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Depletion Interactions in Ionic Micellar Solutions V. Vlachy Faculty of Chemistry and Chemical Technology, University of Ljubljana, P.O. Box 537, 61000 Ljubljana, Slovenia Received January 22, 1996. In Final Form: March 18, 1996X The depletion interactions between macroions caused by the addition of a highly asymmetric electrolyte (micelles) were studied using the integral equation technique. The three-component system contained the following species: macroions (component 1), charged micelles (2), and an equivalent number of small counterions (3). The ions are present as soft charged spheres, and thus the three species differ widely in charge and size. We showed that the model reproduces the major features of reported experimental results (Sober, D. L.; Walz, J. Y. Langmuir 1995, 11, 2352), i.e., (i) the purely repulsive interaction between two macroions in simple electrolyte solutions, (ii) the onset of the attractive minimum above the critical micelle concentration, and (iii) the appearance of structural oscillations at high concentrations of added micelles.
Introduction The depletion force between charged surfaces in micellar solution has been the subject of study in several recent papers.1-6 The theoretical study presented here is stimulated by recent experimental developments in this area of research. Kekicheff and co-workers3,4 reported measurements of the depletion force between two mica surfaces coated with an adsorbed bilayer of cetyltrimethylammonium bromide (CTAB) and immersed in aqueous ionic micellar solutions of CTAB. In a similar study, Sober and Walz6 measured depletion forces between a colloidal particle and a flat plate in the presence of CTAB micelles using a very sensitive optical technique. These experimental results show that at CTAB concentrations below the critical micelle concentration (cmc) the potential energy profiles are repulsive,6 in agreement with those predicted by classical DLVO theory.7 At CTAB concentrations above the cmc, however, the potential energy curves show an attractive minimum. The depth of the minimum increases with increasing micelle concentration. At even higher CTAB concentrations structural oscillations, with nearly constant period, appear. In these papers an approximate theory based on the Asakura-Oosawa type of potential8,9 was used to analyze the experimental results. Intermicellar interactions are most often studied in the framework of the classical one-component model (DLVO theory), where the micellar solution is viewed as an effective one-component fluid in which the macroions interact via the repulsive (screened Coulomb) potential and the attractive, dispersion potential.7 In the onecomponent model the effects of the simple electrolyte and solvent are neglected, except in determining the screening length. The depletion interaction potential of Asakura and Oosawa8,9 may be superimposed on top of the DLVO theory.10 The theory based on this superposition approximation has been applied to study phase separations X
Abstract published in Advance ACS Abstracts, May 15, 1996.
(1) Pashley, R. M.; Ninham, B. W. J. Phys. Chem. 1987, 91, 2902. (2) Bibette, J.; Roux, D.; Nallet. F. Phys. Rev. Lett. 1990, 65, 2470. (3) Richetti, P.; Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1951. (4) Parker, J. L.; Richetti, P.; Kekicheff, P.; Sarman, S. Phys. Rev. Lett. 1992, 68, 1955. (5) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (6) Sober, D. L.; Walz, J. Y. Langmuir 1995, 11, 2352. (7) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of Stability of Lyophobic Collids; Elsevier: Amsterdam, 1948. (8) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (9) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183. (10) Vlachy, V.; Prausnitz, J. M. J. Phys. Chem. 1992, 96, 6465.
S0743-7463(96)00061-3 CCC: $12.00
in solutions of globular proteins and colloids, induced by nonionic polymer.11-14 In the present study we adopt a more microscopic point of view: the micellar solution is treated as a mixture of charged “soft” spheres of different size immersed in a dielectric continuum representing the solvent. Chargesphere models have been successfully applied to study the structure and thermodynamics of highly asymmetric electrolytes.15-20 These models treat all the species on an equal footing; the Coulombic interaction between macroions is screened by available small ions. However, if an additional neutral (or charged) component is added in the solution, the originally repulsive interaction between the macroions may turn into an attractive interaction.10,18 The depletion interaction (attraction) between the large macroions emerges as a result of the multicomponent model and need not be introduced by the Asakura-Oosawa potential. Aim of the present study is to investigate the ability of the three-component charged-sphere model to reproduce the essential features of the experimental results as reported in refs 3 and 6. Multicomponent Model We shall consider a three-component model electrolyte; the ions are described as charged soft spheres embedded in continuous dielectric representing the solvent.17 Two of the species in solution are taken as macroions; species 1 (large macroions) has charge e1 and species 2 (smaller macroions, here termed “micelle”) has charge e2. We assume in this calculation that both macroionic species are negatively charged. The concentration of the remaining ionic component 3 (counterions, ions having charge of opposite sign to the macroions) is given by the electroneutrality condition. The solvent averaged pair interac(11) Gast, A. P.; Hall, C. K.; Russel, W. B. J. Colloid Interface Sci. 1983, 96, 251. (12) Mahadevan, H.; Hall, C. K. AIChE J. 1992, 38, 573. (13) Vlachy, V.; Blanch, H. W.; Prausnitz, J. M. AIChE J. 1993, 39, 215. (14) Kuehner, D.; Chiew, J.; Blanch, H. W.; Prausnitz, J. M. AIChE J. 1995, 41, 2150. (15) Belloni, L. Chem. Phys. 1985, 99, 43. (16) Bratko, D.; Friedman, H. L.; Zhong, E. C. J. Chem. Phys. 1986, 85, 377. (17) Vlachy, V.; Marshall, C. H.; Haymet, A. D. J. J. Am. Chem. Soc. 1989, 111, 4160. (18) Vlachy, V. J. Chem. Phys. 1993, 99, 471. (19) Forciniti, D.; Hall, C. K. J. Chem. Phys. 1994, 100, 7553. (20) Outhwaite, C. W.; Bhuiyan, L. B.; Vlachy, V. Mol. Phys. 1994, 83, 183.
© 1996 American Chemical Society
2882 Langmuir, Vol. 12, No. 12, 1996
Letters
tion between two particles i and j at distance r apart is
uij(r) ) uij*(r) +
uij*(r) )
e i ej 4πr0r
A|zizj|kBT exp[λ(dij - r)] dij
(1)
where r is the relative permittivity, ei ) zie0 is the charge on an ion i (zi is the valency and e0 is the charge of a proton) and dij ) (di + dj). As usual kB is Boltzmann’s constant and T is the absolute temperature. In this calculation the counterions are assumed to be monovalent, z3 ) 1, while z1 ) -26 (macroions) and z2 ) -10 (micelles). The length parameters di used in this work are d1 ) 6 nm, d2 ) 1.5 nm, and d3 ) 0.2 nm, and λ ) 20 nm-1. The value of the Bjerum length LB ) e02/(4πr0kBT) is taken as 0.714 nm, to mimic aqueous solutions at T ) 298 K. The prefactor in the short-range potential, A, is equal to 1.784 nm. The values of λ and A chosen in this study do not model any particular solution. Three-component models have previously been studied using the hypernetted-chain (HNC) equation,10,17-20 the modified Poisson-Boltzmann potential theory,20 and the symmetric Poisson-Boltzmann approximation.20,21 The HNC approximation has been quite successful in describing Coulomb systems (see for example ref 22): a comparison with Monte Carlo simulations for highly asymmetric electrolytes indicates that the theory accurately describes the structure of the solution at higher concentrations, but it fails to produce convergent results for very dilute solutions.15-17,23,24 For the three-component mixture treated here, six coupled integral equations have to be solved numerically. The theory and the numerical procedure used in solving the integral equation has been explained in detail elsewhere17,23 and will not be repeated here.
Figure 1. Reduced potential of mean force, w11(r)/kBT, as a function of the distance r between the centers of particles. The dotted line (curve a) corresponds to c2 ) 3 × 10-4 M, the broken line (b) to c2 ) 0.002 M, and the dashed (c) and continuous line (d) to c2 ) 0.007 M and c2 ) 0.02 M solution, respectively. The concentration of macroions was c1 ) 5 × 10-4 M.
Results and Discussion The principal result of our numerical calculation is the pair correlation function gij(r) as a function of distance r. In particular we are interested in g11(r), that is, the distribution function for large macroions (component 1) under various experimentally relevant conditions. To facilitate comparison with the experimental data,6 we present the results in terms of the potential of mean force, w11(r)/kBT ) -ln[g11(r)], as a function of distance r between the centers of two particles. Figure 1 shows the result for w11(r)/kBT in a mixture of macroions (component 1), “micelles” (2), and counterions (3) under various conditions. The parameter in this calculation is c2, i.e., the concentration of component 2. The concentration of macroions is c1 ) 5 × 10-4 mol/dm3 (d1 ) 6 nm) and the concentration of component 2 (d2 ) 1.5 nm) is (a) c2 ) 3 × 10-4 mol/dm3, (b) c2 ) 0.002 mol/ dm3, (c) c2 ) 0.007 mol/dm3, and, (d) c2 ) 0.02 mol/dm3, respectively. For zero or low concentration of added component (2), the potential of mean force is positive for all distances r, i.e., the force between macroions (1) is repulsive under these conditions. This curve is denoted by a dotted line on Figure 1. The situation corresponds (21) Schmidt, A. B.; Ruckenstein, E. J. Colloid Interface Sci. 1992, 150, 169. (22) Rasaiah, J. C. Theories of Electrolyte Solutions. In The Liquid State and Its Electrical Properties, Kunhardt, E. E., Christophorou, L. G., Luessen, L. H., Eds.; Plenum: New York, 1988; N.A.T.O. ASI Series B, Vol. 193. (23) Ichiye, T.; Haymet, A. D. J. J. Chem. Phys. 1988, 89, 4315. (24) Kalyuzhnyi, Yu. V.; Vlachy, V. Chem. Phys. Lett. 1993, 215, 518.
Figure 2. Macroion-micelle distribution function, g12(r), at c2 ) 3 × 10-4 M (a) and c2 ) 0.02 M (b). Other data are as for Figure 1.
to that below the cmc as presented in ref 6. Under these conditions the classical DLVO theory may be applied.7 For higher concentrations of added micellar component (2), cf. curve b of Figure 1, the w11(r) function shows an attractive minimum. Curves b, c, and d mimic the conditions above the cmc as studied experimentally.3,6 It is clear that the attractive minimum becomes deeper by increasing the concentration of “micelles”, c2. At the same time, the potential curve becomes narrower; this effect has also been observed experimentally. The depth of the potential well, caused by the addition of micellar component (2), was monitored for the concentration range from c2 ) 0.002 mol/dm3 to c2 ) 0.021 mol/dm3 and it is not a linear function of c2.6 Above c2 ) 0.021 mol/dm3 the HNC algorithm becomes unstable, and it is difficult to obtain convergent solutions. At high concentrations of the “micellar” component we obtain oscillations in the potential of mean force; see for
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Langmuir, Vol. 12, No. 12, 1996 2883
Figure 3. Macroion-counterion distribution function, g13(r), at c2 ) 3 × 10-4 M (a) and c2 ) 0.02 M (b). Other data are as for Figure 1.
Figure 4. Counterion-counterion distribution function, g33(r), at c2 ) 3 × 10-4 M (a) and c2 ) 0.02 M (b). Other data are as for Figure 1.
example curve d in Figure 1, where c2 ) 0.02 mol/dm3. The situation corresponds to that shown in Figure 3 of ref 3, that is, for the concentration of CTAB micelles well above the cmc. The oscillations reflect a particle packing effect due to the relatively high volume fraction of the micelles (for c2 ) 0.02 mol/dm3, F2d23 is about 0.041). While the experiment only provides information about the force or the potential profile between the two charged surfaces in micellar solution, the theory is capable of calculating correlation functions, gij(r), between all the species in the system. Some of these correlation functions are shown in Figures 2-4. Figure 2 presents the macroion-micelle correlation function, g12(r), as a function of distance r. The upper curve of Figure 2, indicating strong clustering of “micelles” about the macroion applies to c2 ) 0.02 mol/dm3 and the lower curve to c2 ) 3 × 10-4 mol/dm3. The macroion-counterion distribution function, g13(r), is presented in Figure 3. As expected, an increase of ionic strength (increase of c2) decreases the clustering of counterions around a macroion (component 1). Finally, Figure 4 shows the effect of the increasing concentration of the micellar component on the shape of the counterion-counterion distribution function, g33(r). Our calculation proves that the model asymmetric electrolyte reproduces many essential features of the
experimental results with respect to the depletion interactions. However, there is an important difference between the model and experimental systems studied in ref 3 and 6. The size of the micellar component (species 2) and especially that of the macroions (species 1), chosen in this preliminary calculation, is much smaller than those studied experimentally. We chose to examine here the situation where all the species have a finite size and are mobile in solution. With this respect our calculation has some relevance for understanding the precipitation of globular proteins by the depletion mechanism.11-14 For higher asymmetry in charge and/or size, the HNC algorithm becomes numerically unstable and it is impossible to obtain convergent solutions. In order to study the depletion force between two infinite charged surfaces, other theoretical methods have to be used.25-27 Acknowledgment. This work was supported in part by the U.S.-Slovene Science and Technology Joint Fund 95/8-06 and by the Ministry of Science of Slovenia. LA9600614 (25) Blum, L.; Henderson, D. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992; p 239. (26) Lozada-Cassou, M. In Fundamentals of Inhomogeneous Fluids; Henderson, D., Ed.; Marcel Dekker: New York, 1992; p 303. (27) Feller, S. E.; McQuarrie, D. A. Mol. Phys. 1993, 80, 721.
