Theory of polymer relaxation times in semidilute solutions. I : mode relaxation

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Theory of polymer relaxation times in semidilute solutions. I : mode relaxation

Y. Shiwa

To cite this version:

Y. Shiwa. Theory of polymer relaxation times in semidilute solutions. I : mode relaxation. Journal

de Physique II, EDP Sciences, 1993, 3 (4), pp.477-486. �10.1051/jp2:1993145�. �jpa-00247848�

J.

Phys.

JI France 3 (1993) 477-486 APRIL 1993, PAGE 477

Classification

Physics

Abstracts

61.25H 05.60

Theory of polymer relaxation times in semidilute solutions.

I

_:

mode relaxation

Y. Shiwa

The

Physics

Laboratodes,

Kyushu

Institute of

Technology,

Iizuka, Fukuoka 820,

Japan

(Received 7 July J992, revised 9 November J992, accepted J2

January

J992)

Abstract. The _{rate at} which the intemal mode of motion of the strained

polymer

chains in semidilute solutions relaxes is calculated _as a function of concentration. The excluded-volume and

hydrodynamic

interactions _are taken into account. Gradual

approach

to the behavior of ideal chains due to the screening ^{effect is} demonstrated

explicitly.

1. Introduction.

Over the years a

large body

of theoretical work has

shaped

_our

understanding

of

dynamic properties

of semidilute

polymer

solutions. The first idea _was due _to des Cloizeaux

[I].

He

pointed

out that the semidilute solutions of

polymers

_are those solutions in which the

polymer

Volume fraction is

infinitesimally small,

and

yet polymer

chains _are

sufficiently long.

Even if

the _monomer

density

is

infinitesimally

small in the semidilute

solutions,

_a

specific

chain

overlaps

with _a

significant

number of other chains. Once chains

overlap,

it is

likely

that the chains _{start to}

entangle

one another.

When chains

overlap,

the interactions within a chain and between two chains _are interfered

by

the existence of the other chains. The static consequence is the

screening

of excluded-

Volume

interactions,

_{as was} first

pointed

out

by

Edwards

[2].

The concomitant

dynamic

feature,

first imbodied

by

calculations of Freed and Edwards

[3],

is that the

hydrodynamic

interactions between _monomers of

polymer

^chains are also screened.

As _a consequence, a

theory

of

dynamic properties

of semidilute

polymer

solutions should address itself to the

complicated interplay

of the

gradual screening

effect of both excluded- Volume and

hydrodynamic

interactions and the effect of

entanglements.

For this _reason the

quantitative theory

has _met considerable

difficulties, apart

^from a

(more

_or

less)

successful elucidation of the

asymptotic scaling

behaviors

[4].

In

particular,

the

question

of how the

asymptotic scaling

is attained _as the

degree

^{of chain}

overlapping

is Varied has remained _to be answered.

Theoretical

analysis

^{of such} crossover behavior of

dynamic properties

^has

recently

^been

launched,

and in fact the

gradual screening

of both

hydrodynamic

and excluded-Volume effects is revealed

[5]. Comparison

of this theoretical

prediction

with sedimentation

(cooperative

diffusion)

data is made with credible

agreement [6].

The

entanglement

constraint is also

incorporated

in

working

out the

explicit

form of the _crossover behavior of the self-diffusion coefficient

[7]

and the

polymer Viscosity [8].

Beside the above-mentioned

dynamic quantities

that have been studied _so

far,

relaxation

rates are another

dynamic properties

of great

importance.

The purpose of the present ^and

subsequent

_paper is to examine the rate at which the disturbed

polymer

chain in semidilute solutions relaxes toward

equilibrium.

Our calculation of the relaxation rate

explicitly

shows the

dependence

_on the

degree

^of chain

overlapping (which

will be

loosely

referred _{to as} concentration

dependence).

Firstly

in this paper we consider the relaxation time of the intemal mode of motion of linear chains in semidilute solutions. In section 2 the intemal mode relaxation time is defined. The

theoretical

problem

is then reduced to

calculating

the

equilibrium

statistics of

chain, namely

the correlation function of conformation of _a chain in solutions. In

appendix,

the correlation function is calculated with the aid of the

renormalization-group

method. With the result obtained

there,

the calculation of the mode relaxation time is

completed

to

present

an

explicit

formula for the relaxation time in _terms of the

overlap _parameter.

Section 3 checks the

asymptotic scaling

limit of this formula. We find that in the

asymptotic

semidilute limit both excluded-volume and

hydrodynamic

interactions _are screened out

completely,

and that Rouse- like behavior is obtained. The overall concentration

dependence

to represent the _crossover to this Rouse-like limit is

presented

in section 4.

Given the relaxation _rate of the intemal

mode,

we

study

the

long-time

scale relaxation process in the

following companion

_paper

[9],

and the

question

of the

entanglement

effect is

addressed.

2. Relaxation time of internal motion.

Deferring

discussion of the relaxation of the

global

motion of chains _to the

subsequent

_paper

[9],

_we shall here be concemed with the chain relaxation whose characteristic

length

scale is

shorter than the size of the

polymer

chain. To

represent

^{such intemal motions it is convenient}

to define the relaxation

spectrum

ⁱⁿ terms of the

specific viscosity,

_aJ~~. To each

p-th

normal mode ~p ₌

integer)

_{we may} associate the mode relaxation time r~ ^{via the}

following

relation

[10] (in appropriate units)

:

~= zr~. (2.I)

P

N'1o

~

Here N is the _contour

length

of _a chain and _p is the

polymer

_monomer

density,

and

aJo is the bare solvent

viscosity.

Starting

from the

time-dependent Ginzburg-Landau equations

for semidilute

polymer solutions,

_we have

recently [5]

^{calculated the effective}

specific viscosity

_aJ~~ which is

renormalized

by

the presence of many other chains and the solvent

velocity

field in the solutions. When the excluded-volume and

hydrodynamic screening

effects _are taken into account, the

specific viscosity

is

given by

the

following coupled equations

with the _wave-

vector

dependent

_monomer

mobility,

(~

'(q)

:

'~~P

~~~ ~ /

~)~l' ^~~'~~

f~ ~(q)

=

ti

+

(

~~(~))~, ^(2.3)

1~

(k)

= ~o

k2

₊

p

~j ) _(kR~

»

1)

,

(2.4)

q q

N° 4 RELAXATION TIMES OF SEMIDILUTE POLYMER SOLUTIONS 479

where k and q are

conjugate

momenta to the

spatial

osition vector, r, and the _contour variable

of the _{monomer, n}

respectively

; _m

(2 gr)~~ dk,

m 2 _gr

dq,

and d is the

spatial

k q

dimensionality.

Let

cy

(r )

denote the

position

of the

j-th

chain

=

I,

,

n

) parametrized by

the _contour variable

r(0

« r « N

).

The variable q is related to the mode index p

through

q =

2

grp/N.

The functions

4~(q)

and

Sj(k, q)

_are Fourier transforms of the

equilibrium

correlation functions

(

c~

(r )

_c

j

(

_r

') ~)

^{and V}

^(r

_c~

⁽

r

) (r'

c~(_r

') ))

,

respectively,

V

being

the volume of the system. ^In

equations (2.3)

and

(2.4), to

denotes the bare friction coefficient per segment ^{of the}

chain,

and

R~

stands for the

gyration

radius of the chain. In the

above

equations,

the

many-chain

character of the semidilute

regime

is reflected in the

concentration

dependence

of the correlation functions 4~ and

Sj

of _a

single

chain in semidilute solutions,

rendering

aJ~~ and (~ concentration

dependent.

^We

might emphasize

that the above set of

equations (2.2)-(2.4)

has been obtained without

incorporating

the

entanglement effect,

which govems the

global

motion of chains. With

1~~

^thus

obtained,

the intemal mode

relaxation time

r~

^is

properly

determined from

(2.I).

In

determining _r~,

_we first notice that the

region kR~

» I dominates the

integral

in

(2.3),

where

[5]

~

(k)

= ~

o(k2

+ «~)

,

(2.5) w~~ being

the

hydrodynamic screening length.

Since in the semidilute

regime

the mode-

coupling

contribution

(I.e.,

the second _term in

(2.3))

dominates the bare

part,

we have

~sp/p

₌

(d

i

)-

^'

j _j

(2.6)

q q

with

J(q )

=

~ l~l11 (2.7)

Thus the

p-th-mode

relaxation time is

given by

'1o ~P

(q)

~~ ~

~~~~~ (2.8)

llo

[ilk Si(k, q)lk

_=o

~

Sj (k,

_q

)/ (k~

+ «~)

k

Here and

throughout

we

neglect

the chain end effect

(I),

and have thus assumed translational invariance of the correlation functions with

respect

to

spatial positions

in

arriving

at

equation (2.8).

Thus _we need to calculate the correlation function

Sj(k, q).

This is done with the

renormalization-group

method. The calculation is _a

straightforward

but tedious _one, and is

(')

Since _{we are} concemed with the

asymptotic

limit N

- m (note, however, that the

overlap

paranJeter ^X [see (A. 8)] can still take finite values in this limit), the upper

integration

limit of the

integral

over the _contour variable is extended to

infinity

if it converges and the

integrand

_over the extended

region

is

negligible.

See reference [3] for the _use of this

limiting procedure

in the related context.

presented

in

appendix.

As

given by

the

appendix,

it then follows

that,

to

O(e

m 4

d),

~~~~~~

^~~

(°' ^{dt~i (t}

^;

^x)

^cos

⁽²

^grpt

⁾

~~ ^~

~~ ~P~ j°'do j°'dtio~/(Q~

⁺

^{k~)i s(Q,}

^{t ;}

^x)

^cos

^{(2 arpt)}

~~'~~

o o

Here k~ ^' is the scaled

hydrodynamic screening length,

k

w w

(N/2)~,

_v

being

the excluded- volume exponent v =

(1/2)(1

+ e/8 ). The functions in the

integrands

_are

given

_as follows _:

~i(t;X)

= 2

v(2

_v

I)e~~"~ ~i(t;X), _(2.10)

where

(2)

~i(t;X)

=

t~~~~(1-

^~

^tXF(2t(X+ I))j, (2.ll)

dv -1

with

F

(a)

_«

(2

_v + i

) fo(a)

₊

(2

⁺

) _fj(a)

+

) _{f~(a), (2.12)}

fo(a)= ~~~~~/lna-

^~~~

~eaEj~_~~

(a 1) 2(a 1)~

^{2 '}

and

S(Q,

t

X)

=

e~~~

_exp

(- ^~

G

(Q,

t

X)j

_,

^(2.13)

where

G(Q, t;X)= i~dk j~d9k~sin~9~~~~~ ((1+

^~

)(Jj(k~-2P~kcos ^9)-

o o

k~+a

_k~

i e~~~

(e~

^{~~~ ~°~ ~} l

)

4 P k~

cos~

₉

~ +

~

(2,14)

1+ k k

(I

+

k~)~

In the

above,

c'

= 2 _{y +} In _{gr, y}

being

^Euler's constant

(y

= 0.5772..

),

and

Ei(- a)

is the

exponential-integral

function defined

by Ei(-a)

a~

=

dxe~~/x. The so-called

overlap

a

parameter ^{is denoted}

by

X and is defined

by (A.8)

it may be

roughly interpreted

_as the number

of chains

overlapping

with _a

specific

chain. Also in

(2.13)

we have

put P~w /t~ _Q,

c~ = I _{+ e 2 y} + In _gr

)/8,

^{and am 2 t}

^(X

^{+ I}

⁾

^{; the function}

^Jj(z)

^{is defined}

^{by (A.6).}

(2) ^When one takes the limit _E

~ o of (2.9), it is to be understood that _one puts v ~ l/2 _at the last step of calculation.

N° 4 RELAXATION TIMES OF SEMIDILUTE POLYMER SOLUTIONS 1 481

As is

evident,

the

expression (2.9)

is

quite involved,

and this is _as far _{as we can} go

analytically

in

general.

For Gaussian

chains, however,

_{we can carry}

through

the

integrals

on

the rhs of

(2.9).

In _so

doing,

_we first

identify

the Gaussian-chain _case with the

u*-0 limit of

(2.9),

_as is

usually

done in the

renormalization-group theory [iii

u* is the

fixed-point

value of the

strength

of the excluded-volume interaction. Then _we find that

~~(N/2)~~ k~

+

(2 grp)~

(2.15)

~l

=

grp2 »3

₊

G(2

arp

k~l'

where the

superscript

G indicates the _case of Gaussian chains.

Before

proceeding

_to

numerically evaluating (2.9)

for the

general

_{case, we}

study

in the next section the

asymptotic

behavior of the result

(2.9).

3.

Asymptotic scaling.

For the purpose of

scaling

_argument in semidilute

solutions,

it is convenient _to introduce two

overlap

parameters, ^{static and}

dynamical [5]

_: X

= 4 cu*

(N/2)~~

,

Y

=

4

c(* (N/2)~~

,

(3.I)

where _c

m

p/N

is the

polymer

number

density (

* is the

fixed-point

value associated with the

coupling

constant of the

hydrodynamic

interaction. We shall first consider the

asymptotic

limit of the Gaussian-chain _{case :}

I)

Gaussian-chain solutions

(X

=

0).

In the limit of dilute

solution, I-e-,

Y

- 0 for the Gaussian-chain _case, k tends to zero. We find from

(2.15)

that for d

= 3

r)

~

N~'~ _(3.2)

The familiar Rouse-Zimm behavior is recovered _as

required.

In the

opposite

limit

(the

semidilute

limit),

Y

- cc, we know

[5]

that k Y at d

= 3, _so that

r)~pN~ _(3.3)

We obtain the Rouse-like behavior in the semidilute limit due to the

complete screening

of the

hydrodynamic

interaction.

ii) Self-avoiding

chains.

Similarly,

for

good-solvent

solutions the

asymptotic

limits _can be extracted from

(2.9).

For the dilute limit of

X,

Y

-

0,

rp

N~~

,

(3.4)

thus

reproducing

the

nondraining

result. On the other hand, in the semidilute limit

(X,

Y- _cc

),

the numerator and the denominator of the rhs of

(2.9)

behave _as

X~~'~

_and

k~~~ '~'~, respectively.

Since k

~

Y~'~~~~^~~ in this limit

[5],

_we find

r~~

p-~~~-~~'~d~-l~N~ _(3.5)

It is easy ^to prove that the result

(3.5)

is in accord with the

scaling hypothesis

for

r~. Again

Rouse-like behavior _emerges in the semidilute limit since in this _case both excluded- volume interactions and

hydrodynamic

interactions become

totally icreened

_out.

Having

thus confirmed that the result

(2.9) captures

^the correct

scaling

behavior of the relaxation time

asymptotically,

_{we now} demonstrate the actual _crossover.

4. Numerical result and discussion.

As _{we are} interested in the universal structure of the relaxation times which is

independent

of

microscopic

details of the system, we first construct _a universal ratio from

(2.9). Namely,

_we

take up the ratio of

(2.9)

to its dilute limit.

Moreover,

in the _same

spirit,

_we set

p =

I _to consider

only

^the

primary

relaxation

time,

_rj. Then the universal ratio associated with rj is deduced from

(2.9)

as

a~

a~ a~

dt4 (t)

cos

(2 grt) dQ dtSD(Q, t)

_cos

(2 grt)

Ti _o

o o

~~ ~~~

_{" ~~~}

~""

~~~

"~~

^{~ ~ ~} ~

dQ

^~

dt[Q~/(Q~

₊

k~)] _S(Q,

_t

;

X)

cos

(2 grt)

o o

(4.1)

where

S~(Q, t)m S(Q>

t

0),

and

rf

_is the dilute limit

counterpart

^of rj,

r(z) being

the Gamma function. For the Gaussian-chain

solution,

the universal ratio is

immediately

reduced from

(2.15)

to be

(~

=

~

~

*~ )"~

~

(4.2>

ri ar

[k

+

fir(2

ar k

)]

With the

screening length

k ^I

already

obtained in reference

[5]

_{as a} function of the

overlap

parameter

X,

_we have evaluated

(4.I) numerically by setting

_{e =} 4 d

=

I. The result is illustrated in

figure

I. The upper curve in the

figure _represents

the Gaussian _case

(4.2),

where

the abscissa should be

interpreted

_as Y/4. The

figure clearly

demonstrates that _as the

hydrodynamic screening

effect

(along

with the static

screening effect) prevails,

the

gradual

crossover from the

nondraining

to ideal behavior _occurs with the increase of concentration.

Before

closing

the paper, several comments are in order. In this paper, ^{the relaxation time of}

the intemal mode for semidilute solutions is calculated without

taking

account of the

entanglement

effect. This effect should be

negligible

for the intemal motion whose characteristic

length

scale is much shorter than the size of the

polymer

chain. For

larger length scales, however,

the

entanglements

_{present are}

expected

to be relevant.

Therefore,

_one should

regard

the

foregoing

_arguments in this paper, and the results

(4.I)

and

(4.2)

in

particular,

_as

being

the _ones for _{a more or} less fictitious _case without

entanglements

that enables _one to

assess the

entanglement

effect. The

entanglement

effect is

incorporated

into the

theory

ⁱⁿ

paper II.

In this

connection,

it is

important

to note that _we have obtained the ideal chain behavior in the limit

X(cc Y)

- cc

(which

_we have termed the semidilute

limit)

while

keeping

the

strength

of excluded-volume interactions fixed _{nonzero at u} * This is because the

screening

renders the net

swelling

to vanish in the semidilute limit even if _u * ~ 0. Put in other

words,

the semidilute limit does not

imply

u*

- 0 but the correlation

length

_- 0. The fact u* ~ 0 ensures that

polymer

chains cannot pass

through

each

other,

thus

fulfilling

the _correct condition of

topological constraint, I.e., entanglement. Therefore, although

_as far _as static

properties

are

concemed the semidilute limit may be studied with _a Gaussian chain model

(for

which

u*

=

0),

the

dynamics

is different from that of Gaussian chains. This distinction becomes

important

when _we consider the solution

viscosity

or the

longest

relaxation time

by

incorporating

the

entanglement

^effect.

N° 4 RELAXATION TIMES OF SEMIDILUTE POLYMER SOLUTIONS 483

3

a

~

'

l

2

-I o I

iogx

Fig.

I. Ratio of the

primary

mode relaxation time _vi to its dilute limit

against

the

overlap

parameter ^X.

The upper curve represents ^the

polymer

solution when _no excluded-volume interaction is present.

We stress that in semidilute solutions the _monomer

density

_p is

infinitesimally small,

_so that p is _{not a} natural variable.

(We

have used it in Sect. 3

only

for convenience sake of

comparison

with other related

work.)

For static

quantities,

we now know that the

good

_parameter is the

overlap

parameter

X,

which is

proportional

to

pN~~ '.

_{There is}

a

dynamical counterpart

^of

X,

also

proportional

to

pN~~-~. _Namely,

it is the so-called

dynamical overlap

_parameter Y defined

by (3, I).

The latter parameter ^becomes

particularly important

when _one considers the

dynamics

of Gaussian-chain

solutions,

for which X

=

0

by

definition. The presence of this fundamental parameter ^{in the}

description

of the semidilute solution

dynamics

has _not received due attention in the literature except ^{for reference}

[12].

The fourth remark _to be observed is that the

viscosity

formula

(2.2)

^is extracted from the dressed

equation

for the solvent

viscosity [5].

Hence it does _not contain the pure solute

contribution _to the solution

viscosity.

For this _reason the relaxation time defined via _our

1~~~ does not

yield

at infinite dilution _a

scaling

of the form r~ ^{N ~~~~ ~ ~~ as found in reference}

[13].

Finally,

_we note that Muthukumar

[14]

calculated the intemal mode relaxation time from

(2.2)-(2.4) by

a different method. Without

repeating

the _{comment on} his

formulation,

which has been

given

elsewhere

[5],

_we

just point

out the intemal

inconsistency

of this

theory.

The results _were obtained

by assuming

that the

hydrodynamic

interaction is screened for the characteristic

length

scales of the intemal motion.

Therefore,

for his

theory

to be

consistent,

it is

required

that

w «

RI,

where

RI

is the radius of

gyration

at infinite dilution. On the other

hand,

_we find the

screening length

in his

theory

to scale _as

w~

X

(RI )-~, _{because, using}

his notations,

cAp-

^~ X.

Apparently

the above two relations _are

incompatible

because his

theory

is valid

only

in the

region

where X «1.

Acknowledgements.

I _am indebted to Phil Baldwin for the

clarifying

comments on this work

through

_a

_great

deal of

correspondence.

Appendix.

In this

appendix

_we calculate the correlation function

(3 (r

c~

(r )) (r'

c~

(« )))

with the

renormalization-group

method

[I I]

to derive

equations (2.9)-(2.14).

The Hamiltonian with which _we take the canonical average

(. )

is the so-called Edwards' Hamiltonian

[15]

:

n

N

_dc~

(

_r

)

2

H

(c)

~

=

z

dr ₊

z

dr d«&

(cj ^{(r ) ci(~ )) (A.1)}

2~=j

_o ^d~

2j,1

The parameter v represents the

strength

of the

repulsive

excluded-volume interaction. The

important

static behavior in the semidilute solutions arises from the

gradual screening

of the

excluded-volume effect with the increase of concentration. We must therefore have _a

systematic

formal

procedure

^for

capturing

this static

screening

effect. For that purpose there exists the now-familiar formalism

[16],

_to which the reader is referred for details.

It is convenient to consider the Fourier transform

(in space)

defined

by (neglecting

end

effects)

1(k,

_cr>

=

d(r r') e'~'~~~~'~j& ^(r c~(r

+

cr)) (r'-

c~

(r»j (A.2>

We may also define the function

©o(k, «) by

Do(k,

_cr

)

₌

d(r r') e~~'W~~'~(3 (r

_c~

(r

₊

cr)) &(r'- cj(r)))~, _(A.3)

where the average

(. )

~

is taken with the Gaussian

Hamiltonian, I.e.,

with respect ^{to the first}

term of the rhs of

(A.I).

We _can then

develop

a

diagrammatic

formalism for

calculating I

_{for model}

_(A.I).

_Schematic

representation

of the

diagrams

_necessary to obtain

I(k, _«)

_to

lowest order in v is shown in

figure

2.

Diagrams

are constructed with

©o

as a bare

propagator

and _{v as an} interaction line.

Intermediate

^lines are

integrated; particularly,

for _contour

positions

the

integration

_{range may} be extended to infinities when

appropriate,

since _{we are}

always

interested in the

translationally symmetric

_case.

After

performing simple integrals,

_we then find

I(k, _«)

=

e-~~ (2/«)~'~ «~ iv~(fi)((1+ ^2L-~)e-~~~

_x

L

x

[Jj (

K L ^~

K~) _Jj (L~)]

₊ L- ^~

[e-

^{~ ~ ~} e~ ~~

~~~~])

,

(A.4)

with

m

(2

_gr

)-

^~

dL,

and

L

~~

~

l ₊

Xo J~ (Np~/2 ) ~~'~~

N° 4 RELAXATION TIMES OF SEMIDILUTE POLYMER SOLUTIONS 485

r+~ r ~'~

,i, j. ,i, I,

(a) _{~ = ~ +} ~ ₊ ~

k

,1' '"j .1' (~

+ ~ ₊ ~

/,

(bi ^~ ",

(C) :I' _~l

. . . .

:I' '~

Fig.

2, (a)

DiagranJmatic representation

of the low-order contributions to the correlation function

$(k,

_{~). The solid line labeled}

_by

p and

joining

contour

positions

_p and _{p + ~} denotes the bare propagator

©o(p,

~). The

points

(.) and the intersections between the dressed interaction line [the

double-dashed line, (b)] and the solid lines _are labeled

by

contour

positions along

the chain. The dashed

single

line denotes the bare interaction line _v. (c) Since _{we are} interested in the

translationally symmetric

case, the chain end effects _are

neglected

and these

diagranJs

do not contribute to

$.

In

(A.4)

we have introduced the scaled momenta, Km

fi,

etc. Functions

J~(z),

m ₌

1, 2,

_are defined

by

Ji (z>

=

(i

e- ~>/z

,

(A,6>

j~(z

=

2

(z

i ₊ e-

z>/z~,

J~ being

what is called the

Debye

function. In

(A.5), Xo

_m

cvN~,

and the dressed interaction v~ represents ^{the effect of}

screening

of the excluded-volume interactions.

The above result

(A.4)

obtained

by

the naive

perturbation theory

contains the

ultraviolet, logarithmically divergent integral

at d

=

4. This

logarithmic anomaly

can be

regularized by

the

renormalization-group

method. We have

employed

the dimensional

regularization

method of renormalization. Rather than

describing

the elaborate details of this

procedure,

^{which is} now

quite

standard

[I I],

we

only

mention here that the

analytic

calculation is

greatly

facilitated if

one uses the

following approximants [17]

for the functions

(A.6)

_:

J~(z)

₌ ^"~

,

z m 0

(A.7)

z+m

We also introduce the

following

scaled variables :

Q

=

k(N/2)"

,

X

=

4 cu*

(N/2)~", _(A.8)

where u* is the

fixed-point

value associated with the excluded-volume

parameter

v, and

v is the

corresponding

excluded-volume exponent. ^{Here and henceforth} v and N should be

understood _to be renormalized variables.

Accordingly,

_we denote the renormalized

((k,

~ as

S(Q,

t

X)

with _t

=

«IN.

Then _we

find,

to O

(e

m

4 d

),

S(Q,t;X)=e~~~~l-~( ~-y+Ingr)

4gr 2

2

+t-1~2 4(#.L)~ ~

~~~~

L

2+2X+t~'L~~~~"~~ _(l+L~)~

'

~~~~

where

Y(#, L)

=

(1

⁺

(Jj ^{(L~ 2(# L))}

~

+ e~~~(e~ ^~'~ l

)/L~, (A, lo)

L I ₊ L

with

Pm

_t"

_Q,

v =

(1/2)(1

+

e/8)

and u*

=

(gr~/2)

e. In

(A.9)

the

integral

is _{now over} four dimensional space, ^and yis the Euler's constant, y = 0.5772..

Finally,

the correction of order

e may be

exponentiated

_to

yield (2.13)

in the main text.

Substituting

the

expression (note

that the upper limit of the contour

integration

^is extended to

infinity

due to _our

approximation

of

neglecting

chain end

effects) Sj (k,

q

) a~

w 2 N dt _cos

(2

grpt

)

S

(Q,

_{t ;} X

) (A.

I

I)

o

in

(2.8),

and

performing

_an

integration by

_parts in the _numerator

yields (2.9)

with

~i

(t

_;

X)

=

(1/2 ) 3~[V[ S(Q,

t ;

X)]~ o/3t~

; with

(2.13) inserted,

~i is

easily evaluated,

and

finally

follows

(2. lo).

References

[ii _DES CLOizEAUX J., J. Phys. France 36 (1975) 281.

[2] EDWARDS S. F., Proc. Phys. Soc. 88 (1966) 265.

[3] FREED K. F. and EDWARDS S. F., J. Chem.

Phys.

61(1974) 3626

FREED K. F., in

Progress

in

Liquid Physics,

C. A. Croxton Ed.

(Wiley,

New York, 1978) _p. 343, and Macromolecules 16 (1983) 1855.

[4] _DE GENNES P. G.,

Scaling Concepts

in

Polymer Physics

(Comell Univ. Press, Ithaca, NY, 1979).

[5] SHIWA Y., OoNo Y. and BALDWIN P. R., Macromolecules 21 (1988) 208 SHIWA Y. and OoNo Y., ibid. 21 (1988) 2892.

[6] NYSTROM B, and ROOTS J., J.

Polym.

Sci., Polym. Lett. 28 (1990) 101, and J. Polym. Sci., Polym.

Phys. Ed. 28 (1990) 521.

[7] SHIWA Y.,

Phys.

Rev. Lett. 58 (1987) 2102.

[8] SHIWA Y., J.

Phys.

II France 1(1991) 1331.

[9] SHIWA Y., J. Phys. ^II France

unpublished.

[10] SAITO N., Introduction _to Polymer Physics (Syokabo, Tokyo, 1971) _; GRAESSLEY W. W., Adv.

Polym.

Sci. 16 (1974) 1.

[I Ii See, for instance, OoNo Y., Adv. Chem.

Phys. 61(1985)

301.

[12] STEPANOW S. and HELMIS G., Macromolecules 24

(1991)

1408.

[13] SCHAUB B., CREAMER D. B, and JOHANNESSON H., J. Phys. A

: Math. Gen. 21 (1988) 1431.

[14] MUTHUKUMAR M., Macromolecules 21 (1988) 2891, and references therein.

[15] EDWARDS S. F., Proc.

Phys.

Soc. 85

(1965)

613.

[16] OHTA T. and NAKANISHI A., J. Phys. A Math. Gen. 16 (1983) 4155

NAKANISHI A. and OHTA T., ibid. 18 (1985) 125.

[17] BALDWIN P. R., Phys. Rev. A 34 (1986) 2234.