Statics and Dynamics of the “Liquid-” and “Solidlike” Degrees of Freedom in Lightly Cross-Linked Polymer Networks

HAL Id: jpa-00247328

https://hal.archives-ouvertes.fr/jpa-00247328

Submitted on 1 Jan 1997

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Statics and Dynamics of the “Liquid-” and “Solidlike”

Degrees of Freedom in Lightly Cross-Linked Polymer Networks

S. Panyukov, I. Potemkin

To cite this version:

S. Panyukov, I. Potemkin. Statics and Dynamics of the “Liquid-” and “Solidlike” Degrees of Freedom

in Lightly Cross-Linked Polymer Networks. Journal de Physique I, EDP Sciences, 1997, 7 (2), pp.273-

289. �10.1051/jp1:1997145�. �jpa-00247328�

Statics and Dynalnics of the "Liquid-" and "Solidlike" Degrees

of Freedoln in Lightly Cross-Linked Polylner Networks

S.V. Panyukov

_(~>*)

and

I.I. Potemkin

(~)

(~) ^Lebedev Institute of

Physics,

Russian

Academy

of

Sciences, Moscow, 117924,

Russia (~)

Physics Department,

Moscow State

University, &ioscow,

II

7234,

Russia

(Received

29

May

1996, revised 30

August

1996, accepted 25 October

1996)

PACS.82.

70.Gg

Gels and sols

PACS.62.20.Dc

Elasticity,

elastic constants

PhCS.61.4I.+e

Polymers, elastomers,

and

plastics

Abstract. The

density-density

correlation

function,

that determines the

angular depen-

dence of the

scattering intensity,

is calculated for

strongly

deformed,

lightly

cross-linked

polymer

networks. The

replica

variant of the tube model is used to take into account the

entanglement

effects. It is shown that the contribution of

thermally

driven

(annealed) _density

fluctuations to this correlation function is small with respect to the contribution of frozen-in

(quenched)

fluc- tuations. We also calculate the free _energy functional for such "soft"

solids, taking

into account

liquid- as well _as solid-like

degrees

of freedom _at different spatial scales. The

dynamics

of the

"abnormal

butterfly

effect" is

theoretically

studied.

1. Introduction

"Soft'" solids materials that could be deformed

significantly

without

rupturing.

In the last few _{years an} interest in these materials has

greatly

increased. Some

types

^of

polymer

networks

(rubbers, gels)

and also melts of

entangled polymer chains,

studied at short times after their

deformation,

_are

examples

of such materials. The

physical picture

of mechanical deformations of such substances is

essentially

different _at different

spatial

scales. At

length

scales

greater

than the mesh size of the

polymer

network

(or entanglement quasi-network

of _a melt

iii),

these substances behave _as usual elastic

solids,

but at shorter

length

scales

they

exhibit

properties

of

polymer liquids. Therefore,

for _an

adequate description

of elastic

properties

of the networks and the melts _one has to take into consideration both

liquid

and solid-like

degrees

of freedom.

In

Flory's

classical

theory

of

elasticity

_[2j

only

the solid-like

degrees

of freedom contribute to the entropy of the network.

According

to this

theory

the

isointensity

lines of the

scattering

observed in small

angle

neutron

scattering (SANS) experiments [3-7]

on the

samples subjected

to _an uniaxial tension should be

ellipses.

Such _curves have indeed been observed _at short times in both melts and networks after the instantaneous

stretching [4,5j. Later, however, they

transform into

butterfly-shaped

_curves with _a

peak

in the direction of the network

stretching (the

so-called abnormal

butterfly effect).

The neutron

scattering intensity

is

proportional

to the Fourier component ^{of the}

density-

density

correlation function

Sq

=

(pap-q).

This function

gives explicit

information about

(*)

Author for

correspondence (e-mail: panyukovfl lpi.ac.ru)

©

Les

(ditions

_de

Physique

1997

liquid

and solid-like

degrees

of freedom at different

spatial

scales R

(or

_wave vectors q r-

R~~).

In _{contrast to} usual

polymer liquids,

where there _are

only, thermally

driven

(annealed) density

fluctuations;

in

polymer

networks there _are also frozen-in

(quenched)

fluctuations at

large spatial

scales R. The relative roles of both

types

^of fluctuations have been discussed

intensively

in the literature

[8-16j.

In

[8-1ii

it _was demonstrated that the consideration of

only

annealed fluctuations leads to

butterfly patterns

with maxima in the direction

perpendicular

to the

stretching

direction

(normal butterfly patterns).

In

[12j

an

attempt

to

explain

the abnormal

butterfly

^effect _{as a} consequence of annealed

density

fluctuations has been initiated _on the basis of

phenomenological elasticity theory

of

weakly

deformed

polymer

networks. Onuki

proposed

a

perturbation theory

to the

description

of

point-like

defects frozen-in

during

the

preparation

process of the networks

[13,14j.

This

type

^of

quenched

fluctuations _was shown to lead _to the

abnormal

butterfly

effect. The relative role of annealed and

quenched

fluctuations in

strongly deformed, phantom polymer

networks _was studied in

[is,16j.

The purpose of this work is to construct a

microscopic theory

to describe annealed and

quenched density

fluctuations in

stroilgly

deformed

polymer systems

^{within the framework of} the

entanglement quasi-network

model. The theoretical

description

of both

lightly

cross-linked

polymer

networks where

entanglement

effects _are

essential,

and

polymer

melts studied at small times after their deformation is

proposed.

The traditional

approach

to the

description

of the solid-like

degrees

of freedom _uses the Landau's

elasticity theory iii],

which is based _on the _ex-

pansion

of the free energy of _a solid in terms of the

displacement

_{vector u.} This _vector defines the

displacement

of the

point

of _a

body

with

respect

to its

position

in the

spatially homoge-

neous, "undeformed" state.

Unfortunately,

this

approach

is

hardly applicable

to the

description

of

strongly

deformed

polymers,

because in this _case the strain tensor u~u =

1/2(V~itu

+

Vuu~

is not small. In order _{to overcome} this

problem,

_we take _{as a} "undeformed" _{state an essen-}

tially anisotropic

state of the

ailinely

stretched

polymer

network. The

displacement

vector u

describes

only

the solid-like

degrees

of freedom of the network due to the

displacements

of the cross-links relative to their

positions

ⁱⁿ the ailine state. The

liquid-like degrees

of freedom _are described

by

the _monomer

density c(x)

of the network subchains.

Notice,

that _on

macroscopic length

scales the deformations of the solid _are

always ailine,

_so that the

long-wave components

of the strain tensor _are small and _can be

adequately

modeled

by

the Gaussian

approximation.

In Section I of this paper we introduce the tube model which

desiribes

_{the effect of}

en-

tanglements

in

strongly

and

nonuniformly

deformed

polymer

networks. In Section 3 _{we use} this model _to calculate the

Ginzburg-Landau

functional

(Eq. (17))

for such

"two-component"

systems,

which

depends

_on both the "elastic" deformation tensor and the

"liquid" component

of the

density.

The _source of the

quenched

fluctuations is shown to be the random internal

stresses due to the

irregularities

in the chemical structure of the network. The

dynamics

of

the _occurrence of the "abnormal

butterfly

effect" _is

pursued

in Section 4 in the framework of

the

Ginzburg-Landau

free _energy. We summarize _our main

predictions

in Section 5. Readers who _are

only

interested in _our final results and

comparison

with

experiments

_can focus their

attention

primarily

_on this section.

2. Model of the

Lightly

Cross-Linked

Polymer

Network

The subchains of the

lightly

cross-linked

polymer

networks _are

strongly entangled

and

frag-

ments

thereof,

^which _are

entangled

with each

other,

form the so-called

entanglement quasi-

network

iii.

To describe the elastic

properties

of such

systems,

the

prohibition

of the _cross-over of these

fragments (topological constraints)

has to be taken into _account. We

begin

_{our con-}

sideration with the

study

^of the undeformed network in the

preparation conditions,

which is made

by cross-linking

linear chains in the melt. For

simplicity,

_we

neglect

the chain-end

effects,

and

replace

the

polymer

network

by

a

single strongly entangled

chain of N _monomers where the

degree

of

polymerization

N tends to

infinity

N _{~ cc.} Similar _to the de Gennes' model

iii,

each

fragment

of the network chain _can be considered _as

placed

into _an effective tube of the diameter

r-

aN/~~,

which is formed

by

the

neighboring

in space

fragments

of the chain. Here

a is the _monomer

size,

and

N~

is the average number of _monomers between two

neighboring

entanglements.

For

simplicity,

_we also

neglect

the

cross-linking effects,

_except the fact that these effects

prohibit

the diffusion of the chain

along

the tube. In the context of _our model the network

"topology"

is

completely

characterized

by

the

position

of the tube: _{we say} that _two chains with the _monomer coordinates

(x1°) is))

and

(R(s)) is

is an arc

length

of the

chain,

0 _{< s <}

N)

_are considered to have

equivalent "topology",

if

they

_are both _contained in _a

common tube.

Thus,

_{we can} describe the

topology

of the network

by specifying

the coordi-

nates

(R(s))

of _one of the chains in the tube. In the literature

[18j,

^{it is} more conventional to describe the tube

by

the coordinates

($i(@))

of the center of this tube

(the primitive path).

Clearly,

these coordinates _can be found

by smoothing

of the function

R(s)

at scales _s

r-

N~

and

by changing

the

parameterization

_s ~ i

=

s/N~ jig].

The

topological

constraints select

only

those chain conformations from the conformational _set of the

phantom

network that _are contained in the tube. This

type

^of constraints _can be

represented by introducing

_an additional factor

~

~~~~ ~l

_{~ ~~~}

/ _t~

_~~~~~

^lNl~~~~~

~~~

into the statistical

weight

of the

phantom

network

[19j.

This factor offers _a contribution to the

partition

function of the network

only

if

(x1°1(s) R(s)(

<

aN/~~,

that

is,

when the

fragments

of the chain _are enclosed in the

given

tube. Constant C in

(ii

is estimated _on the basis of

simple scaling

assessments

[20j:

the free energy of the chain in _a tube of di-

ameter

aN/~~

is

given by AF/T

m

N/N~.

On the other

hand, according

to

ii ), AF/T

_m

(N/N~)C ((x(°I R)~) la~N~

_m

CN/N~;

therefore _we find C _m i. In what follows _we shall

use the

equality

C _e i _as the definition of the

parameter N~

for _our model. For

simplicity

_we

assume that

polymer

net~.ork in its

preparation

^conditions

(unswollen)

is

incompressible

with

the _{average monomer}

density p(°I Collecting together

all above mentioned

factors,

_{we can} write the

partition

function of the

network,

the

topology

of which is characterized

by

the set of coordinates

(R(s)),

_as the

integral

_over all the conformations of the chain

z(o) j~j

~

~~jo) z(oj ~joj

_~

_(~)

/

'

where

N

Z(°) (x1°), Rj

= b

Ids

^b

_(x1°) ^x1°) (s))

^p1°1

o

'~

k(oi (~j

²

j~io) (~) R(~)j2

~ ~~~

/~~

a

~

a2Nj

^~~~

~

Here the dot in

k1°)

_denotes the differentiation with

respect

to s. The b-function in

(3)

accounts for the condition of the

incompressibility. Notice,

that for undeformed network the model

under consideration is identical _to the tube model

by

Edwards

[18j.

^{The difference} appears when _we consider the deformation of the network. In the

description

of the network deformed

by

_an amount

l~ along

the coordinate axis

~J ⁼ x,y,z with

respect

to the condition of its

synthesis

_we shall _use the fact that network

topology

^could not be

changed

in the _processes of the deformation. This _means that the

topological

restrictions in the deformed network

should be described

by

the _same

delta-function, equation (ii,

as for undeformed network. The coordinates

x(°I

_{in this}

_equation

can be

expressed throigh

the coordinates ofx ofthe deformed network

by

the relation

~il

=

(x~ it~(A *x(°)))/l~.

Here the vector A

*x1°1

_{with the} coordinates

l~xi~

describes the ailine deformation of the

network,

and the

displacement

vector u takes into consideration the deviation from the ailine behavior.

Substituting

the above relation into

equation (ii

_we observe that the effective

diameter of the tube

affinely changes

with the

stretching

of the

network,

_{as a} result of the invariance of

topological

restrictions with

respect

to the

general

coordinate transformation

x(°)

~ x = A _*

x1°)

₊

u(A

*

x(°)). (4)

The

partition

function of the deformed

polymer network,

the

topology

of which is charac- terized

by

the coordinates

(R(s)),

can be written _as

Z

(x(°I, Rj

⁼

/DUDpZ ^{x1°1, R,}

^u,

pj

,

(5)

where _we take into consideration the fluctuations of the vector u as well _as of the

density p(x),

N

Z

x1°1, R,

_u,

pj

⁼

/Dx

^{b ds b}

^{(x x(s)) p(x)}

0

N ~

x exp

Ids

^{~~~~ B}

/dx ^p~(x)

a 0

~ ~~

^{~~~~~~ ~~}

^{~~ ~~~)~ ~~~~~~~~}

^~

~~~

o

The last

integral

in this

expression gives

^the contribution of the tube to the total

partition function,

the average

position

of the

tube,

A*

k

₊

u

(see Eq. (4)

for the definition of

*),

and its

diameter, aNll~l~,

deform

ailinely

with the

stretching

of the network. The _monomer-

monomer interaction in

equation (6)

is described

by

the second virial coefficient B. The

higher

order _terms in the virial

expansion

_can be

neglected only

for small

enough

_monomer

density,

i.e., in semidilute

regime.

,

It is well-known that in semidilute

polymer

solutions the

(critical) density

fluctuations _are

strong

on scales smaller than the correlation radius

( [ii,

and here the uiean-field

approach

breaks down.

Nevertheless,

these fluctuations _can be taken into account within the framework of the _mean field

theory

if the system ^is described in terms of blobs of size

(,

each

comprising

g monomers

[ii.

When

applying

these ideas to the

description

of _a

polymer network,

_we face

the formal

problem

that the

number, N/g,

of the chain "monomers"

(blobs)

in the

experi-

mental conditions is different from the number N of constituent _monomers in the

preparation

conditions. This

precludes

a direct

application

of the above described model _to the

scaling description

of the swollen

polymer

network.

However,

this

problem

_can be _overcome

by going

to the old structural units

(monomers)

with renormalized parameters ^{instead of de} Gennes' blobs. When

dealing

with the

spatial

scales

larger

than the correlation radius

(,

each blob _can

be

formally

treated _{as a} Gaussian chain

comprising

of _{g monomers,} each of which is charac- terized

by

the renormalized

parameters:

^the size h ₌

(/g~/~

and the virial coefficient 11. The latter _can be calculated from the

following

condition

z e

ljg~/~h~~

₁ 1

(7)

for the well-known

expansion

parameter z introduced in the

perturbation theory

of _a

polymer

chain

[21j.

^{The second}

equality

is correct in the

strongly fluctuating

semidilute solution

regime,

where there _{are no} small

parameters.

For such _a

solution, using explicit expressions (

=

alb~~@

and _g

=

1b~5@ [lj,

_one finds

h ₌

alb~~/~, lj

=

a~lb~&, (8)

where 16 ₌

1~3

_{is the volume fraction of the}

polymer

in

solution,

and I is the

swelling

^coeffi-

cient. We thus conclude that the

polymer

network swollen in _a

good

solvent _can be described

by: _I) substituting

_a for h in the term of

equation (6) containing

the derivative _{over s} and

it) renormalizing

the interaction coefficient in

equation (6) according

to

equation (8).

3. Free

Energy

Functional

Generally speaking,

the

partition

^function

equation (5) gives

the

complete

statistical mechan- ical

description

of the deformed

lightly

cross-linked

polymer

network.

Nevertheless,

the calcu- lation of this function for _a

given

realization of network

topology

is

prohibitively difficult,

and

we use the

self-averaging

_property of the free _{energy to}

simplify

this

problem.

This _means that the free energy of _a

macroscopic

network of _a

given topology

_can be

replaced by

the average

over all

possible

realizations of the network

topology.

To find the

weights

of these realizations let _us consider the

preparation

conditions in _more details.

The

probability distribution, ^P1°1 [Rj,

that describes the conformations

(R(s))

of the chains in a

melt,

has the form

[22j

P1°1[Rj

₌

Dx'1°lP1°1 _x'~°~, Rj

,

(gj P(°1[x1°),R]

₌

Z(°1[x1°),R) /

/Dx'(°IDR' ^{Z(°) (x'1°1,R'j}

,

where the functional

21°)

_{is defined}

by equation (3).

Let _{us assume} that the melt is instan-

taneously

cross-linked

by

irradiation _or

by

other _means. Then the function

P(°I [R] gives

the

probability

distribution of the

given topology (R(s)) (or

tube coordinates

(I(@)),

_see

_previ-

ous

Sect.)

of such network. We also introduced in

equation (9)

the function

P(°) [x(°), Rj,

which

gives

the

probability

distribution of the

conformations, (x(°I (s) ),

of the network whose

topology

is characterized

by

the coordinates

(R(s)).

Using equations (6, 9),

_{we can} write the _average free _energy of the network in the form

[19,22j:

F

= -T

/Dx~°)DR ^{P1°) x1°1, Rj}

^{In Z}

(x(°), Rj

⁼

^~~~

_{dm ,}

⁽¹⁰⁾

~~o

Fm

= TIn

Z~, Zm

=

Dx~°~DR _Z~°~Z'°

=

/Dx(°lDRflZ~~l ~o ⁽¹¹⁾

The

replica

trick

[23j

^{consists of}

calculating Zm

for

arbitrary integer

m and then

performing

analytic

continuation to _m ~ 0. The value

Fm

has the

meaning

of the free _energy of the _so

called

"replica system".

In order to calculate the

partition

function

Zm

of this

system,

^let us

represent ^the b-function in

equation (6)

_as the Fourier

integral

b

p(x)

jdsd(x ^x(s))

⁼

jDh

exp

(I jdx ^h(x)p(x)

^I

Ids

^h

^(x(si)

,

(12)

where ih has the

meaning

of the external field

acting

_on the chain _monomers. Similar

procedure

is carried out with the advent of the field

ih(°I

_in

_equation (3). Furthermore,

let _us

expand

the

expression

^for

Zm (ii

into the series in powers of the variables

u(~l, h(°~, h(~l, (k

=

1,.

,

m),

which _are

displacement

vectors and external fields in each of the

replicas, respectively,

and consider

quadratic

terms in this

expansion.

All the

integrals

_over the micro-variables

R,

x(~~

and the fields

h(~~, (k

=

0,1,. ,m)

in

expression (11)

_are

Gaussian,

and the result of the

simple integration

_can be

represented

in the form

~~ ~~~~~~~~~~~

~~~

II ~~3

~~

^~M

^(~) i~

fLf~fL~~

~

(;~ ^{'"i ~ ~_~}

~

f

^{'PQ + P}

^{) i~»A» (q)Ul~[}

^{l~ m ~}

^~i~~

A(

^+~

~ _(P~~(~+~j

~2

(q) ~j ~(k) ~(l)

~ ~

~"~ #,U

~~

k,i=1

"~ _~

~~

where p =

p(°I /(l~lyl=

is the _average

density

of the stretched

network,

and

u~l, p(~

_are the Fourier components of the

displacement

vectors and of the _monomer densities in k-th

replica, respectively (the

coefficients

A, A~

^and ~J~u are

given

in

Appendix A).

The

non-diagonal (with respect

to the

replica

indexes

k, I)

terms in

equation (13)

_can be

diagonalized

with the _use of the

following identity (which

is also known _as the Hubbard-Stratonovich

transformation)

_:

~~~

II ^~~3 ~k~"~~~~ ~ _~i~~~~-q

M<u

,~

l

/Df ^Wifi

exP

(- IA

~

j Iv _QUI~' )

~i~~

"

/Df ^W[fj

Wifi

_{= exp}

(-( Ii _{~ ~Jii(q)/» q/u} ⁾

,

~ ~

where the

integration

is taken _over the random vector force

fq

^{with the}

components f~

q,

and the matrix

i~~i

is the inverse of the matrix _~J with the components ~J~u. ^{Then relation}

(13)

can be rewritten _as

/Df ^W[fj _/DuDp

exp

(-F[p,

_u,

Ii IT) )~

~~

/Df ^W[fj

^~~~~

Substituting equation (15)

in

equations (11, 10)

we come to the

following expression

for the

free energy

F

= -T

/Df ^P[fj

^In

_/DpDu

exp

(-F[p,

_u,

_f] IT),

(16) P[f]

₌

W[fj / _{/Df l§'[fj}

,

which

provides

_{a way} for

considering

the fuiictional

P[fj

_{as a}

probability-

distribution of the random force f and allows _{us to} make

important

identification

F[p,

_u,

fj / ^d~ _{l~j E~ (q)u~}

qUp _-q

~ _f"

Q"~ _~

~'~~

~

~~~

(q)

with the desired free energy functional.

Equation (ii)

is the main result of this section. The _term

proportional

to

E~(q) gives

the energy of elastic deformation of the network. lve found the

following expression

for the function

E~(q)

valid for

arbitrary

_wave vector q

~"

_{~~~ ~}

~~~~

4 ₊

~Ne~2

(A

*

q)2'

^~~~~

In the continuous

limit,

_{q ~}

0, E~

^{is the} square form of q the coefficient of which _are the elastic moduli of the network. Note that these moduli _are

strongly anisotropic

for

anisotrop- ically

^deformed network with the maximum in the direction of the network

stretching.

^This

anisotropy gives

rise to the normal

butterfly

effect

suppressing

thermal fluctuations in the di- rection of the

stretching.

The elastic contribution _to the free _energy decreases with the rise of

the _wave vector q and vanishes at _wave vectors q »

if (aNll~).

The random Gaussian force f introduced above is characterized

by

the correlation functions

G

=

o, /~

~

/~

-~ ⁼

2~Jv~jq). jig)

It describes the random internal stresses due to the network deformation because of the random distribution of cross-links

(quasi-cross-links)

in the network. It is this force that leads to a

non-affine character of the network deformation and

produces quenched inhomogeneities

in the network, which

stipulates

the abnormal

butterfly

effect. The

expression

for the correlator of the force f valid for

arbitrary

_wave vectors q is shown in

Appendix

A. In the small _q limit

this correlator is

proportional

to

q~

and vanishes for _{q =} 0 since in this limit the total force of the internal stresses

acting

on the whole

sample fq~o

_"

idx f(x)

₌ 0. The correlator of the random forces also vanishes in the limit _q »

i/(aNll~),

_{since there are no} frozen

inhomogeneities

_on such small

spatial

scales.

The third term in

equation (iii

describes the

liquid-like degrees

of the freedom. This state- ment _can be clarified

by introducing

the

liquid

part of the total _monomer

density

c~ = p~ ⁺

ip ~ q»Avjqjuv

q.

12°)

»

In the continuous limit _q ~ 0 the free _energy,

equation (17),

remains finite

only

if the condition cq = 0 is satisfied

(since Aq

+~

q~,

_see

Appendix A). Taking

into account that in

this limit

A~(q

_~

0)

~ 1 _we find from

e~uation (20)

the usual

geometrical

relation between the Fourier components ^of the

displacement

vector uq and the

density

_{pq, pq}

=

-ipqu~.

In the

opposite

limit of

large

_wave vectors q »

I/(aNll~)

we have

A~(q)

_~ 0 and there is

only liquid-like

contribution cq ^{to the} monomer

density

_pq, see

equation (20).

In this limit

A(q)

_m

4p/(a~q~)

and the term

A~~ (q) _(pq(~

in

equation (17) reproduces

the usual

expression

for the free _energy of

polymer liquid inhomogeneities. Thus,

_we conclude that the free _energy of

polymer network, equation ii), gives

correct

description

of both short and

long

_wave vector limits. Such

competition

of solid- and

liquid-like degrees

of the freedom is the main feature of the

polymer

networks and it is absent in the _case of

ordinary

low molecular

weight liquids

and solids.

Since the relaxation processes of

liquid-

and solid-like

degrees

of the freedom _{are govern}

by

different

physical

mechanism it is _more convenient to consider the free energy,

equation (ii),

as the functional of variables _c and _u

~~j~'~~

=

Ii ^~E»(qiu»

qu» _-q

-Li»

qu~ -q

~ ~

2 ~

+B

Cq-PLi~vAv(~)Uvq

+

)j

(21)

p

We

emphasize

that

expression (21)

does not

require

that the

applied

deformation _be small the

components l~

can be

arbitrary large.

The

only assumption

made in the derivation is that the

large-scale

fluctuations of the

density

_are small. This

assumption

is

always

true

by

virtue of the ailine deformation of the network _on

macroscopic

scales. Recall that _we have to substitute the renormalized

parameters

^I and 11,

equation (8),

instead of their bare values _a and B in the above

expression

^for the free _energy.

4.

Dynamics

of Relaxation Processes in Deformed

Polymer

Networks

Liquid-

and solid-like

components

^of the

polymer

network _were shown in the

previous

^Section

to be described

by

the collective variables _c and _u.

Dynamics

of relaxation _processes in such system can be

represented by

the

Langevin equations [24j:

(

_~~~

fit

dfii

^{~ ~~~~~' ~'~}

~'~'~'~'

_~~~~

~~~

^~~

where

ii

_{" u~,}

fi2

_{= uy,}

fi3

_{" uz,}

fi4

_{= c.} The kinetic coefficients

rip

_are

subjected

to

Onsager's

concept of the

symmetry:

_{T~j =}

rj~.

The correlators of _a random Gaussian force

(

have the form

((i(f)I

"

°, ((i(f)(j _(f~))

"

2Tr~jd(f

_f~

). (23)

To find the coefficients

r;j,

we discuss _now the

equations

of motion for all

components ii, (I

=

1, 2, 3, 4) (22).

Consider _a network that is swollen in a

good

solvent

by

_a factor of I with

respect

to the

preparation

state. Let _{us assume} that the solution is _an

incompressible

viscous

liquid.

The

equations

of motion for the

liquid-like _component

of the network _can be written

using

^the Navier-Stokes

equations

where

fl(x), v(x)

and _{~m are} the osmotic pressure, the fluid

velocity

and the

viscosity

of the monomers,

respectively,

and

r(x)

is the external force _per unit volume

acting

on the _monomers.

Since the

density

of cross-links _pores

+~

p/Ne

is small in

comparison

with the _monomer

density

p, we can

neglect

the monomer-cross-link friction with respect to the monomer-monomer one.

Consequently,

the external force

r(x)

is determined

by only

the monomer-monomer friction

acting through

the solution and this _can be written _as

r =

-fl(v vs), (25)

where

vs(x)

is the solvent

velocity

field.

Using (for

the sake of

simplicity)

the model of the

absolutely impenetrable

coil

[20j

^{for the}

fragment

of network chains of the size R

r-

aNll~,

it is easy ^to find the

expression

for the friction coefficient _per unit volume

fl

_r-

6~~sR/R~,

where _~s is the solution

viscosity.

The

scaling

renormalization of this

expression

is obtained

by

the

procedure,

described in Section 2. The first term of the first

expression

in

(24)

_can be

neglected

_on the scales

larger

than R.

Performing

the Fourier transformation of

equation (24) taking

into account

equation (25)

_we find

~~

dF

( _{(vq-vs q)}

₊ ~

~P~bc-q

Perturbation of the solvent

velocity

^field

by

the force

-r(x)

is

represented by

the Navier-Stokes

equations

_as well

((vq-vs _q)

₊

_iqpq

=

o, (27)

a

Ne

where pq ^{is the Fourier}

component

^of the solvent pressure.

Combining

the

expressions (26)

and

(27)

with the

continuity equations

for the _monomers and the

incompressible

solvent

~/

_{~ ~~~~q}

_~'

~~S _q

°' (~~)

we obtain the

equation

of motion for the component _cq

~~~/

~

~~q ~' ~~ ~a~(2Ne

~~~~

When

comparing (29)

with the

corresponding Langevin equation (22),

it becomes clear that

rq

is _a sort of kinetic coefficient

r44

and that all

r41, (I

=

1, 2, 3),

_are

equal

to zero.

Onsager's symmetry

concept

requires

that all the

r~4, Ii =1, 2, 3),

_are

equal

to zero as ~~ell. We thus conclude that the

equations, describing

^the relaxation of the solid-like

components

^{will look} like

(22),

where the indexes

I, j

^take _now

only

the

values,1,2,3.

These

equations

_are the

familiar

iii] equations

of motion for _an elastic medium

Lv»(ai~

₊

_a~ui

=

o,

_{a»~ =}

£,

~, v = x, y, z,

(30)

~ »~

where a~u ^{is the stress}

tensor,

and the

dissipative

stress tensor has the form

~i~ ^~~ ~~)~

~~~ ~j ~~~l~

~~~ ~j ^~~~

^~~~~

Comparing

^these

equations

^with the

Langevin equations (22)

for the components fi~,

(I

=

1, 2, 3),

we find the

expressions

for the Fourier components of the

dissipative

coefficients

r~~

= J~

q21,~

+

q~qj)

+

(q~q~, I,j =1, 2,

3

(32)

where ~ and

(

are the effective

viscosity

coefficients.

Equations (29, 32) give

^the

explicit expressions

for all the

dissipative

coefficients

describing

the relaxation _processes in

polymer

networks.

In order to find the initial conditions for

equation (22),

let _us consider _a network stretched in a~ =

l~ IA

times at _a

given

^instant of time t

= o. Because of the affine character of such instantaneous

deformation,

the desired initial conditions can be written _as

u~(x, +o)

=

a~u~(x, -o) (33)

where the

change

in the network volume before and after this deformation _was

neglected (n~ayaz

=

i).

The Fourier transformations of the

equations (33)

^have the form

'~P

q(+°)

_{" °M~M}

a*q(~o). Cq(+°)

"

Ca*ql~°) (34)

The

Langevin equations (22)

describe the

dynamics

^of the

thermodynamic (annealed)

fluc- tu~tions in the swollen

by

the factor of I with

respect

to the

preparation

conditions and

anisotropically

stretched network at times t > 0.

Analogous equations

for the _case t <

0,

which describe the fluctuations in the swollen network before its

stretching,

_can also be ob- tained from

equation (22),

in which all

l~

should be set

equal

to I

(l~

₌

ii.

The _same

procedure

should be

applied

to the formula

(19)

^{for the} correlation function of the random force

I, characterizing

the internal _stresses due to the network

swellin). Solving

the set of the

linear

equations (22)

for times t < 0 with

l~

= I and for t > 0 with

l~

=

a~l,

and

using

the initial conditions

(34)

for

matching

these solutions at t ₌

0,

_{we see} that for t > 0 the solution

depends

_on f _as well _{as on}

I.

_Now

one has to substitute the solution into the

expression

for the

density-density

correlation function

Sq(t)

_m

(iPq(t)i~)

=

Cq(t) iPjj _q»A»(q)U» q(t) (35)

In addition to the correlators

/~

q

/u

-q

and

f~

q

fu

-q, which characterize the swollen and stretched

(after

the

swelling)

states of the

network, respectively,

the

joint

correlator

/~

q

fu

-q

contributes to the

expression (35).

This correlator is calculated in

Appendix

B.

The behavior of the correlation function

(35)

^{is of} the most interest in two

asymptotic

_cases:

I) ^t ~ cc, I.e. in the

equilibrium

state of the stretched network and

it)

for finite t,

I.e.,

for small times after the instantaneous

stretching

^of the swollen network. In the first _case,

I),

the

expression (35)

can be written _as the _sum of two terms

S~

₌

G~

+ P~

Iii1~ i iv ~/u

-~

~l]ll~~lili~~ ⁽³⁶⁾

Only

the annealed fluctuations contribute to the first term and this term is

responsible

for the normal

butterfly

effect. The correlation function of such

fluctuations, Gq,

is

easily

related to

the correlation

function,

_{gq, of} the

system

^{in which all the} monomer interactions _are "switched off"

(this

relation is well-known from the

physics

of usual

polymer liquids):

The second term in

(36) gives

the contribution of the

quenched fluctuations, originating

^from

the random nature of the random internal _stresses. In the _case of finite t the

expression (35)

can also be written _as the _sum of _two terms. The first describes the

temporal

evolution of the thermal

(annealed) fluctuations,

and the second term characterizes the relaxation of the

density

fluctuations _to their

equilibrium (quenched)

value.

5. Discussions

We

presented

the

theory

of

lightly

cross-linked

networks,

which takes into account both _ex- cluded volume interactions and

entanglements

of the network chains. Whilst the former have been considered

exactly by using

the renormalization of small-scale parameters

(monomer

size and effective virial

coefficient), nobody

knows _any

rigorous

_way to take the

entanglement

effects into consideration. To avoid this

difficulty

_{we use} the

simplest

tube model of

entanglements.

In

obtaining

the tube

parameters

^{for the deformed network} we were

guided by

the idea that the network

topology

should _not

depend

_on the network

deformation.

In the tube model the

topology

of the network is

represented by

the

b-function, equation (i),

which is taken to be

independent

on the network deformation. This condition

inevitably

leads to the

dependence

of the tube diameter _on the network

stretching.

This

point

is the main difference between _our model and that

by

Edwards

[18j.

^We

exploit

the exact

solvability

of _our model to find the free energy functional of the de

formed

network. Unlike the usual

liquids,

the deformations of ~~hich

can be described

by only

_one collective

variable, density

_p, the network must be characterized

by

two

independent thermodynamic

variables: _p and the

displacement

vector u. The usual

geometrical

relation between these

variables, lip /p

=

-Vu,

is ho _more valid for small scale deformations and is restored

only

in the limit of

infinitely

slow deformations. The

interplay

between

liquid-

and solid-like

degrees

of the freedom is

responsible

for the finite-scale

physics

of

polymer gels.

The free _energy

functional, equation (17), depends

_on the force

f,

~.hich is the

quenched

random function in the _sense that it

depends

_on the

spatial position

and does not vary with the time. The

physical

_reason for this force _is that in the process of the network deformation _a

given

^{chain is}

displaced, locally stretching

the network. Thus there is _a

returning

force

acting

on this chain from the network

through

its cross-links. This force leads to the formation of

inhomogeneities

in the network and is

responsible

for the appearance of the abnormal

butterfly

effect under uniaxial

stretching

of the network.

Choosing

the

parameters (the swelling

factor 1

=

2,

the

stretching

factor _a

= 1.5 and

BpNe

=

50) corresponding

to

experimental

conditions

reported

in

[5j,

we

plot

in

Figure

ia the

equal intensity

contours

(Sq

=

const, Eq. (36) ).

For small wave vectors q the

scattering intensity

^is

peaked

in the direction of the

stretching.

For

large

_q ^the

anisotropy

is inverse and the

scattering intensity

has maximum in the direction normal to the

stretching

direction.

The last

patterns

have

really

been observed in

experiments _[5j. Note,

that such behavior is

quite unexpected

since the

only; liquid-like degrees

of the freedom contribute in the

scattering

intensity

in the limit _{q »}

if (aNll~),

where the patterns have _to be transformed into concentric circles.

Indeed, inspection

^of

e~uation (36)

sho~.s that the

picture

^of concentric

ellipses

takes

place only

in the

region

of intermediate _q, it transforms into the

butterfly, patterns

for small _q

qa~

~3 ~2 O Ol

qy

S~

~2

~ ~i

~Y

° q3~

b)

⁰³

Fig.

I. Contour

plot (a)

and 3d

plot (b)

of the static

density-density

correlation function

Sq

in the

(q~,

_qy

plane,

where _q

=

qaNll~

_is the dimensionless _{wave vector.} The

swelling

factor and the

stretching

factor _a for the network

uniaxially

stretched

along

the x-axis

(a~

= a,ay =

n~~/~),

are

taken _as 2 and

1.5, respectively;

and

BpNe

= 50.

and into concentric circles in the limit of

large

_{q, q} » i

/(aNll~).

The three-dimensional

plot

in

Figure

ib demonstrates the _presence of _an

angular singularity

at q = 0 and the enhanced

scattering along

the direction of the network

stretching.

The

scattering

intensities

along

and normal to the

stretching

direction _{versus q are}

plotted

in

Figure

2a.

Notice,

that in the small

q-region

^the

scattering

in the stretched direction

(I)

exceeds that in the normal to the

stretching

^direction

(o)

and that the behavior is inversed for

large

_q. We also show in

Figure

2a the

signal

from unstretched network

(solid)

and that from _a solution

(+)

of uncross-linked chains with the _same

density.

In the

region

of

large

_q both these

curves follow the _same

asymptotic

behavior but for small _q the

scattering intensity

^{from the}

swollen network

considerably

exceed that from the solution. In

Figure

2b the above

scattering

intensities at q = 0 _are

plotted

_{as a} function of the _monomer concentration. In the

preparation

conditions the

signal

is the _{same as} that from the

solution,

it increases

dramatically

with the

swelling

of the network due to the rise of the

amplitude

of the network

inhomogeneities.

The time evolution of the

scattering patterns

^{after the} instantaneous

stretching

of the net- work is shown in

Figure

3.

Immediately

after the

stretching, (t

=

+0),

the

isointensity

lines

are

elliptical

in

shape, Figure 3a,

in

agreement

^{with the classical}

concepts. Later, (t

_>

0),

this

simple

_pattern of concentric

ellipses

is transformed into the abnormal

butterfly

_pattern. The

wings

of the

butterfly

_appears first _on small _{wave vectors q}

(large spatial scales), Figure 3b,

and afterward _are

expanded

into the range of

larger

_q values. The final

stationary butterfly

patterns

are

plotted

in

Figure

ia.

At first

sight

the observation that the

qualitative changes

of the

scattering picture (butterfly pattern)

_appear first in small _q contradicts to the usual

picture

of relaxation processes which

begin

first in small scales. Such small scale

quantitative changes really

take

place

in the

polymer

networks and

correspond

to the motion of outer

ellipses

to the center.

Nevertheless, qualitative

transformations of the concentric

ellipses _pattern

into the

butterfly

_one

begin

at

large ^spatial

scales. All the above results are, ^at

least,

in

qualitative agreement

with the observed _exper- imental results

[3-7j.

The

quantitative comparison

^of _our results with

experimental

data is very

interesting

both for

deeper understanding

the nature of nonaffine local deformations of

entangled polymer

networks and for

verifying

the main features of the tube model.

Appendix

A

Coefficients of the

expansion

of the free _energy functional

(ii)

for the

arbitrary

_q values have the form

Q2

^°~

Q2

A(q)

=

PNe

_exP

I ~j nl

_Au

/

dY exP

~ ~j nl _Al

u ~ ^u

~ _l$ u)

(i ^e~Y/~"

+

(l) i) e~Y/~"j

2 2 2

Q2

x exp

-~jn$lue~Y/~"

-i

,

(A.2)

~

u

aaaaaaaaaa

°aa

~a /~

~a

q

o

a a

a

a a

a

a

a

a

a

a

no oooooo oo oaoo _oo

o

°ooo~ a

° a~

o

~ a

o

~

o ~o

o

a

oi

aj

^q

a ~a

s

^~a

~ ^a

~~

a a

a a

a

~

o a

o

o a

o ~

o

o a

o a

o

a

a a

o a

a ~

o

a

~

o a

a o

a a

o a

+ +

+ o

+ + _{+ o} ^~

+ ₊

+ a °

+ + ~ + + ^{o a}

~ ~ + ^{o a}

+ +

~

o °

+ +

+ + + ₊ ^{o °} ~

~ ~ ~ +^o ₊ ^a

+

~

~ a

b) 16

Fig.

2.

Log-Log

plot of the

scattering

intensities from the swollen and

uniaxially

stretdied net~vork in the direction of

elongation (I)

and normal _to it

(O)>

_as well _as the

signal

from the s~vollen network

(solid)

and from the solution of uncross-linked chains

(+)

~vith the _{same monomer}

density: (a)

the

q-dependence

at 4l

=

pa~

= 0.I and

(b)

the 4l-dependence at q = 0.

qx

.02 0 01 02

qy

al

,

qx

-oi o1

~qy

Fig.

3. Evolution of the contour

plots

of the

density-density

correlation function

Sq(t)

in the

(qx,qy) plane.

Time is measured in the units of the dimensionless time I

=

tpT/Ne(4/3q

₊

(): (al

I

= 0

immediately

after the instantaneous

stretching; (b)

^I

= 0.3 emergence of

"wings"

of _a

butterfly

pattern at small _wave vectors q. Here the

s~velling

coefficient A

= 2, the

stretching

coefficient

o = 1.5, and

BpNe

₌ 50.

~~~~~~ ~~~~~~~ ^~

~

~~~~~~~~

~

~~~

(~.~)

~~

j~e 4(l~

₊

lu)

+

(luNea2(A

_*

q)2'

where unit vector n has

components

_{nu = qu}

/q

and

Q

= Qa.

Appendix

B

To find the

joint probability

distribution P

(f,ij

^{of the forces f and}

^I,

^{it is convenient to}

consider the average of the

product

of the free

energies

of the deformed state

(F)

and of the swollen state

if).

_This

average can be

represented

in the form

(analogs

to the formula

(16) Ffl

=

T2 /DfDi

^P

(f, ij

^In

/DpDu

^exp

^(-F[p,

^u,

^{fj IT)}

jB_i)

x In

/DfiDfi

^exp

^{-fl fi, fi,} ij ^IT

,

where d and fi are the

displacement

vector and the

density

of the swollen

network, respectively.

The _{same average can} be calculated

using

the

replica

trick.

Comparing

the result of such calculations with

equation (B.1)

_we find~ the desired

joint probability

distribution P

if, fj.

^At

the first step ^of this program we write the _average

Ffi _using

_the

_replica

_method

as follows

/Dx(°IDR ^Z(°I jjZ(~l([

=

a~l)jjZ(~)([

=

l)

~ ~~~~

^~

Dx(°IDR

Z(°)

~~

_~~

©J#

where the

partition

^functions

Z(°I

_{and Z}

are defined

by

the

expressions (3)

and

(6),

_respec-

tively.

To calculate this _{average, ~~e} insert the external fields

h1°),h~~)

and

(~l)

_{in each of}

the

replicas

k ₌

1,..,m

and ₌

1,..,n

^and

expand

the

partition

functions in

equation (B.2)

into the _power series in the vectors

ul~),d(~)

and fields

h(°),h(~J,((11, (keeping only

linear and

squared

terms in this

expansion).

We do not go into details of the calculations since _{we use} here the _same

procedure

used to obtain the transformation

(14). Calculating

the

Gaussian

integrals

in

equation (B.2)

_over the coordinates

R, x(~l,

x(11 and the fields

hl~l, I(~), (k

₌

0,..

, m, =

I,..., n)

_as well _as

linearizing

the

non-diagonal

terms

(with

_respect to

replica indexes)

in

equation (B.2) by

the Hubbard-Stratonovich

transformation,

which introduces the random fields f and

I (see

transformation

(16)),

_we show that the

expression (B.2)

_can be

represented

in the form of

(B.I).

The correlator f

I,

_calculated

by averaging

the

product

f

I

with the

probability

P

if, ij,

^takes the form

~" ~~

~"*~ ~

~~ll~ ^{~~ 4}

+

Ne~2

(A

*

q)2

^~

~

~+ ~

4 ₊

(l~

₊

)~Nea2

(A

*

)2~

~~~fl _4(l~

+

1)

~~i~ea2(A

*

q)2

^'

^~~'~~

where A and A _{* q are} the vectors with the coordinates

l~

₌

a~l

and

l~q~, (~

₌

x,y,z),

respectively.