The end vector distribution function of a self-avoiding polymer chain or a random walk for short and large distances : corrections to the scaling

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The end vector distribution function of a self-avoiding

polymer chain or a random walk for short and large

distances : corrections to the scaling

S. Stepanow

To cite this version:

899

Short Communication

The end

_vector

distribution function of

a

self-avoiding

poly-mer

chain

or a

random walk for short and

large

distances :

corrections

to

the

_scaling

S.

_Stepanow

Thchnische Hochschule "Carl Schorlemmer"

_{Leuna-Merseburg,}

Sektion

_Physik,

DDR-4200

_Merseburg,

D.R.G.

(Reçu

le 18 décembre 1989,

accepté

sous forme définitive

le 14 mars

₁₉₉₀₎

Résumé. 2014 La distribution de

_probability

de la distance bout à bout est étudiée à l’aide du _groupe

de renormalisation. Des corrections à la loi d’échelle ont été obtenues _pour les distances courtes et

longues.

Abstract. 2014 The end vector

distribution function of the

_{self-avoiding polymer}

chain is

_investigated

using

the renormalization _group

_(RG)

method. The corrections to the

_scaling

behaviour are found for small and

_large

distances.

1

_Phys.

France 51

₍₁₉₉₀₎

899-904 15 MAI _1990,

Classification

Physics

Abstracts 05.20 - 05.40 - 81.20’S

It is well-known that the end vector distribution function of a

_{self-avoiding polymer}

chain or a random walk

_obeys

the

_scaling

law

_[1]

where d is the

_space

_dimension,

R ~ LI’ is the mean chain end-to-end

_distance,

and v is the critical

exponent.

According

to[2-5]

the function

_{f (x)}

behaves for

_large

and short x as

*

large x [2, 3]

900

* _short

[4,5]

where y

is the critical

_exponent.

In the

_present

letter we will show that

₍₂₎

and

₍₃₎

are

_only

valid

for intermediate x. In the limit x ---> 0 there

_appear

corrections to the

_power

law

_(3).

In the limit

x --> oo,

(2) changes

to the Gauss function.

The distribution function of the end vector,

_p(r),

is defined as follows

where the

_average

is

_performed

with the aid of the Edwards Hamiltonian

_[6].

The

_right-hand

side

of

₍₄₎

is the ratio of the number of states of the

_polymer

chain with the fixed end-to-end distance to

the total number of states of the

_polymer.

The numerator in

_(4),

which we denote

_by

_G(r),

is the Fourier transform of the end-to-end chain correlation function

_{G(p) = (exp(ip(r(L) - r(0)))).}

The denumerator in

₍₄₎

is

_G(p

=

0) .

The

perturbation expansion

of G(p)

in

_powers

of the excluded

volume

_strength

vo can be

_{represented by}

means of

_diagrams

[5].

The result of the calculation of

G(r)

up to the first order of vo is

where z =

Vol-2 (d/27rl)d/2

LE /2

= _vo

LE/2, £

= 4 -

d,

x

= r2d/(2IL).1

is the statistical

_segment

length,

and L is the contour

_length

of the chain.

_r(x)

is the Euler _gamma function and

_03C8(a,

_b,

_x)

is the confluent

_{hypergeometric}

function of second art. The mass

_divergency

has been eliminated from

_(5).

The constant

_vo

is introduced in order to facilitate the _way of

_writing

the formulae. It

offers no

_difficulty

to rewrite the formulae

_through

the variables

_vol-4

and L1. From

₍₅₎

we obtain

up

to the first order of

-* _small

r

where C = 0.5772... is the Euler number.

* large r

We

_begin

with the

_{regularization}

of

_G(r)

for small r. The

term - (41e)

z in

₍₆₎

can be eliminated from the

_perturbation

_{expansion by}

the redefinition of L in the

_prefactor

as follows

The term

_{(2/£) . Z .}

xE /2

forces the

_appearance

of the counterterm X as a factor in

_{G(r) :}

We notice that X does not

_depend

on L. The last two

_singular

terms in

₍₆₎

_change

the

_argument

where

’Ib the first order of e,

Xr

coincides with X and

XL

with

L’/L.

Notice that the

_expansion

parameter

of L’/L

and X are different.

_{Equations (8), (9)}

and

₍₁₁₎

must be

_completed

_by

the renormalization

prescription

of the interaction constant. In accordance with

_[7,8]

we have

Equations (8), (9), (11),

and

₍₁₂₎

enable one to obtain the differential

_equations

of the renormal-ization

_group

_[7-9].The

solution of the latter

_gives

In the

_{scaling regime,}

which is defined

_by

the condition that the second terms in brackets of

₍₁₃₎

and

₍₁₄₎

are much more

_greater

than _one, we obtain

where v =

1/2

₊

e/16

₊ ...

and -y

= 1 +

e /8

+ ... are the critical

exponents

computed

to order e.

G(p

=

0)

behaves in the

_{scaling regime}

as

_[4]

’

The

_exponential

function in

₍₁₀₎

is obtained in the

_{scaling regime}

as

Let us consider the condition

_determining

the

_{scaling regime.}

This condition is different for the

_quantities

_L’,

_XL

and

_X,

_Xr.

For L’ it is

_given

_by

the

_{inequality z}

_{» 1}

_(large

vo and

L). The

scaling regime

for X and

_{X r}

is defined

_by

Independent

of the value of vo this condition breaks down for

sufficiently

small r. In the limit r -->

_0,

X tends _{to 1.}

As a

_consequence,

the number of states of a

_{self-avoiding ring polymer,}

which is

_proportional

to

G(r

=

0),

does not contain the renormalization factor X. The characteristic r*

_separating

the

902

V4’e now show that r*

_given

_by

₍₁₉₎

is

_proportional

to the blob dimension. The number of

_segments

in a blob

Vbj

=

r*2 /12

_obeys

the condition

" ’"

which is

_equivalent

to

_(19).

For

_large

va, r* tends to zero.

Now we turn to the renormalization of

_G(r)

for

_large

r.

The

_{1/e poles}

of

₍₇₎

force the renormalization of

_G(r)

as follows

where

- ... » Il

Th the first order of e there is no difference between

both X

in the

_prefactor

and in the

_exponent.

Analogous

to

₍₁₃₎

and

₍₁₄₎

the renormalization

_group

enables one to obtain

It is _easy to see that in the

_{scaling regime}

that is in accordance with

_(2).

The factor

_1/

_(X

_Ld/2)

in

₍₂₀₎

_gives

in the

_{scaling regime}

re /4 IL d/2

that coincides to the first order of - with the

_prefactor

in

_(2).

In the

_{scaling regime}

our results coincide with that obtained

_by

Oono et al.

_[10].

From

₍₂₂₎

it follows

_that,

in the limit r ---> _oo, X tends to one and as a

_consequence

_{f (x)}

instead of

₍₂₎

_changes

to e-X. The characteristic r = r**

separating

the

scaling regime

from the

nonscaling

one

_obeys

the condition

that can be rewritten as follows

The ratio of r** =

zR2/Ro

to the stretched

_length

of the

_polymer,

rstr =

L,

is obtained as

In the limit of

_large

L it follows that r** > rsir The latter means that r** lies

_beyond

the

applica-bility

of the Edwards model for real

_polymers.

However,

there exists another relevant case. Let

us increase L and

_{keep z}

constant. Then we obtain r** rstr. The latter case is relevant for

_large

polymers

with moderate excluded volume

_strength

vo.

Now we will show how the behaviour of

_G(r)

for

_large

z follows from the

_general

RG

where X and L scale as

The anomalous critical dimensions _yX and _ys are

_{expressed by}

the critical

_exponents

as follows

[11]

In the renormalization scheme with cutoff

_{(,A2-scheme),}

which can be used as the

_parameter

of the

_RG,

_Amin

is the final value of the latter

_(see

_[7,11]).

The Fourier transform of

₍₂₅₎

_yields

Restricting

ourselves to the first term in the

_expansion

of

_Greg(x)

we obtain

The

_application

of the ultraviolet renormalized

_theory

to the

_study

of

_scaling

behaviour

_requires

using

the

_matching

condition

_[11,12].

In the limit of

_large

r,

Amin

has to be identified

_according

to

(22)

as

The

latter,

in combination with the second

_equation

of

_{(26), gives}

an

_equation

for

_Amin,

the solu-tion of which is

From

₍₂₉₎

and

₍₃₁₎

we

_finally

obtain

In summary, we have considered the end vector distribution function of a

_{self-avoiding}

poly-mer chain

_by

means of the RG method. In contrast to the

_{previous investigations}

we have estab-lished the deviations from the

_scaling

behaviour for small and

_large

distances. We have obtained the

_expression

for the characteristic distance r*

_(r**)

below

_(above)

which the des Cloizeaux

(Fisher-McKenzie-Moore) scaling

law is violated.

Acknowledgements.

1 would like to thank Professor G. Helmis for a useful discussion.

References

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_{G., Scaling Concepts}

in

_{Polymer Physics (Cornell University Press, Ithaca, N.Y.)}

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McKENZlE D. S. and MOORE M. _A., J.

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