Ternary Interactions in Dynamics of Θ-State Polymers: A Conformation-Space Renormalization-Group Approach

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Ternary Interactions in Dynamics of Θ-State Polymers:

A Conformation-Space Renormalization-Group Approach

Yasuhiro Shiwa

To cite this version:

Yasuhiro Shiwa. Ternary Interactions in Dynamics of Θ-State Polymers: A Conformation-Space Renormalization-Group Approach. Journal de Physique I, EDP Sciences, 1996, 6 (1), pp.97-108.

�10.1051/jp1:1996131�. �jpa-00247178�

Ternary Interactions in Dynamics of H-State Polymers:

A Conformation-Space Renormalization-Group Approach

Yasuhiro

Shiwa

Division of Natural

Sciences, Kyoto

Institute of

Technology, Matsugasaki, Sakyo-ku,

Kyoto 606,

Japan

(Received

ii

July

1995, received in final form 19

September1995, accepted

6 October

1995)

Abstract. We reconsider the t'Hooft-Veltman type renormalization _treatment of

polymer

properties _near the

theta-point.

We correct other recent studies, and _now demonstrate that the equivalence between this renormalization scheme and _a zero-component field

theory

_{or a} direct

renormalization method for

polymer

chains works

perfectly.

We also

give

_an estimate for the

magnitude

of the renormalized third virial coefficient

using

_{a new} result for the translational diffusion constant at the

theta-point.

PACS. 05.20-y Statistical mechanisms.

PACS. 05.60+w

Transport

_processes:

theory.

PACS.

61.25Hq

Macromolecular and

polymer solutionsj polymer

melts.

1. Introduction

When the

temperature

^of a

polymer _system

of

long

flexible chains diluted in _a

good

solvent is

lowered,

the

repulsive binary

interaction between

segments

on the chain decreases in

strength

and becomes

attractive,

^{while the}

ternary

interaction becomes

relatively

_more

important.

At the so-called

fl-temperature,

the two- and the

three-body

interactions

balance,

and the second

virial coefficient vanishes.

It is

generally accepted

that the

fl-point

is _a tricritical

point

with _upper critical dimension

dc

₌ 3

[ii.

This

implies

that the critical behavior in three dimensions is of mean-field

type

except for

logarithmic

corrections

[2].

^These corrections _come from the

three-body

interactions.

However,

the

precise

form of the

logarithmic

corrections has not been without

dispute.

Kholodenko and Freed _[3] criticized that the field theoretic calculations _[2]

using

the

equivalence

between the

polymer system

^{and the}

O(n)-#~

model in the limit _{n -} o _(~§ is _an J1-component

field)

_{gave wrong} results.

Later,

Freed and his coworkers [4] also claimed that dimensional

regularization

calculations for the continuous

three-parameter

model [5] of the

polymer

would be incorrect in three dimensions and that the cut-off method _must be used

[6].

^{Both these}

points

were refuted

lucidly

in _a series of work

by Duplantier ii, _8].

He showed that the _{J1 -} o

#~ theory

and the

polymer three-parameter theory

_gave

exactly

the _same results. He also elucidated the relation between the cut-off and the dimensional

regularization

schemes to demonstrate the

validity

of the _use of dimensional

regularization.

The above

controversy

was

mainly

concerned _over static

properties

of

polymer solutions,

and the

logarithmic

correction in the

dynamic properties

has not received that much attention. The aim of the

present

paper is to

present

the first _{correction to} the

mean-field

behavior of the

©

Les

(ditions

_{de Physique 1996}

translational diffusion constant at the

fl-point.

For

this,

_we shall

rely

on the

three-parameter

model of

polymer

chains with the

help

of the t'Hooft-Veltman

style

renormalization scheme

(to

be referred

collectively

_as

conformation-space renormalization-group

method

[9]). Actually.

the

problem

_was considered

already by

Freed et al. _{[4] in} the _same framework.

Holn-ever,

the entire calculation is

plagued

with the difficulties described above.

Moreover,

we note that the calculations with use of the

conformation-space renormalization-group

method have _never been

presented

^for

polymer properties

in the

fl-region during

the

dispute (although

the _so- called direct renormalization method of des Cloizeaux

[10]

was

applied

to the

three-parameter model,

and the relation between the direct and the dimensional renormalization schemes _~vas studied

[7]).

In view of this

fact,

it is desirable to _users of other formulations to _recover the

previous

results

directly by

the

conformation-space renormalization-group

method.

This,

too.

~vill be demonstrated in this paper.

The paper is

organized

_as follows. In the next section the

conformation-space

renormalization- group formalism for the continuous

polymer

chains _near the

fl-point

is

given.

We calculate the

leading logarithmic

correction to the

asymptotic

^{law for the} mean end-to-end distance _as well

as the _mean radius of

gyration

of _a

polymer

chain. The results coincide

precisely

,,~ith the

previous

results. In Section

3,

the

foregoing

formulation for the static

polymer properties

is extended to treat

dynamics.

Based _on the kinetic model

[11]

^of

polymer solutions,

we calcu- late the translational diffusion constant at the

fl-point. Computational

details _are

pro,~ided

in

Appendix.

The paper concludes with _a discussion in Section 4.

2. The Model and Renormalization

2. I. THREE-PARAMETER CoNT~Nuous-CHA~N l/IODEL. We consider _a

single

flexible

poly-

mer chain immersed in _a solvent in 3 dimensions with _a conformational

probability _given by

_[5]

P(r)

=

exp[-H(r)],

No

d~j~)

b ^{No No}

Hlrl

₌

_j / _dsl~l~

+

j /

_dS

/ _{ds'blrls) rls'))}

o o o

~ No No No

+ ^° ds

ds' ds"b(r(s) r(s'))b(r(s') r(s")) ii)

6

Here

r(s)

denotes the

position

of the chain

parameterized by

the contour variable s

lo

< ₅ <

No No being

the contour

length(~

of the

chain).

Coefficients bo and co are the two- and the three-

body

interaction

parameters, respectively.

One may have _an

arbitrary sign

for

bo,

while _co is

positive

^for

stability.

In this so-called

three-parameter

model _we have two dimensionless interaction parameters.

They

_are defined

by

zo +

(21r)~3/~boNll~

, yo ⁺

(21r)~~co (2)

2.2. RENORMAL~zAT~ON _{~N THE} fl-REGmN. The model

(I)

is ill-defined since there _are

ultraviolet

divergences appearing

in the

perturbation expansion

of _some

macroscopic

_quan-

tity

^calculated from it. One then has to eliminate these

singularities.

This _can be done

by

renormalization

techniques,

and we use diniensional

regularization _[12].

(~) In

fact,

the parameter No has the dimension of _{an area c~}

(length)~.

This

originates

from the fact that the fractal dimension of _an ideal chain is two. In this paper,

however,

_{we use} the convention of the conformation-space renormalization-group theory to call No the total

length

of the chain.

Let L be _a

phenomenological length

scale,vhich has the _same dimension _as )To We _may then define the dimensionless

coupling

constants _ito and _MO in d _space:

uo ^%

(21r)~~/~b,nL~/~

,

wo +

(21r)~~coR

,

(3)

where _e e 3

d,

and _we have introduced _a

marginal two-body

interaction

parameter

_[7]

bm e

boNll~

,

(4)

which is

marginal

for d

= 3 but not for _e

#

o.

Let _us consider the _mean square end-to-end distance

R2

and the

two-body

osmotic _pa-

rameter g of _an isolated

chain;

in terms of _g, the second virial coefficient

A2

is

given by

A2

"

Ii /2)g (21r-~2)~/2,

^~vhere

^X2

₌

^R2 Id.

Their

expansion

in po~vers of _{~o, MO} and _e may

be evaluated in the

straightforward

_manner

(cf.

^Ref. _[13]

).

^{The results read}

~~ ~'~~ ~~~°°

_~

~~~~°~

~

~°~~ ~~~°°~

~' ~~~

g =

-8wo Ii ~~~wo)

_{+ ~o}

Ii ~~~wo)

+.

(6)

e e

The

ellipsis

denotes third- and

higher-order (in

_uo and

wo)

terms. It should be remarked that in the above _we have

kept only

the dominant contribution in _e to each term.

We _assume that the model

(1)

is renormalizable with the

follo,ving

renormalization

factors,

the

Z's,

and introduce the renormalized

quantities N,

iu and _u:

No"Zp~iv, _wo=Z~vw, uo"Z~u. (7)

Since

singularities

arise from the

three-body interactions,

the Z's

depend only

_{upon w.} Hence

,ve introduce the

expansions

Zi~

"

1+Aw+Biu~+.

,

Z~

₌ I+Ciu+.

,

Z~

= + Diu +

(8)

Because

R2

and g are

directly observable, they

should not be renormalized. Then _we

readily

find that

~

A

= o

,

B

=

~~~

,

C =

~~~

,

D

=

~~~

(9)

3e _{e e}

zw

= 1+

~~~

aJ +

o(w2) (ii)

Z~

=

1+

~~~ MJ + (MJ~) . (12)

(Notice that (10) orrects result of

Ref.

the

lvilson

unctions

[12]

~ to

the

derivative 3/3L being

taken

with

the

fixed

renormalized

parameters

^No

and

Now,

define

the

_w(L)

L

~

AT ₌

Ni(MJ) (16)

Equivalently, introducing

the dilation factor _t with

dL/L

= dt

It

_{[9], we} define

w(t)

and

-V(t) by

The

solution

2.3. AT _THE fl-PO~NT _IN 3-SPACE. In this

subsection,

_we consider _a

system

^{for which}

u = o

(at

^the

fl-point)

and d

=

3,

and calculate

R~

for

large

N. lye

shall,

of _course, obtain the

logarithmic

correction to the mean-field

(ideal-chain)

form.

Going

to the limit _{e -} 0 in

(17),

_we obtain the

equation t~~°

=

-441riu~

,

(19)

which is

easily integrated

to

yield

~°~~~

1 ₊

~j~~1)

In t ~~~~

Combining (18)

and

(20),

_we then find

~~~~ ~~~~ ~~~~~~ +~~~j~~~)In

t~ ^~~~~

The next

step

^{is to}

employ

the

matching procedure _[15]. Namely,

_we continue the solutions

(20)

and

(21)

^until the

perturbation expansion

^for

R~

_can be

trusted;

we shall choose t

= to

at which

iu(to)

<1.

l~Thereupon

the loin-est order

perturbation theory is) gives R~

=

3N(to)(1 41rai(to)) (2?)

With the initial conditions

Nil)

=

N,

_MJ(1)

= y

corresponding

to the

physical

chain of

length

N with _a

coupling

constant y, we choose to

"

lk~la,

where _a is _a small cut-off constant.

Then,

writing

h _% MJ(to) so as to make contact with the notation of reference

[13],

we

finally

obtain from

(22)

~~ _~~~~°~~~~~ ~~~~~

_'

^~°~~~

^~

^~~~

^~~~~

with

~

41rlu(Nla)

~~~~

In the literature h is sometimes referred to _as the renormalized third virial coefficient. We note in

passing

that the condition of

validity

^of

(24)

_{is y}

In(Nla)

»

1/441r.

Similarly,

from the lowest-order

perturbation

result for the _mean square radius of

gyration:

R[

=

~N(to)(i _~~~iu(to))

,

(25)

~~~~

(26)

we

get

~2

=

NAO(l/)(1 W~~

Expressions (23)-(26)

_agree with the results obtained in references

[7,13]

via different methods of renormalization.

2.4. 8-REGmN _~N 3-SPACE. We _{now assume} that _zo is small but finite in order to

study

the

fl-region.

In this _case, too, we can follo~v _a similar sequence of steps as in the

preceding

subsection.

Moreover,

^{it is} necessary to consider the renormalization flow

equation

^of ~ beside

(17)

and

(18).

We first find

~/L~~~°'~°'~"~

(~ ~~~~"~'

~~ ⁺

Ll~

_in z~)N~,co,b,~~ = 167rw +

OIW~ 127)

In

obtaining (27),

let _us recall that it is essential for the L-derivative to be

performed

at fixed bm

iii. Thus,

for d

= 3 and to the lowest

order,

t~~

= -161r~tiu

(28)

This is

integrated

with _use of

(20)

to

yield

~llt)

=

Ji41v)zlh/v)~/~~

,

129)

where _we have set ~c(1) =

Ji4 IV)?(

_z is the renormalized

counterpart

of _zo and

Ji4

_IV) is _some

as

yet unspecified

function of _y.

From the

perturbation expansion

R~

=

3N(to)(1 41riu(to)

+

~~c(to))

,

(30)

we then obtain

R~

=

3NjAo(Y)11 ~~~h)

+

ZA41Y)l~)~~~~l

131)

with

A4(y) +1i4(y)-40(Y). Furthermore,

the radius of

gyration

_can be deduced from

to be

~~ ^~~~°~~~~~ ~~i~~

^~

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

~~~~

Comparing (31)

and

(33)

~vith the first terms of the well-known

expansions R2

=

3N(1+~z+. ^),

RI

=

[Nil

+

fz

+ ~~~~

which _are calculated for _a chain with

only two-body

interactions

(see,

for

instance,

Ref.

[13]),

we find

A4(o)

=1.

(35)

Consequently,

_as advertised in the

introduction,

the results

(31)

and

(33)

_are in exact

agreement

with those obtained

by Duplantier iii

and des Cloizeaux [13]

through

the

zero-component

^field

theory

_or the direct renormalization method.

3. 3lranslational Dilfusion Constant at the fl-Point

Having

formulated the

conformation-space renormalization-group theory

of static

properties,

,ve now

proceed

to examine

dynamics.

In

particular,

_we shall focus _on the translational diffusion at the

fl-point.

~Ve _use the usual set of

Langevin equations describing coupled

chain-solvent

dynamics ii Ii

The solvent

elocity

v(x, t) at a point ix, t)

is

ssumed to _{satisfy the}

pressibility ^ndition, T7 . v =

o.

As a consequence, the

tensor

erator T lects

the trans,,erse

mponents

of the

field it

jf~js,t)f~jsl,tI)j

=

2<piiijs sI)ijt _tI)

,

ifix,t)fixl,tI))

=

~2~~iv2ijx xI)ijt tI) j38)

being

the unit tensor. The parameters

(o

^and _ij~ ,vhich appear ⁱⁿ our model _are the bare friction coefficient _per

segment

of the chain, and the local sol,~ent

viscosity, respectively.

The ratio

(o/q~

becomes the relevant

dynamic

parameter in the model.

Based _on the kinetic model

(36)-(38),

_{we can} deri,~e the formula for the translational diffusion

constant D of the

polymer

chain. This is achieved

by

the elimination of the solvent

degrees

of freedom in favor of _an Oseen tensor

description

of the

polymer _[16, Iii.

The result has the form of the Ilirk,vood formula:

D

=

£ _ii

+

(jj~Tr Ids / _{ds'iTiris) ris'))11}

1391

Here kB is the Boltzmann cunstant, and

T(x)

is the Oseen tensor defined

by (in d-space) T(x)

=

/ _ii kk/k~)e~~'~ (40)

k k

with

f~

e

(21r)~~ f

dk. We remark that the above formula

(39)

is reliable _up to

(~) O((o/~le),

and the average

(.

_may ^{be taken with}

respect

to the full Hamiltonian H of

ii ).

In the

following,

_we calculate the diffusion constant at the

fl-point,

hence _{we may}

put

bo _" ^o

in

ii

from the outset. This is

permissible

if _one

performs (renormalization)

calculations

using

dimensional

regularization;

the dimensional

regularization just

_measures bo with

respect

to its

fl-point

value

[8].

^One may ^use

AppendiK

A to calculate the

right-hand

side of

(39).

One finds D =

~~~ [1+ ~~(o(1 Aiiuo))

,

No (o

9

(41)

to lowest order in _iuo, where

Ai

" o.89582. We have introduced _a dimensionless

(in d-space)

parameter pertaining

to _our model:

IO "

12~l~~/~fmv/~

,

fm *1(0/~/e)N/~~ ₁₄₂₎

Here _we renormalize the

expression (41) by extending

the program

explained

in the

preceding

section. The renormalization factor

Zj

defined

by

lo

"

Z/~( ₁₄₃₎

may be determined with _use of

(lo)

and

ill ). (Notice

that since D

depends

on

No

and _MO, the renormalization of

lo

is affected

by

the renormalized static effects. The result

is,

to

leading

order in _iu and

(,

~

Zj

= 1-

~~

iu~ (44)

e

We _{can now}

proceed

_as before. The renormalization flow

equation

for

(

_{becomes at d}

= 3

t~~

=

(L(

^~ ln

Zj)No,co,j,»

"

-~~~~~iu~ (45)

Notice that the

logarithmic

derivative is taken _at fixed

(m

since

(m

is

marginal

in three dimen- sions

(just

_as the

marginal two-body

interaction

parameter bm;

_see

(27) ).

The solution _to the

equation (45)

is

given by

~~~~

~~~~ ~~~

₁

~~~j~In

t~ ^' ~~~~

where _we have written

(

e

((1).

We observe at this

juncture

that in the limit N _{- oo,}

lit)

-

fexpl-8irv/33) 147)

Since

(

_may take _{on an}

arbitrary Ii-e-, sample-dependent) value,

we

have,

at the

fl-point,

_a

partial draining

^of

varying degrees

_even in the

long-chain

limit

(the scaling limit).

This is to be contrasted with the result of the

renormalization-group

calculation which

employs

the usual

die

4

d) expansion.

The

f-expansion theory predicts

_[9] that in the

scaling

limit one

always

ends _up with the

fixed-point

value for the

draining parameter ((o/~le)L"2

(~)

For

long

chains in

good solvents, (o/~e

_may be taken to be of order @(+4

d).

It is for this _reason that in reference [9] ^it was concluded that there is _no

justification

for the Kirkwood formula

beyond

order _@. It should be

emphasized

that it incurs

no restriction upon the contribution from the static effects

(such

_as the excluded-volume interaction, whose strength ^{is also of the order} @).

Finally,

from

(41)

and

(46)

with the choice

to

_"

Nla

_as

before,

_we obtain the

expression

for D at the

fl-point.

It reads

~)~ ~~D

=

~~ (1+ ~~([Ao(y)]~~/~(1+ Bih))

,

(48)

~~

N

(

9

where

Bi

" 81r

/33 -41.

The second

equation

of

(42) implies

that when the chain becomes very

long,

the second term _on the

right

of

(48)

becomes dominant.

Thus, defining

the

hydrodynamic

radius

RH by

the relation

D=

~~~

,

61r~~RH (49)

we

find,

for

large

value of

N,

Rj~

=

~

[NAO(y))~~/~(1+ Bih) (So)

7r

Equations (48)

and

(So)

constitute the main results of this paper.

4.

Concluding

Remarks

An estimate of the

magnitude

of the

three-body (osmotic) parameter

h is now

possible by comparing

the theoretical results of Section 3 with data from

experiment

and simulations. The

quantity RG/RH

is of interest in this respect. It follows from _our results

(23)

and

(So)

that the ratio at the

fl-point

is

given by

There _are difficulties in the

comparison

of

(51)

with

experiments. First,

the

steady-state

diffusion constant should be measured to determine

RH

_as defined

by (49). However, usually

the values of the

hydrodynamic

radius _are estimated from the initial

decay-rate

of the

dynamic

structure factor

[18].

^{The so-called initial diffusion} constant thus obtained _can be different from _our D in

principle.

Another _reason for the

difficulty

is that the

precise

determination of

RH

from

(49) requires

_an accurate value for the ratio

~~/~~

at the _same time

[19],

^where _~~ ^is the bulk solvent

viscosity.

This is at

present

a serious obstacle in

practically comparing

the

theoretical

prediction

with

experiments.

In reference

[20],

^Bruns

performed

Monte Carlo simulations to

directly

calculate the

equilib-

rium ensemble average of the second term _on the

right-hand

side of

(39). Therefore,

his estimate of

RH

is free from _any such

complication

as described in the

preceding paragraph.

From the

value

RG/RH

= 1.464

given

in this reference

together

with

(51),

_we estimate h

= 4.5 _x

10~3

We _can

separately get

_an estimate from

R~ /R(

that has been measured in the _same simulation.

Comparing

the data

R~/R(

= 5.94 with _our

prediction (see (23)

and

(26)

~

=

6(1

^~~

h)

,

~ (52)

4

we find h

= 4.2 _x

10~~, comparable

with the

dynamical

estimate

given

above. Our values agree

reasonably

well with the value h

=

(1.2

+

o.2)

_x

10~3

which _was estimated

[21]

^{in terms} of other static measurement in real

experiments _[22].

The above

comparison provides

_an evidence of the existence of

ternary

interactions _near the

fl-temperature.

We

emphasize, however,

that

although

_we have _a

good agreement

in order of

magnitude

of

h,

this does not _mean that the

logarithmic

corrections to ideal-chain behavior _as

predicted by

the

theory

have been verified

by experiments

and simulations. In

fact,

_a recent

large-scale

simulation

[23]

^with

supposedly high

_accuracy has found

logarithmic

corrections which _are different from the renormalization theoretical

predictions.

Now that _we have several

different theoretical

approaches

to the

study

of the

fl-region,

which _are

entirely equivalent,

and since this _paper has

provided

the theoretical

expression

for the

dynamic properties,

_we

challenge experimenters

to test these

predictions. Meanwhile,

the renormalization theoretical

calculation of other

transport coefficients,

such _as intrinsic

viscosity

and

decay

rate, ^would be _very useful. In

brief,

it is

hoped

that the renormalization

method,

which has been _so

successful in

elucidating

the solution

properties

of

polymers

in _a

good solvent,

_may resolve those

problems _[24]

that confront the current

theory

of

fl-regions.

Appendix

A

This

appendix

summarizes the calculation of the

right-hand

term of

(39).

We will start with

Tr

/~

ds

/~ ^{ds'jTjrjs) rjs'))j}

₌

id ljs~ / _{Hjkj /k~}

,

jA.I)

where

f~

e

(21r)~~ f

dk and

H[k]

₌

j / _ds' /

ds"(e~~'l~~~'~~~~~")I

e

j / _ds' / ds"h[k;

,

s', s"] (A.2) Throughout

the

Appendix

let S denote

No

for the sake of notational

simplicity.

The function

h[k; s', s"]

_can be written _as

[25]

, ,,

Z(k;S,s',s")

jA.3)

~'~'

_~

'

~

Zi IS)

Here

Z(k; S, s', s")

is _a

partition

function of _a chain with two insertions of _wave vectors k and -k at contour

positions

^s' and s"

along

the

chain,

and hence

Zi IS)

_e

Z(0; S,s',s")

is the

single-chain partition

function.

Diagrammatic perturbation theory

to calculate Z is well established

[13].

^The

diagrams contributing

Z to first order in _{co are} indicated in

Figure

1. As _a

representative example,

_we

illustrate the calculation of the contribution of the

diagram if)

to the

integral Z(S)

_e ds'

ds"Z(k; S, s', s") /k~

which appears in the

expression

for

H[k]. Labelling

the

diagram

_as shown in

Figure _2,

_we find that it is

given by

S s"

s3 s2 si

Z~(S)

_%

(-C0) / / / / _dS" / _dS3 / _dS2 / _dsl / _dS'

k k, k2 0 0 0 0 0

k~~G(0,

S

s")G(k,

s"

s3)G(k k2,

_{s3 s2}

x

G(k ki,

_{s2 si}

)G(k,

_si

s')G(0, s') (A.4)

with

G(q,s)

=

exp(-q~s/2) (A.5)

s

.,, .,,

,~~ ,~_

s'

'~

~j j ~ ⁾

~

s^' ,;~

"'

'''

~ ~

i

(1) (a) (b) (c) (d) (e) (f)

Fig. 1. The diagrams

contributing

to

Z(k;

S,

s', s")

_to first order in three-body interactions. Each

diagram

consists of

one

polymer

line

(solid) representing

_a chain of

lenght S,

and of the

three-body

interaction line

(dashed) connecting

the interaction

points

of the chain. Wave vectors k and -k _are

injected

at the _contour

points

^s' and s".

>. s'

S~ ^'~..

j

k~

S~ 'S)

k~

S, '""'

.>.. s'

Fig.

2. A

diagram contributing

to

Zf(S).

Taking

the

Laplace

transform of

Zf(S)

in

S-space yields

)lip)

=

(-co)P~~ / / / _{k~~ [©(k, p)]~©(k ki, p)©(k k2,}

P)

,

(A.6)

k k, k~

where _p is the

conjugate

variable to

S,

and

©iq,P)

=

iq~/2

₊

P)~~ IA-I)

No~v the

integral

in

(A.6)

is

performed by

the method of dimensional

regularizatioii. Using

standard formulas:

Ii ^{Tin d/2)}

q [q2 +

2(k q)

+

m2]" 2"lrd/2T(o)(m2 k2)°~d/2

^'

~ ^{)~)~(~~ ~} ~[ai~~~~(1 ~~)]~+P

'

~~'~~

we find

jij~)

=

j_~~)~3d/2-7j~~)-3d/2 ^7rld 4)irli d/2)l~

j

_{~ g~}

21d 2) _sin17r13 d/2)lrld/2 1) Performing

the inverse

Laplace

transform of

(A.9)

and

by analytic

continuation to d

=

3,

_we

then obtain

js/~)3/2

~~~~~~~

_~°~

3~7/2 ~~'~~~

Contributions ofother

diagrams

_are evaluated in the _{same manner.} Then

Z(S)

_can be

expressed

as a sum of the contributions associated with the

diagrams (o)-(f)

of

Figure

1:

1

zjs)

=

~j _g~zijs)

,

jA.ii)

imo

1n.here the

prefactor

_{gi is} the

degeneracy (weight)

of the

diagram Ii),

and

zo

~

4j

^S

_)3/2

3 21r Z~

=

-41rMJoZ°, Z~

=

-(Z~

,

Z~ =

Z~

=

In(3/2)

_x Z~

,

Z~

= ln2 _x Z~

,

Zf

=

jZ~ ^(A.12)

Moreover,

for the

single-chain partition

function _we have

Zi(S) s2

=

-(1- _41riuo) (A.13)

Substituting (A.11)

and

(A.13)

into

(A.1),1n-e finally

obtain

Tr

Ids / _{ds'(T(r(s) r(s')))}

=

~~(

^~

)3/2 ^~~~~~°°

,

(A.14)

3 21r 1

4~wo

where

Ci

= 2 +

In(32/81). Keeping

the dominant contributions

brings

_us to the

expression (39)

with

Ai

=

4~(Ci -1).

References

ill

de Gennes

P.G.,

J.

Phys.

Lett. France 36

(1975)

L-55.

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Duplantier B.,

J.

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Renormalization

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Modelling

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[14]

^The

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[3],

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by

_WKF

" 2~w.

[15]

^{Shiwa Y. and Ka~vasaki}

K.,

J.

Phys.

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5345 and reference

[ii

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[16]

^Lee

A.,

Baldwin P. R. and Oono

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6772.

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Y.,

AIP Conf. Proc. 137

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187.

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Y.,

Hirata

M.,

Nemoto

N., Kajiwara

K. and Kurata

M.,

Macromoiec~ties 20

(1987)

2862 and references therein.

[19]

^{Shiwa Y. and Tsunashima}

Y., Physica

A 197

(1993)

47.

[20]

^Bruns

W.,

Macromolecules 17

(1984)

2826.

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B. and Jannink

G., Phys.

Rev. Lett. 70

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3174.

[22]

Duplantier

B~ Jannink G. and Des Cloizeaux

J., Phys.

Rev. Lett. 56

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Boothroyd A.T.,

Reniiie

A-R-, Boothroyd

C.B. and Fetters

L-J-,

ibid. 69

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426.

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P. and

Hegger R.,

J. Chem.

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6881.

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Shiwa

Y.,

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Duplantier B.,

J.

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