A Theory on the Bending Moduli of Thin Membranes by the Use of a Simple Molecular Model

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A Theory on the Bending Moduli of Thin Membranes by the Use of a Simple Molecular Model

Yukio Suezaki, Hiroyuki Ichinose

To cite this version:

Yukio Suezaki, Hiroyuki Ichinose. A Theory on the Bending Moduli of Thin Membranes by the Use of a Simple Molecular Model. Journal de Physique I, EDP Sciences, 1995, 5 (11), pp.1469-1480.

�10.1051/jp1:1995210�. �jpa-00247150�

Classification

Physics

Abstracts

62.10+s 68.10Et 87.22Bt

A Theory

_on

the Bending Moduli of Thin Membranes by the Use of

_a

Simple ^Molecular Model

Yukio Suezaki

(*)

^and

Hiroyuki

Ichinose

Physics Laboratory, Department

of General Education,

Saga

Medical

School, Saga

849,

Japan

(Received

19

May

1995, received in final form and

accepted

16

August 1995)

Abstract. The nature of the

bending elasticity

of

monolayer

membranes _was studied

by

the

use of the free energy

mortel,

in which trie energy of the surface dilations of

head, chain,

aria intermediate surface of the molecule

was assumed. The intermediate surface _was introduced to

reproduce

the frustrated internai _stresses within mdividual molecules. Under

given

curvatures, the _area of the neutral surface and its

position

_were determined _{so as} to minimize the free _energy.

We have shown that there exists _a neutral surface where the saddle

bending

modulus becomes

zero when the intermediate part does not exist. This result is due to the fact that the free energy does not possess any ^term that is resistant to shear and it represents the

liquid phase

of membranes. For frustrated membranes with the intermediate

surface,

saddle

bending

moduli of finite values, but much smaller thon the

cyhndrical

modulus _m

magnitude,

_were

obtained,

the

sign

of which is either positive _or

negative depending

_on the spontaneous area and the

position

of the intermediate surface.

1. Introduction

The elastic moduli of

bending

of

monolayer

and

bilayer

membranes _are

important

for

studying

the various

topologies

of the membranes such _as veside

formation, sphencal microemulsions,

bicontinuous

emulsions,

and _{so on.} Helfrich

iii

described the elastic free energy surface

density, f,

of the

bending

of membranes _as:

f

=

k(c~

+ cy

co)~

+

k'c~cy il

2

where k and k' _are the elastic moduli for

cylindrical bending

and saddle

bending, respectively.

The factors _c~ and cy are the

principal

curvatures of the membrane and _CO is twice the spon-

taneous curvature of the membrane

iii.

This

formula,

and its

variations,

bave been

widely

used _as trie

bending

elastic energy of surfactant membranes

[2-8].

^Several authors have tried to derive trie moduli

according

to their theoretical models

[9-1Ii.

Now _we look into the fact that the shear free condition has not been taken into account for

applying equation (1)

to

liquid

membranes

iii.

One should

keep

in mind the

following

(* ^e-mail: suezaki@smsnet. saga-med. ac.

jp

@

Les Editions de

Physique

1995

example:

when _a bulk

isotropic

solid material which has _two Lame's constants of

elasticity

melts to

liquid,

it then

begins

to possess

only

_a

single

elastic modulus

(the

bulk

modulus)

due to the shear free condition.

According

to the elastic

theory

of thin

membranes,

this fact has

already

been darified

[12,13].

On the other

hand,

_as the old elastic

theory

does not address the

equilibrium

of membranes with

intra~molecularly

frustrated

stresses,

there is _a

possibility

that the saddle

bending

modulus does exist for frustrated

liquid

membranes.

Helfrich discussed

elegantly

the inter-relations _among

k,

k' and _co

by

the moment

expansion

of the stress m the membrane

[14]. According

to his

theory,

k' exists for _any membrane and

is related to the spontaneous curvature.

However,

it _was assumed that the stress is

isotropic

even

though

_c~ and cy are

unequal. Obviously,

theories with this

assumption

are not suitable for the discussion of the delicate difference of

liquid

and solid membranes.

Thus,

_we conclude that the

theory

_on the

bending

moduli of thin membranes has _not

yet

^been

fully

established.

In

1976,

Petrov and Derzhanski

proposed ils]

_a free _energy

mortel, #,

^and later it _was further

analyzed by them,

Mitov and others

[16-19].

As others have

doue,

_we will also refer to the Petrov-Derzhanski-Mitov

(PDM)

mortel. The mortel _assumes the head and tait

(chain)

with each

spontaneous

areas and surface elasticities which will be described later.

Although

the model _is

simple,

it is useful for

studying

the

problems

stated in the

previous paragraphs

because trie model manifests the

liquid phase correctly

In other

words,

the model is

composed

of surface

elasticity

and does not bave any ^term that is resistant to shear. We will reexamine

this model in order to

clarify

the

bending elasticity

of thin membranes in the

liquid phase by correcting

old treatments

[15-19].

In

regards

to the

expenmental

works _on this

problem, only

membranes of ionic surfactants

were examined

[20, 21].

^{Membranes with electric}

charges

_can have _a saddle

bending

modulus

even for

liquid

membranes because of the

long

_range nature of the electric interaction

[22, 23].

Strictly speaking,

the elastic _energy

given by equation (1)

is

postulated

such that the moduli of

bending

do not

depend

_on the

global shape

of the membrane. We will restrict ourselves to the

electrically

neutral membrane _or the membranes with _a

Debye length

much shorter than the radius of curvature m the

following analysis.

In the

following section,

we will revisit the PDM model free energy

defining

_{a new} neutral

surface,

its

position

will be defined _{so as to} minimize the free _energy. The

bending

moduli and the

spontaneous

curvature will be evaluated. The saddle

bending

modulus will be shown to be

zero.

In the third

section,

the PDM model will be extended to possess the intermediate surface between the head and the tait _m order to evaluate the effect of the

intra-molecularly

frustrated

membrane. The neutral surface at which the free _energy is mmimized will be found with numencal calculation inside the membrane _even for

strongly

frustrated membranes. The saddle

bending modulus,

in this _case, possesses finite values much smaller in

magnitude

than that of the

cylindrical bending

modulus.

In the last

section,

a conclusion and discussion will be

given.

2.

Analysis

of PDM Model Free

Energy

^for Bend of Thin Membranes

In order _to examine the

bending elasticity

^{of thin molecular}

membranes,

we will

employ

the PDM model free energy of

elasticity

of the membrane per molecule

Ii?i,

which is wntten as

where

AH

_"

Ail

+

ôHJ

+

à(K) ₍₃₎

Ac

_"

Ail ôc

J +

à(K) ₍₄₎

J =

c~+cy (5)

K ₌ c~cy

(6)

à =

ôH+ôc (7)

In

equation (2), flH, flc, A(, A$, AH,

and

Ac

_are the force

constants,

spontaneous areas and

occupied

_areas,

respectively,

of the

respective sullices,

H and C. H and C _mean the head and the tail

(chain)

of the

molecule, respectively Ii?i

In

equations (3), (4),

and

(7), A, ôH, ôc,

and à _are the _{area per} molecule of trie neutral

surface,

distances from trie neutral surface to trie head and

tail,

and trie distance from head to

tail, respectively.

Trie force constants,

flH, flc,

_are defined

differently

from trie

original

_ones because trie formulation becomes

simpler by

this revision.

Fogden

and others obtained the _zero saddle

bending

modulus _even

though

the treatment of them is different from _ours

[19].

First,

in trie _case of _a flat

plane (J

= K ₌ 0 and

AH

=

Ac

=

A), equation (2)

becomes

The _area A is determined _{so as} to minimize trie free energy, thus:

~"flHlA-Al)+flclA-Al)#° ₁₉₎

The _area

A,

in this _case, is

wntten,

AO, ^and becomes from

equation (9)

_as

The minimized free _energy,

çio,

turns out to be

~~ 2(Î~ÎC ^~~~

~~ ~~

where

AHC

is

A( A$.

Next,

_we will

analyze

the _case in which the membrane _is deformed from the

plane

to _{a curve} with _a finite _mean curvature,

J,

and the Gaussian

curvature, K,

_as are

given

in

equations (2)

to

(6).

Before

proceeding

with further

analysis,

_we will

briefly

state the

physical

_structure of the model free _energy. The

parameters

^{J and K} are the

extemally given

factors of

bend,

A and ôH _are free parameters and others _are theoretical constant terms. The

factors,

A and

ôH,

^{should be determined} _{so as} to minimize the free energy. On the other

hand,

the

previous

researchers identified the _area A _as

Ao

_m

equation (10)

without

minimizing

the free _energy for curved membranes because it _was believed that the neutral surface should be the surface which

conserves the surface _{area upon}

bending [15-19]. Thus,

for curved membranes the treatment

is different from the flot

plane

and the _necessary minimization process of the free _{energy was} absent.

Also,

the curved membranes and the flat membrane _were not treated

equally

_even

though

the flot

plane (J

= 0 and K

=

0)

is not _a

particular point

in the two dimensional _space of externat parameters ^{J and I(. The}

special

point ^{would be J} =

Jo,

and K

=

0,

where

Jo

_is the

spontaneous

curvature if _we define it. The model free energy is per molecule and not per

unit area, so the _area,

A,

could

change

_upon

bending.

Thus,

the area,

A,

is determmed from

equation (2)

as

~~

=

flH(AH A()(i

₊

_ôHJ

₊

à(K)

₊

_flc(Ac A[)(i ôcJ

+

à(K)

= 0

(12)

Inserting equation (12)

in

equation (2),

_we

obtain, j~2

~ ~°

2)

^~~~~

where

1 02 02

~°

2

~~~~~

~

~~~~

_~~~~

D

=

fIHA((i

+

AH)

+

flcA[(i Ac) (15)

fl

_"

flH(1+ ~H)~

+

flC(1 ~C)~ (16)

and where

AH

₌

JôH

₊

Kô(

and

Ac

=

Jôc Kô(. _{Equation (13)}

could also be rewritten

m a similar form to

equation iii),

but _we will _use the form of

equation (13)

because of its

simplicity

and

transparency

^{for further}

analysis.

The definition of the neutral surface is the crucial point ^{of the}

theory.

In the _case of dassical

elasticity,

the neutral surface _is defined _as the center of torque, ^and it comcides with the stress free surface

by

the balance of forces and that of

torques [24]

^and not

using

an assumed free

energy model. The PDM model does _not address the external force

and/or

extemal

torques exphcitly

but is written in terms of

extemally given

curvatures, J and K.

Thus,

the

gmding pnnciple

of the

analysis

_is to mimmize the free energy and not to balance the forces and torques.

The _previous theones defined the neutral surface _so that the free _{energy is}

decoupled

between the surface

elasticity

term and the

bending

term

[15-19]. Instead,

_we define the

position

of the

neutral surface

(ôH)

_{so as} to _mimmize the free _energy, which is written _as

~~

=

Î (flôD 2)Dô)

= 0

(17)

ôôH 2fl2

where

~ô

_"

~

"

~ÎfIHAÎ

+

fICA~Î

+

~~Ù(fIHAÎôH fICA~ôC) (l~)

fié

₌

~~

=

2J[flH(i

+

AH)

+

flc(i Ac

_{)Î +}

4K[flH(i

+

AH)ôH flc(i Ac )ôcl (19) ôôH

From

equations (15)

to

(19),

_we obtain

~

=

~~i~~~ (AHC Al _Ac A$ AH )(J~

2K ₊ JKô

2K~ôHôc (20)

H

fl

Next,

the

spontaneous

curvature,

Jo,

^{will be determmed from}

à#

~0

(~j

~~~~

@

j~ ~~~

where

t

=

)iflJD-2flDJ) ¹²²⁾

ô2j

1

_jfl~2fl _{_~jfl~ _fl} ~j2j j~~j

ôJ2 ~j3

^{~~ ~ ~}

~J

_"

(~) "fIHAÎôH~fICA~ôC (~~)

J=K=0

~~ ~~~J=K=0

~~~~~~ ~~~~~

_~~~~

fIJJ (~~

"

~(flHôÎ

+

flcô~) (~6)

J=K=0

The spontaneous curvature,

Jo,

_is a function of

ôH.

Before

calculating

ôH

by solving

_equa~

tion

(20),

_we obtain the formula for the

bending moduli,

k and k' _as follows:

k =

~~l~)

(27) Ao

ôJ2

j,~=o

k'

=

~~

=

Î(DfIK _{2DKfl) (28)}

Ao

^ôK

j~j~,~~o

2fl2Ao

where

flK l~

"

21flH(1

+

~H)ôÎ

₊

_{flC(1 ~C)ô~Î (~~)}

J=Jo,K=0

~K (~) _fIHAÎôÎ

+

flCA~ô~ (~°)

J=J(,,IC=0

Using equations (15), (16), (29)

and

(30), equation (28)

becomes

k'

=

~~~~~ ^{(AHC A[ AH A(Ac )[à( ii}

⁺

^AH à( ii Ac)Î (31) 2fl Ao

Now,

we notice the

important

fact that the factor in the first bracket in

equation (20)

is identical to the factor in the first bracket in the RHS of

equation (31). Thus,

trie minimization of the free _energy with

respect

to ôH

(Eq. (20)) necessanly

leads k' _to become _zero. This fact is

reasonable because the PDM model consists of surface

elasticity

terms and does not have any term that is resistant _to shear. Dur result is the

contrary

of those obtained

by previous authors,

m which k'

is different from _zero in

general [15-18].

Before

finding

the

position

of the neutral surface

(ôH) by solving equation (20),

_we note that the condition k'

= 0 has another solution for ôH the square bracket in the RHS of

equation (31)

_can be set to _zero and the

resulting

ôH

is almost the center of the membrane and is also _near the center of

torques

^exerted on the head and the chain. This center of trie

torque

^{is the} neutral surface based _on the dassical

theory

of

elasticity [12,13].

The above result states that trie PDM model free energy gives a different neutral surface from that of trie dassical

theory

^of

elasticity _[24].

From

equations (15), (16), (20-26),

_we

finally

obtain ôH and

ôc

_as

~~ fIHÎfIC

~

~

3(Î~~ÎÎ)AOÎ

^{~ ~~~~}

~~ lH~flC ~

3(Î~~ÎÎ)AOÎ

^{~ ~~~~}

The factor

AHC/Ao

in

equations (32)

and

(33)

should be much smaller than

unity, otherwise,

the

system

_can net form stable membranes and the

aggregates

would be

spherical

_or

cyhndrical

micelles. The result of

equations (32)

^and

(33)

is

approximately equal

to that obtained

by previous

authors

[15-18]

which is

expressed

_as

ôHflH

"

ôCflC (34)

Equation (34)

_was obtained based _on the idea that the free energy per molecule is

decoupled

as a function of the _area and the bend

(J

and

K).

Dur result

(Eqs. (32)

and

(33)

is based _on the

principle

that the free _energy should also be minimized with respect to the coordinate of the neutral surface.

In

conclusion,

the modulus of the saddle

bending

of thin

liqmd

membranes has been shown to be _zero

by using

the free _energy model of surface dilation. In other

words,

when _{we wnte}

down the PDM model free _energy in the

following

^form

~

~°~~~~~~ ~

Î~ _{~~ ~~~}

_~

_~ÎÎ

j~j~

~ ~~~~

the last _term of

equation (35)

has been shown to be _zero.

Fogden

and others also obtained the saddle

bending

modulus to be _{zero even}

though

the treatment is different from _ours and

they

were not _aware of the

physical meaning

as stated here

[19].

^{In the} next

section,

_we will examine

the effect of the frustration of the molecules

by mserting

the intermediate surface between the head and the chain.

3. Trie

Bending

Moduli of

Intra~molecularly

Frustrated Membranes

In the _previous

section,

_we revisited the PDM model free _energy

by defining

the neutral surface

so as to minimize the free _energy.

However,

there is _a

possibility

that k' could be finite if the molecule _is frustrated _as Szleifer and others

analyzed _[25]. Also,

the frustrated

bilayer

membranes made of two

monolayers

with their own

spontaneous

curvatures bave been shown to possess a small but finite saddle

bending

^modulus even if the intrinsic saddle

bending

modulus

is zero

initially [17,25].

^From the above

discussion,

we can ask the

following question:

do the

monolayer

membranes in the

liquid phase composed

of intra

molecularly

frustrated surfactants possess a finite saddle

bending modulus,

or not? In order to _answer this

question,

it is worth

estimating

the

bending

moduli

employing

a

simple

model which

correctly

_represents the

hquid

membranes and frustrated intemal stresses within individual molecules.

In order to _examine the effect of the intra-molecular frustration of

monolayer

membranes in the

liquid phase

_upon the

bending moduh,

_we ^{extend the} PDM model

inserting

_an intermediate

surface between head and tail at which the additional surface dilation _energy works _among

neighbonng

molecules. The schematic picture ^{of the molecule} is shown _m

Figure

i. In

Figure i,

the

subscript

G stands for the intermediate surface. The

factors, ôG, di,

and

A[

_mean the distances from the neutral surface to the intermediate surface and that from the head

(H)

to

H

dj ô~

ô~ à

ô~

neubalsulfàce

Fig.

1. Schematic picture of frustrated membrane with intermediate surface.

the intermediate

plane

and the

spontaneous

_{area per} intermediate

surface, respectively.

The

length, ôG,

from the neutral surface to the intermediate

surface, by definition,

_is written _as

ôG = ôH

di (36)

The

length, di,

will be scanned between the head and chain _{as a} theoretical

parameter.

^The free energy per

molecule,

_{çi, of the}

_system

is written _as

The _last term _of

equation

(37) _is the _new term

for

the

intermediate surface and the other

terms

are _the ame as the previous definitions of head (H)

~

= flH

Ii

₊

AH )(AH Al

₊

_flc Ii Ac)(Ac A$)

+

flG(i

+

AG )(AG A$

= 0

(38)

where _we defined the _new factors of the intermediate surface _as follows

AG

_"

Ail

₊

AG (39)

AG

=

JôG+Kô( (40)

The area,

A,

becomes

~

flHli

+

AH)A[

₊

flcli Ac)A[

₊

flGli

+

AG )AS fl j~~~

=

flHli+A~j2+flapi-Acj2+fl~ji+A~)2 j42j

JOURNAL DEPHYSJQIJEi -T. 3.N° ii NOWMBER 1993 38

The minimized free _energy, çio, ^of çi with

respect

to A becomes

çio =

~~Î~ _(A[AH

+

A(Ac _AHC)~

₊

^~~~~ (A[Ac

₊

A[Ac _AGC)~

2fl 2fl

+~~Î~ _{(A(AG A$AH AGH)~ (43)}

2fl

where

AIJ

_"

A( A§ II,

J

=

H, C, G; AjI

=

-AIj) (44)

The form of

equation (43)

is suitable to _see the

physical meaning, however,

the similar formulae to

equation (13)

and the formulae

following

to

equation (13)

_{are more} convenient for _Dur further

analysis.

We will rewrite

equation (43)

_{as m} the

following j~2

çi = 4lo

fi

(45)

2fl

where

~° ~~~~~

_~

~~~~

_~

~~ ~~

_~~~~

D =

fIHA( ii

+

AH

₊

flcA[ ii Ac)

+

fIGA[ ii

+

AG (47)

We will trot repeat the similar formulae made _m the

previous

section.

Instead,

_we will _sum- marize the result of the calculation of the free _energy minimization with respect to ôH and the formula of the saddle

bending modulus, k',

_as follows:

~~ ~~

=

flHflc(ôH

+

ôc)(AHC A[AH Al _{Ac ÎJ~}

_{2K +}

_JK(ôH ôc 2K~ôHôcÎ

D

ôôH

+

flcflG(ôc

_{+ ôG}

)(AGC A[Ac A[AG )ÎJ~

2K ₊

JK(ôG ôc 2K~ôcôGÎ

+

flGflH (ôG ôH)(AGH AIAG A[AH )ÎJ~

2K ₊

JK(ôG _ôH)

₊

2K~ôGôHÎ

(48)

A0fl~k'

_fl~

~~

~

fl~ fl~(ô~(1

₊

~H) ôÎ Il ~CÎ~~~~ ~~~~ ~~~~~

~ ~ ~~

~~~ ^~Î~~

+

~~ _~~~ ~~~~~~~ ~~~ ~~~~

~~~~

Contrary

to the combination of

equations (20)

^and

(31), equations (48)

and

(49)

have _no

common factor which _causes the value of k' to become _zero.

Thus,

the

point

where the

free energy is mmimized does net

necessarily

make the saddle

bending

modulus _zero. Before

proceeding

with further

analysis,

_we notice the

followmg

fact:

equation (49)

_can be

proportional

to the second moment of the stresses _as was defined

by

Helfrich [14] which is wntten

~~~~ z~a(z)dz

_ce

~ô(

+

(à(

+

~ô( ₍₅₀₎

Chain H C G

As the used model free _energy

(Eq. (37))

does not have _terms resistant to

shear,

the above

proportionality

is _a reasonable relation between _Dur treatment and trie moment

expression [14].

In the

following section,

we will evaluate trie neutral

surfaces,

the

spontaneous

curvatures, and the saddle

bending

modulus

numerically changing

theoretical

parameters

based _on the above formulae.

4. Numerical Calculation of the

Bending

Moduli and Other

Quantities

In this

section,

_we will show the results of the numerical calculation of various

quantities assuming

_many combinations of the theoretical

parameters postulated

in the

previous

section

m the _case of frustrated membranes. The chosen combinations of

parameters

are as follows:

first,

the force constants,

flH,

and

flc,

_are chosen to be

equal

and set to be

unity,

1-e-

flH =

flc

= 1

jsij

The evaluation with different combinations of flH ^and

flc

other than

equation là

i may ^not add any

essentially

_new result but will make the _paper

lengthy. Next,

because _{we are}

considering

the frustrated membranes with _a small

spontaneous curvature,

we will fix the _areas of head and chain _as

Al

=

i, Al

= 0.9

(52)

Then,

the

spontaneous

curvature, without intermediate surface becomes

Jo

=

~~)

=

~'~~~ ¹⁵³¹

The addition of

factors, flG, dl, and, A[,

will result in the

change

of the

spontaneous

curvature from

equation (53). Thus,

the theoretical

parameters

^which can be

arbitrarily

chosen _are

flG,

the

spontaneous

_area,

A[,

of the intermediate

surface,

and the

length, dl,

ofthe distance from the head to the intermediate surface. If _we

choose,

for instance,

A[

= 0.95 and

dl

"

0.56,

the membrane _is net frustrated when the membrane takes its

spontaneous

curvature. In the

following,

_we will show the result of the numencal calculations of the neutral surface and other

physical quantities using

the combinations of the above three parameters,

flG, dl,

and

A$.

Also,

the

conditions,

J

=

Jo

and K

=

0,

were taken _{as we concem} with the linear

elasticity

of membranes in the

neighborhood

of

spontaneous

curvature. We will show results of the

combinations of theoretical

parameters

^of

flG

=

o-S,

I.o

(54)

Al

=

0.85, 0.9, 0.95,

1

(55)

For the combinations of force _constants and

spontaneous

areas

given by equations (54)

^and

(55),

the

position

of the intermediate surfaces

Idi)

will be scanned from _zero to à. In

Figure 2a,

the

positions, ôH,

of the neutral surfaces where the free

energies

_are minimized _are

plotted

_as

functions ofthe

position, di,

ofthe intermediate surface. Even

forlargely

frustrated membranes

(broken line),

the neutral surfaces remain within the membrane. In

Figure 2b,

the

cylindrical bending modulus, k,

is

plotted

_{as a} function

ofdi

Because the variations of both

ôH,

and k _are within several _per cent

by

the

change

of

Ag

^{from 0.85 to}

i,

we show

only

the numerical results for

Ag

= 0.85 _as the

representative.

^{The value} of k is

large

when the intermediate surface is

located _near the head _or the

chain,

which is

physically

reasonable if _we consider the stiffness of

o.

l'

/ /

o ,' o

ôH ' ~

ô _,

,'

/ / /

0.4 / 0

/ /

o o.2 o 4 o.6 o.8 1 o o.2 o.4 o.6 o.8 1

dl dl

Î à

ai b)

Fig.

2. Position

(ôH)

of neutral surface _{as a} function of intermediate surface and the

cylindrical bending

modulus, k, _as function of dl

AS

= 0.85," ^" flG

# .5, " "; PG _" 1.

a)

Position

(ôH)

of neutral surface;

b) cylindrical bending modulus,

k.

Jo ,,

~',,

Jo ''

,

, ,

o

, ' '

'' ,'

/ ',

'_

o

o o 2 o 4 o 6 o 8 1 o 0 2 0 4 o.6 0 8 1

dj dj

à à

aj bj

Fig.

3.

Spontaneous

curvatures

(Jo

of frustrated membrane with intermediate

plane

_as function

~~ ~ i< 1' ~0 _{~ ~~} i' ~0 _{~ ~} >'

~0

_~~~ <' i' ~0 _{~_}

~j

p ~,

~j

1. G G G G G

PG _" 1. i

the membrane

against

the extemal

bending torque.

^In

Figure 3,

trie

spontaneous

curvatures

are

plotted

also _~ersus

di

The

mcreasing

_or

decreasing

trends of the

spontaneous

curvature

m

Figure

3 _are

reasonably

correlated to the

geometrical

sizes of

A[.

The convergence of the

spontaneous

curvatures of

equation (52)

at

dl

#

à/2

in

Figure

3 is also reasonable. The saddle

bending

^{moduli calculated}

by equation (49)

for the

respective

theoretical

parameters

are shown

m

Figure

4. We state the

following

two facts _on the numencal calculation of k'.

First,

the saddle

bending

moduli do have finite values with both

positive

and

negative signs.

But

they

are smaller with two orders of

magnitude

than those of

cylindrical

moduli shown in

Figure

2b for

appropnately

chosen theoretical parameters.

Next,

the

sign

of the saddle

bending

modulus is

positive (negative)

when the _area of the intermediate surface _is

larger (smaller)

than the

area of the the _cross section of the _cone made of the head and the chain at the intermediate

surface. The

negative sign

_agrees with the

analysis

made

by

^Szleifer and others

[25]. Although

we have not shown it

here,

_we

proved

that the saddle

bending

modulus remains finite

(not

to

o. o

k' k'

,"' ^",

,,-"~',, o ,'

'~

,l' ', 1'

',

," ' dl ," dl

06 ~e1 ₁ à _8'

1 à

,'

, ^,

-o

",---"

-o ~

,'

' /

, /

-0.oi

a) b)

Fig.

4. Saddle

bending

modulus

(k')

_as function of di " "

AS

= 0.85 " "

AS

= 0.9

" "

AS

= 0.95 " "

AS

= 1;

a)

PG ₌ -Si

b) PG

₌ 1.

be

zero)

_even when the

spontaneous

curvature becomes _zero for frustrated membranes.

5. Conclusion and Discussion

In the

previous sections,

we reexamined the PDM model free _energy of the membrane

elasticity,

the free _{energy was} minimized with

respect

to the _area of the neutral surface per molecule and the distance from the head to the neutral surface. With this _new

analysis,

the saddle

bending

modulus turned _out to be _zero, which is reasonable because the free _energy does not have

terms resistant to shear.

Next,

the PDM free _energy model _was extended to include the

intermediate surface between the head and the

chain,

which _expresses the

intra~molecularly

frustrated membranes. The extended mortel free _{energy was} also minimized with

respect

to the

area of the neutral surface and the distance from the neutral surface _to the head. The saddle

bending modulus, however,

tumed _{out to} possess finite values with both

positive

or

negative

signs. We coula find the neutral surface within the membrane _even for

strongly

frustrated membranes.

But,

the saddle

bending

modulus is

smaller, by

_an amount of two orders of

magnitude,

than the

cylindrical

modulus.

The numerical calculation showed that the sign of the saddle

bending

modulus is

negative (positive)

when the attractive

(repulsive)

forces _are exerted among mtermediate

surfaces, reput

sive

(attractive)

forces _are exerted among

heads,

and

repulsive (attractive)

forces _are exerted among

tails,

which _agrees

qualitatively

with trie numerical result

by

other authors

[25].

The authors

acknowledge

Professor S-A- Safran and Dr. T. Kawakatsu for their critical discussion _on this

problem. They

also thank Mr. D. Stewart for his assistance _in

preparing

the

manuscnpt.

References

iii

Helfrich W., Z.

Naturforsch.

28c

(1973)

693.

[2] ^Faucon J-F-, Mitov M.D., Meleard P., Bivas I. and Bothorel P., J.

Phys.

France 50

(1989)

2389.

[3]

Ljunggren

S. and Eriksson

J-C-,

J. Colloid

interface

Sa. 107

(1985)

138.

[4]

Wang

Z-G- and Safran

S-A-, Europhys.

Lent. Il

(1990)

425.

[Si

Lipowsky

R., Nature 349

(1991)

475.

[6] Safran S-A-, J. Chem. Phys. 78

(1983)

2073.

[7] Safran S-A- and Turkevich L.A.,

Phys.

Reu. Lent. 50

(1983)

1930.

[8]

Gompper

G. and Klein S., J.

Phys.

ii France 2

(1992)

1725;

Leung

R. and Shah D.O., J. Colloid interface Sa. 120

(1987)

320.

[9] ^Nelson D., Statistical mechanics of membrane and surfaces 5

(World

Science,

Singapole, 1988)

12.

[loi

Leibler S., ibid. 5

(1988)

57.

[iii

Kawakatsu T. and Kawasaki

K., Physica

A

167(1990)

690.

[12] ^{Peliti L. and Nelson} D.R.,

Springer

Proc.

Phys.

21,

Physics

of

Amphiphilic Layers p106 (1987).

[13] Kozlov M.M. and Markin

V.S.,

J. Chem. Soc.

Faraday

Trans. if 85

(1989)

277.

[14] ^Helfrich W.,

Physics

of Defects,

chap.

12, R.

Ballian,

M. Kleman and J-P- Poirier, Eds.

(1980)

713.

ils]

Petrov A.G. and Derzhanski

A.,

J.

Phys.

Coll. 37

(1976)

C3-155.

[16] ^Petrov A.G., Mitov M.D. and Derzhanski A., in Adu.

Liq. Cryst.

Res. B

Appls. (Pergamon Press,

Oxford.Acad. Jiado~

Budapest, 1980)

_p. 695.

[17] ^{Petrov A.G. and Bivas} I., Prog.

Surface

SC~. 16

(1984)

389.

[18] ^Leibler S., Statistical Mechanics of Membranes and Surfaces 5, ^D. Nelson, T. Piran and S. Wein-

berg,

Eds.

(1988)

55.

[19]

Fogden

A., Hyde S-T- and

Lundberg

G., J. Chem. Soc.

Faraday

Trans. 87

(1991)

949.

[20]

Farago

B. and Richter

D., Phys.

Reu. Lent. 65

(1990)

3348 [21]

Kellay

H., Meunier J. and Binks

B-P-,

ibid. 70

(1993)

1485.

[22] Wintherhalt M. and Helfrich W., J.

Phys.

Chem. 92

(1988)

6865.

[23] ^{Mitchell D.J. and Ninham} B-W-,

Langmmr

5

(1989)

l121.

[24] ^Niordson F-I-, _m Shell

Theory,

North Holland Series _m

Apphed

Mathematica and Mechamcs

(North Holland~

Amsterdam,

1985).

[25] ^Szleifer I., Kramer D., Ben.Shaul A.~ Gilbert W-M- and Safran S-A-, J. Chem.

Phys.

92

(1990)

6800.