Neutron scattering study of water-soluble polymers adsorbed on surfactant micelles

Colloids Elsevier

and Surfaces, 13 (1985) Science Publishers B.V.,

19-33 Amsterdam

19 -Printed

in The Netherlands

NEUTRON SCATTERING STUDY OF WATER-SOLUBLE ADSORBED ON SURFACTANT MICELLES

POLYMERS

B. CABANE Laboratoire de Physique 91405 Orsay (France)

des Solides,

associL au CNRS,

UniversitB

Paris-Sud.

R. DUPLESSIX CRM, 38042

6 rue Boussingault, 67083 Grenoble Cedex (France)

(Received

30 January

1984;

Strasbourg

accepted

(France)

20 March

and ILL,

156 X,

1984)

ABSTRACT The interaction in water of large, non-ionic macromolecules (PEO) with small anionic surfactant micelles (SDS) has been studied using neutron scattering. In dilute solutions of PEO+SDS, only isolated macromolecules adsorbed on the surfaces of spherical micelles are found. Depending on the amount of SDS in the solution, each macromolecule binds to one, two . . . P micelles. The first few micelles associated with a macromolecule are bound with a free energy of 10-20 kT; subsequently, the gain in free energy, AF, per additional bound micelle decreases as the repulsions between micelles within a PEO + SDS aggregate become more important. As long as AF>kT, all the available micelles are bound to macromolecules and equally distributed among them. Then, beyond a stoichiometric number of micelles per PEO + SDS aggregate, AF becomes smaller than kT; if more micelles are added to the solution, they either remain free or are bound only very weakly to the aggregates.

INTRODUCTION

Polymer

adsorption

on micelles

Many water-soluble macromolecules can adsorb on surfactant micelles [l-5]. Because such micelles are small (spheres of radius 20 a) compared to the dimensions of the macromolecules (radii of gyration in the range 100-1000 a), this adsorption differs from the usual situation where polymer layers adsorb on macroscopic surfaces [6]. In the latter case, the macromolecular chains are known to form a semi-dilute layer whose thickness is determined by a balance between the surface attraction and a repulsion arising from the confinement near the surface. The same forces operate here, but the geometries are very different by virtue of the small dimensions of the micelles. The simplest situation which can be considered is that of one flexible

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20

chain adsorbed on one micelle. Because the dimensions of the micelle are small, it is difficult to confine a large macromolecule near its surface; indeed, Alexander [6] has shown that polymer molecules which are large enough should resist the confinement and remain in the three-dimensional, uncompressed regime. However, this simple situation is not the general case. If enough surfactant is available in the solution, large polymer molecules may escape the confinement by adsorbing much more surfactant than the amount corresponding to one micelle of radius 20 a [3]. The adsorption actually saturates at a fixed value of the weight of surfactant/weight of polymer ratio. This stoichiometry depends on the particular system which is considered (rigidity of the polymer, hydrophile-lipophile balance (HLB) of the surfactant, ionic strength of the solvent), but not on the length of the polymer chain. Materials In the present work, dilute solutions of a poly(ethylene oxide) (PEO) and sodium dodecylsulphate (SDS) in water at high ionic strength (0.4 M NaBr) are considered. A dilute solution of a PEO is one where the macromolecules remain well separated from each other. This puts a constraint on the concentration and molecular weight of the PEO. We use a PEO which has been fractionated by chromatography in water (Toyo Soda SE15); the average molecular weight given by the manufacturer is 150 000, with a dispersity, M,/M, = 1.05. When this product is dissolved in water according to our procedure, a macromolecular solution is obtained whose average molecular weight M,, as measured by viscometry, is 135 000; the average radius of gyration of the macromolecules is 200 a [ 31. For these macromolecules, the crossover concentration where the volumes associated with their radii of gyration begin to overlap is y* = 5 X 10m3 g cm-‘. Dilute solutions are limited to PEO concentrations y
21

on isolated macromolecules, it is also necessary to choose an appropriate range of surfactant concentrations and ionic strengths. In our experiment, the SDS concentration is varied between 6 X 10m4 and 4 X lo-’ g cme3; a high ionic strength is maintained by adding 0.4 M NaBr. In such conditions, most SDS molecules form globular micelles with an average aggregation number of 100; the distribution of their aggregation numbers can be approximated by a Gaussian distribution with a width of 40% [9] ; the concentration of SDS molecules, which do not take part in micelles, can be estimated from the CMC, which is close to 10m4 g cmm3 at this ionic strength. Interaction

diagram for dilute solutions

of PEO and SDS

Figure 1 shows a diagram for the binding of PEO to SDS in water at high ionic strength (0.4 M NaBr). This diagram is common to all PEO polymers of molecular weight larger than 3 X 10m4; on the other hand, it is quite sensitive to the ionic strength. The boundaries which are shown in this diagram do not separate different thermodynamic phases, as the sample is always a one-phase dilute solution of PEO and SDS. Instead, line 1 marks a critical surfactant concentration below which no binding takes place; line 2 marks the saturation discussed above; the SDS added beyond this line forms free micelles

[21.

Fig. 1. Boundaries for the binding of PEO + SDS in water + 0.4 M NaBr. For increasing surfactant (SDS) concentrations, line 1 marks the onset of the binding and line 2 its saturation. The amount of surfactant bound to the polymer at the stoichiometry is when the PEO macromolecules have a molecular weight M, = 135 000, given by x,-x,; the mass of bound SDS per PEO macromolecule is 756 000, much larger than that of a regular SDS micelle, which is about 28 000 for this ionic strength.

22

Also shown in Fig. 1 are the three classical paths across this diagram. Parth a-+a’ represents the progressive saturation of the polymer with surfactant; this is the route usually followed in thermodynamic experiments such as equilibrium dialysis [lo] or surface-tension measurements [ 111. Path b-tb’ represents the dilution of saturated aggregates of PEO + SDS along the stoichiometric line. Through neutron-scattering experiments, we have shown that all the aggregates present in a stoichiometric solution share the same composition, which remains identical with the overall composition of the solution [3]. Notice that the solvent used for dilutions along the stoichiometric line is a solution of SDS at the critical concentration for the formation of aggregates (x,). Finally, path WC represents the dilution of saturated aggregates in a solvent made of excess surfactant micelles; light-scattering experiments along this path, but for a different system, have been performed by Boscher et al. [ 121.

Structure

of stoichiometric

aggregates

The structural model which emerges from the results of neutron-scattering [3] and NMR [2] experiments is as follows. In each stoichiometric aggregate, the surfactant (SDS) molecules form a cluster of small spherical micelles. The polymer is adsorbed on these micelles: 10% of its segments are in direct contact with the micellar surfaces, while the others form loops or strands joining two micelles. The distances between neighbouring micelles

Fig. 2. A two dimensional representation SDS micelles, each containing about 100 adsorbed on the surface of these micelles.

of a PEO + SDS aggregate. The black dots are SDS molecules. The polymer (PEO) is weakly

23

within one aggregate are determined by a competition between intermicellar forces (electrostatic repulsions + Van der Waals attractions) and the free energy of the polymer (surface attraction + configurational entropy). When the second virial coefficient for the intermicellar interactions is zero, the distribution of micelles within one aggregate closely follows the Gaussian distribution of the free polymer. Figure ‘2 shows a tentative picture of such an aggregate. The main purpose of this paper is to examine the validity of this model in non-stoichiometric solutions: what happens when the system is either partially depleted in SDS micelles, or flooded with an excess of micelles? We report here neutron-scattering experiments on solutions located along line a-+a’ of the interaction diagram shown in Fig. 1. The methods are similar to those previously described; however, the present experiments are limited to the range of small scattering vectors which yields the average mass and size of the scattering particles. (The spectrometer was the Dll, at ILL, with a wavelength spread of 5076.) RESULTS

Distribution

of micelles among the aggregates

The first problem which must be considered for non-stoichiometric solutions if the possibility that they contain more than one class of particles. Depending upon the composition of the solution, a variety of nonstoichiometric aggreagtes containing 1, 2, . . . P micelles might coexist with free macromolecules or with excess micelles. In such a case, the intensity scattered at very small angles should be a sum of independent squares corresponding to each class of particles, thus I (Q + 0) = ni (pi- P.J*u~ M,Z+ nj (pi- P~)“u~Mj? + . . . . .

(1)

where ni is the number of particles of mass M and specific volume ui, pi is the average density of scattering length for these particles and ps that of the solvent. By varying one of the parameters which control Eqn (l), and by measuring the corresponding variation of I (Q-to), one can find out whether the sum actually contains many independent terms or only a single one. The classical way is to keep the compositions and numbers of particles constant, but vary ps: this is the contrast variation method, which we used previously for stoichiometric solutions along the path b+b’ of the interaction diagram [3] . Here we use the reverse method, where the composition of the solution is varied along the path a-ta’ of the diagram, and the solvent remains the same. To make things simple, we use a solvent which matches the scattering density of the macromolecules exactly; then the scattering is produced exclusively by the surfactant. Thus, in Eqn (l), all the intensive parameters Pi, ui take the values pm, Um of the surfactant

24

micelles; the extensive parameters tant in each class of aggregates. I (Q -+ 0) = (p,

- ps)2 u,’

[n$‘:

Mi correspond + njMj2 + . . .]

to the

mass

of surfac-

(2)

We shall now use this equation to predict the behaviour of the scattered intensity when the composition of the solution is varied along path a-fa’ of Fig. 1. Consider the following three limiting cases: (a) Over a range of composition, each solution contains only one class of PEO + SDS aggregates. Then Eqn (2) reduces to I (Q -+ 0) = (p,

- pJ*

um2 nMZ

(3)

When the composition of such a solution is varied along path a+a’ of Fig. 1, the mass M of surfactant per aggregate should be proportional to the surfactant concentration X, and the scattered intensity should be proportional to the square of 3~. (b) The solutions contain stoichiometric PEO + SDS aggregates in equilibrium with free PEO macromolecules. This might occur in solutions formed with SDS concentrations x: below the stoichiometry, if, instead of being equally distributed among all PEO macromolecules as in (a), the available SDS were concentrated on a few of them. Only the micelles of PEO + SDS aggregates would contribute to the scattering: indeed, the PEO macromolecules do not scatter in the solvent chosen for this experiment. Thus the behaviour predicted for the intensity would still be given by Eqn (3). However, for such solutions, a rise in the SDS concentration x would increase the number of PEO + SDS aggregates at the expense of the free PEO macromolecules; the mass of each aggregate would remain constant. Thus in Eqn (3) the mass M would remain fixed, while the number II of scattering particles would be proportional to the SDS concentration X; accordingly, the intensity should be proportional to X. (c) The solutions contain stoichiometric PEO + SDS aggregates in equilibrium with excess SDS micelles. This might apply to solutions formed with SDS concentrations x beyond the stoichiometry. In this case, Eqn (2) reduces to 1

(Q- 0) = b, - P,)~u,~b,M; +q M;l

(4)

where n, is the number of stoichiometric aggregates, M, is the mass of SDS in each one, and nf and Mf refer to the free SDS micelles. In such a case, a rise in the SDS concentration x will only cause an increase in the number Izf of the free micelles; accordingly, the intensity should be a linear function of X. Our data taken along a+a’ of the interaction diagram are shown in Fig. 3. Before the stoichiometry (left-hand side) the intensities rise proportionally to the square of the SDS concentration X; this rise covers nearly one decade in composition and two decades in intensity. This is the behaviour predicted above for case (a); thus case (b) is ruled out. Conse-

25

J/o

l/b

Composition

112

1

/Stolchlometry

2

4

8

=

x/x2

Fig. 3. Intensity of neutrons scattered at very small angles by dilute solutions of PEO (M = 135 000) and deuterated SDS in a solvent made of 82% H,O, 18% D,O, and salt (0.4 M NaBr). In this solvent the scattering is produced exclusively by the surfactant. The composition of the solution is varied along the path a-a’ of Fig. 1.

quently, in PEO + SDS solutions located before the stoichiometry, the available SDS is equally distributed among all the PEO macromolecules to form non-stoichiometric aggregates. It then makes sense to calculate the mass M of surfactant in such an aggregate. Figure 4 shows that, before the stoichiometry, M is indeed proportional to the surfactant concentration X. At the stoichiometry, M = 726 000, while the mass of one micelle within the aggregate is in the vicinity of 26 900; thus a stoichiometric aggregate made with a PEO of molecular weight 135 000 contains about 27 micelles. Beyond the stoichiometry, the experimental intensities continue to rise, but at a much reduced rate. This is in qualitative agreement with the idea that the aggregates are saturated with SDS micelles and that free SDS micelles are formed; thus the average mass per particle decreases (Fig. 4). Yet, because the free micelles are of a much smaller mass than the aggregates, they contribute little to the scattered intensity; according to Eqn (4), the rise in the intensity should then be slower than the observed one, and the break in slope at the stoichiometry should be sharper. This is indicated by the dotted line in Fig. 3. Therefore, the aggregates must be

26

I

I

11

1lll,

I

‘12’7 ‘/lb ‘b CornpositIon

‘/4 ‘/2 ’ /S toichiometry

2 x/x2

’

8

Fig. 4. Average mass of surfactant per aggregate in dilute solutions of PEO and SDS. It is argued that, when a solution contains less surfactant than the stoichiometry, all its aggregates hold the same mass of surfactant. The point at x/x, = l/27 is the calculated value for an aggregate which would contain a single micelle, assuming that this micelle had the same aggregation number as the micelles contained in the stoichiometric aggregates. Solvent: 82% H,O, 18% D,O, 0.4 M NaBr. Polymer: Toyo Soda “SE 15”, M, = 135 000.

able to adsorb some additional micelles even beyond stoichiometry where free micelles begin to form. Effect

of

the adsorbed

micelles

on the dimensions

the thermodynamic

of the macromolecules

The next problem to consider for non-stoichiometric aggregates is that their geometry might change with the number of micelles adsorbed on each macromolecule. As mentioned in the Introduction, the adsorption of polymers on macroscopic surfaces results in a compression of the polymer chains near the surface. Here the situation is different; in addition to the free energy of the polymer (surface attraction + configurational entropy) we must also consider the interactions between the adsorbed micelles (electrostatic repulsions + Van der Waals attractions). The balance of these forces determines the maximum number of micelles which can be adsorbed within one polymer chain, i.e., the stoichiometry; but for a fixed number of such micelles, it will also control the actual dimensions of the polymer chain. For example, it is possible to obtain an expansion

27

of the polymer upon adsorption of micelles if the repulsions between micelles are strong. Figure 5 presents our data for the opposite case, where these repulsions are screened by the added salt up to the point where they are approximately balanced by the Van der Waals attractions.

Number

of micelles

per

aggregate

Fig. 5. Radii of gyration R, for aggregates made of PEO (Mv = 135 000) and deuterated SDS. (o), Solvent: D,O + 0.4 M NaBr; scattering is produced almost exclusively by the polymer; the measured radius R,_h is that of the PEO chain in a PEO + SDS aggregate. (*), Solvent: 82% H,O + 18% D,O; scattering by the surfactant only; the measured radius R, is that for the distribution of SDS micelles in an aggregate. (---), Prediction for R,, assuming a random distribution of the micelles along the PEO chain. The composition of the solution is varied along the path a-a’ of Fig. 1; the point at value for one micelle per aggregate. xix, = l/27 is a calculated

Here also we have used the contrast-matching technique to measure separately the radius of gyration of the polymer or that of the surfactant in the aggregate. For the polymer, the measured radii indicate that the PEO chain undergoes a slight compression when it adsorbs a small number of micelles. This is in accord with the low value of the free energy of interaction between PEO and SDS (0.3 kT per SDS molecule, according to Ref. [ 21).

Distribution

of adsorbed micelles within one aggregate

The radii of gyration measured for the surfactant within one aggregate are remarkable in many respects. First of all, for stoichiometric aggregates,

the radius of the surfactant (-210 A) is identical with that of the polymer; thus, the adsorbed micelles are distributed throughout the aggregate. For aggregates containing fewer micelles, the radius of the surfactant remains nearly constant: even when there are only 6-7 micelles per aggregate, the radius of the surfactant is still 198 A (Fig. 5). Thus, changing the number of micelles per aggregate only affects the magnitude of the spatial density of such micelles, but not the shape of their distribution. DISCUSSION

The data presented in Fig. 4 indicate that, before the stoichiometry, all the SDS micelles are strongly bound to the PEO macromolecules and are equally distributed among them; beyond the stoichiometry, they show a weak binding of a few additional micelles. These observations can be rationalized in the following way. The first micelles bound to a PEO macromolecule gain a large amount of (negative) free energy through the adsorption of PEO segments; the drop in chemical potential can be estimated by comparing the critical concentration for the formation of PEO + SDS aggregates, x,, with the CMC of pure SDS, x0; this yields Ap = 0.3 kT per SDS molecule, and AF = 30 kT for a micelle containing 100 SDS molecules. According to theory [ 131, each micelle can only confine a limited number N* of PEO monomers in the vicinity of its surface. A rough estimate yields N*-60 for PEO monomers adsorbed on SDS micelles (the parameters are given in the next section). Each macromolecule contains N-3000 monomers. Thus, after the binding of the first micelle, p additional micelles may be adsorbed with identical gains in free energy until the total number of confined PEO monomers pN * becomes equal to N; then the binding should saturate at p = N*/N- 50. In fact, the binding saturates before this limit is reached: for macromolecules with N = 3000 monomers, we find a saturation at p-27 micelles/ macromolecule for binding in water + 0.4 M NaBr (Fig. 4), and at p-10 micelles/macromolecule when no salt is added. This is caused by the repulsions between the charged micelles. As the aggregate becomes filled with adsorbed micelles, such repulsions become important, and the gain in free energy per additional micelle becomes smaller. The thermodynamic stoichiometry occurs when the free energy of binding becomes comparable to kT: then free micelles are created by thermal fluctuations in the solution. Additional micelles may still be adsorbed within the aggregate, but their binding energies are smaller than kT. The number of such weakly bound micelles will of course depend on the osmotic pressure from the free micelles in the solution. In this respect, each macromolecule acts as a localized well for the chemical potential of the micelles in the solution: when the well is filled beyond the level of the chemical potential of the free micelles, it spills over.

29

Finally, the repulsions between adsorbed micelles also control the behaviour of solutions containing less SDS than the stoichiometry. The important fact is that, for each aggregate, the drops in free energy become smaller and smaller for increasing numbers of bound micelles; consequently, such solutions lower their free energy by distributing the SDS equally among all the macromolecules. The opposite behaviour (coexistence of free macromolecules and saturated aggregates) should be observed for micelles which attract each other.

Structure This last result is important because it makes a discussion of “structure” meaningful: whenever the solution contains a single class of particles, a geometrical model may be constructed to describe them, and it can be checked against the observed scattering patterns. We have already assumed implicitly that all the aggregates formed in PEO + SDS solutions can be described by the general “beads and string” model constructed in Ref. [3] for stoichiometric aggregates. Consider a solution containing less SDS than the stoichiometry: now we know that all its aggregates hold about the same number p of SDS micelles; so, from the theory, we can obtain a precise prediction for the structure of an aggregate with p micelles, and check it against the radii of gyration measured for this solution (Fig. 5). A precise picture of the local structure around an adsorbed micelle can be derived from previous results [2,3] together with the prediction for the local ratio of monomers per micelle. According to Ref. [ 131, the free energy of interaction between the macromolecule and an adsorbed micelle is minimized if the number of monomers which are confined near the micelle is equal to a preferred value N*; we estimate N* = 60 for PEO + SDS (The parameters are: adsorption energy E = 0.2 hT per adsorbed monomer; length of a monomer a = 4 A; radius of a micelle b = 20 A.). The “nat60 monomers in ural” radius of gyration for a PEO subchain containing water would be 18 A [ 31, which is remarkably close to the radius of a micelle. Thus the overall dimensions of the subchain remain unchanged when it is adsorbed by the micelle. The macromolecule also contains some free strands which connect the adsorbed subchains to each other. The average number of monomers in such a free strand is Nf = (N - pN*)/p; its average extension is fixed by the centre-to-centre distance d between micelles. For stoichiometric aggregates in water + 0.4 M NaBr (p = 27, d = 60 a) the average free strand should contain 55 monomers and span a distance of 20 8. Thus it is slightly stretched, obviously because of the repulsions between micelles. For aggregates with less SDS, the free strands will be longer and less stretched. Now consider the overall configuration of the aggregate. In the simplest topology [13], each adsorbed subchain “fully clothes” a micelle with N* monomers; beyond this subchain, the macromolecule goes on

30

to other micelles, but never returns to a previously clothed micelle. Then the configuration of a stoichiometric aggregate is a linear array of p = 27 units, each containing an adsorbed subchain of 60 monomers and a free strand of 55 monomers; the size of each unit is 60 A, i.e. the micellemicelle distance. For an ideal chain of such units, the radius of gyration should be Rch = P’*~ d/ fl= 190 if, whereas the measured value is 210 A. The next step is to consider the radius R, for the distribution of micelles within the aggregate. It may seem obvious that R, should be identical with the radius Rch of the macromolecule. This is correct for large numbers of micelles, but not with small numbers, as shown by the following argument. Let us first assume that, when there are few micelles per aggregate, they are adsorbed at random locations along the macromolecular chain. Then of the distribution of the p micelles in the agthe radius of gyration R, gregate is related to the radius Rch of the chain in a simple way [ 141 : the number of intermicellar distances which contribute to R& is p (p- l), whereas the corresponding norm is p2, thus R2

m

=“-‘) P2

P-l RZh=-R$,

(5)

P

The behaviour predicted by this equation is represented by the dashed line in Fig. 5, assuming that Rch remains equal to 220 a. Because this result may appear surprising, it may be helpful to present it in a more formal, although equivalent way. The radius of gyration for the distribution of scattering centres in a particle may be derived from the Patterson function P(r) of this distribution R R;=’ s R

4nr2r2P(r)dr (6) 4nr2P(r)dr

s0 where R is the largest distance within the particle. If the scattering centres are the monomers of the macromolecule, then P(r) is proportional to the distribution of distance P& (r) within the macromolecule. If the scattering centres are micelles, then P(r) is made up of two terms, one of which contains distances r between two scattering centres located on the same micelle, and the other distances between scattering centres located in different micelles Pm(r) = P PStr) + P(P- l)Pd(r) Because P,(r)

the micelles

= ~~8 (r)

are small, the first term is a delta function

(7) in r (8)

31

whereas the second term is the convolution of distances in the macromolecule Pd(r) = P&r) *P&(r) = P ‘P&(r)

of Ps(r) by the distribution (9)

The “self” term Ps(r) does not contribute to the radius, but it does contribute to the norm in Eqn (6); thus the (p-l)/~~ factor of Eqn (5) is recovered. An identical result would be obtained if, instead of being randomly distributed, the micelles were regularly spaced along the chain, each in the middle of a section containing N/p monomers. In summary, the general beads-and-string model for PEO + SDS aggregates can be refined by using both thermodynamic and structural data. Thermodynamic data (adsorption energy, stoichiometry) determine the number of monomers in those sections of the PEO chain which are wrapped around an SDS micelle, and the number of monomers in the free strands which connect the adsorbed subchains. The dimensions of these subchains and free strands can be derived from high resolution structural data (size of a micelle, distance between neighbouring micelles). The overall topology of the PEO chain is accessible from the measured radii of gyration Rc-, of micelles within the (for the whole chain) and R, (for the distribution aggregate). Validity of the “particle” concept The classical analysis of small angle scattering deals with particles which are supposed to be invariant upon all changes in the solvent. Our scattering data for PEO + SDS aggregates have been treated according to this theory, even though they do not meet this condition. What are the limitations of this theory for supramolecular aggregates? The fact that the composition of the aggregates formed before the stoichiometry will change according to the composition of the solution in itself is not a serious problem: in this range the binding energies of the micelles are large, and the micelles are equally distributed among the aggregates; consequently the scattered intensity follows the laws expected for a solution of identical particles. Problems arise when the binding energy per micelle becomes small, that is beyond the stoichiometry. Then the exchange of micelles between the solvent and the aggregates becomes important; a thermodynamic description based on a preferential solvation of the polymer by the micelles may be more appropriate; such a description has been used by Boscher et al. for the system hydroxyethyl-cellulosenon-ionic surfactant [12]. Similar problems may be encountered if chemically heterogeneous polymers are used, because the adsorption energies of additionally bound micelles will vary through the whole range of compositions. Another process should be considered, by which isolated SDS molecules are exchanged between the aggregates and the solvent. This process may

32

change the aggregation number of the micelles; it becomes important in the vicinity of the critical surfactant concentrations x1 (for the formation of aggregates) or x0 (for the formation of free micelles). Indeed, Zana et al. [4] have found that the aggregation numbers of the first micelles bound to PEO + SDS aggregates are significantly smaller than those of micelles bound at higher surfactant concentrations. Thus, in the region extending from x1 to 5x1, common procedures such as diluting the particles with solvent may lead to unexpected results. Alternatively, micellar growth may be observed at high ionic strength; this is not observed for PEO + SDS aggregates [ 31, but may occur for other systems. The SDS micelles also exchange counter-ions with the solvent; most of these counter-ions are condensed on the micellar surfaces, but some are dispersed throughout the volume of the solution; the corresponding uncompensated charges on the micellar surfaces give rise to the repulsions between the micelles, and also to repulsions between the aggregates. The number of such charges and the range of the repulsions will vary according to the composition of the solution; dilutions may not be safe in this respect either. This problem can be tackled in two ways. Either the micellar charge can be determined from the scattering data through a self-consistent procedure which calculates the intermicellar interaction as a function of charge and aggregation number [ 151, or a large amount of salt is added to the solution to swamp the original distribution of charges and screen the corresponding repulsions; this procedure yields satisfactory results for PEO + SDS aggregates. Using the contrast-matching technique also requires that the particles remain structurally invariant when isotopic substitution is used to enhance or suppress the scattering by one component of the solution. We have checked that PEO + SDS aggregates do remain invariant when hydrogen is replaced by deuterium either in the solvent (H,O/D,O) or in the micelles (C,,H,,SO,Na/C,,D,,SO,Na). This may not be the case for other systems. Finally, the most severe limitation is that the solution may not be described in terms of distinct particles unless it is sufficiently dilute. Indeed, beyond a cross-over concentration y* which depends on the polymer molecular weight and radius, the polymer chains are interpenetrated. In this semi-dilute regime, it is no longer possible to distinguish any individual aggregates, and the stoichiometry associated with the saturation of individual aggregates may no longer exist. CONCLUSION

Polymer + surfactant aggregates in water are labile associations of small amphiphilic molecules with macromolecules, which form spontaneously into a limited variety of compositions and conformations. There is a small region of the interaction diagram (Fig. 1) where these aggregates can be considered as invariant “particles”. To the left, this region stops at SDS

33

concentrations which are a few times larger than the CMC x,; to the right, it is limited by the thermodynamic stoichiometry; and towards higher polymer concentrations, it is limited by the cross-over from the dilute to the semi-dilute regime. Crystallographic methods can then be used to determine a “structure” for the “particles”. These methods transform the scattering patterns into a set of distances and masses; if the experiments cover a wide range of contrasts and scattering vectors, a limited set of geometric parameters can be obtained with a large degree of redundancy [3] ; refinements are of course limited by the information content of the data [ 161. It is fortunate that such a region exists at all in the interaction diagram; such may not be the case for solutions containing heterogeneous macromolecules or polydisperse surfactants. This makes the PEO + SDS system a good model system for the study of interaction between polymers and surfactants. REFERENCES D. Robb, in E.J. Lucassen Reynders (Ed.), Anionic surfactants: Physical Chemistry of surfactant Action, Dekker, New York, NY, 1981, p. 109. 2 B. Cabane, J. Phys. Chem., 81 (1977) 1639. 3 B. Cabane and R. Duplessix, J. Phys. (Paris), 43 (1982) 1529. 4 R. Zana, J. Lang and P. Lianos, in P. Dublin (Ed.), Microdomains in Polymer Solutions, Plenum, New York, NY, 1983. 5 Y. Moroi, H. Akisada, M. Saito and R. Matuura, J. Colloid Interface Sci., 61 (1977) 233. S. Alexander, J. Phys. (Paris), 38 (1977) 977. Y. Layec and M.N. Layec-Raphalen, J. Phys. (Paris), Lett., 44 (1983) L121. C. Strazielle, Makromol. Chem., 119 (1968) 50. B. Cabane, R. Duplessix and T. Zemb, in K.L. Mittal and B. Lindman (Eds.), Surfactants in Solution, Plenum. New York, NY, 1984. 10 (a) K. Shirahama, Colloid Polym. Sci., 252 (1974) 978. (b) K. Shirahama and N. Ide, J. Colloid Interface Sci., 54 (1976) 450. 11 M.N. Jones, J. Colloid Interface Sci., 23 (1967) 36. 12 Y. Boscher, F. Lafuma and C. Quivoron, Polym. Bull., 9 (1983) 533. 13 P.A. Pincus, C.J. Sandroff and T.A. Witten, Jr, J. Phys. (Paris) Lett., 45 (1984) 725. 14 F. Joanny, private communication. 15 J.B. Hayter and J. Penfold, J. Chem. Sot. Faraday Trans. 1,77 (1981) 1851. 16 V. Luzzati and A. Tardieu, Ann. Rev. Biophys. Bioeng., 9 (1981) 1.