Bending tensions and the bending rigidity of fluid membranes

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Bending tensions and the bending rigidity of fluid membranes

W. Helfrich, M. Kozlov

To cite this version:

W. Helfrich, M. Kozlov. Bending tensions and the bending rigidity of fluid membranes. Journal de

Physique II, EDP Sciences, 1993, 3 (3), pp.287-292. �10.1051/jp2:1993132�. �jpa-00247832�

Classification Physics Abstracts

82.65D 87.20C

Short Communication

Bending tensions and the bending rigidity of fluid membranes

W. Helfrich and M-M- Kozlov

Fachbereich Physik, Freie Universit£t Berlin, Arnimallee 14, 1000 Berlin 33, Germany

(RecHved

6 April 1992, revised 29 December 1992, accepted 5

January1993)

Abstract. It is shown theoretically that the lateral tension associated with

bending

_may in effect reduce the elastic resistance _{to curvature} of fluid membranes. In particular, the sponta-

neous curvature of surfactant monolayers gives rise to _a reduction of the bending rigidity.

Stretching

and

bending

_are well-known deformations of fluid membranes such _as

amphiphilic monolayers. They

_are

energetically decoupled

from each other when

expressed

in _terms of the neutral

surface,

I-e- the surface that

keeps

its _area when _a

piece

of membrane is bent from the flat state at _zero lateral tension

ii,

_2]. In real

membranes, bending

and

stretching

cannot be

separated

from each other unless the curvature is uniform. In mechanical

equilibrium

_as well _as in

thermodynamic equilibrium

with respect to matter

flow,

local difserences in

bending

energy are associated with

equal

differences in lateral tension. For

instance,in

membrane

shape equations

the total lateral tension consists of two parts, one uniform and the other

varying

with curvature [3].

Although requiring

elastic _energy, the dilation and

compression by

the tension

ofbending

will be shown _to result in

a decrease of the

bending

_energy, I-e- the total deformational _{energy per} unit _area of neutral surface. The

paradoxical

effect _can be attributed to the concentration and

depletion

of membrane material in

regions

of lower and

higher bending energies, respectively.

In the _presence of _a spontaneous curvature of the

monolayer,

this will be _{seen to} reduce the

bending rigidity.

If there is _no spontaneous curvature,

analogous

corrections of

higher

than

quadratic

order in the curvatures could still

give

rise to _a

significant

reduction of the

bending

energy ^at very strong curvatures.

In

calculating

the effective

bending

_{energy we} consider

only cylindrical

curvature, which should be sufficient to

exactly

obtain the correction of the

bending rigidity

_[3]. We start from

a flat

piece

of membrane with _area AD ^and lateral tension 70. The energy of deformation

expanded

_up to second order in the

change

of _area, A

Ao,

and the curvature,

J,

_may be written _as

~

F "

70(A

_AD) +

)lo ^~~ _~~°~ ^~oJsJAo

+

)~oJ~Ao ^(I)

Here

lo

is the

stretching

elastic modulus of the

membrane,

A the

area after

stretching, (I.e.

288 JOURNAL DE PHYSIQUE II N°3

dilation

or

compression),

_~o the

bending rigidity,

and

Js

the spontaneous curvature. The ab-

sence of _a

quadratic

term

coupling

J and

(A

_{AD) means} that _we have chosen the neutral

surface _to describe the two deformations. Such _a neutral surface _can be defined for _{any 70,} not

only

for the standard _{case 70}

" 0.

The neutral surface

evolving

from the flat state _seems to exist for all

cylindrical

curvatures J and _area differences A AD,

Provided

^the surfactant molecules

are free to shift in the normal direction _or, in other

words,

the neutral surface is not anchored in the material. The full definition of the mobile neutral surface

(which

_seems

particularly

useful for mixed

monolayers)

is

complicated

_as the elastic _energy is transferred to the membrane not

only by bending

and

stretching

but also

by

the normal shift. The additional work is done

by

the _pressure difference

between outside and inside which is _an

integral

part of mechanical

equilibrium. Fortunately,

this work does not contribute to the

quadratic expansion

about the flat state, ^the _pressure difference

varying

as J~ for Js

#

0 and

J~

for J~ ₌ 0 [3].

We _assume the

piece

of membrane to be connected to _a reservoir

which,

for

simplicity,

_may

be _a flat membrane of tension _70. If the

piece

of _area Ao is

bent,

its lateral tension 7 will

change

from 70 ^to

7 " 70 ~JSJ +

)~J~ ⁽²⁾

This

relationship

is _a consequence of mechanical

equilibrium

with the reservoir. It holds not

only

for

equilibrium configurations

but also for

fluctuating

membranes when their

dynamics

are

averaged.

Because of Hooke's law in the form

7"70+lo~ _~~°

,

(3)

which is the derivative of

(I)

with respect to A AD, ^the new tension is associated with _{a new}

area

A =

Ao(I iJsJ

₊

)J~) ₍₄₎

ofthe neutral surface in _terms ofwhich

bending

^and

stretching

are

expressed.

The total _energy of deformation per unit _area of this surface is

given by

Af=~ ^{~°~~ ~°~} ₍₅₎

if the extra _area A AD ^is absorbed

by (or,

if

negative,

taken

from)

the reservoir. Insertion of

(2)

and

(4)

in

(5) yields

~~ ~~~

~

~~~

~~~ ^~~

^~~~

up to second order in J. The last term is the _sum of _a

positive

contribution from the

stretching

energy and _a twice _as

large negative

contribution from the

change

in _area

multiplied by

~oJsJ.

Combining

^{the second and the third} term of

(6),

we find the efsective

bending rigidity

~~j2

~ " ~0

( (7)

0

Let _us

emphasize

that formula

(7)

results from _a

quadratic expansion

about J ₌ 0 and A = AD for

given

Js and _70. We do not restrict ourselves to 70 " 0, the Schulman

limit,

because oil in water and water in oil microemulsions _as well _as flat interfaces _are characterized

by

_some

nonvanishing

₇₀ of the

monolayers.

In most _cases this

positive

tension is rather weak

so that its effect _on the elastic moduli _~o,

lo and, thus,

~ may be

expected

to be

negligible.

The

expansion

of

(5)

_can be continued

beyond quadratic

order. This

gives

exact results if

equation (I)

is valid for

arbitrary

deformations. A continuation _seems

interesting

if the

spontaneous curvature

and, thus,

the difference between _~ and _~o is small. When the

quadratic

correction vanhhes

entirely,

I,e, for Js ₌

0,

the first

nonvanishing

^correction term of the

expansion

is

quartic. Up

to this order _one then obtains from

(S)

~£

f ~~0 j2

_{~ ~}^~~

j4 (~)

A curvature

dependent

effective

bending rigidity

_may be defined

through

@2

/£f

~ _~ j2

~~~~ 0J2 ^~°

~

2

lo

^~~~

The

quartic

correction of A

f

will be

significant only

at very

high

curvatures. In these circum- stances, use of the full formula

(S)

_may ^be

preferable

to an

expansion. Moreover,

there _can be other

quartic

corrections of the

bending

_energy. Even cubic terms _are

possible

since _an

amphiphilic monolayer

is

asymmetric,

Js ₌ 0

notwithstanding.

We introduced the membrane reservoir

mainly

to

simplify

the argument.

Any

other

provi-

sion that

keeps

₇₀ constant has the _same effect. For

instance, equation (7)

^{is valid whenever}

we

produce

sinusoidal

bending

deformations in _a flat membrane. This is because to _a first

approximation

the

accompanying

local dilations and

compressions

of the

monolayer

area

just

cancel each other _so that the total _area remains constant. Similar,

though

less

stringent

_con- siderations

apply

to the

shape

fluctuations of _a

sphere.

We remark that _a

generally

sufficient way of

ensuring

constant 70 is the free

exchange

of surfactant molecules with _an environment of

constant chemical

potential. Stretching

_can still be defined

by considering

_a

piece

^{of membrane}

composed

^of _a ^fixed number of surfactant

molecules,

^and

will,

in

fact,

_occur under the action of the

bending

tension.

The effective

bending rigidity

_can also be derived in _an alternative way which in _{a sense}

requires

_no

stretching.

We bend _a

piece

of

membrane, keeping

_now the _area of the reference surface

strictly

^fixed but

permitting

the membrane _to shift in the direction normal _to the reference surface. As _a consequence the membrane will _assume the

position

of minimum

deformational _energy under the constraint of fixed molecular _cross section _on the reference surface. This _{energy can} be calculated from

(I)

if _one knows the distance _s of the neutral surface _as defined above from the _new reference

surface,

where _s is taken to be

positive

if directed outside _a

cylinder

of

positive

curvature. With J

being

the curvature of the _new reference

surface,

^the effective

bending

_{energy per} unit _area ofthh surface

obeys

~~ ~°~~~

l

~sJ

~

~~° i~l /sJ~

^~

^~°~~~~~

~~~~

where the last _term is the

stretching

_energy.

(Pressure

difserences _can be omitted for small

J;

see

above). Expansion

_up to second order in J and minimization with respect to _s

gives,

for small

enough

curvatures,

~ "

~) _(~~)

o

Inserting

^this in

(10)

_we arrive

again

at the

quadratic approximation (6)

for

hf.

The fact that the reference surfaces _are difserent in the two derivations of the efsective

bending

_energy

290 JOURNAL DE PHYSIQUE II N°3

becomes apparent

only

if

higher

order terms of this _{energy are} included. When

describing

the _same

membrane,

the two reference surfaces have to be at _a distance _s

=

s(J)

from each

other. The reference surface of the second method

is,

of _course, the surface of inextension of the membrane in lateral

equilibrium.

To estimate the efsect of the lateral tension of

bending

_on the membrane

bending rigidity,

_we

use the data from

lipid bilayers

^which are the

only

ones available. The

stretching

moduli and the

bending rigidities

of these

bilayers (at

_room

temperature)

_are about 200

mN/m

and

(2/3)

_x

10~~~ J

[4-6], respectively,

with _a

spread by

at least factor of _two in the latter _case. We will _use half these values in _our estimates. The spontaneous curvature of the

monolayers composing

the

symnJetric bilayers

is _not known for the flat state and _may be

expected

to vary

by

_more

than _a factor of _two between the

lipids investigated. Equating

the spontaneous curvature to the

reciprocal

radius of the

cylinders

in the inverse

hexagonal phase

of _a

phosphatidylethanolamine

in _excess water [7], we have Js ₌

(1/3)

_nm ^~~ With this and the above

numbers,

one obtains

~

)

j2

= 0.037

which, according

to

(7),

_means, that _~ is smaller than _~o

by

less than 4 percent. ^{The result} is

probably

too small since curvatures close to the

geometric limit,

the inverse membrane

thickness,

_are

likely

to be

opposed by

additional forces

deriving

from

bending energies

of

higher

than

quadratic

order in the curvature. A

larger

correction is obtained if _{we use a} theoretical value for ~oJs. Now, _{we assume} the stress

profile

of the

lipid monolayer

to consist of _a surface tension of 50

mN/m

at the

water-lipid

interface and of _a

compensating

_pressure

evenly

distributed _{over a}

monolayer

thickness of 2 _nm.

Utilizing

that

-~oJs equals

the first

moment of the _stress

profile

_{[3, 8], we} compute ~oJs ₌ 5 _x 10~~~ _N. This

gives,

with the above numbers for _~o and

lo

~oJl

~

(~oJs)2

~ ~ ~~

lo ~olo

The

result,

which _may be viewed _as

an upper

limit,

_suggests that _~ is four times smaller than

~o. The actual size of the

rigidity

correction, which is the _same for

bilayers

and

monolayers,

can in

principle

be calculated from molecular models of the

monolayer. Obviously,

the calcu- lations should include the lateral tension of

bending

in order to be accurate.

The efsect of the

bending

tension _on the

bending rigidity

_may be moderate in most

monolay-

ers

consisting

of _a

single

surfactant.

However,

dramatic consequences can be

expected

in mixed

monolayers containing

_a cosurfactant soluble in oil _{or water} and

changing

its concentration in the

monolayer

_{as a} function of lateral tension. The _presence of such _a cosurfactant reduces the

stretching

elastic modulus

and,

because of

(7),

^the

bending rigidity

_~. This is to be examined in detail in _a theoretical treatment of the elastic

properties

of mixed

monolayers published

elsewhere [9]. For _a

preliminary

estimate _we may use the

simple

and

plausible

formula

lo

^~

IT

~~~~

where is the efsective

stretching

modulus and _a the

hydrocarbon

chain _cross section of surfac- tant and

cosurfactant,

both assumed to be one-chain

amphiphiles.

In _a

particular

theoretical model, this

simple

formula without _a numerical factor

applies

to the _case of equal concen-

trations of surfactant and cosurfactant [9].

Inserting

kT

= 4 _x 10~~~J

(room temperature),

a = 3 _x 10~~~ m~

(as

is

typical

of half the _cross section of _a twc-chain

lipid),

and

lo

_" 100

mN/m

_as

above,

_one finds

I _m ~~

cs 10

mN/m (13)

a

which is smaller than

lo by

_an order of

magnitude. Replacing lo

in

(7)

^with this I increases the correction of the

bending rigidity by

the _same factor.

The

quantity controlling

^the

magnitude

of the fourth-order corrections of the

bending

_energy

(8)

and of the _curvature

dependent

part of the efsective

bending rigidity (9)

^is

~oJ~/I.

It has

to be _near

unity

for _a substancial decrease of the

bending

_energy.

Obviously,

this

requires

_ex-

tremely high

curvatures of

nearly

10~ _m ~~. While such curvatures may be out of

experimental reach, they

could be

approached by

thermal

bending

fluctuations. For _an

estimate,

_{we may}

start from _a formula for the _{mean square} fluctuation of the curvature

lJ~)

₌

~ _(")

which is based _on the

equipartion

theorem and the

assumption

ofone

bending

mode _per

hydrc-

carbon chain.

Combining (13)

and

(14)

^leads to the

simple

^result

~o(J~)/I

₌ I.

Accordingly,

there is _a

possibility

that _even in the absence

ofspontaneous

curvature the effective

bending rigidity

ofmixed

monolayers

is reduced

substantially by

the lateral tension of

bending.

The concepts ^here

developed

_can be extended from

monolayers

to

symnJetric bilayers. Easy flip-flop

of surfactant molecules between the

monolayers

is _an additional mechanism to

keep

70 constant when the

bilayer undergoes bending.

This is

possible,

to first order in

bilayer

curvature, because the _curvatures of the

monolayers

_are

opposite. Equation (7) gives

the effective

bending rigidity

of the

bilayer (for

_lo

"

constant)

if _we insert the

bilayer

values for

~o and

lo

but

keep

the

monolayer

spontaneous curvature J~. ^The spontaneous curvature of _a

symmetric bilayer

_{as a} whole is of _course

equal

to _zero.

Finally,

_we note that

Wang

and Safran in _a calculation similar _{to ours} minimize the

bending

energy of _a bent

piece

ofmembrane with respect to molecular _area, thus

obtaining

_a reduced

bending rigidity [10].

^Their

procedure

does _not take into account _any molecular

exchange

with _an external reservoir but minimizes the surfactant chemical

potential.

To describe the membrane deformation

they

_{use an}

arbitrary dividing

surface.

(The dividing

surface is

thought

to divide the surfactant

monolayer

into _an upper and _a lower part; see e-g-

[2]).

An

arbitrary dividing

surface allows _an energy term that

couples

dilation and curvature.

Wang

and Safran's reduction of the

bending rigidity

is

proportional

to this

coupling

term. The effect vanishes if the neutral surface is chosen _as the

dividing

surface.

In our main calculation of the efsective

bending rigidity (Eqs. (I)

to

(7))

_we describe the deformations in _terms of the neutral

surface,

^which _means ^that _our calculation _starts from the reduced

bending rigidity

of

Wang

and Safran.

Moreover,

_we consider molecular

exchange

with _a

reservoir, keeping

the _area of _our

dividing

surface constant instead of the number of molecules. Therefore, the reduced

bending rigidity

obtained

by

_us is _even lower than

Wang

and Safran's result. In _a

parallel

calculation

(see Eq. (10))

_we confirm this result

by

_a minimization of the chemical

potential

with respect to _a normal

displacement

of the molecules

through

_{a new}

reference surface which is _{not a}

dividing

surface. No such normal

displacement

is considered in _our main calculation _or in

Wang

^{and Safran's work.}

Acknowledgements.

We _are

grateful

to R. Germar whose

diploma

thesis raised

problems

that stimulated the present work. Financial support ^{from the Alexander} von Humboldt Foundation for _one of _us

(M.M.

Kozlov)

is

gratefully acknowledged.

JOURNAL DE PHYSIQUE >I -T 3, N'3, MARCH 1993 ^>2

292 JOURNAL DE PHYSIQUE II N°3

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