Polymer chain in annealed random media

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Polymer chain in annealed random media

A. Baumgärtner, B.K. Chakrabarti

To cite this version:

Short Communication

Polymer

chain in annealed random media

A.

_{Baumgärtner(1)}

and B.K

_{Chakrabarti(2)}

(1)

Institut

für

_{Festkörperforschung, Forschungszentrum}

_Jülich,

D-5170

_Jülich,

F.R.G.

(2)

HLRZ,

Forschungszentrum

Jülich,

D-5170

_Jülich,

F.R.G. and Sha Institute of Nuclear

_Physics,

Calcutta

_700009,

India

(Received June

1 st

1990,

accepted June

7,1990)

Abstract. 2014

Self-avoiding polymer

chains

_trapped

in annealed random media have been

_investigated

by

Monte Carlo methods. It is found that the

_{configurational}

characteristics of a

_polymer

are not

affected by

the media and are

_independent

of the concentration of the

_impurities.

This is in _agreement with an

_early

_conjecture,

but in

_disagreement

with recent

_analytical

results.

Classification

Physics

Abstracts

05.40 - 72.90 - 64.70

1. Introduction.

The

_question

on the effect of

_quenched

disorder on the statistics of

_{self-avoiding}

walks

_(SAW)

has been

_quite

_stormy

_{[1, 2] . Compared}

to

_that,

the

_apparently

much more

_simpler

_question

on the effect of annealed

_impurities

on the statistics of SAWs has been addressed

_only

_very

_recently

[3, 4, 5].

Consider the

_problem

of a SAW on a lattice with

_missing

bonds

_(say)

at a

_given

concentration. Assume the

_remaining

bonds to be in thermal

_equilibrium

with the chain.

One would _not, in

_{fact, expect}

_any effect of annealed disorder on the SAW statistics for _any

amount of disorder.

_{Denoting by}

_RN

the

_average

end-to-end distance of an N

_stepped

SAW,

where

_(c)

denotes all

_possible

SAW

_{configurations}

on the

_(perfect)

_lattice,

and

_{r 2 (c)}

denotes the end-to-end distance of the N

_step

SAW for the c-th SAW

_{configuration}

and

Pc =

1 if the

c-configuration

of the SAW is

_possible

on the dilute

lattice; Pc

= 0 otherwise.

P,

=

Pl P2P3.... -PN,

where Pi

are the

_occupation

_operators

_{(Pi =}

1 or

₀₎

of the bonds of the lattice

_through

which the SAW

_{configuration}

c

_(on

the

_perfect

_lattice)

_passes.

For annealed

_impurities,

the

_averaged

1680

end-to-end size of the SAW could be

_{given by}

since

_Pc

>=

p N

_where

_p is the

_average

_occupation

concentration of

_bonds,

because a SAW never visits the same bond more than once. The SAW chain size should thus remain

_completely

unchanged

for any amount of lattice disorder.

Note,

that the above

_independent

disorder _average for the numerator and denominator is

_{perhaps possible}

for annealed disorder

_only,

and was in fact

wrongly

(6, 2]

applied

to the

_quenched

disorder case

_[7].

Annealed

_impurities

_being

in thermal

_equilibrium

with the heat

bath,

are

_quite

often

"con-strained" ;

e.g.,

their total number

may

be

_{given by}

an

_(analytic)

function of the heat bath

pa-rameters or _may be

_kept

fixed. There is considerable literature on the effect of such

_commonly

encountered contrained-annealed

_impurities

on

_magnetic

_systems

_{(n-vector magnetic}

model in

general)

[8-10].

Essentially,

a

_long

range 04

term

_{(with slightly}

different momentum conservation

rule)

appears

[10] ,

apart

from some renormalisation

_(reduction)

of the usual

_4>4

term

_(standard

excluded volume term in the n --+ 0

_limit):

This new v _term,

_arising

from constrained-annealed

_impurities,

is

_responsible

for a crossover to

Fisher renormalised critical behavior

_(v

--->

v/(1 -

_a))

for the

_général

n-vector

_model,

when

_a(=

2 -

_dv)

of the nonrandom model

_(v

=

0)

is

positive.

This Fisher renormalised critical behavior due to annealed

_impurities,

which is so

_commonly

observed in

_magnetic

_systems

_{(for anisotropic}

or

_Ising-like

_magnets

n =

1,

_say),

is,

of course, unrealisable in the SAW limit n ->

_0,

where the

renormalised size

_{exponent v}

_may become

_greater

than

_unity,

which is

_meaningless

_{[4] .}

This

is,

however,

circumvented

_[5]

because of the

_appearance

of a

_{collapsed (v}

=

1/d)

_phase

described

by

a

_{"discontinuity"}

fixed

_point,

when one notices that the Fisher renormalised fixed

_point

_(being

a

_long

_{range interaction fixed}

_point)

is of order

_1/n

and runs

_away

to

_infinity

as n -. 0 in the

SAW limit. Thirumalai

_[3]

_argued

that annealed

_impurities

will

renormalise,

and hence

decrease,

the u term in

₍₃₎

and could

_{eventually (for}

_impurity

densities

_beyond

a critical value

_{pc ),}

make a

crossover to a Gaussian chain behaviour.

_{Duplantier [4] ,}

extended the calculations and

_argued

that

_higher

order

_{(0 3}

_order)

terms would also

_appear

due to annealed

_impurities

and under such cases a tricritical

_0-point

behaviour will be observed at _Pc

_(where

u =

₀₎

and

_collapse

would occur

beyond

that

_point

_(p

_p,, where u is

_negative).

_However,

if such nontrivial

_(averaging)

effects

are introduced

_by

some kind of annealed

_impurities

_(which

do not

_permit

the

_{straightforward}

averaging

in

_Eq.(2))

_then,

the crucial

_term,

would be the v-term

_(not

considered in Refs.

_[3]

and

[4] )

which introduces the Fisher renormalised fixed

_point

for usual n-vector

_magnets

_(n

>

_0).

The

fixed

_point,

of course, runs

_away

to

_give

a

_{"discontinuity"}

fixed

_point

in the SAW

_(n

-->

₀₎

_limit,

giving

[5]

a

_{collapsed phase (v}

=

1/d)

for

any amount of

impurities

(p

1).

In what

follows,

we did not see

_any

such nontrivial effects on the SAW

_statistics,

due to the

_(simplest)

kind of annealed

2.

_Model,

simulation

_technique

and results.

As a model for a

_polymer

_chain,

we consider a

_{self-avoiding}

walk on the cubic lattice. Ensembles

of chain conformations are

_{generated by}

_randomly

_{applying kink-jump}

and

_{reptation algorithms}

during

the simulations

_(for

details see _e.g.

_[11]

_).

A local

_kink-jump

is

_{performed by changing}

the coordinate of the k - th

_polymer

_segment

at rk to

_rk

=

rk+l + rk-i - rk. A

configurational

change according

to the

_{reptation algrorithm}

is obtained

_{by clipping}

one of the two

_segments

at

the ends of the chain and

_attaching

at the other end a new

_segment

_pointing

in a

_randomly

choosen direction on the cubic lattice.

_Any

_attempt

_violating

the self-avoidance is

_rejected.

But even if this

_attempt

is successful with

_respect

to

_{self-avoidance,}

this

_attempt

has to be

rejected

if the new site

_rk

is

_occupied

_by

an

_impurity.

The

_occupation

of

_rk

_by

an

_impurity

at

each

_{attempted configurational change}

of the

_polymer

occurs

_{in q}

_p, where 0

_q

1 is a random number determined anew

_qhenever

the

_polymer

_attempts

to

_occupy

this

_site,

and 0

p

1 is the concentration of the

_impurities.

This fulfills the

_requirement

that the

_(annealed)

impurities

are in "thermal"

_equilibrium

with the

_polymer

chain.

However,

unlike in cases where the total number of such

_impurities

could be

_kept

fixed or

_constrained,

we believe the annealed

impurities

in our model are "unconstrained".

We have estimated the mean

_square

radius of

_gyration

S2

>, the mean

_square

end-to-end distance

_{R2 >,}

the number of

_{nearest-neigbor}

contacts

_per

bead

_E/N,

and the

_{corresponding}

"specific

heat" C

where N is the chain

_length.

Simulations have been

_performed

for

₀

_p

0.95 for N = 300 and N = 600. We found that the

_{configurational}

characteristics are not

_{affected by}

the annealed

_impurities

and are

_independent

ofp.

The results are

_given

below.

The numbers in brackets denote the statistical error.

No

_significant

indication of

_any

crossover in the SAW critical

behaviour,

as

_{predicted by}

the

analytic

theories

_[3-5],

could be observed. The

_"specific

heat" _{doesn’t exhibit any}

_anomaly,

as would be

_expected

at a

_{0-point [4].}

Since critical behaviour of the

_specific

heat has been observed

in numerous simulations

_(see

e.g.

[12] )

of the "classical"

collapse

transition,

we must conclude

that,

at least for the

_present

model of

_{self-avoiding polymers}

in annealed random

_medium,

a

0-point

does not exist.

Since we have not observed _any effect of the

_impurities,

one

_might

distrust the simulations. Therefore we have

_performed

some

_preliminary

simulations of

_polymers

with weaker self-avoidance

_{(Domb-Joyce polymer}

model

_[13]

_),

which in fact

_yield

evidence of an influence

1682

3. Conclusions.

All these observations seem to confirm that _any amount of

_{("unconstrained")}

annealed

_impurities

do not affect the critical behaviour of "strict"

_{self-avoiding polymer}

_chains,

as

_conjectured

orig-inally by

Harris

_[7]

_{(wrongly applied}

to the

_quenched

disorder

_case),

due to the cancellation of the disorder effects on the _average

_polymer

size as discussed in

_{equation (2).}

_{In such cases,}

pre-sumably,

no v

_term,

in Hamiltonian

_(3),

arises and no Fisher renormalised fixed

_point

_appears.

[5]

Our observations do not _{indicate any tricritical}

_0-point

and

_collapse

behaviour

_[4]

for

_polymer

chains in random media with annealed

_impurities.

Acknowledgement.

We are

_grateful

to Professor D. Stauffer for a critical

_reading

of the

_manuscript

and for some useful comments.

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[1]

LEE

_S.B.,

NAKANISHI

H.,

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_A.B.,

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D.,

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