Relaxation of entangled polymers at the classical gel point

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Submitted on 1 Jan 1990

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Relaxation of entangled polymers at the classical gel

point

M. Rubinstein, S. Zurek, T.C.B. Mcleish, R.C. Ball

To cite this version:

Relaxation of

_{entangled polymers}

at the

classical

_{gel point}

M. Rubinstein

_(1),

S. Zurek

_(*),

T. C. B. McLeish

_{(2, *)}

and R. C. Ball

₍₂₎

(1)

Corporate

Research Laboratories, Eastman Kodak

Company,

Rochester, NY 14650-2115,

U.S.A.

(2)

Cavendish

_{Laboratory, Madingley}

Road,

Cambridge,

CB3 _0HE, G.B.

(Reçu

le 6 novembre 1989,

accepté

le 18 décembre

₁₉₈₉₎

Résumé. 2014

Nous considérons la relaxation d’un

système

de chaînes enchevêtrées et

_pontées

au

voisinage

de la transition

_sol-gel,

dans le cadre de la théorie de

champ-moyen

(Flory

et

Stockmayer).

Le calcul utilise le modèle de « tube » des enchevêtrements

_{topologiques :}

la

contrainte diminue à cause de la fluctuation

_{hiérarchique}

des chaînes entre les ponts. Le _temps de relaxation est _{calculé par} une relation _ayant une solution

_analytique

au

_voisinage

de la transition. Une conclusion surprenante

s’impose :

tous les

_polymères

relaxent avant le _temps fini

T~

et donnent une courbe de relaxation

_G(t)

=

G0

03B3-2[a-1 ln (T~/t)]4,

où 03B1 dénombre les

enchevêtrements entre les _ponts, et 03B3 est une constante de l’ordre de l’unité. Pour des _temps

beaucoup plus

courts _que

_T~,

on retrouve une loi d’échelle faible. La relaxation

_rapide

des

polymères

les

_{plus grands}

_(qui

_{n’a pas de} sens

_physique)

est

_gênée

_par une transition vers la

statistique

de la

_percolation.

Les deux domaines sont

_séparés

_par un _temps de l’ordre de

T~.

Abstract. 2014

We examine the relaxation behaviour of an

_entangled

cross-linked

_{polymer gel}

as it

approaches

the

_{gel point}

in mean field

_{(Flory-Stockmayer)}

_percolation.

The calculation is based

on a tube model for the

_topological

interactions in which stress is lost via hierarchical fluctuation of the

_{primitive paths}

between cross-links. The

_decay

time of a _segment is calculated via a

recursion relation which has an

_analytic

solution near the

_{gel point.}

The

startling

conclusion is that all clusters relax in a finite time

_{T~ giving}

a relaxation modulus

_{G (t )}

=

Go

03B3-2[03B1-1 ln

(T~/t)]4,

where 03B1 counts the number of

_{entanglements}

between cross-links and 03B3 is a constant of order

unity.

For timescales much shorter than

_T~

_{this may resemble} a weak power law. The

unphysically rapid

relaxation of the

_largest

clusters is

_{prevented by}

a transition to

_percolation

statistics at

_long

_length

scales. The timescale

_separating

the two

_regimes

is close to

T~.

Classification

Physics

Abstracts 47.50 - 61.25 - 46.30

Introduction.

The

_{dynamical properties}

of

_{polymer gelation}

have

_enjoyed

considerable recent interest both

experimentally

and

_{theoretically}

_[1-7].

Near the

_gel

_point,

at which an infinite cluster first appears, static

properties

of the

_system

take on a

_scaling

form. The molecular

weight

(*)

Permanent address :

Dept.

of

_{Physics, University}

of Sheffield, Sheffield, S3 7RH, G.B.

distribution as

_specified

_by

the number

_density

of clusters of mass

_M,

n

(M),

is a

_power-law

[8-10] :

where

_f

is a cut-off

function,

constant for molecular

_weights

between

_M.,,

which is a low-mass

cut-off,

and

_Mchar’

The latter is the molecular

_weight

of the

_largest

cluster

_present

and

_diverges

at the

_{gel point :}

Here p

is the number

_density

of bonds in the

_system,

or an extent of

reaction,

and Pc is the critical

density

at

percolation.

The zero shear rate

_viscosity

_{qo and recoverable}

compliance

J2

diverge

at _{pc and the} zero shear rate modulus

_Go

_appears

_beyond

it :

The

_{frequency-dependent}

modulus also exhibits

_scaling

behaviour at the

_{gel point}

_{[3-7] :}

There is no direct way of

_determining

the

_dynamic

_{exponents s, t}

and u from the static ones

without further

_assumptions

about the nature of the

_dynamic

interaction

_[2,

_6,

_7].

One of the most

_important

_quantities

_affecting

the

_dynamics

of

_gelation

clusters is their

degree

of

_overlap.

Even without the

_complication

of

_topological

interactions between clusters the

_overlap

determines whether the

_hydrodynamic

interaction between monomers is screened or _{not ;} i.e. whether the

_dynamical

behaviour is Rouse-like or Zimm-like

_[10].

This is seen in solutions of linear

_polymers.

The motion of monomers on

_really

dilute chains is

_strongly

coupled

via the backflow of solvent to distant monomers on the same chain. When chains

begin

to

_overlap,

_however,

_hydrodynamic

contributions from other chains in the

neighbour-hood interfere and

_conspire

to screen the self interaction on

_length

scales

_longer

than the

correlation

_length

for

_density

fluctuations. This «

_{screening length}

» is therefore

dependent

on

concentration. It divides the

_dynamical

modes of a

_polymer

chain into those of

_{large length}

scales which see local friction

_(Rouse-like)

and those of

_{small length}

scales which are

non-local

_(Zimm-like).

Strong

overlap

will

_additionally

mean that

_{entanglements}

_may

_dominate,

_though

these

become

_important

at rather

_higher

concentrations than the onset of

_hydrodynamic

_screening.

This is seen in melts of linear chains where there is a critical molecular

_weight

(which

is

_usually

several

_thousand)

above which

_dynamical

behaviour

_diverges

from the

_{phantom-chain}

predictions. Entanglements

are also affected

_by

_{polydispersity}

so that in a melt of

_polymer

bimodally

distributed

_(say

that a few

_long

chains are

_present

in a bath of much shorter

_ones)

for

_example,

it is

_possible

for the

_long

chains to be

_effectively

_{unentangled (at}

timescales characteristic of their orientational relaxation

_{they only entangle}

with other

_long

_chains)

even

though

the short chains are well

_entangled.

The richness of this

_problem

has been

_investigated

by

Doi et al.

_[11].

For

_gelation

clusters we therefore

_expect

_{entanglements}

to be

_{important only}

when a

of the

_degree

of

_overlap

for a cluster of molecular

_weight

M was introduced

_by

Cates

_[12].

We count the number of

_pieces

of mass M from all

larger

clusters that sit within the volume

spanned by

a

_single

cluster of mass M. The radius of

_gyration

of a cluster scales as

R(M) ’" M1/df

where df

is the fractal dimension of the clusters. The

_degree

of

_overlap

4S (M)

[12]

is then

_{given by}

an

_integral

over the mass distribution

_larger

than M :

Here d the dimension of space. We notice that an entire distribution of clusters may exist at

marginal overlap

(e (M) - const. )

if the

_{exponents df and T obey}

the «

hyperscaling »

[8]

relation :

« Blobs » of mass M

just

_manage to _{fill space}

_{independently}

of the

_scale,

_M,

that we choose

-the

_system

is self-similar. This is satisfied

_by

3-dimensional

_percolation

clusters which have

d f ---

2.5 and T = 2.2

[1],

so are not

_{strongly overlapped.}

A more

_general

result due to Cates

[12]

is that branched

_polymers

with excluded volume which is screened at

_{large overlap}

will

obey

(1.6)

for a _{range of values of} T and

df.

This was used

_by

Rubinstein et al.

_[6]

to calculate the

_{dynamical scaling}

of

_{polymeric percolation}

clusters on the

_assumption

that

_marginal

overlap

was effective in

_{hydrodynamic screening}

but did not cause the clusters to

_entangle

strongly,

results which are confirmed

_{by experiment}

_[3,

_6,

_7].

Near the

_gel

_point

we find that the

_dynamic

modulus satisfies

_(1.4)

with u = 0.69

compared

with the theoretical value of 0.67.

A

_contrasting

case is found in the

_{cross-linking,}

or

_{vulcanisation,}

of dense

_polymer

chains when excluded volume is screened

_throughout

the distribution

_except

_{for the very}

_largest

clusters

_[13].

Now the classical

_theory

is valid and

_gives

the

_{exponents df}

= 4 and

T = 5/2. From

(1.5)

_{we see} that the the clusters _are

strongly overlapped

and that over some

range of timescales at

_least,

_{entangled dynamics}

must dominate. It becomes

_compelling

to ask what viscoelastic behaviour this new

_{assumption implies.}

The

_{viscoelasticity}

of

_entangled

linear

_polymers

is now understood

_using

the tube model of

topological

interactions

_[14]

and its extension to branched

_polymers

has been an active area of interest

_{recently. Applications}

have

_progressed

from treatments of star molecules

_[15]

without

accounting

for « constraint release » to models for more

_complex

molecules

_{[16, 17]}

which treat the

_topological

constraints

_{self-consistently}

_[18].

The main results for

_entangled

tree-like branched

_polymers

_(containing

no closed

_loops)

_{may be} summarised as follows :

1)

as for linear

_polymers,

stress

_(after

a

_step

strain)

comes from the

_anisotropy

of the

polymer

segments.

In fixed networks this is is

_proportional

to the fraction of

_originally

occupied

tube that has not been

_{passed by}

a free chain

end ;

2)

stress relaxes

_by

arm retraction. Most

segments

have an end whose chemical distance to the outside of the tree is shorter

_(by

the molecular

_weight

of the

_segment)

than the distance from the other end. The former fluctuates on a characteristic timescale _Tri in the

entropic

potential

for

_{path length}

that is

_{approximately quadratic}

about the mean

_length,

as for star

Fig.

1. - A

region

of the Bethe lattice on which we _{grow the clusters. Bond i has} a

_seniority

of 3, which

is the

_longest

chemical

_path

to the outside of the cluster of the tree shown in

_(b).

The other tree

_(c)

has a

longer

LCP of 4.

Here

_Mx

is the molecular

_weight

between the cross-links and

_Mei

is the

_entanglement

molecular

_{weight appropriate}

to the timescale at which the i-th

_segment

relaxes.

v is a constant of order one which may be calculated from the tube model’s

_prediction

for the

plateau

modulus of linear

_polymers

to be 15/8

_[11].

Those

_segments

whose ends share the

same chemical distance to the ouside of the tree are the last in their cluster to

relax,

and do so

by reptation

(see

later) ;

3)

relaxation is hierarchical. _{Ti +}

_{1 becomes}

the

_timestep

for the

_{path length}

fluctuations of the

_neigbouring

_segment

nearer in chemical distance to the centre of the tree ;

4)

the

_{large separation}

of timescales on which different

_parts

of a molecule relax means

that all relaxed

_segments

act as effective solvent for unrelaxed

_parts.

If

_Ci

is the concentration of unrelaxed

_segments

at the timescale on which

_segment

i relaxes then the effective dilution

means that the stress is

_proportional

to

C 2 rather

than

_{Ci [19].}

_{This may be} recast in terms of a

dynamic

value of the

_entanglement

molecular

_weight

when it becomes clear that the dilution has a

_dynamical

effect as

_well,

_accelerating

the relaxation of

_neighbouring

_segments

via

_(1.7

and

_1.8).

This acceleration can _{have very}

dramatic effects indeed even in the case of

_simple

star

_polymers

_[18].

_{In this paper} we

_apply

these ideas to

_{Flory-Stockmayer}

_gelation.

Statistics of

entangled paths

at

_gelation.

Consider a Bethe lattice with co-ordination number z

_(such

as shown in

_Fig.

la for

z =

3).

Each bond is

occupied

with

_probability

p and

empty

with

_{probability 1 - p}

independent

of any other bonds of the lattice. This defines the mean field

(Flory-Stockmayer)

percolation

model. It

_produces

an ensemble of branched

polymers

which may be embedded in

concentration is to be

_unity

as it must if we wish to model a

_melt).

We note that a certain

amount of linear

_polymer

is

_present

at _any

_stage

of reaction

_(or

_probability

_p).

In an accurate treatment this fraction must be handled

_separately

from the remainder because

_they

will relax their stress

_{by reptation}

rather than fluctuation.

However,

we

suspend

this consideration for

the moment

_(but

see the discussion

_below),

_{observing only}

that the fraction of linear chains becomes

_{exponentially}

small at

_high

molecular

_weight,

and consider the result of

_fluctuating

dynamics

alone on the model

_system.

We now define the

_concept

of

_seniority

of a bond. If the bond is

_empty,

we _{say that it has}

seniority

0. If it is

_occupied,

then it

_belongs

to a cluster - either finite

or infinite. Consider

bond i

_belonging

to a cluster as shown in

_figure

la. This bond divides the cluster into two trees with bond i as the trunk of each of them

(as

shown in

_Figs.lb

and

1c).

Let

m/l)

and

_mi(2)be

the

_longest

consecutive _sequence of

_occupied

bonds which we call the

_longest

chemical

_path

_(LCP)

in each of these trees

_(in

_Fig.

1

_mi(1)

= 3 and

mi(2)

=

4).

We say that bond

i is of

_seniority

This careful

_prescription

_{above is unnecessary when the}

_polymer

itself is a

_{regular Cayley}

tree

_[17]

_{in which every} branch

_{point i}

is connected to z - 1

_equivalent

branch

_points

with characteristic relaxation timescales _{Ti _} 1, and one with a timescale of Ti + 1. In a tree formed

by

random

_gelation,

_however,

the relaxation times of

_neighbouring

branch

_points

are

_typically

very different. To allow the branch

point

to _move, z - 1 arms must retrace

_completely

within their tubes

_{(Fig. 2),}

_{e.g. in the}

_Cayley

tree

_polymer

we

_require

consecutive

_complete

Fig.

2. - Branch

point

i is mobilised

_by

the

_sequential

retraction of z - 1 arms. In this case

fluctuation of z - 1 arms. In the case where the fluctuation rates of these arms are _very

different,

the

_waiting

time for the

_entanglement

loss of

_figure

2 is dominated

_by

the slowest of the arms

_(excepting

the arm that remains

unrelaxed).

This is

because,

_given

an event of

waiting

time T

the probability

of an

_{(uncorrelated)}

event of

_waiting

time _T2

_occurring

before another of the first kind is

p

(event

2 before event

_{1 ) =}

this tends to

_unity

when

_{Ti >}

r2 which must

_typically

be the case when the

_waiting

times are

exponentially

separated

as in

_(1.7).

Henceforth we assume that the relaxation of a bond is

dominated

_by

the

_neighbouring

tree with the fastest relaxation time. This will be the

_longest

of the relaxation times of the first

_hierarchy

of

_segments

in the tree. This

_segment

in turn will be dominated

_by

the relaxation of the faster of its two

_neighbouring

trees

_(the

slower

_begins

with the the first bond we

_considered)

and so on

_following

the LCP for bond i out to the

_edge

of the cluster. So the

_seniority

of a bond determines its

_dynamics.

The

_seniority

of a bond can

be infinite if it is

_part

of the backbone of the infinite cluster. Bonds of infinite

_seniority

do not relax

by

retraction. Bonds that are not in the backbone of the infinite cluster may relax

by

retraction when their turn comes. Perimeter bonds have

_seniority

one.

_They

are the first to relax in an

_entangled

melt of clusters. Their

_neighbours

which are not

_perimeter

bonds themselves have

_seniority

two and are next to relax after the

_perimeter

bonds,

and so on until either the backbone of the infinite cluster is reached or the

_topological

constraints become ineffective

_{(see below).}

We now need to _{calculate the concentration cm of bonds of}

_seniority

m. Let us first define

two

_{auxiliary quantities :}

_{qm is the}

_probability

that a

_randomly

chosen bond is the base of a

tree with the LCP

_equal

to m and

_Qm

is the

_probability

that a

_randomly

chosen bond is the

base of a tree with the LCP not

_larger

than m so

We can _{calculate qm}

_recursively

in terms of the

_Qm

_noting

that the chosen bond must be

occupied

(with

probability

p)

and that at least one of the z - 1 branches must have a LCP

equal

to m - 1

_(we

exclude the case where

_they

all have branches of LCP of m - 2 or

smaller) :

using

the fact _{that qm} =

Q m - Q m - 1.

These _recurrence relation

together

with the initial values

This

_probability

is nonzero

_only

above the

_percolation

threshold,

i.e.

_{for p}

_{> Pc where}

For

_large

m the finite difference

_{equation (2.4)}

can be

_{approximated by}

a nonlinear second order differential

_equation

in terms of a continuous function

_{Q (m ) :}

(From

henceforth we will use

_subscripted

variables to denote the

_quantities

with discrete

arguments

and standard functional notation to denote their continuous

_{approximants).}

As

m becomes

_large,

we can find the

_asymptotic

behaviour of

_Q(m)

_{by writing}

with

when to second order in

_Q(m),

Away

from the

_{gel point}

the first term dominates

_{giving exponential decay}

of

q(m) =

dQ /dm.

At the

_{gel point,}

however

_{(p = pc = 1 / (z - 1 »}

this

_leading

order term is

identically

zero and the next term becomes dominant. This has a

_power-law

solution and we

have to first order in

_{1 /m}

and for the

_leading

behaviour of

_q(m)

This can be checked

_against

exact evaluation of

_(2.4)

and

_(2.5).

We find that the error

becomes very small for values of m

larger

than

_10,

and that

_asymptotic

forms are reached sooner for

_large

values of

_{functionality}

z.

We are now in a

_position

to calculate the lattice concentration

_{cL (m )}

of bonds with

_seniority

m in the continuous

_{approximation.}

It is

_proportional

to the

_probability

that one of its two trees has the LCP

_equal

to m, while the other has a LCP

_equal

to or

_larger

than m. So the exact discrete result is

with the continuous

_{approximation :}

(once

in each

_tree).

The result

_gives

the lattice concentration of clusters. In the melt the total concentration

_{(volume fraction)}

of all clusters is

_{unity independent}

of the

_degree

of

percolation

so for the

_real-space

concentration we divide

_by

another factor

_{ofp(= l/(z -1)}

at

_{percolation).}

The

_leading

behaviour of

_{c (m )}

for

large m

is determined from

_{(2.11), (2.12)}

and

_(2.13)

to be

We now

_apply

this to the hierarchical relaxation of the ensemble of branched

_polymers

at the

gel point.

Relaxation

_dynamics.

The characteristic times for the relaxation of stress from bonds of

_seniority

m is

_given

now

quite directly by

(1.7)

and

_(1.8)

with the effective concentration C

_(m)

_{given by}

the fraction of bonds of

_seniority

_greater

than m :

and in the continuous

_{approximation,}

where this is a definition of the

_{0 (1)}

_prefactor

_{y (z).}

Between the two characteristic timescales

_{delimiting complete}

relaxation of one

_bond,

stress relaxation

_proceeds

in

_exactly

the same _way as for a

_single

star arm, and follows the function

_{g (t)}

calculated

_by

Pearson and Helfand

_[15].

The final

_expression

for the

_dynamic

_modulus,

where

_Go

is the

_{high-frequency}

plateau

modulus is

(Note

that this is the correct

_expression,

not the one

_{given by}

McLeish in

_[17],

which overcounts contributions from

_higher

_{generations).}

We need

_{only apply}

the

_dynamical

rules summarised in the first section to calculate the series of timescales Tm. We can extract the essential behaviour at the

_gel

_{point by taking}

the continuum limit of the discrete series

Tm

just

as we did for the statistics of bond

_seniority.

This was done for a melt of stars

_by

Ball and McLeish

_[16]

who showed that

_Me

needs to be allowed to _vary over the relaxation of the arms as the effective concentration

_changes.

In our case, we have

where

The

_{dynamical equation}

for T

(m )

can be

_thought

of as a series of _{activated processes in which} the rate

T - 1 (m + dm)

is

_{given by}

an

_attempt

_frequency

T - 1 (m)

and an

_entropic

barrier

energy dU =

kTvMx/Me(m)

dm

(this

is exact for dm =

1).

The

pre-exponential

factor

gives

only

small corrections to the

result,

as _{may be checked}

_by

_comparing

the

_analytic

solution of

(3.3b)

with evaluation of

_(3.3a).

We have to use the current effective value for

M,

at each level in the

_hierarchy

so that all relaxed bonds act as solvent for unrelaxed

_bonds,

while all unrelaxed bonds are effective in

_creating

the

_entanglement

network. This

_requires

the continuous differential

_equation

for

_{r (m)}

to be of the form of

_(3.3b)

rather than

d(log T) = d(U(m)/kT)

in which the function

_Me(m)

is differentiated. For

Flory-Stockmayer gelation

(3.3)

becomes

so that

Here a =

v M xl M eO

and we have

_arranged

the units of time so that

_{T (1 )}

= 1. The

«

accelerating »

effect of the

dynamic

dilution is very

_{strong :}

we recover a finite time T

(1 ) exp {a

_y

(z)}

in the limit m - oo which we call

_{T 00.}

This is

_{exponentially longer}

than the

time to relax the

_perimeter

_{bonds. We may} now calculate the relaxation modulus under the

assumption

that bonds of a

given seniority

relax

_{independently,}

but that the

_density

is

_high

enough

to

_permet

a well-defined

_pre-averaged

effective concentration :

This is a universal function of

_{a -1 log}

_{(T oo/t)}

_up to the

_{functionality-dependent prefactor,}

as

may be seen from

_(3.6).

Just as in the case of

_simple

stars, however for times shorter than

T 00

the

_exponential

term in

_(3.7)

becomes

_effectively

a

_step

function in m and the relaxation

modulus becomes from

_(3.1)

and

_{(3.6) :}

A

_graph

of

_{[G (t) ]1/4}

_{against log t will give straight}

line for timescales dominated

_by

entanglements.

For times t «

_{T 00}

the bracket in

_(3.8)

can be

_expanded

to

_give

a

_regime

over

which

_{G (t )}

has an

_{approximate power-law}

form t-" :

In const with

though

the full function

_{drops steeply}

_{away from any}

_{power-law approximation}

at times

AN EXACT RESULT. - We noted that the

asymptotic

forms for the concentrations

q (m )

and

_{c (m )}

become better

_{approximations}

at _{any finite} m as we increase the

functionality

z. It is therefore no small

_surprise

that

_the

final form for the relaxation modulus

(3.8)

becomes

exact for all

_integer

values of m in the case of z =

3,

the lowest

possible functionality,

provided

we choose a renormalised value for

_Too.

This is

_fairly

_{straightforward}

to

demonstrate,

and is a _{consequence of the}

_simple

form of the recursion relation

_(2.4)

in this

case. The derivation is

_given

in

_appendix

A. The distinction is that

_TOC)

is renormalised to

t (1) exp {3 a}

rather than

_{t (1) exp {16 a }.}

Such a

_change

is to be

_expected

from the observation that the

_asymptotic

calculation overestimates

_C(m)

for small values of

m

_{(Eq. (3.1)).}

An exact solution must

_produce

the faster relaxation that results from

_greater

dilution. The result for z = 3 is useful because it

gives

a bound on the renormalisation of

TOC)

(the

asymptotic

results become more accurate for

_large

_z)

and because it indicates that the form of the relaxation

_spectrum

at low

_seniority

is not

_{strongly dependent}

on

_{functionality}

at

experimentally

relevant values. The

_prediction

of a linear

_dependence

of

_{[G (t) ]1/4}

on

log

(t )

is not

_changed.

Discussion.

The model described above is

_idealised,

and

_requires

careful

_{qualification}

if it is to be

_applied

to

_experiment.

We saw earlier that there are still

_{approximations}

in our treatment _{which may} be

_important,

but even as it stands we know that at

_long

times our

_picture

of a

_highly

entangled

network must fail. We consider this

_regime

first.

TRANSITION AT LONG TIMES. - The existence of

a finite time for the relaxation of

indefinitely

large

finite clusters is

_unphysical,

and we

_expect

the model to break down at times

_(and

corresponding length

scales)

shorter than

_Too

There are a

_priori

two routes

_by

which the model may fail : on the one hand the

_assumption

of classical statistics for the static distribution of the clusters is wrong at

_{large length}

scales because clusters

_{with df}

= 4

eventually pack

in d = 3

[13] ;

on the other we know that

_dynamic

dilution

_eventually

renders the

_largest

clusters

_effectively

_dilute,

after which

_{unentangled dynamics}

are

_appropriate.

In

practice,

whichever

_transition,

_dynamic

or

_static,

occurs first will define the crossover to the

new

_regime.

First we consider

_possible

_dynamic

transitions. At the level of the small modifications

introduced

_by

the term in

_log

_(Mx/Me)

we must

_expect

a

_change

in behaviour when the

effective

_entanglement

molecular

_weight

reaches

_M,,

_[17].

This will occur at a

_high

_seniority

m,,

given by

giving

This cannot affect the main features of the

_entangled

relaxation,

however,

because

segments

there is no

_topological

constraint from the tube

_[22].

We arrive at a

_seniority

for the

_dynamic

transition

_Mdy.

_{given by}

This

_seniority

varies as the square of mx, so in the limit of

_increasing

distance between branch

points, entangled dynamics

may

persist

well

_beyond

_mx.

This

_{potential dynamic}

_{transition may} not be active if our model fails to

_obey

classical statistics at molecular

_{weights corresponding}

to smaller seniorities than

_mdyn.

We must check whether the static

_{configuration}

affects our

_assumptions

at an earlier level in the

_hierarchy.

At

large length

scales and cluster masses fluctuations in the number densities

destroy

the

applicability

of the mean-field

_results,

as

_pointed

out

_by

de Gennes

_[13].

_Beyond

this

_point

percolation

statistics

_hold,

which as we have seen leads to

_{only marginal overlap}

and

unentangled dynamics.

De Gennes casts the result in the form of an interval

_Op

around the

percolation

threshold inside which classical statistics break down and finds

where

_Mk

is the molecular

_weight

of a Kuhn

_length.

We _may use the

_relationship

between the

largest

mass of the distribution and crosslink

_{density Mchar -}

Mx(J1p /Pc)-l/u to

convert this into a critical cluster mass

_{Mstat -}

MX5/3 Mk 213

using

the classical value of u = 1/2.

Finally

_we

use the fact that bonds of

_{seniority m typically belong}

to a cluster of LCP - m and of mass

M - Mx m 2

(the

mass

_scaling

of clusters with

_{path length}

must be half the

_spatial

fractal dimension because the

_segments

are

Gaussian).

This

_produces

When

_Mx

becomes

_large,

this transition dominates because it

_gives

the smallest value for a

transition

_seniority.

In this case, the

dynamics

become « slaved » to the static distribution and must be

_unentangled

at times

_longer

than

_{T (mstat) (which}

can be calculated from

(3.6)).

Which

_type

of crossover wins in

_practice

_{may be affected}

_by

the rather different

_prefactors

to

Mx

in

_(4.3)

and

_(4.5).

The static transition dominates

_only

for

which may be

large

since

_polymer

chains are flexible well below the

entanglement

molecular

weight

in the melt.

However,

the differences of

_seniority

between these two

_possible

transitions

_(and

even the

seniority

mx)

are masked in an

_experiment

which measures relaxation as a function of time because all m » 1

_correspond

to times close to

_{T 00}

from

_(3.6).

The form of

_{G (t )}

observed is

expected

to follow

_(3.8)

for 0 t

_{T 00}

and then cross over to the

_{percolation power-law}

relaxation

_predicted

and observed

_by

Rubinstein et al.

_[6],

_{G(t) -- t-0.67.}

We

_present

a

calculation of

_{G (t )}

for a

_{well-entangled}

ensemble near the

_gel

_point

in

_{figure 3.}

This

illustrates three

_{regimes :}

the

entangled

for 0 t

_{T 00’}

the

_unentangled

for

_{T 00}

t

_{T max}

and the terminal

_regime

for t >

T max

where

T max

is the Rouse relaxation time of the

largest

Fig.

3. - The

_dynamic

modulus

_{G (t )}

of the

_entangled

ensemble

_just

below

_{gel point.}

The molecular

weight

between cross-links is such that

_{T 00}

_exp =

{60}.

The _{crossover to}

unentangled

dynamics

occurs

just

before

_{T 00 , Tmax, corresponding}

to the Rouse relaxation of the

_largest

cluster is

_{exp {70} .}

Also

shown

_superimposed

is the

_{resulting compliance}

_{J (t) (the}

_rising

_curve)

with a vertical shift factor to

allow use of the same axes.

FORM OF THE COMPLIANCE. - Because _{of the very}

_long

terminal times associated with

entangled

branched

_polymers,

_creep

_experiments

on the

_compliance

J(t)

will be more

accurate in

_assessing

the linear

_rheology

than constant-strain

_experiments

such as

_oscillatory

or

_{step-strain rheometry.}

The characteristic behaviour of the different relaxation

_regimes

discussed above is reflected in the

_compliance

_just

as

_strongly

as in the modulus. In the two

regimes

where

_{G (t )}

is

_{approximately}

a

_power-law

_{(0 - t T 00}

when

_{G(t) = t-4/ay,}

and t >

_{T 00}

when

_{G (t) = t - 0.67)}

_{J (t)}

also has a

_power-law

form. This may be seen from the

integral relationship

between

_{G (t )}

and

_{J (t) [20] :}

Taking Laplace

transforms and

_putting

_{G (t )}

=

Go t-U

gives

for 0 u 1 :

So we

_expect

to see

_{J(t )}

_following

the inverse power to

_{G (t ).}

Numerical evaluation of

J(t)

is

_superimposed

in

_{figure 3.}

Below the

_{gel point}

the

_viscosity

is finite

_(the

_largest

characteristic cluster has a terminal time of

_{T max}

of

_Fig.

₃₎

so that the eventual

_slope

of a

4/a y.

Between

_Too

and

_Tmax

the

_slope

is 0.67. The existence of these two

_regimes

of

entangled

and

_unentangled

relaxation divided

_by

a transition near

_{T 00}

is the main

experimental signature

that we

_predict.

EFFECT OF LINEAR AND LINEAR-LIKE SECTIONS. - The

_gelation

_process introduced in section 2 above does not _{exclude the presence of} a fraction of

purely

linear

polymer

at the

_gel

_point.

By

randomly occupying

bonds on the

_Cayley

tree we _may achieve linear sections

_by

_occupying

just

two of the z bonds at each node

_along

a

_simply

connected

_path

between nodes at which

only

one bond is

_occupied

(terminating

the

_chain).

Now when we

_embed

the ensemble into

real space, the fraction of

_segments

which

_belong

to linear chains will not relax their stress

_by

fluctuation,

but

_by

_reptation.

As the molecular

_weight

between cross-links

increases,

so does the ratio of the characteristic times for fluctuation and for

_reptation,

so for

_{well-entangled}

ensembles,

the main effect of these linear

_components

is to a

_provide

a _way to relax a fraction of the

_original

stress

_quickly,

and to renormalise the concentration for the branched

_polymers

whose relaxation follows the

_pattern

described

above,

but

_initially

diluted

_by

the fraction of linear material.

The fraction of linear material of molecular

_weight

_{kM. (consisting}

of k consecutive

_bonds)

we denote

_by

_{L (k, 0 ) (we}

_explain

the notation

_below)

The term _{in p} comes from the necessary

_occupied

_bonds,

in

_{1 - p}

from the

_unoccupied

ones

both

_along

the

_length

and at the ends of the

_chain,

and the term in

_{(z - 1 )}

is a combinatorial

factor

_arising

from the choice of

_paths

on the lattice. The total fraction of bonds in linear

material is

_given

_by

the sum of

_{L (k, 0)}

over k. This is

_dependent

of the

_{functionality}

z and is

_greatest

for small z. At the

gel

point

p = 1 / (z - 1 )

giving,

The total concentration of linear material is the ratio of this sum to the total fraction of

occupied

bonds

_(which

_{fill space in the}

_melt).

This is 1/2 for z = 3 _so at this

functionality

1/4 of

the total mass is in the form of linear chains and 1 -

_{[1 -}

_{(1/4 ) ]2}

= 7/16 of the total initial

stress is relaxed

_by

these linear

_segments.

The relaxation

_spectrum

of these

_reptating

chains will be

_{given by}

a sum over their

_exponential

molecular

_weight

distribution :

The

_reptation

time of k-chains is

_dependent

on their

_length

_cubed,

the well-known result from

linear melts

_[14],

but is also modified

_by

the effective concentration

C,

which is

_defined,

as for

It is clear that this dilution effect reduces the

_reptation

times

_by

9/16 over the

_reptation

régime.

The full relaxation modulus of

_{(4.11 )}

is

_complicated,

but its

_approximate

_{form may be} extracted

_by

_representing

the sum over molecular

_weights

as an

_integral

which is then evaluated

_by

_steepest

_descents,

_giving

a « stretched

_exponential

» :

After this extra short-time _{response, the} branched

_polymers

relax

_by

fluctuation.

_However,

polymers

which are not

_linear,

and

_begin

to relax their stress

_by

fluctuation,

may become

effectively

linear at

_long

times when

_only

_high

seniorities remain unrelaxed. An

_example

is shown in

_figure

_4,

which behaves as a diluted linear

_polymer

after bonds of

_seniority

1 have relaxed. The backbone bonds which have seniorities of two and three

_actually

relax via the

reptation

of the diluted backbone after

_{t (1 ),}

so

_giving

an error to the relaxation

_spectrum

calculated in section

_3,

which assumes that all relaxations

_proceed

via fluctuation. In

_general

we must calculate the number of

_objects

which become linear-like at some

_stage

of the

relaxation. We denote

_by

_{L (n, m )}

the lattice-concentration of

_objects

which

_beyond

a

seniority

m consist of

_just

a linear chain of n

_segments

_{(L (n, 0 )}

counts the

_simple

linear chains

as

above).

We

_find,

after a little

_thought,

that

The first term,

_{q;" + l’}

is the

_probability

that the two end bonds of the linear subsection are of

seniority

m + 1. Then we have the

_probability

that the

_{remaining n -}

2 bonds are

_occupied,

pn- 2multiplied

by

the number of _ways this can occur

_{(z -}

1 )n -1.

We

_multiply

this

_by

the

probability

that all

_{(z - 2) (n - 1)}

side branches have

_seniority

less than or

_equal

to

m,

QJ: - 2)(n -1).

We

should, however,

subtract the

_probability

that the linear subsection is

_part

of a

_longer

linear subsection that was able to

_reptate

at an earlier

_stage.

In this case

_just

one of

the z - 1 bonds at each end of the section must have

_seniority

m -

_giving

the factor

(z -

1)2 q 2,

and the

_remaining

z - 2 must all have seniorities less than m

_giving

the factor

Fig.

4. - The occurrence of linear-like sections at

_long

times. After bonds of

_seniority

one have _relaxed,

this cluster follows

_{reptative dynamics}

within the effective

_entanglement

network because no tree

Q2(,- 2

All n bonds of the subsection must be

_occupied

as

_before,

but now all the side branches must have seniorities

_strictly

less than m. This

_explains

the second term of

(4.14).

Beyond

seniority

m, these

_objects

relax

_{by reptation.}

We need an estimate of the error that

they

introduce into our calculation of the relaxation

_spectrum.

A

_good

measure of this is the

concentration

_{F (m )}

of

_segments

of nominal

_seniority

m that have their actual relaxation

speeded

up

by

belonging

to a backbone that becomes

_effectively

linear,

i. e. that relax

_by

reptation

at an earlier

_timestep.

We observe that

_{objects belonging}

to the fraction

L (n, m’ )

affect bonds with

_seniority

m such that m’ m m’ +

n /2.

The total fraction of

bonds of

_{seniority m}

affected is the sum over relevant linear-like

objects :

The case of z = 3 is

especially

instructive here because _we

_expect

it to

give

the

largest

effect of

linear-like sections

_(only

one _{side branch per node} need relax

_along

a

subsection),

and also

because it is

_possible

to calculate

_{F (m )}

_exactly

in terms of

_{C (m ) (see}

_appendix

_B).

We find that

which means that 1/4 of all relevant bonds are affected. This

_implies

an

_interesting

form of

dynamic

scale-invariance at the

_{gel point :}

the linear-like chains are

_significant

at all scales.

However,

the form of the relaxation modulus is still

_unchanged,

because the

_system

behaves

as if it were at the renormalised concentration

_{Ceff (m) = C (m) - F (m) = 3 C (m)14.}

Repeating

the calculation of

_appendix

A

_using

the effective concentrations

_yields

The

_only

difference

_is,

as before in the case of

_accounting

for

_{non-asymptotic}

_behaviour,

a

renormalisation of

_TOC)

which now has a value of

_t1

_exp

{9 a /4}.

A smaller size of error is

_expected

to arise from effective

_dangling

ends of

_{lengths longer}

than one

_segment.

_Imagine

the linear-like

object

of

_figure

4 connected to a

_large

cluster

_by

one of its ends. Its stress is still relaxed

_by

_fluctuation,

but now the

entropic potential

is that

for a

_dangling

arm four

_segments

_long

rather than one, and allows the intermediate bonds to relax faster than

_they

would in a

_strictly

consecutive

_hierarchy.

Such structures can blur the

time-orderedness of

_seniority,

but occur with a

_frequency

_given

_by

an

_expression

similar to

(4.14).

Moreover,

the

_{re-ordering only}

occurs within the hierarchies of the

_effectively

linear

dangling

end.

_{Large experimental}

deviations from the model

spectrum

are not

_expected

from this source.

AN ALTERNATIVE ENSEMBLE. - Instead of the mechanism of

_creating

clusters on a

_Cayley

tree and then

_embedding

_{in real space,} consider the

_following

_{procédure :}

We

_begin

with a

bath of

_monodisperse

linear flexible

_polymers

with reactive end _{groups of}

_{functionality}

z

_(that

_is,

with z - 1 spare

bonds).

We allow each end group to attach to z - 1 other chains

_(so

that its

_valancy

is

_saturated)

with

_probability

_{p, and} to be

_completely

unreacted with

probability

1 - p.

The reactions are

_independent

or

_mean-field,

and so lead to

Flory-Stockmayer

statistics,

when the chains start

_highly

_overlapped

_(loop

formation is

_suppressed).

If we consider the calculation of

_{probabilities}

of LCPs in trees

_starting

on a chosen

bond,

we

too,

except

that

_{q (0)}

now describes the fraction of linear sections in the final

_bath,

not the

empty

bonds on a

_Cayley

tree. Such a

_system

_{may arise}

_chemically,

and is a little « cleaner »

than the

_Cayley

tree model in that all

_long

linear chains are

_{suppressed, though}

we note that

long-time

linear-like material is still

_present.

BEHAVIOUR AWAY FROM GEL POINT.

_{- Finally}

we _may comment

_briefly

on the behaviour

away from the

gel point.

Below

_{gel point}

the terminal time

_Tmax

of the

_largest

_{cluster may be} less than

_{T 00}

_(rigorously

it must be less than T

(mstat)

_{in which} case there will be no

unentangled

regime

where the

_percolation

_{relaxation appears. The relaxation}

_spectrum

is cut off at

_Tmax.

Above the

_{gel point}

the situation is more

_complicated.

There is a finite fraction of

bonds which contribute to the backbone of infinite clusters. These never

_relax,

and

_give

rise to

a finite modulus. Their existence affects the relaxation of other bonds because now the

effective

_entanglement

molecular

_weight

does not

_diverge

but is bounded above

_by

the

entanglement

of the

_permanent

network. As we move above

_{gel point}

the

_permanent

network becomes denser. There will be a new contribution to the

_dynamic

modulus

_coming

from clusters

_dangling

from the infinite

backbone,

which have an

_exponential

distribution of

dangling

LCPs. The relaxation of such an ensemble was examined

_by

Curro and Pincus

_[21]

in the case of

_strictly

linear

_segments

and led to a

_power-law

stress

_decay.

We defer a detailed

calculation of this

_{complex regime}

to a further paper, but await

_experiments

on

_entangled

gelling

systems

with interest.

Appendix

A.

Here we derive the exact discrete form of the

_seniority

statistics and relaxation

_spectrum

for the case of z = 3. We _start with the recursion relation for the

Q., equation (2.4).

For

z = 3 this becomes

which we can use to _prove a result

_special

to this case for the summed

_quantities

Q,,,:

The result may be

proved by

induction as follows. First we observe that it is true for

m =

0,

for

Qo

= _qo = 1/2 and ql = 1/8. Now suppose it to be true for _a

general

m -

_1,

then

which

_completes

the

_proof.

We also have a relation between the lattice concentration of

seniorities cLm and the one-sided LCP

probabilities :

This is true for m > 0

_only.

For m = 0 _we have CU = 1/2.

The modulus

_depends

on the effective lattice concentration of unrelaxed

_segments

which,

as we saw

_above,

is the

_complement

of the sum of these cm. However

(A6)

makes the sum

very easy to

_do,

so

This can be

_directly

related to the time at which bonds of

_{seniority m relax tm}

via their recursion relation

_(we

are still not at the level of the

_{prefactor) :}

The factor of z - 1 accounts as before for the renormalisation of concentration when we

embed the

_Cayley

tree _{lattice into real space.}

_Finally

we _may sum this relation with

t1 = 1

to show that

_log

_{tm}

=

8 a [Qm - q0 - q1]

and _use

(A5)

_to relate the effective

concentration to the relaxation time :

So the form of the relaxation modulus derived from the

_asymptotic

results becomes exact in the case of z = 3

(for

the discrete _{sequence of}

times tm

with

integer

m),

but with a

renormalised value for

_Too

_{of tl}

_exp

_{(3 a ).}

Appendix

B.

In this

_appendix

we calculate the fraction of linear-like chains at each

_seniority

m,

which we must use in the double sum for

_{F (m ) :}

We may

perform

the sum over n

_using

the two standard results

The first term

_contributing

to

_F(m)

from

_(Bl)

_gives

on

_summing

over n :

using

(A5)

A similar summation can be

_performed

on the second term to

_give,

on

_combination,

The second sum starts from 1 rather than zero because the

_expression

for

_{L (n, 0 )}

does not contain the subtracted

_part

of

_{(B1) .}

Terms cancel between the two sums in

_pairs

_except

for the

largest

element in the first sum, so

which _{proves the result.}

Acknowledgments.

We thank M. E.

_Cates,

R.

_Colby

and L. Leibler for useful discussions. SZ and TCBM were

supported by

ICI

_plc.

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