Topological defects in lamellar phases: passages, and their fluctuations

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Submitted on 1 Jan 1993

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their fluctuations

J. Goos, G. Gompper

To cite this version:

J. Goos, G. Gompper. Topological defects in lamellar phases: passages, and their fluctuations. Journal

de Physique I, EDP Sciences, 1993, 3 (7), pp.1551-1567. �10.1051/jp1:1993200�. �jpa-00246816�

Classification

Physics

Abstracts

05.40 68.10E 82.70

Topological defects in lamellar phases: _passages,

and their fluctuations

J. Goos and G.

Gompper

Sektion

Physik

der

Ludwig-Maximflians-Universitit Minchen,

Theresienstr. 37, 8000 Mfinchen 2~

Germany

(Received

4 March 1993,

accepted

in final form 23 March

1993)

Abstract. In lamellar

phases

of ternary

amphiphilic mixtures~

_{passages are} tube-like _con- nections of

neighboring

oil-

(or water-) layers. Passages

form

spontaneously

due to the ther- mal fluctuations of the monolayers

(or membranes),

which separate oil and _water.

Analogous

membrane conformations also exist in

binary amphiphilic mixtures~

where the membranes _are

amphiphilic bilayers.

We

study

the

shape

and the fluctuations of passages. Our

analysis

is based

on the curvature model of fluid

membranes~

both with and without spontaneous curvature. Pas-

sages have _a very small

bending

_energy~ and show strong fluctuations. The _mean radius~ the fluctuations of the radius~ and the

asphericity

of the passage are calculated. All these

quantities

are

experimentally

accessible.

1. Introduction.

Many complex phases

are formed when

amphiphilic

molecules _are added to mixtures of oil and water ₁₁

2].

^It

depends

_on

temperature

as well _as the concentrations of all

components~

^which of these

phases

is

thermodynamically

stable. One of the most

prominent

and well studied

phases

is the lamellar

phase.

It _occurs in _a

binary system

of water and

amphiphile~

where the

amphiphiles

form

bilayers

which _are

separated by layers

of water. In _{some cases~} the

separation

of the

bilayers

has been found _to be _as

large

_as several thousand

I _[3-5].

_Lamellar

_phases

_also

occur of _course~ in

ternary

systems, ^where

they

consist of

alternating layers

of oil and water~

which _are

separated by amphiphilic monolayers.

Two mechanisms _are

responsible

for the _enormous

separations

between _{mono- or}

bilayers~

which _we will

collectively

call membranes. The first is the

long-range

electrostatic interaction between

equally charged surfaces~

^the second is _an

entropic interaction,

which is due to the thermal fluctuations of the membranes [6]. ^As

long

_as electrostatic interactions _are

present, they

dominate at

large distances,

and _suppress membrane fluctuations

[7]. However, by adding

a

sufficiently large

amount of salt to the system, the electrostatic interaction _can be screened and becomes

effectively

short

ranged.

We will _assume that this is the _case in the

systems

studied here.

Since membranes _are

(nearly)

tensionless

surfaces,

their

shapes

and fluctuations _{are con-} trolled

by

the elastic _curvature Hamiltonian [8]

Rcurv

₌

/

_dS

_{[2n(H Ho)~}

+

kit] (1)

where H

=

(I/Ri

+

I/Rz)/2

is the _mean curvature and Ii

=

1/RiR2

the Gaussian

curvature,

both

given

in terms of the

principle

radii of curvature,

Ri

and

R2. Ho

is the

spontaneous curvature,

which tends to bias the _mean curvature of the membrane. The

integral

in

(I)

is

over the whole membrane. The

bending rigidity

_K and the

saddle-splay

modulus k determine the elastic

bending

_energy of the membrane.

It is the restriction of the membrane fluctuations in the stack due _to the

neighboring

_mem-

branes which reduces the

entropy

of the fluctuations. Each time _a membrane collides with its

neighbors,

it loses

entropy

of order

kB compared

to _a free membrane

[9].

^The

entropy

loss _per unit _area is thus

kB/f(,

^where

_fjj

_-w

^n/kBTd

^{is the}

parallel

correlation

length [10-12],

and d is the average membrane

separation.

This leads to the effective

repulsive

interaction of the

form

l~a

-w

n~~(kBT)~d~~ _[6].

It has been confirmed

experimentally

that there is indeed such

an effective interaction between

neighboring

membranes

[3, 7, 13].

Since each membrane in the stack must collide with its

neighbors quite frequently

due to thermal

fluctuations,

it has been

speculated [14,

_15] ^that some of these collisions lead to the formation of

channels~

_{or passages,}

perpendicular

to the membranes. In _a ternary

system,

^the passages are formed between

neighboring

oiL

(or water-) layers~

thus

connecting neighboring monolayers. Similarly~

in

binary systems they

form _a connection between

neighboring bilayers.

Passages

have been observed

experimentally

in oil-water-surfactant

mixtures,

both in

systems

with ionic

[16]

^{and non-ionic}

[17] amphiphiles. Passages

^{have also been found} in Monte Carlo simulations [18~

19]

of

binary

and

ternary amphiphilic

mixtures.

Finally~

_passages have been

seen in multi-lamellar vesicles of _a

binary lipid-water sy§tem [20].

^{We will often call the channels}

"wormholes~',

_{a name} which is used for _a similar

geometry

ⁱⁿ

gravitation theory.

We want

emphasize

that _we consider here

only

the _case of _a

symmetric

lamellar

phase,

where the _average thickness of the oil- and

water-layers

in _a

ternary system

is identical. In the limit of

a very small oil- _or

water-concentration,

the passages become holes in _a

single bilayer [21-23].

Since the curvature at the

edge

of _a

bilayer

is very

large,

the elastic curvature Hamiltonian

(1)

is not sufficient to calculate the size of the holes in this limit.

The first

attempt

to calculate the

shape

of wormholes with the _use of the curvature model

(1)

is due to Harbich _et al.

[20]. They

found that there _are solutions of the

Euler-Lagrange equations

which have

roughly

the

expected shape.

In

fact,

their solution is _a minimal

surface,

the catenoid.

However,

it _can be

easily

_seen that the catenoid cannot be _a true

wormhole,

be-

cause it does not

satisfy

the

boundary

conditions for two membranes connected

by

_a

wormhole,

which _are embedded in _a stack of other membranes. At

large

radial distances the membranes must be flat and

parallel

to each

other,

which is not the _case for the catenoid.

In this _{paper we} want to discuss the

shape

of _a

single

wormhole. Our calculation is based _on

the Hamiltonian

(1).

^Since the membrane fluctuations stabilize the lamellar

phase,

_a calculation of the

shape

of _a wormhole has to include the effect of fluctuations. To make the

problem

tractable,

_we

study

the fluctuations of two membranes which _are connected

by

the

worml~ole,

but

replace

the interaction of all membranes

by

_an effective

potential _VeR,

which is derived from the steric

repulsion

discussed above.

Thus,

_we have the

partition

function

2 ₌

/

~X _exp

-fl Rcurv (X)

+

/

_dS

_Vea(X) ^~ ₍₂₎

r

ro

ri

S

~d/2

r a)

where

7icurv

is the Hamiltonian

(1)

and

fl

₌

1/kBT.

The

integration

in

(2)

extends _over all conformations of the membrane with the

topology

of _a wormhole consistent with the _appro-

priate boundary

conditions. There _{are no} further

constraints,

in

particular

the _area of the membrane is not fixed.

The _paper is

organized

as follows. In section 2 _we

employ

mean-field

theory,

I-e- _we minimize the Hamiltonian

(1) together

with the effective

potential describing

the steric interaction. As

shown

above,

the

strength

of this

potential

is

temperature dependent. Therefore,

the _mean- field results

depend

_on

temperature.

^Due to the scale invariance of

(1),

the size of the hole is

determined

only by

the

boundary conditions,

I-e.

by

the distance of the two membranes far away from the wormhole. This leads _to wild fluctuations of the

shape

of the _passage.

Therefore,

_we

study

in section 3 the influence of fluctuations _on the size and

shape

of _a wormhole. In section 4 _we discuss the

stability

of wormholes with _non-zero

spontaneous

curvature. A _summary

concludes the _paper in section _5.

2. Mean field

theory.

To calculate the

shape

of _a

wormhole,

_we first consider _a finite

system,

and then take the limit of

large system

^{size. The}

geometry

we consider is shown in

figure

1.

The boundaries of the two

membranes,

which _are connected

by

the

wormhole,

_are two circular

rings

of radius _{ri a} distance d from _one another. For

sufficiently large

_ri

i

the membranes should become

nearly

^flat at

large

radial distances and the

shape

of the wormhole should become

independent

of _ri.

Assuming

rotational

symmetry,

_{we can}

parametrize

the membrane

by

_a surface of revolution of the _curve C

=

(r(s), z(s)),

where _s is the

arc-length

of C measured from _z

= 0

[24, 25]. By introducing

the

angle ~

^{between the ~-axis and the}

tangent

of

C,

the

mean curvature

H,

the Gaussian curvature K and the _area element dS in

(1)

_can be written _as

2H =

~'+

sin

~/r,

It

=

~'sin ~/r,

and dS

=

2irrds,

where the

prime

denotes

d/ds. Assuming symmetry

with

respect

to reflections at _{z =}

0,

_{we can} restrict ourselves to _z >

0,

^{with the}

boundary

conditions

tl10)

=

lr/2 nisi)

_{= ri}

~l"(°)

= °

tlisi)

₌ °

(3)

z(°)

₌ °

z(Si)

₌

d/2

Because the _area of the membrane is not

fixed,

_we also have to minimize the curvature Hamil- tonian with

respect

to si The contribution of the Gaussian curvature to 7i _can be shown to be _a

constant;

this reflects the fixed

topology

of the wormhole

geometry. Therefore,

k is set

equal

to _zero in the

following.

Let _us first

ignore

the

entropic

interaction between the membranes. In this _case, the

shape

of the wormhole is obtained

by minimizing

the functional

£ = ir

/~~

^ds

nr (~'

⁺

^{~~~'~ Ho)}

~

+

7(r'

_cos

~)

+

n(z'-

sin

b)j

(4)

o ^r

with the

boundary

conditions

(3),

and

Lagrangian multipliers 7(s)

and

q(s).

In the limit ri ~ cc and with

vanishing spontaneous curvature,

the

shape

of _a wormhole is found to become

singular;

both the inner radius _{ro "}

r(0)

and the _mean distance between the two membranes tend to zero in this limit.

Indeed,

for _{ro -} 0 the solution is almost _a

catenoid, r(z)

₌

ro

cosh(z/ro),

which is the minimal surface calculated

by

Harbich et al.

[20].

^Because

z(r)

grows very

slowly

with

increasing

_r, the distance between the _upper and the lower membrane is _very small. Therefore the interaction of the two membranes becomes

important

and must be

taken into

account,

if _one wants to calculate the

shape

of _a wormhole. Because the interaction is

repulsive,

it favors

larger

_passsages, and therefore

competes

with the _curvature energy, which

favors

infinitesimally

small wormholes.

Thus,

the steric interaction leads to _a finite inner radius of the

wormhole,

_even in the limit _ri

- cc.

To include the steric interaction _we introduce _an effective

potential l~a,

_as mentioned in the introduction. The form of

l~a

^{is restricted}

by

the

following

conditions:

I)

^{it should} act _on the

projection

of the membranes _onto

a z =const

plane; therefore, Vea

has to be

ir-periodic

in _~b;

it)

^the fluctuations of _a

planar

membrane at _{z =} +

d/2

should be restricted

by

the

potential

in such _{a way} that

((z (z))~)

₌

pd~,

with p of order

unity

_[6]. This leads to an

effective interaction

potential

with

a

d~~-decay.

It has been

argued

in reference [6] ^that p =

1/6

should be used. This value is in reasonable

agreement

with the value _p

= 0.130 obtained from Monte Carlo simulations for _a

single

membrane between two hard walls

[26],

^{and will be used} for all numerical calculations

presented

below. For calculational _reasons

Vea

should also be continuous and differentiable in _z and _~b. The

simplest

_way to fulfill these conditions is to choose

l~a proportonial

to

cos~(irz/d) cos~ ~.

For _reasons that will discussed below _we found that such

a

potential

is not _a

good approximation

if (z( ^becomes

greater d/2. Therefore,

_we chose instead

~~j~ ~)

^{'~ ~}~

(j~ /~)2 /~j2

^~~~2

~ (~)

e , ~4

where

fjj

^{is determined}

by

the condition

(it) above,

which leads _to the

implicit equation

For

Ho

"

0,

this

simplifies

to

fjj

= 2d

pn/kBT. Typical

values for the

parallel

correlation

length

are

fjj Id

₌ 2.8 for

n/kBT

₌

3, Ho

_"

0,

and

fjj Id

= 2.0 for

n/kBT

=

3, Hod

₌ 0.1.

Note that

l~a only depends

_{on z}

Id; therefore,

if

Ho

^is measured in units of

1Id,

the membrane

separation

d _enters the model

only

_{as a} scale

parameter. l~a

is in _{some way}

arbitrary,

of course, but _we believe that other choices should not

change

_our results

qualitatively.

The minimization of

(4), 16)

^{with the}

boundary

conditions

(3)

leads to the

Euler-Lagrange equations

r'

=

cosl~

'

/ ^'

,

/ ',

i-O ^~

~ ⁱ

~ m

C~ 0.5

o-o

o 5 _lo 15

2r/d

Fig.2.

Solutions of the

Euler-Lagrange

equations with flK ₌ 1.19 and _ri

= 7.5 d. The full line is

the mean-field

solution,

with radius _To

= 0.3. The dashed line is calculated with fixed _To

= 1.0.

z'

=

sinl~

7'

_"

~'~

_~~~2

^~

+

HI 2H0~~

+

~

vefl(Z,

_l~)

d2

~

~

q/

₌

~

cos~ ~ z((z/d)~ 1/4) (7)

ill

'~"

~'~~~~

^~

^~~~~~~~

^{~ ~~~~}

~2r~~~~ ^~

+Ho ~°l'~

⁺

$

CDS

~

sin

~ ((z/d)~ i/4)~

jj

A first

integral

of

equations (7)

is

given by

I =

-r(~')~

₊ ^{~~~ '~} ₊

4Hjr

₊ ^~

_{rl~a(z, ~)}

₇

cos ~b q sin _~b.

(8)

n

Minimization with

respect

to the free _upper

boundary

_si

implies

1= 0

[25],

which

gives

the additional

boundary

condition

-(~b'(si))~r(si)

"

7(si).

Solution of the

equations (7)

has

been obtained

numerically

with the _use of _a

shooting

method. In this section _we

only

want to consider membranes without

spontaneous

curvature, I-e-

Ho

_"

0,

_as is the _case for

amphiphilic bilayers.

A

typical

conformation of _a wormhole in this _case is shown in

figure

2. Because

fjj depends

_on

fl,

the

shape changes

with

temperature. However,

the mean-field radius _ro

depends only

_very

weakly

_on

temperature,

as shown in

figure

3. It decreases from _ro

" 0.15 d

at

fin

= to ro " 0.12 d at

fin

₌ 5. The decrease is due _{to a} smaller steric interaction at lower

temperatures;

_as discussed

above,

_ro vanishes if the steric interaction is

neglected. Thus,

the steric interaction increases the radius _ro. The free energy of the wormholes is dominated

by

the steric interaction.

Therefore,

it is of the order of

nd~/f(,

^which can be _as small _as

10~~n

for

fin

= 5.

To check the

assumption

that the

boundary

condiations do not influence the

shape

of the passage, we linearize the

Euler-Lagrange equations (7)

^for

large

_s. This leads to

s2~(4)

+

s~(3)

_~" +

~

s~ = o

jg)

~f(

0.35

n3

~'~~

~",,~

~

,«,~~~

$

0.25

"'~"~~,,~,,,

---~

v cq

0.20

0.15

0 2 4 6

K /

k~T

Fig.3.

Radius (To) ^of a passage, as a function of

K/kBT,

both in the mean-field

approximation (dashed line)

and with fluctuations

(solid line).

and

~

Q" ⁼

j#. ^(lo)

If

fjj

= cc, I-e- if there is _no steric

interction,

the passage is _a

catenoid,

and the curvature

decays

_as

^s~~

But if

fjj

^< cc, the solutions of

(9)

show

asymptotically

_an

exponential decay.

Therefore, #

and the _mean curvature also vanish

exponentially. Together

with

our numerical results for different _ri, this shows

clearly

that _a

sufficiently large

but finite radius _ri does not affect the

shape

of the _passage. This result also

implies

that the lateral interaction between two wormholes

decays exponentially.

It is also

interesting

to determine the

shape

of _a passage with fixed inner radius _ro. We do this

by introducing

_a term

6(s)(r(0) a)~

in the free _energy

density,

where

6(s)

is the Dirac 6 function. This is

equivalent

to

introducing

_an additional constraint

r(0)

₌

ro(a),

with _an associated

Lagrangian multiplier,

_or to

solving

the above differential

equations

with the

boundary

condition

r(0)

_{= ro} instead of

#"(0)

₌ 0. For

large

_ro, the free energy of this

conformation _can be understood _as the line tension of _a linear

edge

of

a folded

membrane,

confined in _a stack of other membranes. In this _{case, one} of the

principal

curvatures becomes very

small,

and _can therefore be

neglected.

Then the curvature Hamiltonian reduces to

s~ s~

7icurv

m ~n ds

r(#')~

_m

_2~nro

ds

(#')~. (11)

The minimization of this Hamiltonian with the

boundary

conditions

(3)

_can be done

exactly.

The result for the free _energy is

F(ro)

" ~K

(B((, ()j ) ₍₁₂₎

where B is the Beta

function,

with

B(3/4, 3/4)

m 1.2.

The conformation shown in

figure

2 exhibits _a

hump just

outside the passage, which grows with

increasing

_ro and

fjj,

^I-e.

increasing PK.

If _{we use a}

z-periodic potential

_as discussed

above,

this

hump

would

"jump"

_over the maximum of the

potential

at _z

= d. This would

correspond

to _a

interpenetration

of two

membranes,

and is

clearly unphysical.

The _same

problem

does

not _occur with the

potential (5). Nevertheless,

if the

hump _gets

to

large,

the

potential

VeR ^is

no

longer

_a

good approximation

to describe the effect of self-avoidance.

In the last

section,

_we have calculated the free _{energy as a} function of the hole-radius _ro.

This result _can be used to

estimate

the fluctuations of the size of the wormhole. We find

Aro/ro

_* 0.6 for _n of order

kBT,

with _{a very} weak

temperature dependence. Thus,

the size and the

shape

of _a wormhole must be

strongly

influenced

by

fluctuations.

To calculate the contribution of fluctuations to the

partition

function

Z,

the

simplest

_ap-

proximation

is to

expand

7i around the mean-field solution to second order in

kBT/n.

This

Gaussian

(or capillary wave) approximation

is valid if the

amplitudes

of the fluctuations

are

small.

However, bra

calculated within this

approximation

is about _as

large

_as the above

estimate,

_so that the condition

bra

< _ro is not fulfilled. The breakdown of the Gaussian

approximation

is due _{to a} soft mode with _an

eigenvalue

of the order of

nd~fjj~

^{This soft}

mode

changes

the size of the

wormhole,

while

keeping

its

shape roughly fixed;

_we will therefore call it the

"breathing

mode". The low _energy of the

breathing

mode is

quite understandable, considering

that the inner

part

^{of the} passage is

approximately

_a

(scale invariant)

minimal surface. We solve the

problem

of this soft mode

by treating

fluctuations in _ro

exactly.

To do so, we insert _a I in the

partition

function

(2),

and then

interchange

the order of

integration,

Z =

/ ~~ da/j/l7Xe~°(~°~~~~exp(-fl7i(X))

~

+

/j _/~~

da

e-PF(a) (13)

~ _~

with fo _#

f/~ r(s

₌

0, #)d#/2~,

where _a is chosen such that the fluctuations of _ro with fixed

a are much smaller than _ro. In _our

calculations,

_{we use a =}

200flnd~~.

The mean-field value of

F(a)

is obtained

by calculating

the _passages with fixed _{ro as} indicated at the end of the last section. The contribution of the

quadratic potential

in

(13)

to the

Euler-Lagrange equations

leads to _a

discontinuity

in the second derivative of the

profile

at z =

0,

because the

potential

acts

only

in the _z

= 0

plane.

Since the

integration

_{over a} extends to -cc,

negative

values of ro are

possible.

Such

unphysical

solutions _are avoided

by introducing

_a hard wall at ro # 0.

Since the free energy of these solutions is _very

high compared

to the mean-field

solution,

their contribution _to the

partition

function is

negligible.

In order to sum

only

_over

physically inequivalent configurations,

we have to select _a

unique parametrization

for the

fluctuating

membrane. Here

physically inequivalent

_means that two

configurations

cannot be transformed into each other

by

_a

reparametrization.

This is done in the

following

_way. All

configurations

not too far from the mean-field

configuration Xa

₌

(r(s)

cos

#, r(s)

sin

#, z(s)) (for

_a

given parameter a)

can be written _as

x(s, <)

₌

xa (s, <)

+

v(s, <)na (s, <) (14)

where

na(s, #)

is the normal vector of the mean-field surface. The

integral

_over l7X is _{now an}

integral

over all

v(s), multiplied by

the

Faddeev-Popov

determinant

[27, 28]

det J

=

fl

det

lgl(S)

₊

a;(v(S)n(S))ajxa(S)I /

det

glj(S) (15)

with

I, j

E

(s, #), g(

is the metric _on

Xa

and

fl~

is _a

(formal) product

_over all

points

_on the membrane. The

Faddeev-Popov

determinant makes _sure that

only

those fluctuations _are taken into account in the functional

integral,

which _are

orthogonal

to

reparametrizations.

Then the

JOURNAL DE PHYSIQUE'-T 1 N'7 JULh (Ml 56

partition

function for fixed _a reads

exp(-flF(a))

=

(16)

/ _{fl dv(S)}

exp

-fl7icurv / _{dS(fll~R log}

_det

J)j

~

The

expansion

to second order in the fluctuations

v leads to

log

^{det J} ₌ 2Hv +

Kv~ (17)

and

g;j =

g( 2vC;j

+

V;vvjv

+

v~C;icj. (18)

where

H,

It _are the _mean and Gaussian _curvature of the mean-field

conformation,

and g;;

and

C;j

_are the first and the second fundamental forms in this

geometry,_respectively.

V denotes the covariant derivative. The result for the

expansion

of the _{curvature energy}

H~

and

potential I~R

_are

given

in the

Appendices

A and B. The linear terms in the

expansion

of

H~

_and

(det g)~/~

vanish

identically

because _we

expand

around _a minimum of the free energy.

Putting everything together,

_we arrive at

-flF(a)

₌

-flfmf(a) (19)

/ _{II dv(S)}

exP

(- /

r ds

d# (vL(S)

+

vO(S, )v)j

where

Fmf

is the free _energy of the mean-field solution

r(s),

L ₌ 2H is due to the linear term in the

expansion

of the

Faddeev-Popov

determinant

(17),

and

O(s, #)

is _a hermitian

operator,

which is obtained

by collecting

all the

quadratic

terms in _v. For the flat

part

of the

membrane,

O reduces to

)fin _((V~)~

₊

f[~j.

To

integrate

_over

v(S),

_we have to define the

measure

fl~ dv(S).

This is done in the usual

way

[29] by expanding

_v in _a set of

eigenfunctions

of the

operator O,

_{v =}

£~anvn,

and

writing fl~ dv(S)

₌

fl~ don.

A further

simplification

arises from the rotational

symmetry

^of

our

problem,

which allows _an

expansion

in terms of

eigenfunctions

of

angular momentum, v(,s, #)

=

~j ~j _{vf (s) (of}

_sin

_ml

₊

_flf

cos

ml). (20)

m ;

The

eigenfunctions vf

_are ^either even or odd functions of _s, due to the reflection

symmetry

with

respect

to the _{z =} 0

plane

of the

operator

^O. The

boundary

conditions for odd

eigenfunctions

are

v(si)

₌

v'(si)

₌ 0 and

v(0)

₌

v"(0)

₌

0,

and for the _even

eigenfunctions v'(0)

= 0 and

v"'(0)

=

(#"(0)(1/ro 3#'(0)) a/ro)v(0).

The

non-vanishing

third derivative in the latter

case is due to the

discontinuity

of the third derivative of the mean-field

profile.

On the flat

part

^{of the}

membrane, vf

_are solutions of Bessel's differential

equation,

with wavevector

kf

and

eigenvalue Ef

₌

n((kf)~ +fj[~)/2.

^{The curvature}

^dependent

^{terms in O} can be

neglected

if

I/kf

< _ro. In this limit _one obtains _a

spectrum

^with

spacing

Ak

=

~/si

After

inserting

the

expansion (20)

in

(19)

and

doing

the Gaussian

integrals

_we arrive at

flF(a)

₌

flfmf(a) (21)

i~~ ^~~°~ i _II '°~ 4

₊

ill

are to

Lf

₌

4~6mo f/~ dsL(s)vf(s).

_sums

in

equation (21)

have _a

divergence

at

high energies.

We therefore

regularize

them

by using

an

exponential decreasing

cutoff

function, £~

;

f(Ef)

-

£~

;

exp(-Ef/Ec) f(Ef).

It is

important

to _use a soft

cutoff,

because the

meiibrane

_{is finite}

ind

_{hence the}

eigenvalues

_are

discrete. Otherwise the free _energy would _vary

discontinously

with the number of

eigenvalues

with

En

<

Ec.

The free energy

F(a)

has not

only

contributions from the

wormhole,

but also from the flat

part

of the membrane. In

fact,

the free _energy of _a

large

membrane is dominated

by

the free energy of the flat part.

Therefore,

in order to extract the contribution of the

wormhole,

_we

subtract the free _energy of _a flat circular membrane with the _{same area,} calculated with the

same cutoff. The

eigenfunctions

of _a flat circular membrane _are

given by

a linear combination of Bessel

functions, vf(s)

=

of Jm(kf's)

+

flf'lm(kfs),

where

of, flf

and

kf

_are ^determined

by

the

boundary

conditions

vf(si)

"

dvf(si)/dsi

"

0,

and the normalization.

From

(13)

_{we can} calculate the _mean size of the wormhole

j dafo exp(-flF(~))

l~°I

"

j

da

exp(-flF(a))

^'

(22)

where

2R ~_~j

fo _"

/ _{-r(s=0, #) (23)}

o 2~

is the _mean inner radius of _a

configuration.

The

eigenvalue equation

is solved

numerically

with

a finite difference method for

fin

₌

1.19,

3.0 and

5.0,

with _zero

spontaneous

curvature. The size of the outer radius _{ri necessary to} avoid finite size effects is determined

by

the ratio

ri/fjj

We find that for

ri/fjj

_>

10,

^the

dependence

of the free energy of the wormhole _{on ri} becomes

negligible.

The free energy of _a wormhole

(relative

to _a flat

membrane)

should be

independent

of the cutoff for

Ec

_{~ cc.} We determine

Ec by

the condition that

(ro)

^should not

depend

_on

Ec.

This

gives

reasonable results for

Ec

-~

~d~/(2ro)~.

It is therefore _necessary to calculate the

eigenvalues

_up to

approximately 10Ec [30]. Figure

4 shows the free energy as a function of the radius _ro of the

underlying

mean-field

conformation,

_up to an additive constant

(which depends

on the

regularization

used in

equation (21)).

The results for the size of the wormhole for different

bending rigidities (or temperatures)

is shown in

figure

3. Thermal fluctuations reduce _{< ro >}

compared

to the mean-field result.

Since _{< ro >} 0.I d in the

temperature

range considered

here,

which

implies

_a diameter in the

center of the wormhole of order 100

1, microscopic

interactions between the membranes in the

hole should be

negligible _[31].

The fluctuations of the radius

bra

₊

((f() _(fo)~)

^~~~

(24)

are about

ro/2 (see Fig. 5)

and

depend only slightly

_on

fin; they

increase with

increasing

stiffness of the membrane. The _reason for this unusual behaviour is

again

the increase in the steric interaction with

decreasing stiffness,

which fixes _ro and therefore

competes

^with the

increasing

thermal fluctuations. We want to

point

out

that, although

^the result for

bra

shown in

figure

5 is very close to the result obtained from Gaussian

fluctuations,

it _now rests on a

solid basis.

Another

interesting quantity

is the

shape

of the wormhole _{at z}

= 0. This

shape

_can be characterized

by

the moment of

gyration

tensor

Q>j

"

j /~~ ^{d# ~(#)~}

⁺

((~l#)) ^j

^~~~

l~'l#)~Jl#) ~'~J) _1~5)

3

2

if

G I

o

O-O O.2 O.4 O.6

2r~/d

FigA.

Free energy of the wormho<e _{as a} function of the radius of the mean-tie<d

configuration,

for

K/kBT

= 3.0.

O.130

~

O.125

~

~ c~

O.120

O 2 4 6

K /

k~T

Fig.5.

Radial fluctuations arc ₌

((f() _(fo)~)~/~

_{of the} radius of _a passage, as a function of

K/kBT.

with _~i

II)

_"

r(z, #)

_cos

#, ~2(#)

_"

r(z, #)

sin

#.

I; is the

I-component

of the center of _mass,

and U is the

perimeter.

We _are

especially

interested in the

"asphericity" [32, 33]

itr14~ii i~~i

~

126)

iitro)~i

where _{£~ "}

Q -1trQ/2 _[34].

Because

A2 depends only weakly

_{on a, we} take in account

only

the fluctuations of the passage with the mean-field radius _ro The

expansion

of the

asphericity

to second order in

kBT/n

is

given

in

Appendix

C. The numerical results for the

mean value

(A2)

are shown in

figure

6.

Again

the increase of

A2

with

increasing

stiffness is due to the steric

interactions,

which allows

stronger

fluctuations at lower

temperatures.

To illustrate _our

results,

and to demonstrate the

importance

of

fluctuations,

_we show two

O.25

A

$

^O.20

v

o.15

O 2 4 6

K /

k~T

Fig.6. Asphericity

A2 _{as a} function of

K/kBT.

typical configurations

of the membrane in

figure

7. The

configurations

_are obtained

by

_gen-

erating amplitudes

for the

eigenmodes

with _a Gaussian

probability

distribution

(where

the

eigenvalue Ef

^{determines the width of} the distribution of

amplitudes

of

eigenmode vf),

and

adding

all fluctuation modes to the mean-field

configuration. Figure

7a shows the whole _sys-

tem;

the

edge

of the membrane is flat due to the

boundary

conditions

(3).

More details of the

wormhole itself _can be _seen in

figure

7b.

4. Wormholes in

systems

with

spontaneous

curvature.

It is well known

[35, 36]

^{that the lamellar}

phase

is also stable for small

spontaneous

_curva-

tures

Ho

Such

a situation _{may occur} in _an unbalanced

ternary

mixture of

water, oil,

and

amphiphiles.

The

phase diagram

of such _a mixture _was calculated in reference

[35],

^with the additional constraints that the _area of the membranes and the oil- and water-concentrations

are fixed. In this _case, the lamellar

phase

is stab<e with _a spontaneous curvature

jHoj s 1/(8d), (27)

where d is the

(mean)

distance between the membranes.

If there _are passages between two

neighboring

membranes the

sign

of the

spontaneous

cur-

vature becomes

important.

A

negative Ho (in

the

parametrization

of the last two

sections)

increases the size of _a

wormhole,

while _a

positive Ho

makes it smaller. If _ro becomes

large,

_we

can

again neglect

terms of order

1/r

in the

Lagrangian (4).

Then the free energy reads

F = ~nro

/ ^~

^~ _ds

_{((#')~ 4Ho1b'+ 4H() (28)}

It is

easily

_seen that the wormhole becomes unstable if the

integral

in

(28)

becomes

negative.

A numerical

analysis

of the

Euler-Lagrange equations (7)

shows that this

happens

for

Hod

<

-0.23 at

fin

₌

1.19,

which is about twice the value

(27)

of the

instability

of the lamellar

phase

as calculated in reference

[35].

^In an infinite

system,

the wormhole would transform into _a

cylinder

at this

point;

in the finite

system

^which we

consider,

_a

jump

in the size of the passage

occurs at this

point,

which

depends

_on the size of the membranes.

a)

b) Fig.?. Typical configurations

of the membrane with

fluctuations,

for

K/kBT

= 1.19 and _ri

" 7.5 d.

The

pictures

_are obtained

by adding

all fluctuations with

eigenvalue

less than 10 _K to the mean-tie<d

configurations

^{of the wormhole. The}

amplitudes

_are chosen

randomly according

to their Boltzmann

weights. a)

Whole

membrane; b)

The

vicinity

of the wormhole.

«, O.75

"',,

~

"',,,

i

_O.50

'~,,,

~

',,,_

v

~"""--,_

~ O.25

'~""'""-

o.oo

-O.2 -O.1 O-O O-1

Hod

Fig.8.

Mean radius fo

(solid line)

and mean-field radius

(dashed line)

of the wormhole at

K/kBT

=

1.18 _{as a} function of the spontaneous curvature Hod.

The mean-fie<d radius of the _{passage as a} function of the

spontaneous

curvature is shown in

figure

8. On the mean-fie<d

<eve',

there is _no

instabi<ity

for

positive Ho

The size _ro of the

wormhole decreases to about 0.07 d at

Ho

"

1/(5 d). However,

when the fluctuations of the

membrane _are

analysed,

_an

instability

is found for

Ho

> 0 also. For

sufficiently large Ho,

the odd

eigenfunction

^{with the} lowest

eigenvalue

becomes _a bound

state;

at

Ho

_*

1/(7.5 _d)

the

eigenvalue

_goes

through

_zero and then becomes

negative

for

large Ho-

O.25

~

O.20

j°

,

°~ O.15

o-lo

-O.2 -O.I O-O

Hod Fig.9.

Radial fluctuations arc

"

((ill (fo)~)~/2

of _a passage ^at

K/KBT

_{as a} function of the

spontaneous curvature Hod.

O.6

O.4

$

A

V

O.2

O-O

-O.2 -O.I O-O O-I

Hod

Fig.10. Asphericity

of the wormho<e at

K/KBT

₌ 1.18 _{as a} function of spontaneous curvature Hod.

For A2

= 1, the circ<e is distorted into _a fine. Thus the

sharp

increase of the

asphericity

at

positive

Ho indicates that the ho<e becomes _more and _more

e<ongated.

The radial fluctuations of the wormhole

(see Fig. 9)

are

a<ways

smaller than the radius of the hole. But the

asphericity

of the

wormhole,

_as shown in

figure 10,

increases

rapidly

with

increasing (positive) spontaneous

curvature. Since the radius _ro of the passage remains

sma«,

membrane co«isions in the inner

part

of the wormho<e wi«

probab<y

become

important

_near

the

instabi<ity; thus,

_{we are}

<eaving

the range of

va<idity

of

our mode<.

5.

Summary

and conclusions.

Shape

and fluctuations of _a passage between

neighboring

membranes in _a <amellar

phase

have been studied in _a model based _on the elastic curvature Hamiltonian. In order to include effects of

self-avoidance,

_we have introduced _an effective

potential,

which describes the steric

interaction of the membranes caused

by

collisions. We found that this

potential

stabilizes wormholes of finite radius _ro, which is of the order 0.15 d.

The effects of thermal fluctuations have been calculated in the

capillary

_wave

approximation.

We find that there _are

large

fluctuations in the radius of the

wormhole,

which _were treated

exactly. Furthermore,

_we have determined the thermal average of the

asphericity,

which _mea-

sures the deviation of _{a z}

= 0 cut

through

the wormhole from _a circle. The _mean size of

a wormhole and its fluctuations _as well _as the

asphericity

_can be extracted from

pictures

of lamellar

phases

obtained

by

^freeze fracture

microscopy [16]. Thus,

_we

hope

that _a

quantitative comparision

of _our results with

experiments

will be

possible

in the _near future.

Finally,

_we have determined the influence of

spontaneous

curvature _on the size and

stability

of wormholes. We found _an

instability

at _a

negative spontaneous curvature,

^{where the lamellar}

phase

becomes

energetically

unstable to _a

phase

of

cylinders.

This

instability

_occurs in _a

region

where the lamellar

phase

is

already

metastable. Another

instability

_occurs at

positive spontaneous curvature,

^{which is due} to

diverging

fluctuations. This

instability

_occurs

roughly

at the _same

point

where the lamellar

phase

becomes metastable.

Therefore,

wormholes _may

play

_an

important

role at the transition from the lamellar

phase

to the

hexagonal phase

of

cylindrical

micelles.

Acknowledgements.

We thank U. Seifert and H.

Wagner

for

stimulating

discussions. This work _was

supported

in

part by

the Deutsche

Forschungsgemeinschaft through Sonderforschungsbereich

266.

Appendix

A.

Expansion

of the curvature energy.

The

expansion

to second order in the fluctuations

gives

for the curvature energy

[37]

(2H)~

₌

4(Hmf Ho)~+

4(Hmf Ho)V~v 8(H$~ _Kmf)Hmf

+

4HoI<mf HjHmfj

v +

(V~v)~

4(Hmf Ho)(C'i _Hmfg[J

₊

_{g[J (6H$~} 41<mf 4HmfHo

₊

H()j

0;v0j

_v +

16H$~

20H$~I<mf

+

41<$~V~(4H$~ 2Kmf)

+

(29)

4Vj (0,(Hmf Ho) (C"J Hog(~) ^v~

where

Hmf, Kmf

_are the _mean and Gaussian curvature of the mean-field

conformation,

and

g[J

^and

C~i

are the first and the second fundamental forms in this

geometry, respectively.

V denotes the covariant derivative.

Appendix

B.

Expansion

of the effective

potential.

In this

appendix

_we want to calculate the

expansion

of the effective

potential I~R

around the mean-fie<d so<ution

Xa, Vea(Xa

+

vna),

where _na is the normal vector _on

Xa.

Due to the rotational

symmetry

^of our

prob<em,

_{we can use} the

expansion (20).

Therefore _we have to

expand

g~~~VeR(Xa(8'~i)

~

"m(~)

C°S ll~~i

~a(8>

~i)

g~~~(~eRl'~a)

+

~~Vm

+

Vm~~"ml' (~°)

are g go

and of the mean-field

conformation, respectively.

Because _we

expand

around _a solution which minimizes the

Hamiltonian,

_we

only

need the coefficients

B?.

The normal vector _on

Xa

is

given by

na(s, #)

₌

(-

sin

#mf(s)

_cos

#,

sin

#mf(s)

sin

#,

_cos

#mf(s)) (31)

where

#mf

is the

tangent angle

of the mean-field

shape. Special

_care has _to be taken in the

expansion

of the

tangent ang<e #.

^Because s is _no

<onger

the correct

arc-<ength

of the

fluctuating membrane,

the

equations

^z'

= sin

#

and r'

=

cos#

do not hold. The correct

expansion

of

#

leads _to

, ,

tan _{~b =}

~~~'~~~

~

"'~f~ ^~)~'~~~

^{~ "}

^"~'~~~

,

(32)

cos ~bmf + v~b~~ ^sin ~bmf v' _cos ~bmf

Then,

after _some

algebra,

_we find

~~

d~ I( ^Z$f

^~

_(-

COS~'fi _f

'~~ ~~~

~~~

~~~

~

~~~

~~

~~~

2

~~j

^{d2 4 ~}

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

~

~~~f~~~~ ~s~~ ^{~ )~~}

^~

^{~~~~~~ ~~~~~'~mf 1)(3z$~ i)}

+ ~ ^Zmf

z$~

ⁱ

~~~

~ _~~~

~~~~ ~~~f ~°~'~mf

_Sin

~~~j

~~ ~~ ~ ^~~~

~~~~

^~

~~~ ~mf)j _0s

₊

~$f

1)

~

~~~~

~ d2 4

II

3

Sin~l~(~~

j)

Here the index "mf'

again

denotes functions

describing

the mean-field

shape

of the passage.

Appendix

C.

Expansion

of the

asphericity.

First _we want to calculate the

gyration

tensor

(25)

in terms of the

expansion

coefficients

of, flf'

of

equation (20).

With the definitions

urn =

~a7v7(0) ₍₃₄₎

;

urn =

~jfl7vf'(0)

;

and

equation (31)

_{we can} write

r(s

₌

0, #)

= rmf

~j(um

_cos

ml

+ _vm sin

ml). (35)

m

Insertion of this

expansion

in

(25) yields

Qii

=

jrg

^{+ A}

Q22

_"

jrg

^li

(36)

with the _square of the radius of

gyration

~( ~$f ~~mf"0

+

"(

_~

)("~

^{~ V~) ~}

~("$

~

V$) (~~)

n=2

For

Q12

we

get

Qi~

=

~

v~(uo rmf)

^~

uibi

+

f (6

+

n(n

+

2)) (unum+~ vnvn+2). (39)

4 8 8

_~

After

calcu<ating

the traces and

doing

the thermal _{averages, we}

finally

arrive at

(~~(4~))

_{" (2~~} ~

Q~2)

"

~~~("() (~("0)("()

^~

~ _(("~)

~

("~)~~

~(('~~) ^l'~()

^~

( ^~

(~

~ ll~(ll~ ^~

~))~ ("$)("$+2) (~°)

m=1

and

+r~

=2

+j

(uo)

and

iv()

do not

vanish,

because of the term linear in _v introduced

by

the

Faddeev-Popov

determinant. Thus the _averages

(u$)

_are

given by

~~$) /~ ~ ^~(~~~~

_~

^~2 ~ ~~~~i~~~~~~~~

I ' ii ^{' J}

j~jj

=

[ _~

~7jj[j(ll~

⁺

^{O((flK)~~) l~~~}

lUSl

₌

jL~~~((()°~~~+°llflK)~~).

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ill Physics

of

Amphiphilic Layers,

J.

Meunier,

D.

Langevin,

N. Boccara

Eds., Springer Proceedings

in

Physics,

Vo<.21

(Springer,

Berlin,

1987).

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Weinberg

Eds.

(World

Scientific,

Singapore, 1989).

[3]

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2718.

[4] ^LARCHE

F.,

APPELL

J.,

PORTE

G.,

BASSEREAU

P.,

MARIGNAN

J., Phys.

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1200.

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R.,

SCHOMACKER

R.,

ROUX

D.,

NALLET

F.,

OLSSON

U.,

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