Disorder induced adsorption of polymers

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Disorder induced adsorption of polymers

A. Baumgärtner, W. Renz

To cite this version:

2641

LE

JOURNAL

DE

_PHYSIQUE

Disorder induced

_adsorption

of

_polymers

A.

Baumgärtner

and W. Renz

Institut für

_{Festkörperforschung, Forschungszentrum}

Jülich, D-5170 Jülich, F.R.G.

(Reçeived

17 _August 1990,

accepted

23 _August

₁₉₉₀₎

Abstract. 2014 The

competition

between

adsorption

and

entropic repulsion

of a

_{single self-avoiding}

polymer trapped

in

_quenched

media

_consisting

of

_{parallel, adsorbing}

rods distributed at random in the

_{perpendicular plane}

are studied

by

simulations and

analytical

arguments. The chain

undergoes

an

_adsorption

transition at finite _temperature which is not observed in a

_regular

medium. Below and above the transition temperature, the chain is

essentially

stretched

_parallel

to

the rods and shrunk in lateral directions with _exponents 03BD~ = 0, 03BD~ = 1 for the

corresponding

components of the radius of

_gyration.

These

_phenomena

are based on two different mechanisms,

localization

by adsorption

at low temperatures and localization

by entropic repulsion

at

_high

temperatures. The _present model is the first where the

_entropic

localization occurs at

_arbitrary

strength

of self-avoidance.

Analytical

arguments suggest, in _agreement with our _data, that the

chain

obeys approximately

conventional

self-avoiding

behavior at the critical

_adsorption

temperature. We _expect that both

_phenomena,

disorder induced

_adsorption

and

_entropic

localization, can be tested

experimentally.

J.

_Phys.

France 51

₍₁₉₉₀₎

2641-2651 1 er DÉCEMBRE _1990,

Classification

Physics

Abstracts _

05.40 - 36.20 - 64.60

1. Introduction.

Flexible

_polymers

chains with

_gradual

self-avoidance

_{(SA) [1-13]}

and strict SA

_{[14-17] trapped}

in

_quenched

random media have attracted much attention very

recently.

These models are

believed to be relevant for

_{problems involving polymers}

in

_{gels (filtration, chromatography,}

catalysis

or extraction from _porous

_rocks).

_Furthermore,

there are

_applications

in

reaction-diffusion

systems

[6]

and

_{non-equilibrium}

surface

_{roughening [5].}

The theoretical interest has increased since these models can serve as

_{prototype systems}

_exhibiting

features of more

general

systems

with

_quenched

disorder.

For ideal chains

_(no

SA)

it has been shown

[1,

2, 5,

6]

that the chain size shrinks for

M>0,

where

_{u 2 =P 0 V2}

measures the

strength

of the

disorder,

p o and v

denoting

the

concentration of the obstacles and the

_{scattering strength}

of an obstacle

_{respectively.}

This is a

consequence of the

entropic repulsion,

which causes the chain to use

_density

fluctuations of

the disorder and to

_{preferentially}

_stay

in

_regions

of low obstacle

_density.

It should be noted that the similar

_problem

of a random walk of a

_{single particle}

in disordered media

(«

ant in a

labyrinth »

[18],

is different from a random walk chain in the same media since different

ensembles are considered

_{[ 1-7].}

The

_{equilibrium configuration}

of such a chain is

strongly

collapsed

for

u2/

(4 - d)

_JN

1,

and for

_spatial

dimension d = 3 the mean _square end-to-end

distance (R2)

obeys

the

scaling

relation

[1,

2]

where f

and N denote the Kuhn

_segment

_length

and the number of

_segments

in the

_chain,

respectively.

The

_scaling

function

_{f (x) -}

const for x ..,c 1

_whereas

_{f (x) - x- 2}

for x > 1. Hence the size of the chain

_becomes,

for

u 2

Ùk >

1,

independent

of chain

_length

N

We call this the

« entropic

localization of a random walk » in

analogy

to the Anderson localization of a

_quantum

_particle.

This has been discovered first

_by

Monte Carlo simulations

[1]

]

and has been discussed

_intensively

later

_[2-13].

In

_{particular, simple}

_arguments

as

introduced

_by

M. E. Fisher

_(estimate

of the

_{Flory-exponent}

vF =

3 / (d +

2 ))

and

by

Imry

and

Ma

_(Ising

model with a

_quenched

random

_field)

have been

_{developed [5, 6] leading}

to the

same results as variational calculations

_[2]

and the use of the results from the

Anderson-localization

_{problem [6].}

So

_far,

the

_entropic

localization has not

_yet

been examined

_{experimentally, although}

some

conjectures concerning

its

_{observability}

in

_{polymer networks [11]}

_]

and at the

_coil-globule

transition in porous solids

[10, 12]

have been made. In

_particular,

there has been

_given

evidence

[3,

10,

12] that,

for

_sufficiently

weak

SA,

the disorder-induced localization should survive. A

_regime

where such weak SA is

_usually

realized is near a

_8-point.

There the influence of disorder reduces to

_shifting

the 9-transition towards

_higher

_temperatures

_(the

SA-side of the

_{transition) According}

to our

_knowledge,

the

_question

if

_quenched

disorder

changes

the tricritical behavior at the

_0-point

has not been examined _up to now.

In this work we

provide

the first

_polymer

model

_system

where the

_entropic

localization is

present

at

_{arbitrary strength}

of self-avoidance. We

_expect

it to

_represent

an

_{experimentally}

realizable situation.

_Moreover,

the

_system

under consideration exhibits a critical

_adsorption

transition which is caused

_{solely by}

the

_quenched

disordered structure of the

_polymer’s

environment.

The main idea to assure the

_independence

on

SA-strength

of the

phenomenon

is to

_study

an

anisotropic

system,

where localization of the chain takes

_place

in d’ - d dimensions and simultaneous

_{unfolding according}

to the SA-constraint is

_expected

in the

_remaining

d - d’ dimensions. An

_appropriate

choice of such a model environment is a disordered array

of

_{long parallel}

_rods,

and hence d =

3,

d’ = 2

(Fig. 1).

As a _consequence, we

_expect

in

general anisotropic

chain

_{configurations}

with !

To validate the model in a realistic situation one has to make sure that the

_lengths

of the rods

are not

_{significantly}

shorter than the

_length

of the

_fully

extended

_polymer

chains.

The second

_{important ingredient}

in our model is assume the obstacles to be

_adsorbing,

which is a common _{situation in many}

_experimental

_systems.

Simultaneous _presence of disorder and

_{adsorption provides}

the

_{following competition :}

at low

_temperatures

the chain likes to adsorb on the rods and thus favours

_regions

of

_{high density}

of obstacles where

they

can

_get

localized. In contrast, at

_high

_temperatures

the chain is

_expected

to move

freely

Fig.

1. - Sketch of two

typical

configurations

of a

_{polymer trapped}

among

randomly

distributed

parallel

rods.

annihilate to lowest order and the chain is

_expected

to behave

_{approximately}

like a

conventional SAW.

These

_questions

and

_predictions

will be worked out in the rest of the paper, which is

organized

as follows : a lattice and continuum version of the model is defined in section 2.

Section 3

_presents

results from a Monte Carlo simulation.

_Analytical

_arguments,

_developed

in section

_4,

_explain

the main results

_given

_{in section 3. Section 5 contains the summary and} conclusions.

2. The model and simulation

_technique.

We consider a

_{single polymer}

chain in an

_anisotropic

medium

_consisting

of M

_{infinitely long}

parallel

hard rods of cross-section

_{a2 pointing}

in the z direction. In the

perpendicular xy-plane

the rods are distributed at random with a concentration p o =

Ma2/L 2 where

L denotes the

linear dimension of the

_system.

The rods are

_impenetrable

and

_adsorbing

with a short _range

attractive

_potential

of

_{strengths 00}

and range a. The

_polymer

chain is modelled

_by

a

self-avoiding

walk

_(SAW)

where each self-intersection costs the free energy

kB

Tw. For

w = 0 the chain is Gaussian whereas it is

strictly self-avoiding

for _{w - oo .}

We are interested in the behavior of the

_system

at small and intermediate rod-concen-trations far from

_percolation

thresholds. In this

_regime,

we

_expect

the same results for chains with small but finite self-intersection

_{probability ps, = exp (- w ), w large,}

as for

strictly

SAW’s.

Furthermore,

in a continuum version of the model we can use the limit

where

_{ruz) = (r 1- (s), z (s) )}

denotes the

_spatial

coordinate of the

_{monomer s and 6}

=

1 /kB

T.

The first term accounts for a free Gaussian random walk in d

_dimensions,

the second term

counts the number of self-intersections and the third term sums _up the disorder

_potential

U

along

the chain. The

_{disorder-averaged}

_{potentials ((/)}

is an

_arbitrary

constant whereas the correlations are

with

_{6 a d’}

_denoting

a d’ = d -1 = 2 dimensional smoothed-out delta function of width a. Here

u2 = po v2(T)

contains

_entropic

and

_energetic

contributions from hard-core

_repulsion

and

short-range

attraction of the rods. This will be further discussed in section 4.

For the Monte Carlo simulation we used a lattice version of the above model on a cubic

lattice with lattice constant a. The rods are

_{represented by}

columns of sites not accessible to

the

_polymer

and _{p o} is the

_{occupation probability.}

For

_{p 0 -- 1 - p c =}

_0.41,

the space in

between the rods is

_percolated

in the

_xy-plane,

i.e. it forms an infinite network of

_empty

space, and hence the

polymer

can move

through

the whole

_system.

As the

_polymer

model we

took that of Domb and

Joyce [19, 20]

on the cubic lattice. Each chain conformation has a

statistical

_weight

where nj

is the number of times the chain visits the lattice

_{site j}

and the sum runs over the full

space. An

adsorption

energy

4>0 >-

0 is attributed to each nearest

_neighbor

contact between a

rod and a chain

_{segment. Hence,}

the lowest energy state of a chain adsorbed on rods is the

straight

conformation surrounded

_by

three

_adjacent

_rods,

where the chain has E = 3 N

contacts.

_(The

_{exponentially}

rare case where the chain was

_trapped

_by

four

_adjacent

rods was

ruled

_out.)

Ensembles of chain conformations are

_{generated by randomly applying kink-jump}

and

reptation algorithms during

the simulations

_(for

details see e.g.

[21]).

Chain conformations

are selected

_according

to the

_Metropolis

scheme : the new conformation is

_accepted

if

exp

[ (Eola -

Enew) 4>O/kB

T]

>- 7], where 0 - 7] 1 is a random number. Since

_{0 0}

is the

_only

energy scale in our

_model,

we henceforth measure the

_temperature

in units of

_4>O/kB.

For all simulations we took w =

1,

which

yields

_on the average about 10

_percent

self-intersections of

the chain. Without

disorder,

the

_{SAW-exponent v}

= 0.588 is well

recovered,

even for short

chains,

N . 20. We used

_{periodic boundary}

conditions in the x

_{and y}

directions. The linear dimension of the basic cell is L _

_150,

which is much

_larger

than the distances travelled

by

the

polymer

chain

_during

the simulations. Without loss of

_generality,

we consider

p = 1 -

p o =

0.7,

disregarding possible peculiar

effects at the

percolation

threshold.

We took ensemble averages over chain conformations at different realizations of the random media. We have limited the number of realizations to

_10,

and have

_{carefully analyzed}

the

_{corresponding}

fluctuations of the mean _{square radius of}

_gyration.

This is discussed in the

next section.

3. Results from Monte Carlo simulations.

Before

_discussing

the effects of disorder on the

_{thermodynamics}

of our

_polymer

_model,

we

first

_present,

_serving

as a reference

_system,

the behavior of the

_polymer

model embedded in a

regular

matrix,

where the rods are distributed

_periodically

on a 2 x 2 sublattice in the

xy-plane

with _{p o} = 1 - p = 0.25. The results for normalized lateral mean _{square radius of}

rods,

E/N,

as a function of

_temperature

T are

_depicted

in

_figure

2. As

_expected,

the

adsorption

transition,

where S, -->

const., occurs at T = 0. At

T =i=

0 one observes

Si -

N

PF.

_{The energy is} a continuous function of T and

_yields

no evidence for an inflection

point

except

very close to T = 0.

Fig.

2.

_{- Polymer}

among

periodically

distributed rods : normalized

perpendicular

component of the

mean square radius

of gyration

,S1

versus temperature T for chain

lengths

N = 80

(A)

N = 140

(0),

and normalized

_adsorption

energy

E(e).

This situation has to be

_compared

to the case where the matrix is disordered. The results for

Si

and

_E/N

are

_presented

in

figures

3 and

_4a,

_{respectively.}

The differences as

_compared

to

figure

2 are

significant.

The

_specific

heat

C,

estimated from the thermal fluctuations of the

energy

E/N

is

depicted

in

figure

4b for N = 140 for several realizations of the disorder. Its

disorder average exhibits a maximum at

_{Tc, ;:-}

2.6

_indicating

_{the appearance of} a

_phase

transition. For

_temperatures

7"

_Tc1

chains of N « 300 are

_strongly

adsorbed such that

S1

is

_independent

_{of N,}

and

_{511 ’-}

N. This is also demonstrated for T =

2.0,

as an

_example,

for various chain

_lengths

N in

_figures

5 and 6. The chain

_{configurations}

are

_strongly

unfolded and

laterally collapsed

with

_exponents

for the

_{corresponding}

_components

of the radius of

_gyration.

The

_{adsorption preferentially}

takes

_place

in

_regions

of

_high

obstacle

_density.

Due to finite size effects we

_expect

Tel

2.6 for N -> 00.

Fig.

3. -

Fig.

4.

_{- (a) Adsorption}

_{energy per monomer,}

_E/N,

versus _temperature for various chain

_lengths

N.

(b)

Fluctuation

C,

of the energy

E/N

versus _temperature for N = 140. Each data

point

_{at one}

temperature

corresponds

to a

_particular

realization of the disordered arrangements of rods.

Fig.

5. -

Perpendicular

component of the radius of

_gyration,

_S$

versus chain

_length

Je at various

temperatures T.

Fig.

6. - Parallel _component of the radius of

_gyration,

_Su2

versus chain

_length

N at various _temperatures

At

_high

_{temperatures,}

_again

the

_polymer

chain is shrunk in the

_xy-plane

such that

S1

is

_independent

of N and

_{511 ’"}

N,

which is

_depicted

in

_figures

5 and

_6,

_{respectively,}

for T = 7.0 and T =00. This effect is observed for all

temperatures

T >

_{Te2 = 3.6,}

where

Te2

is estimated from the maximum of

_{Sl (cf. Fig. 3).}

In this case, the

adsorption

energy is very small

(E/N

= 0.36 for T =

oo),

which indicates a different mechanism for the

_shrinkage

than that at low

_{temperatures.}

In

_fact,

at

_high

_temperatures

the

_entropy

loss due to the presence of obstacles causes this lateral

_collapse

which is a 2-dimensional

analogy

of the

localization of random walks in disordered

_quenched

environments observed

recently

in Monte Carlo simulations of random-walk chains

_trapped

in three dimensional

percolated

media

[1, 7].

In the

_present

case, this localization effect is

imposed

to the chain

only

perpendicular

to the rods due to their disordered

_arrangements

in the

_xy-plane.

It is

_important

to note, that the

_presently

considered case of a

_polymer

_among

hard,

parallel

rods

is,

to our

knowledge,

the first

_example

where the localization effect is observed for chains with full self-avoidance. It seems conceivable that the

present

situation

_might

have an

_experimental

realization. In summary,

according

to

_figures

5 and 6 one observes at

_high

_temperature

Due to finite size effects we

_expect

_Te2

3.6 for N - 00. Since the

_high

_temperature

compression

is a

_purely

_{entropic effect,}

it is still

_possible

that

_{Te2 =i= Tel}

even if

_only

one

_peak

in the fluctuations _{of the energy is} seen, cf.

_figure

4b.

Since our

_{interpretation}

of the lateral

_{compression, P-L = 0,}

is

_independent

of the

SA-induced chain

_unfolding

in the z-direction

_leading

_{to v}

i =

1,

the _same

phenomena

should be

observed in a 2-dimensional model without SA. We have tested this

_{prediction by simulating}

a

random-walk chain

_(w

=

0)

on the square lattice among

_randomly

distributed

_adsorbing

obstacles _{of concentration po} = 1 -

p =

0.3,

at various

_{temperatures.}

The results for

si,

depicted

in

_figure

_7,

are _very similar to our 3-dimensional model

_{(Fig. 3) :}

_si

is almost

independent

of N at

_temperatures

T :

_Tel

and T >

_Te2

and exhibits a maximum near

T = 2.9.

The most

_prominent

feature of

_figures

3 and

7,

the appearance of a N

_dependent

maximum

of

_si

at intermediate

_temperatures

close to the

_adsorption

transition

_Tel

is more difficult to

explain.

Since this effect is observed in two and in three

_dimensions,

we must conclude that

Fig.

7. -

Perpendicular

component of the radius of

_gyration,

_si

versus _temperature T for various chain

lengths

N of a

_polymer

without self-avoidance among

randomly

distributed hard squares in two

SA is not of

_{primary importance}

for these

_phenomena.

Rather a subtle balance between

attraction,

leading

at low

temperatures

to the

_adsorption

of the chain onto the

rods,

and the

repulsion

between chain and

_rods,

_leading

to localization at

_high

_{temperatures,}

seems to be

responsible

for the emergence of conventional SA behavior at intermediate

_temperature

between two

_compressed

states at low and

_high

_{temperatures.}

We come back to a more

quantitative explanation

of this

_point

in section 4.

This

_picture

is

_{supported by}

the

_temperature

_dependence

_{of Sf}

as

_depicted

in

_figure

5. One

observes,

starting

at T = _ao, with

decreasing

_temperatures

an

_increasing

_{range of}

_N,

where

Si

seems to

_approach

SA behavior -

N2 J’P.

Of course, from the data we cannot exclude other

exponents

close to _vF.

_Also,

from our data is not clear whether there is at all a

_temperature

where v #

0 for N --+ oo, which would

_correspond

to

_{Tel:> Te2’}

On the other hand we can,

especially

for the 3-dimensional

_situation,

not exclude that there is a finite

_temperature

window

_Tel

T

_Te2

with

_{S1. ,....,}

N

vF

In the next section we will

_give

an

_analytical

_argument

for the existence of a

_single

critical

_temperature

where

_repulsion

and attraction are balanced such that

and v =

1/2

in d = 2 are

_expected

from the lowest order

_{approximation.}

4.

_Analytical

_arguments.

Here we will

_give

some

analytical

estimates which allow a more

quantitative understanding

of

the

_phenomena

discussed so

far,

_especially

the localization and simultaneous chain

unfolding

by

self-avoidance

(SA)

in presence of disorder

(Eqs. (5))

and the occurrence of a maximum in

figures

3 and 7 with

_{approximately}

free behavior

_{(Eq. (6)).}

This will be done in two

_steps.

Starting

from the

_{scaling properties}

of the Hamiltonian

(Eqs.

(3))

we will

_generalize

Flory-Imry-Ma

type arguments

for the

_present

_anisotropic

model. Two different versions of the disorder term in these

_arguments

have been

_proposed

for the

_isotropic

situation

_{independently}

by

Edwards and Chen

_(henceforth

EC

_[3])

on one hand and

by

Cates and Ball

[5]

and

Nattermann and one of the

_present

authors

[6] (henceforth CBNR)

on the other hand. In a

second

_step

the

_temperature

_dependence

of the disorder correlation

_{(Eq. (3b))}

will be derived in an

_{approximative}

scheme

_leading

to a mean-field

_type

formula for

u 2( T),

which exhibits

exactly

one zero at T =

Tc.

The basic idea of these

_arguments

is to estimate the

_saddle-point

of the

_partition

function

exp ( -

_{f3 F) = L}

exp (-

/377) by using scaling properties.

For our

_system,

the free energy is

config. estimated as

The first and second term on the r.h.s. of

_{equation (7)}

describe the

_stretching

and

compression

of a Gaussian random walk where we have

generalized

the common

_expression

(see

e.g.

[22])

to our

anisotropic

situation.

_(It

is _easy to show that the

_longitudinal

forth term there are the above-mentioned two versions of the disorder contribution.

Extremizing

F with

_respect

to

_Rp

we obtain

Extremizing

F with _respect to

_R,

and

_inserting

_{equation (8)}

we obtain

Before

_discussing

_equation

₍₉₎

for the 3-dimensional case we mention that for a 2-dimensional

random walk without SA

_{( w}

=

0)

the first term in

equation (9)

becomes irrelevant for Nu oo at u # 0 and we obtain the localization result v 1 - 0 with the same

_amplitude

for

both EC and CBNR in

_agreement

with

_[3,

_5,

_6]

Now we retum to the discussion of

_{equation (9).}

Without disorder

_(u

=

0)

the first and third

term balance for N - oo and the usual

_Flory-result

for SAWs

is

recovered,

in

_agreement

with

_{equations (6)}

when

_{anticipating u (Tc)}

= 0 as shown below.

The

_{Flory-solution, equation (11),}

is

_{spoiled by}

either of the disorder-terms as soon as

u #= 0

if N -+ 00.

Furthermore,

the solution

_{RJ.. = Rc, equation (10), only}

survives for

w :: ur 3.

In our

_model,

this condition is

_only

fulfilled at _very low

_temperatures

where the

adsorption

becomes so

_strong

that

_R,

= _a, a

_regime

which is

_hardly

accessible to the Monte Carlo method. For

_strong

_SA,

which we are interested

_in,

SA is relevant on

_{all length}

scales

starting

from lattice constant whereas disorder is irrelevant for R «

_R,

and therefore

W >-

ur 3. Consequently

the

_only

solution in presence of disorder comes from

balancing

the

third and fourth terms in

_{equation (9)}

which

_{yields together}

with

_{equation (8)}

the result

for both EC and CBNR with similar

_{amplitudes (see Eqs. (2))}

_{Al. = w2/f5 u3}

and

A

p

= f4u2/w

for

(CBNR)

and

A 1 -

( / fu 3) i /2

and

A

= fu

for

(EC).

With the result

equation (12)

we have

given

a

_{simple qualitatively}

correct

_explanation

of the simulational result

_{equation (5).}

Of course, it

might

be worthwhile to

_study

the

_prefactors

in detail in

subsequent

work.

Now we consider the second

_point,

the

_temperature

_dependence

of

u 2.

The

_potential

U(r1 )

acting

on a

_polymer

_segment

at

_position

_{r1 is}

_{represented by}

a sum over the

_positions

ri

of

_scattering

sources

(obstacles)

times the interaction

_{potential 0}

_{(r 1 - r i ).}

_{8 e}

contains a

temperature-independent

hard-core

repulsive

and a

_{temperature-dependent short-range}

attractive

_part

with

_amplitudes

_{v o and v}

_{= 4>O/kB T,}

_{respectively.}

The

_simplest

way to

proceed

at this

_point

is to assume a Gaussian

_{probability-distribution}

for the

_scattering

strengths

of the sources, but

altematively

one obtains the same result as a lowest order

After

_performing

the

_integration,

e.g. for

and

_taking

the limit a --+ 0 we obtain

_{equation (3b)}

with

where we have

_{neglected higher}

corrections.

_Independent

of the

_{specific potential}

used for the

derivation,

equation

(14)

represents

a feature which is

_{generally expected :}

For

_simple

potentials

with one form

_parameter,

as in our case, there is

_exactly

one zero

_M(7c)

= 0. This is

basically

our

_argument

for

_{equation (6a)}

and for the appearance of a maximum in

_figures

3 and 7. From

_{equations (10)}

and

₍₁₄₎

we have

with

_{A o}

=

Tfluo

and

T,

=

Ûlluo.

Qualitatively,

the mean-field

_type

result,

equation (15),

describes _very well the observed behavior. In order to demonstrate

this,

we have fitted our

2-dimensional data with

_{u ( T) given}

in

_{equation (14).}

The curves shown in

_{figure 7}

for finite

chain

_length

N are obtained on the basis of the

_scaling

function

_[2]

where

_{Rc( T)}

is

_given

in

_{equation (15).}

All three curves are

_{produced using}

a

_single

set of

parameters

uo,

Mi,

f.

Of course near the critical

_{point Tc}

fluctuations will

_modify

the critical

behavior,

which is

_certainly

an

_{interesting subject}

for future work. The situation

_right

at the critical

_point

will be considered

_analytically

in a

_separate

_{paper in detail.}

5.

_Summary

and conclusions.

In the

_present

work,

we have considered a model

_system

consisting

of a

_{self-avoiding polymer}

chain

_trapped

in a

_quenched

disordered _{array of}

_{parallel adsorbing}

hard rods. The model has

been studied

_by

means of Monte Carlo simulations and

by analytical

_arguments.

It is demonstrated that the chain exhibits an

adsorption

transition at finite

_temperature,

which is not observed if the

_arrangement

of the rods is

_periodic.

Below and above the transition

_point,

the

_polymer

chain is

_essentially

stretched

_parallel

to the rods and shrunk in

perpendicular

direction such that the

_{corresponding}

_components

of the radius of

_gyration

are

vll = 1

and v 1 -

0,

respectively.

It is

argued

and

supported by Flory-Imry-Ma

type

arguments

that the two

_phenomena

at low and

_high

_temperatures

are based on two different mechanisms : at low

temperatures

the localization of the

_{polymer by}

_adsorption

on the rods is

dominating,

whereas at

_high

_temperatures

the localization

_{by entropic repulsion,}

in formal

analogy

to the Anderson localization of

_{quantum systems,}

is

_prevailing.

The

_present

model

seems to be the first one where the

_entropic

localization is

_present

at

_{arbitrary strength}

of self-avoidance. The existence of a critical

_temperature

where the two

_competing

mechanisms

compensate

each

_other,

is

_{supported by analytical}

_arguments.

Moreover

they

suggest,

in

agreement

with the Monte Carlo

_data,

that the

_polymer

chain resumes conventional

self-avoiding

characteristics

_{with VII}

=

V 1.

%.-- 3/5.

be tested

_{experimentally.}

We

_suggest

that also in other

_geometries

the same

_phenomena

can

be

observed,

as

long

as

_they

_providè

all the necessary

ingredients,

disorder,

adsorption

and

anisotropy.

A

good

candidate is a

_layered

structure

_(lateral

dimension d’ =

₁₎

which

_might

be

experimentally

realizable,

e.g. as a _{disordered stack of porous sheets.}

In this context, a remark

_concerning

the

_{thermodynamic}

limit seems suitable : effects which

might

occur in

_{exponentially large}

_systems,

such as Lifshitz tails

_[24]

or slow cross-over

behavior at

_{marginality [25],}

are not considered here. In

_fact,

_they

are _very difficult to find in Monte Carlo simulations as well as in real

_physical

_systems

_{[24, 25].}

The critical

_properties

of the

_adsorption

transition have not

_yet

been examined.

_Also,

the characteristics of the

chain

configurations

at

_T,

are not

_yet

known

_exactly.

The latter

_point

will be considered in detail elsewhere.

In the

_present

_work,

the diffusion of the

_polymer

chain has not been

considered,

since the

applied

Monte Carlo

_{technique (combination}

of

_kink-jump

and

_{reptation algorithm)}

does not

provide

a realistic

_{dynamics. Recently,}

the diffusion of chains has been studied in a similar

(athermal)

disordered medium without

_{adsorption [26],}

where anomalous diffusion has been

observed,

but

_entropic

localization was not

_detected,

_probably

because the chains were too

short.

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