Self avoiding surfaces at interfaces

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Self avoiding surfaces at interfaces

Elisabeth Bouchaud, Jean-Philippe Bouchaud

To cite this version:

Elisabeth Bouchaud, Jean-Philippe Bouchaud. Self avoiding surfaces at interfaces. Journal de

Physique, 1989, 50 (7), pp.829-841. �10.1051/jphys:01989005007082900�. �jpa-00210960�

Self avoiding ^{surfaces at interfaces}

Elisabeth Bouchaud and Jean-Philippe ^Bouchaud (*)

O.N.E.R.A., ^BP 72, 92322 Châtillon Cedex, France

(*) Laboratoire de l’E.N.S., ^{24 rue} Lhomond, 75231 Paris Cedex 05, France and LHMP E.S.P.C.I., ^{10 rue} Vauquelin, 75231 Paris Cedex 05, France

(Reçu ^{le 16} septembre ^1988, révisé le 30 novembre 1988, accepté le 8 décembre 1988)

Résumé.

Nous étudions le problème d’une membrane auto-évitante isolée au voisinage ^d’une

surface faiblement attractive. L’exposant ^de crossover 03A6 caractérisant la transition d’adsorption

satisfait les bomes : 0,6 ~ 03A6 ^~ 0,666. ^Une méthode de renormalisation dans l’espace réel fournit

une très bonne valeur de l’exposant de volume 03BD (~ 0,8) ^et suggère 03A6

^2/3. Cette valeur est utilisée pour obtenir ^en particulier l’épaisseur Da de la couche adsorbée (03B4-3/5) ^et l’énergie par monomère (03B43/2) ^en fonction de l’excès d’énergie de surface 03B4. Le profil de concentration varie en

z-1/6 _{pour a ~} z ~ Da. ^{En solution} semi-diluée, ^le profil ^{s’étend sur} des distances de l’ordre de la

longueur ^de corrélation de la solution ; ^{et varie comme} z-1/2 _pour l’adsorption ^{et comme} z5/2 _{pour la} déplétion, pour des distances ^z correspondant ^{à la} région ^{« centrale »} (Da ^{~ z} ~ 03BE).

Abstract.

We study ^the problem ^{of a} single ^self avoiding membrane in the vicinity ^{of a} weakly attracting interface. The crossover exponent 03A6 ^{needed to} characterize the adsorption transition is shown to lie in the interval 0.6 03A6 0.666. Real space renormalization provides ^{a very} good

value for the bulk exponent ^v (~ 0.8) ^and suggests 03A6

2/3. This value of 03A6 ^{is used to} predict, ^inter alia, ^{the width} Da of the adsorbed layer (~ 03B4-3/5) and the energy per ^monomer (03B43/2) ^{as a function}

of the surface excess free energy ^03B4. The concentration profile ^{is found to} decay ^as z-1/6 for a ~ z ~ Da. For semi-dilute solutions, ^the concentration profile ^is expected ^to extend up to distances of the order of the bulk correlation length, decaying ^as z-1/2 for adsorption, ^and growing ^as z5/2 for depletion ^{in the so} called central region (Da ~ z ~ 03BE).

Classification

Physics ^Abstracts

64.60

05.40

68.42

1. Introduction.

The physics of membranes and fluctuating interfaces is rapidly developing, especially ^{from a}

theoretical point ^{of view} [1, 2]. ^{A fruitful} possibility ^{is to take} advantage of the fact that the

theory ^of polymer solutions is now very well understood, both in the ^« bulk » [3, 4] ^{or near}

interfaces [5]. One of the major breakthrough in this domain in the last decades is surely ^the

consequences of its connection ^to the realm of critical phenomena, ^{and in} particular ^the possibility ^of writing scaling ^laws [3] ^to relate the evolution of the relevant physical quantities.

Combined with physical intuition, ^this approach has proven extremely powerful ^and predictive ; it furthermore provides simple ^and useful-to-think-with images ^{of the} physical reality. One of the most natural for the theoretician (it ^{does not seem obvious to} synthetize

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005007082900

those molecular ^{nets : see} [1] ^{for the} present ^{status of the} art)

generalization ^{of the} polymer problem ^{is to consider} surfaces instead of chains, ^{and to} study ^their properties (shape ^and

structure, phase diagrams, etc.) ^{as the} physical parameters ^are varied. Even if it is clear that these objects ^{have a host of} specific properties (such ^{as for} example ^{a finite} temperature transition between a flat and crumpled phase [1, 6]) which have no counterpart ^{in the} polymer problem, it may be appropriate ^{to further} transpose ^the knowledge ^of polymer physics ^to

describe situations where the two systems ^{should behave} similarly. ^In this paper, ^we analyse, through scaling and real space renormalisation group approaches, ^{the structure of a self} repelling surface of fixed topology (Self Avoiding ^{Tethered Surface} [1, 7]-SAS) ^{near an}

interface (solid wall, liquid-liquid ^or liquid-gas interface). ^In particular ^we analyze ^the

« decrumpling ^» of the SAS as the attraction of the interface is increased and propose scaling

laws for its parallel ^and perpendicular extensions. As for the case of polymers, ^one needs, ⁱⁿ

addition to the bulk exponent v, to introduce a cross-over exponent 0, ^{for which we} give ^both

an upper and ^a lower bound (which ^{are in fact} quite stringent). ^{A small} cell real space RG allows us to obtain quite ^a good ^{value for v} compared ^to the Monte Carlo simulations and the

Flory approximation ^of [7] - thereby confirming ^the prediction v

4/5 ; ^{and an} approximate

value for 0, ^{close to} its upper bound (0 = 2/3). Our results could be checked numerically ;

the use of scaling laws could perhaps ^{also be} justified by field theoretical techniques, ^{as it is}

the case for polymers [8].

2. Bulk behaviour.

The model that we shall study is that of Self Avoiding ^Surfaces which, upon deformation, ^are equivalent ^to planes containing ^{N x N} « monomers ». Thus we do not consider branched surfaces (see e.g. [9]) but rather what was called in [7] ^« tethered surfaces » i.e. polymerized

membranes ^or fixed topology, ^as opposed ^to liquid ^or hexatic membranes [1, 10]. ^These

molecular ^« nets » will hopefully ^{be soon} synthetized ^{and the} experimental investigation ^of

their structure will then be of actuality.

2.1 A SINGLE MEMBRANE IN A GOOD SOLVENT. - A set of N 2 non interacting ^{beads tied} together by ^elastic springs ^{to form a} bidimensional network extends spatially ^{over distances}

instead of the N ^1/2 law holding for ideal chains. The logarithm ^comes from the usual 2-d

divergence of harmonic fluctuations. It also means that a free molecular net is extremely dense, ^{since its} density goes ^to infinity with N in any dimension (p

^N2 (Log N )- dl2) ^which

implies that the self avoiding constraint is always relevant, ^{in contrast} with chains where it becomes irrelevant for high dimensions (d > 4). The excluded volume effect thus swells the surface to R

aN " where v is ^a non trivial critical exponent ^to be determined. A Flory

formula has been proposed ⁱⁿ [7] through ^a scaling analysis of Edwards Hamiltonian ; ^it

reads :

For d

3, ^this predicts v

0.8, which is in good agreement with Monte Carlo simulations [7]

and experiments ^on crumpled papers [7, 11]. ^{Let us} show how the Flory ^formula (2.2) ^{can be}

obtained through ^the general statistical arguments presented ⁱⁿ [12]. ^{Consider a line on the} surface ; the successive displacements along ^{this line are} correlated by the fact that if the n-th

monomer is close to one of the preceeding ^{ones, the next} step ^{is chosen to} expand the surface

radially. The number of statistically ^identical displacements ^is thus, ^{in a mean field} spirit ^and

for a D-dimensional manifold :

If C (n ) is the correlation function of the successive displacements (C (n ) = (Xi Xi + n»’ ^this

number Nid ^is simply ^the integral ^of C (n ), ^since C (n ) essentially ^{counts the} probability ^that

xi + n is equal ^to ri. Thus C (n ) ^{behaves as n - a with a}

vd /D - 1. Then if a is smaller than 1,

the sum of individual displacements ^{within a D} dimensional manifold scales as :

(and ^as Nid Log 1/2 ^{N for D}

2).

Self consistency ^thus requires :

or :

(plus logarithmic corrections for D

2). (2.4) ^{is exact for d}

D, ^and reproduces ^the Flory

result for polymers (D

1). ^This approximation furthermore has a nice feature, already emphasized ⁱⁿ [7] : ^the exponent v ^{of a} 3-dimensional self avoiding walk is the same as that of

a 2 D SAW drawn on a 3 d SAS : 3/5

3/4 . 4/5. The only difference with [7] ^{is that our line of} reasoning naturally ^{leads to} logarithmic corrections to a power law for self avoiding ^surfaces.

It is fairly easy ^to adapt the real space RG procedure developed ^for polymers (which ^is

known to yield ^rather good ^results [13, 14]) ^{to the case} of surfaces. Instead of a link fugacity,

one introduces a « plaquette » fugacity ^k and looks for its evolution under a simple

renormalization transformation. Choosing ^a natural rule (rule ^N° 1) for 2 x 2 x 2 cells (see Fig. 1) gives ^a recursion relation :

which yields ^{a fixed} point ^k

^{0.641 and an} exponent v

0.77. We have also investigated ^a simpler ^rule (rule ^N° 2), ^{easier to} implement ^on larger cells. For 2 x 2 x 2 cells, ^it gives

Fig. ^1. - Cell renormalization of the fugacities k. Rule N° 1 : It is an adaptation ^{of the « corner rule » of} [13] ^we consider all configurations spanning ^{the cube} from ^{left to right starting} from the bottom left

edge. ^{Rule N° 2 : For this rule} only locally Monge ^Surfaces (no overhangs) ^are considered, ^{with the two} following constraints :

no plaquette is allowed on the vertical external sides of the cube. - At least half of the horizontal plaquettes belong ^to the bottom plane. We believe that this rule slightly

overestimates v because of the no overhangs constraint.

v

0.815 ; ^{while for 3 x 3 x 3} cells, for which 3 000 configurations ^{are taken} into account,

one obtains v

0.800. We thereby ^{have obtained an} independent way of estimating ^{the bulk} exponent ^for SAS, ^which fully confirms the results obtained in [7]. ^The quality of this result suggests ^{that this} procedure ^{can be used to estimate} the surface crossover exponent 0 (see

Sect. 3).

2.2 MANY MEMBRANES IN SOLUTION.

Suppose ^{that we now} impose ^{a volume} fraction 0 of

monomers in the solution. The nets will start « seeing ^» each others when (we ^{now consider}

the case D = 2 and d

3)

For 03A6» 03A6*, ^a length scale e appears [3] below which the structure of the surface is not affected by the ambiant pressure : this is expected ^{for an elastic} object ^which distributes the external constraint on the low energy, long wavelength modes ; only strong constraints can

disturb the structure on small length scales. e is defined as :

Thus

This result has also been obtained in [15]. As for the case of semi dilute polymer ^solutions

one can associate a free energy kT _per « blob » of volume ç3, thereby defining ^an energy

density (or ^osmotic pressure) :

This result can also be obtained by writing :

where03A6 /N 2 ^{is the} perfect gas pressure expected ^{for non} overlapping ^membranes (03A6) 03A6 * ) ^and by requiring that II ceases to depend ^{on N} for 0 > 0 *. Note that (2.6) ^shows

the tremendous importance of correlations which screens the excluded volume interaction, reducing ^{the osmotic} pressure from 4>2 (uncorrelated monomers) ^{to 06. It is} interesting ^to

notice that for an assembly ^of flat ^membranes ( v =1 ) ^or for 0 such that gis smaller than the

persistence length, ^{the same} argument predicts II = 03A6 3, which is Helfrish’s result [16].

The structure of a marked net for length ^scales larger than § (e. g. the relative arrangement of the blobs belonging ^{to one} membrane) ^{is more subtle, and we} propose here very

speculative arguments. Analogy ^{with semi dilute} polymer ^{solutions would} suggest ^{in this case}

(below the lower critical dimension) :

which corresponds ^{to a} compact packing of the blobs. However, it is very difficult ^to fold a

surface in a compact way without tearing ^it. Very regular folding, ^{such as} represented ⁱⁿ

Fig. ^2.

Regular folding ^{of a} piece of paper. Folding stops ^when the thickness of the sheet is of the order of its length :

figure 2, ^indeed yields ^{R -} N2J3. This configuration ^{however has a} very low entropy (every

« fault ^» in the folding procedure ^creates large ^{« holes} ») (1) and involves certainly quite ^{a lot}

of bending ^and stretching energies. Two other natural configurations might be considered :

- « Cigar shaped » membranes, ^{folded in one dimension} only ^and fully stretched in the other : this has at least the entropy of 2 d Hamiltonian paths ; ^{however the} rigidity governing

the undulation modes of this ^« rod ^» is very high ^{and this} heavily suppresses this ^source of entropy.

-

« Flat lamellae » of size (N / g) g ^x (Nlg) e x e piled ^{one above} the other. This structure can sustain soft sound waves propagating perpendicular ^to the lamellae.

The probable ^{scenario is} that, ^as concentration increases, ^the _system undergoes ^« liquid crystal » phase transitions, ^{first to a « nematic »} phase ^of rods, ^{then to a « smectic »} phase ^of lamellae, much as in the case of liquid ^membranes [2]. ^Note however that all this paper

concerns equilibrium situations, which may be exceedingly long ^{to reach due to the} strong

topological constraints ; ^one expects ^a viscoelastic behaviour, evolving with time from a very hard solid response ^to that of a very ^« soft » solution (H = 0 6).

3. The problem ^of adsorption.

We now consider one SAS in the vicinity ^{of as} slightly attractive surface. As for polymers [17],

the relevant questions ^{are the} following : how attractive has the surface to be to inforce an

infinite membrane to put ^a finite fraction of its sites in direct contact with it ? How to describe the crossover between the bulk and adsorbed regimes ? What is the structure of the adsorbed membrane ?

(1) ^{It looks} intuitively ^as if the number of « isotropic Hamiltonian surfaces » did not grow

exponentially ^{with N} contrarily ^to the number of Hamiltonian paths. It would be interesting ^to study ^this

number as a function of the anisotropy ratio of the overall imposed shape of the membrane.

3.1 DESCRIPTION OF THE MODEL. - We consider a SAS interacting ^{with an} attractive plane

in the following way : each ^monomer coming ^{in contact} with the wall receives an energy

Ws, while the other monomers have an energy Wb. If the attractive surface is impenetrable,

one must take into account an (orientational) entropy loss for each monomer on the wall :

consequently, ^{an infinite SAS will not « stick » to the} wall unless the difference

Ws - Wb ^{reaches a non zero} negative value 0 W. On the contrary, ^{there is no loss of} entropy ^if the SAS is allowed to cross the surface and thus AW

0 in this case. As for polymers, ^one

introduces [17] ^the parameter ^{6 defined as :}

(kB is the Boltzman constant and T the temperature). Adsorption corresponds ^to

5 :> 0. In all that follows, ^{we assume à « 1} (weak adsorption). ^A finite network will not

undergo ^a phase transition ; there will rather be in that case a crossover region ^defined by :

Adsorption ^then corresponds ^to N2 cf> 03B4 > 1. 0 ^{is the crossover} exponent, the value of which will be discussed in section 3.3, ^and depends ^{on the} position of the threshold AW.

3.2 STRUCTURE OF THE SAS IN THE VICINITY OF AN ATTRACTING SURFACE.

3.2.1 The ^crossover region.

When the surface is « neutral » (i.e. ^condition (3.2) ^is satisfied), ^{the mean number M of monomers in contact} with the surface is precisely ^{of order} [8, 5] :

(3.2) ^{thus means that} adsorption ^or depletion ^{cannot take} place ^{if the excess} surface energy remains smaller than kB ^{T : in this case} thermal fluctuations are dominant. In this region, ^the

membrane remains isotropic, ^{in the sense} that both its parallel (RII ) ^and perpendicular (R1 ) extensions are of order aN ", where v is the bulk exponent discussed above. By analogy

with polymers [18, 19], ^we suppose that the number of ^monomers per unit ^{area at a} distance z

from the wall follows a power law : (a « ^{z «} R )

03A6s ^s is the surface density ^{of monomers in contact with the} wall, and m is the « proximal » exponent [18] ^{which can} be related to v and 03A6. ^{Indeed one has :}

and thus :

The fact that the monomer volume fraction is not a priori ^{constant even} when the surface energy is ^not sufficient to allow adsorption ^{is related to the non zero value of} AW, ^as will be discussed below.

3.2.2 Structure o f ^{the adsorbed} membrane. - As discussed above, N 20 5 is the energy of the

SAS at threshold ; ^this quantity thus appears ^as the natural scaling ^{variable x. For} large ^values

of x, the membrane is adsorbed and the number of monomers in contact with the wall is

proportional ^to the total number of monomers N 2. Thus one can write :

with

so that M , N 2 & (1 - ~)/~ for x > 1. This immediately ^{leads to} the membrane surface free energy

In this adsorption regime, ^{one can} consider the largest subpart ^{of the net} which remains unaffected by ^the attracting wall. Let Da be the scale of this « blob » which contains ga ^monomers. By definition, ^one thus has : Da .-, agâ ^and g2 ~03B4= , ^{1, so that :}

The meaning ^of Da ^{is the} following : ^at length scales smaller than Da, the local behaviour of the net is three dimensional, while for sizes larger ^than Da ^{the adsorbed} membrane behaves as a self avoiding surface in two dimensions. The parallel extension of the SAS is thus given by :

while its perpendicular thickness is simply equal ^to Da. ^{The monomer} volume fraction at a

distance z from the surface still decreases as z- m for z smaller than Da ^and rapidly goes ^{to zero} for larger ^distances.

3.3 THE CROSSOVER EXPONENT. - Until now, ^we have left the value of ~ undetermined.

Nevertheless its actual value is of crucial importance for the evaluation of the physical observables ; this section is devoted to a discussion of the possible ^{values of this} exponent.

3.3.1 The relation between the value of 0 ^{and the} position of ^the adsorption threshold. - Let

us first examine the case of a penetrable adsorbing plane. ^{In this} case, ^one should expect AW

0. Thus at threshold, ^the surface is only ^a fictitious plane intersecting ^a perfectly isotropic SAS. The fraction of monomers on the surface is thus proportional ^to the ratio of the surface of intersection to the volume of the membrane :

This gives [17, 20]

In the case of an impenetrable plane, ^{on the} contrary, ^the plane ^is already attractive at

threshold, ^{since AW 0. Thus we} expect that the fraction of monomers on the wall should be greater ^{than N -} ", or 0 > 1 - v /2 ^{for an} impenetrable surface. This lower bound will be

compared ^{later to our RSRG} estimate. An important ^{remark must be made :} using (3.4) ^with

0 given by (3.7), ^{one obtains for the} proximal exponent m

0 : since at threshold the surface

is absolutely ^neutral (no ^force experienced by ^the monomers), ^no profile ^can build up. On the

contrary, when the surface is impenetrable, ^the attracting ^nature of the wall ^at threshold

induces a concentration profile with enhanced density ^{near the} wall, or m :> 0. This bound is seen, using equation (3.4) ^{to be} equivalent to 4>:> ^{1 -} v /2.

3.3.2 _{An upper} bound for ~. ^{- We now} present ^a plausible argument which, although ^non rigorous ^and perhaps improvable, strongly suggests ^an upper bound of 0 (which turns out to be in agreement ^with all known results). ^The reasoning ^{is based on the two} following assumptions :

a) ^{If one considers a blob of size} Da, ^its adsorption energy is, by definition,

kT. Another contribution to the energy within this blob is the self repulsion Frep ^{due to}

contacts of different parts ^{of the} membrane. At thermodynamic equilibrium, this energy should also be of order kT, independently of the value of 8 and homogeneously distributed among the layers.

b) ^{We assume} that, ^{in a} varying concentration 0 (z ), Frep ^{depends on a local way on the}

function 0 (03A6(z) « 1 ) :

If 03A6 were a slowly varying ^{y g} function of z - Le. 1 aa: g :) 1 1 ^{az 0} 1 ^{« 1- then q would q be equal q}

to the semi-dilute value : q

vd / ( ^{vd -} 2 ). However, ^{in the} proximal region ^{z « D, the}

above inequality ^{is reversed, and} thus, the semi-dilute value for q might ^{not be} appropriate.

It is easy ^to show, however, that the condition :

has only ^{two solutions} in q (depending ^{on which} bound, ^{D or a,} dominates the integral). ^The

second one however does not describe an equilibrium situation, since all the free energy is concentrated in the very first layers. ^{So we must have :}

For an impenetrable surface, ^{one thus has :}

In infinite dimension the mean field value ^v

0 applies, leading ^{to the} equality ^{of the}

lower and upper bound ^to give cp = ^1.

-

For collapsed surfaces, ^v = 2/d ; ^{and the two} bounds coincide again leading ^to 1 - lld.

In two dimensions, ^{the two bounds are} equal ^to 1/2 which is the expected ^result (the

transition is however for infinite surface coupling).

In three dimensions, taking ^v

^{0. 8, we have : 0. 6 0.666..., which} drastically

restricts the possible values of the crossover exponent. ^{We shall see} below that our RSRG result falls within those bounds, and is rather close to the upper ^one.

We also want to mention that analogous ^{bounds can} be found for polymers ^{on Euclidean}

and fractal lattices, ^{which are in} good agreement with all known results [5, 21].

3.3.3 Real space renormalisation calculation of 0. ^{- The} principle of the calculation is an

extension of the one used to compute v in section 2 : indeed one must take into account two

types ^{of cells :}

Bulk cells, for which the critical fugacity ^is equal ^to kb.

Surface cells, which involve three types ^of plaquettes (see Fig. 3) :

-

Surface plaquettes, ^the fugacity of which is equal ^to ks.

-

Vertical plaquettes, ^with fugacity ^k.

Bulk plaquettes ^with fugacity kb.

Fig. ^3. - Cell renormalization with surface fugacities ks, kv.

This surface cell renormalises into a surface cell of half smaller dimensions, defining ^two

new surface and vertical fugacities ks ^and kv. ^As above, ^{we consider} successively ^{two rules :}

Rule N° 1. - We obtain the following mapping ^{in the} (ks, kv) plane :

This mapping generates the flow sketched in figure 4 and possesses three fixed points ^{at finite}

k’s :

-

One of them is ks

^1, kv

^{0 and} corresponds ^{to a surface} demixion, ^{for which some}

membranes are flat on the wall and others are repelled from it into the bulk. Note that the

eigenvalue ^À

⁴ corresponds ^to the bidimensional bulk exponent v

^1.

ks = kv

kb ^{is the} isotropic ^fixed point. ^The study ^{of the} linearized mapping ^{shows that}

Fig. ^4.

Renormalization flow exhibiting ^two phases : ^Adsorbed (A) and Non Adsorbed (NA), ^{and 3}

fixed points : ^{S is the} special adsorption transition and M appears ^{as a} multicritical point ^{for which we}

have no interpretation. Inserts show the effective chemical potential ^exerted by ^{the wall on the}

monomers.

both eigenvalues ^are greater ^than one, i. e. that this fixed point ^is totally unstable and

corresponds perhaps ^{to a} multicritical point ^{in the} phase diagram.

ks

^{0.45 and} kv

0.71 possesses ^one stable and one unstable direction. We guess that this fixed point corresponds ^{to the} special surface transition point. ^The eigenvalue Àg > ^{1 leads to the} following value of the crossover exponent :

which is in perfect agreement with the bounds derived above and suggests ^to take the upper

one as an approximate ^value (03A6 = 2/3 ).

Rule N° 2.

We obtain, for 2 x 2 x 2 cells, 03A6 ^--, 0.66 and for 3 x 3 x 3 cells

~= 0.65. It is as usual very difficult ^to estimate the precision ^{of this} method, ^{but we believe}

that the stability ^{of those} figures serves as a measure of their accuracy ; ^even if a systematic

size dependence ^{cannot of course} be overlooked.

3.4 ADSORPTION OF A SEMI DILUTE SOLUTION. - We now consider a semi dilute solution of

polymeric ^{nets in the} vicinity ^{of an} adsorption plane. Through general arguments provided by

the study of surface transitions in magnetic systems [22], ^{one can} reasonably ^{assume that the}

concentration profile extends up ^to distances of the order of the bulk correlation length e.

Furthermore, following de Gennes’s assumption ^for polymers [22] that in the case of strong adsorption (8 = 1 ) ^the profile ^is self ^similar

^{in the sense that one can define a local}

correlation length 1 (z)

^{z, we obtain,} using (2.5) ;

When 6 is smaller than 1 but larger ^than N -2~ the proximal region extends up ^to distances of order Da ^{from the} adsorbing plane : (using 0 = 0.666, v

0.8) :

while the above profile extends from Da ^to e. Nevertheless, ^{as we have} pointed ^{out in section} 2, semi dilute solutions might ^{be more} complicated ^to describe in the present ^{case than it is} usually ^for polymers, ^{and the} simple assumption ^{of self} similarity ^{referred to above} might

break down.

3.5 DEPLETION. - We ^now consider the case of a repulsive impenetrable ^{surface, or more} precisely ^{à «} 0. The purpose of this section is ^to determine ~ (z) ^{which exhibits two} regimes :

the proximal region ^z Da, ^{where the} profile ^{is still} given by -P (z) , ^{z - 1/6} and the central

region, ^{where e} increases. We shall first study the limit of strong depletion ⁵ = - 1 and then extend the results ^to the weak depletion ^situation.

Assume first that a single ^SAS is attached to the wall by ^{one of its monomers. Since the}

surface is strongly repulsive, the total number of monomers on the surface is of order 1, ^and

thus :

Because the interaction is short ranged, ^the perpendicular extension of the membrane

remains R = N v a. Thus W (z

R) = ^{cP * = N 2 - "d. If} one assumes a power law dependence

for l/J, ^{then the two above} expressions ^{lead to :}

Suppose ^{now that the monomer} previously ^{attached to} the wall is allowed to lie anywhere

between z = a and z

R. In this case the profile ^is simply ^the integral ^{of the} previously

defined CP, ^which yields :

which generalises ^to zD/v the result obtained in [24].

A straightforward ^{extension to} the semi dilute ^case leads to tP(z) = (al e )d (z/a)2/v ^for

a « z « e. When 03B4 1 decreases from 1 to smaller values (but > N - 2 c/», there reappears the

proximal region, and in three dimensions one has (see Fig. 5) :

Fig. ^5.

Concentration profile ^{in the} depletion ^case, showing ^two regions : ^{- the} proximal region, ^still

sensitive to the locally attractive surface ; - the central region ^{sensitive to} the overall thermodynamical repulsion.

4. Conclusion.

In this paper, ^we have investigated the behaviour of a self avoiding ^{membrane near an}

attractive surface. We have argued ^{that this} problem is described by ^{the excess} surface energy 5 and by a crossover exponent 0 ^{related to} the number of monomers on the surface at threshold N . Using geometrical ^and energetic arguments, ^we have shown that in 3

dimensions, 0 lies between 0.6 and 0.666, the lower bound being reached for a penetrable

interface (defect plane). Real space RG which gives ^{a new} estimate of the bulk exponent, ^has been used to obtain 0 ^{= 2/3 for an} impenetrable ^wall. This allows to obtain important quantities ^{such as the monomer} density profile, the width and the radius of the adsorbed

« pancakes ^». These results could perhaps be checked by Monte Carlo simulations and/or

analytical approaches in certain limiting ^{cases. For the case of} strong adsorption ^or depletion

of solutions (which ^should correspond ^to generic experimental situations) ^we suggest

assuming ^{that the} analysis developed ^for polymers is still valid here, ^{which is} by ^{no means}

obvious

the shape of the concentration profile ^{which can be} probed through ^neutron

reflection [25] ^{or neutron} scattering [26, 27] experiments. ^Some points ^cannot be discussed

within the framework presented in this paper, for example ^{the nature} of the semi dilute solution, ^{or the structure} of the ensemble of contact points ^{of a} single adsorbed membrane : do they form closed loops, open lines ^or isolated points ? What is the generalisation ^{of the} loop size distribution defined in the case or linear chains [28, 5, 29] ? ^{It would also be} important ^to understand the meaning of the multicritical point ^which appeared ^{in our RSRG} analysis.

Let us mention that the systematic influence of the bulk concentration Ob ^{on the adsorbed} layer and the full phase diagram ^{in the} plane (4)b’ 8 ) ^{can be} investigated along ^{the lines} developed ^for polymers ⁱⁿ [30, ^5]. Finally, it has been suggested ^{to us} by D. Roux that this

adsorption transition could be related to the unbinding transition [31, 32, 33] if the wall were

replaced by another membrane.

Acknowledgments.

We would like to thank A. Georges and J. Vannimenus for very interesting discussions and Mélodie Bouchaud for systematically crumpling ^our papers. Long, enthousiastic discussions at Cargèse ^{summer school « On} Growth and Form » with D. Roux were most valuable and

helped ^to improve ^the manuscript. E. Bouchaud wants to thank M. Daoud for discussions on

the problem ^of adsorption ^of polymers. ^We finally ^{thank one} of the referees for pointing ^{out a}

weakness in our first derivation of the upper bound of the ^crossover exponent.

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