The size of a single chain in a melt of anchored chains

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The size of a single chain in a melt of anchored chains

M. Brereton, T. Vilgis

To cite this version:

M. Brereton, T. Vilgis. The size of a single chain in a melt of anchored chains. Journal de Physique

I, EDP Sciences, 1992, 2 (12), pp.2281-2292. �10.1051/jp1:1992280�. �jpa-00246700�

J.

Phys.

I France 2 (1992) 2281-2292 _DECEMBER 1992, PAGE 2281

Classification Physics Abstracts

05.40 61.25H

The size of

_a

single chain in

_a

melt of anchored chains

M. G. Brereton

(I)

and T. A.

Vilgis (2)

(1) IRC in

Polymer

Science and

Technology,

The

University

of Leeds, Leeds LS2 9JT, G-B- (2) Max Planck Institut fur

Polymerforschung,

Postfach 3148, 6500Mainz,

Germany

(Received ⁷ April J992,

accepted 28Aagast

J992)

Abstract. This paper considers concentrated polymer systems ^where the translational degree ^of freedom of all the chains _{to move} has been

quenched,

ln

particular

_we consider the

screening properties

of this constrained environment and the

subsequent

effect _on the size of _a

single

chain.

The model consists of many flexible chains, each anchored at one end to fixed

points randomly

distributed in space. The model has

applications

to network

problems.

The

quenched

chain end variables _are handled

by

_a

replica

calculation and the chain

configurations by

_a random

phase approximation

in

replica

_space. After

integrating

_over the

density

fluctuations present ^{in the} constrained environment the

problem

is reduced to a single chain

interacting

with

replicas

of itself through an effective interaction

potential.

Unlike the _case for _a melt the effective interation is _not

completely

screened. In

particular

the inter

replica

interaction, which arises

specifically

from the constraint, is attractive. A

perturbation

calculation confirms that, not

withstanding

the lack of

screening,

the Gaussian nature of the chain is _not affected. The fixed chain ends

only

lead _{to a} contraction of the chain of the order N~ "~, where N is the number of segments ^{in the chain.}

1. Introduction.

In _a

polymer

melt it is well known that the size of _a

single

chain is

given by

^its Gaussian dimension. I-e-

R~

=

b~N,

_{where b is the effective segment}

_length

_{and N the number of}

segments ^{in the chain. This}

original conjecture, by Flory Ii,

was confirmed

mathematically

by

^Edwards

[2]

who showed that the

repulsive

excluded volume

potential

between the

monomer segments ^both on the _same chain and between chains is

effectively

screened

by

_an

attractive interaction induced

by

the

density

fluctuations in the melt. The _same is also _true in _a

compatible blend,

but with _some modifications due to concentration fluctuations

[3].

In both

cases the

density

and concentration fluctuations, _on which these results

depend,

are

given by

the celebrated de Gennes random

phase approximation (RPA) [4].

This

approximation

is valid for dense

polymer

_systems with translational freedom such _as

melts and concentrated solutions. However there _are many other concentrated

polymer

systems, ^such as rubber networks, where the chain

configurations

_are similar to those in _a melt but the translational freedom of the chain to move is restricted. The absence of this

degree

of

freedom negates ^{the immediate} use of the

RPA,

since the translational symmetry ^{of the} system

is broken

resulting

in the

density

fluctuations p~ and

p~ belonging

to different wavevectors k and q

being coupled together.

In effect the

quenched

variables _must be accounted for first

by

means of a _«

replica

» calculation

[5]

in order to restore the translational

symmetry.

^The

coupling

between

density

fluctuations

belonging

to different wavevectors is then eliminated and the RPA _can be used for the

(annealed) replica

chain

configuration

variables.

In _a

previous

_paper

[6]

_we considered the

simple

_case of _a

polymer

melt where _one end of each chain _was anchored to _a fixed

point

in space. For this model

quenched

_{system we} showed how the

density

fluctuations could be

explicitly

calculated and

compared

them _to those in the

unrestricted melt. The

major

conclusion from this work _was that the

density

fluctuations in the

quenched

system _are

virtually

identical to those in the melt _{out to a wavevector} scale q ~

(N~'~R)~

~, where R

= N ^~/~

b,

is the size of _a

single

chain.

Beyond

a wavevector

scale,

set

by

the extended

length

of _a chain

(q

_~

(Nb )~

~), the effect of the anchored chain ends dominate the

density

fluctuations. Similar conclusions have been

reached

by Panyukov [7]

for

topologically

disordered networks.

In this paper we examine the _{extent to} which the additional

density

fluctuations in the

quenched

system ^{interfere with the}

screening

of the excluded volume interactions. We achieve this

by calculating

the size of _a

single

chain with _one end fixed _at the

origin

in the

quenched

system. ^The calculation is

performed using

the

replica

method

[5, 6]

and

gives

rise _{to an} effective interaction between the

replicas.

The

intra-replica

interactions _are similar to those

present

in the melt but differences exist

especially

_{as q ~} 0. The

inter-replica

interactions _are attractive and

depend directly

_on the distribution of the anchored chain

ends, they

_are not screened under any circumstances. The first order

perturbation theory expressions

for the size of _a

single

^chain shows that the

intra-replica

interaction leads to the _same term that Edwards

found for the melt I.e, _a chain

expansion independent

of N, while the

inter-replica

interaction

produces

a

slight

contraction of the chain dimensions but that this vanishes _as N~ ~/~.

The paper ^is

organised

_so that most of the

algebraic

^details of the calculation _are

relegated

to

appendices.

2. A

replica

calculation of the size of _a

single

anchored chain.

Consider the chosen chain _to be the _one with _one end fixed at the

origin,

so that the

position

of any

segments

s on this chain is described

by

^the vector r~, s = i

N,

where N is the number of segments per chain. All of the other chains will be labelled with _a

superscript

a =

I

Nc,

where

Nc

is the total number of chains. The fixed

points

to which each chain end is attached _are described

by R]

and represent ^the

quenched

variables in this

problem.

The annealed variables _are the

configurations (r[)

of the

chains,

where

r)

is measured from the fixed

point R].

The notation is illustrated in

figure

I.

The chains interact with each other

through

_a

repulsive

excluded volume interaction

U(R[ R$).

The average

(end

to end

distance)2

of the

single

chain anchored at the

origin

depends

_on the

quenched

variables

(R])

and is

given by

the usual statistical mechanical formalism in _terms of _a

generating

function

Z(J R]).

in

_,

IRS i

=

~

_{In Z}

_iJ

,

RI ij

o

where

Z

(J

_;

R])

=

exp ~ _£

^~

_{( (r~), (r[,)}

;

R] )

₊

Jr( (2,i)

S[

~~

°

N° 12 SIZE OF A CHAIN IN A MELT OF ANCHORED CHAWS 2283

fixed chain

end chain a

s

r~

R~f

~ ^«

origin

^°

Fig.

i. Schematic illustration of the notation used. The

single

anchored chain of interest has _one end fixed at the

origin

and is shown _as the bold _curve. The other chains _are anchored at the fixed

points

(R~)

12 is the volume of the system ^{and the}

averaging (.. )

~

is done _over the

unperturbed

annealed variables

(r~, r$).

The excluded volume interaction U is

split

into terms

involving

the

coordinates

(r~)

of the

single

chain _at the

origin

and the _rest of the chains

(R[

in the system :

~

( (r~), (R[ )

=

V

(r~ r~,)

+ V

(r~ R$ )

₊ V

(R[ R$ )

where

R)

₌

R]

+

r) (2.2)

Little progress can be made with the evaluation of

(2.i)

^until the average over the

quenched

distribution of the chain ends

(R])

has been

accomplished.

This is done

using

the

replica identity

_:

In Z

=

lim

(Z~ 1) (2.3)

n ~ o ^~

and

writing

Z~ in the

replicated

form

Z~

=

fl

_exp-

£

^~

((r[), (r)") ;@.

(2.4)

~

n kT

«=i ss. o

~

a

The average

(denoted by

_a

bar)

_over the

quenched

variables

(R])

then has the _same status as the

averaged (denoted by angle brackets)

over the annealed variables

(r[").

The

averaging

over the

configurations

^{of all the}

chains,

other than the chain fixed _to the

origin,

is done

using

the Edwards method

[2]

^of

changing

variables from the chain coordinates to collective

density

variables. We

adapt

^this

approach

to the present

replica problem by

a

change

of variables from

R["

to

replica

collective

density

variables

(p( (R]) ),

where

Pi (RI

=

n

Z

_exP

iq (RI

₊

r[" ) (2.5)

as

The details of the transformation and

subsequent averaging

_over both the

quenched

chain end variables

(R])

and the collective

density

variables

p(

are

given

in

appendix

i. The result is that the

problem posed by (2.4)

is transformed _to

~~

~ ~XP

( _{i ~l« (~~ ~$')}

+

~~~~l(2.6)

~)[, ^°

and is

equivalent

to a

single

chain

interacting

with itself and

replicas

of itself

through

an

effective interaction

V$~,.

An

expression

for

V$~,(r~ r~,)

is derived in

appendix

i in _terms of its Fourier

components.

^Set

V$~,(r~

_r~,

)

₌

£ V$~,(q )

_exp

iq. (r~

r~,

q

then _{as a} matrix in

replica

_space

V$~,

is

given by

~h~~~

V*

= V~ I

e~

^U

(2.7)

~

V~

and

~

+

A~ Vq

/~o ~72

e =

q q

~

(i

+

Al vq)~' ^(2,g)

In

(2.7)

I is the unit matrix and U is _a matrix with unit elements

everywhere.

The second _term

(e~)

in

(2.7)

is the _same for all

replicas

^{and is} attractive. The functions

A(

^and

D(

are

given

in

appendix

I _as

Al

= w

Iii _~qi~l~ 117~q>~l~)

and

Di

=

cq I1<~lq>~ l~l ^(2.9)

where _{11~ =}

£

_exp

iq

r~

s

and _{w =}

N~/12

is the concentration of chains

C~

^{is the structure factor of the} distribution of the anchor

points,

which for _a random

distribution of chains is

given _by C~

₌ p.

The end to end distance is

given

from

(2,I)

as

j

~

ij~ ^I

f _r(~

exp

£ ^V$~,(rl' r[")1(2,10)

n ~0 ~

« i

~ ~

s«' o

s. «.

N° 12 SIZE OF A CHAIN IN A MELT OF ANCHORED CHAINS 2285

In the next section _an

explicit expression

is

given

for the

replica-replica

interaction

potentials V$~,

and their

screening properties

discussed.

3. The

replica potentials

and

screening.

A screened

potential V~~(R)

is _one for which the

inegral

_over all space is _zero. I.e.

d~RV~~(R)

=

V~~(q

=

0)

=

0.

(3,I)

The q =

0 limit of the

replica potential (2.7)

is

given by V$~,(q

= 0

)

=

V8~~, Np V~

(~

(3.2)

since from the definitions

(2.9)

^of

A(

^and

D(

A(

= p

7~~(~) (7~~) (~)

^{= 0 at q = 0}

o O

D(

=

C~ _{(7~~) (~)}

=

pN~

_{at q}

=

0.

(3.3)

Consequently

the effective

replica

interaction

potential V$~,(q)

is not screened for the melt with anchored chain ends. To

investigate

the full q

dependence

of the

replica potential

we need

explicit expressions

for the _terms

A(

^and

D(.

In

appendix (2)

_we derive

approximate analytic

forms for

A(

^and

(7~~)

as

A(

=

~

~~(~~

^and

(

_7~~)

=

~

~

(3.4)

(1+Q ) (i

+

Q )

where

Q~

₌ Q~

Nb~/6.

Then from

(2.7

and

2.8)

V$~,(q)

=

~~~ ~~~~

~

~

8~~, ~~~£~~~

^~

^~~~~

~ ~.

(3.5)

(1+ Q

+ 2

CNVQ [(i

₊

Q )

+ 2

CNVQ

For wavevector scales smaller than the chain size I-e-

Q~

»

v1«(q)

= ~

Q~

^V

&~

wv2 N2

~

~ ~ ~~~ ^"

[Q~

+ 2 CNV

]2

~

Q~

b~

_i

~~ ~~ Q~

~~

₊ i

~""

4 CN

iq~ f2

₊

_1j2 ^(3.6)

where

f~

=

b~/(12

cV is the Edward's

screening length.

The first term of

V$~,

i-e-

v~,

^the coefficient of the unit

replica matrix,

is identical to the Edward's screened

potential [2]

for _a

polymer

melt. The second term is due to the anchored chain ends and represents an effective inter and intra attractive

replica

interaction. In terms of the

configuration

variables

(r[)

it has _a

purely exponential

form

w exp

(- (r[ rl')/f).

However the

magnitude

decreases _as N~ and in the next section _we

demonstrate, using

first order

perturbation theory,

that the effect of this

replica

interaction leads _{to a} weak contraction

mN~~/~

_{of the size of the chain. Thus}

our results demonstrate that _not

withstanding

the apparent ^{lack of}

screening

of the

replica _potentials

the

single

chain

properties

_are

only weakly

perturbed by

the anchor constraint.

4. Perturbation calculation of the size of _a

single

chain.

From

(2,10)

the size of _a

single

chain is

given by

W

=

iin~ Z ri~

exp

£ _{Z vim, (rl'} rl")1(4.

i)

n ~ « l s«' _o

s. «"

To first order in V * this becomes

w

=

jr~ j f

ir121~ £ ^r12 _{z vi<}

«~

(ri' i")1) ~.

(4.2)

~ ~ ~~

~~

Since there is _no

replica breaking _symmetry

in the

problem

the _{sum over} each

replica

is the

same and hence

I 2

m2

£

_~7 *

(

^«'

«")

~

~)

~~N) ~~N)o

_{~ f~} ^~N «'«" ~S ~S'

s«' o

s«'

where _« is _{now a} fixed but

arbitrary replica

index. Write

Vi,

~o

(r[ r[" ₎

in terms of _a Fourier transform _as

ZVI, ^«,(rl'- rS")

=

£ Vl,«<,(q) £

_exp

iq(r[ rS")

=

£ Vi, ~,,(q) ~j'~j' (4.4)

SS' q SS q

Then in the _{sum over} «' and «" in

(4.3)

the

following

combinations of

replica

indices lead to inter and

intra-replica

interactions and must be

separately

considered.

Intra-interactions

V$~(q)

= v~

e~, given by equation (2.7) (I)

~r'# ~r"

= ~r

(~)

_~~q

_~q) ((~N "lq( ))o

(11) ^«' # «, «"

= «'

(n

I

(v~ e~) (r(

7~~

~) (4.5)

Inter-interactions

V$~,(q)

= e~,

given by equation (2.7) (iii)

«'= _«, «" # «

12(n

i

)I

Eq

in ~ql~ 11J-q>~

(iV) «'#«,

«"#«'#«

((n

I

) (n 2))

_Eq

(r~)~ l'lq)~( (4.6)

The _terms in the

( )

brackets _are the number of times each _{term occurs.}

The result for

(r()

_can be written in the limit _n

~ 0 _as

w _irii~

=

£ _{z i(vq eq) in}

1~lqi~l~ irii~ Ii _~lqi~l~l

q

+ 2 e~

(r(

_{7~~) (7~}

_{_q)} (rji) l'1q)

~

~) l (4.7)

N° 12 SIZE OF A CHAIN IN A MELT OF ANCHORED CHAINS 2287

The

unperturbed

_averages

(rj _(7~(()~,

_etc,

_occurring

in

(4.7)

_are evaluated

using

the Gaussian chain model in

appendix (2).

For

large

N and

Q~

» i

they

are

given by

~q _~~~ ''lq'~)o _{~~~)o ~q~~~o~} ^~~~ (~ ^~~'~~

B~

=

(r(

_7~~)

_{(7~_~)~} (r()~ _(7~~)~(~)

=

Nb~

^~

~~ (4.9)

o 3

Q

and from

(2.8)

in the

Q

_~ i limit

Q~

V

~~

Q~

+ 2 CNV

(4,10)

~~2 ~2

~~

Q2 jQ~

₊ 2 CNV

j2

The _{sum over} q in

(4.7)

is converted into _an

integral

to

give

@ _jr()

=

~ "

l~ ^{Q~ dQ (v~ A~ e~(A~}

²

^{B~)) (4.}

^Ii

⁾

°

(2

_ar )~

(Nb~)~'~

Q~

The cut-off _wavevector

Qc

^is necessitated because the

expressions

used _are

only

^valid for

Q~

» I. To obtain the

qualitative

features of this result it is sufficient _{to set}

Qc

₌ ^I. The

integrals

are

readily

evaluated to

give,

for VN _~ l

This is the

major

result of this paper. The first _term is the

unperturbed result,

the second

comes from the

intra-potential VI

and is the _{same as} the Edwards' result for the unconstrained

melt,

while the third term _comes from the

inter-replica potential

_e~ and represents ^{the effect of} the fixed chain ends. For VN

~ l this contribution is

independent

of V and leads _{to a} small contraction of the chain. However the effect vanishes _as N~ ^~/~

5. Conclusions.

In this paper we have considered the effect _on the

single

chain

properties

of

quenching

the translational

degree

of freedom of the chain _{to move}

by anchoring

_one end to a fixed

point

in space. The model has direct

applications

to network

problems,

to which _we will _retum in _a

future

publication.

In _a melt the Gaussian character of _a

polymer

chain

(R(~N)

_is

maintained, despite

the excluded volume

interactions, by

the

screening

of these interaction

by density

fluctuations. In _a

previous

_paper

[6]

_we considered the effect _on these

density

fluctuations of

quenching

the translational

degree

of freedom and showed that for small q there _were

significant

contributions to the

density

fluctuations. The

possibility

therefore

arose that

they

could

significantly

interfere with the

screening

^of the excluded volume

interaction.

In this present _paper we have considered the influence of fixed chain ends _on the size of _a

single

chain. The

quenched

chain end variables _were handled

through

a

replica

calculation and upon

integrating

out the

density

variables _an effective

replica-replica

interaction _was obtained.

We showed that the effective

intra-replica

interaction is still screened in the limit

N ~ oo but that the

inter-replica

interaction is _not. A first order

perturbation

calculation however confirmed that the

inter-replica interaction,

which arises

specifically

from the fixed chain

ends,

leads to a contraction of the chain

only

of the order of N~ ~'~.

Our overall conclusion is that for

long,

flexible chains

trapped in,

for

example,

a

network,

the melt random

phase approximation

and

single

chain

properties

_are not

significantly

affected

by

the loss of translational freedom.

Acknowledgements.

MGB is

pleased

to

acknowledge

the

support

^{of the} Max-Planck-Gesellschaft and the

hospitality

of the Max Planck Institute for

Polymerforschung

at

Mainz,

where the work _was

performed.

Appendix

I.

The transformation

(R$"

~ p

()

The

change

ofvariables

(R[")

~

(p(), including

the

averaging

_over the

quenched

chain end variables

(R]),

is

expressed

as

l1Idp(

⁸

_~p) ^£

^exp

^iq R[") ~

=

I

(Al.1)

~~

The delta function is

parameterised

_as

11I ~) ^exp _£

I

#i) ~p)

¹²

^£

^exp

^iq R[")

_~

^(Ai.2)

« as

and the

averaging

^is

accomplished using

the second _moment

approximation lexP El

o exP

[to

+

IS ~l

o

lTl ~) ^{(Ai .3)}

where

£=-112~~£#i)£expiq.R["

« as

The

quenched

_{average over} the

position

of the chain ends _ensures that £ =0 for

q # 0. The fluctuation term

(£~)~

can be

rearranged

as

(E2)~

=

12 ^~

£ #i( #il' £

_exp

iq (R[" ^Rl'" (£

exp

iq (r[" rl'" _(Aj

4)

qq Set

7~/

s

N° 12 SIZE OF A CHAIN IN A MELT OF ANCHORED CHAINS 2289

and treat the

intra-replica

terms

(~r

₌

~r') separately

from the

inter-replica (~r

#

~r')

_ones to

give

:

<a2>~

= D ^-2

z _{~ki ~ki'} Ii _i~Ji

~J

«qi~ i~Jii~ _i~J «qi~) ^3««

+

«« a

q

+

Cq 1llil~ 1ll~ql~l (A16)

where

C~

^{is the} structure factor of the anchor

point

distribution

C~

=

D

£

_exp

iq (R[" Rl'" (Al?)

The _term

(Al.2)

becomes

In ~~j

^exP

Z

i~ki Pi £ _Z

~ki ~kiq Mi"'1(A18)

~

« ««.

q

where

Ml"

₌

Z11 ll~il~"qlo ll~il~ ll~"qlol 3««,

+

Cq ll~ilo ll~"alai (A19)

a

Since the

unperturbed

_average values

occurring

in

(Al.9)

do not

depend

_on the

replica

index the matrix M in

replica

_space can be written _as

M~

=

A(

^I +

D(

U

(Al.10)

where I is the unit matrix in

replica

_space and U is _a matrix with all the elements _set

equal

to I.

A(

^and

D(

are identified from

(Al.9)

and _are

directly

related to the

density

fluctuations for _an

unperturbed polymer

chain anchored _at the

origin by

Al

_{= W}

lll~Jql~l~ 1il~q>~l~)

and

Di

=

Cq II <~q>~l~l (Ai.ii)

The

integrals

in

(A1.8)

_are of _a standard Gaussian form and

give

~ j2

ar

~et

_M ^~~~

i P~ (M ~))"'P it (Al.12)

where M~ is the inverse matrix to M.

The Jacobian

(Al.12)

_{must now} be combined with the Boltzmann factor

occurring

in the

partition

function

(2.I).

To do this the excluded volume interaction energy

occurring

in the Boltzmann factor _can also be written in terms of the collective

density

variables. From

(2.2)

( ((r[), (r[~) R])

=

~~~~ ~~~

~ ~

~~ ~~~~~ ~~'~~

_~ ^~

i ~q PiPiq. (Al.13)

The Jacobian in

(Al.12)

is combined with the

density-density

term in

(Al.13)

I-e- the last term, to

give

exp

£ p)(Mj

""'

+

V~

³

""') pf( (Al.14)

~ ~

««'q

Set

Q~

= M~ ^' ₊ V 1.

The average over the collective variables

(p()

of these combined terms can be written _as

In dPi

exP

£ _{Z Pi(Q~} ~)i"'Pil

₊

Z (vq

_exP

iqrl) Pi (Ails)

~ ««.~ ~»

Each

integral

in

(Al.15)

has _a standard Gaussian forni and the whole

integrates

to

give

j~

"

exp

W""'(rj rj') _(Al.16)

det

Q~

^{2 fl}

where

w««'(ri ri')

=

z iv~j2 z Qj«'z

_exp

_iq(ri ri'). _(Ai.17)

~ _. ~~.

Finally

the therm

(Al.17)

is combined with the

single

chain _term in

(Al.13)

_to

give

for the

replica paltition

function

Z~

=

exp _£

V$~, (r[ r)

+

Jr(~ _(Al.18)

~ ~

j«, °

where

Vj~,(r[ rl')

is

given

ⁱⁿ terms of its Fourier components

Vj~,(q)

_as

V$~,(q)

=

V~ 3~~, V~[~ Q""' _(Al.19)

An

explicit expression

is

required

for the matrix

Q.

Recall that the matrix

Q

is defined

by (Al.14) through

its inverse I-e-

Q~~

=

(M~~

₊ Vi

)

while M~ is derived from

equation (Al.10)

_as M

= A° I ₊ D° U. The matrix

U,

with unit elements

everywhere,

^has the

property U~

=

nU,

which enables matrices of the form M

= A° I ₊ D° U

(Al.20)

to be inverted _as

_1 q D°

~

(Al.21)

~

A° A° A° ₊ nD°

Hence in the limit _{n ~} 0

A° q D°

V~

~

(Al.22)

~

~

i ₊ A°v ^~

(i

+

A°v)2

N° 12 SIZE OF A CHAIN IN A MELT OF ANCHORED CHAINS 2291

and from

(Al.19)

the effective

replica-replica

interaction

potential V$~,(q)

is

given

in matrix forni _as

V

*(q)

=

V~

^I

V~

^~

Q~. (Al.23)

Using (Al.22)

for

Q

this becomes

~ /~o

~2

V *

(q )

=

~ l ^{~ ~}

~

U

(A1 24)

+

A(V~ (I

+

A( V~)

The results

(Al.18)

and

(Al.24)

are the _ones used in the paper.

Appendix

2.

Single

chain

unperturbed

_average values.

In section 4 of the paper,

expressions

were needed for the

following unperturbed, single chain,

average values

(a) j~~)

and

(rj _~~)

(b) (a~~ a~_~)

^and

(r(

a~~

a~_~)

where

(a~

~)

=

£

_exp

iq

_r~ ^and r~ is the chain end _to end _vector

~

The terms

(a)

_can be obtained from the

generating

function

J(p

=

£

_exp

iq

_r~ + p r~

(A2. I)

~ o

and the _terms

(b)

from

H(p

₌

£

_exp

iq (r~

r~,

)

+ p r~

(A2.2)

~~. ^°

The method is the _same for both functions and _we illustrate it for

J(p).

Write

r~ in _tennis of the bond _vectors b~, as

~

r~ =

£

_b~, and

similarly

for r~

s'= i

Then

J(p )

=

(£

exp I

£

_b~,

_(qiY~~,

_{+ p}

) (A2.3)

~ ~, o

where _iY~~,

= I for s'

~ s and _zero otherwise. The

(b~,)

_are Gaussian random

variables,

hence for any ^vector V~

(exp £

_V~

_b~)

₌ exp

£ Vj ^b~/6 (A2.4)

Therefore

2

~~2

²

J(p )

=

£

_exp ^~

^sb~

₊

iq

_{p +}

^~ Nb~ (A2.5)

Convert the _sum into _an

integral

to get

~~p [_

^q~

^j~b2

^~

^iq.

^{p N}

^{~j ~}

^exp

~Nb~

6 3 6

(A2 6) J(R

= 2

b~

(-

^~ + i ~'"

6 3

then

(a~~)

=

J(0)

=

N

~~§~ ^{~~ (A2.7)}

°

Q

where

Q~

=

q~ b~N/6

and

(r(

_a~

_~)

=

~

J(p )

~

(A2.8)

°

~

~

~~ ~~

^~~ _{~ ~} ^~~ _~

Q~~

j~

~~

A

simple parameterised

^{form for}

(a~~)~,

^{correct at} q =

0 and _as q ~ co, and suitable for further

analytic

^{work is}

(

_a~~)

=

~

~

(A2.9)

° +

Q

In section 4

particular

combinations of these terms were needed. For

example _B~ given by equation (4.9)

is

given by

Bq

=

Iii _~qi~ <~-q>~ (rli~ <~q>~(~)

~

2

~3

~2 ~~ ^~ ~~~

~-Q~ ~~e ~~_~

^l -e

~)

3

Q2 Q2 Q4

Foro~l

~ 4 N^~

(A2. 10)

~~

^~~

3

Q~

This is the result used in section 4 of the paper.

The other tennis

(a~~ a~_~)

^and

(r(

a~~

a~_~)

are dealt with in _a similar _manner.

References

[I] FLORY P. J.,

Principles

of

Polymer Chemistry

(Comell

University

Press, Ithaca, 1971).

[2] EDWARDS S. F., J.

Phys.

AS (1975) 1670.

[3] BRERETON M. G., ViLGis T. A., J.

Phys.

France 50

(1989)

245.

[4] _DE GENNES P. G.,

Scaling Concepts

in

Polymer Physics

(Comell

University

Press, Ithaca,

1979).

[5] EDWARDS S. F.,

Polymer

Networks, S. Newmann Ed. (Plenum Press, N-Y-,

1971).

[6] BRERETON M. G., ViLGis T. A., J.

Phys.

France 2 (1992) 581.

[7] PANYUKOV S. V., JETP 69 (1989) 342.