Theoretical progress in polymer adsorption, steric stabilization and flocculation

Colloids and Surfaces, 31 (1988) l-29 Elsevier Science Publishers B.V.. Amsterdam -

Printed in The Netherlands

Theoretical Progress in Polymer Adsorption, Steric Stabilization and Flocculation G.J. FLEER, J.M.H.M. SCHEUTJENS and M.A. COHEN STUART Agricultural University, Laboratory for Physical and Colloid Chemistry, De Dreijen 6, 6703 BC Wageningen (The Netherlands) (Received 12 February 1987; accepted in final form 13 May 1987)

ABSTRACT The range of validity of scaling and mean field models for polymers at interfaces is discussed. According to a recent analysis by Schaefer, mean field theories describe the properties of polymers in solution correctly over a wider range of concentration and solvency than does scaling. Only for flexible polymers in very good solvents at concentrations below a few percent are scaling methods preferred. In polymer adsorption, high segment densities occur and therefore mean field theories are more appropriate than scaling. The most detailed mean field theory for polymers at interfaces has been proposed by Scheutjens and Fleer. The distribution of segments as a function of ranking number and, consequently, end effects (tails) are taken fully into account. Several results are given for adsorbing and nonadsorbing polymers at a surface and for the effect of polymers on colloidal stability (bridging, steric stabilization and depletion flocculation). In some cases, a comparison with the outcome of other theories is made. Experimental results, available so far, corroborate the main predictions.

INTRODUCTION

In recent years, considerable progress has been made in formulating theories for polymers at interfaces and for the effect of polymers, both adsorbing and nonadsorbing, on the stability of colloids. Two main groups of theories can be distinguished, commonly referred to as “scaling” and “mean field”. Both approaches are extensions of earlier polymer solution theories. The first more or less complete account of the mean field theory for polymer solutions appeared in the classical work of Flory [ 11, which formed the basis of numerous elaborations in later years. The central idea is that, at concentrations beyond coil overlap, a solution has essentially a homogeneous segment density. Interactions in such a solution may be treated by assuming a mean field which is the same for any segment because of identical local environment. In the original Flory theory, the chemical potentials contain a logarithmic term in the volume fraction which makes the theory, in principle, applicable to the whole concentration range. In related models, a perturbation approach is de0166-6622/88/$03.50

0 1988 Elsevier Science Publishers B.V.

2

veloped, using second and possibly higher virial coefficients to describe the interactions. Such a treatment is equivalent to expanding the logarithm, and limits the applicability to lower concentrations. Mean field models have been very successful in treating many experimental systems, especially when interactions are weak. However, they fail for very good solvents in which highly swollen coils just overlap. In such a solution the spatial fluctuations in segment density invalidate the mean field assumption. For these systems scaling methods have been used, following largely the pioneering work of De Gennes [ 21. According to this concept, the solution is considered as a closely packed array of “blobs”, which are self-avoiding subchains. In dilute solutions (below overlap), the blob size is equal to the radius of gyration, whereas in semidilute solutions (beyond overlap) it decreases with increasing polymer concentration. When polymers adsorb at a surface, a concentration gradient develops. Theories for this case are usually extensions of solution models. Thus, scaling and mean field approacheshave also been proposed for polymer adsorption. In the present overview, we shall discuss some fundamental aspects of both models. First, we examine the limits of applicability of scaling and mean field methods in solution, thereby following a recent publication by Schaefer [ 31. We then concentrate on adsorption theories. Since we arrive at the conclusion that mean field models are more generally valid than scaling, certainly for the high concentrations usually encountered in adsorbed layers, we treat the adsorption theories based on the former concept in more detail than scaling theories. SOLUTION PROPERTIES IN SCALING AND MEAN FIELD THEORY

It is well known that polymer coils in dilute solution have a dimension which depends on the number, r, of segments per chain. The coil size is proportional to f’, where the exponent v is l/2 in ideally poor or B-solvents and 3/5 in very good solvents where the coils are highly swollen. For example, for a purely random flight chain the mean square end-to-end distance < h2> equals l?, where 1 is the bond length. For real chains of sufficient length in a @-solvent, < h2 > = Cm&, where C, is the so-called characteristic ratio, which depends on the flexibility of the chain. Typical values for C, are around 10. In terms of the radius of gyration, R = ( < h2 > /6) 112,we may write R2 = nl?, where the rigidity index n is defined as C,/6; it is equal to the number of bonds in a persistence length and proportional to the number of bonds in a Kuhn segment [ 31. Hence, in a @-solvent, R=lr~~‘~r”~. In better solvents the chains expand by a factor cy, the linear expansion coefficient. Already in the Flory theory, a was expressed in terms of the solvent quality and the chain length. In very good solvents, cy5 was shown to be proportional to (0.5 -x) nr1j2, where x is the Flory interaction parameter. Hence,

I I I I

ideal

swollen

Fig. 1. Illustration of the crossover between ideal behaviour (R - f’.5) and chain swelling (R - r”.6) of dilute chains in a good solvent. The crossover chain length rr is given by Eqn (Za) .

Re = ln

Ii2 r1/2

R, = lnl/5

(I_

2x) l/5,.3/5

r
(la)

c-r-,

(lb)

Here, the indices 8 and g refer to @-solvents and good solvents, respectively, and r, is the crossover chain length separating the domain of Gaussian coils and that of highly swollen self-avoiding chains. Swelling only occurs for sufficiently long chains, because in a short chain the number of segments that meet each other is low. Obviously, the chain length where self-avoidancy starts to play a role depends on chain flexibility ( C, or n) and solvent quality (x) . Schaefer [ 3 ] pointed out that r, is simply obtained by equating (la) and (lb) : rr =n3/(1-2~)”

(2a)

XT =0.5(1-n3/2r-1/2)

(2b)

Equation (2a) gives the crossover chain length at a given x, Eqn (2b) the crossover value of x at a given r. As expected, rs approaches infinity, for @solvents (x = 0.5) and decreases with increasing solvency (lower x) and increasing chain flexibility (lower n) . A schematic representation of the exponent v in R - f’ is given in Fig. 1. In more concentrated solutions, the chains do not behave as individual coils because, above a certain concentration @‘“, they overlap. The overlap concentration is usually defined as the volume fraction corresponding to r segments in a sphere of radius R: qF”‘= (3/4x) rP/R3. Substituting for R as given in Eqn (1) we obtain immediately:

X’XT

-4tn-

&ov -_

3/5(1_Zx)

@a)

-3/5r-4/5 X
In the scaling picture of a semidilute solution [ 21,a closely packed assembly of subchains (“blobs”) is visualized. Each blob is considered as an internally self-avoiding walk. Hence, the solution is inhomogeneous on the scale of the blob size (or correlation length) c. The concentration dependence of l is obtained from the boundary condition r = R,at $= r and by forcing < to follow power law behaviour according to
3 c(

=47t

l&/4(&2x)

--1/4qj-3/4

(4)

>

Experimental evidence for the #- 3/4dependence of c has been obtained [ 4~51. According to Eqn (4)) 5 decreases with increasing #. However, a blob will not remain self-avoiding if the number of segments in the blob becomes too small. As in isolated coils, self-avoidancy in a blob disappears if the number of segments is smaller than r, ( Eqn ( 2a) ) . The transition occurs when < is equal to the radius of gyration of a coil of r, monomer units, given by substituting Eqn (2a) in Eqn (la) (or ( lb) ) . Equating this value of R with < from Eqn (4) leads immediately to the upper concentration limit $ below which swelling takes place and scaling applies: $= (3/47r) n-3(1-2x)

(5)

Above 3, the solution is again nearly ideal and can be described in a mean field. For # just above 3, a perturbation approach using only a second virial coefficient Bz is adequate_,at higher concentrations higher order terms are necessary. Like r, and @‘“, # depends on solvent quality and chain flexibility. For flexible chains the range where scaling applies is wider than for stiff chains. However, in most experimental systems, 4 is of the order of 0.01. Figure 2 (adapted from ref. 3) gives an example for polystyrene in various solvents. In this diagram the upper limit @‘” of the dilute regime (upper left corner) is indicated as the dashed curve; it is given by Eqn (3). The horizontal line in the dilute regime corresponds to x5 as given in Eqn (2b) ; it separates the ideal domain (Gaussian coils) from self-avoiding chains in dilute solution. The transition 3 between the scaling regime and the mean field domain increases with solvency but hardly exceeds $= 0.01, even in the best solvents available for polystyrene. The dotted curve in Fig. 2 corresponds to the transition between the

cyclohexane

EA, MEK \.. -benzene,

‘j

CCI,

‘... : *t.

mean

--ethyl

benzene

field

Fig. 2. Solvency-concentration diagram for polystyrene, adapted from ref. 3. In the dilute regime the coils are Gaussian (x CX~) or swollen (x >x~), where x7 is given by Eqn (2b). If @> c’, the coils overlap. In the semidilute regime (x
domains where, in the mean field model, a perturbation approach (using only B,) is adequate and where higher order terms are needed. The Flory theory [ 11, using no expansion in terms of powers of 6, could in principle be used over the whole range @>$. Note that in Fig. 2 only v and x5 depend on chain length; the curves drawn apply to a degree of polymerization of 3000. For longer chains, @‘” is smaller and x7 closer to 0.5. It is clear from Fig. 2 that the scaling theory has a rather limited range of validity. Moreover, many exponents in scaling are derived by assuming infinitely long chains. Scaling applies over a wider concentration range only for extremely flexible chains, like polydimethyl siloxane. In most experimental systems an adequate mean field description is more relevant. This conclusion is even more pertinent for the case of polymer adsorption because, in the loop region close to the surface, the segment concentration will nearly always exceed the upper limit $ of the scaling regime.

MODELS FOR POLYMER ADSORPTION

As in the modelling of the properties of polymer solutions, there have been two major approaches in describing polymer adsorption, viz. scaling and mean field. We will outline below the fundamental aspects of scaling and mean field descriptions, treating the latter in slightly more detail.

I

\,proximal \ \_ -,-

t

Q

ten trol

\,proximal \

,L -2

-413

central

distal: \ \

1

__distal :\ R

Fig. 3. Qualitative predictions for the segment concentration profiles according to the scaling theory. For terminally anchored polymer (a) the central regime, extending from z = D to z = L, is characterized by a constant concentration. D is the distance between the anchoring points and L the length over which the hairs extend. For physisorbed polymer (b) the segment density in the central regime decays as .ze413.The segment length, 1, and the radius of gyration, R, are indicated.

Scaling approach The scaling viewpoint of polymer adsorption from good solvents has been developed largely by De Gennes, both for terminally anchored chains [ 61 and for physisorbed homopolymers [ 71. The adsorbed layer is divided into three regions: (i) the proximal regime close to the surface where the segment concentration profile is very sensitive to the details of the segment-surface interactions, (ii ) a ( semidilute) central regime where the profile follows a universal scaling law, and (iii) a distal regime where the chains are so dilute that lateral interactions do not play a role and where the profile decays exponentially towards the density of the bulk solution. The predictions are qualitatively illustrated in Fig. 3. We will focus mainly on the central regime. For a dense array of terminally anchored chains (a “brush”), the profile is governed by the length scale D, the lateral spacing between the anchoring points. The laterally compressed chains tend to elongate in the perpendicular direction as a string of (swollen) blobs. The density in the central regime is then essentially independent of z and is only a function of the graft density: @E (l/D)4’3. For physisorbed chains, a different length scale must apply. De Gennes reasons that at any distance z from the wall, the local correlation length for spatial density fluctuations is equal to the distance z itself (“self-similarity”). Because the correlation length c decays as e-314 in a semidilute solution, @(z) is expected to be proportional to zm413. From our preceding discussion on the validity of the scaling theory we might expect that the profile shapes derived by De Gennes are only valid for #(z) -4.

Mean field models

Whereas the scaling theory provides only global information on the properties of adsorbed polymers (which is, moreover, only valid over a rather limited range of conditions), mean field models can give a much more detailed description of polymer adsorption. Over the last decades, several attempts have been made, both in continuum [ 8-111 and in lattice [ 12-141 theories. The “mean field” arises from the presence of many interacting chains in the vicinity of the surface, and is usually viewed as applying only in the directions parallel to the surface. In the perpendicular direction, z, there is a concentration gradient which gives rise to a field which varies with z. Chain expansion perpendicular to the surface may occur as a result of the spatial variation of the field. Since this field in itself is a function of the (a priori unknown) segment concentration profile, a self-consistent field approach is used in most cases, where the solution is found by iteration, such that the equilibrium profile and the field are mutually consistent. In order to introduce the background of the mean field description, we first outline the essential steps of the most detailed version, the lattice theory of Scheutjens and Fleer [ 14,151. Thereafter, it is relatively easy to make a comparison with the other models [ 8-131. In the Scheutjens-Fleer theory the polymer chains are described as walks in a lattice, where each step is weighted with an entropy factor accounting for the local entropy of mixing and an energy factor describing the nearest neighbour interactions. Each lattice site is occupied by either a polymer segment or a solvent molecule. Let us divide the space adjoining the surface into parallel lattice layers numbered 1,2,...,i,..., M. For adsorption at a single wall, i = 1 is the surface layer, and i= M corresponds to a layer in the bulk of the solution. At equilibrium, there is a certain concentration profile {&} for segments and {@ } for solvent, with & + &’= 1 for any i. The weighting factor for any step in this concentration profile depends on the lattice type (i.e. on the number of neighbouring sites in layer i and in the adjacent layers i- 1 and i+ 1) and on the local concentrations in layers i- 1, i and i+ 1. The weighting factor may be broken down into a bond weighting factor &, or &, and a segment weighting factor Pi. Here, & is the fraction of possible steps within the same layer, and A1that into an adjoining layer. In a simple cubic lattice (co-ordination number 6),&=4/6andil,=1/6. The segmental weighting factor, Pi, depends on the local concentrations because the latter determine the probability of placing a segment in layer i. It can be shown [ 14-161 that Pi is given by Pi=@exp[X(



- <&>

)I exp(xs&J

where 6,,i is unity if i= 1 and zero otherwise, and

(6)

8

<$i>

=Al@i-l

+&@i+~l$i+l

(7)

The factor &’ accounts for the entropy of mixing: in a layer where a fraction &’ of lattice sites is occupied by the solvent dami, = lz In &’ per segment (apart from a constant), giving rise to a factor exp (da&k) = @ in the weighting factor. The Boltzmann factor containing x arises from the polymer-solvent interaction according to the Flory-Huggins model, and that with x8 is related to the net adsorption energy per segment. As usual, x and xs may be temperature dependent because of entropic contributions such as vibrations. The parameter xs was first defined by Silberberg [ 121. The angular brackets in the former Boltzmann factor denote a weighted average over layers i - 1, i and i+ 1, according to Eqn ( 7)) and account for the fact that the fraction of contacts of a segment in i with these three successive layers equals &, & and Ai, respectively. The segment weighting factors Pi and bond weighting factors &, and 1, determine the probability of any conformation occurring in the concentration gradient. A convenient way of sampling the contributions of the many possible conformations to the segment density profile is to consider first the distribution of end segments P ( i,r) , giving the statistical weight that the last segment r finds itself in layer i. The position of all other F- 1 segments is undefined. P ( i,r ) is related to the end segment distribution of chains of one segment less. Ifrisini,r-lmaybei1, i or i + 1. Adding one segment with segment weighting factor Pi and one bond with weighting factor A0 (if F- 1 is in i) or il, (if r-l is in i-l or ii-l) readily gives P( i,F) =Pi.&P( i-l,r-1) + Pin ( i,r - 1) + PihIP ( i + 1, r - 1) . In general, for chains of s segments long we have the following recurrent relation. P(i,s)=Pi

(8)

where the angular brackets have the same meaning as in Eqn ( 7). Using Eqn (8) implies that step reversals are allowed, i.e. segments s and s- 2 may occasionally be in the same lattice site. By repeated application of Eqn (8)) P( i,s) may be related to the distribution P( i,l ) of free segments, where P( i,l ) = Pi as given in Eqn (6). In this way the end segment distributions are expressed in the segment density distribution {$i}. Actually, Eqn (8) is just an alternative representation of the matrix notation introduced by DiMarzio and Rubin [171* There is another relation between {@i}and the end segment distributions {P( i,s) }. Let us consider the distribution #ii(s) of the segment with ranking number s in the chain (1,2 ,...,s ,...,F) . Segment s may be considered as the junction of two chain parts 1,2,...,s and r,r - 1,...,s. The statistical weight of the first chain part ending in i is P( i,s), and that of the second part ending in i is P(i,r-s+l). Now, #i(s) is proportional to the product P(i,s)P(i,r-s+l). However, we have to divide by Pi because the weighting factor for segment s

9

(which is in i) is included in both factors and we have to correct for this double counting. Hence, &(s)=CP(i,s)P(i,r-s+l)/P,

(9)

where C is a normalization constant. In the case of a constant bulk solution concentration &, C is equal to q&$-P: where, analogous to Eqn (6)) P, is the weighting factor for a segment in the bulk solution, given by & exp{X( &, - &) } [ 14-161. Equation (9) is generally known as the composition law. Summation of Eqn (9) over all s from 1 to r gives the overall segment density distribution:

WenowhaveinEqns (lo), (9), (8) and (6) a set of self-consistent equations in M unknown &‘s, which may be solved by a suitable numerical iteration procedure. Once {@i} is known, the set of weighting factors {Pi} and from this the probability of any conformation, the distribution of trains, loops and tails (and, for two surfaces, also bridges), the free energy, etc. can be obtained in a straightforward way. In the model described above, the distribution of the segments as a function of the ranking number in the chain is taken explicitly into account. In this respect the theory is more accurate than in most other models [ 8,10-131 that use.approximations to avoid the cumbersome procedure to keep track of the ranking number. Usually, the effect of the ranking number is assumed to be small for sufficiently long chains and to disappear completely in the limit of infinitely long chains. This assumption boils down to the neglect of end effects (tails). As we have shown recently [ 181, end effects are only small for isolated chains on the surface, but at finite solution concentration tails are very important, even for long chains. In order to illustrate this, it is instructive to make a comparison with the continuum models [ g-111. In such treatments, the statistical weight of walks in the concentration profile is described in terms of Green’s functions which obey a second-order differential equation. The end segment distribution function P( i,s) in the Scheutjens-Fleer model is equivalent to such a Green function integrated over all positions of the starting point of the walk. The recurrence relation ( Eqn (8) ) can also be written as a differential equation by expanding P ( i,s) in a Taylor series up to the second order in i [ 191. For purely random walks where all segmental weighting factors are equal, this leads to a simple diffusion equation in the end-segment distribution function, which then results in a Gaussian distribution for each chain. For a step-weighted random walk, a more complicated second-order differential equation is found. Usually, the polymer-solvent interaction is taken into account through a perturbation approach, equivalent to expanding the exponential in Eqn (6) up to first order. In only one case [ 91 was a numerical solution found for the special case of terminally anchored chains with nonadsorbing segments. In all other cases a

10

further approximation has been made by expanding the distribution functions in terms of eigenfunctions and assuming that the correponding eigenvalues are widely spaced so that the largest eigenvalue, corresponding to the ground state, is dominant. The resulting differential equation in the ground-state eigenfunction can now be solved analytically [ 6,7,10,11]. However, we have shown [ 181 that, in the case of finite solution concentrations, at least two eigenfunctions contribute to the end-segment distribution functions. Of the two corresponding eigenvalues, the largest converges to the second one in the limit of infinite chain length so that both eigenfunctions have to be taken into account. Hence, in the composition law (Eqn (9)) a cross-product of these two eigenfunctions occurs. It is this cross-product that describes mathematically the presence of long dangling tails. Therefore, continuum models in the ground-state approximation do not give a satisfactory representation of polymers at interfaces. It is hoped that in the near future more progress will be made in this important area, because analytical expressions (even if only valid in some limiting cases) may give more insight into the various trends than numerical solutions. Another approach is based upon the square gradient equation introduced by Cahn and Hilliard [ 20,211 and adapted for polymer adsorption by De Gennes [ 71. In this method the excess local free energy is broken down into a concentration-dependent term e representing the free energy difference associated with the transfer of molecules from the bulk solution to the adsorbed layer (with local concentration (6) and a term accounting for the concentration gradient. Since the latter term must be positive and independent of the sign of the gradient, only even exponents of a@/& contribute to the free energy. Neglecting terms of the order four and higher and integrating over z gives the total excess free energy: .

00 F”= IF:($) +K(lagag kq dz 5

\

(11)

0

The parameter ICis assumed to be independent of chain length (hence, end effects are also neglected here) but depends on concentration: IC= (24@) -‘. By minimizing F” the equilibrium profile can be obtained. For some cases an analytical solution has been found [ 7,221. We conclude this theoretical section by pointing out that the Scheutjens-Fleer model can be easily adapted for a variety of systems which have a practical relevance. One example is allowing for polymer polydispersity [ 231, and another is the incorporation of electric charges to describe polyelectrolyte adsorption [ 24-261. Extensions to the case of two interacting surfaces are, in principle, straightforward [ 16,271. Recently, the model has been applied to terminally anchored chains [ 281. Currently, work is in progress to describe

11

::pzy,-+_ 0

20

40

1

60

80 100 10 5 i i Fig. 4. Segment density for terminally anchored polymer (a) and physisorbed polymer (b) in a good solvent (x = 0) according to the Scheutjens-Fleer theory. In (a) the chain length F equals 250 and the coverage is one chain per 25 surface sites (0 = 10) ; profiles are shown for adsorbing (x.=1) and nonadsorbing (x8=0) segments. In (b) a profile is given for ~~1000, x8=0.5 and @*=10-3 (0=0.24).

the adsorption of copolymers, both block and random. A quite interesting feature, which is outside the scope of the present paper, is the application of the model to self-assembling structures such as membranes and micelles of surfactant-type molecules (which are, essentially, block co-oligomers) [ 29,301. In all these cases it is, at present, difficult to visualize how scaling or continuum models could cope with these more complicated and/or multicomponent systems. For the moment, we have to live with the drawbacks of a lattice model giving only numerical solutions. SOME ILLUSTRATIVE

RESULTS

In this section we give some illustrative results obtained with the Scheutjens-Fleer theory. In all cases a hexagonal lattice was assumed. As shown before [ 13,141, the lattice type hardly affects the outcome. In the selection of results we shall emphasize the relevance of long dangling tails, because they are very important for practical applications, determining the hydrodynamic thickness of adsorbed layers [ 18,311 and the capability of polymers to stabilize or flocculate colloidal systems. In addition, we shall pay some attention to the interaction of two particles in the presence of both adsorbing and nonadsorbing polymer. Adsorption Let us first compare the general features of scaling and mean field predictions. Figure 4 gives an example of concentration profiles of terminally anchoredpolymer chains and of physisorbed homopolymers in very good solvents

12

(x = 0). These results may be compared with the analogous predictions of the scaling theory as given schematically in Fig. 3. In Fig. 4a two profiles are given for hairs (terminally attached chains) of 250 segments at high coverage (1 hair per 25 surface sites), for adsorbing ( xs = 1) and nonadsorbing ( xs = 0) segments. Near the wall the segment concentration is much higher for adsorbing segments than for x8= 0, but more distant from the wall the profiles are quite close. For this chain length and coverage, a constant plateau as predicted by De Gennes (Fig. 3a) is not yet found, but for longer hairs the central section of the profile is expected to be flatter. Both scaling [ 61 and mean field [ 281 lead to profiles at low coverage that show a pronounced maximum for nonadsorbing segments. This feature has been corroborated by direct neutron scattering experiments [ 321. Qualitatively, both theoretical models agree nicely. A more quantitative comparison is possible. From Fig. 4a it appears that for long hairs the volume fraction in the central regime is about 0.2. Scaling prewhich, for D = 51, gives 0.12. An argument analogous to the dicts $2: (Z/D)4'3, one used by De Gennes leads to @1: l/Din the mean field viewpoint, corresponding exactly to 0.2. Since the concentration in the adsorbed layer exceeds considerably the upper limit, 3, of the scaling regime, the mean field approach is probably more appropriate. Experimental results for very long hairs at high coverage are not yet available, Figure 4b gives an example of a profile for a homopolymer adsorbing from a good solvent. In order to facilitate comparison with scaling, a low adsorption energy (x,= 0.5) was chosen, because the scaling approach applies only near the critical adsorption energy. Qualitatively, Fig. 3b and 4b look similar. A check for z--3/4 dependence is not quite possible: the profile in Fig. 4b cannot easily be subdivided into a proximal and central regime. It has been shown before [ 14,15 ] that at higher values of xs (e.g. xs = 1) , the profile decay is much steeper than in Fig. 4b. Neutron scattering experiments [ 321 give results that are in agreement with mean field predictions, at least semi-quantitatively. Again, we have to bear in mind that in most cases the local volume fraction exceeds 3. Figure 5 gives a schematic representation of the average conformation of adsorbed homopolymers. In extremely dilute solutions where the chains do not have to compete for surface sites, the adsorption energy usually overcompensates for the entropy loss if a bond in the three-dimensional solution is, upon adsorption, restricted to the two-dimensional surface layer Consequently, the chains lay essentially flat on the surface, unless the adsorption energy is quite close to the critical adsorption energy xsc = -In (1 -A,) -12 [ 14,331. Under these conditions, tails are small and end effects are negligible. Also, the conformation and layer thickness do not depend on molecular weight. In dilute and semidilute solutions, with $* of the order of lo-” to 10-l, the chains compete for surface sites and the extension of the adsorbed layer into

13

(semi)

melt

dilute

isolated

chains

Fig. 5. Schematic picture of average adsorbed chain conformations in extremely dilute solution (isolated chains on the surface), dilute and semidilute solutions, and the polymer melt. The adsorbed layer thickness increases sharply with increasing concentration, mainly due to the contribution of tails. Significant tail formation occurs as soon as the chains begin to compete for surface

trains

1

lo

10 *

trains

10 r

10 *

to 3

to c c

Fig, 6. Chain length dependence of the adsorbed amount (upper curve ) and its contribution due to trains, loops and tails, for a good (athermal) solvent (a) and a @-solvent (b) . r= 1000, xg= 1, $*= 10-3.

the solution increases: loops and tails develop. The loop size and, especially, the tail size distribution broaden greatly [ 15]. The layer thickness is determined largely by the tail size [ 18,311. The loop size increases with increasing chain length but levels off at high values of r, whereas the average tail length

14

is approximately proportional to the molecular weight [ 15,181. The ratio of tail length to chain length depends on solution concentration and varies roughly from 0.05 in dilute solutions (1 ppm) to 0.15 (1% ) . For polymers adsorbing from the melt (@,+ 1) , the average tail length is about l/3 of the chain length. Since in this situation nearly two tails are present per molecule, 2/3 of the segments belonging to chains in contact with the surface are in tails. The remaining middle l/3 of the chain consists of very short trains and relatively long loops in alternating order. The average extension of the adsorbed layer is proportional to the square root of the molecular weight. A more quantitative detail of the contribution of trains, loops and tails to the adsorbed layer as a function of chain length is shown in Fig. 6, for a very good solvent (Fig. 6a) and a @-solvent (Fig. 6b). In this example, $,,= 10m3. For x = 0.5, the adsorbed amount (which is expressed in equivalent monolayers) increases, beyond the oligomer region, linearly with the logarithm of the chain length. With better solvents, the adsorbed amount is much smaller and more weakly dependent on molecular weight, leveling off for long chains. Both trends have been corroborated semiquantitatively by experiments on well-defined systems [ 33-361. It can be seen from Fig. 6 that the number of train segments is, for not too short chains, essentially independent of chain length. The number of tail segments increases, in the chain length range and at the concentration shown, approximately proportional to r, but increases more strongly for higher r and/or &. The fraction of tail segments is roughly constant in the former case but tends ultimately to the limit 2/3 for very long chains, this limit being reached earlier for higher concentrations [ 181. The tail contribution does not depend strongly on solvency. The main difference between good and poor solvents stems from the behaviour of the loops, which extend further from the surface and become longer with increasing r in a poor solvent. However, since loops find themselves in the inner part of the adsorbed layer, the root-mean square and hydrodynamic thicknesses do not depend very much on the solvent quality (at the same bulk solution concentration @,,), but are determined mainly by the tail fraction. Figure 7 illustrates the relative contribution of trains, loops and tails as a function of solution concentration. The ordinate axis is normalized by the total amount adsorbed, which increases slowly in dilute solutions but more strongly above &, N 10m2,approaching a value which is proportional to r1j2 for @*-+ 1. Because of this normalization, the fractions plotted in Fig. 7 reflect the average conformation of an adsorbed chain and give the quantitative background of the qualitative features discussed in connection with Fig. 5. The train fraction (lower curves) decreases slightly in dilute solutions and declines rather rapidly in concentrated solutions; in good solvents it is higher than in poor solvents. The fraction of loop segments (difference between upper and lower curves) is nearly constant over a wide concentration region and

15

---------

--

1 9 Fig. 7. Fraction of segments, in trains (bottom), loops (center) and tails (top) for x =O and x = 0.5, as a function of solution concentration. r = 1000, xa = 1.

8

1o-2

1o-L

1o-6

z

606,H

CO-

1 3 8 Fig. 8. Hydrodynamic thickness as a function of adsorbed amount, for an athermal (X = 0, left) and a &solvent (x=0.5, right) and various chain lengths. The curves are dashed for solution concentrations above ~~=O.Ol. The permeability was taken as c (1 -@i) /pi with c/l’= 1 [ 281. &= 1. 0

1

2

decreases only for high q&+. The contribution of tails is small but significant in dilute solutions and increases rather strongly with increasing &, approaching the fraction 2/3 discussed above. The effect of solvency on the tail fraction is

16

relatively minor. In Fig. 8 some information is collected on the hydrodynamic thickness for several chain lengths and two solvency conditions. The hydrodynamic thickness was computed from the profiles by applying the Debye-Brinkman equation for each layer [ 18,311, assuming a permeability proportional to (1 - &) /$i. The thickness is plotted not as a function of #* but as a function of the adsorbed amount, 13.As discussed above, the thickness at the same solution concentration does not depend very much on solvent quality. At a given adsorbed amount, Sh is considerably higher in good solvents because a much higher concentration is needed to obtain the same number of segments on the surface, and thus relatively many more tails are present. Roughly speaking, the horizontal shift between the curves for x = 0 and x = 0.5 in Fig. 8 corresponds to the difference in train and loop segments, contributing very little to the hydrodynamic thickness. For a given solvent quality, a family of S-shaped curves &.,( 19) is found, one for each chain length. In the initial part, at very low concentrations, the chains lay flat and &, is small and independent of r. As soon as the surface becomes saturated, tails develop and & increases steeply in good and more gradually in poor solvents. In the latter case, loops contribute relatively more to the increase of 8. At a certain point, depending on chain length, &., levels off and increases slowly with adsorbed amount in more concentrated solutions. The curves are dashed where $, > 0.01. In the experimentally interesting region for solutions not too concentrated, &, in a given solvent is only a function of the adsorbed amount 8. At constant $,, 8 and, consequently, & increase with chain length. In one particular case it was shown that, at constant &, & is proportional to r”.8, both theoretically and experimentally [ 311. This suggests that, due to the development of long dangling tails, the hydrodynamic thickness may increase more strongly with chain length than the radius of gyration of a polymer coil in solution. Recently, very detailed calculations by Anderson et al. [ 371 have supported the conclusion that dilute extended tails contribute most to the increase in &. This implies that the form of the permeability function for high segment densities is in fact irrelevant, the l/&-dependence being the most important. In Fig. 9 it is shown how the adsorbed amount, 8, and the hydrodynamic thickness, &, depend on xs. Only the region close to the critical value xX (see the arrow) is shown, For high values of x8, both 8 and & are independent of the adsorption energy (although & reaches its plateau value at much lower x. than 0). With decreasing xs the adsorbed amount becomes gradually smaller due to the disappearance of trains and loops. However, the number of segments in tails remains virtually constant upto very near the desorption point [ 181, and ah follows the same trend, decreasing sharply only in the very last part where x8 -xsc 5 0.1. Hence, the desorption point can be measured accurately using hydrodynamic methods. It has been shown [ 381 that polymers may be

17

156,

lo-

- 1.0

5-

- 0.5

0. 01

Fig. 9. Hydrodynamic thickness Bhand adsorbed amount 0 as a function of the adsorption energy x.. The permeability was the same as in Fig. 8. r= 1000, x=0.5, #,= 10m3. 5

I

1

I

I

c-x1o-3 5 ._ tj e

3-

s

Fig. 10. Adsorption fractionation for a polydisperse sample with a Schulz-Flory distribution (solid curve) with F, = 165 and r,,./r,,= 2.06, for a surface in contact with 120 lattice layers of solution. In this case, 87% of added polymer is adsorbed ( r, = 184, r,,Jl;, = 1.48) and 13% remains in solution ( r, = 40, rW/r, = 1.65). X. = 1, x = 0.5.

18

desorbed by changing the solvent composition, by the addition of a so-called displacer. In effect, this is equivalent to lowering the effective adsorption energy of the polymer in the solvent mixture. From the displacer concentration at which the polymer desorbs, the segmental adsorption energy for the polymer in the original solvent (without displacer) can be calculated using a simple equation [ 381. As far as we are aware, this is the only method available so far to determine x8. Microcalorimetry cannot provide this information, as the entropic contributions to xs are ignored and the demixing enthalpies in the loop and tail region are indistinguishable from the adsorption enthalpy. We conclude this section on adsorption with one example for a polydisperse system [ 231. Figure 10 gives the molecular weight distribution of the adsorbed fraction and of that in the solution for a case where the overall size distribution is of the Schulz-Flory type ( rw/rn= 2.06, r, - 165). It turns out that the solution contains predominantly the smaller molecules ( rw/rn= 1.65, r,= 40) and the surface the large ones ( rw/rn= 1.48, r, - 184). A clear fractionation occurs with the long chains adsorbing preferentially. The transition region is relatively narrow, extending over not more than a factor of about three in the chain length. As we have demonstrated before [ 33,391, this preferential adsorption has very important implications for the adsorption isotherm: the isotherms become much more rounded than for homodisperse samples, depend on the area-to-volume ratio, and lie lower than desorption isotherms obtained by dilution with solvent. This last point especially has caused much confusion in the literature because of the (incorrect) inference that it would imply irreversibility of polymer adsorption. We know now that the differences can be largely explained by fractionation effects. This indicates that, in most cases, polymer adsorption is reversible and can be interpreted with equilibrium theories. Stability Recently, the Scheutjens-Fleer model has been extended to the case of two surfaces in the presence of adsorbing polymer [ 16,27,40]. The first issue to discuss is whether, in practice, a complete equilibrium with the bulk solution can be maintained or do some restrictions apply. In a complete equilibrium the chemical potential of the chains in a narrowing gap between the plates is maintained at the same value as in the bulk of the solution. Because of entropic restrictions the chemical potential would increase if no polymer would desorb upon particle approach. Hence, complete equilibrium implies desorption and transport of polymer towards the bulk solution. In many experimental situations (e.g., Brownian collision of polymer-covered particles, or macroscopic mica surfaces at microscopic separation), it is unlikely that there is time for the polymer to escape. In that case it is reasonable to assume that, as soon as

19

-0.031

I

I

I

__I

Fig. 11. Interaction free energy per surface site between two plates with adsorbing polymer. The dashed curve is for complete equilibrium and #*= 10-s. The solid curves apply to restricted equilibrium where the amount of polymer is fixed at its value at a plate separation of 80 lattice layers. The amounts are 3,3.5 and 5 monolayers, corresponding to equilibrium (at M> 30) with a solution of &= lo-“, lo-’ and lo-*, respectively. r= 1000, x =0.5, xS= 1.

the interaction starts, the polymer is trapped in the gap between the particles and only solvent can leave the gap. For such a restricted equilibrium, with a constant amount of polymer in local equilibrium as a boundary condition, the free energy can also be calculated. Figure 11 shows an example of interaction curves for chains of 1000 segments in a @-solvent, for bulk solution concentrations (at separations beyond, say, 80 layers, where the plates do not interact) of 10-12, 10m6and 10V2. The full curves are for restricted equilibrium (the amount of polymer is fixed at the equilibrium value at M = 80 and equals 3,3.5 and 5 monolayers, respectively), the dashed curve (@,-- 10m6) applies to complete equilibrium. In the latter situation only attraction is found, not only for 1 ppm but under all conditions. This conclusion was also reached by De Gennes [ 411 who derived, in a mean field model, the free energy using the square gradient method discussed in the

theoretical section. Hence, if complete equilibrium were to be maintained, colloidal systems could never be stabilized by homopolymers. Since such a stabilization is known to occur in practice, complete equilibrium during particle approach apparently does not occur. The curves for restricted equilibrium conform (at least) qualitatively to the experimental trends. In (very) dilute solutions there is an attractive well due to bridging, deep enough to induce flocculation, Note that the free energy is given in kT units per lattice site, so that the attraction between two particles, involving probably thousands of lattice sites, can exceed the thermal energy considerably. In concentrated solutions the minimum disappears and only repulsion remains. This corresponds to the well-known phenomenon of steric stabilization. Direct force measurements of polymer between mica surfaces (42-441 give results that are in qualitative agreement with our model calculations. Other models for restricted equilibrium have been formulated by De Gennes [ 411 and Klein and Pincus [ 221. De Gennes found, in a mean field model, an exact cancellation between the repulsion due to volume restriction and bridging attraction in good solvents. A scaling analysis, however, led to a net force that is always repulsive. Klein and Pincus applied the Cahn-De Gennes mean field approach for very poor solvents and concluded that there is an attractive region if the local segment concentration is between the binodal limits in the phase diagram. This attraction results only from the polymer-solvent interaction and not from bridging. The fact that bridging does occur is illustrated in Fig. 12, which gives the total number of segments in bridging, adsorbed nonbridging, and free chains (full curves) and the (total) contribution of trains, loops, tails and bridges (dashed curves) as a function of the plate separation. The numbers of train and loop segments are nearly constant over a large range of in~~a~icle distances. Slightly more trains are formed at the expense of loops only when the plates become very close. For tails the decrease occurs at much larger distances, and their decrease is accompanied by the appearance of bridges. As expected, bridges are formed because tails extending from one surface are caught by the other surface. Figure 13 gives some general trends for the stability of colloids in the presence of polymers. In this figure the depth b&h of the minimum (as shown in Fig. 11) is plotted as a unction of the total amoung 8, of polymer between the plates, for various solvency conditions and (for x =O and x = 0.5 ) two chain lengths. ~A11the curves have the same general shape: with increasing Bt the attractive well first becomes deeper until, at et slightly below one monolayer (i.e. half a monolayer per plate), the attraction reaches a maximum. At-still larger amounts of polymer the attraction becomes weaker and disappears altogether, unless the solvent is extremely poor. In athermal solvents, steric stabilization starts above 8,~ 1.5, in @-solvents 3 monolayers are required, and

21

5e al ” 40

bridging

chains

E 3-

2trains -------------

l-

tails free chains

0

10

20

30

40

50

60

M

Fig. 12. The composition of polymer between two plates as a function of interparticle distance. The polymer is subdivided into bridging, nonbridging, and free (nonadsorbing) chains (solid curves). The dashed curves give the amounts of segments in trains, loops, tails and bridges. 8, =5 monolayers (corresponding to #, = 5. 10m3at large M) . r = 1000, x = 0.5, xS= 1.

(

1

2

3

4

et

5

Afmin kT -0.;

-0.1

---_

r =lO,OOO

-0.c

Fig. 13. The depth of the free energy minimum (see Fig. 11) as a function of the total amount of polymer between the surfaces, at various solvency conditions. If the minimum is zero (e.g. above 13,= 1.5 for x = 0) there is only repulsion. r = 100 (solid curves) or 10,000 (dashed curve for ,y=0.5). For 1 =O the curves for r= 100 and 10,000 coincide. x.= 1.

22

in very poor solvents the tendency of the polymer to phase separate prevents stabilization: the particles will always attract each other due to osmotic forces. The chain length dependence is seemingly rather weak: the solid ( r= 1000) and dashed ( r = 10,000) curves for x = 0.5 nearly coincide. For x = 0 they are even indistinguishable. However, one should realize that the high coverage needed for stabilization can be much more easily reached with long chains than with short ones: high-molecular-weight polymer is a much better stabilizer than low M samples. For example, at x s= 1 the maximum coverage obtainable with r= 100 at x =0.5 is around 3, not quite enough to obtain stability. Chains of 10,000 units can easily give 5 or more monolayers on the plates. Depletion In recent years there has been an increasing interest in the effects of nonadsorbing polymer on the stability of colloids [ 45511. In this case, the loss of conformational entropy close to the surface is not compensated for by an adsorption energy and a depletion zone develops that is void of polymer. The thickness of the depletion zone depends on the solution concentration: in more concentrated systems the osmotic pressure pushes the chains closer to the surface and forces the depletion thickness to decrease. If two particles surrounded by depletion layers approach each other so that there is a separation smaller than twice the depletion thickness, the gap between the particles contains only solvent and an attraction occurs, induced by the osmotic pressure of the outside solution. Several models have been proposed for depletion [ 48-511. Qualitatively, they agree on the main features. We will first discuss some mean field results and then make a comparison with scaling. In the Scheutjens-Fleer theory, depletion can be handled simply by making xs equal to zero (or making it negative). The depletion thickness, A, can be expressed in the (negative) excess amount of polymer, 8,,, which is again given in equivalent monolayers. At an interparticle separation 24, the excess amount equals -2A@, so that 24~ - 6,X/@,. Figure 14 shows an example of interaction free energies between plates for two solution concentrations. The plots are linear over a wide range of plate separations. The extrapolation of this linear region to Af=O gives essentially M= 24. The interaction curves in the range 0
24

understood result. At a concentration considerably below Q>“’(indicated by the arrows in Fig, 15) the coils start to overlap and d decreases. In this region the molecular weight dependence of the depletion thickness also diminishes. The scaling description for depletion produces qualitatively the same results [ 491. Just as the repulsion between polymer (sub) chains in a (semi) dilute solution causes spatial density fluctuations with a characteric size 4, the repulsion between the polymer and wall leads to an inhomogeneity (the depletion zone) of the same size: d = r. This means that in good solvents and for $P”< (b< 3, we expect A/R = ( @*/&,,,) -‘14. A direct comparison with the curves in Fig. 15 is not possible because these apply for x = 0.5. Results for x = 0 show a more gradual decrease of A with &, but follow the same general trends. As stated above, the depletion attraction for spheres is proportional to fld2. The analysis according to the Scheutjens-Fleer theory has been given before [ 511. It is found that the depletion attraction increases with & in dilute solutions (where A is constant ) , but passes through a maximum as a function of $*. In very concentrated solutions the depletion attraction may again become too small to induce flocculation and the system is again stable. Napper [ 50,523 has termed this phenomenon depletion stabilization. Since the stability is not a consequence of some repulsive barrier but simply of too weak an attraction, the term depletion restabilization would be more appropriate. The scaling theory does not predict restabilization: l7, in good solvents is proportional to $*‘I4 [ 21 and A to @**-3’4 [ 491, leading to l7J” - @,,‘j4. Hence, the depletion attraction should increase monotonously with &, which is only true in solutions not too concentrated. Usually, the restabilisation is found above 3, beyond the region where scaling is valid. The results discussed so far are restricted to the stability of hard particles in the presence of a nonadsorb~g polymer. Experimentally, “soft” particles are more often used, e.g. when the particles are surrounded by a layer of grafted hairs. In this case, the definition of the depletion thickness is more complicated because some interpenetration and/or compression of the hairs by the free polymer may occur [ 531. Work is in progress to obtain a theoretical description for this situation as well. Some aspects of the depletion near hairy surfaces have been discussed by De Gennes on the basis of scaling arguments [ 61. CONCLUSIONS

Scaling methods are adequate for weakly overlapping, long flexible chains in good solvents where mean field models produce incorrect results. However, for the adsorption problem scaling is in most cases not a useful alternative because the densities encountered in adsorbed layers are often an order of magnitude too high for scaling laws to apply. In these situations mean field theories give a better description, which is borne out by comp~son with e~erimen~l data.

25

Most treatments have neglected end effects, e.g. by assuming one dominating eigenfunction. Although this enables one to find analytical solutions, this approximation is not warranted for polymers adsorbing from solutions of finite concentrations. One must resort to numerical models, of which the Scheutjens-Fleer theory is the most detailed version. In principle, it is valid for any chain length and any solution concentration and takes into account all possible chain conformations, including those with tails. It has been shown that tails are very important, e.g., for the hydrodynamic thickness of adsorbed layers, for steric stabilization, and for bridging flocculation. Another advantage of this theory is that it can be easily extended to a variety of systems, such as polydisperse samples, mixtures of different polymers, grafted chains, polyelectrolytes, nonadsorbing polymers, copolymers, etc., not only at one surface but also between two surfaces and for other geometries (using spherical or cylindrical lattices).

REFERENCES

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

P.J. Flory, Pinciples of Polymer Chemistry, Cornell University Press, Ithaca, NY, 1953. P.G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, NY, 1979. D.W. Schaefer, Polymer, 25 (1984) 387. M. Daoud, J.P. Cotton, B. Farnoux, G. Jannink, G. Sarma, H. Benoit, R. Duplessix, C. Picot and P.G. De Gennes, Macromolecules, 8 (1975) 804. A. Lapp, C. Picot and C. Strazielle, J. Phys. Lett., 46 (1985) L-1031. P.G. de Gennes, Macromolecules, 13 (1980) 1069. P.G. de Gennes, Macromolecules, 14 (1981) 1637. C.A.J. Hoeve, J. Polym. Sci., C30 (1970) 361. A.K. Dolan and S.F. Edwards, Proc. R. Sot. London, Ser. A, 337 (1974) 509; 343 (1975) 427. I.S. Jones and P. Richmond, J. Chem. Sot. Faraday Trans. 2,73 (1977) 1062. P.R. Gerber and M.A. Moore, Macromolecules, 10 (1977) 476. A. Silberberg, J. Chem. Phys., 48 (1968) 2835. R.J. Roe, J. Chem. Phys., 60 (1974) 4192. J.M.H.M. Scheutjens and G.J. Fleer, J. Phys. Chem., 83 (1979) 1619. J.M.H.M. Scheutjens and G.J. Fleer, J. Phys. Chem., 84 (1980) 178. J.M.H.M. Scheutjens and G.J. Fleer, Macromolecules, 18 (1985) 1882. E.A. DiMarzio and R.J. Rubin, J. Chem. Phys., 5.5 (1971) 4318. J.M.H.M. Scheutjens, G.J. Fleer and M.A. Cohen Stuart, Colloids Surfaces, 21 (1986) 285. H.J. Ploehn, personal communication; see also M.A. Cohen Stuart, NATO workshop Future Directions in Polymer Colloids, in press. J.W. Cahn and J.E. Hilliard, J. Chem. Phys., 28 (1958) 258. J. Cahn, J. Chem. Phys., 66 (1977) 3667. J. Klein and P. Pincus, Macromolecules, 15 (1982) 429. S.P.F.M. Roefs and J.M.H.M. Scheutjens, Macromolecules, in preparation. H.A. van der Schee and J. Lyklema, J. Phys. Chem., 88 (1984) 6661. J. Papenhuizen, H.A. van der Schee and G.J. Fleer, J. CoIIoid Interface Sci., 104 (1985) 540. O.A. Evers, G.J. Fleer, J.M.H.M. Scheutjens and J. LykIema, J. Colloid Interface Sci., 111 (1986) 446.

26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

G.J. Fleer and J.M.H.M. Scheutjens, J. Colloid Interface Sci., 111 (1986) 504. T. Cosgrove, T.G. Heath, B. van Lent, F.A.M. Leermakers and J.M.H.M. Scheutjens, Macromolecules, (1987) in press. F.A.M. Leermakers, J.M.H.M. Scheutjens and J. Lyklema, J. Biophys. Chem., 18 (1983) 353. F.A.M. Leermakers, P.P.A.M. van der Schoot, J.M.H.M. Scheutjens and J. Lyklema, 6th Int. Symposium on Surfactants in Solution, New Dehli, 1986. M.A. Cohen Stuart, F.H.W.H. Waajen, T. Cosgrove, B. Vincent and T.L. Crowley, Macromolecules, 17 (1984) 1825. T. Cosgrove, T.G. Heath, K. Ryan and B. van Lent Polym. Comm., 28 (1987) 64. J.M.H.M. Scheutjens and G.J. Fleer, in TbF. Tadros (Ed.), The Effect of Polymers on Dispersion Properties, Academic Press, London 1982, p. 145. G.J. Fleer and J.M.H.M. Scheutjens, Adv. Colloid Interface Sci., 16 (1982) 341. C. Vander Linden and R. Van Leemput, J. Colloid Interface Sci., 67 (1978) 48. M. Kawaguchi, K. Hayakawa and A. Takahashi, Polym. J., 12 (1980) 265. J.L. Anderson and J.O. Kim, J. Chem. Phys., accepted for publication (1987). M.A. Cohen Stuart, G.J. Fleer and J.M.H.M. Scheutjens, J. Colloid Interface Sci., 97 (1984) 515,526. M.A. Cohen Stuart, J.M.H.M. Scheutjens and G.J. Fleer, J. Polym. Sci., Polym. Phys. Ed., 18 (1980) 559. G.J. Fleer and J.M.H.M. Scheutjens, Croat. Chem. Acta, (1987)) in press. P.G. de Gennes, Macromolecules, 15 (1982) 492. J. Klein, Adv. Colloid Interface Sci., 16 (1982) 101. J.N. Israelachvili, M. Tirrel, J. Klein and Y. Almog, Macromolecules, 17 (1984) 204. J. Klein and P.F. Luckham, Nature (London), 308 (1984) 836. B. Vincent, P.F. Luckham and F.A. Waite, J. Colloid Interface Sci., 73 (1980) 508. J. Clark and B. Vincent, J. Chem. Sot. Faraday Trans. 1,77 (1981) 1831. H. de Hek and A. Vrij, J. Colloid Interface Sci., 84 (1981) 409. P.R. Sperry, H.B. Hopfenberg and N.L. Thomas, J. Colloid Interface Sci., 82 (1981) 62. J.F. Joanny, L. Leibler and P.G. de Gennes, J. Polym. Sci., Polym. Phys. Ed., 17 (1979) 1073. R.I. Feigin and D.H. Napper, J. Colloid Interface Sci., 75 (1981) 525. G.J. Fleer, J.M.H.M. Scheutjens and B. Vincent, ACS Symposium Series, 240 (1984) 245. D.H. Napper, Polymer Stabilization of Colloidal Dispersions, Academic Press, New York, 1983. B. Vincent, J. Edwards, S. Emmett and A. Jones, Colloids Surfaces, 18 (1986) 261.

27

DISCUSSION T. COSGROVE (Bristol University, Bristol, United Kingdom) Recent data by SANS for terminally attached polyethylene oxide on a polystyrene latex with xs xBcare well reproduced by modifications to the SF theory but not the scaling theory. This latter discrepancy is partially due to the low molecular weight of the sample. However, even for physically adsorbed PEO of MW lo6 on latex with a p value < 0.1, there is no indication whatsoever of the scaling prediction of @N z-413. WILLIAM B. RUSSEL (Princeton University, Princeton, NJ, U.S.A.) The arguments about the domains of validity of the scaling and mean field theories are clear. However, the test can be made quantitatively by using a polymer solution theory which is valid over the entire range of concentrations. For example, the screening lengths and solution free energies derived by Muthukumar and Edwards (1982)) essentially interpolate between the dilute and concentrated limits while correctly preserving the scaling law at semi-dilute concentrations. Then comparisons between predictions from the existing formulation and those with the more general expression for the free energy and effective segment length, would settle the question. With respect to the prediction that tails account for two thirds of the chain in the melt, is there an effect of the lattice parameter? G.J. FLEER (Agricultural University, Wageningen, The Netherlands) Yes, there must be a dependence on the lattice type since if 1, is taken to be zero there can be no tails. However, the difference between a cubic and hexagonal lattice is only small. In passing, we note that the 2/3 tail fraction corresponds closely to that calculated by Roe [J. Chem. Phys., 43 ( 1965) 1591; 44 (1966) 42641 for single chains at the adsorption/desorption transition. In both cases there is essentially a homogeneous segment density. Although the critical adsorption energy xsc is a function of the lattice type, the lattice has only a small effect on the chain conformation at the critical point. SANJAY PATEL (University of Minnesota, Minneapolis, MN, U.S.A.) According to you (and Schaefer) scaling arguments should be valid over a very limited (and low) concentration range. On the other hand, measurements of osmotic pressure as a function of polymer concentration by J. Noda et al. show good agreement with scaling predictions for polymer concentrations up to about lo-15%, i.e. they see the 2.25 power rather than the 2.0 power expected from mean field theories. How do you reconcile the two facts?

28

G.J. FLEER (Agricultural University, Wageningen, The Netherlands) We were not aware of this work and would be interested in further details. We note that the cross-over concentrations in Schaefer’s analysis are only approximate because, obviously, there must be a transition region where both models have some validity. Possibly, a more quantitation description could be obtained following Prof. Russel’s suggestions. A. SILBERBERG (Weizmann Institute of Science, Rehovot, Israel) Extremely dilute solutions are only very poorly represented by a mean field approach. This affects the reference system in such cases. It is obvious, moreover, that the contribution from tails in adsorbed layers is most interesting only in case of very high molecular weight and dilute bulk solution when the adsorbed segments beyond the loop region are predominantly in relatively isolated segment clouds arising from the tails. Since the effective hydrodynamic thickness is dominated by the distribution of these outward segments, should not the tails be considered by some other approach? G.J. FLEER (Agricultural University, Wageningen, The Netherlands) The adsorption from dilute solution depends only very weakly on the chemical potential (concentration) of the polymer. Moreover, in the tail region Pi N P, and, hence, the tail segments feel the same field as segments in the bulk solution. The reference system as such is unimportant, it is the difference with the reference system that counts. Tails are not only important for high molecular weights. The fraction of tail segments is considerable, even for relatively short chains, see, e.g. Fig. 6. It is true that tail swelling is not accounted for in our model, except in the loop region where the density gradient is steep. However, tails do not expand when they are shorter than rr. Since the fraction of tail segments is around 0.05r per tail, tail expansion will only occur for r>,20r,. Because r, is of the order of 500 (if x = 0.4, e.g. polystyrene in ethylbenzene), swelling of tails is not important for chain lengths below r N lo4 (MW 1: 106). Another approach might be able to predict the extension of tails more accurately, provided that it could predict the fraction of tails as well. Finally, note that the segment density in the tails region is not low when two polymer layers interact. J. LYKLEMA (Agricultural University, Wageningen, The Netherlands) People who are not very familiar with polymer adsorption theory may wish tom have a few fairly simple guidelines as to how and where scaling theories are appropriate to give a first orientation of distributions in adsorbed layers. Could you formulate such “rules”?

29

G.J. FLEER (Agricultural University, Wageningen, The Netherlands) It will be very difficult to formulate general rules. From Schaefer’s analysis it appears that scaling laws apply for concentrations below - 1% in most practical systems. That would suggest that certainly in the train and loop region the mean field model is more adequate, and that for dilute tails of sufficient length appropriate scaling laws might be developed.