Critical properties of crosslinked polymer melts

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Submitted on 1 Jan 1990

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Critical properties of crosslinked polymer melts

Gary S. Grest, Kurt Kremer

To cite this version:

Critical

_properties

of

crosslinked

_polymer

melts

Gary

S. Grest

_(1,

2,

_*)

_{and Kurt Kremer}

(1)

(1)

Institut für

_{Festkörperforschung, Forschungszentrum}

_{Jülich, D-5170 Jülich,} F.R.G.

(2)

Corporate Research Science Laboratories, Exxon Research and

_{Engineering Company,}

Annandale NJ 08801, U.S.A.

(Received

16 _May 1990,

accepted

24 _{August 1990)}

Abstract. 2014 The critical behavior at the

_percolation

threshold for a

randomly

crosslinked

polymer

melt of linear chains is studied

_by

_computer simulations. We show that the fraction of

crosslinks per

chain pc

at the

_{vulcanization/percolation}

threshold is

_independent

of chain

length

N for

_large

N. Thus for

_long

chains, the volume fraction of crosslinks at the transition decreases as

1/N

in _agreement with

_Flory.

If we allow

_{linkages only}

between different chains and no

self-linkages,

we find that _{pc ~ 0.6 in the} limit of

_large

N. This value in 20 %

_larger

than

_Flory’s

mean

field _{result, pc} = 1/2, as some of the crosslinks do not increase the number of chains in a cluster. When we allow intra-chain

_linkages

as _they occur

_{experimentally,}

the

_scaling

with N remains

unchanged but pc

increases by an addition 20 % for the _present model. We also find in _agreement with de Gennes that the critical _exponents are well described

_by

their mean field values and that the size of the critical

_region

where fluctuations are

_important

is small for

_{long polymer}

chains.

The fractal dimension of the

_percolating

cluster at _pc was found to be 3.

Classification

Physics

Abstracts

64.60 - 64.60A

1. Introduction.

Vulcanization of a

_polymer

melt occurs when

_enough

chemical bonds are formed between

linear

_polymers

to form a infinite network. In

_Flory’s

book on

_{polymer chemistry [1]}

_]

he

developed

a

_{simple theory}

to determine the fraction of crosslinks needed to form the infinite

or

_gel

structure and discussed many of the statistical

properties

of both the sol and

gel phases.

Flory’s description

is similar to the Bethe lattice or tree

_{approximation}

which _{has been very}

successful in

_describing

at least

_{qualitatively}

the

_gelation

_{process in} branched

_{polycondensates}

[2-4]. By assuming

that the

_crosslinking

is

_exclusively

intermolecular and that each crosslink

always

reduces the number of molecules

_by

one,

Flory

estimated the fraction Pc of chains

which are crosslinked at the onset of the formation of an infinite

_(gel)

structure. For a

monodispersed

system of linear chains of

_length

_N,

he showed that

For an

_arbitrary

distribution of linear

polymers,

the factor N in

equation (1)

is

_simply

_replaced

by

the

_{weight averaged degree}

of

_{polymerization}

of the

_primary

_{chains, Z}

_[1].

Since in this model there are two crosslinked chains for each

linkage, Flory’s

result for a

_{monodispersed}

sample

can be

_expressed

_equally

well in terms of the number _p of crosslinks _per

_chain,

Note that one crosslink in the _system

_gives

2 crosslinks _per chain. This later

_quantity

is more

appropriate

for

_comparison

to

_experiment

since it is not

_possible

to restrict each new crosslink

to

_always

decrease the number of free

chains/clusters by

one or to exclude intramolecular crosslinks.

_Obviously

in irradiation

crosslinking,

not all crosslinks are effective in

increasing

the cluster

size,

since many connect two chains which are

_already

in the same cluster or

connect a chain to itself to form a

simple loop.

We find that if we

_randomly

distribute the

crosslinks _{in space yet} allow

only

intermolecular

crosslinks,

then Pc is about 20 %

larger

than

Flory’s

estimate in the limit of

_large

N due to the _presence of « wasted » or double links. In

the case that both inter- and intramolecular crosslinks are

_allowed,

_{Pc is increased}

_by

at least

an additional 20

%,

through

the value

depends

somewhat on the local details of the model as

will be discussed below.

In addition to the remarkable result that _{p ~} scales with

N-1

and therefore becomes _very small as the chain

_length

increases,

the size of the critical

region

also shrinks as N increases. De Gennes

[5]

showed in 1977

using

two different

methods,

one based on a direct estimate of

the fluctuations inside one correlation volume and the other on a

_{generalization}

of the Potts

model

_[6],

that the size of the fluctuations decrease as N increases for concentrated solutions.

De Gennes showed that fluctuations are

_{important only}

in a small

region

_8p

around

Pc

given

by

where d is the dimension of _space. For d = 3 the

right

hand side of

equation (3)

is

N-

1/3.

This means that

_except

for a _very small

_region

around the

_percolation

threshold

p~, the critical

exponents

should take on their classical values as

_predicted

_by

the tree

approximation

[2, 3]

for all dimensions d > 2. A similar reduction in the size of the critical

region

was found

_{by Ray}

and Klein

[7]

for

long

range bond

percolation

where all sites within a

range

RB

can be connected to the central site with a

_probability

_p. The two

_problems

can be

mapped

on to each other

_{by equating}

the

_potential

number of

_neighbors

within a distance

RB,

with

_{N dl 2 - l,}

, the

typical

number of other chains within a volume

Rd

I where is the end-to-end distance of the chain. These results differ from

ordinary

short

range

percolation

and

gelation

for which the critical exponents are

_clearly

not described

by

their classical values

_{[8, 9].}

For d = 2 we have a

_special

situation. There the chains _segregate

and each chain has an _average of 6

_neighbors

_[10].

Thus for the

_{percolation problem}

the chain

structure becomes irrelevant

resulting

in the standard continuum

percolation

result. In this

case

_only

above 6 dimensions are the fluctuations irrelevant and the critical _exponents take on

their classical values. Daoud

_[11]

later

_generalized

de Gennes’ result to semi-dilute solutions

using

the standard

_scaling

ansatz and found that the width of the critical

region

as well as the

gel point

strongly

on

_{concentration,}

as a _consequence of the

_change

in the local structure and environment of dilute chains

_compared

to dense systems.

In this _paper, we

_present

the results of our

_{investigation}

of the critical

_properties

of the vulcanization transition for a dense melt of linear

_polymers.

From our

_analysis

of the

dynamics

of

_polymer

melts

_[12]

we had available a

_large

number of melt

configurations

for

chains of

_length

N = 5 _to 200. Since _{we were} interested in

studying

the

_long

time behavior of

chain as

_opposed

to a more detailed model. As discussed in reference

_[13],

the _computer time

required

to _carry out a

_{comparable study}

for the

_long

time

_dynamics

of a

_polymer

melt and

test the _concept of

_{reptation [14, 15]}

for a detailed model would have been

_impossible.

Fortunately

to test the

_Flory

_argument for the vulcanization threshold and de Gennes’

analysis

for the size of the critical

_region

a coarse

_graines

model is ideal. There is no need for a more detailed model. We have thus carried out such a

_study

of the critical behavior of the

vulcanization _process for our model

_polymers.

Here we will first discuss the vulcanization transition

using configurations

saved

during

our

analysis

of the melt

_{dynamics [12]}

as well as some additional runs on

_larger

_systems made

explicitly

for this

_study.

However the

largest

systems we can run at the _present time is of order

50 000 monomers for short chains

_{(N -- 50)}

and 20 000 for

_long

_(N

=

200)

chains.

Larger

systems take too much _computer time to _generate

_{equilibrated samples.}

This is

particularly

difficult for N

_longer

than the

_{entanglement length Ne,}

since then the

_longest

relaxation time scales as instead of

N2

which is

_expected

for short chains. For our

model,

Ne ~ 3 5.

While these _systems are

_large

_enough

to test

_{qualitatively Flory’s}

_argument,

_they

are

too small for a

quantitative comparison

to

_{equation (2)}

nor to test de Gennes

[5]

analysis

for the size of the critical

_region.

For this we need much

_larger

_systems,

which can be done

_by

constructing

random walk chains with the same radius of

_gyration

and

persistence length

as our melt chains and

_distributing

them

randomly

in a cubic box of the same

_density.

In this way

we could _go to _systems of size 3-4 x 105. For

_chainlength

N =

25,

we could

_study

_up to

M = 12 000 chains while for our

_{longest chainlength,}

N =

200,

M * 2 000. The

principle

limitation is related to the _computer time and memory

required

to enumerate the clusters. The

_complication

arises from the fact that the

_{percolating particles}

are chains and not

_single

lattice sites. The chains then introduce an effective coordination number N since each monomer can

_participate

in the

_{crosslinking.}

Based on

_{Flory’s analysis,}

we _expect that if we

consider

_only

intermolecular

_{crosslinkages,}

the results for the

_percolation

threshold should be

essentially

the same for the two cases since the melt chains have a random walk structure. We

checked this for several systems and found this to be true. However because the local

structure for these two cases are not the same, we find some difference in the

_percolation

threshold when intramolecular

_{crosslinkages}

are allowed. As

_expected

a

_larger

fraction of the

crosslinks connect monomers on the same chain for the random distribution of Gaussian chains than for the

_equilibrated

melt in which monomers do not

_overlap.

We believe this is

mostly

due to the size of the fluctuations in

density

in the former case

_compared

to the latter

case where the

density

fluctuations are reduced

_considerably

due to the excluded volume. For

this reason we concentrate on the case in which

_only

intermolecular

_{crosslinkages}

are allowed.

For all the

_chainlengths

studied

_{(25 ~}

N ~

_{200 ),}

we find that the critical _exponents are well

described

_by

their classical values. For the _system sizes we were able to

_study,

we found no

evidence for a crossover to non-classical exponents which must occur _{very near pc} for _very

large

systems. This result is somewhat

surprising particularly

for the shortest

_chainlength

studied,

N = 25 and

presumably

indicates that the

pre-factors

on the

_right

hand size of

equation

(3)

must be much less than one for the

_present

model. One reason for this _{may be}

the fact that our chains are

_extremely

flexible. Our estimations of the critical _exponents was

determined from a finite size

scaling

analysis

of the

percolation

threshold _pc and from the

divergence

of the

_{weight averaged degree}

of

_{polymerization}

Z of the clusters as _p

_approachs

pc, from which the exponents v and y,

respectively,

can be determined. These two _exponents

were chosen since their classical and d = 3

percolation

values are

_{significantly}

different. The

classical or tree

_{approximation}

results are v =

1/2

and _{y =} 1

compared

to the best numerical estimates for d = 3

percolation, v

= 0.88 and _{y =} 1.74

[16].

At

pc, the cluster size

for

_percolation

in d = 3. Here s is the number of chains in each cluster. An addition

quantity

that one could also use to test the de Gennes

analysis

is the fraction of chains in the infinite

(gel)

cluster above _pc from which the

_{exponent {3}

can be determined.

_{3

also takes on _very

different values in the two cases

_(/3

= 1.0 in the classical

approximation

and 0.45 for

d = 3

percolation),

however our _systems turned out to be too small to determine the

gel

fraction and

_{therefore 8 accurately.}

The fractal dimension of the

_percolating

cluster at _Pc is also of interest. In the classical

approximation,

which is valid above 6

_{dimensions, df}

=

4,

while for short

range

percolation

df - 2.5

for d = 3. If one

_{measures df}

from the

_spatial

falloff of the

_density

correlation function p is not

_possible

since it would lead to a

_diverging

local

density.

Ray

and Klein

_[7]

found for

_long

range bond

percolation

that the value of the fractal or

Hausdorff dimension

depended

on how it was measured.

_They

found that

_{by measuring}

the

spatial dependence

of the

density

they

found that in d = 2 and

3,

that df

=

d,

while

by relating

the mass of the

largest

clusters to the correlation

_{length ~ - (p, - p ) - ’}

that

_{d f}

= 4.

(Ray

and Klein refer to the former measure as the Hausdorff dimension and the later as the fractal dimension. Here we

_simply

refer to this dimension as the fractal

_dimension.)

The difference in

these two results

_[7]

_being

related to the

_interesting

feature of

_long

_range bond

_percolation

that the number of

_spanning

clusters the size of the correlation

_{length ~}

is

_{N~ ~ RB 8 p 3 - a~ 2,}

which is

_greater

than 1. A similar result occurs in the

_{polymer problem}

with

Rd

_replaced

_by

Nd~ 2 -1.

.

Only

as one

approaches

the critical

region,

do these clusters merge and

produce

a

single

spanning

cluster as one would

_expect

for d 6. For the crosslinked

_polymer

_problem

studied

here,

we identified the

_largest

cluster for several values

_{of p}

_{p ~} and determined the

static structure factor

_{S(q ).}

For a fractal in the

scaling regime S(q )

should

decay

as a _power

law -

q df

This is

_equivalent

to

_measuring

the

_density

correlation function in real _space. In

agreement

with

Ray

and Klein

_[7]

we find

_df

= 3. We do not have sufficient data to determine the

_scaling

of the mass of the

_largest

clusters with correlation

length,

but we _{presume from the}

work of

_Ray

and Klein that we would obtain the classical result

_df

= 4 in this case.

This

study

in the first part of an extended simulation

study

of the

_properties

of

_polymer

networks are in progress. Here we concentrate on the critical

properties

near the

percolation

threshold. Above _pc, we have also determined _many of the structural

_properties

of a

crosslinked

_polymer

melt,

like the distribution of

_dangling

ends,

distance between crosslinks and number of crosslinks per chains. Results of that

study

will be

_published

elsewhere

_[17].

Future work will

_investigate

the

dynamic properties

of crosslinked melts and their

_properties

under deformation and

_swelling.

The outline of the _paper is as follows. In the next section we will _very

briefly

review the

model and molecular

_dynamics

methods we used to

_equilibrate

a dense melt. The details are

presented

in reference

_[12].

We will also discuss the cluster enumeration and how we

determined the

_percolation

threshold. In section

3,

we

_present

our results for the

_percolation

threshold for N =

25,

100 and 200 as well as our estimates for the critical _exponents. In

section

4,

we

_present

results for the static structure factor

S(q)

for

large,

finite clusters below pc in order to determine the fractal dimension

_{df. Finally}

in section

5,

we

_briefly

summarize

our results and conclusions.

2. Model and method.

an anharmonic

spring.

The monomers interact

_through

a shifted Lennard-Jones

potential

given by

where r, = 21/6 if

is the interaction cutoff. Since the interaction is

_{purely repulsive,}

for an

isolated chain this models the

_good

solvent limit. For monomers which are connected

_along

the _sequence of the chain there is an additional attractive interaction

potential (called

the

FENE

_potential)

of the form

_[19]

The

_{parameters k}

=

30 e / (T 2 and Ro

= 1.5 _{a are} chosen _to be the _{same as} in reference

[18].

Denoting

the total

_potential

of monomer i

_{by Ui,}

the

_equation

of motion for monomer i is

given by

Here T is the bead friction which acts to

_couple

the monomers to the heat bath. 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 T = 0.5

7- -1

and

k,3

T = 1.0 _e, where T =

~r (m / £ ) 1 ~2.

The

equations

of motion are then solved

_using

either a

third or fifth-order

_{predictor-corrector}

algorithm [20]

with a time

_step

At = 0.006 T . Further

details of the method can be found elsewhere

[12, 18].

The simulations

_[12]

were carried out at a

_density

_{p -}

_MNIV

=

0.85,

where M is the number of chains in the cell. Here we

_present

results for N =

25,

100 and 200. The

largest

systems we could simulate were

_M/N

=

2 000/25, 500/100

and

100/200.

Since the

long

range

excluded volume interactions are screened in a

_melt,

the

_equilibrium

chains are ideal. That

is,

the mean _square end-to-end distance of a chain of N monomers has the

form,

where r

I and rN are the coordinates of the chain ends.

Here I

is the _average bond

_length

between 2 monomers on the chain and

~p

is the

_{persistence length.}

In the

_present

case

~ =

0.97 a and

_{f p =}

1.32.

Ideally

we would like to

_study

vulcanization _{process in very}

_large

_system in order to check

the

_scaling

_{of Pc}

with N and the critical _exponents. However as discussed above we are limited

by

the fact that the relaxation times increase _very

_rapidly

with N. Even for N =

25,

where the

longest

relaxation time is

_{only proportional}

to since N «

_{N e,}

it is not

_practical

at this time

to

_study

_systems

_larger

than M = 2 000. For _systems of size NM > 50

000,

it

simply

takes too

supercomputers

[12].

These systems turn out not to be

_large

_enough

to measure the critical

exponents.

However what we could

do,

following,

the earlier work of

_Leung

and

Eichinger

[21],

is to

_place

M random walk chains in a cell of volume V =

MNp

without

regard

to

overlap.

To make

_comparison

to our melt

studies,

the random walks are constructed

_by

a

simple

Monte Carlo random walk

procedure

with a bond

length I

= 0.97 a and with a

restriction on

_backfolding

so as to

_give

the correct

_{persistence length.}

This is done

_by

requiring

_that

_{r i +1)21}

_I

>

1.02 ~ 2.

For this case we could

_study

_systems as

_large

as

4 x

105,

with the limitation

_being

the time to

_perform

the cluster enumeration. We did most of

our

analysis

for the case in which

_only

intermolecular crosslinks are allowed. We refer to these

configurations

as random walk

_{configuration}

to contrast them with the

_equilibrated

melt in which the chains are Gaussian but where monomers do not

_overlap.

To

_study

the vulcanization process, we introduced crosslinks

randomly

into the _system. The

crosslinking

was carried out as follows. We first located the monomer nearest to the crosslinker. We then made a list of all the

_neighbouring

monomers which were within a

radius rx

of this first monomer. We excluded monomers from the list which were connected

along

the

_original

chain. For the _present

study

we

chose rx

= 1.3 a - but the results were

not sensitive to the choice

_{of rx provided that r,}

is small

compared

to the chain extension. The

largest

value

_{of rx}

we checked was 2.0 ~. For the case with

_only

intermolecular

crosslinks,

we

then

_randomly

chose one of the monomers from the list which was on a different chain and

linked the two monomers

_{together. Occasionally}

there were no

_appropriate

monomers within

r~, in which

case rx was

increased in steps of

0. 1 a 2until

a monomer to connect to was found.

In the most

_general

case, we chose a monomer from the list at random without

_regard

to what chain it was

_part

of. In some cases we also removed second

_{neighbors along}

the chain from the

potential

list of monomers to connect _to, to more

_{realistically}

account for local

_backfolding

in the real

polymer.

In either case, this made

only

an additive contribution to the fraction of crosslinks which were needed to form an infinite structure since

only

the intermolecular crosslinks can do this. The inclusion of more

_neighbors

from the same chain

_simply

increased the number of short

_loops

which do not contributed in

_forming

_larger

clusters. This

_procedure

was

_repeated

until the desired number of crosslinks were added.

A cluster enumeration was then carried out to determine the number of clusters of

size s,

n,,

following procedures

which have been

developed

for lattice

percolation

[16].

This can be

done

_{quite efficiently}

_by

first

_constructing

a list of chains which are connected to chain i. It

was then

_{straightforward}

to determine which chains were in the same cluster. To determine

whether a cluster

_percolated,

we divided the _system into int

_{(L/1.3 a)}

bins,

where

L =

(N M/p )1/3

is the

length

of the simulation cell and int

(x)

is the

integer

_part of x. We then

sorted each monomer into bins

_according

to its x-coordinate. If all the bins have at least one

entry, then the cluster

percolated

in the x-direction. This was done for all three directions. This

_procedure

was then

_repeated

400-2 500 times

_depending

on the size of the

_sample.

The

smaller the

_sample,

the

larger

the number of

samples

needed to obtain reasonable statistics.

We

_{typically generated}

50 random walk

_{configurations}

and 12-50

_independent

sets of

randomly

crosslinks for each. The

_only

other trick that we used to

_speed

_up the

analysis

was to

divide the entire _system into cells so that to locate monomers which were close to the crosslinkers or near

_neighbors

of another monomer, we

_only

had to check a small subset of

the monomers in the _system and not all of them. This followed the

_optimization

scheme for

setting

up the Verlet table for the molecular

dynamics

simulation

[22].

3. Percolation threshold and critical exponents.

The

_analysis

for the

_percolation

threshold is based on finite size

_{scaling [23].}

For each

_sample,

Fig. 1. -

Probability R3(P)

that the system

percolates

in all three directions for N = 25 for various

values of M.

_{(a) Equilibrated}

melt in which the crosslinks are allowed to connect monomers on the same

chain as well as monomers on different chains. However, monomers 1 and 2 chemical units away on the

same chain are excluded from

_forming

a crosslink.

_(b)

Random walk chains in which

_only

intramolecular

crosslinks are allowed.

percolated

and if so in how many directions. In this way we could determine the fraction of times

_{Ri (p )}

that the _system

_percolated

in at least one

_{( i}

=

1 ),

_two

( i

=

2)

or all 3 directions

(i

=

3).

For finite size

systems,

these 3

_{probabilities}

are not

_equal.

A

_typical

result for

R3 (p )

vs. _p for the

_equilibrated

melt and random walk

configuration

is shown in

_figure

1 for several values of M. Note that as M increases

_{R3 (p )}

becomes

_steeper.

In the limit

M --+ oo,

R3 (p )

should become a

_step

function. From this data we

_get

our first indication that

the

_systems

sizes for our

equilibrated

melt data are too small. The curves in

_figure

la shift to

the

_right

as p increases but do not cross as M increases. One would _expect that as one

approaches

the critical

_region

the curves for should become

_steeper

and

_eventually

cross as has been observed for lattice

percolation [23]

and found here for the random walk

configurations,

figure

1 b.

The

_percolation

threshold systems of size L can then be located from data like that in

_figure

1

_by

the fixed

_point

of the

_{equation p}

=

[16].

Because it is easier for finite

systems to

_percolate

in one direction

compared

to all

three,

for finite L. However these three must all be

_equal

as L - oo. The

_{exponent v}

which is defined

_by

the

_dependence

of the correlation

_{length, 03BE ~}

_{(Pc - p)-}

v,

also describes

_{how pci approaches}

Pc as _{L - cc,}

(p~ - p ~Z ) ~ L - i l v

[ 16].

Thus to determine both pc and v we need

_{only plot}

Pci versus L - 1 v or M-

1 / 3 v .

The data should fit on a

_straight

line and

extrapolate

to the same

point

for i =

1,

2 and 3. This is done in

figures

2-4 for several cases.

_Fortunately

the classical

and d = 3

percolation

values of v are

_sufficiently

different that it is not difficult to

_distinguish

the two cases. We have also checked that an alternative criterion for

determining

namely

the value

_{of p}

at which =

1/2 gives

identical results within our error bars

for pc in the limit L - oo

_though

for finite

_L,

the

_Pel’s

differ as one would _expect.

In

_figure

2,

we _present our results for _pci vs.

A4,~- 1/3 "

for an

equilibrated

melt of

chain-length

N = 25 with v =

1/2

and 0.88. As can be

_clearly

seen from this

figure

the results are

inconclusive because of the limited

_sample

sizes available. This should not be too

_surprising

when one realizes that the mean end-to-end distance for N = 25 is R = 6.2 ~

compared

to the

sample

size L = 38.9 for M = 2 000 _or

effectively

only

about 6 across. Similar data for

Fig.

2. -

Pc2 and Pc3 versus the

equilibrated

melt for N = 25 and M = 400, 800 and 2 000.

(a)

v =

1/2,

its classical value

(b) v

= 0.88, its d - 3 lattice

percolation

value. The _system sizes were too

small to determine P c1

accurately.

Fig.

3. _{and p~3} versus for the random walk

_{configurations}

for N = 25 and M in the range M = 300-8 000.

Fig.

4. -

P,,2 and Pc3 versus MI/3 v for the random walk

_{configurations}

for N = ₁₀₀

(0) and

N = 200

(0)

and M in the

range M = 300-4 000 with v =

1/2.

undertook a

study

using

overlapping

random walks. Results for _Pci for N = 25 and

300 ~ M _ 8 000 are shown in

_figure

3. Here we see _very

_clearly

that the data are fit much

better

_by

the classical value of v =

1/2

than 0.88 over the range of M studied. The curves in

figure

3b have considerable curvature and do not

_approach

a

_straigth

line.

_{Extrapolating}

the data to

_0,

we find _pc = 0.735 _± 0.01 for N = 25. We _expect that for

very

large

M,

critical fluctuations must become

_important

and there should be a crossover to non-classical

exponents. However we see no evidence for such a crossover even for N = 25 which indicates

the unknown

pre-factor

on the

_right

hand size of

_{equation (3)}

must be

_quite

small due to the

flexibility

of our chains. Because our

_largest

_systems are

_reasonably

_large,

our estimate for the

extrapolated

value of _pc does not

_depend

_critically

on the value of v. In

_figure

4,

we _present

data for _Pei vs.

M- 1/3 P

for N = 100 and 200 for v =

1/2.

Extrapolating

M- 1/3 P to

0,

we find

p, = 0.64 _± 0.01 and 0.062

± 0.01,

respectively.

To test

_{Flory’s prediction}

_{that pc}

=

1 /2

for

large

N,

we fitted our data for N =

25,

100 and

200 to the

_simple

form,

From this

fit,

we find that pc = 0.60 ± 0.01 for N ~ oo for the case in which there are no intramolecular crosslinks. This

gives

a lower bound to the

_{experimental percolation}

threshold. Thus

_considering

that

_Flory’s

estimate assumes that there are no wasted

crosslinks,

it works rather

well,

underestimating

our result

_{by only}

20 %. While this result is for the random walk

configurations

in which

overlap

is

allowed,

it is

_probably

also a _very

_good

estimate for the

equilibrated

melt. We

conclude this from our results for smaller systems where the

_equilibrated

melts and random walks with

_only

intermolecular crosslinks

_give

the same results for

_{Pei(M) ( i}

=

l, 2 )

within

our error bars. While this

_{correspondence}

_may deviate

_slightly

for

_larger

_systems, it _suggests

as would be

_expected

based on

Flory’s analysis

_{that p,}

for the two cases should

_{approximately}

When one allows intramolecular

_crosslinks,

the value

_{of pc}

is more sensitive to the details of

the model. If the local environment around each monomer is rather uniform then we do not

have to _repeat the entire calculation to determine _p~, all we would need to do is to examine the intermediate

neighborhood

(within

a distance

_r)

to determine the fraction of

neighbors

which are on the same chain and which are not. However if the fluctuations in the numbers of intra- and intermolecular monomers are

_large

then a full calculation is necessary. For the

equilibrated

melt for

large

N where end effects are

_small,

within a

_{distance rx}

= 1.3 a each

monomer has 4.0 ± 1.7

_neighbors

on other chains.

_(Here

the fluctuations

_give

the width of the

distribution and not the error in the determination of the

_mean).

The number of monomers

which are on the same chain at least 1 chemical unit _away is

_{quite large,}

1.7 ±

1.5,

while the

number more than 2 chemical units _apart is somewhat

_smaller,

1.I::t 1.3.

_Increasing

r x will reduce the size of the fluctuations. If we assume that in this case the fluctuations are not too

_large,

we can estimate pc

_by

_{statistically}

_counting

the number of intra- and intermolecular

links which would be

_present

for a random

_placement

of crosslinks. In the two cases discussed

here if we

_neglect

fluctuations,

we find that _pc is

_{approximately}

0.86 or 0.77

depending

on

whether we allow second

neighbors along

the chains to be crosslinked or not. For the random walk case, the fluctuations are much

larger

for the number of

_neighbors

on another

chain,

4.9 ± 4.6 due to increase in the size of the

_density

fluctuations. However because we

constructed our random walks to have the same statistics as in

_melt,

the number of

_neighbors

on the same chain is _very similar. For the random walk case, in the two cases

discussed,

the

number of

_neighbors

on the same chain are 1.7 ± 1.9 and 1.3 ±

1.9,

respectively.

If we

_neglect

fluctuations then

_{pc in}

the random walk model would be

_essentially

the same as for

equilibrated

melt. However this turns out not to be correct due to the

_importance

of fluctuations. For _system sizes where we can determine pci for the

equilibrated

melt and

compare to the random walk case, pci is

actually

a little

_larger,

about

_0.03,

for the random

walk model. For lower

densities,

where the fluctuations are

_expected

to be even more

important,

this difference should increase.

Fig.

5. -

Log-log plot

of

_{weight averaged}

_{polymerization}

index Z versus

(p -

for N = 25 for four

Fig. 6. _{- Log-log}

plot of sns

versus s for p =

Pc = 0.64 for N = 100. Here s is the number of chains in

each cluster. Results are an _average over 800

_{configurations}

for M = 4 000.

Now _{that p~} has been

determined,

it is

possible

to check other critical

_exponents.

As

discussed above the

_{weight averaged}

_{polymerization}

index Z is a

_good

quantity

to measure. It

can be determined from ns,

where

_following

Stauffer et al.

_[16],

the sum is over all clusters in the _system

_except

for the

largest

one. The _{exponent y} is defined

_by

the relation

Results for Z vs.

_{(p - p ~)}

are shown in

figure

5. We see that in the

_scaling

_region

where the

data fall

_nearly

on _top of each

_other,

that _{y =} 1. A non-classical value _{y -} 1.74

_clearly

does

not describe the

data,

though

very close to _Pc for very

large

M,

we _expect the data to crossover

to this

_larger

value of y. However instead of

becoming

steeper

as would be

required

for

y - 1.74, the curves break _away and saturate to a finite value

_{at p~}

due to the finite size of the

system. Finite size

scaling

results for Z at p =

Pc will be discussed in the next section since it

can be related to the fractal dimension of the clusters.

Results for the cluster distribution function _sns at _pc are shown in

_figure

6 for

N = 100 and M = 4 000. The data fit a _power with -r - 2.34 ± 0.2 for the range of s accessible.

Since the classical and mean field

_predictions

are _very similar for these two _cases, this result

cannot be used to rule out the non-classical result T = 2.2 as we were able to do for Z. Here

again

however,

we _expect that for much

_larger

values of s, the non-classical exponent should

apply.

4. Fractal dimension.

There are a number of ways to determine the fractal dimension of the

_percolating

cluster. However since

_hyperscaling

is not valid in the mean field

_regime,

one must be careful not to

d f

== d - f3

/ v

are

_only

valid in the critical

_regime

for d = 6. One

_simple

_way to determine the

fractal dimension is to measure the

_density

correlation p

(r)

for

large

clusters in the

vicinity

of

pc. In the

scaling regime

p (r) - r df- d

.

Equivalently,

one can measure the static structure

factor

_for

each cluster

S (q),

since

_{S(q) - q -}

the

_scaling

_regime.

Both measure the internal

structure of the cluster. Because our

polymer

clusters are made of random walk

chains,

which

themselves have

_{df = 2,}

one

_expects

that

_{S(q )}

will contain at least four

regimes.

For

q 2 7T

where Rc

is the radius of the

cluster,

S(q ) approaches

N,

the size of the cluster.

For 2 _q 2

_{7T IR,}

_{S(q) -}

_q

df while for 2 7T

_/R

-- _q -- 2 7T _(T,

_{S(g ) ~}

q - 2.

This latter

regime

is due to the internal structure of each individual chain.

Finally for q >

2 7T

_{l a,}

one is

sampling

distance scales shorter than the bond

length

and

_{S(q )}

is outside of the

_scaling

regime.

Thus to determine

_df,

we must be in the limit where

_Rc

is

_large

_compared

to

R ~ N

_~~2

1 yet N is still

large

enough

to be in the mean field

regime.

For the

systems

sizes

presently

accessible to us, these two constraints can best be satisfied for a

_system

containing

M = 12 000 chains of

length

N = 25. To calculate

S(q )

after

crosslinking

the

system,

we

identified the

_largest

cluster and unfolded it until it was a continuous

object

in _space without

regard

to the

periodic boundary

conditions. For

_highly

crosslinked

clusters,

this cannot be done without

_breaking

some of the

_crosslinks,

since the

_object

is multiconnected

_through

the

boundary

conditions. However since we are below p~, this was not a

_problem

_except for 3

configurations

at p

= 0.70 in which one crosslink had to be removed in order to _generate the unfolded

spanning

cluster. The results for

S(q )

shown in

_{figure 7}

were

_averaged

over 50

clusters and 20 random

_q’s

for each

_{I q I.}

For

_large

_{q, q3 S(q) -- q as}

_expected

from the random walk chains.

_{For q}

2 7T

_/R,

we

_clearly

see a

regime

where df

= 3 in _agreement with

Ray

and Klein’s result

[7]

for

long

range bond

percolation.

For

longer

chains,

say N =

100,

the

largest

system

we can

_study

at _present is M = 4 000. In this _case the

scaling

regime

where

S(q) -

q df

is reduced

considerably

in

_{size, making}

it difficult to determine

_df.

A second measure

_{of df}

can be obtained from the finite size

scaling

of the

_largest

cluster at

p~

[16],

M, -

L

df --

ifrl3.

Here

Mc

is the number of chains in the

largest

cluster. This relation

also enters into the finite size

scaling

of Z at _pc. At _{p~, the summation in}

_{equation (9)}

is cutoff

Fig. 7. -

Fig.

8. -

Log-log plot

of the

_largest

cluster

_Mc

and

_{weight average}

molecular

_weight

Z versus M at

p = Pc for N = 25 and 100.

by

the

_largest

cluster in the

_{sample. Substituting}

the mean field result

equation

(9)

and

_converting

the sum to an

_integral

with an _upper cutoff of we find that

In

_figure

_8,

we

_plot

our results for

_M,

and

Z(Pc)

versus M on a

_log-log

_plot.

For

Z(Pc)’

we find that the

_slope

of the line is

_{approximately}

_0.5,

_indicating

that df

= 3 consistent with the results for

_{,S(q ).}

However results for the

largest

sized cluster at _Pc

_give

a value for

df -

2.4,

somewhat smaller than

_expected.

This is

_{presumably only}

a crossover effect due to

the

_relatively

small systems which we are able to

_study.

_Apparently

_{Z(P,) is}

not as

_strongly

effected since it is a

_{weighted averaged}

over all clusters.

5. Conclusions.

In this paper we have described the results of our simulations of the critical behavior of

randomly

crosslinked

_polymer

melts. As

_expected

the volume fraction of crosslinks at the

transition decreased as

_{1/ N}

_[1].

However we found that the

_pre-factor

is somewhat

_larger

than the

_Flory’s

estimate due to _{the presence of} « wasted » links which do not increase the cluster size. In the case that we

_only

include intermolecular

_{interactions,}

we find that the

number of _{crosslinks per} chain at the

_{percolation/gelation}

threshold _p, = 0.60 _± 0.01

compared

to

_{Flory’s original}

estimate of

_1/2.

Inclusion of intercluster

_crosslinks,

which of

course are

_always

_present

_{experimentally,}

increases this number

_by

an additional 20 %

though

this result

_depends

somewhat on the local details of the model. In

_addition,

we also verified de

Gennes’

_{prediction [5]}

that the size of the critical

_regime

is _{very small} for a melt of

_long

chains

and that the critical _exponents should take on their classical values. This reduction in the size

of the critical

_region

is in _agreement with

_Ray

and Klein

_[7]

who studied a

_closely

related

model,

long

range bond

percolation.

While some of the _present studies were carried out

_using

persistence length

as our melt chains were distributed

_randomly

at the same

_density

without

regard

to

_overlap.

This is done in order so that we could

_study

_very

_large

_systems which were

needed in order to be able to

_perform

finite size

scaling

analysis

for Pc and the critical

exponents.

Due to the _very slow relaxation times of

polymer

melts,

it is not

_possible

at the

present

time to

_equilibrate

melts of more than 20 000 monomers for

_long

chains.

In this _paper we have concentrated on the critical

_properties

of

polymer

networks.

However this is

only

one _aspect of the

interesting problem

of

polymer

networks and rubber.

Most rubbers of

_practical

_importance

are

_{highly crosslinked, with p}

far above _{pc. In a}

_separate

study [ 17],

we have measured _many of the structural

_properties

of a crosslinked

_polymer

melt

starting

from our

_equilibrated

melt

_{configurations.}

This

_quantities

included the distribution of

dangling

ends,

distance between crosslinks and number of crosslinks _per chain. Future work

now in _{progress will} address the issues of network

_dynamics

and deformation and

swelling.

Acknowledgments.

We thank D.

_Stauffer,

K. Binder and W. Klein for

helpful

discussions. GSG wishes to thank

the IFF-KFA for their

_{hospitality during}

his visits while most of this work was done. We

would also like to

_acknowledge

_support

from NATO travel

_grant

_86/680.

~

References

[1]

FLORY P.,

Principles

of

_{Polymer Chemistry (Cornell University, Ithaca)}

1953.

[2]

FLORY _P., J. Am. Chem. Soc. 63

₍₁₉₄₁₎

3083, 3091.

[3]

STOCKMAYER W., J. Chem.

_Phys.

11

₍₁₉₄₃₎

45 ;

GORDON M., SCANTELBURY G., Trans.

_Farady

Soc. 60

₍₁₉₆₄₎

604.

[4]

MILLER D. _R., MACOSKO C. _W., J.

_Polym.

Sci. B :

_{Poly. Phys.}

25

₍₁₉₈₇₎

_2441, 26

₍₁₉₈₈₎

1.

[5]

DE GENNES P. _G., J.

_Phys.

Lett. France 38

₍₁₉₇₇₎

L-355.

[6]

FORTUIN C., KASTELEJN P.,

Physica

57

₍₁₉₇₂₎

536.

[7]

RAY T. S., KLEIN W., J. Stat.

_Phys.

53 ₍₁₉₈₈₎ 773.

[8]

STAUFFER _D., J. Chem. Soc.

_Faraday

Trans. 2 72

₍₁₉₇₆₎

1354.

[9]

DE GENNES P. G., J.

_Phys.

Lett. France 37

₍₁₉₇₆₎

L-1.

[10]

CARMESIN L, KREMER K., J.

_Phys.

France 51

₍₁₉₉₀₎

915.

[11]

DAOUD _M., J.

_Phys.

Lett. France 40

₍₁₉₇₉₎

L-201.

[12] KREMER _K., GREST G. S., CARMESIN _I.,

_Phys.

Rev. Lett. 61

₍₁₉₈₈₎

_{566 ;} KREMER K., GREST G. _S., J. Chem.

_{Phys. (1990)}

_92, 5057.

[13]

KREMER K., GREST _{G. S., Computer} Simulations of

_Polymers,

Ed. R. J. Roe

_{(Prentice-Hall,}

Englewood

Cliffs, NJ,

1990) p. 167.

[14]

EDWARDS S. F., Proc.

_Phys.

Soc. 92

₍₁₉₆₇₎

9.

[15]

DE GENNES P. _G., J. Chem.

_Phys.

55

₍₁₉₇₁₎

572.

[16]

STAUFFER D., CONIGLIO _A., ADAM _M.,

_Polymer

Networks, Ed. K. Du0161ek, Adv.

_Polymer

Sci.

(Springer-Verlag, Berlin)

44

_{(1982) p. 103.}

[17]

GREST G. S., KREMER K., Macromolecules

_{(in press,}

_1990).

[18]

GREST G. S., KREMER K.,

Phys.

Rev. A 33

₍₁₉₈₆₎

3628.

[19]

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

Dynamics

of

_{Polymeric Liquids}

_(Wiley,

New

York)

Vol. 1

_(1977).

[20] GEAR C. _W., Numerical Initial Value Problems in

_Ordinary

Differential

_Equations

_{(Prentice-Hall,}

Englewood

Cliffs, NJ) 1971.