Self-avoiding-self-attracting model of polymer collapse and adsorption

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Self-avoiding-self-attracting model of polymer collapse and adsorption

S. Cattarinussi, G. Jug

To cite this version:

S. Cattarinussi, G. Jug. Self-avoiding-self-attracting model of polymer collapse and adsorption. Jour-

nal de Physique II, EDP Sciences, 1991, 1 (3), pp.397-419. �10.1051/jp2:1991176�. �jpa-00247526�

Classification

Physics

Abstracts

05 50 64.60 68 42 87 10

Self-avoiding-self-attracting model of polymer collapse and

adsorption

S. Cattannussi

(')

and G

Jug ('>~)

(')

Intemational School for Advanced Studies, Strada Costiera II, 34014 Tneste,

Italy

(2)

Theory

and

Computational

Science

Group,

AFRC-IFRN,

CoIney-Lane,

Norwich NR4 7UA,

UK

(Received

24 July 1990, reviled12 November 1990,

accepted

6 December 1990)

Rksumk. Nous

pr6seutons

l'dtude ddtaillde d'un _nouveau moddle

g60m6tnque

_pour la

transition vers un ktat

globulaire

des

polyrndres

_en solution En l'absence d'une interface _nous

retrouvons le

diagramme

de

phase

correct et la valeur approximative des exposants critiques. Une

structure de

phase

plus none, montrant la trds

probable

coexistence des transitions

d'adsorption

et d'effondrement,

apparait

_en prdsence d'un substrat exergant une attraction I courte

portde

Les exposants

critiques

et le

diagramme

de

phase

ont dtd

dvaluds,

_{pour un espace} I 2 et I

3dimensions,

I l'aide d'une mdthode de renormahsation _sur rdseau supportant ^la conjecture selon

laquelle

la

prdsence

d'un substrat adsorbant

produit

une augmentation de la

tempdrature

d'effondrement

Abstract. We present a detailed study of _a new

geometncal

model of

polymer phase

transition

in the bulk and _near interfaces For the

coil-globule

_{transition m} the bulk _we recover the _correct

phase diagram

and approximate exponents ^When an attractive substrate _is present a ncher

phase

structure appears with the strong

possibility

of coexistence of

collapse

and adsorption We

estimate size exponents ^{and the} fixed point structure

by

means of cell-renormalization methods _in both d

=

2 and d

=

3 dimensions,

providing

some evidence for the enhancement of the

collapse

transition temperature m the presence of the

adsorbing

substrate.

1. Inwoduction.

Studies

_{of the} conformational properties ^of macromolecules _m solution often

begin

with the

InvestIgatIon

of

single-chain behaviour, corresponding

to the _very dilute regime in realistic

expenments on solutions. Theoretical studies make _use of random walk statistical models

[1, 2],

often _set up on

lattices,

and charactense the macromolecular state _or

phase through

the set of

asymptotic exponents

^attached to _a

polymer

chain of

diverging length. Easily

accessible

from

scattering

expenments ^{is the} size exponent _v, defined

through (R~)

_cc

^N~

^~ for _a

single

chain of

polymensatlon

index N

- co. In order to deal with steric and electrostatic

forces, interacting self-avoiding

walk models _are

usually employed

in

conjunction

with exact

enumeration methods

[3]

_or Monte Carlo calculations _on finite lattices

[4].

398 JOURNAL DE

PHYSIQUE

II M 3

The N

- co

polymer

statistics has often been shown to map on a

magnetic cntical-point problem.

A precise

correspondence

has been achieved for the treatment of the SAW

model,

as this has been

recognised

to be

equivalent

to the

zero-component

O

(n )

model of _a

magnetic system [5-8].

Thanks to this mapping, ^it becomes then

possible

to

replace

the N

- co limit

by

the requirement that the

polymer

chain's

partition (or generating)

function must exhibit _a cntical

singularity.

In setting up a model fop the calculation of this

generating

^function one must be

guided by

the

knowledge

of the

effective

monomer-monomer interactions

(monomer- solvent,

solvent-solvent and monomer-monomer interactions _can all be reduced to _an effective monomer-monomer interaction

[8])

In _a

good

solvent the effective interaction _is

repulsive

and the chain will be _{in a} coil state, as

predicted by

the

simplest

SAW model. In _a poor ^solvent the effective interaction is attractive and the

polymer

molecule will be either _{in a} coil _{or in a}

collapsed

state

(referred

to as the

globule) depending

_on ^whether _or not the

configurational

_entropy will be sufficient to prevent

collapse.

In its

collapsed

state the chain is

characterized

by

_{a new size exponent v~}

= ,

d

being

the space

dimensionality.

A _new model d

is

required

in order to describe the _crossover between the coil and the

collapsed phases

In the

interacting-SAW

model

[9-12]

_an interaction energy is

assigned

to every

pair

of _monomers which _are located _on

neighboring

sites but _are

non-adjacent. along

the chain sequence. The

statistical

analysis

then shows the existence of three fixed points ^with three

corresponding

_size exponents vs~w, v~ and Pa which charactenze the different states of the

molecule, namely

the Coil, Globule and 8

phases, respectively,

the latter

being

associated with the

coil-globule

transition

point

^itself

A

novel, entirely geometrical approach

to the _same

problem

has been

proposed by

_one of the present authors

[13]

and

might

be referred to _as the

Sef-Avoiding-Sef-Attracting

Walk

(SASAW)

model. In the d= 2 _version of this model the chain's microscopic ^states are

represented by

walks _{on an}

appropnate tnangular lattice,

chosen _{so as} to disallow _more than

single

contacts

amongst

the chain _{monomers, a}

geometndal

_way of

taking

into account three- and

higher-body repulsions

which _are believed to be

always

_present when

collapse

^takes

place

Lattice _{sites can} be visited twice at the most and each monomer-monomer contact _is then

weighted by

_a

probability

factor

f

which _can represent

temperature changes

in the

solvent _via the

plausible

functional

dependence f

= I exp

(E/kT) (E

< 0

being

the effective

monomer-monomer

interaction) Alternatively,

_a suitable

dependence

_on the

pH

of the

solvent _can be constructed for each

particular chemico-physical

situation. A small-cell real- space renormalization scheme _is then

implemented

and shows three non-tnvlal fixed

points

for

f

=

0, f

= I and

f

=

f°

which _are associated with the

coil-, globule-

and

fJ-point respectively.

A bnef _account of this work _can be found _in reference

[13] Recently,

_a related

self-attracting-self-avoiding-trail

model has been

proposed

to further investigate the

question

of the

universality

class of the d

=

2

B-point [14],

also

by

_means of small-cell

real-space

renormalization _group methods

In

parallel

with the

study

of the

fKpoint problem, polymer adsorption by

_a wall

exhibiting

_a short _range attractive

potential

towards the macromolecule has also stimulated _some

considerable interest. The _case of chains _in

good

solvent has

already

been

extensively

studied

[15-19]

and it has been established that the

problem

_is

analogous,

_in the limit of very

long churns,

to that of the, cntical behaviour of _a magnetic

system

^{with modified}

couplings along

the

edge

of the semi-infinite bulk The

«special»

cntical point

(in

_magnetic

language) corresponds

to the

adsorption

transition in which the fraction of adsorbed _{monomers goes} from _zero to a finite value

[15]. Moreover,

when the

adsorption temperature

T~ is

approached

from

below,

the characteristic _size of the macromolecule _in the direction

perpendicular

to the wall _is

expected

to

diverge

with _{a new} critical exponent _v~ ^{which has been first}

calculated,

_in

the context of

real-space

renormalization

methods, by

Kremer

[16].

Fig

I Schematic representation ^{of the SASAW model of}

adsorption

of _a

collapsing

macromolecule

(d

₌ 2 space dimensions) In the window, _{a more} realistic, continuum space representation of _a part ^of

the

polymer

chain _{is gJven}

Less attention has been devoted to the

adsorption

of _a

polymeric

macro-molecule _{in a poor}

solvent The

problem

_is nevertheless of fundamental importance both for theoretical

statistical

physics

and for the

modelhng

of actual

physical

and

biophysical

_processes

[20]

In

particular

the situation _in which both

collapse

and

adsorption

transitions take

place, possibly simultaneously,

becomes relevant to colloidal stabilization and membrane

biophysics, modelhng

what

might happen

to a protein ^molecule

sitting

at a fluid-solid interface _{or on} the surface of _an

impenetrable biological

membrane. The question of the exJstence of _a multicntical fixed

point corresponding

to the coexistence of both

adsorption

and

collapse

transitions has

already

received _a

preliminary

affirmative _{answer in} the work of Bouchaud and Vannimenus

[21]

and of Veal et al.

[22].

In both papers an interacting ^SAW model has been

studied,

but with different

techmques

Whllst the former authors have

looked,

_{using a} deterministic

(but

not very

realistic)

fractal lattice _space, at _an

exactly

solvable

model,

the latter have used _a

powerful

transfer matnx method

(as

well _as exact results

[23])

to treat the

particular

_case of directed

polymers.

In addition and previous ^to the work above

mentioned,

we recall that _van Dieren and Kremer

[24]

and

Eisenriegler

and Diehl

[25]

have examined the

400 JOURNAL DE

PHYSIQUE

II M 3

interesting

^{and subtle}

aspect

^{of the}

logarithmic

corrections to mean-field behaviour for this

problem

in d= 3. Within the exact solution for the

anisotropic

models

[21, 23]

the multicntical

adsorption-collapse point

occurs at the bulk

temperature,

whilst for the

isotropic polymer field-theory study [25]

this result _is

directly implied by

the assumed one-to- one mapping on the

zero-component

^tncritical

magnetic problem

with modified surface

potential.

In this article _we

re-present

^{and extend the SASAW model} in order to include the

treatment of

polymer adsorption

_{in poor} solvent. The main

results,

discussed _in detail _m

section

3,

will be the

prediction

of _a rich

phase diagram,

_as well _as the evaluation of the cntical exponents related to each

phase.

The

specific conjectural prediction

of _a

globule

stabilization effect due to

adsorption

has been

briefly reported

_{in a previous}

publication [26],

and will be _more

extensively

discussed here with _some additional

arguments

^This

prediction

of _a shift _in the

b-temperature

at

adsorption

with respect to its bulk value is at variance with

the exact results of references

[21, 23]

and the treatment of reference

[25] However,

the

adsorption

₊

collapse

fixed point must be

regarded

_{as a}

surface

fixed point, so that _{it is} natural to

expect

^that a non-zero fraction of the N

polymer

_monomers will feel the effect of the

surface

potential

in the

thermodynamic

N

- co limit Thls

corresponds

to the consideration that

precisely

at

adsorption

the surface order parameter

(fraction

of adsorbed surface

bonds)

becomes _non-zero,

putting

the present

polymer problem

in _a somewhat different

physical

context than the associated

magnetic

^situation. It _is rather natural to

expect

a shift _in the b- temperature at

adsorption,

_{except in} the

special

_case of directed

polymers [22, 23],

as the

following argument by

Dill and Alonso

[27] convincingly

suggests. ^The presence of the

ngld

wall entails _a reduction _in the

polymer

chain's entropy

(linked

to its allowed

spatial configurations)

which _is much _{more severe} for the coil

phase

than it is for the

collapsed

_one. A stabilisation of the

globule

thus _{ensues upon}

adsorption.

This _{paper rums} at

giving

quantitative support ^{to this} argument

and, hopefully,

at

stimulating

further work _on the

problem

_as it is clear that _a subtle _crossover

phenomenon

must occur

along

the

collapse

line _as the

adsorption _point

_is

a§proached. Furthermore,

the field-theoretic mapping on the

magnetic

^{tncntical surface}

problem

_appears to require careful re-consideration _in the

light

of the argument ^of reference

[27]

and of the results obtained _in this work.

The most promising feature of this work _is

certainly

the relative

simplicity

of the SASAW model which should allow for further

developments,

ⁱⁿ

particular

towards _{a more} realistic

study

of

protein folding

and

adsorption

which includes the effects of chain disorder.

Elaborating

_on the

original

work

[13],

the microscopic ^states of _a

long (infinite) polymeric

chain _{near a} wall will be

represented,

in d

=

2

dimensions, by

walks _{on a} semi-infinite

tnangular

lattice and, in d

= 3

dimensions, by

_a

semi-infinite

_stacked

tnangular

lattice with the surface either _on the basal

triangular

lattice

plane

_{or on} the side square lattice face. A bond

fugacity _k~

_is associated with each monomer-monomer bond _in the

bulk, whereas, following

Kremer

[16]

^and in order to take into account the attractive _nature of the

wall,

the bonds _on the

limiting

surface _or

edge

will be attributed _a different

fugacity

_k~ A third

parameter f,

_as defined

already,

_is then introduced to

weight

the monomer-monomer contacts.

In order to

implement

_a

real-space

renormalization _group calculation _we must be able _to enumerate all the

walks, compatible

with the

model,

_on a bare and renormalized lattice cells.

Th1s is

only possible

if _we restrict our considerations to

appropriate

small cells. All the different renormahzations _we have

performed

_are shown _in

figure

2 In d

=

2 dimensions the three renormalization factors _we have chosen _are b

=

2,

b

= 3 and b

=

(.

In d= 3 dimensions _we have

performed

two renormahzations both with b

=

2 but with different _non-

equivalent

locations of the wall The renormalization process gives nse to three _recursion

b-3f~2

b=3 -

~

a)

b-2

C)

Fig. 2

-(a)

Bare and renormalized cells for the three renormahzations

performed

_in the two- dimensional _space with the

rescaling

factor b

equal

to

~,

_{2 and 3}

_(b)

_and

(c)

As _m

(a),

but for the

2

three-dimensional _case The shaded _areas show the two

possible, non-equivalent,

positions for the surface.

relations which present several fixed

points Lineansing

the renormalization transformation

in the

nelghbourhood

of these fixed points we obtain the cntical

exponents

^{associated with the} different

phases

of the molecule. The remainder of the article _is

organised

_as follows. In section 2 _we present ^the

model,

_recursion relations and

ensuing

fixed

points,

^critical

exponents and

phase diagram

for _a macromolecule _{near a} wall. The model _is then

specialized

to a chain _in the bulk and the results of the

onginal

work _are recovered for d

=

2,

whilst novel results _are

presented

for d= 3. Section 3 contains _our discussion of the results and _a

conclusion,

whilst _in the

appendix

_we

present

a

description

of the enumeration

algonthm

_as

well _{as a} table of the

generated

walks.

2. Cell renormalization _: results for surface-induced transitions.

One of the features of the present ^work is

certainly

the relative

simplicity

of the SASAW model which should allow for further

developments,

_in

particular

^towards _{a more} ^realistic

study

of

protein folding

and

adsorption

which includes the effects of chain disorder.

402 JOURNAL DE

PHYSIQUE

II M 3

Elaborating

on the

original

^work

[13],

the

microscopic

states of _a

long (infinite) polymeric

chain _{near a} wall will be

represented,

in d

= 2

dimensions, by

walks _{on a} semi-infinite

triangular

lattice

and,

_in d

= 3

dimensions, by

a semi-infinite stacked

tnangular

lattice with the surface either _on the basal

tnangular

lattice

plane

_{or on} the side _square lattice face. A

bond

fugacity k~

is associated with each monomer-monomer bond _in the

bulk, whereas, following

Kremer

[16]

and in order to take into account the attractive nature of the

wall,

^the bonds _on the

limiting

surface _or

edge

will be attributed _a different

fugacity

_k~ A third

parameter f,

_as defined _in the

Introduction,

_is then introduced to

weight

the _monomer-

monomer contacts

In order to

implement

_a

real-space

renormalization _group calculation _we must be able to enumerate all the walks

compatible

with the model _{on a} bare and renormahzed lattice cells.

This _is feasible if _we restrict _our considerations _to appropriate small cells. All the different renormalizations _we have

performed

are shown _in

figure

2. In d

= 2 dimensions the three renormalization factors _we have chosen _are b

=

2,

b

= 3 and b

= In d

=

3 dimensions _we 2

have

performed

two renormalizations both with b

= 2 but with different

non-equivalent

locations of the wall. The renormalization _{process gives nse} to three _recursion relations which present several fixed

points. Lineansing

the renormalization transformation in the

neighbour-

hood of these fixed points we obtain the cntical exponents associated with the different

phases

of the molecule.

Our SASAW model's

parameters, k~,

k~ and

f,

_require three

separate

recursion relations.

Two of these _are derived

by equating

bare and renormahzed cell

partition

functions for both the _case of _a macromolecule _in the bulk and _{near a} wall

zib(~i _{fi )} zb(~ f) (j)

0 b, 0 b,

zis(x)(~i

~i

fi) zs(x)(~

_~

_f) (~)

g b, s, 0 b, _s,

The third _recursion

relation,

which defines the contact

probability f',

_{is given in} terms of the fraction of

weighted

walks

containing

at least _one contact in the bare cell.

f _fm(

_j

f)"max

^~'~

Zj©~~~

"

~~~

, m.

~

i

~~n~ax zllltn~

₊

"f _{f'~(' f)~'~~~}

'~

~~~~~

~

The cell

partition

functions _are given

by

Z( _{(k~, f )}

=

(

I

f )"~~ Z)(k~

₊

z f'~ Z$ (k~ (4)

Z[~~~(k~, k~, f )

=

(

I

f

_)~~~~

_Z(~x~(k~, k~) / ~~ _{£ f~} Z$~~(k~, k~) (5)

where

Zo

and

Z~

enumerate the walks with _no

(m

= 0 _or

SAIV~

_{or m} contacts in the

cell, respectively. They

_{are given}

by

z~

₌

z

_Cm

(~b,

_~

s) k(~ k]~, (6)

Spanmngw~k~

n~

and'n~ being

the number of random walk steps ^{in the} bulk and _on the surface

(if present),

respectively,

^and

c~(n~,

n~) the number of SASAW with

(n~,

_n~)

_steps

and

having

_m contacts.

n~~~ is the maximum number of contact sites available _m the chosen lattice cell and _x represents the _minimum allowed fraction of surface bonds _{m a}

partition

sum like

equation (6) (that

_{is, we sum}

only

_over walks

spanning

in the direction of the surface and

having nJ(n~

+

nJ

_{m x} if the surface _is

present).

For the renormalized cell part1tlon ^function we have

used the modified construction

~, nax

zl

~ rim

zl

(~)

0 Gm ,

m =0

since we must allow for the

possibility

of

globule-like

walks

(m

_» 0

)

to be renormahzed into

coil-like walks

(m

= 0

)

when

f

=

f'

= I. In

equation (3),

J~~~~ refers to the _minimum allowed adsorbed fraction _in order _to take _as many

configurations

_as

possible

into account. As discussed _in

[13],

the _recursion relation for

f,

_equation

(3),

takes _{into account} the

probabilistic

nature of this parameter in the

simplest possible

_way,

identifying f'

with the fraction of

suitably-weighted

bare-cell walks that would turn into a renormahsed-cell walk

containing

_a

contact.

A schematic

representation

of the SASAW model _is

proposed

in

figure I,

whilst _some

spanning

coil and

globule configurations

_on 3 _x 3 and 3 _x 3 _x 3 cells _are

presented

_in

figure

3.

~/~ /~

f

^{~ ~}

f

v s

D=8 n=I

n~=11

ⁿ⁼⁰

~=12

n=2

n~=19

n~=1

~

m=o~ m=1~ m=2~

_m=6

n~=I

I

n~=6

^m=6

n~=

^{I I}

n~=7

^m=5

Fig 3 Some

spanning configurations

_in 3 _x 3 and 3 _x 3 _x 3 cells In the three-dimensional _{case we} present a continuum version of the

proposed configurations

which allows _{one to} follow the SASAW

path

404 JOURNAL DE

PHYSIQUE

II M 3

In

figure

4 _we

present

a sketch of the location of all the fixed

points

_in the three-dimensional

_parameter

space.

Table§I aqd

II coqtain ^{the values and the relative}

exponents

^{for the} relevant

txed

_{points of all the different} renormahzations

presented in,figure2

for d

= 2 and d

= 3

dimensions, respectively

In each

table,

_a

study

_{as a} function of the _minimum adsorbed fraction _{x is}

reported

in _{one case.}

The

phase diagrams reported

_in

figure

5 _are two-dimensional sections of the three- dimensional parameter space, in which the renormalization flow is

represented by projected

and normalized _arrows Some

interesting

^{features of these}

diagrams

are discussed _in the next section.

Setting

_k~

=

k~

^and

k(

₌

k(, equations (I)

and

(2)

become

equivalent.

The _minimum fraction of surface bonds _x loses its meaning and _in

equation(6)

the coefficients

c~(n~, n~)

are

replaced by c~(n~)

Then _we

obtain,

for d

=

2,

the

two-parameter

renormali- zation scheme

presented

_in the

original

^work

[13]

with fixed points and cntical

exponents

as

shown _in table I. The bulk fixed points ^and cntical

exponents

^{for d} ₌ ³ have been obtained with "the _same

procedure

and _are

presented

_in table II

Figure

5a and

figure

5b show the

collapse phase diagrams,

in both d

=

2 and d

= 3

dimensions, respectively,

obtained with the condJtion k~ =

k(

=

0. We note that this condition _is

equivalent

to a bulk

situation,

_setting k~ =

0 _{in our} renormalization scheme

simply

disallows the chain to have any bond _on the

surface,

_an alternative way ^to recover the bulk

problem

Table I. Fixed point characterization

for

the _recursion relation _in d

=

2.

Reported

_are the

SAW,

8 and G

fixed

_point values both with and without

surface

interactions Results

refer

to 2 _x 2 _~ l _x

1,

3 _x 3 _~ l _x I and 3 _x 3

~ 2 _x 2 cell renormalizations A

study

_{as a}

function

of

the _minimum adsorbed

fraction

_{x is}

reported

_{m one case.} Stars denote

complex eigenvalues

_;

m brackets _are the

expected

_exponent

values,

when known.

SAW

B

_Globule

316 vtp ⁷⁸² (3/4) _, 498 vtp ⁵⁴² (4/7) k~z 451 _v y 560 (1/2)

-

~~~~

j w0 t 663 vim1945 t =I

x

-

.316 v ~ 782 (3/4) 518 v ~ 580 k

~z 451 v tp 560 (1/2)

X ₌ 1/4 ₄₃₂

vs« ¹⁴⁶⁵ (3/2)

°°5v~«

060 k ~ ⁸⁶² v~W i coo ji

~

w0 f 480 vim ^{2 466} ~i

° k~ 316 v~ ⁷⁸² (3/4) 516 vtp ⁵⁷² k~,451 vb~ 560 (1/2)

~

_X

= k~- ³⁸⁴ v~«1613 (3/2) 795 v~«1 241 km 687

v ,1 155 (1j

~

f «0

w 510 vim 2 521 j

~i

^~

m ~

~ 316 v~ 7821 (3/4) 514 v~564 k~z451 vb~ ⁵⁶⁰ (1/2)

m

X _" 1'6

377 v~- ¹⁶³⁰ (3/2) ⁶¹⁵ v~«1 507 k~« 553 vs»1 365 (1)

w0 f _w 534 vim ^{2 563} ~i

- ~~j~ ^{298 vdm772 j3/4) 376 v ~ 615 (4/7) 338 v} ~ 632 (1/2)

~

j ~o 535 vjW3 056 j ~i

SUfface k~.298 _{v tp} 772 (3/4)

~m 392 _{v b~} 618

~z 338 _{v b~} 632 (1/2)

m ~ ₌ ij6 ⁴¹³ vs»1547 1312) ⁶¹⁷ vs~ ¹²⁷¹ ~ ⁵⁹² vs. ²⁴¹ ii

# _{t W0}

w 404 vim ^{6 063} ~i

il

~~j~ ^{284 v ~.757 (3/4)} k,.449 _v

~ 552 14'71 k_bm 356 _{v b~} 487 j1/2)

"

-0 f 256 vim ^{622 j} ,i

SUrface k~m,284 vtp.757 (3/4) k~z ⁴³⁵ vb~.. . 356 v~> 487 (1/2)

'~

i j~

kr.453

v~«1490 j312) kr ⁸⁴⁷ _v~» ^{i109 675} v~m ⁹⁹⁹ Ii

#

t -0 f «,180 vim ^{. . .} i ~i

Table II. As _m table

I,

but

for

d

=

3,

2 _x 2 _x 2

~ l _x I _x I cell renormalization.

SAW e Globule

~~j~ ^{222 V~ 635 588) 319 Vb- 410 (1/2)} kb" 281 Vb" ⁴²⁵ (1/3)

j

^{1w0 .524 VI" 2.383 f ml}

~ 222 Vb" ⁶³³ (588) 341 V~- 454 kb" 281V~-.425 (1/3)

~

X ₌ l14 256 v~» ^{911 379} v~w ⁶⁸⁸ k~« 306 v~« ⁶⁴³

~ f -0 » 268 VI" 3 616 t »1

@

U k~- 222 vbw ⁶³⁵ (588) 335 vb" 418 281 v~» ⁴²⁶ (1/3)

m ~ ~ ij~ k~« ²⁴⁵ v~m ⁹⁷² v~- ⁷⁴¹ 291 v~» 711

~

_{t «0} 372 VI" 6 '~3 =1

~

Ol 222 v~« ⁶³⁵ (588) 333 v~- 415 .281 vb" 425 (1/3)

=li12 245 v~- ^{972 342 v~w 753 288 v~« 733}

~

_~o t -.394 VI* 5 894 ~1

"

j~~ ~~~ vb- ⁵³⁷ (588) ³⁷⁰ vb" 344 327 vb" 352 (1/3)

~

₂₉₂

v~« ^{953 39°} Vs* ^{735 344} vs" 717

X ₌ 1/12

~ ~~~ vim 2 226 =1

(1) ^Sutlace as in fig ^2b (2) ^Sutlace as in fig ^2c

4

5

6 o

f

Fig

4 A schematic representation ^{of the fixed}

point

locations _in the three-dimensional parameter space. The exact coordinates of the

adsorption-desorption

fixed points 6, 7 and 8 _{are given in} tables I and II The fixed points 9, 10 and 11 represent the adsorbed molecule _in its Globule-, t9- and SAW- phase,

respectively,

whilst the fixed points 3, 4 and 5 represent ^{the different}

phases

of the bulk chain

3. Discussion of results and conclusions.

3, I SIZE EXPONENTS AND PHASE DIAGRAMS.

LookJng

at tables I and it _{we can see} that the

size

exponents

^{obtained with} our SASAW model _are close to the

expected values, reported

_m brackets when known. As for the novel exponents, we

point

out that

they

take reasonable values. In

particular

the _v~

exponent

^{decreases when} _going ^{from the SAW-} to b- and

globule-

phases,

as is

expected

for

increasingly

_{more compact} structures.

TakJng

into account the relative sm~ii _size of the cells

conmdered,

_{we can} conclude that the model

gives satisfactory

406 JOURNAL'DE

PHYSIQUE

II M 3

~

,

8

.6

~ f

~ ~/~~~~~~~~

_~~~~

_~~~<sl

~ ~

~~~~~~~ ~~ ~ / j ^{~ ~~}

~~~~~~~~~~ ~ ~ ~ j

_j~

~~~~~~~~~~/~~~ ~

~

sir<si«rrrrrrrrrr<ij

/ ~~~~~~~~~~~~~~~~~

^i~

/ ~~~~~~~~~~~~~--~~/

~~~~~~~~~--~---&-

_-.i

~~~

^{-- --}

o

0 Z 4 6 .6

f 3)

8

.6

j

~

~ ~~~~~

'~

))jjll~~~sll/ji~~

o

0

Z 4 .6 8

I b)

Fig

5. Two~dimensional sections of the full parameter space (see Fig 4) _in which the renormali-

zation flux _is

represented by projected

^{and normal12ed} arrows For _a discussion of these

figures,

see

section 3.

"

__~_

~

~~~~~~~~~~~

<ai ~/~~

~~~~~fil ~~~~~~-~-~->

~

~ ^~~~~~ ~ ~ / j ^{~ ~~~}

~~~~~~~~~ / jj ^~~~~

~~~~~'~~~~~~ ~ j ^~~

~~~/~~~~~~~~~

0

° 2 4 6 6

ks C)

i-

Id ~°

~

~ ~ j' _j~

~$

8

j7' j7' ~

~ -7

j7f ~ ~

m ~

~ ~

j~7'

~ _~/~ ~j7 _,,-~j

6

fi § _{$ $}

_j~ $~'~

~

~ ~ ~ /"~~/~,/

~

/~'~ _~~~-ll/

~

~

~ ~ j _{j j}

_~ ~

~~~ _~~'~ -~-

~~~ ~

__ __ _~___, , ,

~ ~ ~

~~~

~

o z 4 6 .8

k.~

d)

Fig. 5

(coniinued)

408 JOURNAL DE

PHYSIQUE

II M 3

l~

'

8

0

0 .2 4 6 8

e)

11

Fig

5 (continued).

~

~~)))ll

~ s

~~~~~

'~

j4~U

> > > >->->-

~

~~/~~~~/~ / /~ / ~ ~ ^{~ ~~}

(((§§SSSSSSS~~T<£

z

j~~~~~~~~~~~~~~~~~

j~~~~~~~~~~~~~~~~~

~~~~~~~~~~~----~~

~~~~~~~~~---~~

0

0 2 4 6 6

ks g)

"

~

~ _~ $~~

~

~ ~ j j ) _)$"~

~

~

~ ~ ~ ,i l~~~~~~

~

~ ~~ ~ ~

j~

~

~ ~ ~ ~ ~ / ~~

j~ ~~~~ ~ ~ ~

~~~~~~

~

0

0 2 4 6 6

ks h)

Fig 5 (continued)

410 JOURNAL DE

PHYSIQUE

II M 3

1)

~

Li

Fig

5

(continued).

6

~ _{~ ~}

~

~

~ ~ ~ ~ $~~~~

~ ~~~~~~~~~~

~ ^~~~-

~

/ _g~ ~ ~~~~~~~~ j

~ / / j ~

/ / _g~ ~ ~~~~~~~~~~~

j / _{j~ ~} ~ ~~~~~~~~~~~~/

j / j~ ~~~~~~~~~~~~~~~

~

/ ~ ~ ~~~~~~~~~~~~~~i

~ ~ ~~~~~~~~~~~~~w~w-w~

j~ ~~~~~~w~w~w~w-w-w-w-w-~<--,---

~~~~~~~----+-+-+-+-+-+-+-

0

0 .2 4 6 6

ks i)

"

fi ^~

~

~ ~~

)j

~

~

~

~~~

~

~~~~~~

~

i ~$~$

~ j / ^(~~

j~~~~~~~~

s

~

~~~~~~~~ ~ ^~

0

0 2 4 6

6

ks m)

,

Fig

5

(continued).

412 JOURNAL DE

PHYSIQUE

II M 3

quantitative

results _even if _it does not allow _us to make _a

particular

claim for any new

precise

exponent

To

complement

the numencal values of the critical

exponents,

^the

phase diagrams

shown _in

figure

5 give a more

complete

_view of the results obtained within _our model

They

make the companson with previous

results,

_as well _as the check for internal

consistency

^of the

model,

more

complete.

For this _{reason we}

analyse,

_in the

following,

_some different

two-parameter

sections of the

three-parameter phase

_space We first consider _a macromolecule _{in a}

good

solvent

by looking

at the

phase diagram

_on the

plane

defined

by f

= 0. For the two-

dimensional

polymer

_we note

(Fig. 5c)

that _we

exactly

_recover the

adsorption phase diagram

structure obtained

by

Kremer ₁₁

6].

^{For d}

=

3 the structure _is s1mllar

(Fig. 5d), except

that the fixed

point (k~

=

0,

_k~

=

k~)

_{is no more} at k~ = I but at k~ =

0 316 This _can be

easily

understood _since for

k~

=

0,

the chain _is constrained _to the

(d

I

)-dimensional

surface. If d

=

2 the molecule _occupies

entirely

the available space and the cntical

fugacity

must be

equal

to I. If d

= 3 the chain is constrained to the two-dimensional surface and _{we recover} the bulk two-dimensional _case, k~,

replacing _k~,

with _a coil fixed point at k~ = 0.316.

Hence, relaxing

the condition

f

= 0 but

keeping k~

=

0,

_we obtain for the d

=

3

problem

the two- dimensional

collapse phase diagram

obtained _in

[13],

with the variables

(k~, f) replacing (k~, f) (Fig. 5e)

It _is worth

noting that, depending

_on its position ^{in the} three-dimensional cell

(Figs.

2b and

2c),

the surface has either _a

triangular

_{or a square} lattice symmetry. ^{In the}

latter _case the model does _not allow for _a

collapse

transition and the

only

fixed

point (Fig 50

with _a

physical

_meaning

corresponds

to the

coil-phase (f

=

0)

For

f

=

I, figures 5g

and 5h _give the

adsorption phase diagrams

for the

collapsed

chain _in d

=

2 and d

=

3, respectively Finally,

the

diagrams

_in

figures

51and 5k represent the _cross- section taken at

kb

= ko~ for ^{the d} =

2 and d

= 3

three-parameter phase diagrams (Fig 4).

These

give

the

phase

_structure of the macromolecule when both

collapse

and

adsorption

take

place

and

represent

part ^of the _main results of _our work. It _can be _seen that for d

=

3 all four

possible phases, free/coil, free/collapsed, bound/coil

and

bound/collapsed,

_are

present ^and are

separated by

a

single

fourth-order cntical point

However,

for d= 2 the bound

(or adsorbed)

chain becomes one-dimensional and the

collapse

transition _{can no}

longer

take

place.

This last feature _{is in} agreement with the results obtained

by

the authors of references

[21, 22, 23].

3 2 ENHANCEMENT _{oF THE} coIL-GLOBULE TRANSITION TEMPERATURE. An

important

new feature in the behaviour of _a

polymeric

chain when _a wall _is present is

suggested by

the

analysis

of _our results _in tables II and III We have found that the value of

fj

is

always significantly depressed by

the presence of the

attracting

surface

[26], corresponding (for example)

to _an enhancement of the

fKpoint

transition temperature

(collapse temperature)

Alternatively,

_we could say that _a

collapse

transition _can be induced

by adsorption,

_or that

desorption

_is induced

by

_a

globule

to coil transition in _a bound macromolecule. Some of these features have been observed _in recent

neutron-scattering

experiments on

proteins

^adsorbed

on a

suspension

of latex

spheres [28]

Since _a finite _size

analysis

_is

beyond

the

capabilities

of _our small-cell renormalization

method,

one could argue that the observed shift is due _{to a} finite _size effect This would be the

case in a magnetic

problem,

where the modified surface

exchange

cannot shift the bulk transition

temperature

in the

thermodynamic

limit. To refute this argument for the

polymer problem

at hand _we have

performed

the

following

calculation. In the bulk 3 _x 3 to 2 _x 2 renormalization

scheme,

_we have disallowed the chain to have any bond _{on one} of the

edges

of both the bare and renormahzed cell We obtain _in this way a 8

fixed-point

with f(U~k = 0

240,

instead of

f(~'~

= 0.256

(see

Tab

I)

Since

jn

the _case of _a

jemi-infinite

lattice the fraction of sites on the

edge

vanishes _m the

thermodynamic limit,

~we expect this shift to

Table III.

-Class#ication of

the SASAW walks

(coefficients c~(n~,n~))

_on the

dfferent

cells

(a)

2 _x 2 _x 2 cell with the

surface

_{as m}

figure 2b, (b)

3 _x 3

cell,- (c)

2 _x 2

cell, (d)

I _x I _x I cell with the

surface

_{as m}

figure 2b, (e)

I _x I cell. The spanning direction is in all

cases

parallel

to the

surface

mU

m m=

n~

~S m= ~~

m=7

414 JOURNAL DE

PHYSIQUE

II M 3

Table IIIa

(continued)

~=8

disappear

in that limit. It is therefore

plausible

to consider this shift _{as an} approximate

measure of the finite _size effect. But _we observe from table

I,

for the related renormalization

scheme,

that this finite _size shift of the value of

fo

is

considerably

smaller than the surface- induced decrease of the _same parameter ~f(~~~~* ₌

0.180), which,

therefore _we

feel,

cannot be

attnbuted to a finite _size effect alone.

Another

possible

criticism of _our results

might

be the _use of _{a recursion}

relation, equation (I),

which _{is a} bulk renormalization

equation

containing a parameter

fthat

_{is in} fact renormalised

through

the _use of surface

partition functions,

_see

equation (3).

In order to

remedy

tills

partial inconsistency,

_we have

repeated _part

of _our renormalization calculations

by

_using two separate contact

probabilities, f~

and

f~, referring

to the situation with and without the surface

present, respectively.

We then need four recursion relations _:

equation (I)

containing

f~ only, equation (2)

_containing

f~ only, equation (3) containing

bulk

part1tlon

functions for the renormalization of

f~

and

equation (3) containing

surface

partition

functions for the renormalization of

f~.

This _ensures bulk _vs surface

consistency,

and the results obtained _are-

reported

in table IV. It _can be _seen that _again the surface

f~ parameter

is

depressed

at the coexJstence between

adsorption

and

collapse,

which _we thus

interpret

_{as a}

_surface

^fixed

point.

We §tress that the _more involved calculation

requiring

_an

f~

for bulk contacts and _an

f~

for surface contacts leads to a somewhat

problematic physical

interpretation and appears ^to have reasonable chances to work

only

for much

I%rger

^cells

It _remains for _us to recall the limitations of the renormalization scheme used _m order to

attain _{our mmn} conclusion.

Real-space

cell-renormalization _is and remains _a

poorly

controlled method of

charactensing

the

asymptotic

^{behaviour of} a lattice model.

Also,

_a

systematic study

of the results _{as a} function of the surface fraction parameter x is

beyond

the

capability

of _our calculations. Nevertheless _we believe that the

systematic

shift _in the b-

temperature

observed in all renormalizations upon

adsorption

does

provide _support

to the conjecture ^contained in the

argument

^{of Dill and Alonso}

[27].

The result _is of great

importance

^{for surface} macromolecular science and deserves further theoretical attention with the

techniques

of statistical mechanics. We believe there is no

disagreement

with the

exact results obtained for directed

polymers [23],

_as the entropy

argument [27]

for the shift in

b-temperature

^{fails in this} case

(the collapse

_{occurring in} the direction

parallel

to the

surface).

For-the

field-theory mapping [25],

_we

plan

to conduct _a closer look _at the

origin

of all the

appropriate

bulk and surface operators starting

from

_the statistical

mechinics

_{of the}

_polymer problem.

In

conclusion,

_we have

presented

_a detailed

study

of _a

geometncal

model of

collapse

and

adsorption

of _a macromolecule _{near an} attractive wall.

Simple

cell-renormalization calcu- lations in both d

=

2 and d

= 3 dimensions

yield

the

complete phase diagram

and reasonable

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