Spectral dimension of fluid membranes

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Spectral dimension of fluid membranes

Shigeyuki Komura, Artur Baumgärtner

To cite this version:

Short Communication

Spectral

dimension

of

fluid

membranes

Shigeyuki Komura(*)

and Artur

_Baumgärtner

Institut für

_{Festkörperforschung, Forschungszentrum}

Jülich,

D-5170

Jülich,

F.R.G. 1

(Received 13August

1990,

accepted 20August

1990)

Abstract. 2014 The

spectral

dimension ds of

_polymerized

and fluid

_{self-avoiding}

vesicles are

investi-gated

by

Monte Carlo methods. For both cases we obtained ds=2, which indicates that these surfaces

belong

to the same class of "microcanonical" surfaces. Classification

Physics Abstracts

05.40 - 36.20 - 68.10

Properties

of flexible sheet

_polymer

networks have been

_explored

in recent theoretical

investi-gations [1]. Polymerized

membranes have a nonzero shear modulus in the

_plane

and are said to

be

_solid-like,

whereas fluid membranes are not

_rigid

in the

_plane

and have zero

_in-plane

modulus.

It has been shown

_using

_computer

simulations

_[2,3]

that

_{polymerized self-avoiding}

membranes

are

_essentially

flat and the

_{corresponding in-plane squared}

radius of

_gyration

is

_proportional

to

the number of monomers on the

surface,

Very

recently,

a model

_{of fluid self-avoiding}

membranes has been

_proposed

which

_exhibits,

in con-trast to

_polymerized

_membranes,

_{crumpled shapes}

with v m 0.8

_{[3] .}

One

_{important question}

with

respect

to the differences between

_polymerized

and fluid membranes is related to their internal

connectivity

which is

_commonly

characterised

_by

the

_spectral

dimension

ds

[4,5] .

The

_spectral

dimensions of various classes of membranes have been discussed in detail

_by

Cates

_{[6] .}

For the

case

_{of polymerized}

surfaces with random

_{connectivity,}

it is

_expected

d,

= 2.

Very

recently,

it has

been shown that even for fractal surfaces with hierarchical

_connectivity

_[7]

the

_spectral

dimension

is,

somewhat

_{unexpectedly,}

not

_changed

and also

ds

= 2.

In the

_{present Communication,}

we

_report

on Monte Carlo studies of the

_spectral

dimension for

fluid

membranes. For

_comparison,

we also include the results of

_ds

for the

_{corresponding}

model of

_polymerized

membranes.

The standard _way to measure the

_spectral

dimension is related to a random walk on the

_given

surface

_{[4, 5] .}

Given a

_particular

realization of the

membrane,

the

_{corresponding}

_mean-square

2396

displacements

at time t of a random walk on the surface is

_expressed

as

where

_df

=

2/v

is the fractal dimension of the model membrane. In

_practice,

r2(t)

_{has to be}

aver-aged

over various walks on one

_particular

frozen realization of the surface as well over different realizations. For

convenience,

we used a

_spherically

closed surface

_("vesicle")

in order to take into account the

_{periodic boundary}

conditions for the random walk

_properly.

The simulation

_technique

for

_generating

various conformations of

_{self-avoiding}

vesicles is the

same as has been used

_{previously [3] .}

In a

_polymerized

_vesicle,

the

_connectivity

at each monomer

is fixed. For a fluid

vesicle, however,

we relax the restriction on the fixed

_{connectivity;}

we allow the

monomers to

_exchange

their

_neighbors,

but

_keeping

the rule that the

_topology

and the

_integrity

of the stucture should be

_{preserved ("triangulation" procedure).}

This

_procedure

_provides

for a

_given

monomer to _escape after several bond

_exchanges

from its

_{original neighborhoods}

of monomers,

and hence

_represents

a "fluid"

_particle.

The

_averaged

_mean-square

_displacement

of a random walker on various sizes of vesicles are

analyzed by

using

the crossover

_scaling

form

_(similar

to that of diffusion on

_percolating

_clusters,

e.g.,

Ref.[8]

),

where

_{f (x)}

is a

_scaling

function with

_{f(z) -}

x ds/df

for x _{« 1 and}

_{f (x)}

= const. for x » 1.

With this

_scaling

function,

the

_mean-square

_displacement

behaves as

(r2( t») "-1

ids/df

for t « T,

and

_{(r2 ( t») "-1}

Nv for t » r, where T =

N 2/ ds

_gives

the crossover time. For _very

_long

times

(t

»

_T),

(r2(t») saturates

and remains constant

_reflecting

the fact that the

_displacement

of the

random walker are bounded

_by

the finite size of the vesicles. The

_{exponent v}

has been estimated

according

to the relation

₍₂₎

at _{x »}

_{1, i.e.,}

_{(r2(t)) -}

_NV,

which

_{yields v}

= 0.98 f 0.02 and

0.83 f 0.02 for

_polymerized

and fluid

vesicles,

_{respectively.}

Since we

_expect

that under the

_scaling

of

₍₂₎

all data should

_collapse

to a

_single

_curve, we have estimated the

_spectral

dimension

ds

from several

_attempts

to obtain

_{optimal overlap}

of the curves for all N. This is

_presented

in

_figure

1,

where y =

(r2(t»)

IN"

is shown as a function of x =

t/N2/ds.

Fixing

v =0.98 and 0.83 for

the two

_types

of

_vesicles,

we obtained the best fit

_using

ds

= 1.96 + 0.05 for

polymerized

and

ds=

2.02 ± 0.04 for fluid vesicles. Of course, these estimates of v

= 2/d f

and

_cas

are consistent with the

_{slope f(z ) -}

xds/df

for x « 1. Our

_conjecture

is that

ds

= 2 for both

polymerized

and

fluid vesicles.

It is worthwhile to

_point

out the next consideration on the crossover time r. The return

prob-ability

P(t)

that the random walker comes back to the

_{starting point}

at time t scales as

_[4,5]

Therefore the number of accessible sites

_E(t)

increases as

_{E(t) -}

(P(t) ) - 1 -

tds/2.

When t comes close to the crossover time

_{r, t N}

N 2/ds

holds. This means that the number of accessible sites amounts to the order of E

_(t)

~

tds/2~N.

This _can be

interpreted

that once the random

walker has visited all N

sites,

it will start to access the sites it has

_already

visited and therefore the

mean-square

displacement

begins

to saturate for t > T.

Finally,

it should be noted that

_ds

= 2 of the

_present

model for fluid membranes

_implies

that this

Fig.

1. -

Scaling plot

for _average

_{mean-squared displacement}

of a random walker on

_polymerized

and fluid

setf-avoidingvesicles.

Here x =

tIN 2/d -1

and _y

= r2(t)>

IN"

with ds = 1.96 and v = 0.98 for

_polymerized

vesicles ds = 2.02 and v = 0.83 for fluid vesicles.

to

_polymerized

random surfaces

_{[2, 3, 9] subjected}

to the constraints of constant surface area, S =

const., and fixed "local"

_{connectivity.}

With

_respect

to the

_present

results for fluid

_membranes,

df

=

2.5,

it seems to be reasonable to extend the class of "microcanonical" surfaces

_by

_including

fluid surfaces with S = _const., but with the weaker constraint of fixed

"global" connectivity,

or

in other

_words,

constant numbers of

_vertices,

faces and

_edges.

Moreover,

with

_respect

to the

recent

_findings

of

_ds

= 2 for deterministic fractal surfaces with

df

= 2.33

[7] ,

it is

suggestive

to

attribute

ds

= 2 to the class of "microcanonical" manifolds in

_{general, including (so}

far as we know

currently)

random

_polymerized,

fluid and fractal surfaces.

Acknowledgements.

One of us

_(S.K.)

is _very

_grateful

for the

_hospitality

of IFF at

_{Forschungszentrum}

Jülich. He also would like to

_express

his

_appreciation

to Dr. M.E.

_Cates,

Dr. D. Kroll and Dr. Y.

_làguchi

for useful discussions. This work was done on CRAY

X-MP/416.

References

[1]

See for instance

_Proceedings

of the Fifth Jerusalem Winter

School,

Statistical Mechanics of Mem-branes and

_Surfaces,

Eds. D.R.

Nelson,

T. Piran and S.

_{Weinberg (World Scientific)}

1989.

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