Molecular dynamics simulations of the structure of gelation/percolation clusters and random tethered membranes

(1)

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Molecular dynamics simulations of the structure of

gelation/percolation clusters and random tethered

membranes

Gary S. Grest, Michael Murat

To cite this version:

(2)

Molecular

_dynamics

simulations of the

structure

of

gelation/percolation

clusters and random tethered membranes

Gary

S. Grest and Michael Murat

Corporate

Research Science

_{Laboratories,}

Exxon Research and

_{Engineering Company,}

Annan-dale, NJ _08801,U.S.A.

(Reçu

le 28 décembre 1989,

accepté

sous

_{forme définitive}

le 28

_{février 1990)}

Abstract. 2014 We

study

the

_equilibrium

structure of isolated

_(diluted)

three dimensional

percolation

clusters and site-diluted tethered membranes

using

a molecular

_dynamics

simulation. We find that the

_percolation

clusters swell upon dilution and the fractal dimension

changes

from 2.5 to about 2, in _agreementwith recent

_Flory-type

theories and neutron

_{scattering experiments}

on

_gelation

clusters. The

_{equilibrium configuration}

of the site-diluted membranes is found to be

rough

on small

_length

scales but flat

_{asymptotically.}

We measure the

_{ratio 03A6 = 03BB3/03BB03}

where

03BB

3 is

the

_{largest eigenvalue}

of the interia matrix at

_equilibrium

and

_03BB03

is its value when the membrane is

_perfectly

flat. We find

_{that 03A6}

decreases

_linearly

with

_decreasing

p, the fraction of sites _presentin the membrane. The membranes are shown to have

anisotropic scattering

functions.

_Only

when

_{p ~ p+c,}

where

_{pc is}

the

percolation

threshold, do the membranes become

isotropic, indicating

that random site-dilution is not sufficient to

_produce

a

_crumpling

of

two-dimensional membranes. Classification

Physics

Abstracts

82.70G - 64.60 - 05.40

1. Introduction.

In

_1984,

Cates

_[1]

introduced the term

_{« polymeric}

fractal » to describe the class of fractals which are made of flexible

_polymer

chains at short

_length

scales but have an

arbitrary

self-similiar

_connectivity

at

_large

distances.

_Examples

of such fractal are

ordinary,

linear

polymers,

branched

_polymers

and swollen

_{gelation/percolation}

clusters

_generated

near the

percolation

threshold _pc.Such fractals are

_interesting

because

_they

have no inherent

_rigidity.

Because such

_systems

are

_locally

_very

_flexible,

after

_they

are allowed to relax and take on

their

_{equilibrium shape, they}

_{very often}have a _verydifferent fractal dimension than

_they

had when

_they

were first constructed. Cates

_[1]

worked out a

_{Flory-level theory [2]}

in the _presence of excluded volume interactions to determine an estimate for the fractal dimension

d f

which relates the mass or number of monomers N of the fractal to the size

Rg,

Rgdf‘ ~

N.

_{By balancing}

the elastic

_(entropic)

free energy of the «

_{phantom » object}

without

self-avoidance with the mean field estimate of the excluded volume

interaction,

he showed

that df depends only

on the

_spectral

dimension

[3] d

and the dimension of

_{space d}

(3)

This is a _very

_important

relation since it

_{predicts that df}

is

_independent

of the

_spatial

configuration

of the

_object

and

_{depends only}

on its

_{connectivity.}

Despite

the fact that the

_Flory

_argument

is

_only

a

_simple

mean field

_theory

it is known to

work _verywell for a number cases, often

producing

estimates

_{for df}

which differ from the exact result

_{by only}

a few

_percent

or less. One familiar

_example

is that of linear

_polymers,

for which

i

= 1.

Flory

[2]

first worked out this case and found

_{that v == 11df}

=

3/5.

This result is

very close to _{the best renormalization group estimates}

_[4]

which

_{give v}

= 0.588. Percolation

clusters at the

_percolation

_{threshold pc}are also

_{good examples}

of

_polymeric

fractals. Since

d = 4/3

in all dimensions

_[3,

5,

6] equation

(1)

predicts

that a

_single,

isolated

_percolation

cluster has a fractal

_{dimension df}

=

2 (d

₊

2 )/5,

independent

of the dimension in which it _was

generated.

Thus a

percolation

cluster made of flexible

_segments

with two dimensional

connectivity

embedded in 3 dimensions is

_expected

to have the same fractal dimension as a

percolation

cluster with three dimensional

_{connectivity,}

_{namely df}

= 2. This result was also

found

_by

Daoud et al.

_[7]

who studied the

_properties

of

_gelation

clusters

_{generated just}

below the

_sol-gel

transition.

_They

showed that

_gelation

clusters swell after the excluded volume

interactions from the

_nearby

clusters are screened

_(removed)

and their the fractal dimension

which

_initially

was 2.5 in three dimensions is reduced to 2. This

_swelling

is

_{produced by}

the excluded volume which builts _upfor intra cluster interactions after the other clusters are

removed,

exactly

as occurs for linear

_polymers.

Their

arguments

were identical to Cates’

though they

did not

_present

their results in terms of the

_spectral

dimension

i

Gelation and

percolation

clusters are believed to be in the same

_universatity

class

[8].

Adam et al.

_{[9], using}

light scattering,

found that the undiluted

_gelation

clusters

_{have df}

= 2.5 ± 0.09 while

Bouchaud et al.

_[10]

have verified

_using

small

_angle

neutron

_scattering

that the diluted clusters

just

below the

_sol-gel

transition

_{have df}

= 1.98 ± 0.03. In this paper we

_present

the first

detailed numerical simulations for diluted

_percolation

clusters with either two or three dimensional

_connectivity

embedded in 3 dimensions and show

_{that df}

is indeed 2 within our

numerical resolution. Thus we find in

_agreement

with the neutron

_scattering

results

_[10]

that if there are deviations from

_{Flory theory, they}

are _verysmall.

Another

_example

of a structure which is not a

_polymeric

fractal but for which

_{equation (1)}

was believed to hold is that of a

_tethered,

_{self-avoiding}

membranes in which the monomers are connected to form a

_regular

D dimensional _arrayembedded in d dimensions. Kantor et al.

[11-13]

have

_{recently analyzed}

these structures

_using

a

_variety

of

_{techniques including Flory}

theory,

renormalization group calculations and Monte Carlo simulations. It turned out that the

_{Flory theory}

is identical to that discussed

_by

Cates

_[1]

_provided

one substitutes

i

= D in

equation (1).

For D = 2 and d =

3,

the _casemost often

studied,

this

theory predicts

that df

=

2.5,

in which _casethe membrane would be

crumpled.

The fact that membranes should

_crumple

is

_{supported by}

renormalization group calculations

[11,

12, 14,

15]

and is consistent with the

_early

Monte Carlo simulations

_by

Kantor et al.

_{[11, 12]}

on small

systems.

However recent more detailed simulations on

_larger

_systems

_by

Plischke and Boal

_[16],

Abraham et al.

_{[ 17]}

and Ho and

_{Baumgârtner [ 18]}

indicate that these tethered membranes do

not

_crumple.

Two of _{these groups}

_{[16, 17]}

examined the

_shape

of their tethered membranes

by studying

the

_eigenvalues

of the inertia tensor.

_They

found that

_they

have the

_scaling

behavior

_{À 1 ~}

L 2 il

and

_{A3 ~}

L 2

_t3

_{where À1}

_and

_{A 3}

are the smallest and

_{largest eingenvalues,}

respectively.

These

_exponents

were found to be consistent with

_{v,3 = 1}

and _{vl -}

_0.7,

corresponding

to a

_rough

but

_{asymptotically}

flat

_object.

(4)

constructed. In the Monte Carlo simulations of Kantor et al.

_[11,

12]

and Plischke and Boal

[16]

and the molecular

_dynamics

simulations of Abraham et al.

_[17],

a

_triangular

_arrayof

monomers was considered in which all the nearest

_neighbors

are tethered. Ho and

Baumgârtner

[18]

considered two models. The first is similar to that in references

_{[11, 17]}

except

that it has a _squarelattice

_topology

while the second is for a

_triangular

_{array of}

infinitely

thin,

flexible bonds in which the

_length

of the tethers are limited within 1 and

J3.

The latter simulation is carried out

_using

a Monte Carlo method in which bead

displacements

are

_{accepted only}

if no

_crossing

of bonds occurs, thus

_providing

a model for an

impenetrable

surface. In all of these cases, the motion of a

_single

monomer is

_highly

constrained since it is connected to four or six nearest

_neighbors.

This

_{high connectivity}

apparently gives

rise to some

_type

of local

bending

which suppresses the

crumpling

transition.

While this is one _wayto

_produce

a tethered

membrane,

another _wayone could

_imagine

is a

network of crosslinks which is

_{topologically}

2-dimensional but which is made up of

long

flexible linear

_polymers

between the crosslinks. Such a structure would still

_satisfy

the conditions of self-avoidance and have D =

2,

yet

be closer to Cates’

polymeric

fractal.

Obviously

such an

_object

could be _veryflexible

_locally.

Whether it would

_crumple

for

_system

sizes much

_larger

than the

_length

of the

_polymer

_segments

between crosslinks is an _open

question

which remains to be answered. Work in this direction is

_{presently underway by}

Abraham and Nelson

_[19].

Another

_possible

way to introduce a local

_flexibility

into the

original highly

connected membranes is

_simply

to remove some of the monomers

randomly.

As the fraction p of monomers

_present

decreases,

the average coordination number will decrease. The

_question

that arises then is whether the membrane will

_crumple

before it falls

apart at Pc

or

_{simply roughens slightly}

_{but remain flat. In this paper,}we show that at least for a

site diluted

_triangular

_lattice,

the membrane remains

_{asymptotically}

_{flat for all p >pc- At}

p,, there appears to be a discontinuous transition from a

_flat,

_{highly anisotropic object}

to an

isotropic

one.

_However,

as discussed above since diluted

_percolation

clusters have

d f

= 2 in d = 3 within

Flory theory,

_wehave the curious result that the fractal dimension does

not

_change

even

though

the character of the membrane

_{changes considerably}

at _pc. While _{tethered membranes in the presence of excluded volume interactions}are not

_yet

_very

well

_understood,

_{« phantom »}

membranes in which the excluded volume interaction is

neglected

are. In this case one can use a Gaussian

_{approximation [1,}

2,

13]

and one finds that the fractal

_{dimension df depends only}

on the

_{object’s connectivity through}

its

_spectral

dimension

_J,

Here the

_subscript

o indicates that the excluded volume interaction is absent. This

_equation

reproduces

the well-known result for Gaussian linear

_{polymers, namely dfo =}

_{1/ v 0 = 2.}

Duering

and Kantor

_[20]

found that it also works very well for

Sierpenski gaskets.

For

percolation

clusters,

if we assume

d = 4/3, equation (2) gives d fo

= 4. In this paper, we

confirm this result for

_topogically

2 and 3 dimensional

_percolation

clusters. For tethered

membranes with D =

i =

2,

dfo

= 00. A more detailed

analysis by

Kantor et al.

[11,

_12]

showed that in fact

This

_logarithmic

behavior has been confirmed

by

Kantor et al.

_{[ 11, 12] using}

Monte Carlo

simulations.

In this _paperwe examine the

_properties

of

percolation

clusters

generated

in both 2 and 3

(5)

tethered membranes

using

a molecular

_dynamics

simulations in which the monomers are

weakly coupled

to a heat bath. This method has been very successful in

studying

the static and

dynamic properties

of linear

_{polymers [21, 22],}

star

_{polymers [23]}

and the interaction between

grafted polymeric

brushes

_{[24, 25].}

In

_agreement

with the

_{Flory theory}

and earlier neutron

scattering experiments

for

_gelation

clusters we

find df =

2 in the _{presence of}excluded volume

interactions. With no excluded volume interaction we find

_{df. =}

4. We also studied a number of 2 dimensional random tethered membranes. With no

_{dilution, p}

=

1,

_westudied

systems

of

size 3 L :5 55 on a

_triangular

lattice and find in _agreementwith earlier studies

_[17]

that the membranes are

_rough

but flat and that v3 = 1

and Pl c

0.7.

For p

1,

we could not do finite

size

_scaling

in the same _wayon

_usually

does

_{for p}

= 1. The

difficulty

is that to

produce

reliable

results for the radius of

_gyration

and the 3

_eigenvalues

of the inertia tensor for each size L for random

_membranes,

we would need many realizations.

However,

since the

_{equilibration}

times for these membranes are very

long,

of order several million time

_steps

for N of order a

few

thousand,

we are not able to

_study

more than a few realizations for each _p.

_Instead,

we

made several very

large

membranes,

N from 1 to 3 thousand and ran each for of order

3-6 x

106

time

_steps.

Then to determine the fractal

_dimension,

we measured both the

spherically averaged

structure

_factor,

_S(q),

as well as the

_anisotropic

structure factor first defined

_by

Plischke and Boal

_{[ 16] .}

In the

scaling regime, S( q ) q df

Thus

_{by varying}

_q,we can

_study

the

_properties

of a few membranes on

_{all length}

scales. We find that as _p

decreases,

our site diluted membranes

_roughen

but do not

_crumple

before

_falling

_apart

at

p.

Following

Kantor and Nelson

[26],

we measured the order

_parameter

defined as the ratio

of the

_{largest eigenvalue}

of the inertia tensor

_{l 3}

to its value in the initial flat

_{configuration}

We find

_{that 0}

decreases

_{approximately linearly}

as _pdecreases and reaches a value of about

0.31 1

_{at p’.}

_{Then 0}

must

_{drop discontinously}

to zero. Thus the introduction

_{of randomness,}

in this case site

dilution,

does in fact

_{reduce 0}

but not _{all the way}to zero which is the

necessary condition for the membrane to be

_{crumpled. Though}

these results are

_encouraging,

if one would like to find a

_crumpled

_membrane,

it does not _{appear that random site}dilution

alone is insufficient to

_produce

the desired results.

_Possibly

some combination of random

and/or

correlated disorder

_[19]

will cause the membranes to

_crumple,

but this remains to be

seen.

In section

_2,

we describe the model and the molecular

_{dynamics technique}

used to

_perform

the simulations. Because we are

_using

a molecular

_dynamics

instead of a Monte Carlo

method,

the

_potential

between the monomers is continuous. It differs in its detail from that chosen

by

Abraham et al.

_[17]

In section

_3,

we

_present

our results for

_percolation

clusters,

while in section

_4,

we describe our results for site diluted tethered membranes.

_Finally

in

section

_5,

we

_briefly

summarize our results and indicate some future directions.

2. Model and method.

The

_{equilibration}

of the

_polymeric

membranes is achieved

_using

a molecular

_dynamics

method

_[21]

in which each monomer is

_coupled

to a heat bath. Each membrane consists of N monomers of mass m connected

_by

anharmonic

_springs.

The monomers interact

_through

a

(6)

with r,

=

21/6

eT. 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

_[27]

The

_{parameters k}

= 30

e / a 2 and

_Ro

= 1.5 u are chosen to be the same as in reference

[21 ].

Note that the form of the

_potential

is somewhat different from those used

_by

Kantor et al.

_[11,

12],

Plischke and Boal

_{[ 16]}

and Abraham et al.

_{[ 17]}

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 U;,}

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.

Wl (t)

describes the random force

_acting

on each bead. It can be written as a Gaussian white noise with

where T is the

_temperature

and

_kB

is the Boltzmann constant. We have used

r = 0.5 - 1.5

T -1

and kB

T = 1.0 e. The

_equations

of motion are then solved

_using

a

fifth-order

_{predictor-corrector algorithm [28]}

with a time

step

àt = 0.006 T. Here T

= (m e ) 1/2.

The program was vectorized for a

_Cray

_{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 are added. The cpu time per

_step

increased

_{approximately linearly}

with N. For the three dimensional

_percolation

cluster with

N = 3

966,

the program took about 17 hours of

cpu time for

106

time

steps

with excluded volume interactions and 12 hours without. With this choice of

_parameters,

_{the average}bond

length

between nearest

_neighbors

that are tethered is found to be 0.97 a. The maximum

extension of the bonds are monitored

_during

the course of the simulation in order to assure

that no

_{bond-crossing}

occured. Further details of the method can be found elsewhere

_[21].

We simulated

_systems

whose initial

_{configurations}

were that of a three dimensional

percolation

cluster very near the

_percolation

_threshold,

_Pcas well as two dimensional clusters for a _range

_{of p}

_{> p . The 3d clusters}were

_generated

on a

_simple

cubic lattice with open

boundary

conditions

_{by diluting}

the lattice down to a

density of p =

_0.33,

slightly

above the

percolation

threshold

_[30]

_{of Pc}

= 0.31. We

_preferred

not to simulate

systems

exactly

at

p, as those

_systems

would have very

non-representative

structures. The sizes of the

_systems

that we simulate at _p= 0.33 are still smaller than the correlation

_length

at the same _p.We then eliminated all the clusters

_except

for the

_largest

one. The

_clusters,

which were

subsequently

equilibrated

and studied in

_detailed,

were of size N =

622, 856, 980,

1 136 and

3 966.

_Typically

1-3 million time

_steps

were needed to

_equilibrate

the clusters. After

equilibration,

the simulations were run for an additional 4-6 x

106 _steps,

_during

which the

(7)

Table 1. - The

occupation

probability,

p, lattice size, L, the number

the

_equilibrated

membranes and order parameter

~,

equation (4) for

p >_ 0.6.

objects

were

constructed

on a

_triangular

lattice. A section in the form of a

_hexagon

_{[16, 17]}

with sides of

_length

L was diluted to an

_occupation

fraction _p,with _p_{> p c}= 0.5.

Again,

all

but the

_largest

cluster were eliminated. The number of monomers, N and the lattice

size,

L

for the cases that were simulated are shown in table I. Also shown in the table are the total

duration of the runs

_{(after equilibration)}

in units of T, the

eigenvalues

of the inertia matrix of

the

_equilibrated

membranes and

_cp.

_Typically

1.0-2.5 million

_steps

were needed to

_equilibrate

the

_systems.

The

_{equilibration}

of all the

_systems

was checked

_{by monitoring}

the

autocorre-lation function of the radius of

_gyration

of the membranes.

We also

_performed

simulations of

_{initially planar}

undiluted membranes

(p = 1 )

for a _range

of L

_values,

13 _ L _ 55. For these

membranes,

the number of

sites,

N,

is

_{given by}

N =

(3 L2+

1 )/4.

The simulations were run for 1-6 x

106

time

_steps

after

_{equilibration.}

Finite size

_{scaling analysis}

for the moments of interia tensor _{agree with Abrahams et al.’s}

results that _P3= 1.0 and

v

==0.7.

Percolation clusters

_{at p}

= _Pcwith no excluded volume interactions were also studied. This was done

_{by simply eliminating}

the Lennard-Jones

_repulsion

between all monomers which

were not tethered. The interaction between the tethered monomers remained

_equal

to the

sum of the Lennad-Jones

_potential,

_{equation (5)}

and the anharmonic

_{spring, equation (6).}

The bond

_length

between tethered monomers remained the same as in the excluded volume case.

_{Equilibration}

of these membranes

_required

much

_longer

runs

_(up

to 4 x

106

time

_steps),

as the

_starting

states for these membranes were far away from

equilibrium.

To

_analyze

our results were calculated several

quantities. Typically

every 100

steps

we

calculated the inertia

_matrix,

its

_eigenvalues

_{À i (À 1 : À 2 : À 3)}

and the radius of

_gyration

squared,

_Rg2

given by

their sum. These

_quantities

were also used to check the

_{equilibration}

of the

_system

_through

their autocorrelation. The

_{spherically averaged}

structure

_factor,

(8)

we also calculated. The

_angle

brackets

represent

a

_{configurational}

_average

_typically

taken

every 1 000 time

steps

in the

_present

simulation. For

_{each q}

₌

_{1 q 1}

, we

averaged

over 20

random orientations. In the

_{scaling regime, S(q) - q}

df

_Following

Plischke and Boal

_[16],

we

also considered the

_{anisotropic scattering intensity,}

_{Si (q ),}

where the

_q’s

were restricted to be

parallel

to the

_{eigenvector êi}

of the inertia matrix

_{corresponding}

to the

_eigenvalue

À i.

Since the

_{topologically planar}

_{membranes with p > P c turned}out to

_{have k 3 =A 2}

_{> A 1,}

instead of

_{simply measuring S2 (q )}

and

_{S3 (q )}

_separately,

we calculated

such that q 1 is

perpendicular

to

_êl.

We

_averaged

over 40 random orientations of

q 1 for

each q

=

Iq1- 1.

If in the future it becomes

possible

_it,

i.e.

S1 (q )

and

_{S1 (q ).}

As will be shown in the next two

_sections,

these

_{anisotropic scattering}

functions were _{very useful in}

_{demonstrating}

that for p > P c’ the tethered membranes were

anisotropic

and not

_crumpled,

while those close to _pcwere

_isotropic.

3. Dimensional

_percolation

clusters.

Figure

1 shows the initial

_{configuration}

of the 3d

_percolation

cluster

_{at p}

= 0.33 with

N = 3 966

particles

and a

_snapshot

of the same cluster after 6.5 x

106

time

_steps.

The linear

dimensions of the cluster have increased

_considerably

after the relaxation and it has a more

open structure. In order to

_quantify

the

_change

in the structure of the

_cluster,

we show in

Fig.

1. - The initial

configuration

and an

_{equilibrated configuration}

of a 3d

_percolation

cluster of size N = 3 966. At t = 0,

Rg

= 13, while after

equilibration

it has swollen

(9)

Fig.

2. - The

spherically averaged scattering intensity multiplied by

q 2 for

3d

_percolation

clusters of size

(0)

N = 3 966,

(A)

N = 1 136 and

(e)

N = 856. The radii of

gyration

of these clusters after

equilibration

are 18, 11 and 10,

respectively.

figure

2

_{S(q ) q 2 for}

3 clusters of size N =

856,

1 136 and 3 966. Here

S(q )

is the

spherically

averaged scattering intensity.

We find that on

_length

scales smaller than the radius of

gyration,

_Rg,

of each cluster

_(that

is,

for q

> 2 ir

_/Rg),

and down to several bond

_lengths,

S(q )

scales

_roughly

as

q - 2,

_leading

to a fractal dimension of

_{d f =}

2.0.

_{By plotting}

the

scattering intensity

scaled with different _powersof _q,we find an

_uncertainty

of about 0.1 in

df.

Thus the initial fractal dimension of 2.5 is

_clearly

ruled out.

_Furthermore,

data from the three different clusters

_overlap

in this

_{region, showing}

that the structure at these

_length

scales is not affected

_by

the finite size of the

_systems.

The oscillations seen in the « flat »

_region

seem to diminish with increased

_averaging

as we found

_by

_{comparing figure}

2 with

_plots

made after shorter simulations. The overshoot in

_{q 2 S(q )}

for

_{small q}

in

_figure

2 has also been seen for

ring polymers [21]

and stars

_[32].

This

_swelling

effect is similar to the

_{experimental findings}

of Bouchaud et al.

_[10]

that the

fractal dimension of

_gels

near the

gelation

threshold is reduced

[1, 7]

form 2. 5 ± 0.09 to

1.98 ± 0.03 _upon

_dilution,

also in

_agreement

with the

_Flory

_{theory, equation (1) using}

i

=

4/3

for _a3d

percolation

cluster. Within _our

accuracy, we do not see _any

_significant

deviations from the

_{Flory theory}

result.

We also carried out simulations to

_investigate

the structure of several of the 3d clusters for

which the

_repulsive

interaction between non-bonded monomers was turned off. We show in

figure

3 the

_{resulting ,S(q ) multiplied}

_{by q4}

for clusters of size N = 1 136 and 3 966. One _can

see the trend to a flat

_region

_{at q}

> 2

_{Tf / Rg.}

However,

as the radius of

_gyration

for these

clusters is

_only

several bond

_lengths,

the

scaling region

is very small. We note that the smaller

cluster,

N = 1 136 has been

equilibrated

for _atotal of 7 _x

106 Ot

and shows very

good

agreement with the

_prediction

_{S( q) ~ q - 4.}

The

_larger

cluster was not run as

_{long (only}

(10)

Fig.

3. - The

spherically averaged scattering intensity

multiplied

by

q4 for

3d

percolation

clusters of size

(0)

N = 3 966 and

(A)

N = 1 136 with the excluded volume interaction turned off. The radii

of gyration

after

_{equilibration}

are 5 and 3.5,

respectively.

was much slower than with excluded volume.

_Though

the

_{scaling region}

is not

_large,

our data

is

_clearly

consistent with a fractal dimension of

4,

in

_agreement

with the

_{prediction given by}

equation (2).

4. Random tethered membranes.

Before we discuss our results for site-diluted tethered

membranes,

let us first review our

results for _p= 1. We studied

_systems

of size 13 s L _s55

using

the interaction

potential

described in section 2. In

_agreement

with the results of reference

_[17]

we find that

À 3 -- L 2

and

_{À 1 ~}

L

1.4.

For our

_largest

_system,

L =

55,

_wefind that the value of the order

parameter

0,

defined

by

equation (4),

is 0.85. Thus for this

_{particular potential,}

which

locally

is

_probably

a little less flexible than that used

_by

Abraham et al.

_[17]

since it contains no _range

where the force

_vanishes,

there is

_relatively

little

_roughening

in the transverse direction for the

_triangular

lattice. This

_roughening

increases as the membrane is diluted as seen below.

Figure

4 shows

_{typical configurations}

for out site-diluted membranes after several million

time

_steps

for

_{p > 1.}

Even

_though

the membranes are

_rough,

all but the ones at p =

Pc maintain their two dimensional nature.

_Figure

5 shows the

_{spherically averaged}

S(q )

for the membranes

_{with p}

=

0.9,

0.7 and 0.6. The

decay

is consistent with

df =

2.0 ± 0.1.

_However,

in contrast to the results for the three dimensional

_percolation

clusters which exhibited the same fractal dimension due to the

swelling

of an

isotropic object,

these membranes are

_{quite anisotropic.}

As can be seen from table

_{I, A}

1 appears to decrease

slightly

as L increases

_{for p}

= 0.60. We believe this is not an indication that the

_system

is not

completely equilibrated

but is

_probably

a finite size effect. For small

_L,

sites removed near the

edge play

a more

_important

role in

_{locally roughening}

the

_system

then interior sites

(see

Fig. 4).

In this case

_A

_{1 could}

first decrease as L increases before

_crossing

over to its

_expected

(11)

Fig.

4. -

Snapshots

of the diluted membranes after

equilibration. (a)

p = 0.9, N = 3 422 ;

(b)

p = 0.7, N = 2 _{641 ;}

(c) p

= 0.6, N = 2 149 ;

(d) p

= 0.51, N = 2 093. All the _structures_are

plotted

_on

the same scale.

Fig.

5. -

The

_{spherically averaged scattering intensity}

for site diluted membranes with

₍₀₎

p = 0.9, N = 3 _{422 ;}

(A)

p = 0.7, N = 2 641 and

(e)

p = 0.6, N = 2 149. The data for _p= 0.9 and

p = 0.6 have been shifted

upwards

and downwards

respectively

for the sake

of clarity.

The _arrowsshow

(12)

(see

Tab.

_I)

indicates that

_they

become

_rougher

as _p

_decreases,

but A

_{1 remains}

much smaller

than A2 - A

3. As one reaches p =

0. 51,

very close to _Pc= 0. 5 the membranes become _more

isotropic

and the values of

_{A 2}

_{and A 3}

are no loner

_{approximately equal.}

This is seen most

clearly

from the behavior of the

_{anisotropic scattering}

functions.

_Figure

6a shows

S(q ),

S1 (q)

and

_{S1 (q )}

for a membrane

_{at p}

= 0.51 with 2 093 _monomers.In the

scaling

regime (q > 2

_{TT / Rg}

= 0.4),

the three

_scattering

_{functions converge. In}

_figure

6b the same

functions for a p = 0.7 membrane are shown. The three

_scattering

functions behave very

Fig. 6. - (a) S(q ) (o), Sl (q ) (0)

and

_{S1 (q ) (A)}

for the membrane diluted to _p= 0.51 with 2 093

(13)

differently

and converge

only

at the

_q-values

that

_correspond

to the

_length

of a

_single

bond.

The

_{corresponding scattering}

intensities for the membranes at _p= 0.6 show a similar

behavior. The oscillations in

_{S1 (q )}

arise from

_scattering

off a 2d

_planar

object

with a

relatively sharp

interface. This has been discussed in detail

_by

Abraham and Nelson

_[17]

for tethered membranes. A similar effect was also seen

_by

Grest et al.

_[23]

in

S(q )

for many-arm star

_polymers.

To

_quantify

the amount of

_roughness,

we measured the order

_parameter

_4,

equation (4).

Since for a

_uncrumpled

membrane

_{k 3}

and

_{A 3}

both scale as

L 2,

,

4

will be nonzero. For a

crumpled

membrane 4 -

0 as L --> oo. In

_figure

7 we

plot 4

for our

samples

as a function of

p. Since we were able to

_{study only}

a few

samples

for each value

of p

due to the slow

equilibration

of these

membranes,

we do not know the

_magnitude

of the finite size corrections

to

_0,

_though

we do not

_expect

them to be

_large

since our

_samples

are all

_{reasonably large.}

As

expected, 0

decreases as _p

_decreases,

_indicating

that the membranes indeed

_roughen.

This is consistent with the

_plot

of the

_{configurations}

in

_figure

4.

_{As p approaches p~ 4}

has decreased

by

almost a factor of 3 from its value at _p= 1. Since

À 3 ~ N 2/

df and

À 3 0 ~ N 2/

1.9 for

p = p c

(1.9

being

the fractal dimension of the

_original

2d

_{percolation cluster),}

we

_expect

4 -

0 as L --> oo, albeit very

slowly.

However,

since the

dependence of 4

on L is

weak,

much

larger

systems

than we can simulate would be needed to check this

_explicitly.

Thus,

assuming

that for _p=

pc,

4

=

0,

_wearrive _atthe conclusion that there _mustbe _afirst order

discontinuity in 4

as _p

_approaches

_pc.

_{For p}

_{p c,}

_only

finite size clusters

_(membranes)

are

present.

Thus,

site dilution

alone,

on the

_triangular

lattice is not sufficient to

_produce

a

crumpled

membrane,

though

it does have the effect of

_producing

a

_rougher

membrane.

Although

we have not carried out an

_explicit

finite size

_scaling,

it would be _very

_surprising

if

for 1 _{> p}_{>P c the membrane would}

_crumple

for very

large

L and not

_{for p}

= 1. This

suggests

that our results will not

_{change significantly}

if we had been able to

_study

even

_larger

membranes.

Let us return to the

_properties

of the tethered membranes at _pc.

_Figure

6a showed that the

objects

are

_isotropic

as one would

_expect.

The

_{only question}

which remains is whether it is

Fig.

7. -

(14)

Fig.

8. - The

spherically averaged scattering

function

multipied by

q2

for 2d

_percolation

clusters at

p = 0.51 and

(0)

N = 832,

(A)

N = 1 395 and

(e)

N = 2 093. The _averageradii of

gyration

for these

clusters are 11, 14 and 17,

respectively.

Also shown is the average

S(q )

of the three clusters

₍₃₎

with the

same

_scaling

shifted

_arbitrarily

downwards.

Fig.

9. - The

spherically averaged scattering intensity multiplied by

q4

for membranes diluted to

(15)

possible

to

_distinguish

the observed result

_{for df}

from the

_{Flory theory}

_prediction

d f

=

2,

equation

(1).

In

figure

8 we

_present

results for 3 membranes at _p= 0.51.

Unfortu-nately

the oscillations in

_{S(q )}

in each individual

_sample

is

_large

and it is difficult to determine

df accurately.

In the same

_figure

we also

_present

_{S(q ) averaged}

over the three

_samples.

This is somewhat like what occurs

_{experimentally}

_[10]

when the small clusters are removed and the

scattering

involves an _averageover the

_largest

clusters.

_{Clearly averaging}

over

_only

3 clusters

is not

_equivalent

to what occurs

_{experimentally,}

but it does indicate that one would

_expect

most of the oscillations in

_{S(q )}

for individual clusters to be eliminated

_{by averaging}

over a

large

number of clusters. From either the individual or the average

S(q )

it is not

_possible

to

obtain a _very

_good

estimate for

_df.

The

_{scaling region}

for these clusters should extend from qmin

= 2 TT / Rg to

qmax = 2

TT /3

=

2,

where for the

upper limit we consider

_{only length}

scales

larger

than 3 lattice

_spacings.

Here

_Rg

= 14 is the average radius of

gyration

of the clusters

and qmin = 0.4. In this

regime

q 2 S( q)

is

approximately

flat,

suggesting d f

= 2 ± 0.1. The

upturn

for

_{larger q corresponds}

to

_length

scales smaller than a few lattice

spacings

and is

outside the

_{scaling regime.}

For

_completeness

we also studied the 2d

_percolation

clusters imbedded in d

= ’3

with no

excluded volume interactions. Results for

_{q4 S(q)}

for N = 832 and N = 1 395 clusters are

shown in

figure

9. Not

_{surprisingly,}

in

_agreement

to the 3d

_percolation

clusters and

equation

(2)

as discussed in section

3,

we find

_df

= 4.

5. Conclusions.

In this paper we have

_presented

results for the

_{spherically averaged}

and

_{anisotropic scattering}

function

_{S(q )}

for

_{gelation/percolation}

clusters and random tethered membranes. _In

agree-ment with

_{Flory theory}

and neutron

_{scattering experiments,}

we find that the 3d

_percolation

clusters swell when the excluded volume interactions from their

_neighboring

clusters is removed. This increase is

_significant

and

_corresponds

to a new fractal dimension

df

= 2 whereas its

original dimension df

was 2.5

_prior

to dilution. We also showed that a

two-dimensional

_percolation

cluster embedded in three dimensions also

_{has di -}

2. We then showed that as one

_randomly

site-dilutes a tethered membrane with a 2-dimensional

topology,

the

_{objects roughen}

but remain flat. The amount of this

roughness

increases

linearly

as the fraction of sites

_{present, p}

decreases.

However,

the membranes fall

_apart

at

Pc before

they crumple.

By

comparing

the

_{anisotropic scattering}

functions

_{Si (q )}

defined first

by

Plischke and Boal

_[16]

_{and S1-}

_{(q )}

with the

_{spherically averaged S(q),}

we showed that 2d

tethered membranes at _Pcare

_isotropic

while those for p > P c are

_anisotropic.

A

_complete

treatment of the

_scattering

function

_{S(q )}

from

_{anisotropic objects}

has been discussed

_by

Abraham and Nelson

_[17]

and our results are consistent with their

_analysis.

Gelation/percolation

clusters and linear

_polymers

are but two

_examples

of the

_general

class of «

_polymeric

fractals »

_{[1] which}

have no inherent

_rigidity.

These are

_objects

which are made

of

_polymeric

_segments

at short

_length

scales but have

_arbitrary

self-similar

_connectivity

at

larger

distances. There are _{many other}

_objects,

for

_example,

diffusion limited

_aggregates

and

cluster-cluster

_aggregates

_(both

reaction and diffusion

limited)

which

_satisfy

the second

criterion,

that of self-similar

_{connectivity,}

but do not swell and

_change

their fractal dimension

because

_they

are not made of flexible

_segments.

Most

_experimental

_aggregate

_systems

_[33],

particularly

those made from

_gold

or

_silica,

are

locally

very

rigid.

It is for this reason that

experimentally

one does not have to _{worry about these}

_{objects swelling}

and

_changing

their fractal dimension.

_However,

because d

is very near 1 for _manyof these

_aggregates

_{(as they}

contain very few

loops),

if one could find a _wayto

_change

the

_{persistence length}

after

_they

(16)

Though

we do not know

d precisely

for these

aggregates,

it is

_probably

close to 1.1 and

_surely

less than the

_percolation

value

_{of 4/3.}

The

Flory

theory

would

_predict

that

they

should swell and have a new fractal

_{dimension df}

in the

vicinity

of 1.8. Since this is very near the fractal

dimension of diffusion limited cluster-cluster

aggregates,

these

_aggregates

are not

_expected

to

swell _verymuch even if the

_{persistence length}

of their

_segments

is reduced. While it may be

difficult to

_change

the

_{persistence length experimentally}

after these

_aggregates

are

_generated,

this is

_quite

easy to do

_using

computer

generated

aggregates.

We

_plan

to _carryout such a

study using

the molecular

_{dynamics technique}

described here.

Acknowledgments.

We thank D. R. Nelson for several

_helpful

discussions and the

_suggestion

that we measure the

order

parameter 0

given

in

_{equation (4).}

We also thank C.

_Safinya,

T. A. Witten and D. A. Weitz for

_helpful

discussions. MM is a

_recepient

of the Chaim Weizmann

Rev. B 30

₍₁₉₈₄₎

4090 ; ZABOLITZKY J. _G.,ibid. 30

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Ho _J.-S.,BAUMGÄRTNER A.,

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BIRD R. B., ARMSTRONG R. _C.,HASSAGER O.,

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