J. Phys. Chem. 1980, 84, 2418-2422
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(2) G. S. Wartley in “Micelllzation, Solubillzation, and Microemulsions”, Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977, p 23. (3) C. Tanford, “The Hydrophobic Effect: Formation of Micelles and Biological Membranes”, Wiley, New York, 1973, p 36. (4) E. I. Franses, W.D. Thesis, University of Minnesota, Minneapolis, MN, 1979. (5) L. E. Scriven in “Micelllzation, Solubilization, and Mlcroemulsions”, Vol. 2, K. L. Mittal, Ed., Plenum Press, New York, 1977, p 877. (6) P. Mukerjee in “Mlcellization, Solubilization, and Mlcroemulsions”, Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977, p 171. (7) P. Debye and E. W. Anacker, J. Phys. Colb&Chem., 55, 644 (1951). (8) E. J. Staples and G. J. T. Tiddy, J. Chem. Soc., Faraday Trans. I , 74, 2530 (1978). (9) N. A. Mazer, 0. B. Benedek, and M. C. Carey, J. Phys. Chem., 80, 1075 (1976). (10) (a) C. Y. Young, P. J. Missel, N. A. Mazer, G. B. Benedek, and M. C. Carey, J. Phys. Chem., 82, 1375 (1978); (b) P. J. Mlssel, N. A. Mazer, G. B. Benedsk, C. Y. Young, and M. C. Carey, ibM., in press; (c) N. A. Mazer, M. C. Carey, and 0. B. Benedek in “Micellization,
(11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21)
Solubillzatlon, and Microemulsions”, Vol. 1, K. L. Mittal, Ed., Plenum Press, New York, 1977, p 359. M. Corti and V. Degiorglo, Ann. Phys. (Paris), 3, 303 (1978). M. Corti and V. Degiorglo, to be submltted. K. J. Mvsels and R. J. Otter. J. Colloid Sci.. 16. 482 11961). E. I. Franses, H. T. Davis, W. G. Miller, and L. E. Scrlven, ACS Symp. Ser., No. 91, 35 (1979). R. C. Mwray and G. S. Hartley, Trans. Farachy Soc., 31, 183 (1935). E. I. Franses, J. E. Puia. Y. Talmon. W. G. Miller. L. E. Scriven. and H. T. Davis, J. Phys. >hem., In press. L. M. Kushner and W. D. Hubbard, J. Col/o/d Sci., 10, 428 (1955). H. F. Huisman, Proc. K. Ned. Acad. Ser. Wet., Ser. B, 67, 367, 375, 388, 407 (1964). K. W. Wagner, Arch. flektrotech. (Berlln), 2, 371 (1914), clted by T. Hanai in “Emulslon Science”, P. Sherman, Ed., Academic Press, New York, 1968, p 379. R. T. Roberts and C. Chachaty, Chem. Phys. Lett., 22, 348 (1973). Y. Talmon, H. T. Davis, L. E. Scrlven, and E. L. Thomas, Rev. Sci. Instrum., 50, 698 (1979).
Interactions and Aggregation in Mlcroemulsions. A Small-Angle Neutron Scattering Study R. Ober and C. Taupin” Physique de la Mati&e CondensOe, E R A . 542 du Centre National de la Recherche Scientlfique, Coll6ge de France, 75231 Paris Cedex 05,France (Received: January 22, 1980)
We performed a study of various systems of well-defined water-in-oil microemulsions by means of small-angle neutron scattering. The equation of state, which is deduced from the scattered intensity at zero angle, shows that the interactions in these systems are essentially of the hard-sphere type. Some systems behave as pure hard-sphere liquids, but a small attractive term has to be added in most cases. The order of magnitude of this attraction is not compatible with van der Waals forces. The analysis of the scattered intensity as a function of the momentum transfer shows that the hard-sphere type systems behave as isolated spheres. On the contrary, the attractive systems present several characteristic features of doublets of spheres. The physical origin of the attraction is discussed.
Introduction In the past few years, much interest has been raised by microemulsions. These transparent fluid systems, which were identified as colloidal systems by Schulman in 1943,l have the capability of solubilizing water and oil in nearly all relative proportions. Various theoretical and experimental articles2 have been published in an attempt to understand their structure, stability, and occurrence as a function of chemical composition. In a previous paper,3 we determined the structure of typical microemulsion systems (water, cyclohexane, sodium dodecyl sulfate, and 1-pentanol) in the oil-rich region. The main conclusions were the following: The microemulsion is well described by spherical water droplets dispersed in an oily medium, the polydispersity in size being remarkably low. At constant ionic strength, the size of the water core of the droplets is determined by the area per polar head of the soap molecule which remains almost constant when the ratio of soap to water is varied. These conclusions are in good agreement with the experimental findings of other authors on similar ~ y s t e m s . ~ In our previous studies, we explored several systems the characteristics of which are reported in Table I. This table shows that besides decreasing the area per polar head of the soap molecule, which was expected, the increase of the ionic strength of the aqueous phase induces a decrease of the hydrodynamic volume. In fact, the hydrodynamic thickness rh - rw decreases from >23 to 16.5 A, revealing that the continuous phase has been expelled from the interfacial film. 0022-3654/80/208~-2418$01.OO/O
TABLE I: Structural Data for Four Types of Microemulsionsa A
B
C
1.25 0.18 34.5 52 46.5 12 18
2.5 0.087 56.5 68 65.5 9 23
3.75 0.045 83 66 92 9 27
NaCl mol/L
w/sb’
P/Ci rw, A A,g At rwe ‘4 r C - r W ,A rh - r w , f A
B1M 1 2.5 0.026 66.5 52 74.5 8
16.5
a Systems A, B, C correspond to increasing amounts of solubilized water.3 System B1M (1M NaCl) corresponds to unpublished results of J. P. L e Pesant. W/S is the weight ratio of water to soap. cP/C is the volume ratio of pentanol to cyclohexane in the continuous phase.j rw is the radius of the water core. e rc is the radius which is not penetrated by the continuous phase (see ref 3). f % is the hydrodynamical radius. g A is the area per polar head of the SDS molecules.
One of the puzzling questions which is raised by the wide domain of existence of the microemulsionsis the possibility of structural changes associated with an increase of the amount of the dispersed water phase. It is known that a good approximation to obtain systems of variable droplet concentration is to maintain the water-to-soap ratio, WIS, constant and to add to the water-soap-cyclohexane mixture the minimum quantity of alcohol which is necessary to obtain a clear phase. The “titration curve” (pentanol volume vs. cyclohexane volume) shows that the pentanol 0 1980 American Chemical Society
Interactions and Aggregation in Microemulsions
The Journal of Physical Chemistry, Vol. 84, No. 19, 1980 2419
partitions between the continuous phase and the interfacial film, in agreement with the low water solubility of the 1-pentanol, The composition of the continuous phase remains constant. In ref 3, we found that a very sensitive test of the constancy of the droplet is given by the variable contrast technique. We used this technique in concentrated systems to show that it is possible to increase the number of droplets without modifying their comp~sition.~ This technique6 also revealed that above a volume concentration of dispersed phase of 0.6, which corresponds to the random packing fraction: the microemulsions undergo a structural inversion, water becoming the continuous phase. In this article, we investigate the interactions between droplets in the concentrated water-in-oil region by studying the osmotic compressibility of these liquid systems. I t will be seen that, depending upon which system is studied, two types of equation of state are derived, which correspond to different shapes of the scattering curves (as a function of the momentum transfer q). Neutron Scattering Experiments Experimental Section. Small-angle neutron scattering experiments were performed at the Institut Laue Langevin (Grenoble, France) on the D11 apparatus. We used cold neutrons with a wavelength of 11A and a spread Ah/X = to 0.1. (The momentum transfer q ranged from 4 X 4X A-l where Iql = 4a sin @ / A with 6 ' the scattering angle.) In order to enhance1 the scattered intensity, we used the contrast pattern referred to as type a in Figure 6 of ref 3, i.e., deuterated water, 1-pentanol-0-d,protonated cyclohexane, and sodium dodecyl sulfate (SDS). As the scattering length per unit volume of the sulfate part of the SDS molecule is not very different from that of heavy water, the scattering voluma includes both the water and the polar heads. The microemulsions were contained in quartz cells either 1 or 0.5 mm thick. This thickness was chosen as a compromise between the desire to maximize the scattered intensity and the need to keep the multiple scattering as low as possible. The scattered intensities were corrected for the proton incoherent scattering, the continuous phase scattering, and the attenuation of both primary and scattered beams. Theoretical Section. For small momentum transfer the coherent scattered intensity per unit volume is proportional to i(q,c) = KcuS&) (1 .t c / u
1[g(r)
- 11 exp(-i+it)
d7)
(1) where K is the contrast factor, u is the scattering volume of a particle, c is the volume fraction of the dispersed phase, SO(q) is the average over all orientations of the square of the one particle structure factor F(q'), (S,(O) = l), and g(r) is the pair correlation function for the center of mass of the particles. In these experiments we did not obtain the absolute scattered intensities, but, as the experimental settings were kept constant in all the experiments, the intensities are determined to within a constant factor. We examine eq 1 for two limiting cases. Limit q = 0. For a solution of monodisperse particles of volume fraction c, i(q = 0,c) is related to the equation of state of the system by eq 2, where II is the osmotic
(2)-l
i(q = 0,c) = KkTc -
pressure of the particles in solution, k is the Boltzmann
constant, and T i s the absolute temperature. i(q = 0,c) is obtained by the extrapolation to q = 0 of i(q,c) for small 4.
In the case of very low concentrations ci-l(c) 1 + Bc (3) where B is the second virial coefficient. Limit c 0. Equation 1 becomes i(q,c 0) = KcuSo(q) (4) It is well-known8that the analysis of this function of q gives the radius of gyration of the scattering particle at very low q and information about the shape of the particles at higher q.
-
-
-
Study of the Osmotic Compressibility Theory. Vrijg performed the first study, of osmotic compressibility in microemulsions by means of light scattering. The data were analyzed semiempirically,using a two-term potential. The hard sphere term is the most important one. The hard sphere radius is found to be 9 A larger than the radius of the water core; i.e., the hardsphere radius is very similar to the radius r, determined by neutron scattering (Table I). The second, smaller term is attractive. One might suppose van der Waals forces to be the physical origin of this term; however, in the benzene-water system studied by Vrij, the Hamaker constant value was found to be too small to fit the observed order of magnitude of this attractive interaction. We performed a similar theoretical analysis by using both hard-sphere and pair-type potentials. We also tried to explain the small attractive term by an electrostatic potential following the method of Landau,'O as electrical conductivity measurements5 have shown that the dropleh bear a small residual charge of the order of one electronic charge per droplet. Recall that the basic expressions for the osmotic pressure ll are
II = n H S + Uatt where we use for llHs the semiempirical expression proposed by Carnahan et al? I I H= ~ kT(c,/v,)(I y y2 - y3)(1 - y)-3 (5) where y is the unknown volume fraction of hard spheres and c, and u, are respectively the volume fraction and the volume per droplet calculated from the radius r,; the volume u, contains water and cyclohexane nonpenetrated interphase (soap plus excess alcohol) and was shown to be constant up to c, = 0.5.5 We set y = ac, with a = uHs/u,. IIattcan be due to either of the following causes: (1)van der Waals potentials; following Vrij we used
+ +
= -kTpc,2/2~, is related to the Hamaker constant A by uvdW
where
A 6kT
s-1
s - s3 In
(6)
-
where s = rHs/r,, (2) the screened electrostatic potential between the charged particles (absolute mean charge ze):
natt= ne*
--Tp'Cz/2
(8)
VC
where
where c is the dielectric constant of the continuous phase. Two points should be noted at this stage. First, this expression for the electrostatic term is valid only for very
2420
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Ober and Taupin
The Journal of Physical Chemistry, Vol. 84, No. 19, 1980
Flgure 1. Fit of the osmotic compressibility curves. Microemulsions A (-), B (.* s), and C (- - -) are fitted with a hard-sphere plus van der
Waals potential, and microemulsion BIM is fitted with a HS potential alone: the maximum intensity was set to 1 for each system. TABLE 11: Values of Various Parameters
A O1a
P Bcb
Bm Aexptbd
ergs
B
1 1 923 1123 -2.9 -1.2 -15 - 11 3x 1.7 X 10-11 10-11
C 0.98 13*3 -5.5
- 13
1x lo-”
B1M 0.97 3+5 4 ->0 -0
a 01 3 u H s / u , . B , the second virial coefficient which is deduced from the entire concentration range. ‘ B , is its value when measured on diluted samples (Zimm analysis). Aexptl is the Hamaker constant value deduced from p . The value of this constant given in ref 12 is A , , , = 1.5 X erg.
low concentrations.1° Second, because of the screening effects, the contribution of the electrostatic interactions is important only at very low concentrations, as a result of which ci-l(q = 0,c) diverges as c-1/2as c approaches zero. We shall examine successively the fit of the osmotic compressibility curve for the entire concentration range and the second virial coefficient deduced from systems at low concentrations. Fit of the Entire Experimental Curve. Figure 1shows the fitted curves i(q = O,cc) for various systems. The maximum intensity was set equal to 1 for purposes of comparison. With one exception, the curves look very similar. We fit these curves with two theoretical models by a least-squares fitting procedure minimizing x2 with respect to each fit parameter: (a) HS potential alone; one parameter a;(b) HS vdW potentials; two parameters (a,p). In both cases, the parameter a fixes the concentration a t which the scattered intensity is maximum; in the second case (b), determines the width of the peak. Figure 2 shows that the best fit with the HS model is not satisfactory except in the B1M case (Figure 2b). The addition of the van der Waals type attractive term improves the quality of the fit as shown in Figure 2a. Table I1 summarizes the values of the various parameters. The parameter a is found to be close to 1to within a few percent in all cases. This confirms our interpretation of the rc radius as being equivalent to the hard-sphere radius. In order to determine whether nettis due to van der Waals forces, therefore, we compare the experimentally derived value of the parameter ,B to its value derived from the Hamaker constant by using eq 7 and ref 12. The comparison of the two p for the various A, B, C systems shows that the experimental attraction is 100 times larger than the expected van der Waals one. This conclusion is in agreement with the conclusion drawn by Vrij in ref 9.
+
.C
Comparison of the experimental data ( 0 )with both HS fit (broken curve)and HS plus vdW fit (full curve) for (a)A microemulsion and (b) the BIM system. Flgure 2.
Another point excluding van der Waals interaction should be noted. As in our systems both dispersed and continuous phases are kept constant in chemical nature (respectively, water and cyclohexane), we expected a similar van der Waals attractive interaction to occur in all of them. Clearly, this is not the case since B1M systems do not exhibit this attraction. Study of the Second Virial Coefficient ( B ) of the Osmotic Pressure. Two values of this coefficient for each system are given in Table 11. One is deduced from the fit over the entire concentration range, and the other is measured from low concentration by using the technique developed by Zimm.13 Except in the case of the salinated B1M system, where the B value of +4 is not very far from the +8 value corresponding to a pure HS system, these two values disagree. This suggests that the attractive term is responsible for this discrepancy. If we recall that electrostatic forces would appear only at low concentrations, we can calculate the 0’term (see eq 9) in the case of droplets having a radius of 75 A and 3.5. The bearing a residual charge of he; this gives p’ consequent modification of the measured B is approximately plc;l12 where c, is the mean concentration used in the Zimm plot analysis. The deduced value is ca. -35, only twice as large as the measured values. Equation 9 shows that p’ varies as the third power of the droplet charge, and as a change of 30% of the charge would fit the data, we cannot exclude such an interaction in our analysis of the A, B, C systems. Moreover, as pointed out by one of the referees, the precise location of the charge inside the droplet would influence these forces. The question remains: what is the origin of the different behavior of the B1M systems which bear similar residual charges?l4
-
The Journal of Physical Chemistry, Vol. 84, No. 19, 1980 2421
Interactions and Aggregation in Microemuisions
I (q) (a.u.
U
(a1
i
2
1
0
Figure 4. Comparison of the scattered intensity curves for varlous systems with identical radius of gyration: (-) monodisperse spheres; (- -) spheres with a polydispersity of A R / R = 0.4; (- -) ellipsoid ( v = 2 or v = 0.166); (---) ellipsoid v = 3. ell
U
1
0
1
2
3
4 t
-
Flgure 3. Plot of the scattered intensity as a function of 9RQ for diluted systems (c, 0.02) for (a) B microemulsion and (b) B1M microemulsion. (0)Experimental data. The full curve is the calculated intensity in the case of a spherical particle with the same radius of gyration. The r a d b of gyration was determined in experiments with a smaller 9 range (9RQ < 1). The broken line shows the small effect of the second virial coefficient.
Study of the Shape Function S 0 ( q ) Figure 3, a and b, shows the angular variation of the scattered intensity at large q for B and B1M type microemulsions. (A and C type microemulsions would give curves similar to that of B.) Recall that one can deduce the radius of gyration R G of the scattering particle from such a curve. The intensities are given in Figure 3 plotted as a function of qRG so as to be comparable with the theoretical curve of the scattered intensity due to monodisperse spheres of the same radius of gyration. Clearly, the two types of microemulsions look different, and only the B1M curve is very close to the theoretical curve for spheres. The difference between the experimental curve and the theoretical one could be due either to polydispersity or to anisotropy of the scattering particles. l5 The polydispersity of these systems as determined in 0.1. ref 3 by analytical centrifugation is low: A r / r Figure 4 shows that even a larger polydispersity, A r l r = 0.4, is clearly not sufficient to explain the experimental curve.
-
Flgure 5. Variation of the scattered intensity at 9 = 0 as a function of the anisotropy ratio v of ellipsoids of identical radius of gyration.
If we consider now the possibility of anisotropy of the scattering particles, we can perform two experimental tests to check such an anisotropy, examining both the shape of the scattering curve and the scattered intensity which is determined by the true volume of the scattering particle. Figure 5 shows the variation of the intensity i(q = 0) scattered by an ellipsoid (a, a, ua) as a function of the ellipsoid anisotropy u, the radius of gyration being kept constant. This intensity, which is proportional to the volume of the ellipsoid, has a sharp maximum for u = 1. If we consider that B1M systems are well modeled by spheres (u = l),we can locate on this curve the intensities scattered by the A, B, C cystems. This procedure leads to two possibilities: the systems are represented by either oblate (u = 0.4) or prolate (u = 2.3) ellipsoid. We eliminate the first possibility of 0.4 because it does not modify sufficiently the scattering curve;16on the other hand, it is shown in Figure 4 that a prolate ellipsoid (u 3) is in excellent agreement with the experimental curve. It should be noted that the two tests lead to approximately the same anisotropy (2 < u < 3). At this point, a comment should be made about such anisotropic particles in A, B, C systems. The behavior of concentrated dispersions of anisotropic objects is usually very different from that of spheres, giving rise to ordered structures and high viscosities which were never observed
2422
The Journal of Physical Chemistry, Vol. 84,
No. 19, 1980
in these systems. Moreover, for the same volume of dispersed water, ellipsoids would lead to larger areas per polar head of the soap than spheres: 85 A2 instead of 68 A2. Such a high value was never observed in lyotropic films. In the following discussion, we present a model which reconciles all the different kinds of behavior of the various systems.
Discussion We performed a systematic study of interactions in four water-in-oil microemulsion systems. The chemical nature of oil and alcohol was kept constant in order to maintain the same value of the Hamaker constant in the various systems. In fact, the composition of the (oily) continuous phase is slightly variable because of the variable amount of 1-pentanol (see Table I), but it is reasonable to predict that the effect of the 1-pentanol will be to decrease slightly the value of the Hamaker constant. The conclusions of the experimental study are the following. Whatever the experiment, two types of behavior appear-one corresponds to A, B, and C systems, the other to the B1M (NaC1) system. The B1M microemulsion behaves as a pure hard-sphere liquid, either in the concentrated range or in the diluted one. The scattering curve S,(q) is very close to that of a sphere, and the second virial coefficient deduced from the fit of the entire curve agrees with that deduced from the diluted range analysis (see Table 11). The A, B, and C systems exhibit a scattering curve similar to that of prolate anisotropic particles. As such anisotropic particles do not seem to be in agreement with other experiments, we suggest that this apparent anisotropy could be due to the presence of aggregates and, in the case of low-concentration systems, essentially to doublets of spheres (mean distance 2r, between the centers). The number of these doublets should increase with concentration depending on the process of aggregation. In the case of a diluted system (3% in volume in dispersed phase), it was possible to show that less than 10% of dimerized droplets would lead to the observed apparent anisotropy. Moreover, it was shown by Tanford17that the presence of dimers apparently decreases the second virial coefficient, which would explain the discrepancy between the two different determinations of B in Table 11. Recently, the aggregation due to electrical charges was studied theoreti~a1ly.l~It is important to note that this aggregation of droplets occurs only in the systems which exhibit attractive terms in their equation of state. The question of the origin of attractive forces in microemulsions remains open. These forces are too large to be van der Waals forces. The order of magnitude of electrostatic forces is more difficult to discuss but, as we pointed out, these forces have to be similar in the various systems.
Ober and Taupin
Another possibility is that these attractive forces originate from the structure of the interfacial film, in correlation with the current studies on “structural or solvation forces”.18 Up to now, we do not have clear experimental evidence in favor of such an explanation. Nevertheless, Table I shows that the interfacial structure of the B1M system is different from that of the A, B, C systems since at constant radius the decrease of the hydrodynamic thickness reveals an expulsion of the continuous phase from the interfacial film in the pure hard-sphere system. Acknowledgment. It is a pleasure to acknowledge the people who contributed to this study. The neutron experiments were made possible by the allocation of beam time on the D 11apparatus at the Institut Laue Langevin (Grenoble, France). We are grateful to Dr. A. Dianoux, Dr. R. Duplessix, Dr. H. Goeltz, Dr. J. Haas, and Dr. M. Roth for many discussions about the use of the apparatus. We have benefited from fruitful discussions with P. G. de Gennes, B. Ninham, M. Lagues, D. Levesque, and J. J. Weis. This investigation was supported by a research grant from the DBlBgation GBnBrale 5 la Recherche Scientifique et Technique. References and Notes T. P. Hoar and J. H. Schulman, Nature(London), 152, 102 (1943). J. H. Schulman and J. B. Montagne, Ann. N . Y. Acad. Scl., 92, 366 (1961); L. M. Prince, Ed., “Microemulsions Theory and Practice”, Academic Press, New York, 1977; E. Ruckenstein and J. C. Chi, J. Chem. Soc., Faraday Trans. 2, 71, 1690 (1975); A. Skoulios and D. Guillon, J . Phys. (Paris), Lett., 38, L-137 (1977); D. 0. Shah, R. D. Walker, S. P. E. of AIME, paper number SPE 5815 (1976); S. Friberg, I. Lapczynska, and G. Gillberg, J . Colloid Interface Sci., 58, 19 (1976); A. Gracia, P. Chabrat, J. Lachaise, L. Letamendi, J. Rouch, C. Vaucamps, M. Bourrel, and C. Chambu, J. Phys. (Paris), Lett., 38, L-253 (1977); L. E. Scrlven, Nature(London), 263, 123 (1976). M. Dvolaitzky, M. Guyot, M. Lagues, J. P. Lepesant, R. Ober, C. Sauterey, and C. Taupin, J. Chem. Phys., 89, 3279 (1978). H. F. Eicke and J. Rehak, Helv. Chim. Acta, 59, 2883 (1976). M. Lagues, R. Ober, and C. Taupin, J . Phys. (Paris), Lett., 39, L-487 (1978). H. B. Stuhrmann, J . Appl. Crystallogr., 7, 173 (1974). J. L. Finney, Nature (London), 266, 309 (1977). A. Guinier and G. Fournet, “Small Angle Scattering”, Wiiey, New York, 1955. W. G. M. Agterof, J. A. J. Van Zomeren, and A. Vrlj, Chem. Phys. Left., 43, 363 (1976). L. Landau and E. Lifchitz, “Physique Statistique”, MIR, Ed., Moscow, 1967, p 275. N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51, 635 (1969). J. Visser, Adv. Colloid Interface Sci., 3, 331 (1972). B. Zimm, J . Chem. Phys., 16, 1093 (1948). M. Lagues and C. Sauterey, private communication. W . Beeman, P. Kaesberg, J. W. Anderegg, and M. B. Webb, “Encyclopedia of Physics”, Voi. 32, “Structural Research”, SpringerVerlag, New York, 1957, p 21. We also examlned the effect of the negative experimentally found virial coefficient on the sphere scattering curve; this Increases the difference between the experimental curve and the theoretical one. C. Tanford, “Physical Chemistry of Macromolecules”, Wiiey, New York, 1961, p 203. D. Y. C. Chan, D. J. Mitchell, B. W. Ninham, and B. A. Pailthorpe, Mol. Phys., 35, 1669 (1978); M. E. Fisher and P. G. de Gennes, C. R . Hebd. Seances Acad. Sci., Ser. 8 , 287, 207 (1978); M. J. Grimson, P. Richmond, and G. Rickayzen, Mol. Phys., in press. W. Ebeling and M. Grigo, Ann. Phys. (Lelpzig), 37, 21 (1980).
