Self-avoinding tethered membranes embedded into high dimensions

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Self-avoinding tethered membranes embedded into high dimensions

Gary Grest

To cite this version:

Gary Grest. Self-avoinding tethered membranes embedded into high dimensions. Journal de Physique

I, EDP Sciences, 1991, 1 (12), pp.1695-1708. �10.1051/jp1:1991237�. �jpa-00246447�

J P%ys. Ifmnce 1

(1991)

1695-1708 DtCEdBRE1991, PAGE 1695

Classification

PhysicsAbnnwti

64.60C-05.40-64.60A

Self-avoiding tethered membranes embedded into high ^dimensions

Gary

S. Grest

Corporate

Research Science Lab~ratories, Exxon Research and

Engineering

Company, Artnandale, NJ 0881, U.S.A.

(Received 7Jkne lwl,

twcepted

5

September

lwl )

Abstract. The

equilibrium

structure of

self~voiding

two-dimensional tethered membranes em-

bedded in_a space ofdimension d _> 2 _are studied

using

molecular

dynamics

simulations for 3 < d < 8.

For

embedding

^dimension d > 5, the membranes crumple while they remain flat for d < 4. In the crumbled

phase,

^{the radius} of

gyration Rg

^increases as L", where L is the linear size of the _uncrum-

pled surfhce, and _v is

significantly larger

than the

simple Flory

estimate _vp

=

4/(d

₊

2).

For d

= 4,

the membrane remains flat, _even in the presence ofsite-dilution,

indicating

that as for d

= 3, random site-dilution alone is _not sufficient _to

produce crumpling

of txvo-dimcnsional membranes.

I. Introduction.

The statistical

properties

of

objects

of

arbitrary

fractal

connectivity

embedded in _a space of di- mension d has been the

subject

of great ^{interest in} recent years

[Ii.

Besides the well-known _ex-

ample

of linear

polymers,

other

systems

which have been

investigated recently

are swollen

gela- tion/percolation

^clusters

generated

_near the

percolation

threshold _pc

[2], Sierpiiiski gaskets [3,4]

and two-dimensional tethered membranes

[5,6].

^Gates

iii

was the first _{to use} the _term

'polymeric

fractals' to describe the class of fractals which _are made of flexiwe

polymer

chains at short

length

scales but which have _an

arbitrary

self-similiar

connectivity

_at

large

distances. Such fractals _are in-

teresting

because

they

are

expected

to have _no inherent

rigidity.

Because such systems are

locally

very

flexible,

after

they

_are allowed _to relax and take _on their

equilArium shape, they

_very ^often

have _a very different fractal dimension than

they

had when

they

were first constructed. Gates

[Ii

worked out _a

Flory-level theory

[7] ^{in the} presence ^{of excluded volume} interactions to determine

an estimate for the swollen fractal dimension

df

which relates the _{mass or} number of _monomers N of the fractal to the size

llg, R(t

~ N in the presence ^{of excluded volume} interactions.

By balancing

the elastic

(entropic)

free energy ^{of the}

'phantom' object

vithout self-avoidance with

the _mean field estimate of the excluded volume

interaction,

he showed that

dj depends only

on

1696 JOURNALDEPHYSIQtJEI N°12

the

spectral

dimension

I _[8]

_{and the dimension of space d}

Kantcr et al [5] ^derived a simifiar result for D-dimensional manifolds

(D interger)

such _as linear

polymers (D

₌

I),

tethered surfaces

(D

₌

2)

and

gels (D

₌

3).

^{In this} case the

spectral

dimen- sion and the the dimension of the manifold _are

equivalent (I

=

D).

Fbr networks with

regular connectivity,

the exact value of

df depends only

on the value of D and the

embedding

dimension d

[5].

^{However for networks vith}

irregular connectivity, particularly

for

objects

which _are inher-

ently

fractal and cannot be

expressed

ⁱⁿ terms of _a Euclidean

metric,

it is not clear whether the

exact value of df

depends only

_on

I

_{or on other details such}

as

topological

constraints

[I].

Despite

the fact that the

Flory _argument

is

only

a

simple

mean field

theory,

it b known to work very ^{well for} a number _cases, often

producing

^estimates for

df

which differ from the exact result

by only

a few

percent

or less ford _< due, Here due ₌

4il(2-I) [I,9,10,11]

is the upper ^{critical dimen-}

sion above which self-avoidance becomes

irrelevanj Examples

where the

Flory theory, equation (I),

works

extremely

well include linear

polymers (d

₌

I)

for d < due ₌ 4,

gelation/percolation

clusters

(I

~

4/3

in all dimensions

[8])

^{embedded into d} ₌ 3

[2,12,13]

and

triangular Sierpifiski gaskets (I

= 2In

3/

Ins _m 1.365

[14])

embedded into dimension 3 < d < due m 8.6

[4].

^The

Rory

result for

percolation/gelation

clusters is

expected

tc be _a

good approximation

for all d < d~c ₌ 8, but it has

only

been checked in d

= 3. In the two cases I _{am aware} of which do not

satisfy

_equa-

tion

(I),

an additional

bending rigidity

is believed to be present. ^These are the

Sierpifiski gasket

in d

= 2

[23]

^and two-dimensional tethered membranes

(D

₌

^I

₌

2)

in d

= 3. lvhile the

early

simulations [5]

suggested

that tethered membranes

satisfy equation (I), later,

_more detafled _work found that

they

remain flat

[15-18]

^{and do} not

crumple.

This result _was

initially

_very

surprbing

since renormalization group calculations

[5-11]

also

suggested

that the flat

phase

was unstable.

The lack of _a

crumpling

^{transition in} d ₌ 3 has

recently

been

explained

in terms of an

implicit bending rigidity

which is induced

by

the self-avoidance

requirement

_even when no such term is

present

^{in the}

microscopic

Hamiltonhn

[18].

^If

bending rigidity

is

relevant,

then _{one cannot ex-}

pect

^the

Flory theory

to work

Recently

^Boa( et al

[18]

^{studied rather small} tethered membranes

(L

<

17)

embedded in d ₌ 4 and 5 dimensions and

suggested

the existence of _{a new}

'rough' phase,

^{in wliich} the membranes _were neither flat nor

crumpled.

However in thin paper, ^{I vill} show that simulations _on much

larger systems suggest

that tethered membranes

actually crumple

for d > 5 and that the value of dfis

significantly

smaller than

predicted by equation (I).

In d

= 4,

the membranes remain flat,

It is

interesting

to consider in _more detail

why

some systems, ⁱⁿ

particular

those for which the

spectrdl

dimension is close to

I, crumple

and others do not. Abraham and Nelson

[18] suggest

that _a

bending rigidity proportional

to the temperature ^b

generated

for

entropic

reasons

by

the excluded volume interactions,

They

found that second

neighbor

interactions alone are sufficient to

produce

a flat

phase

even when furthur

neighbor

interactions _are tumed oft In _a

separate study, they _[20]

also found that _a network of flexible linear

polymers

that are crosslinked _to form

a two-dimensional tethered membrane also remains flat _even

though

^this

object

is very flexible ^lo-

cafly [21].

^lhh suggests ^{that the} interactions

inducing

the

bending rigidity

whatever

they

are must be relevant under renormalization. Had thb _not been the case, then above a critical

length

^of the

linear

polymer chain,

the induced

bending rigidity

would fall below the critical value necessary ^to

keep

the membranes flat

[22]

^{and the membrane would}

crumple.

lb understand

why

excluded volume interactions do not

generate

a

comparable bending rigidity

for

percolation/gelation

clus- ters near pc or

Sierpifski gaskets,

^{it b} informative to consider the

relevancy

of

higher

interaction

terms in the Hamiltonian. Since renormalization group

approaches explicitly

take into account

N°12 THETHERED MEMBRAN& EMBEDDED IN HIGH DIMENSIONS 1697

the

two-body repulsive interactions,

it _seems reasonable to assume that

higher

^order terms are

responsible

for

generating

the

implicit bending rigidity

tllat

keeps

a tethered membrane flat, At least vithin

Flory theory [10,23]

^{it b}

straightforward

to determine for _a

given

value of

land

_d

whether the

n-body

interaction _vn b relevant _or not,

According

to the

Flory theory [10,23]

I= ~) j~, (2)

b the fine where

n-body

interactions _vn become relevanL

two-body

terms are irrelevant for d _>

dw

₌

4il(I ₂₎

_as mentioned above,

Three-body

terms are irrelevant for

I

>

4d/(6

₊

d),

For

'polymeric

fractals'

[21]

embedded into

3-dimensions, three-body

terms are irrelvant for

I

_<

4/3,

This

explains why

there b _no induced

bending rigidity

for linear chains and

percolation/gelation

clusters _{near pc,} Since the value of

ifor Sierpifiski gaskets

is close to

4j3,

it is not too

surprising

that

they crumple.

_{It may} also

explain why

these

systems approximately satisfy equation (I)

_so well.

For tethered membranes,

however, Rory theory

suggests ^that

four-body

interactions _are relevant below 4-dimensions and

three-body

terms are relevant below 6-dimensions and must be taken into account

[23].

^In

fact,

in d

= 3,

3-,

4- and

5-body

interactions _are relevant and

6-body

terms are

marginal according

to

Rory theoly.

If

three-body

terms are sufficient to generate an effective

bending rigidity,

then

Rory theory

would

predict

that above d

= 6 membranes should

crumple,

while if

four-body

terms are necessary ^to generate ^the

bending rigidity,

then the

crumpling

would occur above d

= 4. lvhile there

ii

no reason to take the

Roly theory

_very

seriously

as to the exact dimension at which _a tethered membrane vill

crumple,

it does

suggest

^that

crumpled

membranes may exist, albeit in dimensions

higher

than 3, In this paper, ^I present ^{the results of} a detailed

molecular simulation in 3 < d < 8 dimensions which show that for d > 5, tethered membranes do in fact

crumple, lvhy

the _crossover is between d

= 4 and 5is _not clear, In

addition,

I find that these

crumpled

membranes do not

satisfy equation (I)

very wefl~

Though

there b no real _reason

why

the

Roly theoly

^{should work} so well for fractals ~vith

I

near

I,

it is somewhat

surprising

that

equation (I)

does not

give

a reasonable estimate for

crumpled

membranes in ₅ < d _< 8, Renormalization

group calculations

[23]

suggest ^that at least for tethered membranes, df should

satisfy equation (I)

as d

- oo, Additional simulations for

objects

with

spectral

dimensions between 413 and 2 would be vely useful to

clarify

the _crossover dimension d from the flat to the

crumpled phase.

In d = 4, 1find that tethered membranes remain flat and do not

crumple.

^As was done in d

= 3

[13,20,24],

^{it is}

interesting

to introduce local

flexibility

into these

highly

^{connected membranes} to check whether

they crumple,

^In an earlier

study,

Murat and

[13]

studied

randomly

site-diluted

membranes in d

= 3. We found that membranes

simply roughen

as the dilution is increased but remain flat until

they

fall apart ^{at the}

percolation ihreshold.

Plischke and Fourcade

[24]

^found a

similiar result for bond-diluted membranes, In thin paper, ^{I show that} site-diluted membranes also remain flat in d ₌ 4. One convenient way ^to characterize the

properties

of diluted membranes is

to measure the order

parameter [22]

^defined

by

^{the ratio} of the

largest eigenvalue

^{of the} moment

of inertia tensor Ad to its value in the initial flat

configuration 1[,

#

_"

ld/1$. (3)

In d =

3,

we found that

#

decreased

approximately linearly

as p decreased and reached a value of about 0.31 _at

pi,

In d

= 4, find that

#

also decreases _as p decreases but the

dependence

is not as linear _as in d ₌ 3, At

pi, #

reaches _a value of about

0.15,

then

# drops discontinously

to

zero, Thus _as in d

= 3, random site-dilution alone b _not sufficient to

produce

a

crumpled

state, In

light

of the above discussion _on the

relevancy

of

higher

order terms this result should _not be

too

surprising.

1698 JOURNALDEPHYSIQtJEI N°12

The outline of the paper ^is as follows. In section 2, ^describe the model and the molecular

dynamics technique

used to

perform

the simulations. In section 3,

present

_my results for the

eigenvalues

of the moment of inerth tensor, the radius of

gyration

and the static _structure func- tions for two-dimensional tethered membranes embedded into dbnensions up ^{to 8. I will then}

present

^{results for} site-diluted membranes. In section

4,

1will

briefly

summarize the results and

indicate _some future directions,

2. Model and method.

Each membrane consbts of N _monomers of _{mass m} connected

by

anharmonic

sprillgs,

The

monomers interact

through

_a shifted Lennard-Jones

potential given by

U~(r)

_#

~~ ~~~~~

~~~~ ^~

~j

^{~~ ~ ~ci}

(4)

0 _{r > rc,}

with _{rc =}

21/~a,

_This

purely repulsive potential represents

^{the excluded volume} interactions that

are dominant in the _case of _monomers immersed in _a

good

solvent, For _monomers which _are tethered

(nearest neighbors)

there is _an additional attractive interaction

potential

of the form

[25]

~

~bond~~) ~

-0.5kR(

In I

(

ifr < Ro;

(5)

oJ r > Ro.

The parameters ^k =

30ela2

and Ro ₌ 1.5a _are chosen _to be the _{same as} in

[26].

^{Note that the} form of the

potential

is somewhat different from those used

by

Kantor et aL

[5],

Plhchke and Boa(

[15]

and Abraham _et al

[16]

^{in that there exists} no range of

separation

_r in which the force

vanishes. Also note that this

potential

does not include any

explicit bending

terms.

Denoting

the total

potential

of _monomer I

by Ui,

the

equation

of motion for monomer I is

given by

~~~ ~~ ~~~

_~

~~~~'

_~~~

Here r is the bead friction which acts to

couple

the _monomers to the heat bath.

Wi(t)

describes the random force

acting

on each bead. It can be written _{as ;i} Gaussian white none vith

(W;(t) W; (t'))

₌

b;;b(t t')6kBTr, (7)

where T is the

temperature

^and kB is the Boltzmann constant. I have used r ₌ r-I _and

kBT

₌

1.0e. Here

r =

«(mle)1/2.

The

equations

of motion _are then solved

using

_a

velocity

Verlet

algo-

rithm

[2~

^with a time

step

At ₌ 0.012r

[28].

^The program was vectorized for _a

supercomputer following

the

procedure

described

by

Grest _et al.

[29] except

^{that the}

periodic boundary

conditions

were removed and additional interactions between _monomers which _are tethered _was added. The cell method was used to check whether non-tethered _{momoners were} within the range of inter- action

by projecting

the membrane into 3 dimensions. The axh of the 3-dhnensional cubc _were choosen

parallel

to the

eigenvectors 6; (d

2 < I <

d)

^{of the} moment of inertia tensor with the

largest

three

eigenvalues.

^Since the membrane _can rotate

during

the _course of the

sitnulation,

the direction of the cube _was

updated

_every lW time

steps.

^{A similiar}

procedure

was llsed

by

^Boal et al

[19]

^{in their} simulations for d _> 3

except

^that

they projected

the membrane _{onto an}

infinitely

long cydinder.

^The _cpu titne per

step

^increased

approximately linearly

with dN. For the tethered

N°12 THETHERED MEMBRANES EMBEDDED IN HIGH DIMENSIONS 1699

membrane of size N ₌ 2437 embedded into d

= 5,1 million

timesteps

took about 17 hours of cpu time _{on our}

Cray

XMP 14/se. Wth thb choice of

parameters,

^the average ^bond

length

between

nearest

neighbors

that _are tethered is found to be 0.97«. Further details of the method can be

found elsewhere

[26].

The sitnulations _were

performed

on two-dimensional

triangular

_arrays of monomers in which

all the nearest

neighbors

were tethered. The

shape

^of these finite

systems

were in the form of _a

hexagonal

sheet of linear dimension L. Values of L in the range ^{13 < L < 57} were studied for

embedding

dimensions 3 < d < 6 while for d ₌ 8 the

largest _system

studied _was L =43. For these membranes the number of

sites,

N, is

given by

N ₌

(3L2

+

1)/4.

For d

= 4, simulations _were also carried out on site-diluted membranes with _an

occupation

fraction _{p > pc}

= 0.5. In this _case all but the

largest

duster _was eliminated and N _was in the range ^{1676 < N < 3070.}

l~pically

0.5 1.0 _x 10~ time

steps

were needed _to

equilibrate

the systems. ^The

equilAration

of all the

systems

were checked

by monitoring

the autocorrelation function of the radius of

gyration

of the membranes. The simulations _were then _run for 1.5 2.0 _x 10~ time steps ^after

equilibration.

Note that due to the

larger

time

step

used here

compared

to our earlier simulations _on tethered membranes in d

= 3

[13],

^{the total}

length

of the runs are about twice _as

long.

lb

analyze

our results _we calculated several

quantities. Every

lW steps ^{the inerth}

matrix,

its

eigenvalues I; (li

<

12

< Ad) ^{and the radius of}

gyration squared, R(, given by

their sum were calculated. These

quantities

were also used _to check the

equilibration

of the system

through

their autocorrelation. In the

crumpled phase,

all d

eigenvalues

are

expected

to scale _as

L2",

_where

v =

2/df.

In the flat

phase,

the

largest

two

eigenvalues

should scale _as L2"" with vjj = I, while the

(d 2)

^smallest

eigenvalues

should scale _as L2(

The

spherically averaged

structure

factor, S(q), given by

S(q)

₌

j _L e'~'~~.~~'~ ), (8)

;J

was also calculated. The

angle

brackets represent a

configurational

_average

typically

^taken ev-

ery ^{10W0 time} steps. ^{For each} _q = q], I

averaged

_over 30 random orientations. In the

scaling regime, S(q)

-J

q-df.

As

pointed

out

by

Plischke and Boal

[15],

^{it is also}

important

to consider the

anisotropic

scatteri ng

intensity, S;(q),

where the

q's

are restricted _to be

parallel

to the

eigen-

vector d; of the inertia matrix

corresponding

to the

eigenvalue I;.

Since the

topologically planar

membranes have

Ad'-~ Ad-

i »

I; (I

< d

2),

instead of

simply measuring Sd(q)

and

Sd-i(q) separately,

it h useful to measure

sj(q)

₌

( _i£ e.~i~r,-r,> ), (9)

;,;

such that qjj ^h

parallel

to ad6d + _ad- i&d-i, where _a; are random numbers. For d > 4, also calculated

S12(q),

which is

equivalent

to Sjj

(q), except

that _q is

parallel

to al Al + a262. For these later two

quantities,

I

averaged

_over 30 random orientations for each _{q = q (,}

thereby improv- ing

the statistics

compared

to

S;(q). S12(q)

and

Si (q)

were found to

give equivalent

results. As will be shown in the next two

sections,

these

anisotropic scattering

^functions are very useful in

demonstrating

that for d < 4, the tethered membranes _are

anisotropic

and

flat,

while for

larger

dimensions, they

_are

hotropic

and

crumpled.

1700 JOURNAL DE PHYSIQUE I N°12

1

d-« ' d-s ^'

j

^' _~ 3

4 s

' s

ii ml

#A #» 3~ #A #» ~# a-o 4A

In L InL

i-1 dm0

.>

s

~ ^'

fi 5

0

<cl

>a >» s-a am «»

lnL

Fig. I. Eigenvalucs of the moment of inertia tensor Ai venus L for a)^d ₌ 4,

b)

d

= 5 and

c)

d

= 6.

3. Results.

One

simple

way to

distingush

the flat

phase

bom the

crumpled phase

^is to examine the

eigenval-

ues of the moment of inertia matrix and the mean-square ^{radius of}

gyration given by

the sum of its

eigenvalues.

^In

figure I,

results for the

eigenvalues

of the moment of inerth _tensor

ii

versus L for

tethered membranes embedded in d ₌ 4, 5 and 6 ditnensions _are

prqsented.

The lines _are least square ^fits to the data for 13 < L < 57. In d ₌

4,

find that the

largest

two

eigenvalues

scale as 1;

-J L2vll, _{with Jij =} 0.95 + 0.05, conshtent with the membrane

being

flat. The two smallest

eigenvalues

scale as

L2',

with

(

₌ 0.84 + 0.05. As _seen bom

figure la,

the

magnitude

of the

eigenvalues, 14

-~

13

>

12

-~

ii,

_are also consistent what _one would

expect

^for a flat membrane.

However,

the behavior of the

eigenvalues

is

significantly

different for d > 5 _{as seen} in

figures

16 and _c. In thb _{case, as} for d

= 8

(not shown),

the d

eigenvalues

scale with

approximately

the same

slope.

^{From fits} to the

data,

_{v =} 0.85 + 0.05 in d

=

5,

0.74 + 0.04 in d

= 6 and 0.60 + 0.05 in d = 8. In

figure 2,

results for the mean square ^{radius of}

gyration (R~)

venus L for membranes

embedded into ditnension

d,

3 < d < 8, are shown~ For d

= 3 and

4, (R()

_-J

^L2

as one

expects

for the flat

phase.

^{However for}

larger

^{values of}

d,

the

slope

^decreases as d increases. Fits to the data

give

results for _v consistent with those obtained from

scaling

Ad as one would expect. ^While these results

give

the first clear evidence that the membranes

crumple

for d >

5,

one should not

N°12 THETHEREDMEMBRANESEMBEDDEDINHIGHDIMENSIONS 1701

take these estimates of the

exponents

too

seriously.

From earlier

experience

in d

= 3

[16,19~0]

^it

was found that values for

(

determined from

scaling

of

li

varied

widely,

^bom about 0.65 to 0.80.

More reliable estimates _were obtained

by

^finite size

scaling

of the

anisotropic

structure function

Si(q)

which will be discussed below.

dm3

«

s

« o

iT

s

o

m a-o «»

~ L

Flg.

~ Radius of

gyration (R()

«mm L for membranes cmbcddcd into dimension d,3 < d < 8.

° O j S

W~

~ 4

5 *

s X

11 +

4A -3A

In q

Flg. ^{3. The} spherically averaged structure function

S(q)

for the largest membrane studied

(N

=2437 for d = 4,5 and 6 and N=1387 for d

=

8)

venus q.

The

spherically averaged

strucuture function

S(q)

h shown in

figure

3 for the

largest

membrane

studied,

N=2437 for d

= 4,5 and 6 and N=1387 for d ₌ 8. In the

scaling regitne,

_{qm~ =}

2~/llg

_<

q ^<

2~/3

_m

2, S(q)

-J

q-df.

The most accurate way ^to determine

df

from data of thin

type

^h to

replot qdfs(q)

vernu q on a

log-log plot

for different values of

df.

In the

scaling regitne,

data for different values of L should

overlap. df

_can then be obtained bom those values which

give

17o2 JOURNAL DE PHYSIQUE I N°12

d ₌ 4

o o o

o K

~ ,'

W _.~

b I

_~

~ / ₍ ~ ~.,~.

, V

.

-3.0

In q

d =

o,

o#

,

" , j .

c ..

2.0

~

/

'

o

-3.0

In

d = 6

e ox

if~" '_

m

, ?'

$ i '

~ '

#

'.J

In

Fl g. 4. - The pherically

averaged 12(q)

(o)

N°12 THETHERED MEMBRANES EMBEDDED IN HIGH DIMENSIONS 1703

a

reasonably

flat set of _curves in the

scaling regime. Following

thb

procedure,

reasonable fits _to the data _were obtained with

df

₌ 2.2 in d ₌ 4, 2A in d

= 5, 2.8 in d

= 6 and 3.3 in d ₌ 8.

The error in df is

approximately

+0. I. Since _v

=

2/df

for tethered membranes, these results

give

v =

0.91, 0.83,

0.71 and 0.61 for d ₌ 4,

5,

6 and

8, respectively.

In the flat

phase,

the membrane is _not

isotropic

and

S(q)

is _a mixture of

scattering

contributions

parallel

and

perpendicular

to the

plane

of the membrane. Thus it is _not

surprising

that

df

^is

slightly larger

than 2 in d ₌ 4. This is

seen more

clearly

in

figure

4a, where

q2S(q)

is shown for N =2437. As discussed

by

Abraham and Nelson

[18],

^{the kink and} extra structure in

S(q)

for a flat membrane is related to the different

scaling regimes

for

scattering

in the

plane

of the membrane and

perpendicular

to it. For d > 5, the results for _{v are} consistent with those obtained from the

scaling

of

ii.

Because the flat

phase

is

anisotropic,

it is very useful ^to consider

scattering

functions for ori- ented membranes. One convenient way ^{to do} this is to measure the

in-plane

structure function Sjj

(q) given by equation (9).

In d

=

3,

the fluctuations

perpendicular

to the

plane

of the _mem- brane in the flat

phase

were studied

by choosing

_{q ii} ki,

Si (q) [15,18,19].

For d > 4, one can also

study S;(q) (I

< d

2),

_{or average over} a number of

is,

_as

long

as

they

are all

perpendicular

to both Ad ^and Ad-

I In order to

improve statitics,

it is useful to determine

S12(q),

ⁱⁿ which q was

choosen to be in the

plane

^formed

by

Al and

62.

Results for

S(q),

_Sjj

(q)

and

S12(q)

are shown in

figure

4 for _a L =57

(N=2437)

membrane in d ₌ 4, 5 and 6. The three

scattering

functions

behave very

differently

and converge

only

for

q-values

which

correspond

to the

length

of a

single

bond. The oscillations in Sjj

(q)

arise bom the

scattering

off _a 2d

planar object

with _a

relatively sharp

interface. This effect has been discussed in detail

by

Abraham and Nelson

[18]

^{for tethered}

membranes in d

= 3. Note that these oscillations _{even occur} for d > 5, where the membranes

are

crumpled

and not flat. This is because the membrane still has _a well defined characteristic size and

shape

when

projected

unto the

plane

determined

by

the two

largest eigenvectors

of the

momemt of inertia matrix. This

projected

membrane is

roughly

the

shape

of _a disk

(ld

-~ Ad-

i

and as a result oscillations in the

'in-plane'scattering

occur. As shown in

figure 5,

the first and

deepest

minimum _occurs for qjjL _m 10 as in d ₌ 3

[18,19].

^However the

larger

the value of d, the fewer of these oscillations _are present ^and

they

come less well-defined- In d

= 8,

only

a small shoulder remains in Sjj

(q)

with not well

developed

minimum.

The clearest evidence that the membranes _are

crumpled

for d > 5 _comes from finite size

scaling

of the

partial scattering

functions S;

(q),

_Sjj

(q)

and

S12(q).

^In the flat

phase

_Sjj

(q) IN

= al

(qL"ii

with vjj ⁼ I, while

S;(q) IN

_-J

S12(q)/N

=

02(qL() (I

< d

2).

^{However in} the

crumpled phase

all of these structure functions should scale _as

qL".

In

figures

5 and 6, show the structure

factors

plotted

in the

scaling

^{form for 4 values} of L in the range ^{21 < L < 57 for d} ₌ 4, 5 and 6.

Results for

Si(q)

show _a similiar

scaling

with _the

essintialJy

the same

exponent (

as determined from

S12(q).

In d

= 4, the results

clearly

_support the conclusion that the membrane is flat. The value of

(

₌ 0.77 + 0.04 while vjj is very near I. In d

= 5,

(

and vjj differ

slightly

but are the same within their error bars. In d

= 6 and

8,

the two

exponents

agree very well. A summary of the exponents ^{obtained from}

scaling

_Sjj

(q)

and

S12(q)

are

given

in table I. The _error bars _are

subjectively

determined

by replotting figures

5 and 6 for several values of the exponents ^and

qual- itatively determining

which

give

^{the best fit. For flat}

membranes,

the

in-plane

structure function Sjj

(q)

^should not

satisfy

the

scaling

relation _{over a} wide range ^of intermediate wavevectors

[18].

This is

dearly

_seen in

figure

5a.

The exponents ^{determined from the}

scaling plots

ofvarious _structure factors _are shown in

figure

7. Also shown is the

prediction

of the

Flory theory,

_{v =}

2/df

₌

4/(d

+

2).

For d

= 3, simulations

on finite membranes of the

type

^{studied here found}

(

₌ 0.64 + 0.04

[18,30,31].

^two more recent studies

[32] suggest

^{that this value is too}

high

and the actual value is closer to 0.5. The difference

being

^attributed to a crossover effect due

partially

to the open

boundary

conditions.

However,

there remains _some

question

as to whether the

breaking

of rotation invariance which _occurs when

1704 JOURNAL DE PHYSIQUE I N°12

lhble I. Values

for

the crkkal e~ponenti

(

and Jij

for

3 < d < 8

for

nvo-dbnensknal tethered membranes _as detemihed

Jtom

the

scaling of _Sjj(q)

and

S12(q) (Si(q)

ih d

=

3).

Results

for

^d ₌ 3

Jtom r~fierence flf3lJ.

d

(

_vjj

3 0.64+0.04 1.0

4 0.77 + 0.04 1.0

5 0.77 + 0.03 0.82 + 0.05

6 0.69+0.03 0.69+0.05

8 0.60+0.03 0.60+0.03

~_~ d=5

z ~

)

_e

Of ot

~ <

t'~ <hi

I-o 3.0 4.0 5.0 _{-in on in 2.0 3A 4A 5A en}

in q L In q L°*'

d=6

(c) -9.0

-2.0 -1.0 o-o I-o 2.0 3.0 4.0 5.0

In qL"'

Fig. 5. The in-plane structure function Sjj(q)

IN

_wnus qL"ii for 4 values of L in the range ²¹ < L < 57 for

a)

d

= 4,

b)

d

= 5 and

c)

d

= 6. The values for

vjj ^are given in table I.

one introduces

periodic boundary

conditions does _not reduce the size of the

height

fluctuations and

change (.

At this

time,

the

precise

value of

(

in d

= 3 remains _an open

question.

lb first order in _I

Id _[33], (

₌

(1 1Id)

₊

O(1/d2),

which is in remarkable

agreement

^{with the simulation} results in both d

= 3 and 4 for membranes with free boundaries. While this agreement may

only

be coindence, it is

intriguing.

In

figure 7,

have used 0.64 for

(,

since this is the value which has been measured for

comparable

^{finite size} systems ^{with free}

boundary

conditions _as llsed in the

present study.

While the free ends may ^{effect the value for}

(

in the flat

phase, they

are less

likely

to effect the behavior of the system ^{in the}

crumpled phase

or the _crossover dimension between the

N°12 THETHERED MEMBRANES EMBEDDED IN HIGH DIMENSIONS 1705

dm4 d«5

§

~i ^~~~

~

-7.0

("1

~bl -9A

-2 -1 0 1 2 3 4 5 -2.0 -in lo in 2.0 3A 4A 5.0

in qLan in q L'"

dins

tc)

-2.0

In qL'°'

Rg.

6. The strucuture function

S12(q)/N

_wnus

qL(

for various values of L in the range ^{21 < L < 57}

for

a)

d

= 4,

b)

d

= 5 and c)^d = 6.the values for ( are given in table1.

1.2

1-o .

,

~~

#

~ 0.6

o.4

o~

o-o

2 4 6 8

d

Fig.

7. The exponents ( (o ^and _vjj

(.)

_as a function of the

embedding

dimension d determined from the

scaling

ofs(q), S12(q)

and Sjj

(q).

li~r d > 5, _v

= vjj ⁼ (,

indicating

that the membranes are

isotropic.

The

Flory

exponent up = 4

/(d

+

2)

is shown for comparison.

1706 JOURNAL DE PHYSIQUE I N°12

o>

w

B O

°

o-S 0.7 OS 0.9 1

P

Fig.

8. The order parameter #

(Eq. (4))

as a function of _p for random site-diluted membranes in d

= 3

and 4. Each point represents the results for _one simulated membrane. The results for d

= 3 _are from

reference [13].

flat and

crumpled phases.

One

interesting point

to note is that in the

crumpled phase,

the values of _{v are}

significantly larger

than

predicted by

the

simple Flory theory.

In terms of their fractal

dimension,

this means that the

crumpled

membranes are

flatter

or less

crumpled,

then

predicted by equation (I).

From the

large

d

expansion [23],

one

expects

that the _v should

approach

_VP as

d - c~J. This

unexpected large

derivation from the

Flory prediction,

^which does not occur for

polymeric

fractals with smaller values of

I,

_h

_presumably

_related

_jo

_{the fact that d and du~}

are

well

separated

for tethered membranes

compared

to

systems

with d _near I.

In reference 13, Murat and I studied the

properties

ofsite-diluted tethered membranes in d

= 3.

We found that site-dilution alone _was not sufficient to

produce

a

crumpled

membrane,

though

it did have the effect of

producing

a

rougher

membrane. For the

triangular

lattice used in the

present

simulations, the value of the order parameter

#

was

quite large, 0.85,

in d ₌ 3 for _non- diluted membranes

(p

₌

I).

The introduction of disorder reduced this value to about 0.31 before the membranes fell apart at pc. Because of the extra

degrees

of freedom in d

= 4, the value of # iS

significantly

reduced _even for _p

= I. Thus it seemed reasonable to

explore

the

possibility

that additional site-dilution may ^{have the} desired effect of

producing

a

crumpled phase

in 4-dimensions

even

though

it is not sufficient to do _so in d ₌ 3. In

figure 8, plot

# for _a number of

samples

in both 3 and 4 dimensions. Since it is

possible

to

only study

a few

samples

for each value of _p due to the slow

equilibration

of the membranes, the exact

magnitude

of the finite size corrections to

#

is _not

known, though

I do not

expect

them to be

large

since the

samples

are

reasonably large.

For each value of _p

typically

2-3

samples

was used to determine

#.

In d

= 3, there was very ^little

sample

to

sample

variation in the value of

#

for each value of_{p while} in d

= 4 the fluctuations _were much

larger.

As

expected #

decreases as p decreases, but does not go ^to zero for _{p > p~} in either d

= 3 _or 4. Since site-dilution alone _cannot effect the

relevancy

of the

higher

order interactions

~n for _{p >} pc, it not

surprising

that these membranes remained flat.

4. Conclusions.

In thb paper, ^{I have shown from extensive molecular}

dynamics

simulations that two-dimensional tethered membranes remain flat when embedded into 4-dimensions but

crumple

ⁱⁿ

higher

dimen- sions. These results suggest ^{that the induced}

bending rigidity

which

keeps

these membranes flat in d = 3 is also

present

ⁱⁿ 4-dimensions. If one assumes that this

bending rigidity

is

generated

N°12 THETHERED MEMBRAN& EMBEDDED IN HIGH DIMENSIONS 1707

by higher

order interaction terms, then it is

possible

to

interpret

^{the transition} from _a flat state to _a

crumpled

state in terms of whether these

higher

order terms are relevant or noL A

simple Flory theory

_suggests that the

four-body

interaction terms are irrelevant above 4 dimensions and

three-body

terms above 6.

Depending

whether

three-body

terms alone _are sufficient to

generate

the

required bending rigidity

or

four-body

terms are needed,

Flory theory predicts

^d ₌ ⁴ or 6 _as the dimension above which the membrane should

crumple.

The fact that the transition _occurs for d _< 6 b consistent with the measured value of _v in the

crumpled phase being larger

than

Flory theory predicts [34].

The

larger

the value of _v, the smaller the fractal dimension df. This leads to fewer

interactions,

_as the

crumpled

state is less

compact

^than

predicted by equation (I),

_sug-

gesting

that the _crossover dimension

being

less than 6 and _v > vp are related.

However, why

the

crossover dimension b between d

= 4 and 5 remains _an open

question.

2A O . .

,'

' CWfiI@@d

'

1.5 SA l@~0V@~t

D/.

. .

"

d '

t~ X X

SA 1rlelevant

oa

o-o

0 2 4 6 8 lo 12 14

d

Fig. 9. Phase

diagram

for

polymeric

fractals. The _crosses represent ^data for linear polymers I

= 1, ^the

squares are for

Sierpifiski

gaskets [3,4] ^{and the circles} are for two-dimensional tethered membranes. The open symbols _are in the flat phase while the closed symbols ^and crosses are in the crumpled phase. The dashed line separates the flat and crumpled phase. SA stands for self-avoidance which is irrelevant in the lower fight of the

diagram.

In

figure 9,

a

phase diagram

for

'polymeric

^fractals'

[21]

^{determined from all available data}

is shown. This

plot

is similiar to _one

recently published by

Levinson [4] except ^{that his contains}

an intermediate

'rough' phase ((

< tij ^<

I),

which Boal et al

[19]

^{found for small membranes}

(L

<

17)

in d

= 4 and 5. The

present

simulations _on

mqch larger

_systems

suggest

that this

phase

does not exist. lbthered membranes in d

= 4 are flat while those in d

= 5 are

crumpled.

Combining

results for linear

polymers, triangular Sierpifiski gaskets

and tethered

membranes,

_one

sees that the

phase diagram

b

quite

rich. There h _a flat

phase

^{in between} the stretched

regime

and the

crumpled regime.

The

crumpled regime

b divided into two

regions,

one where self avoidance

(SA)

b relevant and the other where it is _noL Additional data for other values of

I

_{would be very}

useful to fill _out this

phase diagram. Systems

with values

of1

_>

_2,

_for

example gels,

embedded in d _> 3

woulfbe

^useful to examine the

phase boundaly

between the flat and

crumpled regimes.

Systems

with d in the range 4/3 _to

2,

would be very

helpful

in

understanding why Flory theory

is

a

good approximation

for _v for values of

I

near I, but not so

good for1=

2. I

plan

to carry out some of these studies in the future.