A Field Theory for Polymeric Networks with Excluded Volume

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A Field Theory for Polymeric Networks with Excluded Volume

Thomas Vilgis, Michael Solf

To cite this version:

Thomas Vilgis, Michael Solf. A Field Theory for Polymeric Networks with Excluded Volume. Journal

de Physique I, EDP Sciences, 1995, 5 (10), pp.1241-1246. �10.1051/jp1:1995194�. �jpa-00247133�

Classification

Physics

Abstracts

5.20-y 61Al+e 64.60Cn

Short Communication

A Field Theory for Polymeric Networks with Excluded Volume

Thomas A.

Vilgis

and Michael P. Soif

Max-Planck-Institut für

Polymerforschung,

Postfach 3148, 55021

Mainz, Germany

(Received

5

May1995,

received in final form 24

July

1995,

accepted

16

August 1995)

Abstract. In this _{note we} present a field

theory

for Gaussian

phantom

networks with _ex- cluded volume interaction.

Contrary

to earlier work which _was formulated

solely

_m terms of chain

variables,

the Deam-Edwards Hamiltonian of _a

polymenc

network is transformed to _a

special

_case of _a dn-dimensional

anisotropic O(m)

field

theory

_m the limit _n,m

-

0.

Using Flory-type

arguments we predict the _size of the network. Its

density

and the vulcanization

threshold _are

computed

from the saddle

point approximation.

For _an elastic affine deformation

we find from _our mean-field calculation that the essential parts of the free energy agree with the result

predicted

earlier

by

Deam and Edwards.

1. Introduction

It is

by

_{now a} well

accepted

fact that

randomly

crosslinked macromolecules _are

examples

for

equilibrium amorphous

solids

il, 2]. Still,

after _more than

twenty

_years ^{of research and}

despite

the undoubted _success of

phantom-type models,

a

microscopic understanding

of

polymeric

networks based _on statistical mechanics and

many-body theory

_seems to _{remain an} elusive task. Theoretical progress in this direction _was

mainly

hindered

by

the fact that

permanent

crosslinks in networks take _on the rote of

quenched

random variables. It _{was soon} realized that

a proper statistical formulation of the

problem requires replica

field

theory,

and this would lead to formidable mathematical difliculties

(indeed

the famous

replica

trick _{to treat}

polymeric

networks _was introduced

by

Edwards _{many years} before the

spin glass problem _[3]).

Recently

there has been _enormous theoretical interest in

polymeric

networks

mainly

based

on the Deam-Edwards model

[1-9].

In this paper we

report

_{on some} recent progress and

new

development

in the field theoretical formulation of the network

problem.

Dur

approach

utilizes the Edwards Hamiltonian and

replica

field

theory. However,

unhke the standard model

[1-3, 5,6]

where _an effective Hamiltoman _is

expressed

in terms of

generalized density variables,

the

problem

_is transformed to a field

theory

similar to the _one used in

polymer

solution

theory [loi.

This note is the first in _{a serres} of _papers which will contain _more mathematical details

and extensions. Parallel _to this

publication

_we will

report

on a new formulation that avoids

replica theory completely _[4].

This novel

approach

introduces _new

concepts

^{for the}

theory

^of

©

Les Editions de

Physique

1995

1242 JOURNAL DE

PHYSIQUE

I N°la

randomly

crosslinked

polymers

_[4] that allows

quantitative predictions

which _are not easy ^to obtain within the "classical"

replica

formalism.

2. Field Theoretical Formulation of

Polymer

Networks

The

breakthrough

in the theoretical and mathematical formulation of

polymeric

networks _came with the work of Deam and Edwards

[Ii.

We start from the

(dimensionless)

Deam-Edwards

Hamiltonian in

replica

_space for _a Gaussian

phantom

network with excluded volume interaction

~i ₌

~iw

+ _{7~~ +}

7~c

-

1( _£~

dS1°li~~~

~

(i)

+

~'

f _/~

ds

/~ _{ds'ô(R~(s) R~(s'))} ^~' /~

ds

/~ ds' jj

à

(R~(s) R~(s'))

2

a=i o o 2

o o a=1

The first _term in

(1)

^{is the} Wiener-measure that models the

connectivity

of Gaussian chains.

Monomer

positions

_are described

by

d-dimensional _vectors

Ris),

where the

arclength

_{s is}

assumed _{to run over} ail _monomers in the network. The index _a denotes diiferent

replica

sectors. The second _term

represents

^{the exduded volume} _energy ^which is

approximated by

_a

pseudo-potential

in form of _a Dirac

à-function,

and v' is the exduded volume

parameter.

^The last term is the most essential _one for the network

problem.

It is the eifect of

vulcanization,

and ~t' ^{is the} chemical

potential

for the crosslinks. It has been shown

il,5,8]

that

integration

of the

partition

function _{over ~1'} is dominated

by

the

steepest

^descent and that _~1' in

equation il

has the

meaning

of _a crosslink

density

_{per monomer.} An _extension of the above Hamiltonian to

many-chain

^networks is

straightforward.

For convenience _we consider here

only

_a ^network

formed from _a

single

chain of

macroscopic

size. It is

generally accepted

that this

simplification

has little _{or no} eifect _on the elastic

properties

of

highly

crosslinked

systems [Ii.

In

writing

down

equation (1)

_we have further assumed _an uniform distribution of crosslinks

[5].

^{To mortel}

a more reahstic vulcanization process from _a

liquid

_state, it is sometimes convenient to add

an extra

replica

to

generate

^the crosslink distribution

il, _6].

This introduces _an additional

parameter

ⁱⁿ the

model, namely

^{the exduded volume}

parameter

^{of the network} at fabrication.

To

keep things

_as

simple

_as

possible

_we will consider Hamiltonian

(1)

_as minimal

representation

of _a Gaussian

phantom

network with exduded volume. For a proper derivation of the above

equation

and lucid discussion _see for

example

reference [5].

The

important

observation in

(1)

is that the excluded volume term separates ⁱⁿ ail

rephcas,

whereas the crosslink term is

highly coupled.

It is this additional

coupling

between ail n

replicas

which leads to _enormous technical difliculties and

keeps

recent theories from

making quantitative computational predictions

without further variational methods _or other crude

assumptions.

In

previous

theories the Hamiltonian

(1)

has been treated

by

vanational _means

iii,

where the excluded volume and crosslink term have been modelled

uniformly by

_a harmonie triai

potential.

In this note _{we use a} diiferent route of

thought.

Guided

by

_our

previous

work _[7]

we atm for _a field theoretical

representation

of the

problem.

In contrast to an earlier

publication

where _we started from first

principles

to set up the field

theory,

_we work here in close

analogy

to the _case of _a

single polymer

_in solution. The

computational

method has been formulated in detail in references

[10-12],

and _{we can} be brief. The basic idea goes back _to Edwards

[13]

^and has also been used

by

Goldbart and coworkers [5] who introduced

generalized density

fields

Rif)

₌

/

_ds

à(F É(s))

,

(2)

~

in which ail _n

replicas

_are contained in _a

super-vector

F

=

(ri

_r2,

...,

rn).

A suitable order

parameter [5,

6] ^{is the Fourier transform} of

Rif)

Q(É)

=

/~

ds _exp

(iÉ É(s)) ⁽³⁾

The essential

point

to realize is that the crosslink _term in

equation il

_can be written in terms of _a dn-dimensional

super-vector

₌

(ki,

.,

kn)

in

replica

_{space, i-e-,}

7ic

₌

-~ /

_dé

_Q(f)~

=

-~ ~j _Q(É) Ri-É)

,

(4)

2 2

~

where ~ln =

2M/(N~V" ),

M is the total number of crosslinks and V the volume. It is

important

to note that the exduded volume term does not

require

the introduction of other collective fields such _as the classical densities. These _are

already cjntained

^{in the definition of the order}

parameter Q(É),

that is the

physical density p~(k)

=

/

_ds

exp

(ik R~(s))

_can be

expressed

o

by

_a

corresponding

field

p~(k)

₊

Q(É~)

₌

Rio,.

,

0, k,0,.

.,

0).

Here _we have introduced _one-

replica

wavevector

É~

=

(0,

...,

0, k, 0,..

,

0),

where the

only

_non-zero

component

_appears at the a-th

position.

We _{are now} in _a

position

to set up the field

theory.

To do

this,

the Hamiltonian is rewritten

by

_means of the definitions

(3)

and

(4). Setting

v =

v'IV, equation il

becomes

7i =

f _/~

ds

~§~~~~

^~

+

i f _£

Q(É~) Q(-É~) ) £ _Q(É) Ri-É) ₍₅₎

2

~=i o S 2

~=i ç ç

The

one-replica

sector of this Hamiltonian has been used

by

_us

previously _[14]

to

compute stability

criteria for

semi-interpenetrating

networks and crosslinked

polymer

blends.

By

one-

replica

_{sector we mean}

equation (5) considering only one-replica

wavevectors

É~.

In this _case it

is

easily

_seen that the network

problem

is

equivalent

to _a

polymer

_in solution with _an effective excluded volume _v~~~

= v ~li that is

responsible

for the

thermodynamic phase

behavior

(see

references

[8,14]

for _more

details).

So far most of the _more recent works _on

polymer

networks have used

equation (5)

as a

starting point

^{for further}

approximations

and

investigations.

In contrast to

this,

the above Hamiltonian is _now turned into _an exact field

theory

in the standard _manner. To derive the field

theory

for

randomly

crosslinked

macromolecules,

_we

decouple

the _pair interactions

by

random fields

using

^the Hubbard-Stratonovich transformation.

Obviously

there _{are two} fields needed: _one for the excluded volume term and the other _one for the crosslink contribution.

The basic diiference between the two is that the crosslink term _is

attractive,

whereas the exduded volume term is

repulsive.

This _{poses some} well-known

problems

in the field

theory

of networks. The addition of crosslinks to

polymers

in solution creates _an attractive interaction between _monomers and tries to shrink the

polymer

to _a

collapsed

bail of certain size. This

collapse

is

prevented by

the

repulsive

^exduded volume forces. In ail classical theories exduded

volume forces _are

ignored

and _a

density

constraint has to be

employed

to

keep

the

phantom

network at finite _size with finite

elasticity _{[2, 8].}

Applying

the above transformations

[loi

the

two-point

Green function of the functional

integral corresponding

to the Hamiltonian

(5)

_can be written _in terms of _a

generalized

diffusion

equation

lE ^Vi +1~j

_4l

_(fa

_{+ flf}

_if)

G

(F, F'; E)

₌

à(f f')

,

(6)

2

~i

1244 JOURNAL DE

PHYSIQUE

I N°la

where E is the

Laplace conjugate

to the total chain

length N,

and

é«

₌

(0,.

.,

0,

_r«,

0,..., 0).

Here 4l

if)

is the Hubbard-Stratonovich field needed _to

decouple

the excluded volume interac-

tion,

^and flf

if)

is the field that

decouples

the crosslink term. The latter

equation

^resembles in- deed _very

closely

the Green function of _a

polymer

in solution

[10-12]. Alternatively

G

(f, f'; E)

can be

represented

_as functional

integral

_over Gaussian fields

by

_use of the

operator

^defined via

equation (6).

After

employing

this standard

procedure

for

setting

_up

polymer

field

theories,

we arrive at the

following

e~act network Hamiltonian

7i ₌

Il

dé_q~

if) (2E i7()

q~

if) ii)

4

+ v

/

_dé

q~(f)

q~(P)

/ _{dé'q~(P') q~(f')} ^f _{à(r« r'«)}

~ln

/

_dé

q~ if)

q~

if) )

,

~_~

where

q~(é)

^is now a

m-component

vector field of _a dn-dimensional variable é.

Contrary

to earlier work [9] in which _a similar but scaiar field

theory

for

polymer

networks _was

constructed,

the

amsotropic O(m)

field

theory

contained in

ii)

is _an exact

representation

of the Deam- Edwards mortel. In the remainder of the note we will discuss _our

findings briefly.

Mathematical details will be discussed in _a

forthcoming lengthier publication.

3. Discussion of Results

To

get

a

feeling

of the contents of the field

theory,

it is first turned into _a

Flory theory.

For this it is useful to note the

correspondence

of the

polymer density

Q and the field

q~

[10,15]

RIF)

= v7

lé)

_v7

lé) ₁₈₎

The first

part

of the Hamiltonian

ii) involving

the

Laplace conjugate

E fixes the number of monomers, I.e.,

J

dé

nié)

= N. The free

part

of the field

theory

in

equation ii)

is

generated by

the Wiener _measure and thus

corresponds

to _an "elastic" chain

entropy.

The second term

contains information about the exduded volume of the network and is

repulsive,

whereas the third term is _an attractive interaction due to the preseuce of crossliuks. The three contributions

can in _a

Flory

_sense be written _as

n(+nv(~-~ln$, _j9)

where R is the

macroscopic

size of the network.

Inspection

of the

Flory

free _energy in the

replica

limit hm à

Ian yields

the

following expression

for the free _energy

n-o

F _m

(

+ v

((

₊

_{2~1odN~ log}

_R

₍₁₀₎

The first observation _is that the crosslink term creates a

logarithmic

contribution. Such

loga-

rithmic _{terms are}

typical

for disordered

polymer systems [16,1?i

and do net appear in solutions

or blends. In the uncrosslinked _case when _~lo

" 0 the

polymer

is swollen in solution. When-

ever the crosslink

density

_is

sufficiently large

the elastic

part

^{of the}

Flory

free _{energy con} be

neglected,

and the network size is R _cc

iv _/~lo)~~~

which is obtained

by minimizing

the

Flory

free _energy with

respect

to R. Thus the size of the network is determined

by

the balance of the excluded volume and the crosslink

density

_{in a}

physical

sensible way. It is rather remark- able that this crude estimate

by

the

Flory argument

agrees with the

expression

^of Ball and

Edwards _[8] obtained

by

_a variational calculation.

It is also

interesting

to note that the field

theory

defined

by equation ii)

_agrees for _v

= 0

with that derived

by completely

diiferent _means in references

[7,13]

for Gaussian networks with

functionality

four. Thus trie method described here

yields

the _same field

theory

_as obtained

previously.

In

iii

_{we were}

mainly

interested in mean-field solutions to the field

theory

with _a

constant

density. Similarly

_we

expect

here that the

partition

function _can be estimated

by

the saddle

point approximation.

The

steepest

^{descent of trie field}

equation ii)

is

given by

E9'i~) Vi _9'i~)

₊

_9'i~) V É / _d~'9'i~') '9'i~') ôi~a ~'ai ~9'(~) '9'(~)

" °

(~~)

This

equation

is

impossible

to solve in

general,

but in close

analogy

with

[7,13]

we look for _a constant

density

solution. To obtain _a criterion for

phase stability

it is sufficient to consider trie

one-replica

sector that has been shown to be relevant for

thermodynamic quantities

such

as

scattering amplitude

and

phase

behavior

[14].

^{This is} so because the attractive crosslink term has to be counterbalanced

by

the exduded volume in the

one-replica

sector to avoid

collapse.

In this _case it is easy ^to show that the mean-value of the

physical density

p ^is

given by p

=

1/(N

_[~li

vi ).

^This is _a well-known result first derived

by

Ball and Edwards [8]

by completely

^diiferent _means. In the latter

equation

_we have used the fact that the

Laplace conjugate

in the

steepest

descent is E _cc

1IN. Stability analysis

of the Hamiltonian shows that _a critical crosslink

density

~L~ for the

liquid-solid

transition

exists,

for which the

theory

becomes unstable. It is given

by

~1~= +v

(12)

Nfl

which _can be

interpreted

_as vulcanization threshold. This result makes

physically

_sense. The first _{term agrees} with the vulcanization threshold for

long polymer

chains

[18]

^{which is} propor-

tional to

1IN.

The _reason is that in the vulcanization _process

considered,

each _{monomer can}

participate

in the

crosslinking

_process. Thus the effective

functionality f

is of order

f

_cc

1IN.

In the absence of the exduded volume interaction the

system gels earlier,

_1-e-,

~1~ is reduced since the

probability

for crosslink formation should be

larger

when the chains

experience

no

excluded volume interaction. In _a melt situation the excluded volume interaction

gets

^screened

to _{v cc}

1IN,

and _{we recover} the _case discussed earlier

by

de Gennes

[18].

Zeroth order fluctuations _cari be estimated from the saddle

point equation. Using

the _con- stant

density approximation

to linearize the saddle

point equation,

the correlation function

Go

in

replica

_{space is} found _to be

Go(É)

_oc

(13)

~2 ₊

(-2

For the correlation

length (

we obtain

(2

oc

1/[~Li vi

which is similar to the _case of _{a concen-} trated solution

[8].

The full

replica

Hamiltonian

il)

_can be

easily generalized

to treat the _case of _an

affinely

deformed

polymer

network.

Employing

the affine deformation

assumption [1,9]

the mean-field free _energy

density

F is calculated _up to _an irrelevant constant. Flom

equation ii)

we find

that

F =

v'p~

₊

~L'~j À)

,

(14)

where

Ài's

are macroscopic

stretching

ratios in the1

= ~, y, z directions. This result is very similar to the _ones derived

previously by

Deam and Edwards

iii

and

Panyukov _[9].

The first term is the total exduded volume energy, whereas the second term

yields

the well-known

Neo-Hookian law for _an

elastically

deformed network.

1246 JOURNAL DE

PHYSIQUE

I N°la

4» Conclusion

In summary, we have

presented

the field theoretical formulation of the Deam-Edwards mortel for

polymeric

networks. The

Flory

estimate of the e~act field Hamiltonian contained the dassical

theory

of solutions and

predicted

the size of the network _{as a} detailed balance between exduded volume forces and

crosslinking

constraints. This

prediction

_was not contained in the dassical

phantom-type

theories in _a

quantitative

manner. The field

theory presented

here is

an extension of the field

theory

derived earlier

[7, 9,13]

and contains this work _{as a}

special

case.

Using

^mean-field

arguments

we obtained _an

expression

for the vulcanization threshold

of crosslinked

polymers

and the free energy of _an

elastically

deformed rubber. Dur

predictions

agree with those of Ball and Edwards obtained

by

variational

path integral

calculation. In the

present

note _we

reported briefly

_on recent

results,

but mathematical details and extensions will be

published

in _{a more} extended paper.

Acknowledgments

The authors have to thank Y. Rabin for

sending

us

preprint [19]

^{after this work} was

completed.

The authors _are also

grateful

to the _anonymous Referee who

brought

reference _[9] to _our attention. Financial

support by

the Deutsche

Forschungsgememschaft,

SFB

262,

is

gratefully acknowledged.

References

iii

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F.,

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A.

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