Polymer adsorption : bounds on the cross-over exponent and exact results for simple models

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Polymer adsorption : bounds on the cross-over exponent and exact results for simple models

E. Bouchaud, J. Vannimenus

To cite this version:

E. Bouchaud, J. Vannimenus. Polymer adsorption : bounds on the cross-over exponent and exact results for simple models. Journal de Physique, 1989, 50 (19), pp.2931-2949.

�10.1051/jphys:0198900500190293100�. �jpa-00211114�

Polymer adsorption : ^{bounds on the} cross-over exponent ^and

exact results for simple ^models

E. Bouchaud (1) and J. Vannimenus (2, 3)

(1) ONERA, ^{29 av. Division} Leclerc, B.P. 72, 92320 Chatillon Cedex, ^France

(2) ^Institut Laue-Langevin and Laboratoire Louis-Néel (*), 38042 Grenoble Cedex, ^France

(3) Laboratoire de Physique Statistique ^{de l’ENS} (**), ^{24 rue} Lhomond, 75231 Paris Cedex 05, France

(Reçu ^{le 28} février ^1989, accepté ^sous forme définitive ^{le 5} juin 1989)

Résumé.

Nous donnons des bomes supérieure ^et inférieure pour l’exposant 03A6 ^de cross-over, dans le problème ^de l’adsorption ^d’un polymère ^{sur une} surface attractive impénétrable. ^Ces

bornes sont bien vérifiées pour des modèles exactement résolubles, ^où le polymère ^{réside sur un}

réseau fractal. Dans l’un de ces modèles nous observons également ^un point multicrique, ^{où une}

transition d’effondrement en volume coexiste avec une transition d’adsorption ^en surface. Une méthode simple de renormalisation dans l’espace ^{réel est} présentée pour le réseau carré, ^elle donne 03A6 = 0,48, ^{en bon accord avec la valeur} présumée exacte 03A6 = 1/2, qui ^{coincide avec notre}

borne supérieure.

Abstract.

We provide ^an upper and ^a lower bound for the cross-over exponent 03A6 ⁱⁿ polymer adsorption ^{on an} impenetrable attractive surface. These bounds are shown to be verified for

exactly solvable models where the polymer ^{resides on a} fractal lattice. In one of these models we

also find a multicritical point ^{where a bulk} collapse transition and a surface adsorption ^transition

coexist. A simple real-space renormalization scheme is presented for the square lattice, it gives 03A6

0.48, ^{in close} agreement ^{with the} presumably ^{exact value} 1/2, which coincides with our upper bound.

Classification

Physics ^Abstracts

36.20E

64.60A

82.65

Introduction.

Polymer adsorption ^{on a} rigid, impenetrab’le ^substrate provides ^{a nice} example ^{of a situation}

where contact can be made between a domain of practical ^and technological importance [1]

and the powerful methods of modern statistical physics. ^{This has motivated much} fundamental work in the recent years [2, 3], ⁱⁿ particular for the limit of high dilution in a good solvent,

where only ^{one chain} interacting ^{with an} attractive wall is considered. This case is well

(*) Laboratoire propre du CNRS.

(**) ^Permanent address : Unité de recherches associée au CNRS et à l’Université Pierre-et-Marie Curie.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500190293100

understood physically by ^{now as a} surface critical phenomenon : there exists an unbinding temperature, analogous ^{to a} tricritical point, and in its vicinity a cross-over regime is observed where simple scaling laws hold. The polymer ^{forms a} self-similar adsorbed layer ^{with a} slowly decaying (power law) density profile ^{and the} corresponding ^critical exponents ^{can be} computed using renormalization methods [4] ^{or numerical} approaches [5, 6]. Recent work has made this picture ^{still more} quantitative, ^with conjectures ^{for the exact values of some critical} exponents ^{in two} dimensions [7, 8] and detailed studies of the case of a 0-solvent, where subtle

logarithmic corrections appear [9].

In the present paper ^we contribute several more pieces ^{to that} general ^{frame :} first, ^we provide ^a lower bound on the cross-over exponent 0 which governs the region where critical

scaling holds, by comparing ^{the cases of} penetrable (e.g., fluid-fluid) ^and impenetrable (solid wall) surfaces. For a space of fractal dimension dF ^{and an} adsorbing surface of dimension

ds, ^{this bound is} ~> ^{1 -} (dF - ds ) v , where v is the gyration ^radius exponent ^{for the} polymer

in the swollen coil state in bulk solution. An upper bound, ~ ds/dF, is obtained as a

consistency condition for adsorption ^{to occur when the} self-repulsion energy is of the ^same order as the attractive energy due ^to the interface. That bound is reached in the case of chains in bad solvents which form collapsed globules ^{in bulk solution. For} adsorption ^{at the} (Euclidean) surface of a standard (Euclidean) space of dimension d, the upper bound is

(d -1 )/d, ^{and in two} dimensions, it coincides with the value 0 ^{= 1/2} recently ^obtained through ^a conformal invariance argument by Burkhardt et al. [8], ^thus giving ^some insight ^into

the physical meaning ^{of this} striking ^result.

Two models of polymer adsorption with excluded-volume effects are solved exactly ^{in the}

second and third parts. ^The only solvable models of adsorption ^{until now are} restricted to Gaussian chains [10] ^{or to} directed walks [11], ^so they ^{do not} take excluded-volume effects into account and cannot be used to test some of the basic physical features of the scaling approach. In the models considered here the polymer ^is represented by ^a self-avoiding ^walk (SAW) restricted to the bonds of a fractal lattice, ^{such as a} Sierpinski gasket. ^{We find that an} adsorption transition does occur on these lattices, ^{when one} boundary of the fractal is made attractive, ^{and the} cross-over exponent satisfies the bounds derived in the first part. ^These bounds turn out to be rather narrow, ^so these results provide ^a stringent ^{test of the} scaling assumptions made in their derivation

e.g., 0.7925 0.7481 -- 0.720 for the 3-d

Sierpinski gasket.

Moreover we are able to study the behaviour of the adsorbed polymer ^{at a} 0-point, ^{when a} collapse transition occurs in the bulk phase ^{due to} attractive monomer-monomer interactions

[12]. ^The surprising finding is that the standard symmetric multicritical point which would be

naively expected is unstable. The unbinding transition at the 0 temperature is associated with

a novel type ^of anisotropic multicritical point.

In the last part ^a simple real-space renormalization group (RSRG) approximation ^is presented ^for adsorption ^on the square lattice, using ^three parameters ^{in the recursion} relations. This scheme has the advantage ^{over the} previously proposed two-parameter approximations [13] ^of being ^more physically transparent ^{and it} gives ^a result 0

^{0.48 close}

to the presumably ^exact value 0 ^{= 1/2} [8]. We discuss briefly the relation of that approach

with the study ^of real-space renormalization groups for surface problems ^made by ^Burkhardt

and Eisenriegler [14].

1. The cross-over exponent : physical meaning and bounds.

1.1 THE PROBLEM OF ADSORPTION. - In this section, ^{we extend} previous theories of

polymer adsorption [6, 17] ^{to the case where the} polymer ^is constrained to lie on a fractal

lattice of fractal dimension dF, ^{in the} vicinity ^{of an} attractive surface of fractal dimension

ds. The usual situation for a d-dimensional Euclidean lattice is recovered for dF

^{d and} ds = d - 1.

1.1.1 Description of the model.

The surface may be either impenetrable (solid wall) ^or penetrable (fluid-fluid interface). ^{Each monomer} coming ^{in contact} with the wall receives an

energy Es (Es 0), while all the others have an energy Eb. ^An infinite coil with one end attached to the surface will not

stick » to the surface (i.e. ^the fraction of monomers lying ^on

the surface will not be finite) unless the difference Es - Eb ^{reaches a} critical value 0394E. As we shall argue farther, this value is zero in the case of a penetrable interface, while it is

strictly negative ^{in the case of an} impenetrable wall. In any ^{case we} define the dimensionless parameter ^{6 as follows} [15] :

where kB is the Boltzmann constant and T is the temperature. Adsorption corresponds ^to positive values of 8.

A finite chain will not undergo ^a sharp phase transition. The cross-over between the

adsorption ^{and the} depletion regimes ^{will take} place for values of 8 verifying :

where N is the total number of monomers (see Fig. 1). ^This inequality defines the cross-over

exponent 0. Adsorption corresponds ^{to à} > N -~

Fig. 1.

^Schematic diagram showing ^{the different} types ^of polymer behaviour, according ^{to the}

number N of monomers and the interaction energy Es with the wall. In the adsorption regime, ^the

number of monomers in contact with the wall is proportional ^to N, while in the cross-over region ^it

behaves as N ~. ^A phase transition occurs in the limit N , oo, ^{for a} critical value Es - Eb

^dE.

1.1.2 Structure of the adsorbed chain.

*

The cross-over region.

^{In this} region, the number M of monomers in direct contact with the surface can be estimated in the following way. In the cross-over region defined above

(Eq. (2)), ^no

surface order

can take place because thermal fluctuations are dominant. This

means that the total excess surface energy ^{M 8 kT} is smaller than kT in this region, being precisely ^{of order kT on the} cross-over line. Then :

on this line, ⁱⁿ agreement with the results of Eisenriegler et ^al. [6].

A usual and intuitively reasonable assumption is that in this region the chain is roughly isotropic, having ^both parallel ^and perpendicular extensions of the order of the bulk

(dF-dimensional space) gyration radius R - aN ’ (where a is of the order of the monomer size or, equivalently, of the lattice spacing). ^{This is} indeed verified in the solvable models studied below. Thus the surface occupied by the chain is of order N vds

and the average number of

monomers at a distance z from the surface may be written ^as (N Jlds tP _{(z ) ).} Let us also define p (z), which is the number of accessible sites ^on the fractal set, ^{at a distance z} from the fractal interface :

with

In fact, the well-behaved quantity ^{would in} general rather be the integrated density

(f p (z ) dz ), but this does not change ^the argument.

By analogy ^{with the case} of standard lattices, ^{we assume that 0} (z ) ^{has the} following power- law behaviour

where 0 s is the surface fraction of monomers in contact with the adsorbing subspace :

Writing ^{that N} is the total number of _monomers, we get the relation

which leads to a relation between m and 0

consistent with the condition m +x * 1 assumed in its derivation. Note that when

dF ^is equal ^to (ds ⁺ 1), we recover de Gennes and Pincus’

proximal exponent » [16, 17].

*

The adsorption regime.

In the adsorbed regime, the number M of monomers which are

in direct contact with the surface is proportional ^to the total number N. In order to estimate M in the two regions, ^{we make the} following ^standard scaling assumption

which implies ^{in the adsorbed} regime

Let us consider the largest subpart of the chain which remains unadsorbed [18, 19]. ^{Let D be}

the size of this

blob », and g ^{the number of monomers} it contains (Fig. 2). By definition, ^its

surface excess energy is of order kT :

Fig. ^2.

^A polymer ^{on an} attractive surface (hatched area) ^{in the} adsorption regime : ^flat regions ⁱⁿ

contact with the wall (« trains ») coexist with unadsorbed isotropic

^blobs

(delimited by ^dashed circles).

Being ^{in the crossover} region, ^this subpart of the chain is still isotropic, ^so

implying

The meaning of D is the following : ^at length scales smaller than D, ^{the chain is} isotropic, exhibiting ^{a local} dF-dimensional behaviour, ^{while at} larger length scales the behaviour is

ds-dimensional. ^{Thus the} perpendicular extension of the adsorbed

pancake

is of order D :

while its parallel ^extension RII is the radius of a ds-dimensional polymer having ^{monomers of}

size D :

where v’ is the ds-dimensional ^critical exponent.

1.2 UPPER AND LOWER BOUNDS FOR THE CROSS-OVER EXPONENT.

Until _now, the value of the exponent 0 has remained undetermined. However, ^{we have seen} that it is of primary importance because it is needed to evaluate all the physical observables of the problem. ^In

this section, ^we determine the accessible values of 0 [19, 20].

1.2.1 Comparison between the cases o f ^the penetrable ^{and the} impenetrable surfaces : ^{a lower}

bound for 0.

^{We now come back to the} comparison between the penetrable ^{and the} impenetrable surfaces. In the latter _case, even when there is no gain in energy for ^{a monomer}

on the surface, ^{i.e. when} Es

Eb, ^{there is an} entropy loss per ^{monomer on} the wall because it is impenetrable : ^{once a monomer lies on the wall, the} following segment ^{has a restricted} orientational choice. In the former _case, on the contrary, configurations spanning through ^the

surface are allowed, and there is no loss of entropy per ^{monomer to} take into account. Thus

Es

Eb ^also corresponds ^{to a zero} gain ⁱⁿ free energy per monomer on the surface and the

adsorption threshold in this case is just

In the impenetrable ^case, the energy difference (Es - Eb) ^must compensate ^{for the} entropy loss :

JOURNAL DE

PHYSIQUE. -

50,

19,

We now evaluate the number of monomers on the surface in the two situations. In the case of the penetrable interface, because AE is strictly ^{zero, the} surface, ^at threshold, ^is only ^a

fictitious ds-dimensional subspace intersecting ^an isotropic dF-dimensional ^SAW. The fraction of monomers lying ^on it is then proportional ^to the ratio of the surface of the intersection to the volume of the chain :

or :

where 0 p ^{is the} cross-over exponent ^{relative to the} penetrable interface, ^{so that} finally

Note that this value corresponds ^to the result of Bray ^{and Moore} [21] ^when dF ^is equal ^to (ds + 1 ), ⁱⁿ particular for Euclidean spaces.

Furthermore, using ^relation (6), ^{we note that this value} of 0 ^{leads to a} proximal exponent

m

0, i. e. , ^{to a flat} density profile (per accessible site), ^{which is a natural} consequence ^{of the} presence of ^a perfectly neutral interface. On the contrary, ^an impenetrable ^wall being already

attractive at threshold, ^one expects ^the density ^{to start} decreasing away from the wall, _,so ^the exponent ^{m should be} positive (or zero) ^{in this case. This} implies

This lower bound on 0 ^{will be} compared ^to the various results we further obtain by ^{the exact}

real space renormalization procedure ^{on fractals.}

Note however that this result appears ^{not to} be valid for a 0-polymer, according ^{to the} expansion ^{in E}

3-d obtained by Eisenriegler [9]. ^{In fact a bulk} 0-polymer remains swollen at the surface (the two-dimensional 6-temperature being smaller than the 3-d one) and this effect alone would lead to a density profile increasing with distance to the wall. Thus in this case we cannot predict ^{how the} density ^varies

i.e., ^the sign of m - when a 0-polymer ^{is in the} vicinity ^{of an} attractive surface.

1.2.2 An upper bound for 0.

^{We have seen} that in the case of a penetrable ^{interface we} know 0 exactly ^{as a function of v. We now return to the} impenetrable ^{case to determine an}

upper bound. For this purpose, let ^us first define P (z ) ^{as the monomer surface fraction at a}

distance z from the wall, where the average is performed ^over the surface of a blob, ^{i.e. over}

Dds. The average number of ^{monomers at a} distance ^z from the surface is then

(D/a )dS 0 (z). ^{Note that on the} cross-over line D becomes of the order of R, ^{and this} definition of 0 (z) coincides with that given in section 1.1.2.

We now assume that the self-repelling energy FD per blob of the adsorbed coil ^can be

expressed ^{as a function} of P (z) ^and p (z), ^where p (z ) ^{is the} number of accessible sites defined in equation (4). By analogy ^{with the case} of bulk semi-dilute polymer ^solutions [12],

we write

where ^we make no a priori assumption ^{on the} exponent q. This expression should be valid as

long as W (z) « ^{1. At} equilibrium, ^the self-repelling energy should be of the order of the

adsorption energy per blob, which is itself of order kT, ^{so we have to solve the} equation

Furthermore, ^at equilibrium, this energy should be homogeneously distributed in the adsorbed layer. ^{This means that the} integral ⁱⁿ equation (19) ^must be dominated by the upper

bound, i.e. that the exponent m obeys the condition : x + mq -- 1. ^{In this case,} equation (19)

has ^a solution only ^if

which is the simplest generalization of the classical result [12, 22] ^on bulk semi-dilute

solutions : q

v d / ( v d -1 ), ^{to the case of a} fractal lattice.

Using the relations x = 1 + ds - dF (Eq. (4)) ^and m = dF - ds + (0 - 1)/1, (Eq. (6)), ^the requirement x ⁺ mq * 1 ^then implies

In summary, the cross-over exponent 0 should lie between Pp ^{and 0. :}

For Euclidean lattices, this double inequality ^reads

Note that the two bounds ^are equal if the chain is in the form of ^a collapsed globule, ^since

v =1 /dF ^{in this case.} This leads to the value cp coll

ds/ dF, ^{and in} particular CPcoll

(d - 1)ld ^for standard Euclidean spaces. A simple way ^to interpret this result is to introduce the gyration ^radius RG ^{of the} (isotropic) globule ^{and write} equation (3) in the form

This means that the points ^{of contact} between the globule and the wall occupy ^a finite fraction of its projection ^onto the wall : the contact is

dense

The two bounds ^are also equal ^{in one} dimension : 0 1-d

0, which is consistent with the fact that no transition can occur in this case. For d

2, v

^3/4 [23], ^{and one must have} 1/4 -- 0 - 1/2, ^but according ^to the results of Burkhardt et al. [8], the upper bound is reached.

This is rather surprising ^at first, ^since 0 = 1/2 is also the value for ideal Gaussian walks, suggesting ^that adsorption and self-avoidance effects somehow compensate. ^Our argument points ^{to a different} physical intepretation of this result : it corresponds ^{in fact to the}

saturation of the limit x + mq

1 in equation (18) (here, x

^{0, m} = 1/3, q

3). ^{For a} larger

value of 0 ^the self-repelling energy would be concentrated in the region very close ^to the wall,

rather than being distributed in the whole adsorbed layer, ^{and the} polymer ^{would not be in} equilibrium.

Finally, ^{for d}

3, using ^the Flory approximation [12] v ^{= 3/5, we obtain :}

.

which is in agreement with the result of numerical investigations, 0 ^{= 0.6} [5, 6].

Our derivation is based on plausible scaling assumptions ^and physical reasoning rather than

rigorous arguments, ^{so it is} important ^{to assess its} reliability ^{on the} exactly ^{solvable cases}

which are studied in the following ^sections.

2. Adsorption ^on fractal lattices : 2-d gasket.

The asymptotic behaviour of large ^{SAW can} be obtained exactly ^on Sierpinski gaskets [24, 25], and the various usual exponents (gyration ^radius, number of closed loops, ...) ^{exist and}

are close to their Euclidean counterparts. ^The non-integral ^effective dimensionality ^{of these}

fractal lattices does not introduce pathological features, and these models are valuable to

study various bulk polymer properties ^{or as crude} representations ^of polymers in disordered environments like porous media [26]. ^{Here we show how to} extend this approach ^{to surface} problems.

2.1 DERIVATION OF THE RECURSION RELATIONS.

The first model we consider is described in figure ^{3 : a} self-avoiding ^{walk of N} steps ^is restricted to the bonds of a Sierpinski gasket,

one side of which is an impenetrable attractive wall. As will be discussed below, it is necessary to introduce two new parameters in addition to the bulk fugacity ^{in order to} explore ^{the full} phase space of the polymer. ^{Here the} parameters ^{chosen are the} energies Es ^and Et denoting respectively the interaction with the wall of monomers in the surface layer ^{and in}

the adjacent ^one (the energy Eb ^{of a monomer} in the bulk is taken as the energy ^zero and an

attractive energy is negative). ^No monomer-monomer interaction is considered apart ^from the excluded-volume effect, because this brings ^no qualitative change ^{in the case of the 2-d} gasket.

Fig. 3.

^A self-avoiding ^{walk of N = 10} steps (wiggly line) ^{on a 2-d} Sierpinski gasket ^{of order}

k

2, in the presence of ^an attractive wall. The links lying ^{on the wall} (thick line) ^{have an} energy

Es, ^{those on the} layer ^{next to the wall} (dotted lines) ^{have an} energy E,.

Following ^the approach first used by ^{Dhar to} study ^{the bulk} properties ^{of SAW on fractals} [24], ^{a closed system} of recursion relations can be obtained between three functions (Fig. 4) :

-

the generating ^function

where x is the fugacity ^{and 1B (k)} (N ) is the number of configurations ^such that the SAW joins

two given ^{vertices a k-th order} gasket ;

Fig. 4.

Diagrams representing ^{the bulk} generating ^function (B) ^{and the two surface} partition

functions (C ^and D), ^for polymers ^{on the} Sierpinski gasket.

- the partial partition ^functions

where w

exp (- ES/T) ^{and t}

exp (- Et/T) ^{are Boltzmann factors, and the summations}

bear on repeated indices. e (k)(N, M, R ) (respectively, 1) (k) (N, M, R» is the number of

configurations where both extremities of the polymer ^{are fixed on} the attractive surface at the vertices of a k-th order gasket (resp., only ^{one fixed on that} surface), and such that M

monomers lie on the attractive surface, ^{R monomers} being ^{in the} adjacent layer. ^{It is useful to}

note that the average number of ^monomers in contact with the wall, ^{for SAW} spanning ^{a k-th}

order gasket, is

whereas the number of these SAW is

If the leading singularity ^{of C and D} depends ^{on w} only through the critical fugacity xc(w), at ^{fixed t,} the fraction of adsorbed monomers is for large ^N

where Xc is the critical fugacity ^{where the} iterations start diverging. Using the relation between

conjugate variable N = (xc - x )-1, one recovers equations (2), (3) ^and (7b), ^{if Xc is constant}

for w w *

exp (DE/kT ), ^{and has a} singularity of the form

with a critical exponent ^{related to the} cross-over exponent by C

1/ cp .

It is technically convenient to use Dhar’s 3-simplex geometry where each vertex is split ^into

two, ^so there is no need to forbid the configurations ^{where the} polymer visits twice the vertex

joining ^two triangles, ^without intersecting itself. The asymptotic ^{behaviour is not modified}

and the recursion relations are then easily ^obtained [25]. ^For example, ^to join ^{the two vertices}

on the attractive wall belonging ^{to a} (k ^{+ 1} )-th ^order triangle, ^{the SAW can} either go through

the vertex common to the two k-th order triangles adjacent ^{to the} wall, ^or visit the third

«

bulk » triangle

^so the recursion for W (k)(N, ^M, R ) ^is

where for clarity ^the primed (resp., unprimed) quantities ^{denote order k + 1} (resp., k). ^Such convolution-type ^relations yield simple algebraic relations between the three generating functions, ^{which can} be obtained directly using simple graphical rules for the diagrams ^of figure ⁴ [24] :

The first relation is the bulk equation ^studied by previous authors and is decoupled ^{from the}

other ones. Note also that D

D’

0 satisfies equation (27c), this reflects the fact that, ^{if the} surface is decoupled ^{from the bulk,} it remains so under iteration.

The initial conditions on the first-order gasket (an elementary triangle) ^{are obtained} by considering ^{the two} possible ^chains (with ^{one or two} steps) and their energy : for instance,

one has e (1)(1, 1, 0 )

1, C (1)(2, ^0, 2 ) ^{= 1. This} gives

(Note ^{that the} present conventions ^are slightly different from those of Ref. [25]). ^More general initial conditions could be considered by allowing longer-ranged interactions with the

wall, but that would not alter the qualitative behaviour of the system.

2.2 PHASE DIAGRAM. - In the following ^{we use as} independent variables the parameters w and t, rather than the temperature T and the ratio Et/ES, this choice is convenient and does not alter the qualitative aspects ^{of the} phase diagram. The numerical study of relations (27)

and (28) shows that for an attractive surface (w > 1 ) ^an unbinding transition at a finite temperature ^exists only ^{if t 1, i.e.,} when the interaction potential ^{in the} layer adjacent ^to

the wall is repulsive. ^{For t} > 1, ^the polymer ^is always adsorbed : this may be understood by noting ^that impenetrable walls exist in the bulk of the gasket, and the presence of the attractive wall does not entail a loss of conformational entropy ^{for the} polymer

^the

situation is analogous ^{to the case of a} penetrable (fluid-fluid) interface. A repulsive part ^{in the} potential is essential to favor the desorbed state and restore a transition (an entropic ^{effect is}

then present because the number of available bonds varies with the distance to the wall).

For any fixed value of t > 1, the behaviour of the critical fugacity Xc (w ) ^{is the} following (see Fig. 5).

i) ^{For w} w*(t), ^{i.e., at} high temperatures, xc is ^constant and equal ^to the bulk critical

fugacity : Xc (w) = xb

0.4316834. The relevant fixed point is the bulk SAW point

Fig. 5. - Critical fugacity Xc ^{as a} function of the attraction parameter w

exp (- ES/kT ), ^{at fixed}

t

exp (- Et/kT), ^{for t}

^1.

The fraction of monomers in contact with the wall vanishes (Eq. (25)), the free energy per

monomer is independent ^of T, ^{so the} polymer is in the desorbed state.

The gyration ^radius exponent v ^{can be obtained} by noting ^that B(k), ^{C (k) and}

D(k) are in fact (unnormalized) correlation functions for distances Lk

2k and are expected ^to decay ^as exp (- Lkl e ) ^for large Lk. The correlation length e (x, T ) maya priori depend ^{on the}

function considered and diverges ^as (xc - x )- v ^{close to} the critical line. Consider now the recursion system very close ^to the fixed point (b *, ^0, 0 ) : ^{at first,} the distance to the fixed

point varies with the order k as (A b)k SI’ where À b ^{is the} largest eigenvalue ^{of the linearized} system ^and 81 ’" (xc - x). ^This regime ^lasts until the distance becomes comparable ^to b *, ^and for k > K

In (b */81 )/ln A b the recursion enters the regime where B’ = B2 . B(k)

then decays ^as exp (- Lkl ), ^with

so

(the subscript b denotes bulk properties).

ii) ^{If w}

w * (t ), xc(w*) ^{is still} equal ^to Xb, but equations (27a-c) iterate towards a new

fixed point

where the three generating ^{functions are} equal ^{and which} corresponds ^{to the} « special »

transition in the conventional terminology. The existence of such an isotropic ^fixed point

follows from the requirement that when there is no surface potential (here, w ^{= t} = 1) ^one

must recover bulk behaviour and the three generating ^{functions are identical} by symmetry.

The relevance of this fixed point ^{for a line in} parameter space is in agreement ^{with the} conventional physical picture : ^the unbinding temperature corresponds ^{to an exact} (asymp- totic) compensation between the attractive and repulsive (or entropic) ^parts of the wall

potential ^{and the} polymer, though ^{bound to the} surface, ^{behaves as} in the bulk solution.

The largest eigenvalue of the linearized system remains the one associated with the bulk

relation, equation (28a) : A 1= A b, ^{so the} gyration ^{radius is} govemed by ^{the bulk} exponent.

The transverse correlation length g 1- (x), associated with D, ^{and the} parallel ^one gll (x), associated with C, ^are equal, showing ^{that the} polymer ^is isotropic. ^{This is} again ⁱⁿ

total agreement ^{with the} accepted picture for non-fractal systems.

iii) For w > w * (t ), ^one reaches first the fixed point corresponding ^{to an adsorbed} polymer

with vi = 1 and v_L

0, ^since 03BEII (x) = (xc - x )-1 ^and 03BE_L (x) = ^{C t.}

The critical fugacity ^now depends ^{on w and close to} the tricritical point ^{one has} (Eq. (26))

where the value of the cross-over exponent 0 ^{can be obtained} through ^the following argument. ^Consider initial conditions (Xi’ Wi, ti ) very close ^to the tricritical f.p. (b *, b *, b * ),

but that finally ^{iterate towards} (0,1, 0 ). ^{As the} dependence of B (1), ^{C (1) and D (1) on} x, w and t is invertible in this region ^and non-singular, the iterations define through equation (27) ^a

sequence (xk, wk, tk ). ^{The relation for} B (k) depends only ^{on x, so one has}

(Xb - Xk) - (k b )k (Xb -Xi), while the variation of w is dominated by ^the eigenvector

associated with the second largest eigenvalue ^at (b *, b *, b * ) : (Wk - W * ) - (À2)k(Wi - w*),

with À2 = {B/5 ^{+ (37 - 16} 0)1I2} ^{/2 =1.67096.} The condition that the iterated points

remain on the critical line implies

This result can be checked numerically using the recursion equations. ^The free energy per

monomer is [27]

with

The value obtained for the cross-over exponent ^{is not} surprising : it is close to the value

cp = 1/2 for regular 2-d lattices [8], ^{and it} obeys the bounds derived in the first part. ^The fractal dimension of the gasket ^is dF

^In 3 /In 2 = 1.585 and the dimension of the adsorbing

surface is ds = 1, ^so equation (21) yields ^{for the} present ^case

Two remarks are worth making :

-

another non-trivial fixed point exists, namely the « Surface-Bulk » point (b *, 1, 0 ), ^but

it plays ^no role in the phase diagram because it is too repulsive (all ^three eigenvalues ^are larger ^than 1). ^{This means} that the coexistence of an adsorbed polymer ^{and a free one in}

solution is unstable as soon as there is a non-zero coupling between the surface and the bulk ;

-

an interesting ^{behavior occurs, for w}

w * (t ), ^{on a line} x(w) ^{where the} generating

function C diverges but B goes ^{to zero} in such a way that the product BC --+ 1. The physical interpretation ^{is the} following [28] : ^{if one} forces the polymer ^{to make a} given angle w with ^the wall, ^it is necessary ^to apply ^{a finite tension, until a} critical line is reached which depends ^on 03C8.

By studying the relevant generating ^{functions one sees} that the line (BC ) * ^{= 1} corresponds

to an angle 4f = 7r/3, ^{the line} (BC 2) * ^{= 1 to} 03C0 /6, ^etc.

Finally, it would be very instructive ^to compute ^{the monomer} density profile ^{to check} directly ^the scaling form and the relation between the exponents m and 0 (Eq. (6)). However,

this raises a technical problem because the number of sites at a distance z from the wall

decreases exponentially ^{and with} strong oscillations, ^{and one has to extract a} correction to this

leading dependence. ^The calculation of the exponent yl associated with the number of open SAW of fixed length ^{with one} end attached to the surface [7, 29] is also cumbersome because twelve partial generating functions have to be taken into account.

3. Adsorption ^{on the 3-d} gasket.

Adsorption ^{on a 3-d} Sierpinski gasket ^{with one} attractive surface is more complex than for the 2-d _case, but it presents ^an interesting ^new feature : when a monomer-monomer attraction

v

exp(- EiIT) ^is introduced to represent ^{the effect of a} bad solvent [12], ^a collapse

transition may ^occur in the bulk for a critical value v,,. Adsorption ^{close to such a} 8-point ^has

been studied in great ^detail using renormalization methods for Euclidean _spaces [9] ^{and a rich}

multicritical behaviour is expected for effective dimensions lower than dc

3. Here the fractal dimension is dF

^{2 and} that of the adsorbing ^surface (itself ^{a 2-d} gasket) ^is ds

^In 3 /In ^2.

3.1 RECURSION RELATIONS.

Without loss of generality ^one may restrict the ^monomer-

monomer interactions to act only within the first-order tetrahedra [25]. It is then sufficient to consider 5 partition ^{functions to obtain a} closed recursion system :

a) ^{the two bulk} functions, ^{P and} Q, introduced in previous ^work [24, 25],

where IS (k)(N, J) and a (k)(N, J) ^enumerate respectively ^the configurations ^{where one} N-step self-avoiding ^chain (two chains) spans ^a tetrahedron of order k with J self-interactions. These functions obey ^two coupled ^recursion equations, independent of the interaction v

which posssess three non-trivial fixed points [25]

-

a SAW f.p. :

-

a « collapsed globule » f.p. :

-

a « 0-chain

f.p. :

b) their surface counterparts Pw, Pt and q, which ^enumerate the three types ^of polymer configurations ^{close to the} wall, weighted according ^to their energy (see Fig. 6). ^These depend a priori ^on 5 variables and obey ^{three new relations}

The notation is chosen to emphasize ^{the link with the} surface interaction parameters w ^{and t} introduced in part 2, ^{and to make} apparent ^the correspondence between the terms in the two groups of equations

ⁱⁿ particular, if pt

pw

P and q

Q initially, ^then obviously ^the equalitiés remain verified on iteration.

The system ^{of 5} equations ^{is rather} complicated ^to analyze and has many fixed points, ^but

Fig. ^6.

Diagrams representing the five basic partition functions for the 3-d Sierpinski gasket.

several of them can be identified by simple inspection

^for compacity ^{we denote a fixed} point (f.p.) by its coordinates (P, ^{Q ;} ;P w, Pt, q ) :

-

if P

Q

⁰ (no polymer ^{in bulk} solution), ^a f.p. exists where Pw + pW ^{= 1} (i.e.,

pw

b *, the critical fugacity for the 2-d gasket) and pt = q

0. It describes an adsorbed

polymer behaving asymptotically ^{as a SAW on a 2-d} gasket, and is denoted

-

if P and Q ^{are at one} of the three bulk f.p., ^one may have pt = q = 0 ^and

pw

b * (adsorbed ^SAW coexisting ^{with a 3-d} polymer) :

-

if P

0 and Q

QG (bulk collapsed polymer), equation (31c) ^is decoupled ^from (31d- e) and there is in addition a non-trivial f.p., where pw

b * but q

QG ^{is non zero. This}

would describe coexistence of adsorbed SAW, ^adsorbed globules ^{and free} globules :

-

a symmetric tricritical f.p., ^which corresponds ^{to the usual} special transition :

-

a symmetric

^{adsorbed and} collapsed » f.p. :

In order to discover which of these fixed points, ^or possibly ^other ones, play ^{an actual} role, it is necessary ^to iterate equations (31) numerically, starting from initial values given ^{in terms}

of the bare physical parameters. The initial conditions on the first-order tetrahedron are

In fact one should take five independent parameters ^into account to explore ^{the whole} phase

space. For instance the monomer-monomer interaction v could have a different value vs

in the surface layer, but this should not modify ^the qualitative behaviour of the system

(unless ^a collapse transition occurs in the surface layer itself, which is excluded for the 2-d

gasket).

3.2 SAW REGIME.

For weak monomer-monomer interactions (v « vo = 1.8532), ^the

relevant fixed point ^{is found to be} (Ps, Qs ; ^{0, 0,} 0), ^{i.e., the} polymer remains in the bulk SAW regime. ^{There is} only ^{one relevant} eigenvalue A b

2.79656, ^so

vb = ln 2/ln A b = 0.6740 (note that this value is different from the value vb

0.7294

incorrectly quoted ^{in Refs.} [24, 25]).

When the surface attraction parameter ^increases (at ^fixed t), the critical fugacity

x, reniains independent of w until w = ws (t ), where the recursion relations flow towards a new fixed point : (Ps, Qs ; Ps, Ps, Qs ). This is the expected symmetric « special » f.p., ^which

describes SAW at the unbinding transition, ^{with a} gyration ^radius exponent equal ^to

v b. The second largest eigenvalue ^is A 2

2.158360, ^{so the} cross-over and specific ^heat exponents ^are

in very good agreement ^{with a} direct numerical determination of the line Xc ( w) ^for

w

w,. In the latter ^case the relevant fixed point corresponds ^to the adsorbed SAW :

(0, 0 ; b *, 0, 0 ), ^as expected.

Here again the bounds on the cross-over exponent ^{are verified}

These bounds are quite ^narrow, and the lower one is very close ^to the exact value. Note that the value of 0 is very different from the accepted value for Euclidean 3-d systems,

~ = 0.6: this is reasonable since the 3-d gasket ^{is a} very open ^structure with a rather dense

adsorbing surface, and the bounds show that these two aspects ^are important ⁱⁿ determining 0.

3.3 0 REGIME.

The collapse transition in the bulk solution occurs when the initial conditions are such that P (1) = Q (1) = 1/3. Solving equations (31a-b) for the critical par- ameters one finds xe

0.1681105..., v e = 1.8531999..., ^{and a} precise ^numerical study ^{of the}

recursion can be made for these values, ^{as a} function of the attraction parameter ^{w, for fixed t.}

For small w, the critical fugacity ^{is constant and the} polymer remains in solution, ^{under the}

form of a 0-chain. Its gyration ^radius exponent ^is given by ^the largest eigenvalue

A e = 100/27 : _ve

In 2/ln ^{À (J}

0.52939. Note that the exponent ^{for a} purely random walk is

v

ln 2/ln ⁶

^0.387 [30], ^{so the} polymer ^{at the} 8-point is very different here from a

Gaussian chain. The second « thermal » eigenvalue, k 0(2)

20/9, yields ^the specific ^heat

exponent of the bulk collapse transition :

The surprise ^{is that at a critical value} we (t ), the iterations go ^{to a new} fixed point ^{which is}

not the symmetric

adsorbed and collapsed » point (1/3, ..., 1/3) identified above, ^{as one}

would naively expect ^from experience ^{with the SAW} regime. ^In fact, ^this symmetric point ^has four eigenvalues larger ^{than one,} which makes it too repulsive ^to appear in the phase diagram, except may be ^{as a} higher-order multicritical point. The relevant fixed point ^has coordinates

and the linearized system has three repulsive directions, ^{with a surface} eigenvalue A s = 2.27115, ^{in addition to the two bulk ones. As} AsÀ03B8, ^the gyration ^radii RII ^and Rl remain both of the order of the bulk radius, ^{but the} anisotropy ^induced by the wall in the correlation functions pw(k) Pt (k) ^and q (k) ^{does not} disappear completely ^at large ^distance (k - oo ). ^{A similar} asymmetric ^fixed point ^has recently been found in a RSRG study ^{of self-} avoiding ^surfaces [20]. The ratios pwlp, ptl P ^and q 1 Q =1= ^1, play ^{here a role} analogous ^to

universal amplitudes in usual critical phenomena.

For w > we (t ), the relevant f.p. is the adsorbed SAW point ^{but one} expects ^{that the critical} line _{xo -} Xc ( W ) = (w - w 0)’ is characterized by ^{a new} exponent C = 1/~03B8. The connection

between ~0 ^{and the} eigenvalues ^{is not} direct, however, ^{because the} simple argument leading

to relation (29) ^{is no} longer valid, ^{due to the existence} of the second bulk eigenvalue

A J2). Numerically, ^{we find} 1/~03B8,

^{1.3-1.6, the} large uncertainty being ^{due to} oscillations and to the difficulty ^of approaching very close ^{to a} strongly repulsive ^fixed point.

3.4 GLOBULE REGIME.

If the monomer-monomer interaction v is made still more

attractive, ^an unbinding transition occurs between an adsorbed SAW state and a free

collapsed globule phase ^{for a} critical value WG, but its ^nature changes. ^{The fixed} point ^{at the}

transition is (0, QG ; ^{b *, 0,} 0 ), signalling the coexistence between the adsorbed SAW and the

globule, ^{and the} slope dxc/dw ^{of the critical} fugacity is finite for w

WG ^{+ E, so} the transition is first ^order (Fig. 7).

The situation changes again ^for larger ^values of v, where one recovers a continuous transition governed by ^{the fixed} point

The two largest eigenvalues ^are À 1 = ^{4 and} À 2

3, ^{so v}

v G = 1/2 at the transition and the

cross-over exponent 0

^In 3 /In ^{4, in} agreement with the direct numerical determination and with the general result derived in part 1, ~

dsIdF in a globule regime ^where

v

=1 /dF. ^The phase diagram ^{in the} plane (v, w ) ^{at fixed t 1 is} summarized in figure ^8.

4. Real _space renormalization for the square lattice.

Contrarily ^{to the} previous examples where the recursion relations obtained from the real- space renormalization ^were exact, the method is only ^an approximation ^{in the case of}

Euclidean lattices. However, ^this type ^of approach ^{has often} proved useful for many polymer problems [31], ^{and we} propose here ^a simple procedure for the square lattice, which is very

analogous ^{to the} previous calculations and give good ^results.

Fig. 7.. Fig. 8.

Fig. 7.

^Critical fugacity for the 3-d gasket, ^{when the} monomer-monomer interaction v::> vo, the critical value for bulk collapse. The finite slope ^{at w}

WG signals ^a first-order transition.

Fig. ^8.

^Phase diagram ^{in the} plane (v, w ), ^{at fixed} t (= 0.5 ).The ^bulk collapse transition occurs at v

v 03B8 and the ^crosses indicate a first-order transition line.

Real-space renormalization schemes for polymers ^are usually expressed ^{in terms of}

«

effective weights

^{for the various} types ^of steps ^{that a SAW} may take. Here it is natural ^to retain a weight ^{u for} steps along ^{the surface, and a} weight t ^for steps perpendicular ^{to the} surface, in addition to the bulk fugacity x (Fig. 9). ^{To make the} analogy with the calculation

on the 2-d Sierpinski gasket precise, it suffices to remark that these weights may be viewed ^as

a short-hand notation for the generating functions introduced in part ^{2. The} correspondence

is the following with the bare quantities denoted xo, uo and to ^{to avoid} confusion,

the bulk generating function for walks spanning ^{a cell} (see Eq. (23a)),

the two surface generating functions for walks having ^both extremities or only ^one extremity

on the attractive wall respectively (see Eqs/ (23b-c)).

Fig. ^9.

Renormalization scheme for the surface weights ^{u and t, on a} (2 x 2 ) cell of the square lattice.

If we now consider the renormalization of a (2 x 2 ) ^{cell, as shown in} figure 9, the recursion relations read :

These equations ^{have a} pure bulk fixed point : xb

0.466, ^{u = t}

0, ^{and a} pure surface ^{one :}

x = t = 0, u, = 1. The only ^{non trivial fixed} point describing ^an adsorption transition is here the isotropic ^fixed point :

As in the previous examples, ^the cross-over exponent 0 is obtained from the two largest eigenvalues, À b ^and A S, corresponding ^to the bulk relation (33a) ^{and to} the surface ones (33b- c) respectively :

which is in good agreement with Ishinabe’s numerical result (~ = 0.53 ) ^and is very close ^to the exact value 0 ^{= 1/2} given by Burkhardt et al. [8].

This result is also somewhat better than the value 0 e--- ^{0.55 ±} 0.15 obtained by ^Kremer [13], ^{with a} two-parameter renormalization including ^{some extra} constraints. Though only

one extra surface parameter appears ^to be relevant in the RG _sense, the introduction of the third parameter t ⁱⁿ equations (33) is rather natural physically, and it is also justified by ^the

exact calculations on fractals. A detailed study ^on larger cells would be necessary ^{to see} whether our method converges rapidly ^{to the value} 1/2. Extension to 3-dimensional cells would also be of great interest, ^{but in this case the} configurations would have to be enumerated by computer.

Finally, ^{we note} that for the magnetic analogue ^{of the} adsorption problem ^a real-space

renormalization scheme using ^three parameters ^{has been} proposed previously by ^Burkhardt

and Eisenriegler [14]. ^{However, their} approach is different from _ours, in the sense that one of their parameters controls the penetrability ^{of the} interface, ^{while we work} directly ^{with the} impenetrable ^{case. A} full calculation including ^{also the} penetrable ^case would then require

the introduction of a fourth fugacity ^{in our} description.

Conclusion.

We have presented ^a generalization ^to fractal spaces of the theory ^of single-chain polymer adsorption, obtaining non-trivial bounds on the cross-over exponent valid in any space dimension

in particular, ^{for 2-d} systems, ^{the exact value} 0 ^{= 1/2 saturates} the upper bound. We have checked these bounds on several models, showing ^{how exact recursion}

relations can be obtained and solved to yield ^this exponent. ^Similar types of bounds also hold for adsorption ^of self-avoiding ^surfaces (SAS) [20], ^this suggests ^{that the} approach presented

here may be fruitful in a variety of situations. Another interesting finding is the existence of

an asymmetric ^fixed point for the 3-d Sierpinski gasket, ^{when the} polymer ^{in bulk} solution is at its 0-point. ^This surprising ^fact might ^{be a} peculiar feature of the fractal structure, but a

similar result is also found for the SAS adsorption problem ^{in d}

3 dimensions [20]. ^The possibility ^{that such a} situation is more general ^{should be} explored ⁱⁿ detail, using for instance

real-space renormalization schemes more sophisticated ^{than the} simple ^one presented ^{in the}

last part.