Short-time dynamics and statics of free and confined tethered membranes

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Short-time dynamics and statics of free and confined tethered membranes

J.H. van Vliet

To cite this version:

J.H. van Vliet. Short-time dynamics and statics of free and confined tethered membranes. Journal de

Physique II, EDP Sciences, 1994, 4 (10), pp.1737-1753. �10.1051/jp2:1994229�. �jpa-00248074�

Classification

Physics

^Abstracts

68.10 82.70 87.20

Short-time dynamics and statics of free and confined tethered membranes

J-H- _van Vliet

(*)

Institut fur

Festk6rperforschung, Forschungszentrum Jiilich,

Postfach 1913, 52425 Jiilich~

Germany

(Received

1

February1994,

revised 17 June 1994,

accepted

5

July1994)

Abstract. Monte Carlo simulation and theoretical results for the short-time

dynamics

and the statics of

hexagonal self-avoiding

tethered membranes in free _space

or confined in _a slit _are

compared.

In particular ^the initial

decay

rate of the

dynamic

structure factor and the static structure factor _are determined. It 15 shown that for small

scattering

vector

lengths

the tethered membrane has the

shape

and short-time diIfu5ion constants of _a flat disc. Further it 15 shown that the

decay

rate for intermediate

scattering

vector

lengths

suggests that the membrane

primarily

relaxes

through

its

out-of-plane

modes,

implying

that the membrane behaves _{a5 a} rather 5tiIf sheet. For _a membrane confined to _a slit both the structure factor and the

decay

rate with the

scattering

vector

parallel

to the membrane surface _are in better agreement ^with

theory

than for _a free membrane due to suppression of

edge

fluctuations. The influence of

hydrodynamic

interactions of the membrane with the slit walls _on the

decay

rate15 discussed.

l~ Introduction~

Tethered membranes _can be looked upon as a two-dimensional

generalization

^of linear

poly-

mers~ I.e.~ ^network

polymers. Experimental

realizations of such network

polymers

_{are, e-g-,}

the

spectrine

network in the cell membranes of

erythrocytes iii,

_a

graphite

oxide

crystalline

membrane [2]~ and cross-linked

monolayers

_[3].

The

properties

of tethered

membranes,

_as for

polymers _[4,

_{5]~ can} be discussed _on

essentially

three

length

scales. The small

scattering

vector q

length

scale

qL

< 2~r, ^the intermediate _q scale

2~r/L

< q <

2~rla

and the

high

_q scale _{qa > 2~r~} where L is the diameter of the tethered

membrane,

and _a the radius of the beads connected

by

tethers in the membrane. The diameter L of _a

particular

membrane conformation is defined here _as the distance between the _centres of

mass of the two membrane beads which _are at the maximum distance for that conformation.

(*)

^{Present address: Cavendish}

Laboratory, University

of

~cambridge, Madingley

Road,

Cambridge

CB3 OHE, United

Kingdom

The

largest

_progress in

understanding

the

physics

of

(tethered)

membranes

by theoretical,

simulation and

experimental

methods _11,

2,

_6]

sofar,

has been made for the statics at intermedi- ate q. Simulation studies

suggest

that

self-avoiding

tethered membranes with excluded volume

between beads and tether

length

lo <

2v5a

_are flat

[6-8].

^{The earlier} theoretical studies _on the

dynamics

^{of tethered membranes} concentrated _on

crumpled

membranes

[9, 10].

^More recent theoretical work concentrated _on the

dynamical properties

^{of flat}

(tethered)

membranes and the effects of

hydrodynamic

interaction

[11, 12].

^To my

knowledge

sofar _no simulation work has been

presented dealing explicitly

with the influence of

hydrodynamic

interactions

(HI)

_on

the

dynamics

of tethered membranes.

A

question dominating

_a

large part

^{of the} _papers ^{which have been}

presented

to

date,

is if self-

avoiding

^{tethered membranes} are flat _or

crumpled.

For _a

crumpled

membrane the square radius of

gyration

should scale _as

R(

~-

N~/~,

and for _a flat

membrane, R(

~- N

[6, 9].

^{A related}

scaling analysis

_can also be

performed

for the

eigenvalues

of the inertia tensor

(see,

_{e.g., [13]}

).

A reliable

scaling analysis

of simulation data is

particularly dependent

_on

large

membrane sizes since the

edge

fluctuations of the membrane _are

large

_[8].

Although by

finite size

scaling

the situation

can be

improved _[7],

the _most

straightforward approach, I-e-, simulating

tethered membranes for a scale of

large diameters, spanning preferably

_some decades in

length,

is not very realistic

considering

the

large

number of beads and

large

relaxation times

[14]

^needed to

equilibrate

membrane conformations and to

get independent

membrane

conformations,

_as

compared

to

scaling problems

studied for linear

polymer

chains. Another

approach

to determine if the membrane is flat _or

crumpled

is the

analysis

of the static structure factor for its

scaling

with _q for the intermediate q scale. However this

approach

has not

presented

_very conclusive evidence to

date,

since the observed

scaling exponent

^for a flat membrane is not very different from that of _a

crumpled membrane,

and for _a flat membrane the

apparant

value

depends crucially

_on

the details of the membrane model

[15,

16] as well _{as on} the _presence of

edge

fluctuations

[6].

Although

now the

self-avoiding

membrane is

generally

believed to be flat it _seems worthwhile to

provide

_a criterion for flatness which is

simpler

and does not

depend

_on the intermediate

length

scale

fluctuations,

_as discussed later _on in detail.

Frey

and Nelson [11] worked

out,

^{within the framework of}

Langevin dynamics,

the behaviour of the

dynamic

structure factor

parallel

and

perpendicular

_to the membrane surface. In contrast to the short-time behaviour dealt with here

they

discussed the

long-time

behaviour. As

argued by Frey

and Nelson and

Lipowsky

_[12] the relaxation rate

Tjj~

^{due to} fluctuations

perpendicular

to the membrane surface with

hydrodynamic

interactions scales for the intermediate _q scale _as

Tjj

~

~- Q~~~~

(1)

where

(

is the

roughness exponent.

^For the relaxation rate due to fluctuations

parallel

to the membrane surface

Frey

and Nelson showed

T£~

~-

q~~~ (2)

where the

roughness

exponent

(

is connected _{to uJ}

by

the

scaling

relation

[8, 17]

(

"

((2 ^{+1°) (3)}

In this paper Monte Carlo

(MC)

simulation and theoretical results _are

presented

on the

dynamics

of of tethered

self-avoiding membranes,

ⁱⁿ

particular

for the initial

decay

rate

r(q)

of the

dynamic

structure factor. For

S(q)

results _are

presented

in

particular

for

qL

< 2~r, which offers _an alternative way ^to look at the flatness of _a membrane.

Further,

data _are

provided

for the situation in which _a membrane is confined between two walls. The first

objective

of

this confinement is to suppress the

edge

fluctuation of the finite-size membranes

simulated,

in order to make _a

comparision

with the

theory, developed

for tethered membranes

neglecting edge fluctuations,

easier [8]. Confinement restricts the

out-of-plane

fluctuations

[18] whereby

the

relatively large edge

fluctuations _are

expected

to be

suppressed effectively.

The second

objective

is to demonstrate the effect of confinement _on the

hydrodynamics

of _a tethered

membrane.

2~

Theory~

2~1 STATICS. The model of the tethered membrane used in the simulation

study presented

here is the well-known network of

triangulated

beads and free

perimeter,

first introduced

by

Kantor et al.

[14].

^{Here the} customary

hexagonal shaped

membrane is used

[19].

The structure factor is defined _as

S(q)

₊

j _{~ iexP(iq}

rv

)) (4)

where N is the number of beads and _r~ the vector

connecting

beads I and

j.

The

angular

brackets indicate _{an average over} all conformations.

Averages

of

S(q)

_can be taken with

respect

to different orientations of q: over all

orientations,

_over orientations

parallel

to the membrane surface

(perpendicular

to the smallest

eigenvector

of the inertia

tensor)

_or orientations _per-

pendicular

to the membrane surface

(parallel

to the smallest

eigenvector

of the inertia

tensor).

These averages will be indicated

by

the connotations

"isotropical", "parallel"

and

"perpen-

dicular"

respectively.

The two latter connotations _are also indicated

by

the indices "I" and

"))" respectively.

If _a membrane confined in _a slit is concerned

defining

_averages

parallel

and

perpendicular

to the slit makes _sense. This is indicated

by

indices "XY" and "Z"

respectively.

The

isotropically averaged S(q)

calculated for the simulated membrane conformations is calculated with the usual formula _[5]

S(q)

=

p ~j l~ ₍₅₎

~~

~~~i~~

For the simulated membrane conformations the _average

S(q)

with the

scattering

vector

parallel

to the membrane surface

S(qi)

is _an average over at least 30 random orientations of qi

(20].

For small values of _q with

respect

to the diameter L of the tethered membrane

qL

< 2~r the

shape

of the tethered membrane _can be

judged by comparing

it to the

S(q)

obtained

by

_a

model calculation. For _a flat membrane with

qL

_< the size of the fluctuations

perpendicular

to the membrane surface rjj ^{< because of the}

scaling

relation rjj ^~-

L~ [12, 21].

^{Therefore if}

the membrane is flat its

isotropically averaged S(q)

should be similar to that of _an

infinitely

thin disc

is, 22]

~~~~ q~)2 _~

~)~~~~

^~~~

where R is the radius of the disc. The

symbol J~

denotes _a Bessel function of the order _x.

For _a flat disc I calculated the ratio of the diameter of the disc and the radius of

gyration

~2 fl

8 ~~~

This ratio _can be used _{as an} additional criterion to

judge

how flat the membrane

actually

is.

Abraham and Nelson _[8] evaluated the structure factor

analytically by approximating

the membrane

by

_a disc. The basic elements in this derivation _are:

I)

fluctuations

parallel

and

perpendicular

to the membrane surface _are assumed

Gaussian;

it)

the summation _over all distances in

equation (4)

is

replaced by

the

integration

_over the

prob- ability density

function

(16/~r) f(x),

for the normalized distances _{on a}

disc, I-e-, ) _£~~

is

replaced by (16/~r) f/ _fix)

_dx.

The function

fix)

is

fix)

=

x(arccos(x) xfi) ₍₈₎

1

(16/~r) f(x)dx

= 1

where _x

=

1/(2R)

and the distance between two

points

within the disc. This distance function

f(x)

_was

long

_ago introduced

by Kratky

and Porod [22] ^{to calculate} the structure factor for

an

infinitely

flat disc

(see Eq. (6)).

Since _a summation _over discrete

points

is

replaced by

_an

integration

it is _a

priori

clear that this

approach

will

only

be succesful for values of _q with

ql

_<

1, I-e-,

^for the intermediate _or small _q scale.

With the identification 2R ₌ L and the _use of

equation (8)

and the

equation

Fiqjj,

_qi, L

=

Jo (qi

XL

expi- (qj (xL)~~ exp(- (q j _{(XL )~ (9)}

Abraham and Nelson _[8] derived from

equation (4)

S(qjj,qi, L)

"

(16/7r) / f(X)Flqjj,qi, L)dX 11°)

Actually

the version of

equation (10) presented by

Abraham and Nelson is

slightly

different in that the

argument

of

Jo

is

multiplied by

an additional fit

parameter.

As will become clear in section 4 _a

satisfactory

fit with simulation data _can be made without this additional

parameter.

2~2 SHORT-TIME _DYNAMICS. The

dynamic

structure factor is defined here _as

S(q, t)

%

j _~

(exP(iq ^in(t) rj(°)i) ^ill)

The initial

decay

rate of

S(q, t)

is defined _as

riq)

₊

it j

_in

(Siq, t)) ⁱ¹²⁾

Akcascu and Giirol

[23]

^{showed that}

r(q)

within the framework of the Smoluchowski

equation

can be calculated _{as an} ensemble _average in

configuration

_space.

£~~ (q D~

_q

exp(iq

_r~

))

~(q)

"

~

_~~

⁽¹³⁾

~ exp iq r~

where

D~

is the diffusion tensor. A _more detailed account _on the derivation of the

previous

equation,

the relation of

r(q)

to

scattering experiments,

and its

application

to various

polymeric

systems

can be found in the _papers of Akcascu et al.

[23],

^several text books and review articles

[4, 5, 24, 25].

Equation (13)

_can be used to

probe

for

2~r/L

< q <

2~rla

the

decay

rate of the internal fluctuations of the

polymerized object.

In the limit

qL

< I the short-time diffusion

constant, D,

_can be extracted from

equation (13)

since

[4,

5]

iirn

rjq)/q2

= D

j14)

The

resulting expression

is

~

2~/2 ~ _{~~ ~"}

_{~~ ~~~~}

~J

An

assumption underlying

the _use of the Smoluchowski

equation

is that the relaxation times of the fluid modes _are much shorter than the internal relaxation times of the

polymerized object.

Therefore

equation (13)

for the initial

decay

rate is useful in the

description

of the

dynamics

for

scattering experiments designed

to

study only

the internal and translational motions of the

polymerized object.

For

polymerized objects

with constrainted coordinates the relaxation

times associated with those coordinates will decrease and

eventually

become

widely separated

from the relaxation times of the other slower internal modes. If the

scattering experiment

is

designed

to

probe

these slower modes

only,

then the initial

decay

rate must be calculated with

a modified diffusion operator ^{in which}

appropriate

constraints _are

imposed

at the outset as

argued by Stockmayer

et al.

[26]. If, however,

one is still interested in the relaxation of the fast internal modes associated with the constrainted coordinates

equation (13)

for the initial

decay

rate must be used.

As

argued by

Abraham and Nelson _[8] the excluded volume interactions between the beads of _a tethered membrane induce _an

entropical bending rigidity.

As

pointed

_out in the

previous paragraph

this constraint

might

influence the initial

decay

rate measured in _a

scattering

ex-

periment

not

probing

the faster relaxation times related to such _a

constraint, necessitating

_a different

expression

for the initial

decay

rate. The interest here is in the

situation,

_as considered

by Frey

and Nelson

[11]

^{in the derivation of}

equations (I)

and

(2),

_were the solvent relaxation times _are assumed much shorter than the internal relaxation times _on the intermediate q scale

without

making

_any further distinction for internal relaxation time scales, This

assumption

is

likely

to be most valid for _a flat tethered membrane at the lowest

possible entropic

bend-

ing rigidity, I-e-,

for _a tether

length

lo

=

2v5a _just

_small

_enough

_to

_prevent

the membrane from

crumpling

_[8]. In order to evaluate

equations (13)

and

(15)

_an

explicit expression

^{for the}

diffusion tensor is needed. The diffusion tensor used here is the Oseen tensor

[4,

5]

Dzj

_#

^~~~ 6zj1

+

~~~~ ⁽¹

^6~j

1

+ ~~ ~~~

II16)

irqa irqr~ r~~

or its

improved

version the

Rotne-Prager-Yamakawa (RPY)

tensor

[25, 27]

JJ~

-

16vI

+

~iii~11 ^61J) III

⁺

~~i~"

⁺

ill III ~"l~" II

~~~~

where r~ ^{> 2a.}

It is also

possible

to include HI of the membrane beads with the two walls

by

modification of the

diagonal

elements of the diffusion tensor. An account of the calculation of the

diagonal

elements of the diffusion tensor in the presence of walls and the

assumptions underlying

this

calculation is

presented

in the

Appendix.

The simulation results _on

r(q)

_are obtained

by evaluating equation (13) using

the RPY tensor as the diffusion _{tensor. For the simulation}

results HI with the walls is either included _or not.

I

analytically

calculated the initial

decay

rate with the _q

parallel

to the membrane surface from

equation (13) by approximating

the membrane

by

_a disc _as done

by

Abraham and Nelson in their calculation of

S(qjj,qi,L) according

to

equation (10).

The diffusion _tensor used to

calculate

r(qi)

is the Oseen tensor.

~~~~~ ~~~~ ^~~~~~~~

^~

^~~~~~~ ~~/L ~~~~~ _(~0(qiXL)

₊

@@)

~~~/~) II fl~)J0(~IXL)Gdz

~~~~~ ~~~

j18)

C

=

/ _f(x)dx

r = ri +

rjj

ri " XL

~/L

o =

/I

+

A(xL)2(-2

G

= exp

~- ~q( ^xL)~) _r(

=

A(xL)~~

2

The normalization constant C for the numerator is _a consequence of

replacing

_a summation

over indices I

# j by

an

integration. Replacing

_a summation _over all indices I and

j

_as for the

denominator,

results in

16/~r

_{as a} normalization _constant.

In _a similar way I evaluated

equation (15). Particularly simple analytic

solutions _can be obtained

by

introduction of the

following

elements:

I) modelling

^of a flat

discj

_r

= ri "

XL;

it) large

disc

size; 2a/L

_-

0j

iii)

_use of the Oseen tensor

(Eq. (16)).

Further the

isotropically averaged

diffusion constant~

D,is

obtained

by averaging

^the r-h-s of

equation (15)

over all orientations and the diffusion _constants

parallel~ Di,

and

perpendicular~

Djj,

^{to the disc surface} are obtained

by averaging

the r-h-s- of

equation (15)

_over the orientations

parallel

and

perpendicular

to the surface

respectively.

The result is

16kBT

~

9~r2qL

2kBT (19)

~~ @

4kBT

~'~ 3~r2qL

The relation between the diffusion _constants is

D

=

)Di

+

(Djj ⁽²⁰⁾

It is _a

priori

clear that these

equations

will

predict

the diffusion of tethered membranes

correctly only

in the limit

qrjj

^« 1~ which however is

automatically

satisfied since

qL

< 1 for

equation (15)

_as noted before. The

expression

of

Frey

and Nelson

ill]

for D has _a

slightly larger prefactor

D

=

2kBT/(3~rqL)~

which _can be

explained by

the _use of _a circular

integration

domain instead of the correct

integration

domain

prescribed by equation (8)~

ⁱⁿ

averaging

the r-h-s- of

equation (15).

Note that for

equation (18)

and

equation (19)

HI with the walls is not included. Further it should be noted that

equations analogous

to

equations

_(18~

19)

_can be obtained

using

the

RPY tensor~

equation (17),

in _a

straightforward

_manner. The

resulting equations

however _are

more

complicated

and

give

rise to

only

small corrections _{or, as} for the

isotropically averaged D,

to

exactly

the _same result.

3~ Model and simulation method~

For the Monte Carlo

simulations~

the tethered membrane

considered,

is the well-known

hexagon consisting

of _a

triangular

network of N

spherical

beads _[14~

19].

^{The radius of} a bead is

a = 0.5.

Neighbouring

beads in the network _are linked

by

tethers of

length

lo < 1.6. Pairwise hard-core

repulsion

of all beads exists. The choice of tether

length

^{and the}

repulsion

_assures

self-avoidance of the membrane. No

explicit bending rigidity

is introduced in the simulations.

Along

the

diagonal

of the

hexagon Lo

+1 beads _are

placed

connected

by Lo

tethers. The total number of beads is N

=

(3(Lo +1)~ +1)/4.

The size of the membranes simulated is in the range

Lo

₌ 10-50. To _my

knowledge

the

largest hexagonal triangularly

tethered membrane

simulated sofar has _a size of

Lo

+1 ₌ 75 in _a MD simulation

[6, 8].

Membranes are simulated both in free _space and confined _to

a slit bounded

by

two

repulsive

walls. The walls of the slit

are

perpendicular

to the Z-axis and restrict the

Z-component

of the centre of each bead to be in the interval

[0, 2d],

with _a

corresponding

^distance between walls H ₌ 2d ₊ 2a.

A MC _move is defined

by

_an

attempted randomly

selected

displacement

for each bead. This

displacement

is

produced by selecting

a

step-size

_{so "} 0 0.2 for each of the three

orthogonal

directions in space; the average size of _a MC _move

resulting

is _s

=

0.svsso.

_A MC step ^is defined _{as a}

cycle

in which all N beads

experienced

_one MC _move.

For each MC _move

overlap

with other beads and

eventually walls,

_as well _as the

tethering

constraint has to be checked before

accepting

or

rejecting

the _move. The beads _can be divided in four

classes;

the characteristic of each class

being

^that a bead is tethered

only

to a nearest

neighbour

bead of another class. In this way it is

possible

to check the interactions of _a bead

with the _nearest

neigbour

beads and the next nearest

neigbour

^beads

(connected

to the bead

which is moved via a

neighbour bead, I-e-,

two tether

lengths away)

^after all beads of _a class have been

subjected

to a MC _move.

All the other interactions _are checked

only

if the bead

positions

before the MC

step

are such that _a

danger

exists that beads

might overlap

in this MC

step.

^This is

implemented by keeping

on record a list of those

beads,

other than

neighbour

and _nearest

neighbour beads,

which _are at a distance < 2a +

2viso.

_{The distances between these} beads _are checked in every MC step. ^{This list is}

updated

when the _sum of the

magnitudes

of the _two

largest displacements

since the last

update

for _any two beads

Ii-e-, including

those not on the

list)

exeeds

2vsso.

A related

algorithm

is the automatic

updating

of _a Verlet

neighbour

list

[28].

To

update

the

list,

distances between beads _are checked

by using

_a variant of the cell index

algorithm _[28].

The membrane is

projected

_on the

XY,

X Z _or YZ

plane.

The actual

plane

used is the

plane

for which the membrane surface area

projected

_on it is the

largest.

The difference

with the usual scheme

being

that _one of the three coordinates

X,

Y _or Z is not used to

assign

a bead to a cell. This method is efficient due to the flatness of the membrane. A related

algorithm, whereby

the cell orientation is determined

by

the orientation of the

eigenvectors

of the membrane's inertia tensor is described

by

Grest

[13].

^{This scheme of classified}

beads,

_a list of beads and cells _can be

easily programmed

in vectorizable code.

The relaxation time _TR is defined _as the time needed for the tethered membrane to diffuse

its _own radius of

gyration.

In _a Monte Carlo simulation _{run a} membrane _was

equilibrated

for

a number of MC steps at least five times

larger

than the relaxation time _TR. The membrane conformation _was

sampled

at intervals of MC

steps

at least twice _TR. For _a flat tethered membrane in free space TR *

N~/(as)~ _[29].

Here the value _TR

"

N2/(0.svsso)~

_{is used.}

For _a confined membrane the relaxation time is

expected

to be smaller

[18].

^{For the}

largest

membrane

(Lo

_"

50)

2

independent

_runs where

performed.

^For the membranes of smaller size 3

independent

_{runs were}

performed. Quantities

of interest _were calculated for each conformation

sampled

in _{a run}

(at

least 10

samples

_per

run)

and

averaged

afterwards _over all

samples

in that _run. Final _averages and statistical _errors therein _were obtained

by averaging

the averages

per run over all _runs.

The simulations _were

performed

_{on a} CRAY

Y/MP

M94 vector computer or a Silicon

Graph-

ics

Indigo

workstation. The CPU time in seconds has been

computed

for _{runs on} the CRAY computer ^{and varies from 0.020} s per MC step for

Lo

" lo to 0.055 _s per MC

step

for

Lo

_" 50.

4~ Results and discussion~

4~l FLATNESS _{OF THE MEMBRANE.} The idea that _a tethered

self-avoiding

membrane is

flat is confirmed

by comparison

of the

isotropically averaged

structure factor

S(q)

for _a rather

large

tethered membrane in free _space,

Lo

_"

50,

with

S(q)

for _a disk with _a diameter

equal

to that of the membrane for small _q in

figure

I. It is clear that for the _{q range} of

interest, qL

< 2~r,

S(q)

for the membrane is in

good _agreement

with

S(q)

of _a disc.

Further confirmation of the membrane's flatness _can be found in the

comparison

of the ratio

R(/L~

^for a disk with that of _a tethered membrane _as obtained from the simulations and

presented

in table I. It is clear that for _a tethered membrane in free _space the ratio

R(/L~

is less than the theoretical ratio of 0.125

provided by equation (7). However, by confining

the

membrane to _a slit with H

= 3.5 the simulation result for

R(/L~

is within _error

equal

to the

0.8

0.6 S(q) o

off

o

0.2

0

0.02 0.08 0.12 0.16 0.2

q

Fig.

I.

Dependence

of the

isotropically averaged

structure factor

S(q)

_{as a} function of _q for small values of _q. Data points:

S(q)

for

a tethered membrane with Lo ₌ 50. Solid line:

S(q)

for

a disc with

radius 2R

= 46

according

to equation

(6).

The diameter of the disc is set equal to the diameter L of the tethered membrane obtained from the simulation.

Table I. Radius of

gyration

and diameter.

environment

Lo L~ R2 j~2/~2

free _space 10 79.5 9.81 0.12

free _space 20 295 29.4 0.10

free _space 30 740 82.2 0.11

free _space 50 2125 235 0.11

slit;

H

= S-S 20 371 41.9 0.113

slit;

H

= 3.5 20 396 49.2 0.124

Table II. Short-time diffusion constants; 11

=

D/kBT.

~~~i~~~~~~~ j~

jjsimulation jjtheory jjsimulation jjtheory j~simulation jjtheory

~ l I

it

free _space 10 1.71E-2 2.02E-2 1.78&2 2.27&2 1.56E-2 1.52&2

free _space 20 1.00E-2 1.05E-2 1.04&2 1.18&2 9.41E-3 7.85&3

free _space 30 6.34E-3 6.62&3 6.69&3 7.45&3 5.61E-3 4.97&3

free _space 50 3.85E-3 3.91E-3 4.12E-3 4.40E-3 3.30E-3 2.93&3

slit;

H ₌ S-S 20 8.99E-3 9.33E-3 9.62&3 1.05&2 7.62E-3 7.00&3

slit;

H

= 3.5 20 8.52E-3 9.05E-3 9.24E-3 1.02&2 6.90 E-3 6.79&3

theoretical value. It should be noted that the statistical _error for free membranes in the data of table I. is

approximately 10i~ higher

than for the confined situation. This

suggests

^that

by

confinement the

large edge

fluctuations _can be

effectively suppressed.

This

suppression

of

edge

fluctuations also increases the diameter of the membrane _as

compared

to _a free membrane of the _same

size, I-e-, equal Lo.

A

comparison

of the short-time diffusion constants for _a disk and the simulation results for

a tethered membrane in table II

give

further

support

to the idea of _a flat membrane. The diffusion constants for _a disk _are calculated

by

substitution of the L values from the simulation

data

presented

in table I in

equation jig).

There is _a

good agreement

^between the diffusion constants from the simulation and the results for _a disk.

Noteworthy

is that the trend observed for the

disks, Di

> D >

Djj,

^is

clearly

^confirmed

by

the simulation data and in accordance with the

expectance

^that a flat

object

will

experience

^less friction

moving parallel

_to its surface than

perpendicular

to it. The smaller diffusion _constants for the confined membranes _are due to the

larger diameter,

_see table

I,

of these membranes due to

suppression

of

edge

fluctuations.

4~2 RELAXATION _{ON THE} INTERMEDIATE q ^SCALE. The relaxation modes of the membrane

are

investigated

for the intermediate _q scale

2~r/L

< <

2~rla.

The

edge

fluctuations

clearly

dominate the

isotropically averaged

initial

decay

rate _as shown in

figure 2, resulting

in _a

scaling

relation

typical

for _a

crumpled object,

r

~- Q~. A similar

explanation

has been offered

by

Abraham and Nelson for the

"crumpled"

_appearance of the

isotropically averaged

structure factor

[8].

^The

large

influence of the

edge

fluctuations shows also _up in the

large

data scatter for

r(qjj perpendicular

to the membrane

(results

not shown

here).

The initial

decay

rate

parallel

to the membrane surface is _as the

corresponding

structure factor _[8] less disturbed

by edge

fluctuations. Therefore

r(qi)

_can be used to

get

an

insight

in the relaxation modes of the membrane. In

figure

3 _a

scaling plot

for

r(Qi)/q~~~~

_versus

QIL

with

(

= 0.65 for two membrane sizes shows

clearly

that

r(qi)

'~ Q~~~~ for the

larger

values of the intermediate _q scale

(first peak)

in accordance with

equation Ii _).

It

suggests

that relaxation is dominated

by

the

out-of-plane

relaxation

(bending modes) [11], I.e.,

^the membrane behaves lik4 _a rather stiff sheet. This statement derives further

support

^{from the} fact that

in-plane

relaxation

(phonon modes)

of

r(Qi) according

to

equation (2)

is

surely

not

recognizable.

4~3 PARALLEL STRUCTURE FACTOR AND DECAY RATE. The

suppression

of

edge

fluctua-

tions

by

confinement between two walls leads to _a better

agreement

^between the simulation results for the

parallel

structure factor for _not too

large

_q values and the theoretical result

according

to

equations (9, 10)

than found before

by

Abraham and Nelson [8] ^as shown in

figure

4~

+ +

~ +

0.1 ₊

~

+ ~

~ + +

+

~

+

~~/

+ +

i(~)

~~/

+

ooi

oooi

o-i i to

q

Fig.

^2.

Log-log plot

of the initial decay rate

~(q)

₌

r(q)/kBTq~

_{as a} function of _q for both the

isotropic

average of

r(q)

(lower

curve)

and

r(q i) (upper curve)

for _a tethered membrane with Lo

= 50.

~

i

~

~~_

+~ o

~

~

~ +o oi >

~

x~q~> _oi

ii

11~i$& ^' _~

'~/ ^~ ' ~~

°

~ ~ " ^'

o-o

i ^o

~ L

Fig.

3.

Log-log

plot of the initial

decay

rate

x(qi)

_"

r(qi)/kBTq )~~'

_{as a} function of qiL ^with ( ₌ 0.65:

(+)

Lo

= 30,

(0)

Lo

= 50.

o.oi S(q~)

w

o.oooi

o-i i lo

q~

Fig.

4.

Log-log

plot of the structure factor

S(qi)

_{as a} function of _qi Data

points: S(q)

for _a tethered membrane with Lo

" 20 in

a slit of diameter H ₌ 3.5. Solid line:

S(q) according

to

equations (9), (10)

with L

= 19.9, B

= 0.05 and qjj = 0. Value for L obtained from the simulation.

4. The

corresponding

initial

decay

rate,

r(Qi),

shows _a similar

good agreement

^{with the}

decay

rate calculated from

equation (18) presented

in

figure

5. Another

possibility

is to compare the theoretical _curves for the confined membrane with the structure factor

S(q~y)

and

decay

rate

r(q~y) parallel

to the walls of the slit. However there is _no notable difference with

S(qi)

and

r(qi)

and therefore these results _are not

presented

here.

lo

i

Do $

o ~

1lq4

_o-i ^° _S

o

Slj$

h/ W _~9

o.oi

o.ooi

o-i i lo

q~

Fig.

5.

Log-log

plot of the initial

decay

rate

~(qi)

"

r(qi)/kBTq (

as a function of _qi The HI interaction with the wall is not included. Data points:

~(qi)

membrane with Lo

= 20 in _a slit of

diameter H

= 3.5. Solid line:

~(qi) according

to

equation (18)

with L

= 19.9, B

= o.05,and A

= 0.62.

Value for L obtained from the simulation.

4~4 HYDRODYNAMIC INTERACTION wiTH THE WALLS. Sofar for the membranes confined

to a slit the HI with the walls has not been included in the simulation results discussed. The

reason for this is that the presence of walls sofar has been intended _to diminish the

edge

fluctuations and not to represent the membrane _as

being physically

confined.

However,

^if the walls _are considered _as real

physical objects

_present to

study

the influence of confinement

on membrane

dynamics,

the influence of HI with the walls is relevant.

Further,

the results obtained _may

depend

_on the orientation of _q,

I-e-, parallel

_or

perpendicular

to the membrane

surface

_or to the walls.

For the short-time diffusion _constants there is _no difference if HI with the walls is included

or not. This is due to the

tendency

of the membrane's centre of _mass to avoid the

walls,

since the

neighbourhood

of _a wall reduces its translational

entropy, thereby minimizing

the HI with the walls. The influence of HI with the walls should be _even less for

Di

than for

Djj

^since translation

parallel

to the membrane

surface,

which is _{on average}

parallel

to the

walls,

^results in smaller HI with the walls than translation

perpendicular

to the walls.

However,

the latter effect is of

secondary importance

since both

Di

and

Djj

are not influenced

by

HI with the walls.

Also for the initial

decay

rate

parallel

to the membrane

surface, r(Qi),

_or the

walls, r(q~y),

introduction of HI with the walls does not make _a notable difference

(results

not shown

here).

However for _q

perpendicular

to the membrane _as

presented

in

figure 6,

_or

perpendicular

to the walls _as

presented

in

figure 7,

there is _a difference. It is clear that for _{q >} 1 the presence of HI with the walls decreases the

decay

rate. The effect is

stronger

^for

r(Qjj)

^{than for}

r(qz).

Further,

for _{all q} values

r(Qjj)

<

r(Qz).

These results _can be

easily

understood in

physical

terms

realizing

that the medium between _a wall and the membrane is

pushed against

the wall

by

_a local _move of the membrane towards the

wall, causing

backflow of medium and

thereby

_a disturbance of the

velocity

of the medium at the

positions

of membrane beads. On the other hand for _{a move}

parallel

to the wall the

displaced

medium flows almost

freely parallel

to the walls

thereby causing considerably

less disturbance

o-i

T~q,i

o.oi

o.ooi

o-i i lo

ql

Fig.

6.

Log-log plot

of the initial

decay

rate ~(qjj =

r(qjj /kBTq(

_{as a} function of qjj for a tethered membrane with Lo

" 20 in _a slit of diameter H

= 5.5:

IQ)

HI of the membrane with the walls

included;

(+)

HI of the membrane with the walls not included.

1iq4 _o-i

o.oi

o-i i lo

qi

Fig.

7.

Log-log plot

of the initial decay rate

~(qz

=

r(qz /kBTq)

as a function of _qz for _a tethered membrane with Lo ₌ 20 in

a slit of diameter H

= 5.5:

IQ)

HI of the membrane with the walls included;

(+)

HI of the membrane with the walls not included.

of the

velocity

at the

positions

of the membrane

beads,

_as reflected

by

the

insensitivity

^of

r(qi

and

r(q~y

to HI with the walls. The effect is

larger

for

r(Qjj

^{than for}

r(Qz ),

because _an

instantaneous orientation of the membrane surface will not

always

be

parallel

to the walls of the slit and

r(qz

contains _a component

perpendicular

and

parallel

to the surface

and,

as discussed

before,

the latter

component

is

hardly

influenced

by

HI with the walls. The faster diffusion of

the membrane

parallel

than

perpendicular

to its

surface, Di

_>

Djj,

ⁱⁿ combination with the

previous

^notion that the orientation of the membrane surface is

only

_{on average}

parallel

to the slit

walls, explains

that for all _q values

r(Qjj)

^<

r(Qz).

5.

Concluding

remarks.

It is shown that the simulation data for the

isotropically averaged

structure factor for _a self-

avoiding triangularly

^{tethered membrane} are in

agreement

with the theoretical results obtained for _a disc. This

strongly suggest

that _a

self-avoiding

tethered membrane is flat and not _crum-

pled.

The

advantages

of

using

the structure factor for small and not for intermediate _Q, are

its

insensitivity

to

edge fluctuations,

which _are rather

large

for the

relatively

small membrane diameters used in

computer simulations,

and the

possibility

to compare with the

simple

model

structure factor of _a disc. Under the

right conditions, I.e.,

for membrane sizes

being

neither too

large

nor too small _as

compared

to the available _wave

length

of the radiation

used,

this

criterion

might

also be useful in

experimental

_cases,

especially

since _a

scaling analysis

of data to make _a distinction with _a

crumpled

membrane is far from trivial.

The short-time diffusion _constants obtained from

simple

model calculations _{on a} disc and simulation data show _a close

agreement, giving

further support to the statement that _a self-

avoiding triangularly

tethered membrane is flat.

The

comparison

between the simulation data and theoretical _curves for structure factor and initial

decay

rate of _a confined tethered membrane shows that the

edge

fluctuations _can be

effectively suppressed by

the walls.

It is shown that the

self-avoiding triangularly

tethered membrane in the intermediate scale is

likely

to relax

primarily by out-of-plane (bending)

modes _even in the direction

parallel

to its

surface, suggesting

that the tethered membrane behaves _{as a} rather stiff sheet. It is clear however that to _use this result in the

interpretation

^of a

scattering experiment

is _a subtle matter. It

especially requires

that the relaxation times for the fluid _are

considerably

less than for the internal relaxation modes of the membrane and that all the internal relaxation modes of the membrane _can be

probed

in the

experiment.

The latter condition is

probably

best

guaranteed by

the _use of _a membrane which has _a

bending rigidity

which is not too

large.

It should be realized that the effect of

hydrodynamics

with the walls _on the

decay

rate and short-time diffusion _constants of _a confined membrane is based _on crude

approximations

as stated before in the context of

polymer hydrodynamics _[30].

HI of _a membrane bead

is modelled _as the HI of _a

single

solid

sphere

with _a wall.

Further,

in

superimposing

the

single

wall contributions to model HI with two

walls,

is _a rather

strong approximations

which will

definitely

fail if the walls _are too close

together _[31]. Furthermore,

the HI of _a tethered

membrane with _a wall is dealt with _as the HI of _a collection of N

independently moving spheres,

that is not a trivial

assumption. Therefore,

results _on HI with the walls of _a tethered

membrane _are

likely

to be valid

only qualitatively, especially

for small slit widths.

Having

said

that,

the effects of HI with the walls _on the

dynamics

of the membrane _seems

interesting enough

to warrant further _more

quantitative

and detailed

investigations.

Acknowledgements.

This work _was

supported by

the Commission of the

European Community

within the framework of the "Human

Capital

and

Mobility"

_programme. The author thanks K. Kremer for the

critical

reading

of the

manuscript. Stimulating

discussions with K.

Kremer,

R.

Lipowsky,

R-R-

Netz,

M. Kraus and U. Seifert _as well _as the _comments of _{an anonymous} referee _are

gratefully acknowledged.

Appendix.

The effect of HI _on the diffusion tensor of _a solid

sphere moving parallel

to a

single

wall is known

[31, 32]

~~~

~~~ ^{~ 6~}

^~

^~~

^{~ ~°~ ~ ~ ~~~~}

D""

=

) (-(

_In

l~ ~)

⁺

0.9588)

~

for _y

i

1

7rna l/

where _y

=

a/h,

with h the distance between the centre of the

sphere

and the

wall,

and _a

= x, y.

For the HI of _a solid

sphere moving perpendicular

to a wall the

corresponding equation

is

[31, 33]

D~~

=

) (I ^(y

⁺

jy~

^{+ for} y <

(22)

7rna

D~~

=

~~~

_~

ln

~)

⁺

0.971264)

~

for _y

I

6~rqa

1- y 5 _y

For y = 0.76980 the short and

long

distance

approximations

for the

sphere moving parallel

to a wall _are

equal.

For _y

= 0.4250959 the short and

long

distance

approximations

for the

sphere moving perpendicular

to _a wall _are

equal. Labeling

the two

parallel

walls with _a and b

respectively

and

superimposing

corrections to the HI of _a

sphere

due to a

single wall,

the

diagonal

elements of the diffusion tensor for _a

sphere

between two walls _are

expressed

for ya ^<

0.76980,

_yb < 0.76980 _as

D~~

"

$ (1 (lva

+

vb)

+

jlva

⁺

_vb)~) 123)

for _ya §

0.76980,

_yb > 0.76980 _as

~~~

~~~

~6~~

~

~~

~

5

~~

b~~

~ ~

~~~~ ~

for _y~ §

0.4250959,

_yb I 0.4250959 _as

D~~

"

) (1 (lva

⁺

^vb)

⁺

jlva

⁺

_vb)~)

and for _y~

§ 0.4250959,

_yb > 0.4250959 _as

D~~

=

) ^-(y~

+

jy(

+ ~~

In ^~~ +

0.971264)

)

7rna yb l/b

Due to the

symmetry

in the indices _a and b of the

equations,

_a and b _can be

interchanged

where dictated

by

the

position

of the

sphere

with respect ^{to the walls. The}

superposition

of the corrections of the two walls to

get

^{the overall correction} to the diffusion tensor is _an

approximation suggested by

Oseen. This

approximation

_assumes that the

hydrodynamic

in- teraction with _a wall is not influenced

by

the presence of the other wall. As stated

by Happel

and Brenner [31] ^this can

only

be _correct if the walls _are not close

together

and have _a finite extension.

They

showed

explicitly

for _a bead

moving perpendicular

to _a wall that the _presence of _a second wall leads to _a difference in the correction _as

compared

_to the correction obtained

by

the

superposition approximation.

Nevertheless the differences _are

only small, although

_one

might

expect ^that for

a

sphere

which has _a short distance to both walls the

superposition approximation

is

highly

erroneous. That situation however is not encountered for the tethered

membrane studied here.

To include the HI of the membrane beads with the walls in the diffusion tensor, either Oseen

or RPY tensor, the

diagonal

elements _on the r.h.s. of either

equation (16)

_or

equation (17)

should be modified _as

D$"

e D"* and D]~~ ^e D~~

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