Rigid polymer network models

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Submitted on 1 Jan 1990

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Rigid polymer network models

J.L. Jones, C.M. Marques

To cite this version:

Rigid

polymer

network models

J. L. Jones

₍₁₎

and C. M.

_Marques

_(2,*)

(1)

Cavendish

_{Laboratory, Cambridge}

CB3 OHE U.K.

(2)

Ecole Normale

Supérieure

de

Lyon,

69364

Lyon

Cedex 07, France

(Reçu

le 6 novembre 1989,

accepté

sous

_{forme définitive}

le 20

février 1990)

Résumé. 2014 Nous

présentons

un modèle décrivant l’élasticité d’un réseau de chaînes

rigides.

La

variation du module

élastique

avec la fraction

volumique

de

polymère

est déterminée dans les deux cas où :

(a)

les chaînes sont

_rigidement

connectées entre _elles, c’est-à-dire que les seules

déformations

possibles

du réseau sont celles obtenues en courbant les chaînes

_(réseau

énergétique)

et

_(b)

les chaînes _peuvent s’articuler librement autour des

_points

d’interconnection

et une

_entropie

est alors associée à ces

_degrés

de liberté. Nous définissons une concentration

03A6c

de raccordement entre ces deux _{comportements.} Un renforcement du réseau

rigide

peut être effectué en introduisant une

_petite

concentration cP de bâtonnets de taille P

(beaucoup plus

grande

que la distance moyenne L entre

_points

de réticulation. Le module

élastique

se voit dans ces conditions

_augmenté

d’un facteur,

GP

~ _{cP P.} Nous discutons en conclusion le

_gonflement

et

le «

_{dégonflement »}

des réseaux

rigides

par le

changement

de la

qualité

du solvant. Nous

calculons une forme

_possible

du taux de

_gonflement

en fonction de la taille de la maille du réseau.

Le

_système

est instable pour un taux de

_{gonflement supérieur}

à _{v2 eq} =

(5/4)3.

Abstract. 2014 A model is

presented

for the

_elasticity

of networks

_composed

of

_rigid

chains. The elastic modulus as a function of

polymer

volume fraction is determined for two cases :

_(a)

the

chains are

_rigidly

crosslinked to each other i.e. the network may only deform

by bending

the chains

_(energetic

network)

and,

(b)

the chains are allowed to rotate

_freely

about the crosslink

points

so that there is _entropy associated with the crosslinks. A crossover concentration between the two

_{behaviours 03A6c}

is defined. The reinforcement of a

_rigid

_network,

_{by introducing}

a small concentration cp of

straight

rod-like elements of

length

P

(much

greater than the mesh size of the

lattice)

is found to increase the modulus

_by

a

_{factor, GP}

~ cP P.

Finally

we discuss how

rigid

networks may swell, and « de-swell » if the solvent

quality

is

changed

and calculate a

_possible

form of the

_swelling

ratio, as a function of the

_length

of the network chains. The _system is found to

be unstable for a

_swelling

ratio _greater than _{v2 eq} =

(5/4)3.

Classification

Physics

Abstracts

62.20D - 81.40 - 03.40D

1. Introduction.

Rigid

networks are

_present

in _many

_guises

in _nature, for

_instance,

silica

_gels,

and the fractal

structures

_arising

from irreversible

_aggregation

are often

_{rigid. Polymers}

which are

_naturally

stiff,

such as helical molecules

_{(PBLG), polydiacetylene, kappacarrageenan, polyamides}

and

peptides,

can

_aggregate

to form networks. We will model these networks as stiff elements

joined

at crosslink

_points.

The

_elasticity

of the network will

_depend

_upon the stiffness of the elements and the

_degree

of

_mobility

of the chains at each crosslink

_point.

It is

_important

to

_quantify

the

_rigidity

of the chain. Even classical flexible

chains,

entirely

dominated

_by

the

_entropy

associated with their

_myriad

of

_{configurations,}

are stiff on a short

enough

length

scale

_(that

of individual

_bonds).

Such chains

_behave,

for short

_enough

deformations,

as

springs

with

spring

constants which scale with the

_{temperature T,}

i.e. k

oc T

_(where

N is the

_degree

of

_{polymerisation}

of the

_chain).

The

_scaling

with T indicates

N

that the déformation is

_restricting

the number of chain

_{configurations, causing}

the

_entropy

to

drop.

Our chain will be

_rigid

in the sense of the

_spring

constant

_{being independent}

of

T,

the distortions are

_paid

for with bend _{energy. We} will first review a model of Ball and Kantor

_[1]

which

_quantifies

what level of stiffness is

_required

before the network is

_energetic

rather than

entropic.

Other discussions

_dealing

with the cross-over from a

_rigid

chain to a flexible one

may be found in references

[2]

and

[3].

Then we _{will go} on to calculate the

_spring

constant of a frozen chain with an

_arbitrary

_shape

in the limit of small deformations. From the

_assumptions

of affine déformation and

_rigidity

of the

_crosslinks,

_{will emerge laws}

_governing

the

_scaling

of the elastic modulus with volume fraction of the chains in the network.

Even in the case where the network is built from

_rigid

_chains,

its

_elasticity

may be dominated

_by

_entropic

rather than

_energetic

effects. This is

_possible

if the elements can rotate

freely

around the crosslink

_points

and the

_topological

constraints are « soft »

_enough.

This

problem

has been considered

by

Edwards et al.

[4].

The

scaling

of the elastic modulus with the volume fraction can be different from the

system

with

_rigid

crosslinks. These two different behaviours can be

_{simultaneously displayed}

and we will discuss the conditions under which each of them can be observed. We will

_mainly

concern ourselves with

_modelling

networks with

_rigid

_crosslinks,

so there is no

_entropy

associated with the

system

due to the free rotation of stiff elements about

_{hinges, although}

we will discuss

_entropic

contributions to the modulus in section 4.

Reinforcement of the

_{rigid gels}

can be achieved

_by

the introduction of extra chains in the network. We will calculate the increase of the elastic modulus of the

_network,

as a function of

the added concentration of

_rigid

rods and their size.

We shall conclude with a calculation of the

_swelling

and «

_{de-swelling »}

of

_rigid

networks with

_changing

solvent

_{quality. Again}

molecular

_weight

between crosslink

_points

is

_important.

2. Chain

_rigidity.

In order to arrive at a criterion for the

_rigid

chain

_limit,

in terms of

_polymer

chain

_parameters,

Ball and Kantor

_{[1] consider}

a chain in the form of

_{a frozen}

random walk. See

_figure

1. The chain is

_{represented by}

a walk of N

_steps,

each of

_length

a. The

_{equilibrium angle}

for the

_joint

between the i-th

_step

and the

_(i

+ 1

_)-th

_step

_{is 0 i,}

_{where 0}

_{may take any value} between 0 and 2 7T. The random walk is frozen in the sense that an _energy, U =

B ( P i -

() i)2 /2

is

required

to _{go from the}

_{equilibrium angle 0}

_to

_{0 i. B}

is the elastic force constant. If we then

consider

_applying

a force F to the ends of the chain the distortion can be

found,

enabling

the modulus to then be determined. In this paper we are

_{essentially only}

interested in two

particular limiting

values of the above

_quantities.

Let us consider the limit for which

Fig.

1. - A chain in the form of a frozen random walk of N _steps each of

length

a. The

_angles

between

consecutive steps are

_only

allowed to

_{change by}

small amounts when the force F is

_applied

to the ends of the chain.

which leads to the Gaussian chain result for the modulus and is to be

_expected

at

_high

temperatures.

Now

_taking

the

_high

_B/N,

low T limit i.e. when

_NkT/B

1,

the distortion is

which has a

_dependence

on B but no T

_dependence,

unlike the Gaussian chain result. It should be noted that an alternative _way of

_discussing

the two limits is in terms of the thermal

persistence length

of the molecular chain

_Qp

=

Ba /kT.

The

high

_temperature

limit

corre-sponds

to

_lp

L

_(L

is the contour

_length

of the

_chain),

so that the chain is flexible and we not

surprisingly

obtain the Gaussian chain result. The low

_temperature

limit,

however

corre-sponds

to

_{fp » L and}

so the chain is

rigid.

In the

_{following·discussion}

we shall limit ourselves to networks

_composed

of elements which are

_energetic

rather than

_entropic

in their

_elasticity.

These

elements,

e.g. the molecular

chains,

may be conceived as frozen chains with some fractal dimension

_{1 / v}

connecting

the size R to the mass distribution : R =

N’a,

where a is a

_length

of the order of the monomer

size. As an

_example

when v = 1 _we have _a network of rod-like chains.

3. Elastic response of

rigid

chains.

3.1 BENDING A RIGID ROD. - The deformation of

objects

with

_large

_aspect

ratios,

for instance rods or frozen chains with a fractal dimension close to

_unity,

is achieved most

_easily

for a

_{given applied}

force

_by

_bending

rather than

stretching

or

_compressing.

This notion is the basis for the

_elasticity

of cellular structures

_(Ashby

and Gibson

_[5])

and has been

_applied

to tenuous fractal

_{elasticity (Kantor}

and Webman

_[6]).

Applying

a force F

_transversely

to the end of a

_single

rod

_produces

a deflection X which

scales as

L3 for

a fixed F. See

_figure

2. In the limit of small deformations the rod

behaves,

as

expected,

like a

_spring,

with a

_spring

constant of

where E is the rod material

modulus, t

the cross-sectional

dimension,

and the constant

depends

on

_boundary

conditions and on the

_shape

of the cross-section. We

_ignore

details since we are interested

_only

in

_scaling.

In the case of a rod-like molecular

chain,

it is not

Fig.

2. - The rod bends when a force F is

_{applied, producing}

a deflection X.

3.2 BENDING A FROZEN CHAIN WITH AN ARBITRARY SHAPE. - The

_assumption

that the

polymer

chains are

_straight

rods _may be severe, and it is worthwhile

_looking

in more detail at

the conformation of the chains. A chain which is

_typical

of the stiff

_synthetic

_systems

we have

in mind is

_given

in

_figure

3. This has many benzene

rings

joined together,

and

_gives

rise to a

structure, which is not linear see

_figure

_4,

but has an

_{overall linearity.}

Stricter

_linearity

can be

obtained

_{by using}

molecules with

_linkages

of the

_type

shown in

_figure

5 which are discussed in

reference

_[7].

Fig.

3. - Part of a molecular chain which will deform

_by

rotation about the bonds.

Fig.

4. - A schematic

diagram

of a molecular chain formed from the _type of molecules in

_figure

3.

Fig.

5. - Linear chains can be formed

using

the _type of molecules shown _above, as in Husman and

As we have

discussed,

if the force constant

_preventing

deformation is

_{quite large,}

we have an

energetic

network. It may be the case

_though

that in the real

system,

over

_long

time

periods,

i.e.

_{long compared}

with the time for the modulus to be

measured,

the bonds _may

_flip

around and the structure exhibit behaviour more characteristic of flexible chains. The chains

may then have

entropic

contributions to _{their elastic energy and have} a

_temperature

dependent

modulus,

for

_long

time measurements.

Returning,

to short time modulus measurements, where

_{large changes}

in the

_shape

of the chain are not allowed it is

_possible

that the chain has a

_rigid,

but random structure. For chains which are not

_straight

we consider a more

_general

case i.e. that of a

_rigid

but

_{arbitrary shaped}

chain. This

_problem

has been considered before

_by

Kantor and Webman

_[6]

for fractal structures, Ball and Kantor

_[1]

_]

and Brown

_[8].

A

_simple

form of the

_type

of calculation is

given

below. Consider the

_{arbitrary shaped}

element in

_figure

6 with forces

_applied

to each end

(the

ends

_being

the crosslink

_points

that

_{mechanically couple}

the chain to the

network).

The

torque

on the i-th element is

_{B à 0}

_{i =}

Frr

where

_â8

_i is the

angle through

which the i-th

element of the chain bends

_through

as it is strained

and r/-

is the

_{perpendicular}

distance from

the

_point

of

_application

of the force to the i-th element of the chain. The

_change

in

_length

in the x-direction is where

àr§’

is the

_change

in the

_length

of the i-th element in the direction

_parallel

to the x-direction.

The

_{arbitrarily shaped}

chain

_behaves,

under small

deformations,

as a

_spring

with a

_spring

constant

It can be seen .that for v =

1/2

this

gives equation (2),

_so that

equation (2)

is

just

the

special

case of

_{equation (6),}

where the chain has the

_{configuration}

of a random walk. In the limit of

the fractal dimension

_being

_one, we recover the rod

_spring

constant of

_{equation (3) (using}

Ba =

Et).

For fractal dimensions

higher

than _one, the

equivalent spring rigidity

is smaller

by

a factor of

_{R/N ~ N v -1.}

As in the

_{preceding paragraph}

the calculation

_ignores

the

extensibility

of the

_rigid

elements

of the

chain,

since this leads to a less

_compliant

elastic

response.

4.

_Elasticity

of a network of

_rigid

chains.

4.1 FROZEN CROSSLINKS. - We first consider

_systems

where there is a fixed number of

chemical crosslinks which are

_permanent

and

_rigid

i.e.

they

are not

_{freely hinged.}

We now

want to determine the modulus of a network of this

_type.

A

_qualitative

argument

to derive the shear modulus is

_given

first and a more detailed and more

_general

_argument

is

_given

later.

If we

_apply

an external force to the network it will be transmitted

_by

the

_rigid

_system.

We make the

_assumption

that the

_system

deforms

_affinely

down to a

_{length-scale -}

0

_(R),

the distance between crosslinks.

_Although

it is not clear that this

_assumption

should hold for

_rigid

chains for the

_types

of deformations that we are

_considering,

we use it here as an

_attempt

to

obtain a

_simple,

first order model _{of the processes} involved when

_rigid

networks deform. If there is a force F

acting

on each

chain,

and a total of n chains in a volume

R3,

then the stress carried is a =

nF /R2.

Because of the affine

assumption

the strain of the

network will be

_{given by}

e =

X / R.

Young’s

modulus

G r

for small deformations is then

given

by,

We define the volume

fraction 0

of

_polymer

in our

_system

as follows. If there are n chains in a

volume

_R3,

each of volume -

Na3

(where a

is the

_typical

monomer dimension and the

cross-section t is taken to be the monomer

_{length a) then qb}

is

the làter relation

_relying

on R - N " a. In our

_modelling

we assume that n is small so that the

effect of

_{entanglements}

can be

_ignored.

The volume fraction may be varied

by

holding n

fixed

and

_allowing

N to _vary, or

by fixing

N and

allowing n

to

_change.

The later of these can be achieved

_{by controlling}

the

_{functionality}

of the crosslinks. If N is constant then we obtain the

following scaling

of the modulus with concentration.

If we fix n and allow N to _vary then we obtain a different

scaling

of modulus with

concentration,

as

_given

below

There exist more

_general

_arguments

to calculate the stress in a network which do not

_rely

on

defining

an area and a volume for each molecular

_chain,

see Doi and Edwards

_[9].

The stress across a

_given

_area, see

_figure

7 is then

_{given by}

the

_following

argument.

where

_Rm

is the

_position

of the m-th

_junction

and

_Fmn

is the force exerted

_by

the

_{point n}

on the

Fig.

7. - Definition of the stress tensor

U afJ. The component of the stress _{U az’} is the a _component of

the force

_{(per area)}

that the material above the

_plane

of area A and a distance h from the

origin

exerts

on the material below the

_plane.

not allow us to obtain a

_precise

result for the

modulus,

it is worth

_{following through}

the

argument

since the

_{underlying assumptions}

are drawn

_sharply

into focus.

Considering figure

8 there are three rods

_exerting

forces and

_torques

at m. The

_analysis

is

extremely

involved and so we introduce some

_{simplifying assumptions.}

These involve :

_(a)

Taking

only

the

_{reaction, -}

_Fmn

to evaluate the distortions of strut mn. This

_represents

the

action of the

_{counter-couples}

at n

_{by clamping}

and vice versa for the

_couples

at m.

_(b)

Assuming

all of the

_Fmn

is effective in

_bending

mn.

_(c)

_Assuming

that the deformation

ORmn

is affine. Then if we consider the deformation of one

_rod,

we obtain

where

_{(A - 1 )}

is the strain and

_{following (b)}

above we

_get

If we

_impose

a uniaxial strain

_{À zz’}

_inserting

the above into

_{equation (11)}

we obtain

Averaging

this

_using,

_{Rmn, Z}

= L cos 0 and

(cos2

0 >= 1/3

(which

_{assumes an}

isotropic

chain

distribution) yields

Fig.

8. - Derivation of the stress tensor

U zZ. The component

of Fmn

in the z-direction causes a

_change

in

For the modulus

_ÔUzz/ÔEzz

with _Ezz =

(Àzz -

1 )

we

_get

a result

_{differing only by unimportant}

factors from

_{equation (7).}

For a

_straight

chain Ba =

Et 4 and

in

equation (15) NS

is defined as

the number of chains in a volume V.

Returning

to

_{equation (10),}

we can choose to follow the

_{scaling properties by varying}

_cp,

the volume fraction of the

_{polymer, by allowing}

N to _vary, at fixed n. When the chains are

_straight

(fractal

dimension

_{1 / v}

=

1)

the elastic modulus scales _as the second

power of the

polymer

volume

fraction,

as is found for a wide class of cellular structures as shown in

_[5].

This is in

_good

_agreement

with the

_experimental

results of Rinaudo et al. on the elastic

modulus of

_{kappa-carrageenan gel}

which

_gives

a _power law

_dependence

of

G - o 2 --t

0.01 as

discussed in

_[10].

The chains of this

gel

are believed to have a _very

large persistence length

and,

indeed,

electron

_micrographs

show a structure which could be

described

as an

entanglement

of almost

_straight

chains see reference

_[11].

Another

_system

which could

_provide

a

_convincing

test of our

predictions

is

the

silica

gel

system.

These

_gels

have been characterized

_by

several

_experimental

groups.

By

varying

the

pH

of the reaction bath which

_produces

the

_gel

it is

_possible

to

_modify,

at least to some extent, its fractal dimension. Schaefer et al.

_[12]

_report

fractal dimensions obtained

_by

SAXS

_varying

from 1.7 to 2.3. We are not aware of _any

_systematic

measurement of the elastic modulus of the

_gel

as a function of its fractal dimension but results

_{reported by}

Dumas et al.

_{[ 13] give}

a

power law

dependence

of

_{G - 0 5 ±}

0.5,

consistent with the fractal dimensions of order of two

reported by

Schaefer.

The value of B in the above

_equation

may be estimated for a

_particular

_system,

that of

figure

3. It can then be used to determine the order of

_magnitude

of the modulus

_expected

for such

_systems,

and a

_comparison

be made with

_{typical experimental}

data. It is assumed that for

a molecular chain such as

_figure

_3,

the bend is achieved

_by

rotation about molecular bonds. Rotation about the bond shown causes the chain to bend out of the

_plane

_{of the paper. This is} shown in the more schematic

_{diagram figure}

9. This mechanism is different in detail from that of

_figure

1. We assume

_coplanarity

of the two successive benzene

_rings

and their connection is

a

quadratic

minimum of the energy, U = C m

2/2,

where m is the rotation about the CD axis

(systems

not

_obeying

this

_type

_{of potential}

are

beyond

the scope of this

paper).

The effect is to move end E from

_{coplanar position}

_El

to the elevated

_{position E2.}

For small cv the

displacement

is

_{perpendicular}

to the

_plane

and is

_{given by - w}

sin

yDE,

and the effective

angle

bent

_through

is a - w sin

yDE/CE.

The rate of

_change

of

_angle

with distance moved

along

the chain is a

/CE,

and the bend constant is Ba = B. CE. The energy for

bending

the

chain is then

_{given by}

U =

B (CE )2

( a

/CE

)2/2.

The

expression

for the bend constant

being

Ba = C . CE

( CE /DE

sin

y )2.

Then

by using

data

giving

the variation of U with w we can

calculate C and thus Ba.

_Obviously

the exact value of Ba

_depends

on the

_geometry

of

particular

molecules,

and so we

_{only give}

an order of

_magnitude

estimate for a

_typical

_system,

that of

_figure

3. We thank Dr. A. H. Windle for

_supplying

the

_potential

data.

_Assuming

that

DE/CE - 1/4,

sin y =

B/3/2

and that CE = a -

10-9

m we obtain Ra -

10-27

Jm. Then

_using

N -

_10,

_{and ~ - 10-}

we obtain a value for the elastic modulus of G -

106

Pa. In the work of Aharoni and

Èdwards

_[14]

the modulus is found to be of the order

106

_Pa,

and over the range

of

_temperature

_{investigated,}

it is

_independent

of

_temperature,

which fits the

_energetic

network model.

4.2 REINFORCEMENT OF THE GEL. _{- It may be of} some

_practical

interest to consider a

system

into which there has been introduced a small concentration _{c p of chains with} a size

Fig.

9. -

Diagram

to show how the bend of a molecular chain may be achieved

by

rotations about

bonds. The _segments BC, CD and DE are _part of a chain. When E is in

_{position El}

there is a

_linearity

a

long AEI.

If the _segment DE rotates about the axis CD,

through

an

_{angle w}

out of the

plane

of the

paper so that the end moves from

_{position El}

to

_position

_E2

then the line

_connecting

C and E is no

longer parallel

to

_AEI

and has rotated

_through

an

_{angle a}

from the direction of

AEI.

Therefore

successive rotations

along

the molecular chain

give

rise to

_bending

of the chain.

discuss the case

_{of straight}

chains

_{(of length P) being}

introduced into a network of rods’of size

L

_{(L «}

_{P ).}

When a force is

_applied

to the network there are two

_types

of chain-chain

contacts which transmit the stress. The first ones are the

_permanent

crosslink

_points

that have

been discussed in the

_preceding

section i.e. chemical crosslinks

_consisting

of covalent bonds. In addition to these ones there will be rod-rod contact

_points

which are

formed

when a P rod

meets other L rods

_along

its

_length,

we will call these contact

_points

extra links.

_They

are extra

in the sense that as the network is deformed the rods will form more of these contacts. This

type

of contact has been considered before in relation to

_rigid

networks

_by

Doi and Kuzuu

[ 15].

In

_{[ 15]}

the

_only

_type

of links in the

_system

are

_physical

_ones, formed

_by

rod-rod contacts

occurring during

deformation. In such a network it is found that the

_elasticity

is

_highly

non-linear. In the network which we discuss this non-linear

_elasticity

is not

_present,

at a first level of

_{approximation,}

since the stress carried

_by

the extra links is transmitted via the network of fixed crosslinks.

_{Non-linearity}

would

_only

occur at

_higher

concentrations of added P rods.

The extra _{links increase the free energy} of the

_system

and;

following [ 15]

we write this extra

free energy per unit volume as

where e (x)

is the amount of deflection

_imposed

to one rod

_by

the deformation of the

network. A

_rough

estimate of the extra _energy can be obtained from

_{d ),}

the

_averaged

deformation

imposed

on the P

_rods,

and

_dx2),

_the

_squared

_{average distance between} two

points carrying

the stress. The

_assumption

of affine deformation leads to

modulus of the

_gel

_by

a factor which increases

_linearly

with concentration and with the size of the extra

_rods,

_giving

a contribution to the modulus of

4.3 FREELY HINGED NETWORK. - We consider in this

_paragraph

a

_system

which is made from _{chains of average} size R

_meeting

at

_permanent

crosslink

_points

of

_{functionality}

z. The chains are

_{freely hinged}

in the sense that if one isolates from the network the z chains

meeting

at some crosslink

_point,

there will be no constraint on the

_angles

between the chains.

Thus,

in

such a

_network,

the constraints

_acting

on the crosslinks are

_only

derived from the

topology

of

the

_system.

Other work on

freely hinged

_systems

has been done

by Thorpe

and

Phillips [16]

and also Edwards et al. in

_[4].

We

_expect

that a _very

_{high functionality}

will

_completely

freeze the

_degrees

of freedom of the

_system.

This

_problem

has been considered before

_{by Vilgis et}

al. in reference

_[17].

In order to estimate the critical

_{functionality}

which would freeze the

_angles,

we describe the

_system

as an ensemble of N

_points

- the crosslink

_points

- each of them

connected to z

neighbours.

The

description

of the

dynamics

of these N

points

in a

d-dimensional _{space needs Nd} coordinates. There are, in a _very

_large

_system,

_Nz/2

constraints of the

_type

. The total number of

degrees

of freedom per crosslink

point

is then

If one wants to

_keep

some

_mobility

in a three dimensional

_system,

the

_{functionality}

needs to

be less than six. For z = 4 there will be a

_degree

of freedom per crosslink

_point.

We now assume that the

_{functionality}

is low

_enough

in order to

_keep

some internal

_mobility

in the

system.

At non-zero

_temperature,

the thermal fluctuations of the

system

determine its

susceptibility

where Y is the average vector

_joining

the two

_points

which

_represent

the extreme

_positions

of the end of the chain as the chain

undergoes

thermal fluctuations. Let a be the

angle

formed

by

two chains

_meeting

at a crosslink

_point.

The fluctuations in the

_positions

of the end of any

particular

chain

_depend

upon the fluctuations 6 a of the

angle

between the two chains about its average value of a. Therefore since R is the end to end distance of the chain we obtain the

following equation

for (

y2)

The

_{susceptibility y}

is the inverse of the force constant k. The elastic modulus can thus be

written as a function of the

_angle

fluctuations

8 a 2)

and the distance between crosslinks R :

The exact value

_{of (&a 2>}

_depends

in a _very

complicated

way on the

_topology

of the network and will remain as an unknown

_parameter.

From

equation (8)

we

_get

the

scaling

of the elastic

The

_elasticity

of this

_entropic

network of

_rigid

chains has a very different

scaling

from the

system

with frozen crosslinks

_(compare

with

_Eq.

_{(10)) :}

The

_entropic

and

_enthalpic

_type

of elastic behaviours can be

simultaneously

_present

in

actual

_systems,

_depending

on the

_rigidity

of the chains and on the

_magnitude

of the fluctuations of the

_angles.

At a

macroscopic

level there is an

imposed

stress u and the two

different contributions to the elastic modulus act in

_parallel.

The

_resulting

modulus G is then

given by

Comparing

the two contributions it is natural to define the crossover concentration

0 c

that can be written as

for

_straight

chains

_{( v}

=

1 ).

For volume fractions

larger than 0,,,

the elastic behaviour has an

entropic origin

and the elastic modulus follows the

_scaling

of

_{equation (24).}

For volume fractions smaller

_{than c/J c}

the deformation of the network is

_{accomplished by}

the

_bending

of the rods

_leading

to the

_scaling

of

_{equation (10).}

In

_practice

a

_{large rigidity}

of the chains

_(large

B)

and a

_large

thermal fluctuation of the

angles

(large 5 a

2) ) lead

to a very small crossover

concentration and the

_displayed

behaviour is

_entropic.

5.

_Swelling

of

_rigid

networks

When networks come into contact with solvent three main effects can exist. First there is the

preference

of the chains for each other or for

_solvent,

which

_{following Flory [ 18]}

and

_Huggins

[19]

is

_{conventionally expressed}

in terms of mean field

_theory

as

_giving

an _energy

_change

_per monomer site of

_{k TX 0 ( 1 - 0 )}

when

_mixing

occurs. y is the solvent

quality

and is

negative

for solvent

_absorption

into the network to be

favoured,

and

_positive

for

_{expulsion. Secondly,}

if solvent is

imbibed,

the chains must extend so

_reducing

their

_entropy,

and there will be an

elastic

_restoring

_force,

_opposing

extension. In the networks described in _{this paper, then} at

least at short

_times,

these forces will be

_energetic.

At

_long

times it is

_possible

that the chains

display entropic

behaviour due to bond

_{flips occurring.}

The third contribution is an

_entropy

of

mixing

when solvent is absorbed. This is

_only

_present

when the chains are

_flexible,

that is in

the

_long

time

_entropic

limit,

where

_{configurational changes}

are allowed. In the

_rigid

_system

no

such

_entropy

of

_mixing

term exists. In the

_general

case these three effects

_compete

with each

other and the

_system

reaches

_{swelling equilibrium}

_{when the total energy of the}

_system

is a

minimum with

_respect

to the amount _{of solvent taken up.}

In this section we first want to consider the case of a

rigid

network made of

energetic

chains,

and to ask the

_question

of how the network swells. The

_swelling

process

brings

into focus

several

problems

still to be resolved in the

_modelling

of

_rigid

systems.

- How much does

_rigid

chain

_overlap

effect classical

_swelling,

to what extent is Doi and Kuzuu treatment

_{required ?}

Over what

_length

scale is the chain deformation affine with the bulk ? A

_large

amount of work has been done

_by

Bastide et al. on the de-

_swelling

of

_gels

and their deformation

mechanism,

In

_rigid

_systems

the

_entropy

of

_mixing

term discussed above vanishes

_(because

no

configurational

changes

are allowed to

_occur),

and therefore we can

_only

consider the

k Ty 0 ( 1 - 0 )

term to be

_driving

the

_swelling.

Thus

_swelling

will

_only

occur

_{if X is negative}

so

that the energy can be lowered.

We make the

_assumption

in our model that for the

_swelling

_{process the} network is

_phantom

i.e. the chains may pass

through

each other. This is

_admittedly

a dramatic

_{simplification}

and

ignores

such difficulties as

_log-jams.

On the other hand we assume contacts between chains and solvent are described

_{by kTXc/J (1 - c/J),}

which

_implies

that the number of rod-rod

contacts scale

_{as 02.}

We shall use it however to

_give

a first

_{approximation}

to the

_swelling

mechanism. The total _{free energy of}

_mixing

of the network is written as

The elastic free energy is

_{given by}

where

_{h Ro}

= R and

_Ro

is the unstretched value of the end to end chain

_length,

and ns is the total number of chains.

Also,

for

isotropic swelling

the extension ratio À is

i.e. the

_quantity

_{V2 is} defined as the ratio of the initial and final volumes of the network.

Therefore,

if n

solvent

monomers mix with a fixed

_number,

_n2

_{of polymer}

monomers in the

network,

then _V2 =

(n2

₊

no)l(n2

₊

ni)

where no

is the initial number of solvent monomers.

The volume fraction of

_polymer

is

_{then cp = n2/ (n2 + nI),}

and the initial value is

00

=

n2/(n2

₊

ni) ,

_so that V2

= cp /CPo.

The total solvation energy

when n

1 monomers of

solvent are absorbed is

_{given by}

In order to find the

_{equilibrium swelling point}

the total _{free energy}

_Ftot

must be minimised with

_respect

to _{n1. This}

_gives,

It can be seen that if we consider the small

swelling

limit,

so that

_{V2 eq}

= 1 - 8 where 8 1 then we obtain

If we take the

large swelling

limit i.e. then we arrive at the result

It is

_interesting

at this

_point

to _{compare these results with the}

_{corresponding}

ones for a flexible network. For a classical Gaussian chain the elastic _{free energy is}

_{given by}

The energy of

mixing

of an

_entropic

network with solvent contains both an

_{entropic mixing}

term and an _{energy of} solvation due to solvent interactions. For the

_entropic

network and

taking cb o

= 1,

we have

For this

_type

of network the

_equilibrium

condition can be determined and is found to

_be,

In the

_{large swelling}

limit i.e.

_{V2 eq}

« 1 we

obtain,

This shows a

_significant

difference in the N

_depedence

of the

_swelling

ratios for

_entropic

networks and

_energetic

networks.

We can also look at _{how the network may « de-swell} », i.e. how solvent may be

expelled, by

using

a _poor solvent and

_allowing

the network to decrease in volume. This

_type

of behaviour will be seen in a

rigid

network

_{only if x >}

0. We have

already

seen that

swelling equilibrium

is

achieved when the _{osmotic pressure} ir

vanishes,

and ir is

given by

When X >

0 it can be seen that there is a

_region

in which the -

_{y v2term}

dominates and the

system

will be unstable. This will occur when

and

_substituting

in the condition at

_{equilibrium (32)}

we obtain

Experimentally,

the

_swelling

of

rigid

networks has been

_{performed by}

Aharoni

_{[14] by}

changing

the solvent

_quality.

The

_swelling

was

_performed

over several months and factors of

order two for the

_swelling

ratio were observed. The

_experiments

are done over

_long

time

periods

in order to allow the solvent to enter into the networks

_slowly,

otherwise the networks

are

destroyed by

the process. We have mentioned

_already

in section 3 that in addition to the

energetic

behaviour of these networks there could also be a different

_long

time scale behaviour. This would occur because of bonds

_flipping

over

_long

time

_periods

and cause

configurational changes leading

to

_entropic

behaviour. These

_long

time

_{(slow) flips}

would

give

rise to a

long

time

_temperature

_dependent

_modulus,

and in

addition,

because

configurational changes

can occur, the network could

_display

a conventional

_swelling

mechanism as described

_by

_FLory

_[18]

and

_{Huggins [19],}

if the time for

_swelling

is

greater

than the time for bond

_flips

to take

_place.

4. Conclusion.

We have

_presented

a model for the deformation of a network made of

_rigid

elements and

found

_that,

if the deformation is restricted to the

_bending

of the chains then the

_scaling

behaviour of the modulus with concentration under certain conditions

_depends

on the fractal

dimension of the chains. For

_straight

rod-like chains this

_{scaling being çb}

₂

and for chains which are frozen random

walks 05 .

The other feature of this elastic modulus

being

that it is

temperature

independent,

since it is derived from

_energetic

rather than

_entropic

effects.

We also consider an

_entropic

contribution to the modulus for the case where the crosslinks are

_{freely hinged}

and obtain a new set of power laws for the variation of modulus with

concentration. The two moduli i.e.

_entropic

and

_energetic,

have different

_scaling

with concentration and so we can define a crossover

_{concentration Oc}

above which the

_entropic

effects dominate the

modulus,

and it becomes

_temperature

_dependent.

If the network is reinforced

_{by introducing}

a concentration

_cp

of rods of size P into the

system,

then for P > L

_(where

L is the distance between crosslinks in the

_network)

the modulus is found to be increased

_by

a factor which is

directly proportional

to

_cp

and P.

The

_swelling

of

_rigid

networks

_{by imbibing}

solvent is a _very

_complex

problem

and we have

given only

a first

_{approximation}

as to _{the processes}

_{taking place.}

The

_{equilibrium swelling}

ratio

_{v2 eq}

is determined for the case where the crosslinks are

_rigid

for a

_general

fractal

dimension v of the

_chains,

however it is difficult to visualise how the network can swell for

v =

1,

and it is

possible

that the

analysis

breaks down before v = 1. If the network «

de-swells » it is found that an

_instability

exists when the ratio of initial to final

volumes,

V2 eq is

greater

than

(5/4)3.

Although,

there is

only

a limited amount of

experimental

data

currently

available for such

_systems,

the results of

_[12]

and

_[13]

_give

the modulus versus

concentration power law. The

_dependence

of this _power on the fractal dimension of the

chains,

agrees

qualitatively

with our

_predictions.

In

_[14]

the networks

_synthesised

have a

temperature

independent

modulus and to an order of

_magnitude

the modulus measured agrees

quantitatively

with what is

predicted

for very similar

systems

by

our model.

Acknowledgements.

The authors would like to thank M.

_Wamer,

J. F.

_Joanny,

R. C.

Ball,

S. F. Edwards and S. M. Aharoni for many

helpful

contributions to this work. One of us

_{(J. J.)}

is

grateful

for a

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