Fluctuations of a polymerized membrane between walls

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Fluctuations of a polymerized membrane between walls

G. Gompper, D. Kroll

To cite this version:

G. Gompper, D. Kroll. Fluctuations of a polymerized membrane between walls. Journal de Physique

I, EDP Sciences, 1991, 1 (10), pp.1411-1432. �10.1051/jp1:1991211�. �jpa-00246425�

J Phys. I Fmnce 1

(1991)

1411-1432 OmOBRE1991, PAGE 1411

Classification

PhysicsAbs+acts

68.45Gd-05.70Jk-64.60Fr

Fluctuations of

_a

polymerized membrane between walls

G.

Gompper(~)

and D.M.

Knoll(21*)

(~ Sektion

Physik

der

Ludmig-Maximilians-Universit3t

Monchen, Theresienstr. 37, 80W Monchen 2,

Germany

(~) AHPCRC,

University

of Minnesota, 11W

lAhshington

Avenue South,

Minneapolis,

MN 55414, U.S.A.

(Received3 June 1991,

accepted

3Ju~yI ml)

Abstract. The fluctuations of _a

self-avoiding, polymer12ed (or tethered)

membrane which is _re- stdcted by the presence ^of two

parallel

^hard walls are studied

using

Monte Carlo simulations and

scaling

arguments. ^tile

scaling

behavior of the _area, the pressure, ^{the normal vector}

susceptibility,

as well as the _mean curvature

suceptibility

_are

investigated.

lAh calculate the

scaling

functions and show that the critical behavior follows the predicted scaling laws vith an exponent _q = 0.55 + 0.10.

1. Iutloductiou.

OrR of the

major

current

challenges

in theoretical

physics

b to understand the statbtical _me- chanics of surfaces and membranes

(see,

_e.g. Refs.

[1,2],

^{and references}

therein).

This field has

attracted considerable attention

recently

because of its

importance

in

understanding

such

diversq problems

_as the structure of lamellar

liquid crystals, microemulsions,

block

copolymers

and other

self-assembling

structures

[3,4],

cell membrane interactions in

biology

[5] ^{and biomimetic} systems, and world-sheet

dynamics

in

string theory [6].

The membranes of

physical

interest in chemical and

biological applications

can be

regarded

as

two-dimensional

self-avoiding

_su~fiaces imbedded in three-dimensional space.

They

are built up ^of

monomers which may ^be either free to diffuse in the membrane

surface,

or are frozen in

place

and thus form a network with fixed coordination. In the first _{case one}

speaks

of

fluid

membranes

[7-14],

and in the second, of tethered _or

po~ymeri2ed

membranes

[11,15-33].

Another

important possibility

are hexafic

membranes,

with extended bond orientational order

[34,35].

^{Because their}

microscopic

surface tension is small _or vanishes

altogether,

the

strength

of the

out-of-plane

fluc- tuations in membranes h controlled

by

the _{curvature or}

bending

_energy

_[7,8].

^Since the _corre-

sponding

elastic _constant is often of the order of

kBT,

membranes exhibit wild fluctuations. The

importance

and effect of these fluctuations b determined

by

the internal state of the membrane.

(*)

Permanent address; Institut for

Festk6rperforschung,

KFA Jolich, Postfach 1913, 5170 J01ich, Germany

1412 JOURNAL DE PI IYSIQUE I N°10

Polymerized

^self-avoid

ing

membranes have been shown to be flat at

long length

scales

[23,25-27],

whereas fluid membranes _are believed _to be

crumpled objects

with _a nontrivial fractal dimension

[7-14].

The fact that membranes are

self-avoiding

surfaces

complicates

_any theoretical

investiga-

tion of _even the

simplest questions regarding

the conformation of

single

bolated

membranes,

and is one of the

primary

reasons that Monte Carlo simulations have

played

a

particularly important

role in this field of research

[23,25-27].

Since _a tethered membrane is

always

flat, its

physical properties

are very

anisotropic.

In order not to loose

important information,

it is therefore necessary to

study

oriented

samples.

The sim-

plest

_way to achieve this

objective

is to restrict the rotational

degrees

of freedom of the membrane

by placing

it between two

parallel

walls. However, this not

only

orients the membrane, but also restricts the

out-of-plane

fluctuations. This _tums out to be

quite useful,

since

by varying

the wall

separation,

_one can control the

strength

of fluctuations and thus

study

^their effect on different

length

scales

[36].

The

primary

motivation of the present paper ^{b to} obtain _a better

understanding

of the elastic

properties

of

polymerized

membranes as well _as the interactions of membranes with surfaces or

other membranes. This is necessary ^{if future}

experiments

_are to be

interpreted correctly. Indeed, X-ray scattering experiments

on lamellar

phases

^of these membranes

provide

the most

promising

method for

measuring

the

bending rigidity

and

in-plane

elastic constants, as well _as

determining

the form and

strength

of the steric interaction between membranes

[37-50].

The

X-ray scattering

factor in this _case is characterized

by power-law singularities,

and the exponent

describing

this behavior is related to the membranes' elastic constants; simulations are

required

to determine the universal

amplitudes [44,45]

^which appear ^{in this}

exponent.

In the

present

paper we

present

a detailed

analysis

of the

long-length-scale

behavior of the

bending rigidity

^{and various}

susceptibilities.

^{Some related work has been described in} a recent letter

[50].

^{The outline of the} paper ^is as follows. In sections 2 and 3, _we introduce the continuum model

commonly

used in the theoretical

analysis

of

polymerized

membranes and summarize the details of the simulation

procedure.

Section 4 contains _a discussion of various effects which make

a direct

comparison

of simulation data with

theory

difficulL In

particular,

_we have found that

boundary

effects _are

particularly pronounced

for the free

edge boundary

^conditions we

employ,

and that _care must be taken to

separate

the 'bulk' behavior from

edge

or

boundary

effects. The

technique

of submembrane

averaging

we

employ

is discussed in this section. Our results for the critical behavior of the _area

fluctuations,

the pressure, ^the

susceptibility

of normal _{vectors, as} well

as mean curvature

susceptibility

are then

presented

in sections 5-8. We close the paper ^with a

comparison

of our results with those of other simulations of

polymerized

membranes.

2. Continuum model for

nearly

flat membranes.

In the flat

phase,

deviations of the _monomers from the flat _zero

temperature

^reference state can be described in terms of

in-plane phonon displacements u(z1, z2)

and

out-of-plane displacements z(z

i,

z~),

where

(z

i_,

z~)

_are internal membrane coordinates. In terms of these

displacements,

the free energy reads

[15]

F(z, u)

_{= 2}^~

/d~z(V~z)~

⁺ ₂

/d~z[2pu(

⁺

^{lu);], (1)}

where the strain matrix u;j is ^{related to} z and _u

by

_{u;j =}

)[0;uj

+

0j

_u; +

(0;z)(0j z)].

^{The first}

term in

(I)

describes the elastic energy ^of

bending,

and the second the elastic

stretching

_energy,

with the Lamd coefficients I and _p. Renormalization group

theory

shows

[18]

that the interaction

N°10 FLUCTUAIIIONS oF A POLYMERIZED MEMBRANE BETWEEN WALLS 1413

of the

in-plane phonon

modes and the

out-of,plane

undulation modes leads to renormalized _wave,

number-dependent coupling

constants pR,

lR,

and KR:

pR(q), lR(q)

~

q°~

(2) KR(q)

_~

q~~,

with

"

((2 ^{Qi). (3)}

Using

these

results,

we can calculate the

dependence

of several

quantities

on the

distance, 2d,

between the walls

[47]. By comparing

<

~~(~)

>C"

kBT /

~ ~

°'f(~~, ⁽⁴⁾

q>j~l

^{R q}

where < >c denotes the

cumulant,

with _< z~ _>c~w

d~,

we obtain the

parallel

correlation

length

_fjj

~w

d~/(~-n)

_{This can be inserted back into the} renormalized elastic constants to

give KR(d)

_~w

d~n/(~-n),

and

pR(d)

_~w

d-~n>/(~-n).

_{These elastic constants describe the elastic re-}

sponse ^{of stacks of}

polymerized

^membranes on

length

^scales

larger

than the correlation

length _fjj [47].

3. Monte Carlo simulation of tethered membranes.

We consider a

triangular

network of N

spherical

beads of diameter _a

=

[51]. Neighboring

beads in the network are linked

by

tethers of

length

lo. Self-avoidance is

generated by

the

pair-

whe hard-core

repulsion

of all beads,

together

with _a choice of tether

lengths

lo <

V$

_{and a}

sufficiently

small

'stepsize'

_s for each trial _move

(we

use lo ₌ 1.6 and _s < o.15 in _our simula-

tions).

For

simplicity,

we do not include an

explicit bending

_energy term into the Hamiltonian.

This does not

imply,

however, ^that no such _term appears ^{in the}

corresponding

model

(I):

the excluded volume

interactions, together

with the

tethering constraints,

_generate this _term

[31]

on

intermediate

length

scales

(larger

than _a, but much smaller than the correlation

length

_fjj

).

^The

global shape

of the network is

hexagonal,

with _a diameter of L monomers, and _a 'radius' R such

that L ₌ 2R + 1. Such _a membrane consists of N ₌

(3L~

+ 1)

/4

=

3R(R

+

I)

+ I monomers, and

of Na ₌

(L

_1)~

= 6R~

triangles.

We have simulated membranes from size L

= 17 to L

= 49,

and for 10~ _to 2 _x 10~ Monte Carlo

steps

per monomer

(MCS),

the

longer

times

corresponding

to the

larger

membrane sizes _or

larger

wall

separations.

The walls _are oriented

perpendicular

to the z-axh and restrict the

z-component

^{of the} center of each bead to lie in the intervall [0,

2dj.

^{A few}

typical

membrane

configurations

are shown in

figure

I.

Although

the

long wavelength

undulation

modes _are

clearly suppressed by

the walls, for the

larger

wall

separations they

look

quite

similar to the

configurations

of a free membrane shown in reference

[31].

The results of reference

[26]

^{indicate that} the relaxation time _rR of _a membrane of size L with

tether-length

lo ₌

vi

_and

stepsize

_{s =} 0.20 _can be

approximated by

_{m = To} +

TiL4~n,

with _To

= 1585, _{Ti =} 0.79, and q = 0.7.

Assuming

that thin relation remains

approximately

valid for the somewhat smaller tether

length

and

stepsize

used in _our simulations and that _we

can

extrapolate

^{this result} to the

larger

system sizes considered here, wc find _{rR =} 300, 000 MCS for the

largest

membrane considered

(L

₌

49).

^For a membrane between walls, _we would have to insert the

parallel

correlation

length

_fjj for L in this relation. For

large _fjj

we therefore have rR ~w

d~(4-n)/(~-n)

_{so that the relaxation time decreases}

extremely rapidly

with

dccreasing

^{d. This}

1414 JOURNAL DE PHYSIQUE I N°10

Fig.

I.

Configurations

of _a

self-avoiding

tethered membrane between two walls of

separation

^2d ₌ 9.0

(a)

after 12 _x 10~ _MCS and

(b)

2 x 10~ MCS later. In both cases, the

projections

_{are on} the _zy,, the _zz- and the

yz,planes. lbthering

bonds _are drawn between

neighboring

_monomers, whose hard _core size is _not

shown.

N°10 FLUCTUATIONS oF A POLYMERIZED MEMBRANE BETWEEN WALLS 1415

Fig. 1.

(continued).

makes _us

reasonably

confident that we have not

only

reached

equilibrium

in our

simulations,

but have also

averaged

over a sufficient number of

independent configurations.

4.

Boundary eTects,

finite size

eTects~

and corrections to

scaling.

In _our

simulations,

the finite extension of the model membrane manifests itself

through

both

boundary

and finite size effects.

Bounda~y effects

influence the monomers in the

vicinity

of the

1416 JOURNAI- DE PI IYSIQUE I N°10

membrane

edges,

where, due to the

missing neighbors,

the fluctuations _are much

stronger

^than

in the interior of the membrane. We will _see several indications of thin effect below. The width of this

boundary region

has been

conjectured [31]

to be of the order of

f(~~~l/~

~w d. One way to

distingubh

bulk from

boundary

behavior in the simulations is to consider submembranes

[11]

of

hexagonal shape

with _a dhmeter L,

(radius

R, which _are smaller than the diameter L

(radius R)

of the simulated membrane. The

boundary

effects _can then be identified from _a

change

in

behavior _as L,

approaches

L.

The

finite

size

effects,

on the other

hand,

are due to the lower

cutoff,

_q~n;n

= 2x

IL

in all Fourier

integrals.

On

length

scales

larger

than L, the membrane acts as a

rigid body. Finally,

there b the effect that for wall

separations

of the order of L the orientational

degrees

of freedom are restored.

This,

however,

is not very

important

for the

present

simulations because finite size effects _set in for much smaller wall

separations,

and we make _no

attempt

to

study

our systems

beyond

that

poinL

Because it is

possible

to fulfill the

inequality

d _« fjj ^«

L,

the walls

provide

an effective way for

keeping

the

boundary

^effects under control.

There is another limitation _on the range over which

scaling

laws _can be observed. The

scaling

laws

only

hold when the correlation

length

is the

only

relevant

length

scale in the system. ^Thb can

only

be the _case if it b much

larger

than all

microscopic length scales,

like the tether

length.

Fbr finite correlation

lengths

we therefore get corrections to

scaling.

All

thermodynamic quantities,

like the variance of the

z-distribution,

must have the

general

form

< Z~ >C"

f~' "(L/ill, '°/ill, ~/ill ), (5)

with

(

₌

(2 q)/2,

and a

scaling

function E which will in

general depend

on the

boundary

conditions. The results of the continuum model

(I)

can

only

be

compared

with the simulations in the limit lo

/fjj

_- 0 and

a/fjj

_- 0.

In

general,

there will be

non-negligible correction-to-scaling

contributions in _our simulation data so that _we expect, ^for

example,

< z~ _>c= Eo

d~'/(~-n)(I

₊

_Eid-~

₊

_), (6)

with _a

correction-to-scaling

_{exponent w.} For local

quantities

such as < z~ _>c, these corrections

originate

from the the finiteness of the

arguments

of the

scaling

function E _or the presence ^of irrelevant

operators

ⁱⁿ our discretization.

The corrections _to

scaling

_are most

easily

identified for _a

quantity

^which has _a known

scaling

behavior for

large d,

like the smallest

eigenvalue, 13,

of the moments of intertia tensor N

lap

₌

p Llra(I) fallrp(I) fpl, (7)

where _r is the coordinate vector of _monomer I, _a,

fl

E

(z,

_y,

z),

and f is the center of _mass of the

particular configuration

under consideration. _<

13

>

obviously

scales _as d~.

However,

when we

plot

<

13

> d-~ _{versus d-}

I,

_see

figure 2,

we do not

get

a constant, but _an

approxhnately

straight

line with finite

slope:

<

13

_>

d-~

= ao +

aid-~, ₍₈₎

with _ao

= 0.0177, _{al =} 0.0824. A

comparison

with

(6)

shows that the

corrections-to-scaling exponent

w ci I.

Another

quantity,

which should also scale _as

d~,

is the variance of the distribution of _z -values. It can be seen from

figure

3 that corrections to

scaling

are infact almost identical with those found in

N°10 FLUCTUA3IONS oF A POLYMERIZED MEMBRANE BETWEEN WALLS 1417

o-1

_~ «

<l~>d

°.°~

+ 17

X 25

o 33

D 49

o

o,o 5 1-o

1/2d

Fig. ~ The scaled average ^{of the smallest}

eigenvalue,

13, of the moments of inertia tensior I,

equation (7),

_versus

1/2d.

m6

~

_O.05

m

A L

+ 17

m~ _X 25

~ ~

u 49

2 4 6

1/(2d)

Fig. 3. The variance, < z~ _>c, of the distribution of height variables, as a function of the inverse wall

separation.

0.05 + + + + + + + + + + +

~i

~ ~

cQ

o o o o

=

° o o o o

~

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

~

D

My

2d

/

+ 2.5

M

~ ~~

V Q

~

D 9.o

~° ~ 40

Fig.

4. The variance, < z~ _{>c, of the} distribution of

height

variables, as a function of submembrane size, obtained from _a simulation of _a membrane of diameter L

= 49.

1418 JOURNAL DE PHYSIQUE I N°10

equation (8).

This is

perhaps

not too

surprising,

because <

13

> and < z~ _{>c are both measures} of the width of the membrane. It is

interesting

to note that < z~ _>c, when calculated for circular

pieces

of membrane of diameter Ls, is

essentially independent

^of Ls

(see Fig. 4).

Submembrane averages suppress

boundary effects,

but _are different from

periodic boundary

conditions. In both cases, the finite size

scaling

function E can be calculated. At the level of Gaus- sian

fluctuations,

with the

phonon degrees

of freedom

integrated

out, an effective Hamiltonian

Hlzl

₌

(° L(q~-n

₊

fj[~~-~~)zqz-q, ⁽⁹⁾

q

can be used

[52],

^where _{q; =}

fm;

with _{m; =} 0,1,..

,

N;, I ₌ 1,2. In the _case of

pmodic bbundaJy

conditions _we have

< z~ _>c=

~~~(~))~ £

_~~_~~

(10)

~°

9 ~~ ^~ ~

~ll

For q = 0, the calculation of the

scaling

function is

possible analytically,

with the result

E(y,0,0)

₌

j~~

₊

(ii ^? arctan(j~~)j, (II)

where

§

₌

)

=

L/(2xfjj),

and we have absorbed a factor _no

/kBT

in the

scaling

function.

Here _we have treated the _q

= 0 mode

separately,

and

replaced

the sum over all other modes

by

an

integral [53].

^{In the} submembrane case, on the other hand, _we consider _a small

part

^of a very

large

membrane of size L

(infact,

we will

usually

take the

thermodynamic

limit L

-

cc).

For local

quantities,

like _< z~ _{>c, there are}

no finite size effects associated with

Ls,

but

only

with the size L of the whole

membrane;

these will appear ^for

Ls/L

_- I, ^if at all. This can be seen

clearly

in

figure

4.

5. Real and

projected

_area.

In the

simulations,

both the real area, A, and the area

projected

onto the

wall,

Ah, fluctuate. The infinitesimal area element is

d~z/@,

_where

_g(x)

_{is the} determinant of the metric tensor, so

that the _mean area, < A >, and the mean

projected

_area, < Ah _>, are

given by

< A _>=

/ _d2z

<

AR

_>

_(12a)

< Ah _>=

f _d~z

<

li _nz(x)

_>,

_(12b)

where _nz is the

z-component

of the unit surface

normal,

_n. In the

Monge

_gauge, the

configu-

rations _are

parameterized

in terms of _a

single-valued

function

z(zi, z2)

of the Cartesian coordi- nates of _a reference

plane.

In this _{case n} has the form _n

=

(-01z, -02z, 1)/

+

(Vz)2,

_so that

< Ab >=

f d~z.

For small undulations the

square-root

can be

expanded,

to

yield

~ ~

A~>~~

~

"

~~~~ /))~ ^~~KR(~)q~'

^~~~~

with _qm;n

=

2x/L

for free membranes, and qm;n =

2x/fjj

^{for mcmbranes between walls}

^(for

fjj ^<

L).

Since the

integral

contains _an upper ^{momentum cutoff} qmax which k

proportional

to the inverse size of _{a monomer,}

(13) implies

that for _a free mcmbrane the ratio of

projected

to

N°10 FLUCnJAIIIONS oF A POLYMERIZED MEMBRANE BETWEEN WALLS 1419

real _area

approaches

a finite constant as L

- cc. This should be

compared

with the _case of fluid

membranes,

where the ratio

(13) diverges logarithmically [9,10]

^with system ^size

(for

L «

fp,

where

fp

^{is the}

persistence length [54]).

^{It follows from}

(13)

that the

asymptotic

ratio of _< A _>

and <

Ab

> is

approached

with the power ^law

fin

~w

d-~n/(~-n)

_for

a membrane between walls.

Biological

or artificial membranes are believed _to have a fixed _area per

amphiphile

_or

lipid headgroup,

so that the real membrane area does not fluctuate. In the

simulations, only single

monomers are moved at a

time,

so that the area of the membrane cannot be constanL

However,

the average area fluctuates very

little,

_as shown in

figure

5.

Furthermore,

the average area and its fluctuations _are almost

independent

of the wall

separation. Therefore,

_we will consider the

membrane area to be constant in the

following analysb

^of our data.

<1.O

Z~

~$ +».mwm . m . o X D

5 +

"A

~~~*X8

. e . O + X

<

i ~

< ^{L1725 3349}

m~ V + X o D

I

o.o ^{~ °'~}

o 5 to

2d

Fig.

5. Average membrane _area per elenlentary

triangle,

_< A >

/Na (upper

part), and its fluctuations,

(<

^A~ _{> <} A

>~) /

_< A _> (lower

part),

as a function of wall separation, 2d.

o. 5

+

~ ^{o X}

<n

~ ⁺

< ^L

A ~

+ 17

<

V X 25

o 33

u 4g

o,o

o 5 to

Fig. 6. Relative deviation of the average

projected

_{area, < AI~ >,} from the real membrane area, < A >,

versus the wall

separation,

2d, for membranes of various sizes. The _curve shows the

expected

power law,

v4th _{q =} 0.60

(see text).

The relative deviation of the average

projected

_area from the real _area is shown in

figure

6.

For

larger

values of d there are substantial finite size effects.

Nevertheless,

the

large

L data are

1420 JOURNAL DE PIIYSIQUE I N°10

in

quite satisfactory agreement

^{with the} power ^{law behavior}

(<

A _{> <}

Ab >)/

_{< Ab >=}

ko

ki(2d)-~n/(~~n),

with ko

= 0.40,

ki

= 0.30, and q = 0.60.

6. The pressure.

The pressure, p, which the membrane _{exerts on} the walls, can be obtained from the

entropic

interaction

[37,42,52,28,47]

where

[5~28]

_{T =}

4/(2 q).

^One has

p =

-oini/od

~-

d~~~~ (15)

In the simulations, we determine the number

density

of _monomers at the surface, _nw, and

employ

the well-known sum-rule for

hard-sphere

systems near a hard

wall,

flP

= nw,

(16)

where

fl

= I

/(kBT),

to calculate the pressure.

Explicitly,

we use N

nw =

~ ~

~~

~

Lib< (z;)

+

b<(2d z;)1, (17)

where

with _{c =} 0.1.

10° _~

~P

^~

j

~

i

l+~eff

~

i

+ 3

o 25

lo

x 33 ^~

D 49 °

lO°

~~ lO~ _{0 0.05}

~,~ Ol

N'

Fig.

7.

(a)

The pressure flp, which the membrane _{exerts on} the walls, as a function of the wall separation.

(b)

The effective exponent _T~« obtained from

(a)

as a funtion of membrane size. tile full line represents a

naive

extrapolation

of the data to

large N-1/~

~w L.

N°10 FLUCTUAIIIONSOFAPOLYMERIZEDMEMBRANEBETWEENWALLS 1421

The pressure ^is

expected

to decrease with the power ^law

(15)

as the wall

separation

^increases.

Thb is indeed the case, as shown in

figure

7a. The effective

exponent

T~jj

(L)

for membranes of different size b shown in

figure

7b. An

extrapolation

to L _{= cc} is difficult, _as

long

as there b

no

guidance

_as to what the

L-dependence

^of the correction term should be. From

figure 7b,

we

conclude that _{T =} 3.2+0.5, which

implies

^via the

scaling

relation _T

=

4/(2-q)

_{that q =} 0.75+0.20.

We _can get a better estimate of _T when _we

try

to eliminate the

boundary

effects and the effect of the finite _monomer size. The first _can be avoided

by calculating

_averages for submembranes of

increasing size,

Ls, obtained from _a simulation of _a

single large

membrane of size L ₌ 49. We

include

again

the corrections to

scaling

to minimize the later, I.e. we write

p=

pod~~(I+pid~~+.. ), (19)

assuming

that the

corrections-to-scaling exponent

^{is close} to

unity,

as in

(8).

The pressure as a function of

Ls

for four different wall

separations

is shown in

figure

8a. We take the values of Ls ₌ 29 to calculate the critical

exponent,

^because

they

should be least affected

by boundary

_or

finite size effects. A fit to the form

(19)

for 2d ₌

2.5,

4.0, and 6.0

yields

r = 2.54, _{po =} 6.04, and pi = -0.52

(see Fig. 8b).

Thb

implies

_{q =} 0.43. The pressure ^{for 2d} = 9.0 b somewhat

larger

than

expected

^from the

fit;

_we attribute thb to the onset of finite size elects.

lO°

iP

+

+ + + + + + +

(~l~[

_io-~

10'

~ 4.0

tip

o o o o o o o o o o

6.0

~Q-Z ^X X X X X X X X X X ~

g-o

° D a a u a a a u a a

~~-3

O 20 40 10~

L~ 2d

Fig.

8.

(a)

The pressure flp, _{as a} function of submembrane size Ls, for various wall

separations,

_as indi- cated.

(b)

The pressure flp for Ls

= 29 _as a function of the wall

separation.

The curve is _a fit of the data for 2d ₌ 2.5, 4.0 and 6.o to equation

(19)

(see text).

7. Nornlal,nornlal correlations.

Another

quantity

we considered is the normal-normal

susceptibility

x"Na[<it~>-<fi>~], ₍₂₀₎

where

fi ₌

) _{En; (21)}

b

JOURNAL DE PHYSIQUE i T I,M 10, OCTOBRE 199I 56

1422 JOURNAL DE PHYSIQUE I N°10

is the average ^of all normal vectors of _a

particular configuration

of the membrane. Thb quan-

tity

b very easy ^{to calculate} in the simulations. For _a membrane oriented

parallel

to the walls

(perpendicular

to the

z-direction,

so that < fi~ >=<

by

>=

0),

the

susceptibility

_can be

split

into the two contributions _X

= Xz + xii, where

Xz =

Na[< fi)

> < hz >< fiz

>],

~

(22)

xjj = Na < fijj ^>,

and fi ₌

(fijj,

^fiz

).

^In the

following,

_we will

only

consider the

parallel

component ^of x,

namely

Xii

In the continuum

limit,

_Xii becomes

Xii ⁼

/ _{d~zGjj(x), (23)}

where Gjj

(x)

=< njj

(x)njj (o)

>. lb

leading

order, we can write Gjj ^{in the}

Monge

_gauge as

Gi(x)

Ci<

Vz(x) Vz(°)

_>,

(24)

so that for a

portion

of membrane of _area D

(with

a linear dimension

ll~),

we have

Xii ^"

/ _d~z

<

Vz(X) Vz(O)

>

(25)

The

leading

contribution _to the

integral (25)

comes from the behavior of the correlation func- tion at

large

dhtances. For _a free

membrane,

the correlation function in

(25) decays asymptotically

as

<

$7z(~) $7z(Q)

>= ~~j' ~~~

eiq'X

~ ~-q

(~~)

/

(2x)2 KR(q)q~

Therefore,

for _a membrane between the

walls,

_Gjj must have the

scaling

form

Gll(~)

" ~ ~Bll

(~/ill). (27)

The

scaling

^function

8(y) decays exponentially

^for _y » I. This

implies

that

Xi "

/d~ZGj(X)

^"

f(~~*(l~s/fj ^). (28)

For

large

d

(large fjj),

_Xii ^becomes

independent

of d, so that

il(y)

_-

y~-°

for _{y -} 0. The

scaling

function lP can

again

be calculated in the Gaussian

approximation.

For

periodic boundary conditions,

Xii ⁼

Gii(q

=

0)

% 0.

(29)

For submembranes of circular

shape

^{with radius} ll~, one has

Xii ⁼

(2x)~R, /~ _{dqGjj (q)Ji (ql~s), (30)}

o

where

Gjj(q)

=

kBTq~/[Ko(q~~°

₊

f[~~~°~)]

^{and Ji denotes} a Bessel function. For _q

= 0 the

integral

_can be done

analytically,

^{with the} result

N° lo FLUCTUAIIIONS oF A POLYMERIZED MEMBRANE BETWEEN WALLS 1423

~V(v) =

(2K)~v kef(v), ₍₃₁₎

where

kei'(z)

=

£kei(z),

and

kei(z)

denotes _a Kelvin function.

Here,

as in

(11),

we have ab- sorbed _a factor _no

/kBT

in the

scaling

function. From

(31),

we obtain the behavior oflP for small and

large _argument.

For _{y -} 0, we have

*(v)

₌

-(2K)~v~lCE

+

In())1, ^(32a)

with the Euler _constant GE ₌

0.577215;

^for _{y - cc,}

*(v)

₌

-2K~/~v~/~ exP(-

fi)ICOS( fi) _Sin(fi)1. (32b)

The results of _a numerical evaluation of

(30),

for _{Q =} 0.7, and

(31),

_{for q =} 0.0, are shown in

figure

9a. Note that the

scaling

functions for both values of _q become

negative

for

large

ar gum ent, in agreement ^with the

asymtotic

behavior

(32b).

q

~ o.i

o 5 z-

~

2d

a ^{+ 2.5}

(

o 4.o

,y

(

_{X 6.o}

7J ^O 9.0

JD

,

X

&

O. 'i °

Q

^°~ + + + +

~ ~

R~ /(~~

° ~

n

s/ Iii

^~

m

~

0,I

~ 2d

fl~ + 2.5

$ ~

_o 4.0

I

~ ^{~ ~~}

' 005

f~D

^{o 9.0}

~ ^fii~

^x

'i

~~°~o~

^°

~ 0 ^~~

- 0 5

n~/(j,

Fig. 9. The scaling function fit of the susceptibility _Xii of normal vectors: (a) calculated from equation

(31)

for _~

= 0.lJ (dashcd line), and from

Eq.(30)

for _~

= 0.70 (full

linc).

(b) MC

scaling

function, for co = 0.667, _cl

= 1.1.5, and _~

= lJJ5. (c) ^MC

waling

function, for _co

= fi.40, _cl

= lJ.43, and _~

= lJ.30. In

(b)

^and

(c),

thc

scaling

variablc 15 proportional to

Rs/(jj.

1424 JOURNAL DE PIiYSIQUE I N°10

The

susceptibility

_Xii also contains conUibutions from the short distance behavior of the corre-

lation function. The local contribution should scale _as

where _{co =}

O(I)

and _{ci =}

Oil)

are constants. For d

- cc, this contribution

approaches

_a

constant, ^and therefore does not

change

the

scaling

behavior of xjj.

However,

for the

interpre-

tation of the simulation data, it b

important

to take

(33)

into acccunL Another correction scales like the energy

density,

and therefore goes like [eo ⁺ ei

(2d)-~(~+°)/(~-°)]

for

large

d. The

leading

correction _term is therefore

given by (33).

The

scaling

function, obtained from the data

by subtracting (33),

^{b shown in}

figure

9b. All data fall _{onto a}

single

curve for _q

= 0.65, _{co =} 0.667 and _ci

= 1.15.

However,

the _same data also scale for _q

=

0.30,

_{co =} 0.40 and _ci

=

0.43,

see

figure

^{9c. The}

collapse

^of the data onto a

single

_curve b

obviously

not very sensitive _to the exact value of _q.

However,

the behavior of the

scaling

function for

large

argument

changes

with _q: lvhereas the

scaling

function b

positive

over

the whole range for _q

= 0.30, it becomes

negative

for

large

_y ^for _{q =} 0.65. Since the

scaling

function for the effective Hamiltonian is also

negative

for

large

argument, we believe that the estimate _q

= 0.50 + 0.10 is reasonable.

By comparing

^the

scaling

function calculated from the effective model

(9)

and the MC

data,

we can get an estimate for the

nnipl1tlide

of the correlation

length

_{fjj =}

to (2d)~/(~~°) Figure

9

implies to

₌ 0.30 + 0.07.

8. Renonnalized

bending rigidity.

The renormalized

bending rigidity,

_KR, b of fundamental

importance

^for an

understanding

of the behavior of

fluctuating

membranes. We would therefore like to calculate it

directly (as

_an inverse

susceptibility).

The

required expression

must involve _a correlation function

[50,55]

^of the mean

curvature,

H(x)

₌

)lt[K(x)]

₌

)[I/Ri(x)

₊

1/R2(x)],

where K is the

(local)

curvature tensor, and RI and

R2

_are the

principal

radii of _curvature. We therefore define

jjj

_

j j _~~~i / _d2z'i

<

H~x~~~~'~

>~ ~~~~

where

g(x)

is the determinant of the metric tensor.

Using

the

Monge

_gauge and

expanding

for small

undulations,

one obtains, to

leading

order,

H(x)

_m

)V~z(x)

so that

~~~

=

/d~z / _d~z'

<

V~z(x)V~z(x')

_>c

(35)

K~fl ^< A _>

in this limit. Consider _now the contribution _{to K~a} from _a circular

piece

of membrane of radius

R,

« R. We _{can use} Gauss' _{theorem to} write the

integrals

in

(35)

as line

integrals

around the

perimeter

C of this domain. Since _no

point

_on the

boundary

of _a disc b

singled

out, we have

/d~z ^d~z'

<

V~z(x)V~z(x')

_>c= 2xll~

j~

ds _<

er(s) Vz(s) er(s') Vz(s')

_>c,

(36)

N°10 FLUCTUAIIIONSOFAPOLYMERIZEDMEMBRANEBETWEENWALLS 1425

where _s and s' _are located _on C, and _er b the 2-dimensional radial unit vector in the space ^of internal coordinates.

Using

the

scaling

form for the correlation function derived above, _we arrive at

)~

=

<~f(~°r(R, /fjj), (37)

where

r(y)

_{- const.} for _{y - cc.} Fbr _a free membrane, where

fjj

- cc, the

right-hand

side of (37~ ^{must become}

independent

of

fjj,

so that

r(y)

_-

vi

-n for _y

- 0. This

implies K~a(l&)

_{~w II~Q}

for _a free membrane.

We _can

again

calculate the

scaling

^function r in the Gaussian

approxhnatioll~ Ignoring

^the

complication

of the circular

boundary

in

(36),

and

replacing

it

by

a square

boundary

of side

length Ls

₌

2xll~,

we have

In the _case of

periodic boundary conditions,

it h easy ^{to show}

(compare Eq.(29))

^that

Kj~, equation (35),

vanikhes

hientical~y. By

^{virtue of}

(36),

this

implies

^that the line

integral

around the

perimeter

also vanishes. Nevertheless, a non-zero result is obtained

along

^other closed contours as well _as

portions

of the

perimeter.

lb determine r in the latter _{case, we} consider _a membrane of

quadratic shape

and linear dimension L ₌ 2x

ll~.

^The contribution from _one

edge

of thb square

can be obtained

by replacing

the _{sum over} the _non-zero

q-modes

in

(38) by

an

integral

with lower cutoff _qm~

=

2x/L

₌

I/ll~.

In thb way we find

~2 (~~~

~(Y)

_"

~~i~~~ ~;,

~~

q~~~ ~

~(i~~~~~'

For _q

= 0,

r(y)

=

jjInj

⁺

(Y

⁺

⁽ 2arctanj ~

i)

2

arctanj ~

+

i)

+

2xj. j40)

2

2y

_{+ y Y}

For _{q >} 0, the

integral

has to be evaluated

numerically.

The

scaling

function b shown in

figure

10a.

For the submembrane _{case, on} the other

hand,

we obtain from

(38),

with R >

ll~,

r(y)

₌ 4

dqi )

dq~~~~~~~'Y~

~

~

(41)

%~ ^~

qi q

Q~+

I For _q

=

0,

the

q2-integral

can be carried out, so that

This function b

plotted

in

figure

10b. For _{q >} 0, _we find it convenient to write the

integral (41)

in the form

r(v)

= 4K

/~~

^da

/~ _{dqqGjj (q)Jo(aq), (43)}

1426 JOURNAL DE PI IYSIQUE I N° lo

~~i

r

lo~~ lo° lo' 10~~

10°

R~/(~~ R,/(~~

Fig.

lo. The scaling ^function r,

(a)

calculated from

equation (39), (40)

^for the _case of

periodic boundary

conditions; _~b) calculated from

equations (42), (43)

for the subnlembrane _case.

where Jo denotes _a Bessel function. In thb

form,

the

integrals

are

easily

evaluated

numerically.

The result b shown in

figure

lob for q = 0.7. Note that in contrast to the

periodic

case

~fig. 10a),

the

scaling

function is non-monotonic in thb _case.

We have used three different discretizations of

(34)

to calculate _K~«. The first involves express- in g the ^{curvature H} in terns of correlations of the

gradient

of the surface unit normal _vector. If

n; is the unit vector normal to

triangle

I, _we define

[50]

$j~~~

"~

ILL

~'J

(~'~ ~J)llL L

~km

(ilk ~m)1

_>,

(44)

e& I j(I) ^k m(k)

where e;j =

(Il~ Rj Ii

^Il+

Rj

b the unit vector

along

the line

joining

the centers of mass, Il~

of

trhngles

I and

j.

In order to better understand this

expression,

note that if _we choose _a metric in which all intemal

lengths

_are

equal,

I.e. all

triangles

_on the surface _are

equilateral

and of _area one, e;j

(n;

_nj is

just

the normal curvature in the direction e;j in ^the continuum limit. Now if

ki,

,

k3

are three normal _curvatures at a

point

_z of the surface in directions which intersect _at

2x/3 radians,

then

ki

₊k2+

k3

₌ ^H

[56]. Thus,

^{with thb}

metric, (44)

is the natural dhcretization of

(34).

Itzykson

describes how to discretize fields _on

triangulated

random surfaces in reference

[57].

Discretizing

the _mean curvature in thin way, one has

~~~'

j~~

^~ =<

ii

_fi;

i _{j'J (r;}

_rj)i

ii

_fi~

j _{j~~ (r~ rm)1}

_>,

₍₄₅₎

~efl _{I j(I)} 'J k m(k) ^~~'

where the sums over I,k _{are over} all _monomers, and the sums over

j(I), m(k)

over their

neigh-

bors.

Furthermore, I;j

^{is the dbtance between the} two monomers I,

j,

and a;j ^{b the}

length

of

a bond in the dual lattice.

Finally,

i1; is the surface normal at monomer I, which is obtained

by averaging

over the surface normals of all

Uhngles

which have I _{as a corner.}

If _{we assume} that there _{are no}

overhangs,

we can

again

^{work with} the

Monge parametrhation,

and

expand

to

leading

order in derivatives of _z. This leads _to the

simpler expression

~

~"

i~ ^~~~

=<

IL L _{I'( (z; zj)I} IL L )~"'(zk

^{zm)1 >}

⁽⁴⁶⁾

~e& _{I j(I)} 'J k m(k) ^~~'