a

diagram

with _no

self-energy

insertions in its internal lines _a skeleton

diagram.

^The contributions of all proper

I-diagrams

to

Z(k)

_are obtained

by drawing

all

skeleton

I-diagrams

and then

making

all

possible self-energy

insertions into the internal lines.

This is

equivalent

to:

Z(k)

=

~j [self-energy

contributions of all

topologically

distinct

skeleton

I-diagrams,

with

So(k) replaced by S(k)] (18)

Generally

this is an

integral equation

^for

Z(k).

A

problem

with the

expansion

of the free energy ^is that the bare correlation

length lo

enters via the propagator So

(k).

This

length

will be

strongly

renormalized

by

the anharmonic terms of the Hamiltonian contained in

7ii[z(k)]

_{as soon as} the

amplitude

of the undulations d becomes of the order of the

decay length

^~~~ of the interaction

potential. Clearly

it would be better if

only

the renormalized correlation

leng~h

entered the

expansion

of the free _energy. This _can be

accomplished

if somehow

So(k)

_can be

replaced by S(k).

To obtain the free _energy in terms of

S(k),

one cannot

simply

_sum over all skeleton

0-diagrams

with

So(k) replaced by S(k),

since this would result in _an

overcounting

of

diagrams

[14]. ^{Lee and}

Yang

[13] ^and

Luttinger

and Ward [14] ^{have shown how to}

perform

the proper resummation. Because there _{are some} differences between this calculation and the quantum _one, the

analysis

in _{our case} is

given

in detail in

Appendix

A. It is found that:

~ "

~(~)

+

~~(~

^~

(~)/~~()

+

~(~)~(~)

~j [contributions

of all

topologically

distinct

skeleton

0-diagrams,

^with So

(k) replaced by S(k)] (19)

In

Appendix

^A it is also shown that

~~

@

^{~ ~~~~}

Hence, the functional F has _an extremum for the function

Z(k)

that satisfies

equation (18).

As is mentioned

by

Lee and

Yang,

this _can be used _{as a} variational

principle generating

the

equation

for

Z(k)

from the free _energy functional.

Now consider the correlation function of the system ^with the trial Hamiltonian:

~~~~

K(k(

+

-4(/)4

sin~(ki/2))

^~~~~

For the function

A(k)

for which the

right-hand

side of

(15)

has _a minimum, this will be _an

approximation

of the correlation function of the true system.

Therefore,

_an

approximation

of the _proper

self-energy

of the true system ^is

#ven by

Z(k)

₌

K(Ap~ A~~(k))4sin~(ki/2) ₍₂₂₎

In terms of

Z(k)

thus

defined,

the

Gibbs-Bogoliubov inequality

_can be written _as:

F <

V(D)

+

/ ln(S~~(k)/Kk()

+

/ _S(k)Z(k)

2 ~ 2

~j [contributions

of all

topologically

~ distinct I-vertex

0-diagrams,

with

S(k)

_as

propagator] (23)

where

Z(k)

is to be found from

(20). Thus, using

this

inequality together

with the trial Hamiltonian

7itr, naturally

leads to the first order term

(I.e. only including

I-vertex

diagrams)

of the

expansion (19),

in terms of powers of

V(D).

However, the full expansion is essential for

calculating higher

^order terms.

The Gaussian

approximation.

From the

equation

for

Z(k)

it _can be _seen that to first order in

V(D)

the bare correlation func- tion So

(k)

is

only

renormalized

by "tadpole diagrams" (I.e. diagrams

with interaction lines

starting

and

ending

at the _same

vertex).

This

merely

amounts to _a

k-independent

renormal- ization of the bare correlation

length lo Thus,

in this _case the free _{energy can} be calculated

using

_a correlation function

S(k)

with _a

k-independent

correlation

length

A. The contribution

bfi

to the free _energy of the first order skeleton

0-diagrams,

^shown in

figure I,

is:

bfi =

~j ^~~~~~f~ /

S(k)4sin~(ki/2)) ⁽²⁴⁾

~~

_2'~m.

k

~

Because the correlation

length

is

k-independent,

the relation between the _average

amplitude

of the undulations d and the correlation

length

is

again #ven by

d~

=

A~/2xK.

The free energy ^to first order in

V(D)

_{can now} be calculated. In terms of

d,

F ₌ ~

~ +

V(D) exp((~d)~ /2) ⁽²⁵⁾

In

principle

the value of d must be found from the

equation

for

Z(k). However,

since

Z(k)

_can

be considered to be _a function of

d,

the functional derivative in

(20)

reduces to _an

ordinary

one:

dF/dd

= 0

(26)

which _expresses the renormalization of the bare

amplitude

of the

undulations,

do

(~d)~ exp((~d)~/2)

=

(~do)~ (27)

This represents

essentially

the Gaussian

approximation,

the

only

difference between the

Odijk expressions

and the

expressions presented here,

is in the constants. Because the actual

repulsion

is stronger ^than is assumed in the harmonic

approximation,

the renormalized

amplitude

of the undulations is smaller than the bare _one. If ~d _<

I,

the renormalization of do can be

neglected

and the harmonic

approximation

is sound.

in

2

4

Fig. I. The _one vertex skeleton 0-diagrams contributing to the free _{energy, m} > 2.

It is difficult _to evaluate the free _{energy to}

higher

order in

V(D),

since

A(k)

_{can no}

longer

be assumed to be

k-independent.

Even if this _were not the _case,

evaluating

the contributions of

higher

order

diagrams

in _an accurate way poses

problems.

For

example,

in order to estimate the contribution of the second order term, we make _an additional

approximation, eliminating

the difficult

ki integrations.

A stack of membranes is considered in which there _are

only

two

perpendicular modes,

_an

in-phase

and _an

out-of-phase

mode. This _{amounts to}

setting

N

= 2

in equation

(3)

for the

Hamiltonian,

instead of

letting

N _{- cxJ.}

Then,

_as is shown in

Appendix B,

the second order contribution to the free _energy is

roughly

&>~ _-~

>2v2(D)~~P)~jj(~~~

(28)

provided (~d)~

» 1. The second order contribution

bF2

will be

extremely

small

compared

to the first _as

long

as

(~d)~

is

appreciably

smaller than

2/3 (~D).

We therefore

tentatively

conclude that the first order

approximation

will be valid _as

long

_as

(~d)~

< cl

(~D) (29)

with _a constant cl

equal

to about

2/3. Showing

that the _sum of all

higher

order terms will be

small if this bound is

fulfilled,

is very difficult.

Nevertheless,

in

Appendix

B it is shown that if the second order contribution is _no

longer

^small

compared

to the first order term, ^then the third order _term will not be small

compared

to the second order term

either,

I-e-,

beyond

the _range of

validity

of the first order _or Gaussian

approximation,

convergence of the

expansion

of the free _energy is poor. This

might

indicate that _{we are}

entering

_a

regime

^where the electrostatic

potential

starts

looking

like _an effective hard-wall

potential,

for which _an

ordinary perturbation expansion

of the free _energy

simply

does not work. In view of the _poor convergence of the

expansion beyond

^the _range ^of

validity

of the first order _or Gaussian

approximation,

it does not seem worthwile to carry the

perturbation expansion

of the free _{energy any} further.

Discussion.

For

highly charged membranes,

the effective surface

charge density

Ze~ reduces to _a constant

x~~ _{and the} electrostatic contribution to the elastic

bending

modulus is

predicted

to reduce to I

/x~Q (10, iii. Assuming

K ₌ Ko + I

/x~Q

and Ze~ ₌ x~~ _and

fixing

the value of the intrinsic

bending

modulus

Ko,

the

amplitude

of the undulations d is _a function of the dimensionless variables ~D and

~Qi

~~~~~~~~~~~~~~~~~

~j~~~

~~i~~o

^~~~~

For _an aqueous monovalent

electrolyte

of concentration _ns

(in M)

at _room temperature, ~~~

=

0.30ni~/~nm

_and

_Q

" 0.71 _nm, whence

~Q

"

2.37n(/~

In table I values of ~d have been

tabulated _{as a} function of ~D and the salt concentration _ns

(in M),

for

Ko

₌ I. Since ~d

can be much

larger

than unity, the enhancement of the electrostatic

repulsion, proportional

to

exp((~d)~ /2),

_can be substantial. Now consider the ratio of the electrostatic to the confinement free energy,

given by

As

expected,

^{this decreases if the} fluctuations increase.

Only

if

(~d)~ /2

« 1, does the electro- static free _energy dominate _over the confinement free _energy. If

(~d)~/2

_{» 1 it is} the other

Table I. ~d _{as a} function of _ns

(in

M

)

and ~D for

Ko

" I.

~D

ns

(in M)

10 20 30 40 50

1.0x10~~ _{1.3 3.7 5.5 7.0} 1.0x10~3 _{0.9 3.3 5.2 6.7} 1.0x10~~ _{0.6 2.8 4.8 6.4 7.7}

1.0x10~5 _{0.3 2.2 4A 6.1 7A}

way around. For values of

(~d)~

_m

2/3 (~D)

where the first order _or Gaussian

approximation presumably

breaks

down,

the electrostatic free _energy is

only

_a small fraction of the total free

energy. Whereas the free _energy

decays roughly exponentially

in the

electrostatically

dom- inated

regime,

it

decays roughly algebraically

in the

regime

dominated

by

confinement free energy. It must be noted however, that in the

regime

dominated

by

confinement free _energy, the

amplitude

of the undulations is still _very much affected

by

the electrostatic

repulsion.

Beyond

the range of

validity

of the first order _or Gaussian approximation a

regime

is entered in which the electrostatic interaction

only

affects the

amplitude

of the

undulations,

but does

not any

longer

contribute

significantly

to the free _energy. In this

regime,

the effects of steric- and

hydration repulsion

_can

probably

_no

longer

be

neglected. Although

it is

straightforward

to estimate the order of

magnitude

of the

repulsion

in this

regime, dealing

with the

interplay

between the effects of steric-,

hydration-

and electrostatic

repulsion

in _{an accurate} way is

extremely

^difficult.

Nevertheless,

the

strongly fluctuating regime,

I-e- the

limiting

_case of great

separations

_{or very}

high

^salt concentrations, _can be

analyzed qualitatively

_[2]. In this limit the situation is described

by

Helfrich's

theory

_[7], except for the existence of _a

boundary layer

of thickness

b, independent

of

D, preventing

the membranes from

touching

each other. It _was

argued by Odijk

_[2] that in this

limit,

the

typical

_area of interaction of two

colliding membranes,

will be of order K~~~ The thickness b of the

boundary layer

_can then be estimated from

K~-2v(&)

=

o(1) (32)

And the free energy will be

given by:

~

K(D b)2

^~~~~

where _c2 is the _constant

appropriate

to the Helfrich free _{energy, c2 "} 0.10. This is

expected

to be valid down to

separations

for which ~b _m

1/2

~D. From the fact that the

strongly

fluctuating regime

cannot

overlap

with the

re#me

^for which the Gaussian

approximation

will be

valid, Odijk

concluded _[2] that

(~d)~

<

ci(~D) (34)

with _{cl m} I, is _an approximate bound _on the

validity

of the Gaussian

approximation,

in fair agreement ^{with the value} cl " 2

/3

obtained from the estimate of the contribution of the second order _term.

From _a bound like

(34)

it is

actually possible

_to understand the _structure of the second order _term in _a

qualitative

way.

Obviously, bF2

°J

-V~(D),

with _a minus

sign

^because the free energy in the Gaussian

approximation

is _an

upperbound

_on ^the true free energy. To obtain the proper

dimensions,

this must be

multiplied by

an area A~, since the characteristic

lengthscale

of the system is

given by

the

correlationlength

>. Similar _to the first order term, ^{this will be} enhanced

by

_a

dimensionless, increasing

function of

(~d)~,

since the

sign

of ~d is irrelevant. As in the _case of the first order term, ^{this function will be dominated}

by

_an

exponential.

Thus _we

arrive at

bF2 °J

->~V~(D)exp(n(~d)~/2) (35)

As

long

_as

(~d)~

<

ci(~D),

the first order term should be the dominant _term.

This, finally, implies

_{n =} +

2/ci

For I-dimensional systems,

notably

for

hexagonal DNA,

the

importance

of undulation enhan- ced-electrostatic forces has _now been

clearly

demonstrated [15,

16]. Unfortunately, published

osmotic stress and

scattering experiments

_on lamellar

phases

of

charged membranes,

have not been

performed

in the

re#me

where the electrostatic

repulsion

_may be

strongly enhanced,

while the contribution of the electrostatic free _{energy to} the total free _energy is still

apprecia-

ble.

Nevertheless, recently Odijk

showed [2] that his

theory,

if combined with the Lindemann

melting rule,

_can be used to

interpret experimental

data _on the

melting

of _a lamellar

phase

of

charged

membranes [18]. ^{In the}

regime

of membrane

separations

_up to about lo

Debye lengths,

_{some cases} have been

reported

where it _was found that the electrostatic

repulsion

_was

somewhat

larger

than

expected

_[3,

17].

^Other osmotic stress and

scattering experiments

have been

performed

at ionic

strenghts

either too

high

_{[19] or} too low [20] ^to see the enhancement of the electrostatic

repulsion.

It would therefore be

interesting

to see whether further

experi-

ments _can be

performed

in the

regime

which has been indicated in table _I. In this

regime,

the situation cannot be described

by

Helfrich'

theory,

_nor

by

pure electrostatic

repulsion

between flat membranes.

However, equation (25)

should

apply.

Acknowledgements.

would like to thank T.

Odijk

for _many

stimulating

discussions.

Appendix

A.

The

expression

for the free _energy.

The

proof

that

equation (19)

^for the free energy ^{in terms} of the correlation function is correct,

closely

follows the _papers of Lee and

Yang

_[13] and

Luttinger

and Ward [14]. ^To

be#n with,

a parameter is needed that _counts the number of internal lines of _a

given 0-diagram.

This parameter is introduced

by making

the substitution

z(ki)

_-

»i12z(ki) ₍₃₆₎

in the equation for 7iI. If _p

= 0, then 7i ₌ 7io. It will be shown that for all _p,

including

_p

= 1,

>(»)

=

>s(») (37)

where

F(p)

=

V(D)

+

/ _{ln(Sp~(k) /Kk()}

+

F'(p) ₍₃₈₎

2 k _~

F'(p)

₌

~ [contributions

of all

topologically

distinct

0-diagrams] (39)

and

?s(»)

=

V(D)

₊

/(S~~ ^{(k, »)/Kk()}

⁺

/ _{S(k, »)z(k, »)}

+

?1(») (40) F](p)

=

£ [contributions

of all

topologically

distinct skeleton

0-diagrams,

with

So(k)

relaced

by S(k,p).] (41)

At _p

= 0 both free

energies

reduce _to Fo. To _prove their

equality

for _p

#

0, it is _necessary to show that

~~~~ ~~i~

~~~~

In

F, only

F'

depends

_{on p.} Consider the contribution _to F' of all

0-diagrams

with _u internal lines. If _{we open up any} of these _u

lines,

_{we get}

I-diagrams

of u-th order in p,

contributing

to the correlation function. Let _us call the

self-energy

contributions of these

diagrams Z~(k, p),

the total u-th order

self-energy,

which has _{proper as} well _as

improper

contributions. In terms of

Z~(k, p),

the free energy F' is

#ven by:

?'(»)

=

f ) / So(k)zu(k,») (43)

The factor

1/2u

in

(43)

is due to the fact that

by closing

all

possible I-diagrams

with _u I internal

lines,

the

resulting 0-diagrams

_are counted 2u times. This will _now be shown in _a little

more detail. Consider _an

arbitrary 0-diagram having

_u internal lines.

Opening

_{up any} of the internal lines leads _{to a} number of

topologically

distinct

I-diagrams. Suppose

that _a

particular

one can be obtained in _n ways. Then the symmetry number So of the

0-diagram

is related to the symmetry ^number

Si

of this

particular I-diagram

_[13]:

So _" nsi

(44)

where it has not been decided which of the external lines will be associated with the _wave

vectors k _or -k. In the

0-diagram

there must be _a group of _n lines

playing

_an

equivalent

role

(since _opening

_{up any} of these lines leads to the _same

I-diagram).

In the

I-diagram

this group has been reduced to n I internal lines

playing

_an

equivalent

role. For the symmetry ^numbers of the

diagrams

this

implies

So

/Si

=

nl/(n I)I

= n, whence relation

(44). Using

this relation it _can be shown that

by closing

all

I-diagrams

that _can be obtained

by opening

_{up any} of the

internal lines of _a certain

0-diagram,

the latter is counted _u times.

Similarly

all

0-diagrams

_are

counted _u times

by closing

all

I-diagrams having

_u I internal lines.

However,

since for the

I-diagrams

obtained

by opening

_{up an} internal line of _a

0-diagram,

there _are still _{two ways} of

associating

the external lines with k _or

-k,

in fact all

0-diagrams

will be counted 2u times if all

I-diagrams, contributing

to the correlation

function,

with _u 1 internal

lines,

_are closed.

The derivative of F' with respect to p is

given by

p°1'(»~

=

f / so(k, p)z~(k, p) (45)

»

u=~ k

Let _us call the _sum of the u-th order total

self-energies,

the total

self-energy, Z'(k, p).

This

quantity

_can be written in terms of the _proper

self-energy Z(k, p):

Z'(k, p)

₌

Z(k,p)

₊

Zjk, p)So(k)Z(k, p)

_+..

~~(~)~~~(i~j)

~~~~

Using

this

equation,

it is found that

p~jj~~~

⁼

-j / S(k, p)Z(k, p) (47)

P _k

The p

dependence

of the free _energy Fs is

partly implicit,

^due to the _p

dependence

^of

Z(k, p)

and

partly explicit

due to the

explicit

interactions

occurring

in the skeleton

0-diagram

_con-

tributing

to

F]. By

the rules of

implicit differentiation,

the derivative of Fs with respect to p is

given by:

~~~~ ~~~~~~Z(k,p)

~

~~~~~)~

p

~~~~~

^~~~~

The skeleton

0-diagrams contributing

to

F]

_can be obtained

by closing

all skeleton

I-diagrams.

Let

Z$(k,p)

be the total u-th order

self-energy according

to

(18).

Contributions to

Z$(k, p)

come from all skeleton

I-diagrams

with _u I internal lines where

only

the

p's occurring explicitly

_are used to determine the order. Since all skeleton

I-diagrams

_{are proper,}

Z$(k, p)

is _a proper

self-energy. By closing

all skeleton

I-diagrams

with _u I internal

lines,

_as in

(43),

every skeleton

0-diagram

with _u internal lines will be counted 2u times.

Therefore,

_we have

Fl(P)

"

f ) / S(k, P) ES (k, P) (49)

u=2

and hence

»

1°ll~~1

~~~~,

=

( _{S(k »)~S (k »)}

=

/ S(k, p)Z(k, p) (50)

~ ~

Thus,

to prove the

equality

of the derivatives of F and Fs, ^it is

enough

to show that

())~~~

=

)S~(k, ^{p)Z(k, p)}

⁺

(())~~~

^{= 0}

⁽⁵¹⁾

Differentiating

_an

arbitrary

skeleton

0-diagram contributing

to

F]

with respect to

Z(k,p)

amounts to

opening

_{up any} of its internal lines.

By differentiating

all skeleton

0-diagrams,

all skeleton

I-diagrams

_are obtained with the _proper symmetry

number,

because of

equation (44),

_apart from _a factor 2. The factor 2 is

again

due to the fact that for

I-diagrams

obtained

by opening

_{up an} internal line of _a

0-diagram,

it has not yet been determined which of the external lines wil be associated with either k _or -k.

Therefore,

_we have

~

[contributions

of all skeleton

0-diagrams]

bZ(k, p)

=

)[contributions

^{of all skeleton}

I-diagrams] (52)

whence

li~)~)

,

~

~~

~~' ~~~~~' "~

_~~~~

This

completes

the

proof

of

equation (37). Finally,

note that, _upon

setting

_{p =} I in

(51),

_we

~~~~

~ik)

~ ~~~~

which _can be used _{as a} variational

principle generating

the

equation

for

Z(k)

from the free energy functional.

Appendix

B.

An estimate of the second order term.

In the

approximation

N

= 2, _an

in-phase

mode

zi(x)

and _an

out-of-phase

mode

z~(x)

_are

defined _as

z;(x)

₊

)(zi(x)

+

z2(x)) (55)

zo(x)

₊

)(zi(x) ^{z2(x)) (56)}

Only

the

out-of-phase

mode contributes to the relevant free _energy. Thus _we arrive at a

Hamiltonian in terms of

z~(x):

~ijz~j

₌

/ _{d2x )(az~(x))2}

+

v(D

₊

vsz~(x))

₊

_v(D vsz~(x) ₍₅₇₎

To find the free _energy of _two

membranes,

_per unit area, with respect to free

membranes,

the free _energy of _a free _zo mode _must be subtracted. In this _case there is _no

perpendicular

Fourier

transform,

and the two-dimensional

in-plane

wavevector will be denoted

by

k. Since _a

possible k-dependence

of the correlation

length

is not taken into account, ^the correlation function is given

by:

~~~~ ~~°~~~~°~ ~~~

K(k4 4A~4)

^~~~~

and the

amplitude

of the undulations

by

jd~

⁼

/ _S(k)

=

~j ₍₅₉₎

k 16

"~~~~~

j _j°°

^d~k

₍₆₀₎

k

~ -co

(2~)~

The vertex factor of _an n-th order vertex

(n

= 4, 6,

8,..)

_now becomes

2"/~+~V~"'(D) ~

^~

₍₆₁₎

ki,k~,.. k«

where the asterisk indicates that the k summation is restricted to the values of k for which

ki

+ k2 + + kn

= 0. The second order contribution

bF2

to the free _{energy, per} unit _area, per

membrane,

_can be written _as

b?2

=

-A~V~(D)) exP((~d)~)f((~d)~) (62)

The part

exp((~d)~) /2

is due to the

tadpoles

of the second order

diagrams

shown in

figure

2a.

The dimensionless function

f((~d)~)

is due to the middle part of the second order

diagrams.

It is

given by:

~~~~^~

2

~~~

k,

,

k~

k

~~~~~ ~~~~

1 1

2 2

nJ 2m _n2

1 1 1

2 2

~J

order

diagrams

shown in

figure

2b

(these

_are not all the

diagrams contributing

to third

order).

Obtaining

these

diagrams

from the second order

diagrams

of

figure

2a is similar to

obtaining

the second order

diagrams

from the first order

diagrams

^of

figure

I.

Therefore, bF3/bF~

will be

roughly proportional

to

bF2/bfi,

I-e- if

(~d)~

_>

2/3 (~D),

not

only

will the second order term be

large

^with respect to the first order term, but also the third order _term will be

large

with respect to the second order _term.

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