Application of the Baxter Model for Hard Spheres with Surface

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Application of the Baxter Model for Hard Spheres with Surface Adhesion to SANS Data for the U(VI)-HNO3, TBP-n-Dodecane System Renato Chiarizia,*,† Ken L. Nash,† Mark P. Jensen,† Pappanan Thiyagarajan,‡ and Ken C. Littrell‡ Chemistry Division and IPNS Division, Argonne National Laboratory, Argonne, Illinois 60439 Received April 11, 2003. In Final Form: August 25, 2003 Small-angle neutron scattering (SANS) data for the tri-n-butyl phosphate (TBP)-n-dodecane, HNO3UO2(NO3)2 solvent extraction system have been interpreted using the Baxter model for hard spheres with surface adhesion. The increase in the scattering intensity in the low Q range observed when increasing amounts of HNO3 or UO2(NO3)2 are transferred into the organic phase has been interpreted as arising from interactions between solute particles. The SANS data have been reproduced using a 12-16 Å diameter of the hard sphere, dhs, and a 5.6kBT-7.1kBT stickiness parameter, τ-1. When in contact with an aqueous phase, TBP in n-dodecane forms small reverse micelles containing three TBP molecules. Upon extraction of water, HNO3, and UO2(NO3)2, the swollen micelles interact through attractive forces between their polar cores with a potential energy of about 2kBT and an effective Hamaker constant of about 4kBT. The intermicellar attraction, under suitable conditions, leads to third-phase formation. Upon phase splitting, most of the solutes in the original organic phase (water, TBP, HNO3, and UO2(NO3)2) separate in a continuous phase containing interspersed layers of n-dodecane.

Introduction Third-phase formation in liquid-liquid extraction of metal species is observed when, at high metal or mineral acid loading of the organic phase under suitable experimental conditions, the organic phase splits into two layers. The light layer contains most of the diluent and low concentrations of extractant, metal, and acid, while the heavy or “third” phase is a highly concentrated solution of extractant, metal, and acid. This phenomenon has been investigated in a number of studies and excellent reviews have summarized the most important aspects of thirdphase formation in solvent extraction.1,2 The main focus in previous works has been on the conditions under which third-phase formation is observed or avoided. Preventing third-phase formation is of paramount importance in solvent extraction systems of nuclear interest, like the tri-n-butyl phosphate (TBP)-n-dodecane system, to avoid the potentially catastrophic consequences of accidentally assembling a critical mass of fissionable materials in third phases. The composition of the species present in the heavy organic phase has also been the subject of several investigations. Very little information, however, is available on structural aspects of third-phase formation. Third-phase formation has traditionally been attributed to insufficient solubility of the metal-ligand complex in the nonpolar organic-phase diluent.3 To explain this phenomenon, however, other approaches have been followed based on the similarity of solvent extraction systems * Corresponding author. E-mail: [email protected]. Fax: 630252-7501. † Chemistry Division, Argonne National Laboratory. ‡ IPNS Division, Argonne National Laboratory. (1) Vasudeva Rao, P. R.; Kolarik, Z. A Review of Third Phase Formation in Extraction of Actinides by Neutral Organophosphorus Extractants. Solvent Extr. Ion Exch. 1996, 14, 955-993. (2) Kertes, A. S. The Chemistry of the Formation and Elimination of a Third Phase in Organophosphorus and Amine Extraction Systems. In Solvent Extraction Chemistry of Metals; McKay, H. A. C., Healy, T. V., Jenkins, I. L., Naylor, A., Eds.; McMillan: London, 1965; 377-399. (3) Marcus, Y.; Kertes, A. S. Ion Exchange and Solvent Extraction of Metal Complexes; Wiley-Interscience: New York, 1969, p 715.

with surfactant solutions. This is not surprising because reagents used in solvent extraction, as a general rule, are amphiphiles; that is, they possess both a hydrophilic and a hydrophobic nature. The hydrophilic nature is due to the functional group that specifically interacts with the target metal species, while the hydrophobic part of the molecule ensures solubility in water-immiscible organic diluents. As a consequence, commonly used extractants exhibit surface-active properties,4 and a comparison of the behavior of their solutions to that of conventional surfactants is, therefore, justified. The presence of reverse micellar structures and the formation of a microemulsion have been recently proposed for TBP solvent extraction systems.5-8 According to the microemulsion approach, the third phase observed when alkane solutions of TBP are used to extract mineral acids or metal salts from highly acidic aqueous solutions corresponds to the middle phase of a Winsor III system, in which the microemulsion (middle phase) is in equilibrium with both the aqueous and the oil phase. The propensity of TBP to originate a microemulsion under these conditions is ascribed to the significant surfactant character exhibited (4) Cox, M.; Flett, D. S. The Significance of Surface Activity in Solvent Extraction Reagents. In Proc. Int. Solv. Extr. Conf. ISEC 77, CIM Special Volume 21; Lucas, B. H., Ritcey, G. M., Smith, H. W., Eds.; The Canadian Institute of Mining and Metallurgy: Montreal, 1979; vol. 1, pp 63-72. (5) Osseo-Asare, K. Volume Changes and Distribution of HCl and H2O in the Tri-n-butyl Phosphate-H2O-HCl Liquid-Liquid System: A Reversed Micellar Phenomenological Model. Colloids Surf. 1990, 50, 373-392. (6) Osseo-Asare, K. Aggregation, Reversed Micelles and Microemulsions in Liquid-Liquid Extraction: The Tri-n-butyl Phosphate-DiluentWater-Electrolyte System. Adv. Colloid Interface Sci. 1991, 37, 123173. (7) Osseo-Asare, K. Third Phase Formation in Solvent Extraction: A Microemulsion Model. In Metal Separation Technologies Beyond 2000: Integrating Novel Chemistry with Processing; Liddel, K. C., Chaiko, D. J., Eds.; The Minerals, Metals & Materials Society: Warrendale, PA, 1999; pp 339-346. (8) Osseo-Asare, K. Microemulsions and Third Phase Formation. In Proc. Int. Solv. Extr. Conf. ISEC 2002; Sole, K. C., Cole, P. M., Preston, J. S., Robinson, D. J., Eds.; S. African Institute of Mining and Metallurgy: Johannesburg, 2002; pp 118-124.

10.1021/la030152i CCC: $25.00 © 2003 American Chemical Society Published on Web 10/11/2003

Application of the Baxter Model

by TBP when water molecules, protons, or metal ions are bonded to its phosphoryl group.7,9-11 Third-phase formation in solvent extraction strongly resembles cloud-point phase separation, a reversible critical phenomenon usually observed for aqueous solutions of nonionic surfactants.12,13 When aqueous micellar solutions of these substances are heated to a certain temperature (the cloud point), they become turbid. Upon a further increase of the temperature, the system splits into two phases, one almost surfactant-free, the other surfactant-rich. The same phenomenon can occur for relatively concentrated aqueous solutions of ionic surfactants upon addition of salts,12,13 an effect making the analogy with solvent extraction even more striking. Particularly relevant for the present study is the observation that these critical phenomena are triggered by a large growth of the micellar aggregation number, that is, the size of the aggregates.13 Several authors, in fact, have reported an increase of the size of the micelles as the system approaches critical conditions.14-20 Other authors, however, have maintained that phase separation is not preceded by aggregate size growth but, rather, by increasingly long-ranged spatial correlations between small micelles, characterized by critical concentration fluctuations due to attractive intermicellar interactions.21-30 Also, in the solvent extraction of metal species by a variety of extractants in nonpolar diluents, recent reports have recognized that the phenomenon of third-phase formation is preceded, in general, by extensive aggregation or polymerization of the metal-extractant complexes in (9) Tomoaia, M.; Andrei, Z.; Chifu, E. Tension Interfaciale Dans le Syste`me Solution Benzeńique de Tributyl Phosphate/Solution Aqueuse de Nitrate Cuivrique. Rev. Roum. Chim. 1973, 18, 1547-1554. (10) Sagert, N. H.; Lee, W.; Quinn, M. J. The Adsorption of Tri-nbutyl Phosphate at the n-Dodecane-Water Interface. Can. J. Chem. 1979, 57, 1218-1223. (11) Shmidt, V. S.; Nikitin, S. D. Influence of the Nature of the Components of the Extraction Systems Used for the Extraction of Radionuclides on the Interfacial Tension. 1. Influence of the Nature of the Diluents in Systems with Tributyl Phosphate. Sov. Radioch. 1983, 25, 399-404. (12) Rosen, M. J. Solubilization by Solutions of Surfactants: Micellar Catalysis. Surfactants and Interfacial Phenomena, 2nd ed.; Wiley: New York, 1989; pp 171-206. (13) Hinze, W. L.; Pramauro, E. A Critical Review of SurfactantMediated Phase Separations (Cloud Point Extractions): Theory and Applications. Crit. Rev. Anal. Chem. 1993, 24, 133-177. (14) Hoffmann, H.; Kielman, H. S.; Pavlovic, D.; Platz, G.; Ulbricht, W. Kinetic Investigations at the Cloud Point of Non-Ionic Surfactants. J. Colloid Interface Sci. 1981, 80, 237-239. (15) Kjellander, R. Phase Separation on Non-Ionic Surfactant Solutions. J. Chem. Soc., Faraday Trans. 2 1982, 78, 2025-2042. (16) Brown, W.; Johnsen, R.; Stilbs, P.; Lindman, B. Size and Shape of Non-Ionic Amphiphile (C12E6) Micelles in Dilute Aqueous Solutions as Derived from Quasielastic and Intensity Light Scattering, Sedimentation, and Pulsed-Field-Gradient Nuclear Magnetic Resonance Self-Diffusion Data. J. Phys. Chem. 1983, 87, 4548-4553. (17) Ravey, J.-C. Lower Consolute Curve Related to Micellar Structure on Non-Ionic Surfactants. J. Colloid Interface Sci. 1983, 94, 289-291. (18) Blankschtein, D.; Thurston, G. M.; Benedek, G. B. Phenomenological Theory of Equilibrium Thermodynamic Properties and Phase Separation of Micellar Solutions. J. Chem. Phys. 1986, 85, 7268-7288. (19) Kato, T.; Seimiya, T. Study of Intermicellar Interaction and Micelle Size Distribution in Aqueous Solutions of Non-Ionic Surfactants by Measurements of Mutual Diffusion and Self-Diffusion Coefficients. J. Phys. Chem. 1986, 90, 3159-3167. (20) Lindman, B.; Wennerstro¨m, H. Non-Ionic Micelles Growth with Increasing Temperature. J. Phys. Chem. 1991, 95, 6053-6054. (21) Triolo, R.; Magid, L. J.; Johnson, J. S.; Child, H. R. Small-Angle Neutron Scattering from Aqueous Micellar Solutions of a Non-Ionic Surfactant as a Function of Temperature. J. Phys. Chem. 1982, 86, 3689-3695. (22) Zulauf, M.; Rosenbusch, J. P. Micelle Clusters of Octylhydroxyoligo-(oxyethylenes). J. Phys. Chem. 1983, 87, 856-862. (23) Toprakcioglu, C.; Dore, J. C.; Robinson, B. H.; Howe, A.; Chieux, P. Small Angle Neutron Scattering Studies of Microemulsions Stabilized by Aerosol-OT. Part 2. Critical Scattering and Phase Stability. J. Chem. Soc., Faraday Trans. 1 1984, 80, 413-422.

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the organic phase.31-41 The extractants used in these works, namely, octyl(phenyl)-N,N-diisobutylcarbamoylmethylphosphine oxide (CMPO),31,32 di(2-ethylhexyl)methylene-, -ethylene-, and -butylenediphosphonic acids,33-35 and N,N′-dimethyl-N,N′-dibutyl-2-tetradecylmalonamide (DMDBTDMA)36-41 are all bifunctional molecules, containing one PdO and one CdO, two PdO, or two CdO donor groups, respectively. In these systems, polymeric species in the presence of high metal concentrations are not surprising because metal ions can promote polymerization by bridging functional groups of different extractant molecules. In an attempt to verify that the formation of large aggregates in the organic phase before phase splitting is (24) Corti, M.; Minero, C.; Degiorgio, V. Cloud Point Transition in Non-Ionic Micellar Solutions. J. Phys. Chem. 1984, 88, 309-317. (25) Hayter, J. B. Determination of the Structure and Dynamics of Micellar Solutions by Neutron Small-Angle Scattering. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti M., Eds.; Elsevier: Amsterdam, 1985; pp 59-93. (26) Degiorgio, V. Non-Ionic Micelles. In Physics of Amphiphiles: Micelles, Vesicles and Microemulsions; Degiorgio, V., Corti, M., Eds.; Elsevier: Amsterdam, 1985; pp 303-335. (27) Zulauf, M.; Weckstro¨m, K.; Hayter, J. B.; Degiorgio, V.; Corti, M. Neutron Scattering Study of Micelle Structure in Isotropic Aqueous Solutions of Poly(oxyethylene) Amphiphiles. J. Phys. Chem. 1985, 89, 309-317. (28) Howe, A. M.; Toprakcioglu, C.; Dore, J. C.; Robinson, B. H. SmallAngle Neutron Scattering Studies of Microemulsions Stabilised by Aersol-OT. J. Chem. Soc., Faraday Trans. 1 1986, 82, 2411-2422. (29) Cummins, P. G.; Hayter, J. B.; Penfold, J. Staples, E. A SmallAngle Neutron Scattering Investigation of Shear-Aligned Hexaethyleneglycolmonohexa-decyl ether (C16E6) Micelles as a Function of Temperature. Chem. Phys. Lett. 1987, 138, 436-440. (30) Magid, L. J. Structure and Dynamics by Small-Angle Neutron Scattering. In Non-Ionic Surfactants, Physical Chemistry; Schick, M. J., Ed.; Surfactant Science Series Vol. 23; Marcel Dekker: New York, 1987; pp 677-752. (31) Thiyagarajan, P.; Diamond, H.; Horwitz, E. P. Small-Angle Neutron Scattering Studies of the Aggregation of Pr(NO3)3 - CMPO and PrCl3 - CMPO Complexes in Organic Solvents. J. Appl. Crystallogr. 1988, 21, 848-852. (32) Diamond, H.; Thiyagarajan, P.; Horwitz, E. P. Small-Angle Neutron Scattering Studies of Praseodymium - CMPO Polymerization. Solvent Extr. Ion Exch. 1990, 8, 503-513. (33) Chiarizia, R.; Urban, V.; Thiyagarajan, P.; Herlinger, A. W. Aggregation of P,P′-Di(2-ethylhexyl) Methanediphosphonic Acid and its Fe(III) Complexes. Solvent Extr. Ion Exch. 1998, 16, 1257-1278. (34) Chiarizia, R.; Urban, V.; Thiyagarajan, P.; Herlinger, A. W. Aggregation of Complexes Formed in the Extraction of Selected Metal Cations by P,P′-Di(2-ethylhexyl) Methanediphosphonic Acid. Solvent Extr. Ion Exch. 1999, 17, 113-132. (35) Chiarizia, R.; Urban, V.; Thiyagarajan, P.; Herlinger, A. W. SANS Study of Aggregation of the Complexes Formed by Selected Metal Cations and P,P′-Di(2-ethylhexyl) Ethane- and Butane-diphosphonic Acids. Solvent Extr. Ion Exch. 1999, 17, 1171-1194. (36) Erlinger, C.; Gazeau, D.; Zemb, Th.; Madic, C.; Lefranç ois, L.; Hebrant, M.; Tondre, C. Effect of Nitric Acid Extraction on Phase Behavior, Microstructure and Interactions Between Primary Aggregates in the System Dimethyldibutyltetradecylmalonamide (DMDBTDMA)/ n-Dodecane/Water: A Phase Analysis and Small-Angle X-ray Scattering (SAXS) Characterisation Study. Solvent Extr. Ion Exch. 1998, 16, 707738. (37) Lefranç ois, L.; Belnet, F.; Noel, D.; Tondre, C. Third Phase Formation: A New Predictive Approach. In Solvent Extraction for the 21st Century. Proceedings of ISEC ‘99; Cox, M., Hidalgo, M., Valiente, M., Eds.; Society of Chemical Industry (SCI): London, 2001; Vol. 1, pp 637-641. (38) Lefranç ois, L.; Belnet, F.; Noel, D.; Tondre, C. An Attempt to Theoretically Predict Third-Phase Formation in the Dimethyldibutyltetradecylmalonamide (DMDBTDMA)/Dodecane/Water/Nitric Acid Extraction System. Sep. Sci. Technol. 1999, 34, 755-770. (39) Erlinger, C.; Belloni, L.; Zemb, Th.; Madic, C. Attractive Interactions Between Reverse Aggregates and Phase Separation in Concentrated Malonamide Extractant Solutions. Langmuir 1999, 15, 22902300. (40) Mandin, C.; Martinet, L.; Zemb, Th.; Berthon, L.; Madic, C. Caracte´risation de l’Auto-Organisation du Phosphate de Tributyl en Solvent Organique. Report CEA-R5930, Commissariat a l’Energie Atomique, France, 2000. (41) Lefranç ois, L.; Delpuech, J.-J.; He´brant, M.; Chrisment, J.; Tondre, C. Aggregation and Protonation Phenomena in Third Phase Formation: An NMR Study of the Quaternary Malonamide/Dodecane/ Nitric Acid/Water System. J. Phys. Chem. 2001, 105, 2551-2564.

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Table 1. Sample Compositions and Results from Ellipsoid of Rotation Fit sample

[TBP]a M

[U],b M

[HNO3],b M

[H2O],c M

ηd

a, minor semiaxis, Å

b, major semiaxis, Å

1 2a 2b 2c 3a 3b 3c 3d 3e 4a 4b 4c (LOC) 4d (third phase) 4e (light phase)

0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 0.73 1.9 0.51

0 0.032 0.16 0.34 0 0 0 0 0 0.032 0.22 0.23 0.93 0.17

0 0 0 0 0.027 0.30 0.48 0.69 0.74 0.72 0.28 0.27 0.94 0.21

0.096 e e e e e e e 0.091 0.081 0.086 0.12 0.35 0.14

0.202 ( 0.018 0.202 ( 0.015 0.211 ( 0.015 0.224 ( 0.016 0.201 ( 0.015 0.212 ( 0.016 0.219 ( 0.016 0.227 ( 0.020 0.231 ( 0.023 0.232 ( 0.022 0.228 ( 0.018 0.229 ( 0.020 0.63 ( 0.10 0.163 ( 0.029

5.1 ( 0.4 5.6 ( 0.2 6.8 ( 0.1 7.8 ( 0.1 5.6 ( 0.3 6.8 ( 0.1 6.6 ( 0.1 7.0 ( 0.1 6.7 ( 0.1 6.7 ( 0.1 7.4 ( 0.1 7.5 ( 0.1 6.5 ( 0.3 7.1 ( 0.2

17.6 ( 0.3 19.4 ( 0.3 23.5 ( 0.2 28.6 ( 0.2 18.3 ( 0.3 25.8 ( 0.2 27.2 ( 0.3 28.1 ( 0.3 25.9 ( 0.3 27.1 ( 0.2 28.8 ( 0.3 32.1 ( 0.3 21.4 ( 0.4 25.2 ( 0.4

a Estimated accuracy ( 2%, except for samples 4d,e (( 5%). b Estimated accuracy ( 2%. c Estimated accuracy ( 10%. d Solute volume fraction given by the sum of the TBP, UO2(NO3)2, HNO3, and H2O volume fractions. The molar volumes of the components were taken as equal to 273.79, 70.70, 39.93, and 18.02 cm3, respectively. From the available density of UO2(NO3)2‚6(H2O) (2.807 g/cm3),48 the density of uranyl nitrate was estimated to be 5.566 g/cm3. e Not determined.

a general feature shared by most solvent extraction systems, including those based on monofunctional extractants, and to solidify the links between solvent extraction systems and surfactant solutions, we have recently revisited the HNO3-UO2(NO3)2, TBP-n-dodecane system from the standpoint of third-phase formation.42,43 In this study, small-angle neutron scattering (SANS) measurements were performed on TBP solutions loaded with only HNO3 or increasing amounts of U(VI). The SANS data, interpreted using an ellipsoidal particle model, revealed the presence, before phase splitting, of aggregates of the reverse micelle type with the major and minor axes up to about 64 and 15 Å, respectively. For the interpretation of the SANS data reported previously,42,43 intermicellar interactions were not considered. In that study, however, a high concentration of TBP (20% (v/v) or 0.73 M) was used. The high extractant concentration was dictated by the need to generate a relatively large amount of the third phase that could be easily separated and characterized. Under these conditions, it is possible that attractive interactions exist among the swollen TBP reverse micelles and that these interactions may play an important role in the organic-phase splitting. Similar interactions between small reverse micelles in the DMDBTDMA-dodecane-HNO3 solvent extraction system, measured through small-angle X-ray scattering and modeled with Baxter’s model for hard spheres with surface adhesion,44-46 were shown to result mostly from the van der Waals forces between the polar cores of the reverse micelles.36,39 The objective of the present work was to determine whether the HNO3-UO2(NO3)2, TBP-n-dodecane SANS data, previously interpreted solely with a micellar growth model,42,43 could be successfully interpreted by using the (42) Jensen, M. P.; Chiarizia, R.; Ferraro, J. R.; Borkowski, M.; Nash, K. L.; Thiyagarajan, P.; Littrell, K. C. New Insights in Third Phase Formation in the U(VI)-HNO3, TBP-Alkane System. In Proc. Int. Solv. Extr. Conf. ISEC 2002; Sole, K. C., Cole, P. M., Preston, J. S., Robinson, D. J., Eds.; S. African Institute of Mining and Metallurgy: Johannesburg, 2002; pp 1137-1142. (43) Chiarizia, R.; Jensen, M. P.; Borkowski, M.; Ferraro, J. R.; Thiyagarajan, P.; Littrell, K. C. Third Phase Revisited: the U(VI), HNO3 - TBP, n-Dodecane System. Solvent Extr. Ion Exch. 2003, 21, 1-27. (44) Baxter, R. J. Percus-Yevick Equation for Hard Spheres with Surface Adhesion. J. Chem. Phys. 1968, 49, 2770-2774. (45) Menon, S. V. G.; Kelkar, V. K.; Manohar, C. Application of Baxter’s Model to the Theory of Cloud Points of Nonionic Surfactant Solutions. Phys. Rev. A 1991, 43, 1130-1133. (46) Goyal, P. S.; Menon, S. V. G.; Dasannacharya, B. A.; Thiyagarajan, P. Small-Angle Neutron Scattering Study of Micellar Structure and Interparticle Interactions in Triton X-100 Solutions. Phys. Rev. E 1995, 51, 2308-2315.

Baxter sticky hard-spheres model to quantify intermicellar interactions, leading to an explanation of third-phase formation based on intermicellar interactions rather than on micellar growth. Investigated Samples The preparation and analytical characterization of the TBP samples in deuterated n-dodecane containing various amounts of HNO3 and UO2(NO3)2 have been reported in detail in previous works.42,43 The SANS measurements were performed at the time-of-flight small-angle neutron diffractometer at the Intense Pulsed Neutron Source of Argonne National Laboratory.47 The characteristics of the diffractometer, the procedure for placing the data on an absolute scale, and the fit of the data to the form factor for an ellipsoid of rotation have been reported previously.43 Table 1 summarizes the compositions of the organic phases investigated in this work, together with the results from the ellipsoid of rotation fit. Sample 1 contained only TBP. The samples of series 2 were prepared by dissolving increasing amounts of solid UO2(NO3)2‚6H2O in the 20% TBP solution. The samples of series 3 were prepared by extracting HNO3 from aqueous solutions having progressively higher HNO3 concentrations (0.5, 2.5, 5.0, 7.5, and 10 M). Although TBP is known to form species with HNO3 having the composition TBP‚(HNO3)i, with i ) 1-4,49 the analytical data in Table 1 indicate that under the experimental conditions of this work HNO3 exists in the organic phase predominantly as the 1:1 complex with TBP. Finally, the samples of series 4 were prepared by contacting the TBP solution with aqueous 10 M HNO3 containing progressively increasing concentrations of UO2(NO3)2. Within the experimental uncertainty, the data in Table 1 show that for each 1 mol of UO2(NO3)2 entering the organic phase, 2 mol of HNO3 were displaced from it. This is consistent with the hypothesis that the uranium salt forms the UO2(NO3)2‚2TBP adduct with TBP,50 while the excess TBP remains in the form of TBP‚HNO3. One of the samples of this series (4c) contained the U(VI) limiting organic concentration (LOC), that is, the highest metal concen(47) Thiyagarajan, P.; Urban, V.; Littrell, K. C.; Wozniak, D. G.; Belch, H.; Vitt, R.; Toeller, J.; Leach, D. J.; Haumann, J. R.; Ostrowski, G. E.; Donley, L. I.; Hammonds, J. P.; Carpenter, J. M.; Crawford, R. K. The Performance of the Small-Angle Diffractometer SAND at IPNS. In ICANS XIV, Proc. 14th Meeting on the International Collaboration on Advanced Neutron Sources; Carpenter, J. M., Tobin, C., Eds.; National Technical Information Service: Springfield, VA, 1998; Vol. 2, pp 864878. (48) Handbook of Chemistry and Physics, 64th ed.; Weast, R. C., Ed.; CRC Press: Boca Raton, FL, 1983; p B-153. (49) Schaekers, J. M. Extraction of Inorganic Acids. In Science and Technology of Tributyl Phosphate; Schulz, W. W., Navratil, J. D., Kertes, A. S. Eds.; CRC Press: Boca Raton, FL, 1991; Vol. IV, pp 71-204. (50) McKay, H. A. C. Tri-n-butyl Phosphate as an Extracting Agent for the Nitrates of the Actinide Elements. In Proc. Int. Conf. Peac. Uses Atomic Energy; United Nations: New York, 1956; Vol. 7, pp 314-317.


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µ ) ηλ(1 - η)

(8)

γ ) η(1 + 0.5η)/[3(1 - η)2]

(9)

) τ + [η/(1 - η)]

(10)

λ ) (6/η)[ - (2 - γ)0.5]

(11)

Chart 1

tration in the organic phase that could be achieved under the selected experimental conditions without phase splitting. When contacting the 20% TBP organic phase with an aqueous solution that was 0.66 M UO2(NO3)2 and 10.5 M HNO3, a third phase was obtained having a volume of about 16% of the original organic-phase volume and a density of 1.21 g/mL (sample 4d). The light organic phase in equilibrium with the aqueous phase and the third phase is represented by sample 4e. The data in Table 1 indicate that the third phase is highly concentrated in TBP, U(VI), and HNO3 and also has a higher water content than the other samples. The U(NO3)2/TBP/HNO3 concentration ratios in the third-phase sample are about 1:2:1, in agreement with previous results obtained under similar conditions.51

Interparticle Interaction Model A convenient way to calculate short-range interactions between colloidal aggregates is offered by Baxter’s model for hard spheres with surface adhesion.44 According to this model and with reference to Chart 1, the hard-sphere sticky potential, U(r), expressed in kBT units (kB ) Boltzmann constant) is given by the following equations:

U(r) ) ∞ for r < dhs

(1)

[ (

(2)

U(r) ) lim ln 12τ δfdhs

)]

δ - dhs dhs

for dhs < r < δ

U(r) ) 0 for r > δ

(3)

where r is the distance between two particles, dhs is the hard-sphere diameter, that is, the distance from the origin below which the particles become incompressible, and δ - dhs represents the width of the square-well attraction potential. Equation 1 implies that when the separation r between two particles equals dhs, the repulsion becomes infinite. Equation 3 indicates that, for separations larger than δ, the attraction between the particles vanishes. Equation 2 allows the calculation of the interparticle attraction potential energy for very small separations [(δ - dhs)/dhs e 0.1], if the parameter τ is known. τ-1, also expressed in kBT units, is the “stickiness parameter”, and its value is higher when the adhesion between particles is stronger. An important advantage of the Baxter model approximation is that analytical expressions have been derived for the structure factor S(Q) in terms of the parameter τ. The previously reported39,45 expressions have been used in this work. They are repeated here for the sake of clarity (with x ) Qdhs and η ) particle volume fraction):

S(Q) ) [1 - C(Q)]-1 -6

Figure 1 (experimental points) shows, as an example, the SANS data obtained for the first four samples in Table 1 as the scattering intensity versus the momentum transfer Q ) (4π/λ) sin θ, where θ is 1/2 the scattering angle and λ is the wavelength of the neutrons. The data indicate that, as more UO2(NO3)2 is added to the TBP solution, the scattering intensity at low Q values increases. This has been interpreted previously as due to changes in the particle form factor P(Q), implying micellar growth.43 The same effect can arise, however, from changes in the structure factor S(Q) due to intermicellar interactions (see eq 12 below). Because of the relatively high TBP concentration in our samples, interactions between micelles are highly likely and have been calculated using the model outlined above. The scattering intensity for a monodisperse system of particles can be written as

I(Q) ) (∆F)2NpVp2P(Q) S(Q) + Iinc

C(Q) ) -24ηx {Rx (sin x - x cos x) + βx2[2x sin x - (x2 - 2) cos x - 2] + 0.5ηR[(4x3 24x) sin x - (x4 - 12x2 + 24) cos x + 24]} 2η2λ2(1 - cos x)x-2 + 2ηλx-1 sin x (5) 4

R ) (1 + 2η - µ) /(1 - η)

(6)

β ) -[3η(2 + η)2 - 2µ(1 + 7η + η2) + µ2(2 + η)]/[2(1 - η)4] (7)

(12)

where Iinc is is the incoherent scattering background (cm-1), Np is the number of scattering units per unit volume (cm-3), Vp is the particle volume (cm3), and (∆F)2 is the contrast factor (cm-4; in our experiments the contrast between the solute and the solvent was determined by the different neutron scattering properties of the H atoms of TBP and the D atoms of deuterated n-dodecane). By defining the TBP average aggregation number, n, as the number of TBP molecules present in each aggregate and by neglecting the volume of the polar core of the micelles, it follows

Np ) NTBP/n

(13)

Vp ) nVTBP

(14)

where NTBP and VTBP are the number of TBP molecules per the unit volume and the TBP molecular volume, respectively. By substituting eqs 13 and 14 into eq 12, one obtains

I(Q) ) (∆F)2NTBPVTBPVpP(Q) S(Q) + Iinc

(15)

Equation 15 has been used for the Baxter model calculations as described in the following. Given the preliminary values of Iinc, dhs, and τ, Vp was calculated as the volume of a sphere of radius dhs/2, P(Q) was given by the form factor of a sphere52 with radius R ) dhs/2,

(4)

3

2

Results and Discussion

P(Q) )

{

}

3[sin(QR) - QR cos(QR)] (QR)3

2

(16)

and S(Q) was obtained from eqs 4-11. NTBP was provided (51) Solovkin, A. S.; Povitskii, N. S.; Lunichkina, K. P. Formation of a Third Phase in the System UO2(NO3)2-HNO3-H2O-Tri-n-butyl Phosphate-Kerosene. Russ. J. Inorg. Chem. (Engl. Transl.) 1960, 5, 10261028. (52) Pedersen, J. S. Analysis of Small-Angle Scattering Data from Colloids and Polymer Solutions: Modeling and Least-Squares Fitting. Adv. Colloid Interface Sci. 1997, 70, 171-210.

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Figure 1. SANS data and I(Q) curves calculated for samples 1 and 2a-c using the fit parameters reported in Table 2.

Chiarizia et al.

Figure 3. SANS data and I(Q) curves calculated for sample 2c using the fit parameters reported in Table 2 with a (10% variation of the τ and dhs values.

Figure 2. SANS data and I(Q), P(Q), and S(Q) curves calculated for sample 2c using the fit parameters reported in Table 2.

by the known TBP concentration, 4.55 × 10-22 cm3 was used for VTBP (weight of one TBP molecule divided by the TBP density), and 4.35 × 1021 cm-4 was the value of the contrast factor.43 The calculated I(Q) values were compared with the experimental ones, and the three parameters Iinc, dhs, and τ were optimized using the “Optimizer” tool of the spreadsheet program Quattro Pro 5.0 (Borland). The best fit for each data set was obtained by minimizing the sum of the squared differences between the experimental and the calculated I(Q) values divided by the experimental error associated with the I(Q) values. The calculations were also repeated using the nonlinear curve fitting features of the Origin program (Microcal Software, Inc.). The two programs provided the same best values for the three parameters Iinc, dhs, and τ. The Origin program also provided the uncertainty associated with the fitting parameters. Figure 1 (continuous lines) shows the fit of the SANS data for samples 1 and 2a-c. Figure 2 shows, as an example, the calculated P(Q) and S(Q) curves together with the experimental and calculated I(Q) values for sample 2c. The S(Q) curve exhibits a pronounced maximum at Q ) 0.5 Å-1, which is barely discernible in the experimental points and calculated I(Q) curve (as a result of the very low values of P(Q) in the high Q range). The Q value corresponding to the maximum in S(Q) can be used to obtain the correlation distance between interacting particles, d, through Bragg’s law in the form d ) 2π/Qmax. Figure 3 shows how the I(Q) results calculated for sample 2c change for 10% variations in either direction of the best dhs and τ values. The I(Q) curve is extremely sensitive in the low Q range to changes in dhs and τ. However, the figure shows that an increase in dhs is partly

Figure 4. SANS data and I(Q) curves calculated for samples of series 3 using the fit parameters reported in Table 2. Data for samples 3b-e are shifted vertically by 0.5, 1, 1.5, and 2 cm-1, respectively.

compensated by a decrease in τ. Despite the correlation between the dhs and τ parameters, the wide range of Q used in the fit procedure (from 0.005 to 1 cm-1) and the fact that the same results were obtained by using two different algorithms lend support to the validity of the best fit results. Figures 4 and 5 show a comparison of the experimental and the calculated intensity values for the samples of series 3 and 4, respectively. In these figures, some of the data have been translated vertically to avoid overlapping (see details in the figure captions). Table 2 summarizes for all the samples the values of Iinc, dhs, and τ provided by the Baxter model calculations. The table also includes the TBP aggregation numbers calculated using eq 14, the TBP aggregation numbers provided by the ellipsoidal fit43 for comparison, and the correlation distances calculated from the position of the S(Q) maximum. In the table, the errors reported for the dhs and n values are determined by the uncertainty on the solute volume fractions (η, see Table 1), which, in turn, depends on the experimental errors for the analytical determination of the various solution components. In the


Figure 5. SANS data and I(Q) curves calculated for samples of series 4 using the fit parameters reported in Table 2. Data for samples 4b, 4c(LOC), 4d(H) (third phase), and 4e(L) (light phase) are shifted vertically by 0.5, 1, 2.5, and 3 cm-1, respectively.

fitting procedures used, variations of the value of η within the experimental uncertainty increase only the errors on dhs and n, while the Iinc and τ parameters consistently exhibit very small uncertainty intervals. From the data in Table 2, it appears that dhs, the diameter of the particles, increases when increasing amounts of UO2(NO3)2 (samples of series 2) or HNO3 (samples of series 3) are introduced into the organic phase. This reflects the swelling that occurs when polar substances are accommodated in the polar core of the micelles. The effect is reversed in samples 4a-c, where HNO3 is progressively replaced by UO2(NO3)2, indicating that the former is more effective than the latter in promoting micellar swelling. The TBP aggregation number for sample 1, which contains only TBP without U(VI) or HNO3, is identical to the n value provided by the ellipsoidal fit of the data. For this sample, the two models, that is, interparticle interaction and micellar growth, are equivalent because both provide TBP aggregation numbers consistent with literature information regarding solutions containing only TBP (a dimerization constant of TBP in n-dodecane of 2.6 has been reported).53 When polar solutes (H2O, HNO3, UO2(NO3)2) are introduced into the TBP solution through solubilization or extraction from an aqueous phase, TBP micellization takes place. For the other samples, an increase in dhs translates into a small increase in the TBP aggregation number. The n values, however, remain significantly smaller than those provided by the ellipsoidal fit, as expected, because in the latter case the entire increase in scattering was interpreted as arising from particle growth. The average value of the TBP aggregation numbers for all the samples of series 2-4 in Table 2 is equal to 3.2. It is reasonable to conclude, therefore, that each micelle contains three TBP molecules, in partial agreement with the conclusions reported earlier based on an analysis of previous literature data reinterpreted from the point of view of TBP micellization.6 Application of the three-dimensional Molecular Modeling Software Alchemy III (Tripos Associates, Inc.) to the

Langmuir, Vol. 19, No. 23, 2003 9597

TBP molecule in a vacuum provided a length of the molecule of 7.11 Å (average distance from the last carbon of each butyl group to the oxygen atom of the PdO group). A spherical micelle containing three TBP molecules plus a small polar core should, therefore, have a diameter g 14.2 Å, which is higher than the 12-13 Å dhs provided by the hard-sphere model for the samples with the lowest U(VI) or HNO3 concentration (2a,b and 3a,b). This is an indication that the micelles are somewhat permeable, and some interpenetration occurs between two interacting micelles. This conclusion is also supported by the values of the interparticle correlation distances reported in Table 2. The values of this distance are systematically about 2 Å smaller than the corresponding dhs values, which may be taken as an indication that two interacting particles overlap by about 2 Å. The third-phase sample (4d) represents a unique case. Attempts to fit the SANS data in the same way as that for the other samples led to a very poor fit with a meaningless negative τ value. This failure is probably due to the fact that the sticky sphere model is only valid for solute volume fractions well below 50%.39,44 A much better fit (shown in Figure 5) was obtained by considering the third-phase sample, which has much higher concentrations of TBP, UO2(NO3)2, and HNO3 than the other samples, as a continuous phase of average composition UO2(NO3)2‚2TBP‚HNO3‚0.35H2O containing interspersed layers of n-dodecane as the solute. A similar picture of the third phase was given earlier for the third phase of our TBP system,43 as well as for the third phase formed in the DMDBTDMA system.36 For the calculations, the ndodecane volume fraction in the third-phase sample (0.37 ( 0.1) and a contrast factor of 3.42 × 1021 cm-4 were used. This value of the contrast factor was obtained by comparing the scattering length density of deuterated n-dodecane with that of a medium having the UO2(NO3)2‚2TBP‚HNO3‚0.35H2O composition. A dhs value of 13.3 ( 0.7 Å was obtained for the thirdphase solution, while the value of n was 3.3 ( 0.6. In this case, n does not represent an aggregation number; rather, it represents the number of molecules contained in the dodecane layers interspersed in the third phase. An Alchemy III three-dimensional model of n-dodecane indicated that this molecule can be considered as a cylinder 15.85 Å long and 2.77 Å thick. The volume occupied by three stacked n-dodecane molecules is 862 Å3. The radius of the equivalent sphere is 5.9 Å, which corresponds to a dhs value of 11.8 Å. Considering the approximations involved in the calculations, this value is in good agreement with the value provided by the model. The τ values in Table 2 decrease when more HNO3 or UO2(NO3)2 molecules are transferred into the organic phase. This reflects an increase of the short-range attraction forces between the polar core of the micelles due to dipole/dipole interactions. The τ value for the LOC sample is 0.140. Attempts to introduce more HNO3 and UO2(NO3)2 into this solution, which would further decrease the τ value, causes organic-phase splitting. Therefore, the critical τ value can be taken as 0.140. After phase splitting, the heavy phase exhibits a τ value of 0.127, while the light phase has a τ value of 0.150, that is, higher than the critical one. From the values of τ for each sample, it is possible, using eq 2, to calculate the attractive potential energy, U(r) in kBT units (at the temperature of the SANS (53) Vandegrift, G. F. Diluents for TBP Extraction Systems. In Science and Technology of Tributyl Phosphate; Schulz, W. W., Navratil, J. D., Eds.; CRC Press: Boca Raton, FL, 1984; Vol. I, pp 69-136.

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Chiarizia et al.

Table 2. Results from Baxter Model Calculations sample

Iinc, cm-1

dhs, Å

τ, (kBT)-1

na

n from ellips. fitb

correlation distance,c Å


0.113 ( 0.005 0.101 ( 0.005 0.083 ( 0.007 0.069 ( 0.008 0.112 ( 0.005 0.097 ( 0.006 0.110 ( 0.005 0.110 ( 0.006 0.128 ( 0.007 0.131 ( 0.007 -0.003 ( 0.007 0.022 ( 0.006 0.175 ( 0.007 0.019 ( 0.005

11.6 ( 0.3 12.2 ( 0.3 13.2 ( 0.4 14.1 ( 0.4 11.7 ( 0.3 13.4 ( 0.4 12.9 ( 0.4 12.9 ( 0.5 16.1 ( 0.6 16.3 ( 0.6 14.6 ( 0.5 14.7 ( 0.6 13.3 ( 0.7 13.9 ( 0.7

0.180 ( 0.004 0.172 ( 0.003 0.156 ( 0.003 0.143 ( 0.002 0.174 ( 0.003 0.151 ( 0.002 0.146 ( 0.002 0.142 ( 0.002 0.164 ( 0.002 0.163 ( 0.002 0.148 ( 0.002 0.140 ( 0.002 0.127 ( 0.031 0.150 ( 0.004

1.8 ( 0.2 2.1 ( 0.2 2.6 ( 0.2 3.2 ( 0.3 1.9 ( 0.1 2.8 ( 0.2 2.5 ( 0.2 2.5 ( 0.3 4.8 ( 0.3 5.0 ( 0.6 3.6 ( 0.4 3.7 ( 0.4 d 3.1 ( 0.5

1.8 ( 0.2 2.2 ( 0.2 3.3 ( 0.2 4.7 ( 0.3 1.9 ( 0.2 3.8 ( 0.2 3.7 ( 0.2 3.8 ( 0.2 5.0 ( 0.3 5.4 ( 0.4 4.9 ( 0.3 5.8 ( 0.4 d 4.7 ( 0.5

9.7 10.5 11.2 11.6 9.8 11.2 11.0 10.8 13.4 14.0 12.1 12.1 11.0 11.4

a Weight-average aggregation number of TBP. b Values taken from ref 43. c Estimated from the position of the maximum in each S(Q) versus Q plot through the relation d ) 2π/Qmax. d See text for discussion of the third-phase sample.

Table 3. Stickiness Parameter (τ-1), Potential Energy of Attraction (U(r)), and Effective Hamaker Constant (Aeff) for the Investigated Samples sample

τ-1, (kBT)

U(r), (kBT)

Aeff, (kBT)


5.56 ( 0.12 5.81 ( 0.10 6.41 ( 0.12 6.99 ( 0.10 5.73 ( 0.10 6.63 ( 0.09 6.86 ( 0.09 7.06 ( 0.10 6.09 ( 0.07 6.15 ( 0.08 6.73 ( 0.09 7.14 ( 0.10 7.87 ( 1.8 6.64 ( 0.18

-1.53 ( 0.02 -1.58 ( 0.02 -1.67 ( 0.02 -1.76 ( 0.01 -1.56 ( 0.02 -1.71 ( 0.01 -1.74 ( 0.01 -1.77 ( 0.01 -1.62 ( 0.01 -1.63 ( 0.01 -1.72 ( 0.01 -1.78 ( 0.01 -1.88 ( 0.25 -1.71 ( 0.03

3.68 ( 0.16 3.78 ( 0.14 4.02 ( 0.16 4.23 ( 0.18 3.75 ( 0.14 4.10 ( 0.17 4.18 ( 0.19 4.25 ( 0.23 3.90 ( 0.21 3.92 ( 0.21 4.14 ( 0.20 4.28 ( 0.23 4.51 ( 0.70 4.11 ( 0.29

measurements, 23 °C, kBT ) 4.09 × 10-21 J) between two particles separated by a distance equal to 10% of the hardsphere diameter. These values are reported in Table 3 and shown in Figure 6 together with the values of the stickiness parameter, τ-1, as a function of the UO2(NO3)2 concentration (samples of series 2), HNO3 concentration (samples of series 3), and HNO3 concentration plus three times the UO2(NO3)2 concentration (samples of series 4), in the organic phase. (In the latter case, the abscissa values in Figure 6, corresponding to the ionic strengths of aqueous solutions having the same compositons as the organic solutions, are used only as a convenient way to differentiate among the samples of series 4, without making any assumption on the actual state of the solutes in the organic phase.) In Figure 6, the data for the samples of series 4 clearly show a steep increase of τ-1 and of the attractive potential energy (more negative U(r) values), as the LOC condition is approached, leading to third-phase formation. The light organic phase in equilibrium with the third phase, on the other hand, exhibits τ-1 and U(r) values comparable to those of samples that are relatively far from phase splitting. The increasing trends of the τ-1 and U(r) values are less pronounced for the samples of series 2 and 3 because, under the conditions used for the preparation of these samples, third-phase formation did not take place. We have no explanation for the out-of-line behavior of sample 3e (highest HNO3 concentration), whose τ-1 and U(r) values seem rather low. The interpretation of the SANS data for the TBP-ndodecane, HNO3-UO2(NO3)2 system according to the Baxter model for hard spheres with surface adhesion provides a simple explanation of the phenomenon of thirdphase formation. When the LOC condition is not exceeded,

the increase in the stickiness parameter and intermicellar attraction energy, observed with increasing solute concentrations, has no chemical effect on the solvent extraction process. The small TBP micelles, however, are subjected to two contrasting physical forces. On one hand, the thermal energy, kBT, keeps the micelles dispersed in the solvent. On the other hand, the energy of intermicellar attraction makes the micelles stick together. As long as these two energies are comparable, the organic phase is stable. The separation of most of the solute particles in a new phase is energetically favorable and, therefore, expected when the energy of attraction between the particles in solution becomes substantially larger (about twice) than the thermal energy. To our knowledge, this is the first time that a quantitative explanation of thirdphase formation in TBP solvent extraction systems has been successfully attempted on the basis of an intermicellar interaction model. Intermicellar interactions can also be estimated using the Hamaker constant, A. This is a material property that represents the strength of van der Waals attraction (positive A) or repulsion (negative A). Typical values54 of A for simple materials, such as water, acetone, and toluene, are about 10kBT. For two spherical particles with the same radius, approximate values of the Hamaker constant (in kBT units) can be obtained from the U(r) values using the relation54

U(r) ) -AR/12ds for R . ds

(17)

where R is the radius and ds is the separation between the surfaces of the interacting particles. Table 3 and Figure 6 report the values of the effective Hamaker constants (Aeff) of the TBP micelles calculated with eq 17 using the dhs values of Table 2 and ds values equal to 10% of the hard-sphere diameter. Again, for the samples of series 4, the values of Aeff exhibit a significant increase when third-phase formation is approached, reaching the value of 4.3kBT for the LOC sample. The values of the effective Hamaker constant for the TBP micelles in Table 3 are somewhat higher than those reported previously for the DMDBTDMA micelles under comparable conditions.39 Nevertheless, the similarity between the two systems, when their solvent extraction behavior is interpreted with the same micellar interaction model, is remarkable. It is possible that this similarity may extend to many other solvent extraction systems. (54) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997; chapter 10, pp 462-498.


Langmuir, Vol. 19, No. 23, 2003 9599

Figure 6. Values of the stickiness parameter, τ-1/kBT, attraction potential energy, U(r)/kBT, and effective Hamaker constant, Aeff/kBT, versus the HNO3 or UO2(NO3)2 concentration in the organic phase.

Conclusions Prompted by literature reports on the behavior of DMDBTDMA36-41 reverse micelles in n-dodecane, we have interpreted SANS data for the TBP-n-dodecane, HNO3-UO2(NO3)2 solvent extraction system using the Baxter model for hard spheres with surface adhesion. According to this model, the increase in scattering intensity in the low Q range experimentally observed when increasing amounts of HNO3 or UO2(NO3)2 are transferred into the organic phase can be ascribed to interactions between solute particles. By using the available analytical expressions for the structure factor, the SANS data have been reproduced reasonably well by means of the independent parameters dhs, the diameter of the hard sphere, with values between 12 and 16 Å, and τ-1, the stickiness parameter, with values between 5.6 and 7.1. From these calculations, we conclude that TBP at high concentrations in n-dodecane, in contact with the aqueous phase containing nitric acid and uranyl nitrate, forms small reverse micelles containing three TBP molecules. These micelles swell when polar solutes such as water, nitric acid, and uranyl nitrate, bound to the PdO group of the TBP molecules, are incorporated into their polar core. The swollen micelles interact through attractive forces between the polar cores with a potential energy of about 2kBT and an effective Hamaker constant of about 4kBT. The stickiness parameter, potential energy of attraction, and effective Hamaker constant increase

significantly when, by introducing more HNO3 or UO2(NO3)2 in the organic phase, third-phase formation is approached. Upon phase splitting, most of the solutes in the original organic phase (water, TBP, HNO3, and UO2(NO3)2) collect in a separate phase containing interspersed layers of n-dodecane. The sticky hard-sphere model has been successfully applied also to the interpretation of the SANS data obtained for a third-phase sample. As a general conclusion, the present study strongly supports the opinion expressed by the authors of previous seminal works on micellization of solvent extraction reagents.5-8,36-41 According to these authors, complex phenomena often observed in solvent extraction, such as thirdphase formation, can be understood and predicted through an approach based on the synergism between coordination chemistry and colloid chemistry. Acknowledgment. This work was funded by the U.S. Department of Energy, Office of Basic Energy Science, Division of Chemical Science (for the part performed at the Chemistry Division of ANL) and Division of Material Science (for the part performed at INPS), under Contract No. W-31-109-ENG-38. The authors express their gratitude to Denis Wozniak (INPS) for the help provided in the SANS measurements. LA030152I

Application of the Baxter Model for Hard Spheres with Surface

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