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Langmuir 1999, 15, 2290-2300
Attractive Interactions between Reverse Aggregates and Phase Separation in Concentrated Malonamide Extractant Solutions C. Erlinger,†,‡ L. Belloni,‡ Th. Zemb,*,‡ and C. Madic† CEA, DCC/DRRV/SEMP CEA-Valrhoˆ (Marcoule), B.P. 171, 30207 Bagnols-sur-Ce` ze, Cedex, France, and CEA, DSM/DRECAM/SCM CEA-Saclay, 91191 Gif-sur-Yvette, Cedex France Received March 18, 1998. In Final Form: November 2, 1998
Using small angle X-ray scattering, conductivity, and phase behavior determination, we show that concentrated solutions of molanamide extractants, dimethyldibutyltetradecylmalonamide (DMDBTDMA), are organized in reverse oligomeric aggregates which have many features in common with reverse micelles. The aggregation numbers of these reverse globular aggregates as well as their interaction potential are determined from absolute scattering curves. An attractive interaction is responsible for the demixing of the oil phase when in equilibrium with excess oil. Prediction of conductivity as well as the formation conditions for the third phase is possible using standard liquid theory applied to the extractant aggregates. The interactions, modeled with the sticky sphere model proposed by Baxter, are shown to be due to steric interactions resulting from the hydrophobic tails of the extractant molecule and van der Waals forces between the highly polarizable water core of the reverse micelles. The attractive interaction in the oil phase, equilibrated with water, is determined as a fonction of temperature, extractant molecule concentration, and proton and neodynium(III) cation concentration. It is shown that van der Waals interactions, with an effective Hamaker constant of 3kT, quantitatively explain the behavior of DMDBTDMA in n-dodecane in terms of scattering as well as phase stability limits.
I. Introduction 1. Nuclear Fuel Reprocessing. Hydrometallurgy is the preferred separation method used in nuclear fuel reprocessing. The so-called PUREX process, based on the use of tri-n-butyl phosphate (TBP) diluted in an aliphatic solvent is used worldwide in all existing reprocessing plants.1 New strategies of managing nuclear wastes using the implementation of the PUREX process are under study in several countries with a particular aim to separate longlived radionuclides such as those of the minor actinides: neptunium, americium, and curium. Ultimately these are transformed into short-lived fission products. Among the new separation methods under development in this framework, the hydrometallurgical processes are preferred. The TRUEX2 and the DIAMEX3 processes are some examples and are based on the use of the extractants octyl(phenyl)-N,N-diisobutylcarbamoylmethyl phophine oxide (CMPO) and dimethyldibutyltetradecylmalonamide (DMDBTDMA), respectively. The chemical structure of DMDBTDMA is represented in Figure 1. All forms of these liquid-liquid extraction processes are faced with the problem of phase separation where a third phase splitting the organic phase occurs. This phenomenon is considered a major drawback: for safety concerns, in the case of the occurrence of criticality of risk * To whom correspondence should be addressed. E-mail: zemb@ nanga.saclay.cea.fr. † CEA, DCC/DRRV/SEMP CEA-Valrho ˆ. ‡ CEA, DSM/DRECAM/SCM CEA-Saclay. (1) Science and technology of tributyl phosphate; Schulz, Wallace W., Navratil, James D., Talbot, Andrea E., Eds; CRC: Boca Raton, FL, 1984; p 1991. (2) Horwitz, E. P.; Shulz, W. W. Solvent Extr. Ion Exch. 1985, 3, 75. (3) Madic, C.; Blanc, P.; Condamines, N.; Baron, P.; et al. In The Fourth International Conference on Nuclear Fuel Reprocessing and Waste Management. RECOD′94. Proceedings-Volume 3- London, (UK), 24-28 April 1994, and CEA-CONF-12297.
Figure 1. Dimethyldibutyltetradecylmalonamide (DMDBTDMA), extractant molecule used in the DIAMEX process.
when a third phase occurs in plutonium extraction, and for process implementation within the extracting. To avoid the occurrence of third phase, several solutions have been put foward. For the PUREX process, a highly branched aliphatic diluent, such as hydrogenated tetrapropene (TPH) was selected for COGEMA’s La Hague plants. Moreover, the flowsheet design limits the concentration of Pu(IV) nitrate within the organic phase.4 In the case of the TRUEX process, another solution, proposed by Horwitz, consists of the addition of the polar modifier, TBP, to the CMPO/aliphatic solvent solution. For the DIAMEX process, we show that the pahse separation can be avoided by optimizing the structure of the diamide extractant. Up to now, the studies of third phase formation within liquid-liquid extraction processes consisted principally in the determination of third phase boundaries according to several parameters, such as the concentration of extractant and solutes extracted, the nature of the organic diluent, and the temperature. A review of studies related to the PUREX process was published recently by Rao and Kolarik.5 However, few studies of the third phase formation in liquid-liquid extraction considered the problem from a basic physicochemical point of view. (4) Baron, P.; Boullis, B.; Germain, M.; Gue, J. P.; Miquel P.; Poncelet, F. J.; Dormant, J. M.; Dutertre, F. GLOBAL 93, Seattle, (USA), 12-17 September 1993. (5) Rao, P. R. V.; Kolarik, Z. Solvent Extr. Ion Exch. 1996, 14 (6), 955.
10.1021/la980313w CCC: $18.00 © 1999 American Chemical Society Published on Web 03/09/1999
Phase Separation in Extractant Solutions
The aim of this paper is to demonstrate that organic solutions of extractant molecules can be treated as reverse micelles and that the prediction of the composition where the third phase appears is possible using liquid state theory.6 In comparison with systems studied in reverse micellar physics,7 the systems considered here are concentrated, i.e., the micellar volume fractions exceed 10%. However, typical industrial plant applications such as PUREX8 are still more concentrated, and the systems studied here would be labeled as “dilute” in terms of the extractant molecule concentration. In the language of self-assembled surfactant systems, the initial “two phase” equilibrium is equivalent to a Winsor II microemulsion: a concentrated surfactant solution in oil is in equilibrium with an excess water phase. A dissociated component, in our case nitric acid at a concentartion of several moles per liter, is subject to Donnan exclusion between the small quantity of water extracted in the oil phase and bulk water.9,10 If the ionic strength in the water phase is too high, a phase transition from two phases in a “Winsor III” equilibrium occurs. All these phenomena have been previously described with water-in-oil reverse micelles.7 2. Modeling the Phase Separation. The most commonly studied surfactant system is the ternary sodium bis(2-ethyl) sulfosuccinate (AOT)/water/alcane system11 (AOT has 10 carbon atoms in two branched hydrophobic chains and can be used to model small extractant molecules). In the oil-rich region of the reverse micellar system, net attractive interactions between micelles are observed.12-14 Phase separation with excess water occurs at temperatures less than T*, since the radius of the micelles would exceed the maximum allowed radius. At temperatures higher than T*, phase separation gives a phase of excess oil, due to a liquid-liquid phase separation. The phase separation can be modeled using sticky spheres.13 The generality of these phase transition effects in the case of AOT induced by temperature, salt, oil nature, or ionic strength has been shown by Lang, Zana, and coworkers.15-18 Reverse aggregates formed by larger surfactant molecules are subject to shape changes under the influence of volume and curvature constraints.19 The most studied model system is the didodecyldimethylammonium bromide (DDAB)/water/oil system (DDAB has 24 carbon atoms in two long chains, melting below room temperature). The aggregates connect and do not remain globular. Neither scattering nor conductivity of the reverse micelles in this system can be successfully modeled with a sticky sphere model.20-22 The shape changes observed are the same as those of the classical sodium-based AOT molecule (6) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. (7) Pileni, M. P. Structure and Reactivity in Reverse Micelles; Elsevier: Amsterdam, 1989. (8) Bourgeois, M. In Techniques de l′Inge´ nieur, Traite´ de Me´ canique et Chaleur-B3650 1-36, 1994. (9) Israelachvili J. N. Intermolecular and surface forces, 2nd ed; Academic Press: London, 1991. (10) Israelachvili, J. N.; Mitchell, D. J.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 2 1976, 72, 1525. (11) Chevalier, Y.; Zemb, Th. Rep. Prog. Phys. 1990, 53, 279. (12) Cassin, G.; Badiali, J. P.; Pileni, M. P. J. Phys. Chem. 1995, 99, 1241. (13) Pitre´, F.; Regnaut, C.; Pileni, M. P. Langmuir 1993, 9, 2855. (14) Cassin, G.; Duda, Y.; Holovko, M.; Badiali, J. P.; Pileni, M. P. J. Chem. Phys. 1997, 107 (7), 2683. (15) Jada, A.; Lang, J.; Zana, R. J. Phys. Chem. 1990, 94, 381. (16) Jada, A.; Lang, J.; Zana, R., Makhoufi, R.; Hirsch, E.; Candau, S. J. J. Phys. Chem. 1990, 94, 387. (17) Hou, M. J.; Shah, D. O. Langmuir 1987, 3, 1086. (18) Menon, S. V. G.; Keklar, V. K.; Manohar, C. Phys. Rev. A 1991, 43, 2, 1130. (19) Zemb, Th. Colloid Surf., A, in press.
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after it has been ion exchanged with aluminum trivalent counterions.23 Experimentally, surfactant solutions containing globular aggregates can be distinguished easily from connected systems by performing scattering experiments along a dilution line.24,25 One of the aims of this work is to identify if DMDBTDMA/oil/water/nitric acid quaternary systems belong to the class of interacting globular aggregates or disordered connected structures described by the disordered open connected (DOC) model.26 Once the category of the reverse micelles (interacting globules or connected aggregates) is determined, standard procedures27 of interpretation allow the determination of the aggregation number as well as the magnitude and the decay length of the attractive or repulsive, interaction between micelles.28 The interactions between reverse micelles may have several origins including excluded volume effects, hard core repulsions, or van der Waals forces.9 Steric interactions29 may also play a role owing to the penetration of the solvent30 into the surfactant hydrophobic chains. We aim to distinguish the main types of interaction31 acting in the system and investigate the effect of an added ionic species (for technical reasons, we investigate nitric acid and not the effect of sodium chloride). Some pioneering work by Osseo-Asare32 has established that concentrated extractant solutions of tri-n-butyl phosphate (TBP) form nanometric size water-in-oil microemulsions (i.e., reverse micelles). The presence of reverse hydrated aggregates (TBP‚H2O)n was inferred on the basis of proton NMR, vapor pressure osmometry, and infrared data consistent with current notions of the aggregation of nonionic surfactant.33-36 Moreover, the 1:1 TBP/nitric acid complex is highly susceptible to aggregation as demonstrated by the dramatic increase in viscosity associated with the formation of this complex.32 Finally, the 1:1 TBP/nitric acid complex is similar to some wellestablished cationic surfactants such as protonated amines, which are also known to form reversed micelles.33 As Osseo-Asare noticed in his review32 “once the amphiphilic nature of TBP-HA complexes is accepted, we can now take advantage of the theoretical and experimental tools available for the study of reversed micelles and microemulsions. Such a marriage between (20) Ninham, B. W.; Barnes, I. S.; Hyde S. T.; Derian P.-J.; Zemb, Th. Europhys. Lett. 1987, 4, 561. (21) Ninham, B. W.; Barnes, I. S.; Hyde S. T.; Derian P.-J.; Zemb, Th. J. Phys. (Paris) 1990, 51, 2605. (22) Barnes, I. S.; Hyde S. T.; Derian P.-J.; Ninham, B. W.; Drifford, M.; Zemb, Th. J. Phys. Chem. 1988, 92, (8), 1988. (23) Bardez, E.; Ngyen, C.; Zemb, Th. Langmuir 1995, 11, 3374. (24) Barnes, I. S.; Hyde S. T.; Derian P.-J.; Ninham, B. W.; Drifford, M.; Warr G. G.; Zemb, Th. Prog. Colloid Polym. Sci. 1988, 76, 90. (25) Zemb, Th.; Hyde, S. T.; Derian, P -J.; Barnes, I. S.; Ninham, B. W. J. Phys. Chem. 1987, 91, 3814. (26) Zemb, Th.; Barnes, S. T.; Derian, P.-J.; Ninham, B. W. Prog. Colloid Polym. Sci. 1990, 81, 20. (27) Hayter, J. B. In Proceedings of the International School of Physics Enrico Fermi; Degiorgio, V., Corti, M., Eds.; North-Holland: Amsterdam, 1985; p 59. (28) Belloni, L. In Interacting monodisperse and polydisperse spheres; Lindner, P., Zemb, Th., Eds.; Elsevier Science Publishers B.V.: Amsterdam, 1991; p 135. (29) Napper, H. In Polymeric stabilization of colloidal dispersions; Colloı¨d science, 1983. (30) Evans, D. F.; Mitchell, D. F.; Ninham, B. W. J. Phys. Chem. 1986, 90, 2817. (31) Huruguen, J. P. The`se de l′Universite´ de Paris VI, 1991. (32) Osseo-Asare, K. Adv. Colloid Interface Sci. 1991, 37, 123. (33) Kertes, A. S.; Gutmann, H., Surf. Colloid Sci. 1976, 8, 193. (34) Shinoda, K.; Lindman, B. Langmuir 1987, 3, 135. (35) Zana, R. In Surfactant Solution, New methods of investigation; Zana, R., Eds.; Marcel Dekker: New York, 1987. (36) Lindman, B., Wennerstro¨m, H. Topics in Current Chemistry; Springer-Verlag: Berlin, Heidelberg, New York, 1980; Vol. 87, p 1.
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surfactant science and solvent extraction stands to benefit both fields.” II. Experimental Methods Chemicals. The extractant DMDBTDMA (99% pure) was obtained from PANCHIM (Lisses, France) and used as received. n-Dodecane (purum) and nitric acid (53%) were from OSI. Deionized, double distilled water was used throughout the study. Techniques. Preparation of Multiphase Systems. Two milliliters of the aqueous phase, containing nitric acid and/or metal nitrate at the chosen concentration, and 2 mL of n-dodecane containing DMDBTDMA were thoroughly mixed for several minutes using a Vortex mixer. The mixture was then centrifuged with a thermostated Sigma 3K30 centrifuge for 5 min. The samples were thermostated in water bath at the desired temperature. Third Phase Limits. Determination of the maximum solvent loading without the formation of a second organic phase was determined in the following way: 1.5 mL of solvent (DMDBTDMA diluted in n-dodecane) was added to (in a thermostated water bath) 1.5 mL of concentrated nitric acid in order to induce the separation of a third phase. Then the system was diluted by equal volumes (0.05-0.1 mL) of the original solvent and deionized water. The mixture was mixed for 5 min and centrifugated. The procedure was repeated until the third phase disappeared. The two phases were then analyzed after separation. The estimated accuracy for the third phase concentration boundaries is about 10% (and will be shown as horizontal error in subsequent plots). The Karl Fischer method was used to determine the water content of the organic phases. The apparatus used was a Metrohm 684KF coulometer. It can detect between 10 and 100 µg of water. Potentiometric titrations were conducted with a Mettler DL 77 titrator equipped with combined glass electrodes. The concentration of protons transferred from the aqueous phase to the organic phase was determined in two different ways: (i) titration of the aqueous phase, after contact with the organic phase, and evaluation of the variation in proton concentration compared to the initial concentration; (ii) back extraction of the protons into an aqueous solution of sodium hydroxide followed by revese potentiometric titration. Potentiometric titrations of DMDBTDMA were performed using HClO4 in acetic anhydride with XG900 and XR900 Tacussel electrodes specialized for acetic acid media. Conductivity measurements were performed with a Tacussel CD810 conductimeter. Small-Angle X-ray Scattering. Small-angle X-ray (SAXS) experiments were performed on the Huxley-Holmes, High Flux camera of the small-angle X-ray laboratory in the Service de Chimie Mole´culaire, at CEA-Saclay (France). The X-ray source was a copper rotating anode operating at 15 kW. KR1 radiation (λ ) 1.54 Å) was selected by the combination of a nickel-covered mirror and a bent, asymmetrically cut germanium 〈111〉 monochromator. A Q range from 0.01 to 0.4 Å-1 is usually covered with this instrument using a two-dimensional gas-filled chamber 0.3 m in diameter. The sample to detector distance was 2.15 m. After exposure, the isotropic two-dimensional spectra were radially averaged and the intensity of scattered X-rays was expressed as a function of momentum transfer Q. The scattering cross sections (1/V)(δΣ/δΩ) on the absolute scale in cm-1 were determined by the mean of a calibrated internal monitor proportional to the number of unscattered X-ray photons. The absolute data were then corrected for the background produced by the sample cell and the solvent and the electronic background from the detector.37,38 Detailed descriptions are available of the characteristics of the camera and its detectors, including the procedure for determining the scattering cross sections on the absolute scale.39 (37) Le Flanchec, V.; Gazeau, D.; Taboury, J.; Zemb, Th. Appl. Crystallogr. 1996, 29, 110. (38) Ne´, F.; Gazeau, D.; Lambard, J.; Lesieur, P.; Zemb, Th. J. Appl. Crystallogr. 1993, 26, 763. (39) Stuhrmann, H. B. J. Appl. Crystallogr. 1978, 11, 325.
Erlinger et al. Measurements of the Monomer Concentration. In this system the scattering objects are the molecular aggregates. However not all of the DMDBTDMA molecules participate in these aggregates. The concentration of “monomers” not included in the aggregates is usually approximated as the critical micellar concentration (cmc) and, by extension, designated by the same name. Since volume fractions and the density of the objects n are very sensitive to the concentration of molecularly dispersed extractant molecules, it is important to determine the cmc on the same samples with the best possible accuracy. We took advantage of the available data to determine the DMDBTDMA concentration at which the X-ray scattering intensity vanishes, i.e., the concentration at which aggregates are no longer observed in solution. Since the shape of the scattered intensity curves, I(Q) vs Q, may be concentration-dependent, the only modelindependent method for determining the amount of aggregates present in solution is to determine the invariant Q* 40
Q* )
∫0∞Q2 I(Q) dQ
(1)
To allow the integral to converge, the high-Q part of the scattering curve must be extended as a “Porod plot”,40 and the low-Q part is assumed to be less divergent than Q2.41 These conditions are satisfied in this series of experiments. A similar procedure has been applied by Warr et al. to detect the increase of the monomer concentration observed with micelles upon heating.24 Determination of Hard Core and Stickiness Parameter. The X-ray intensity is defined as27
I(Q) ) n v2 ∆F2 P(Q) S(Q)
(2)
where n is the scattering particle concentration (in particles‚cm-3). For surfactant aggregates, the density n of micelles is calculated as n ) (c - cmc)/N, where N is the mean aggregation number of the surfactant aggregate and cmc is the concentration of surfactant molecules which do not participate in the scattering objects. This cmc can be measured directly from the second moment of the scattering curve, as explained above. v and ∆F are, respectively, the volume of the scattering particles (in cm3) and the difference in the scattering length densities of particles and solvent (in cm-2), also called “the contrast term” of the scattering experiment. ∆F is not a free parameter but is calculated, as suggested by Hayter,27 from the known electronic density of the solvent, the N extractant headgroups, the extracted water, and nitric acid and ions. The X-ray scattered intensity I(Q) has the dimension of area per unit volume and is also called the “scattering cross section” of the particles. P(Q) is the form factor of a sphere defined as40
P(Q) )
[
]
3 sin(QRpolar) - QRpolar cos(QRpolar) (QRpolar)3
2
(3)
where Rpolar is the radius of the scattering spheres (in Å).The radius of the polar core is calculated by adding the known volumes of the N extractant headgroups, the extracted water, and the nitric acid and ions (see potentiometric titration section). The size of the scattering particles may be different from the effective size of the particles, determing the range of the steric interactions. The term “scattering particles” refers to the highlighted parts of the particles with respect to the type of scattering radiation (light, X-rays, or neutrons). S(Q) is the structure factor which accounts for interparticle interactions. A satisfactory description of the interactions between colloidal aggregates in the case of shortrange interactions, i.e., when particles only feel neighbors at close proximity, may be obtained from the model proposed by (40) Guinier, A.; Fournet, G. In Small Angle Scattering of X-rays; Wiley & Sons: New York, 1955. (41) Zemb, Th. In Scattering of connected networks; Lindner, P., Zemb, Th., Eds.; North-Holland: Amsterdam, 1991; p 177.
Phase Separation in Extractant Solutions
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Baxter,6 based on the so-called hard sphere sticky potential U(r) defined as:
U(r)/kbT ) limδfdhsA(r) where
A(r) ) +∞
for r < dhs
) ln [12.τ.(dhs -δ)/ dhs]
for dhs < r < δ (4)
)0
for r < δ
r is the distance between two particles and dhs is the “hard sphere” diameter. It represents the separation below which the particles become incompressible, and the repulsive force between two particles suddenly becomes infinite. The difference (dhs - δ) represents the extent of the effective attractive interaction modeling the different molecular interactions between the two particles. Baxter’s model is exact only for a vanishing extent of the attractive potential. In practice, calculations of compressibilities, as explained below, are valid approximations as long as (dhs - δ)/dhs < 0.1. The potential chosen here includes only “ short range ” attraction. We could have used a modified version of the DLVO screened potential, such as introduced by Hayter and Zulauf.42 However, since we will need analytic solutions to determine the phase separation line as well as expressions for S(Q), we had to use a model which gives an expression for both quantities. This is the case of the Baxter model when a shortrange attractive well surrounds a hard sphere core. In other words, this model corresponds to the hard sphere repulsive interaction together with a surface adhesion (hard sphere sticky potential). The reciprocal of the parameter τ is called the “stickiness parameter”. τ-1 represents the strength of the adhesion and is easily expressed in kT units using eq 4. The higher the value of τ-1, the greater the strength of adhesion, and the deeper the attractive potential well. S(Q) is easily calculated in a closed analytical form at any concentration and temperature:
S(Q) ) [I - C(Q)]-1
(5)
where C(Q), the Fourier transform of the direct correlation function, is given by
C(Q) ) -24ηx-6{Rx3(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 (6) the η is the particle fraction volume and x ) Qdhs (dhs is the hard sphere diameter), setting
R ) (1 + 2η - µ)2/(1 - η)4 β ) -[3η(2 + η) - 2µ(1 + 7η + η2) + µ2(2 + η)]/ [2(1 - η)4] µ ) λη(1 - η) γ ) η(1 + 0.5η)/[3(1 - η)2] ) τ + [η/(1 - η)] λ ) (6/η)[ - (2 - γ)0.5]
(7)
As mentioned above τ-1 reflects the stickiness and λ is the number of first neighbors surrounding a given reverse micelle. At large scattering angles, the scattered intensity is dominated by the shape and size of the scattering particles via the form (42) Hayter, J. B.; Zulauf, M. Colloid Polym. Sci. 1982, 260, 1023.
Figure 2. Schematic phase prism representing the limit between the two-phase and three-phase regions (curved dashed line). The lines drawn in the triangular section of the tetrahedron represent approximately the paths I and II followed for the SAXS analysis (see text section 5). factor P(Q), while at small Q, interactions dominate scattering via the structure factor S(Q). A divergence of S(Q) toward high values indicates high osmotic compressibility and hence the proximity of a phase transition. Dilution experiments are an easy way to check whether interactions or shape, or both, change with particle concentration. If the size of the aggregates does not vary with concentration c, pH, or temperature, the scattering curves I(Q)/(c - cmc) merge into a single curve at high Q. If interparticle interactions are present in solution and shape does not change upon dilution, the low Q part of the scattering curve reflects the effect of these interactions and allows the estimation of the hard sphere radius as well as the strength of the interaction potential. Fitting the experimental scattered intensity curves I(Q) with eq 2, using τ and dhs (and hence Rhs) as “almost” independent parameters (see discussion in section III.3), we can determine the size and shape of the scattering particles from the form factor P(Q), together with the structure factor S(Q) derived from the potential depicted in eq 4. The criteria used to obtain the best fit is to minimize the sum of the square of the differences between the model and the experiment. This helps to determine the stickiness parameter τ-1,Rhs, and λ (eq 7).
III. Results 1. Sample Preparation for SAXS Measurements. We investigate the quaternary system DMDBTDMA/ water/nitric acid in n-dodecane and the quintenary system DMDBTDMA/n-dodecane/water/nitric acid/neodyme nitrate. The extractant molecule is surface active, as recently shown by surface tension determination at the air/water (or oil/water) interface.43 The phase prism is schematically shown in Figure 2. We investigate the oil-rich region with typically 3% to 30% of extractant in oil equilibrated with a nitric acid solution of known concentration. Water is coextracted with nitric acid in the oil phase.43 Four types of dilution denoted paths I to IV are considered: Path I (Figure 2 and Figure 3) consists of increasing the acid content of the excess water phase, thus approaching and finally crossing the Winsor II and Winsor III phase limit. Path II (Figure 2 and Figure 3) is a dilution of sample no. 10 (Table 2a) with n-dodecane in order to vary the extractant concentration while the ratio [HNO3]org/[DM(43) Erlinger, C.; Gazeau, D.; Zemb, Th.; Madic, C.; Lefranc¸ ois, L.; Herbrant, M.; Tondre, C. Solvent Extr. Ion Exch., in press.
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Erlinger et al. Table 1. (a) Sample Composition along Path I
total [DMDBTDMA] oil phase initial [HNO3] equilibrium equilibrium equilibrium sample organic phase density aqueous phase [HNO3] organic [H2O] organic [HNO3]org/ no. (mol/L) (g/cm3) (mol/L) phase (mol/L) phase (mol/L) [DMDBTDMA] φaggregatesb Rtotc (Å) Rpolar core (Å) 1 2 3 4 5a 6 7 8 9
0.5 0.5 0.5 0.5 0.5 0.72 0.72 0.72 0.72
1.03 1.03 1.03 1.03 1.03 1.16 1.16 1.16 1.16
0 0.5 2.5 3.5 5 0 0.1 2.3 3.7
0 0.01 0.26 0.39 1.37 0 0.014 0.35 0.7
0.1 0.11 0.25 0.35 1.2 0.2 0.2 0.47 0.5
0 0.02 0.52 0.78 2.74 0 0.019 0.49 0.97
0.22 0.22 0.23 0.24
10.5 10.5 10.7 10.8
6.1 6.1 6.7 6.9
0.32 0.32 0.34 0.35
11.0 11.1 11.3 11.4
6.5 6.5 7.1 7.5
(b) Samples Analyzed with Baxter Model along Path I sample no.
N
Rhs (Å)
R
τ-1
U(r)/kT
A/kT
1 2 3 4 5a 6 7 8 9
5 6 6 6
6.2 7.1 6.7 7.0
0.98 0.8 1 1
4.9 5.2 9.1 12.1
-1.40 -1.46 -2.02 -2.31
2.33 2.35 3.05 3.45
7 8 7 7
6.8 8.6 8.2 7.4
0.9 0.3 0.7 1
3.7 5.5 8.2 9.8
-1.13 -1.52 -1.92 -2.10
2.25 2.80 3.30 3.50
a “Third phase sample”, all values on this line refer to the third phase. b φ aggregates ) sum of water, acid, and aggregated diamide volume fractions. Molecular volumes of the components are 799.5 Å3/molecule for DMDBTDMA, 152.8 Å3/molecule for the polar core of the DMDBTDMA, 30 Å3/molecule for water, and 66.3 Å3/molecule for nitric acid. c RTot ) total radius of micelles (including the polar core and the hydrophobe chains).
Figure 3. Representation of a planar cut at constant water to n-dodecane ratio in Figure 2. The lines represent the paths I and II followed for the SAXS experiments and the points show the different samples which have been investigated. Numbers refer to sample numbers in Tables 1 and 2.
Figure 4. Representation of the paths III and IV, followed for the SAXS experiments (the points show the different samples which have been investigated). Numbers refer to sample numbers in Table 3. The ratio of water to n-dodecane has been kept constant.
DBTDMA] is kept constant. This procedure is classically used in order to keep the aggregate volume fraction constant.44 Along path III (Figure 4), the DMDBTDMA concentration is kept constant (0.54 M) and the neodynium(III) nitrate concentration increases. The organic phase does not contain any nitric acid in this case. Along path IV (Figure 4), DMDBTDMA and nitric acid concentrations are kept constant (0.52 and 0.24 M, respectively) and the neodynium(III) nitrate concentration increases. For the quaternary system DMDBTDMA/water/nitric acid in n-dodecane the corresponding paths in the phase prism are schematically shown in Figure 2. We now consider more precisely the cut at constant water to n-dodecane ratio. The result is the plane shown in Figure 3 (note that composition of the solutions in equilibrium
after separation in two or three phases are not necessarily in this plane). Composition of the samples investigated are indicated in Figure 3, together with the experimentally determined three-phase separation. Tables 1a, 2a, and 3a give the exact composition of the samples, including the values of coextracted water determined by Karl-Fisher titration as well as final nitric acid concentration. As previously reported43 a significant amount of water is coextracted. 2. Critical Micellar Concentration. We have performed standard SAXS measurements on the 15 samples located on the cut of the phase prism (Figure 3). In the q-range observed, a molecular dispersion of extractant molecules would produce a low scattered intensity, analoguous to n-dodecane. This constant scattering is direct proof of the heterogeneity of the solution and of the presence of molecular aggregates. Moreover, since measurements are on an absolute scale, scattering being measured in cm-1 along the convenient scale defined by Stuhrmann,37 the intensity of the scattering is proportional
(44) Graciaa, A.; Lachaise, J.; Martinez, A.; Chambu, C. C. R. Acad. Sci., Paris B 1976, 282 B, 547.
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Langmuir, Vol. 15, No. 7, 1999 2295 Table 2. (a) Sample Composition along Path II
total [DMDBTDMA] oil phase equilibrium [HNO3] equilibrium [H2O] sample organic phase density organic phase organic phase equilibrium [HNO3]org/no. (mol/L) (g/cm3) (mol/L) (mol/L) [DMDBTDMA] φaggregatesa Rtot (Å) Rpolar core (Å) 10 11 12 13 14 15
0.70 0.51 0.44 0.34 0.30 0.12
1.15 1.04 1.0 0.94 0.92 0.81
0.33 0.24 0.21 0.16 0.14 0.06
0.36 0.28 0.21 0.14 0.15 0.12
0.47 0.47 0.48 0.47 0.47 0.50
0.33 0.24 0.20 0.15 0.13 0.038
10.0 9.5 9.6 9.2 8.5 8.5
6.2 5.9 6.0 5.7 5.3 5.6
(b) Samples Analyzed along Path II sample no.
N
Rhs (Å)
R
τ-1
U(r)/kT
10 11 12 13 14 15
5 4 4 4 3 3
6.2 6.3 7 6.7 6.6 7.1
1 0.9 0.7 0.7 0.6 0.5
8.6 8.9 8.6 9.1 9.5 12.7
-2.00 -2.05 -2.02 -2.04 -2.09 -2.36
3 a φ aggregates ) sum of water, acid, and aggregated diamide volume fractions. Molecular volumes of the components are 799.5 Å /molecule for DMDBTDMA, 152.8 Å3/molecule for the polar core of the DMDBTDMA, 30 Å3/molecule for water, and 66.3 Å3/molecule for nitric acid. b R Tot ) total radius of micelles (including the polar core and the hydrophobe chains).
Table 3. (a) Sample Composition along Paths III and IV total equilibrium equilibrium [DMDBTDMA] oil phase equilibrium [H2O] equilibrium [Nd(III)] [HNO3] sample organic phase density organic phase organic phase organic phase [HNO3]org/no. (mol/L) (g/cm3) (mol/L) (mol/L) [DMDBTDMA] φaggregatesa Rtot(Å) Rpolar core (Å) (mol/L) 16 17 18 19 20 21 22 23 24
0.54 0.54 0.54 0.54 0.52 0.52 0.52 0.52 0.52
1.05 1.05 1.05 1.05 1.04 1.04 1.04 1.04 1.04
0 0.004 0.009 0.01 0 0.01 0.015 0.022 0.03
0 0 0 0 0.24 0.24 0.24 0.24 0.24
0.090 0.090 0.097 0.15 0.25 0.26 0.26 0.24 0.27
0 0 0 0 0.46 0.46 0.46 0.46 0.46
0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24
11.1 11.0 11.0 11.0 10.7 10.7 11.2 11.2 11.2
6.5 5.7 5.7 6.4 6.6 6.6 7.0 6.9 7.0
(b) Samples Analyzed along Paths III and IV sample no.
N
Rhs (Å)
R
τ-1
U(r)/kT
A/kT
16 17 18 19 20 21 22 23 24
7 5 5 7 6 6 7 7 7
7.5 6 6 7 6.0 6.6 7.0 7.0 7.0
0.72 0.75 0.75 0.66 1 1 1 1 1
10.9 4.8 6.6 6 9.1 9.0 9.1 9.9 10.8
-2.21 -1.39 -1.71 -1.61 -2.03 -2.01 -2.03 -2.11 -2.20
2.98 2.44 2.95 2.5 3.1 3.1 2.9 3.2 3.3
a φ 3 aggregates ) sum of water, acid, and aggregated diamide volume fractions. Molecular volumes of the components are 799.5 Å /molecule for DMDBTDMA, 152.8 Å3/molecule for the polar core of the DMDBTDMA, 30 Å3/molecule for water, and 66.3 Å3/molecule for nitric acid.
to the aggregation number. Measurements of the mass of the reverse aggregates formed are therefore known once the effects due to interactions are properly taken into account.45 The existence of scattering is a direct proof of the presence of reverse micelles. We make use of a general theorem46 to evaluate, by a direct method, the value of the cmc. Plot of the invariant Q* (see Experimental Methods) versus the total DMDBTDMA concentration gives the value of the cmc. As shown in Figure 5, the concentration at which the aggregates of DMDBTDMA in n-dodecane disappear is determined as cmc ) 0.05 ( 0.01 mol/L. The concentration (45) Chen, S. H. Annu. Rev. Phys. Chem. 1986, 37, 351. (46) Warr, G.; Zemb, Th.; Drifford, M. J. Phys. Chem. 1990, 90, 7, 3086.
of monomers in equilibrium with micelles will be considered equal to the cmc. The volume fraction of the aggregates was always determined from molecular volumes after subtraction of the cmc ()0.05 M of extractant). We note that NMR allows indirect determination of cmc and aggregation numbers from the derivative of the NMR peak position versus concentration.36 This method is only valid if chemical shifts are independent of aggregation numbers, an assumption clearly not valid with ionic surfactants.47 However, for the case of DMDBTDMA in n-dodecane the direct,47 but imprecise, method coincides with the NMR determination using peak shift values. For the present work, we will assume that the cmc is independent of the nitric acid content.48 (47) So¨derman, O.; Guering, P. Colloid Polym. Sci. 1987, 265, 76.
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Figure 5. Variation of the invariant Q* with the DMDBTDMA concentration. The zero intercept is the concentration where reverse aggregates disappear.
Erlinger et al.
Figure 7. Calculated scattered intensity (s), structure factor (- -), and form factor (- - -) by modeled small spherical oligomers (N ) 4) of DMDBTDMA compared to the SAXS signal measured for DMDBTDMA of 0.34 M (see sample 13 in Table 2) in n-dodecane (+). The nitric acid concentration of the organic phase is 0.16 M.
3. Measurements in Organic Phase along Path II. Figure 6 shows the scattering for a series of dilutions along path II (the scattering spectra are shown in reduced units). The existence of a common limit at high scattering vectors for all the samples in the series is direct evidence of an approximately constant area per aggregate (the samples have the same Porod limit) while the increase of scattering at low angles can be explained by changes in the shape or interaction between the aggregates.
We have interpreted all our data self-consistently by using a sticky sphere model: the initial observation is that scattering increases when the phase limit is approached (see the discussion by Lum Van and Warr49). The scattered intensity has been analyzed using a standard procedure to separate the form factor P(q) from the structure factor S(q), which is available in a convenient analytical form in the case of a sticky sphere model.6 A typical result of the final fit is shown in Figure 7. The scattered intensity obtained from the model is shown by the thick line while the scale for S(q) is shown on the right side of the figure. The parameters used for this fit were the aggregation number N, the average number monomers of DMDBTDMA included in each reverse micelle, the hard core radius (Rhs), and the depth of the sticky sphere attractive potential expressed in kT/micelle. The sensitivity of the best fit (see Experimental Method for the best fit criteria) to the attraction and aggregation number is illustrated in Figure 8. A variation of 10% of one of the two parameters can be easily detected since the model curve calculated differs from the measured one. However, we note that the two parameters N and τ-1 are not completely orthogonal: an increase in the attraction can be partially compensated by a decrease in the aggregation number. This problem can only be avoided by increasing, as much as possible, the q-range where data are measured to high and low q. The limitation at large q is the low intensity of the scattering while the low q limitation is the quality of the focusing, which affects the scattering detected near edge of the beam stop in the absence of the sample. As can be seen in Tables 1b, 2b, and 3b, the calculated hard core radius and the polar core (consisting of the extracted water and nitric acid, and the carbonyl and methyl groups of the diamide extractant) are very close. Consequently, only a small fraction of the aliphatic chains participate in the hard core. The excluded volume is constituted by almost 95% of the chain volume. 4. Modeling the Experimental Scattering Intensity with the Baxter Model. Figure 9 shows the two general results of this study:
(48) Nigond, L. The`se de l′Universite´ de Clermont-Ferrand II, 1992, CEA-R 5610.
(49) Lum Wan, J. A.; Warr, G.; White, L. R.; Grieser, F. Colloid Polym. Sci. 1987, 265, 528.
Figure 6. Scattered intensity I(q) normalized by the DMDBTDMA concentration minus its critical micellar concentration versus the momentum transfer q measured along path II (see Table 2a), for a fixed ratio [HNO3]org/[DMDBTDMA] ) 0.47. The DMDBTDMA concentration in the organic phase ranges from 0.12 to 0.7 M: 0.12 M (0); 0.3 M (O); 0.34 M (]); 0.44 M (+); 0.51 M ([); 0.70 M; (b).
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Figure 10. Variation of the aggregation number (b) along path II (see Table 2b) versus the aggregate molar fraction, for a fixed ratio [HNO3]org/[DMDBTDMA] of 0.47. The standard deviation to the rate of change of the mean aggregation number is (0.88N)1/2. Figure 8. Calculated scattered intensities of DMDBTDMA compared to the scattered intensities measured for DMDBTDMA of 0.34 M (see sample 13 in Table 2) in n-dodecane (+) (the nitric acid concentration of the organic phase is 0.16 M).
dent of concentration, as expected for a thermodynamic field variable. The standard deviation of the rate of change of mean aggregation number with respect to micelle concentration (S - X1) is given by10
σ2 )
∂N h ∂N h )N h ∂ ln X1 ∂ ln(S - X1)
(8)
with σ ) standard deviation of the distribution of sizes which measures polydispersity, N h ) mean aggregation number, X1 ) monomer concentration, and S ) total amphiphile concentration. A rapid change of N h with concentration is evidence of a large distribution of polydispersity in micelle size (at a given concentration). The slope of the straight N h ) f(ln(S - X1)) curve calculated from Figure 10 is 0.88. It follows from eq 8 that Figure 9. Variation of the sticky sphere interaction potential U(r)/kT ([) and the aggregation number (b) calculated with eq 6 along path II (see Table 2) versus the aggregate volume fraction, for a fixed ratio [HNO3]org/[DMDBTDMA] of 0.47.
(i) The aggregation number is increasing with concentration. This has already been observed with micelles of low aggregation number.50 Micelles of large aggregation numbers have stronger sterical constraints and can only turn suddenly from spheres into giant aggregates.51 (ii) The attractive potential between micelles does not depend on concentration. The model of sticky spheres holds much better here than in any other micellar system we are aware of. This result is very encouraging because, in the case of AOT reverse micelles, the sticky sphere potential is a parameter which had to be adjusted for each concentration,13 indicating that variations in topology cannot be neglected as assumed by the DOC model.19 We have now established that the small aggregates of radius less than 1 nm, and with an aggregation number less than 10, interact via short-range attractions, i.e., attractions extending over distances shorter than one aggregate diameter. The order of magnitude of the attraction is 2 kT/micelle at room temperature, indepen(50) Hayter, J. B.; Hayoun, M.; Zemb, Th. Colloid Polym. Sci. 1984, 262, 798. (51) Porte, G. In Micellar Growth, Flexibility and Polymorphism in Dilute Solutions; Gelbart, W. M., Shaul, A. B., Roux, D., Eds.; Springer: Berlin, 1994; p 105.
h σ ) x0.88N
(9)
The polydispersity of micelles is very slight from N h 1/2. This may demonstrate a large polydispersity in the system. To our knowledge this is the first time that mass polydispersity of reverse aggregates has been determined with the single underlying approximation that water and surfactant molecules are assumed to be a pseudobinary system (i.e., the fluctuations of the water-to-extractant ratio in the aggregates can be neglected). 5. Study of Attractive Interactions. The next step in our investigation was to determine the variation of the attractive interaction with temperature for samples along the dilution path IV (see compositions in Table 3a). As seen in Figure 11, the reduced magnitude of the attractive interactions U/kT is independent of temperature. In accondance, van der Waals interaction only depends on A/kT (A is the Hamaker constant) and geometric terms. We deduce from this invariance that among the three mechanisms described in the Introduction, dispersion forces which vary as A/kT, provide the dominant mechanism. Such dispersion forces appear between the polar cores (extractant headgroups, water, and extracted nitric acid) separated by the extractant molecule chains and the solvent (n-dodecane and extractant monomers). We will come back to this point later. Measurements in Organic Phase along Path I. We approach the Winsor II-Winsor III phase transition limit
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Figure 11. Variation of the interaction potential U(r)/kT versus temperature for fixed DMDBTDMA and nitric acid concentrations in the organic phase of 0.52 and 0.24 M, respectively. The neodynium(III) nitrate organic phase concentration ranges as follows: 0 M (9); 0.01 M (b); 0.015 M (2); 0.022 M (×); 0.03 M ([). The dashed line (- -) is a guide for eyes.
Erlinger et al.
Figure 13. Variation of the interaction potential U(r)/kT (full symbol) and the aggregation number (empty symbol) versus the ionic force for a fixed DMDBTDMA and nitric acid concentration in the organic phase of 0.52 and 0.24 M, respectively. The neodynium(III) nitrate organic phase concentration range is as follows: 0 M ([, ]); 0.01 M (9, 0); 0.015 M (9, 0); 0.022 M (9, 0); 0.03 M (9, 0) (see Table 3). (2) and (4) represent the interaction potential U(r)/kT and the aggregation number for 0.52 M DMDBTDMA concentration without nitric acid in the organic phase, respectively. The neodynium(III) nitrate organic phase concentration is 0.04 M.
Figure 12. Variation of the interaction potential U(r)/kT (full symbol) and the aggregation number (empty symbol) along path I versus [HNO3]org for two fixed DMDBTDMA concentrations of 0.5 M ([, ]) and 0.72 M (9, 0). The dashed line is only a guide for eyes.
by increasing the acid content of the water phase at fixed room temperature. The results are shown in Figure 12. The aggregation number is constant while the attractive potential increases with addition of acid. This well-known experimental fact may now be explained quantitatively: as water and nitric acid molecules in the polar core of the reverse micelles increase in concentration, the attractive dispersion force between cores increases. Measurements in Organic Phase along Path IV. Finally, we investigated the effect of a diffferent ionic species, in approching the phase separation by addition of neodynium(III) nitrate (path IV, Figure 4). This trivalent cation is known to be very efficient in inducing a third phase separation even at low concentration. As can be seen in Table 3, the attractive potential between the aggregates is not sensitive to the presence of neodynium(III) once the dispersion forces mechanism is turned on by the presence of 0.24 M of nitric acid. The picture of the microstructure of the reverse micelles formed by DMDBTDMA is now complete: (i) The aggregation numbers, taking values between three and eight, of aggregate increase with extractant concentration but are insensitive to temperature or the presence of nitric acid. (ii) The associate hard core volume does not include the volume of apolar chains. (iii) The polar cores, including ionic headgroups of the surfactant molecule, water, and nitric acid interact via mainly dispersion forces.
Figure 14. Conductivity measurements (b) and number of closest neighbors λ (O) versus [HNO3]org concentration for a fixed DMDBTDMA concentration of 0.5 M.
Is prediction of macroscopic behavior possible from this picture of the microstructure? IV. Discussion 1. Prediction of Conductivity. The modeling of the extractant solutions, as set of polarizable hard core spheres dominated by an attractive interaction, allows the observed behavior of conductivity to be rationalized. Indeed, the conductivity of this water-in-oil solution is strikingly high and varies by order of magnitudes depending on the location of the sample in the phase diagram.43 The simplest approach is to consider that conductivity is proportional to the number of first neighbors λ of a given reverse micelle (determined by eq 7). The comparison between the calculated and observed conductivity is shown in Figure 14. It can be concluded that the conductivity of extractant aggregate solutions is due to ions exchanging between polar cores of micelles. As can be seen in Figure 14, there is an increase of conductivity near the formation of the third phase. 2. Prediction of Third Phase Limit. A prediction of the onset of demixting (i.e., the maximum extracted nitric acid concentration) is possible using the sticky hard sphere Baxter model.6
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Langmuir, Vol. 15, No. 7, 1999 2299
Figure 15. Variation of the sticky parameter τ-1 along path I versus [HNO3]org for two fixed DMDBTDMA concentrations of 0.5 M ([) and 0.72 M (9). Experimental points can be fitted with straight line of slopes 7.1(- -), 6.2 (s), 5.7 (- -).
Figure 17. Baxter model potential (s) and van der Waals interaction (- - -) curves describing the interaction between two DMDBTDMA aggregates, for 0.5 M DMDBTDMA concentration, 0.39 M organic nitric acid concentration, and 0.35 M water concentration in the organic phase. The polar core and the hard core radius (Rpolar and Rhc) are very similar (6.9 and 7 Å, respectively), and the aggregate radius is about 10.8 Å. The van der Waals potential has been calculated using 3.45 kT as an effective Hamaker constant.
Figure 16. Comparison of experimental and theoretical (calculated using the Baxter model) limits between the twophase and three-phase regions of the phase prism. (b) represents the experimental data with n-dodecane. The predicted limit (s) has been calculated taking into account different slopes determined in Figure 15 (7.1 (- -)), 6.2 (s), 5.7 (- -)).
The analytical Percus-Yevick solution gives the limit of the two-phase region (liquid-gas transition or, in our case, a concentrated solution-diluted solution transition) characterized by a critical point located at Φc ) 12% and τc ) 0.097 (Φ is the volume fraction of the extractant molecules and τ is a dimensionless measure of the temperature). In our case the reciprocal of τ, τ-1, can be related to extracted nitric acid concentration. Indeed, for any extractant concentration, τ-1 varies linearly with the extracted nitric acid concentration (Figure 15). One must keep in mind that above Φc the liquid branch of the Percus-Yevick solution is a spinodal line (infinite compressibility S(0)) while below Φc the gas branch of the Percus-Yevick solution does not present any divergence of the compressibility (a mathematical solution can be calculated6 but it does not correspond to a physical state of the system). Under these conditions a prediction of the third phase limit (the two phase to three phase transition) is possible. On Figure 16 the experimental (full circles) and theoretical (full line) third phase limits are presented. Figure 16 proves the self-consistency of the modeling approach described here. The position of the third phase limit can be predicted without any parameter from liquid state theory, once the magnitude of the attractive interaction has been determined via scattering experiments. As can be seen in Figure 16, the prediction of phase boundary is precise up to 0.6 mol/L of DMDBTDMA, corresponding to a volume fraction of aggregates lower than 33%. At higher concentration of aggregates, the
prediction is below this observed phase separation. This is probably due to the fact that chains of neighboring aggregates are directly in contact with each other. Consequently, the modeling of the system as spheres in a continuous solvent is not valid for an aggregate volume fraction of the order of 50%. However, the main demonstration is still valid: the prediction includes the reentrant, non-monotonic path of the phase limit in the ([DMDBTDMA], [HNO3]) cut of the phase prism.
V. Conclusion In the first section of this paper we proposed three main molecular interactions to illustrate the physics of the interacting reverse micelles: (i) repulsive interactions between hard cores, (ii) sterical stabilization due to the surfactant hydrophobic chains, and (iii) effective attractions due to van der Waals interactions between water droplets dispersed in oil. By modeling the experimental data with Baxter model, we have been able to picture the microstructure of the reverse micelles formed by DMDBTDMA (Figure 17). In summary: (i) The aggregation number is between three and eight and increases with extractant concentration but is insensitive to temperature or the presence of nitric acid. (ii) The associated hard core volume does not include the volume of the apolar chains. (iii) The polar cores, including ionic headgroups of the surfactant molecule, coextract water and nitric acid and interact mainly via dispersion forces. The integral of the Baxter step is the sum of a steric stabilization due to chains and a van der Waals attraction of known decay lengh but with an unknown effective Hamaker constant Aeff. The dispersion forces between
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polar spheres of radius Rpolar separated by oil are given by52
U ) kT Aeff 2Rpolar2 2Rpolar2 f1(Rpolar, d) + + ln 6 f1(Rpolar, d) f2(Rpolar, d) f2(Rpolar, d)
[
(
)] (10)
with
R1 ) R2 ) Rpolar f1(Rpolar, d) ) d2 + 4Rpolar d f2(Rpolar, d) ) d2 + 4Rpolar d + 4Rpolar2 We have demonstrated that among the three mechanisms described above, the dispersion force is the dominant mechanism. Thus, as we know for each sample the value of the integral of the Baxter step, we can calculate the corresponding Hamaker constant by modeling the sterical stabilization by a wall taken at 0.3 Å of the hard core radius. As can be seen in Figure 18, the effective Hamaker constant, and hence the potential between aggregates, is not sensitive to the presence of neodynium(III) once the dispersion forces are turned on by the presence of nitric acid. This observation has two important consequences: (i) A given ion, to be extracted from the nitric acid aqueous phase, always interacts with about four to seven (52) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Deckker, Inc.: New York, Chapter 10.
Figure 18. Effective Hamaker constant Aeff calculated for three fixed DMDBTDMA concentrations of 0.5 M (O, samples 1 to 4), 0.72 M (s, samples 6-9), 0.54 M (2; sample 16-19), 0.52 M (×, samples 20-24) versus density of the polar core Fpolar. The effective Hamaker constant has been calculated for a 0.3 Å minimal approach distance between the two hard cores of interacting micelles.
extractant molecules and not just one. This should be taken into account in molecular modeling of the extraction. (ii) A depletion layer, i.e., a nanometric layer depleted in concentration of extractant solution, should always be present near the oil-water interface. This local decrease over a few nanometers of concentration of surfactant molecules has been observed in other reverse micelles53 and may be responsible of kinetic slowing of the extraction.54 LA980313W (53) Meunier, J. J. Phys. Lett. 1985, 46, L-1005; J. Phys. (Paris) 1987, 48, 1819. (54) Tondre, C.; Hebrant, M. J. Mol. Liq. 1997, 72, 279.
