Article pubs.acs.org/JPCB
Critical Exponents for Solvent Extraction Resolved Using SAXS Ross J. Ellis* Chemical Sciences and Engineering Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439, United States ABSTRACT: The solvent extraction of an ionizable solute (H3PO4) from water into a water-in-oil microemulsion, and subsequent organic phase splitting (known as third phase formation), has been recast as a critical phenomenon by linking system structure to solute concentration via a critical exponent. The transuranic extraction (TRUEX) system was investigated by extracting increasing concentrations of H3PO4 into a microemulsionconsisting of two extractant amphiphiles (CMPO and TBP) and water in n-dodecaneand taking small-angle X-ray scattering (SAXS) measurements from the resulting solutions. The H3PO4 concentration at which phase splitting occurred was defined as the critical concentration (XC), and this was related to the precritical concentrations (X) by the reduced parameter ε = (XC − X)/XC. The scattering intensity at the zero angle I(0), relating to the interaction between reverse micellar aggregates, conformed to the relation I(0) = I0ε−γ, with critical exponent γ = 2.20. To check γ, SAXS measurements were taken from the organic phase in situ with variable temperature through the point at which third phase formation initiates (the critical temperature), giving I(0) = I0t−γ, where t = (T − TC)/TC and TC and T are the critical and precritical temperatures, with critical exponent γ = 2.55. These γ values suggest third phase formation is a universal phenomenon manifest from a critical double point. Thus, solvent extraction is reduced to its fundamental physical roots where the system is not defined by detailed analysis of metrical properties but by linking the fundamental order to thermodynamic parameters via an exponent, working toward a more predictive understanding of third phase formation.
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INTRODUCTION The self-assembling properties of amphiphiles and their ability to marry the hydrophobic to the hydrophilic give them a rich mesostructural chemistry that produces diverse physical properties on which a range of technological applications depend in both specialized (e.g., nanosynthesis) and universal (e.g., cleaning detergent) fields. In the ubiquitous separations process known as solvent extraction, amphiphilic “extractants” are used to remove hydrophilic entities from an aqueous phase into a hydrophobic organic oil. The amphiphilic properties of extractants can cause them to self-assemble into reverse micelles (RMs) that influence the extractive properties of the system as well as the physical properties of the organic phase.1−4 In many solvent extraction systems, a problematic consequence of this self-assembling behavior is a spontaneous phase transition that manifests in the splitting of the organic phase into two domains when high concentrations of ionizable solute (Brønsted acids or metal ions) are extracted. The technological consequences of this phenomenon, known as “third phase formation”, are serious, limiting the economic viability of commercial solvent extraction systems as well as risking environmental contamination.5 These problems are exasperated by the unpredictability of third phase formation, which is drastically dependent on the system conditions. Therefore, a fundamental platform is sought to understand how extraction drives the physical properties that control the phase change in order to move toward a predictive understanding. Over the past decade, progress has been made in understanding the fundamental physical mechanisms that drive third phase formation in solvent extraction systems. © 2013 American Chemical Society
Using small-angle X-ray scattering (SAXS) and small-angle neutron scattering (SANS), investigators demonstrated that the phenomenon arises because of interactions between small spherical aggregates of reverse micelle type.6 When the attractive potential reaches a certain limit, the aggregates spontaneously condense into a separate phase.2,3,8 This model is concurrent with the extensive wider literature on phase transitions observed in oil-rich microemulsions, which is summarized in several reviews.9−15 In some cases, phase transitions in microemulsions have been shown to behave like critical phenomena that conform to principles of universality.10,24 Critical phenomena are defined, not by detailed material parameters, but by the fundamental collective ordering of the system that is controlled by interactions between particles. This means that thermodynamic properties of completely different critical systems (such as an Ising ferromagnet or a boiling liquid) often show the same dependence on, for example, temperature. This dependence can be described using a critical exponent that defines the nonanalyticity of various thermodynamic functions so that phenomena as different as the liquid/gas and ferromagnetic transitions can be described by the same set of critical exponents and are said to belong to the same universality class. The critical exponent defines how the thermodynamic driver (or field variable) of the phase transition is connected to system order in relation to the critical point, so that the system Received: August 12, 2013 Revised: December 11, 2013 Published: December 13, 2013 315
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developed for separating used nuclear fuel and comprises a mixture of two extractant amphiphiles (Figure 1): octyl-
behavior in response to various thermodynamic stimuli can be better predicted and cast in a wider physical context. Different theories have been postulated to explain the physical origin of universal behavior, one of the most important being the Ising model, which predicts theoretical values for critical exponents and serves as a comparison for experimentally derived values.25,26 Most studies that seek critical exponents in microemulsions exploit temperature variation because it is a known field variable that can easily be independently controlled. These use the “reduced temperature”, t, to define a vector about the critical point:
t = (T − TC)/TC
Figure 1. The TRUEX solvent: 0.2 M octyl(phenyl)-N,Ndiisobutylcarbamoylmethylphosphine oxide (CMPO, left) and 1.2 M tri-n-butyl phosphate (TBP, right) in n-dodecane.
(1)
(phenyl)-N,N-diisobutylcarbamoylmethylphosphine oxide (CMPO) and tri-n-butyl phosphate (TBP) in n-dodecane. In our previous studies of the TRUEX system,27 phosphoric acid (H3PO4) was shown to instigate phase splitting through an easily manageable concentration range with negligible coextraction of water, making the TRUEX−H3PO4 system a good candidate for this study.
where TC and T are the critical (relating to the temperature at which phase splitting occurs) and precritical temperatures, respectively. However, some studies have shown that the water-oversurfactant ratio in microemulsions also behaves like a field variable akin to chemical potential.10,16 In these studies, the critical field variable was defined as a vector about the critical point: [(WC − W)/WC], where WC and W are the critical and precritical water-over-surfactant ratios. In solvent extraction, third phase formation is driven by the extraction of an ionizable hydrophilic solute (such as Brønsted acids or metal salts) into a water−amphiphile−oil system. Thus, the phase change is also driven by increasing the concentration of a chemical component (in this case metal salt or Brønsted acids instead of water) and may have an associated critical exponent. In the same way as the water-over-surfactant ratio studies,10,12 a vector about the critical point can be defined:
ε = [(XC − X )/XC]
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EXPERIMENTAL SECTION Solvent Extraction. CMPO was recrystallized from previously synthesized stocks as described in the literature.28 n-Dodecane (99%) and TBP were obtained from Sigma-Aldrich (Milwaukee, WI), and TBP was further purified using vacuum distillation before use. H3PO4 (Optima grade) came from Fisher Scientific (Pittsburgh, PA). All of the other reagents used in this study were of analytical grade and used without further purification. The aqueous solvent extraction solutions containing increasing concentrations of H3PO4 were prepared using ultrahigh-purity water (18.2 MΩ cm). Aliquots (2 mL) of aqueous phases containing varying H3PO4 concentrations were stirred to equilibrium with aliquots (2 mL) of organic phase (0.2 M CMPO and 1.2 M TBP in ndodecane) at 23 ± 0.5 °C by vortexing in screw cap glass test tubes for 30−40 min. The phases were then separated using a centrifuge (5 min at 3300 rpm), after which samples of the various phases were withdrawn for analysis. For the determination of the critical H3PO4 concentration (concentration at which phase separation initiates), 1.5 mL of aqueous and organic phases were contacted under conditions where the organic phase splits into two, forming the third phase. Very small volumes (10−20 μL) of fresh organic phase and water were added to this three-phase system until the third phase disappeared. The H3PO4 concentration in the organic phase was determined by stripping into water (three contacts at 1:2 organic to aqueous phase ratio) and analyzing the aqueous strip solutions. Complete stripping was confirmed in all samples by a simple mass balance. Potentiometric titrations with 0.1 or 1 M NaOH solutions were used to establish H3PO4 concentration in all aqueous phases. Water concentration in the organic phase was determined using Karl Fischer titrations (Metrohm 756/ 831 KF coulometer (Riverview, FL)). Collection of Small-Angle X-ray Scattering (SAXS). SAXS was performed at the Advanced Photon Source (Argonne National Laboratory) at beamline 12-ID-B where the 12 keV incident X-ray beam is monochromatized with a Si ⟨220⟩ double crystal monochromator and double focused by flat vertical and horizontal Pd-coated mirrors to produce a spot size of around 200 (H) by 50 (V) μm2. Samples were contained in 2 mm diameter quartz capillaries (Charles Supper Co., 20QZ). The 1/e attenuation length of the pure diluent, n-
(2)
where XC and X are the critical and precritical solute-overextractant ratios. Critical phenomena are investigated by observing how thermodynamic field variables drive changes in the interparticle relations that produce system order and lead to the phase transition. Third phase formation stems from interactions between reverse micellar particles that are altered when ionizable solutes are extracted into their cores.6 X-ray and neutron scattering is sensitive to these interactions, particularly in the low-q region. Therefore, to define changes in system order in the solvent extraction organic phase, we can calculate the scattering at zero angle I(0) where scattering intensity is most sensitive to interparticle interactions and least sensitive to the shape of individual particles. To explore whether ε behaves like a critical field variable, SAXS was used to investigate the dependency of I(0) on solute extraction (ε). I(0) is proportional to osmotic compressibility and related to correlation length (ξ, a measure of system order in statistical mechanics) by the Ornstein−Zernike relation I(q) = I(0)/(1 + q2ξ2), where q is the scattering vector. Therefore, according to phase transition theory, if the extraction of ionizable solute into the microemulsion behaves like a critical field variable, I(0) variation should be related by a simple power law:10,12 I(0) = I0ε−γ
(3)
where ε defines a distance from the critical point related to solute-over-surfactant ratio, γ is the critical exponent for compressibility, and I0 is a scale factor. The present study is focused on the technologically important transuranic extraction (TRUEX) system, which was 316
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dodecane, is 12 mm at the incident photon energy, which is 6 × longer than the path length of the capillary tube. Scattering profiles were acquired in 0.5 s exposures on a Pilates 2M counting detector with pixel size 0.172 mm and dynamic range 2.20 The sample-to-detector distance was such to provide a range for momentum transfer 0.01 ≤ q (Å−1) ≤ 0.3. Calibration of scattering vector (q) was achieved using a silver behenate standard.29 The 2D scattering images were radially averaged, producing plots of scattered intensity, I(q) vs q, where q (Å−1) = (4π/λ) sin θ and λ is the wavelength of the X-rays and 2θ is the scattering angle, as described previously.30 The I(q) data were normalized to absolute scale (cm−1) through calibration with SAXS taken from water (18.2 MΩ cm). Background subtraction of solvent and capillary was performed with a single capillary tube to obtain data free from capillary-to-capillary variation. The capillary was subtracted from the sample data using a scale factor of 1, and the scale factor used for the subtraction of the background dodecane solvent was adjusted to reflect the volume ratio of dodecane in each of the individual samples (calculated from metrical analysis of the solutions). In this way, background subtraction removed the capillary and solvent scattering at the appropriate ratios so that the SAXS waves are representative of the aggregates themselves and not the solvent or vessel in which they are contained. Application of the Generalized Indirect Fourier Transformation (GIFT). The GIFT method31−33 was employed to produce p(r) functions from SAXS data across the entire q-range (0.01−0.3 Å−1). GIFT interprets the SAXS data assuming it is a globular particle system, I(q) = nP(q)S(q), where P(q) is the average form factor (particle shape and size), S(q) is the average structure factor (interparticle interaction effects), and n is the number of particles per unit volume. P(q) is equivalent to the Fourier transformation of the corresponding real-space function, p(r), according to eq 4: P(Q ) = 4π
∫0
∞
p(r )
sin Qr dr Qr
with the PY closure relation to interpret SAXS data from interacting worm-like nonionic surfactant reverse micelle systems in nonpolar media (similar chemically to solvent extraction organic phases).41−44 A detailed discussion of the treatment of SAXS data using GIFT is given in a recent review article by Glatter.40 Temperature Variation SAXS. The critical temperature (temperature at which third phase formation initiates) of a TRUEX organic phase containing 0.440 M H3PO4 and 0.95 M water was determined by cooling the solution and taking SAXS measurements in situ while observing the sample using a ccd camera. A microcapilliary containing the microemulsion was held within an aluminum block and cooled using a Peltier cooler at a rate of 0.5 °C/min, and SAXS measurements were taken every second with 0.1 s exposure times. The temperature of the capillary/microemulsion and the block were checked to be consistent using an IR camera so that the sample came to equilibrium before each shot was taken. Cooling the solution induced third phase formation which was heralded by the solution turning opaque, and the SAXS results showed simultaneously a collapse in intensity in both the high- and low-q regions following a strong −2 power-law response. This power law agrees with the power law observed at the critical H3PO4 concentration and helps to identify the critical temperature. Repeating the critical temperature measurement gave results that agreed within 1 K. The plot showing the dependence of I(0) on temperature was constructed from SAXS measurements taken in situ when heating an aliquot of fresh 0.440 M H3PO4-TRUEX microemulsion. The same heating rate was used, and an IR camera showed that the sample came to equilibrium before each shot was taken. For both heating and cooling, background subtractions were performed in the same way as described above but using SAXS data from capillary and dodecane solvent within 1 K of the sample temperature.
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(4)
RESULTS AND DISCUSSION In solvent extraction, the mole ratio of amphiphilic extractant to oil is fixed, and their concentrations do not change significantly. Only the concentration of extractable solute and sometimes water change appreciably during the process, so the “phase space” is explored by tracking the concentration of extracted solute (in this case H3PO4) and water in the microemulsion as a function of aqueous phase concentration. In this investigation, the concentrations of CMPO and TBP in n-dodecane were fixed at 0.2 and 1.2 M, respectively, which are the optimal concentrations used in the technological process. The point of phase splitting was approached by stirring this organic medium with aqueous solutions containing increasing concentrations of extractable solute, in this case H3PO4. By incrementally increasing the concentration of H3PO4 in the aqueous phase and monitoring H3PO4 and water concentration in the organic phase, the macroscopic (extractive) behavior of the system was defined (Figure 2). As H3PO4 concentration in the aqueous phase increases, more H3PO4 reports to the organic phase (open markers) until reaching a concentration of 0.472 M (the “critical” concentration, marked with the dotted line), after which phase splitting occurs. Of note is the change in slope after an organic H3PO4 concentration of about 0.284 M, which suggests a change in the properties of the solvent. The water concentration stays approximately level at around 1 M. The high water concentration, remaining relatively unperturbed during H3PO4 extraction, suggests that the organic solvent is a
Therefore, to deduce p(r), the inverse Fourier transformation (IFT) of the experimental SAXS P(q) must be performed. For concentrated interacting systems, such as in the solvent extraction organic phase, the structure factor, S(q), must also be modeled and subtracted from the scattering data, I(q), to give P(q) that corresponds to the scattering particles alone. Structure factor model selection is important in achieving coherent p(r) functions using GIFT, and the model selected in this study was the Percus−Yevick (PY) closure relation.34 This particular structure factor model has been shown previously, in numerous solvent extraction studies, to closely approximate the interaction effects.4,35−38 The PY closure relation solves the Ornstein−Zernicke equation by assuming that the hard sphere potential is zero if there is no particle overlap and infinity if there is, approximating the excluded volume effect. The resulting structure factor is dependent on particle volume fraction and radius, with an in-built modification that includes polydispersity within a distribution function. The advantage of the PY closure relation structure factor is that it is stable for many systems as many potentials behave similarly to effective hard spheres.39 As stated by Glatter,40 the PY structure factor model also holds true for deviations from “hard” and spherical particles and is applicable to modeling interactions between elongated particles. This is demonstrated in a number of recent publications where the GIFT method was used in combination 317
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approximated by the structure factor model for hard spheres, which has been successfully applied to a range of solvent extraction3,36 and amphiphile−oil41,43,44 systems. Here, the hard sphere model in conjunction with the generalized indirect Fourier transform (GIFT) method was used to deconvolute the interaction effects and generate real-space functions representative of the aggregate structure, as described in recent publications.27,50−52 The GIFT-generated p(r) functions for the SAXS data in Figure 3a are shown in Figure 3b and give an incremental picture of the rate of change of aggregate structure with respect to H3PO4 extraction. As H3PO4 concentration is increased from 0.03 to 0.220 M, relatively little change is observed with the same general p(r) function showing a pronounced peak at approximately the same position. This indicates globular particles of approximately constant morphology in this H3PO4 range,53 agreeing with the large body of scattering literature on solvent extraction systems.6 Increasing the H3PO4 concentration from 0.220 to 0.284 M results in a shift of the peak position and an elongation of the function to higher r, suggesting a change in morphology that continues up to 0.436 M where the function extends to above 200 Å. The development of elongated particles at higher solute concentrations agrees with previous work on the TRUEX27,48 and related systems.54 Beyond this, a small increase in H3PO4 concentration from 0.436 M to near the critical point at 0.472 M results in a large change in aggregate morphology, where the “ski slope” shaped functions in Figure 3b give way to a large broad peak that suggests 2-dimensional or interconnected structures.33,55 Such a sudden change in micelle size/shape close to the point of phase splitting as well as the jump in scattering intensity in the low-q region of Figure 3a is suggestive of critical behavior and comes about when the usually modest, short-ranged van der Waals attraction becomes large and long-ranged.15 In addition to giving a general perspective of the evolution of particle morphology, the p(r) functions can also provide a quantitative estimate of average particle diameter by locating the second infection point of the curve.33 These values were calculated from the p(r) functions in Figure 3b using the second differential and are plotted in Figure 4. In the H3PO4 concentration range of 0−0.284 M, the particle diameter remains approximately constant. After this concentration, the diameter increases first steadily and then rapidly, betraying a significant change in particle morphology in this range. The change in morphology likely causes the change in the macroscopic extractive properties discussed above in relation
Figure 2. Organic water (solid square markers) and organic H3PO4 (open diamond markers) equilibrium concentrations (Y-axis) versus aqueous equilibrium H3PO4 concentrations (X-axis) for extraction of H3PO4.
water-in-oil microemulsion45 where water molecules retain in the center of the reverse micelle aggregates46 that are known to persist in many solvent extraction systems containing TBP1,4,36,37,47 and mixtures of TBP and CMPO.27,48 H3PO4 drives the phase change in this system when it is incorporated into the microemulsion structure, with little change in the relative concentrations of the other constituents. In this way, H3PO4 concentration is a virtually independent factor driving the phase change in this system and was thus investigated as a critical field variable. Before exploring critical behavior in a microemulsion, it is important to define how the structure changes when the phase transition is approached because I(0) values cannot be used to derive critical exponents in regions where a large change in particle morphology also occurs. SAXS data were collected from the organic solutions with increasing H3PO4 concentration. H3PO4 extraction perturbs the background subtracted scattering waves (Figure 3a), with increased scattering in the low-to-medium q-range. As more H3PO4 is extracted, the SAXS waves tend toward a straight line of gradient −2 on the log−log plots at the highest H3PO4 concentration of 0.472 M, showing a power-law behavior of q−2. Power-law behavior in the SAXS data can give qualitative insight into the structure of the solutions; q0 indicates globular reverse micelles (at low H3PO4 concentrations), whereas and q−2 indicates 2-dimentional superstructures.49 The structure of the aggregates in concentrated microemulsions such as solvent extraction organic phases can be difficult to discern from SAXS data because repulsive interactions between the particles, arising from the excluded volume effect, influence the scattering.40 Fortunately, the scattering contribution produced by these interactions can be
Figure 3. (a) Background subtracted, normalized SAXS waves with organic H3PO4 (acid) concentration increasing from light blue to dark green and (b) the corresponding GIFT-generated p(r) functions. 318
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the critical concentration of solute (in this case H3PO4). According to phase transition theory, if H3PO4 extraction is a true critical field variable, the γ value derived using temperature variation should be approximately equivalent to that obtained using H3PO4 concentration variation. As discussed in the Introduction, temperature has been used to derive critical exponents in numerous microemulsion studies. These use the “reduced temperature”, t, to define a vector about the critical point according to eq 1. If the phase transition is a critical phenomenon, then the plot of I(0) against t should be linear on the log−log plot according to
Figure 4. Plot of average aggregate diameter (d) estimated from the second inflection point on the p(r) functions versus organic H3PO4 concentration.
I(0) = I0t −γ
(5)
To determine the critical exponent of third phase formation in the TRUEX system using temperature, a high-H3PO4 solution was selected (0.440 M), which undergoes third phase formation upon cooling. The critical temperature of the solution (temperature at which third phase formation is induced) was accurately determined by taking SAXS measurements in situ while slowly cooling a microcapilliary containing the microemulsion in a Peltier-cooled aluminum block. Figure 6a shows
to Figure 2 as the physical properties of microemulsions are often tethered to their structure. Having identified the region in which particle morphology remains constant with increasing H3PO4 concentration (0− 0.284 M) the next stage was to plot the I(0) values in this region against the field variable parameter ε (eq 2). The GIFT fits to the SAXS data were used to estimate the scattering at q = 0 for each H3PO4 concentration between 0 and 0.284 M and these were plotted against the corresponding ε values. On the log−log plot of I(0) versus ε (Figure 5) it is seen that I(0) and ε follow a power law according to eq 3.
Figure 5. Plot of log I(0) versus log ε for the SAXS waves in Figure 3a show the linear behavior expected from a critical field variable. Exponent γ = 2.20 was calculated from the least-squares fit to the data (R = 0.984, dotted line). This is close to the doubled universal Ising value of 2.48 shown by the gray line (offset from the data for comparison purposes). The SAXS waves from the solutions where significant changes in particle morphology occur do not yield linear data and are omitted from this plot. Figure 6. Background subtracted, normalized SAXS waves collected from the organic phase containing 0.440 M H3PO4 during: (a) Cooling from 280 to 275 K. Scattering increases in the low-q region during cooling from 280 to 279.8 K (blue waves) until it approaches a straight line of gradient −2 on the log−log scale at the critical point of 279.6 K (green wave). Cooling beyond the critical point (red waves) shows decreasing scattering intensity in both the high and low q. (b) Heating of a fresh solution from 286 K (dark blue) to 350 K (dark red) causes a decrease in scattering in the low q and constant scattering in the high q.
A least-squares fitting to the data (shown by the dotted line in Figure 5) gives the value 2.20 for critical exponent γ. This is close to double the universal Ising value for correlation length (2 × 1.24 = 2.48), which is shown by the gray line in Figure 5 and offset from the experimental data for comparison. It should be noted that the data for the H3PO4 concentrations where significant changes to particle morphology are observed (0.284 M and higher) do not follow the linear power law on the log− log plot. The linear log−log behavior of H3PO4 extraction against I(0) suggests that the biphasic extraction of H3PO4 by the TRUEX solvent behaves like a critical field variable and that third phase formation has a critical exponent of approximately double the universal Ising value. To corroborate this result, temperature variation was exploited to probe the phase transition point. Temperature is a well-known critical field variable, and third phase formation can be induced in solvent extraction organic phases by changing the temperature of a solution with close to
selected SAXS results collected from the solution as it was cooled from 280 K and through the critical temperature. For the data in the precritical temperature range (shown by the blue lines), the intensity of the scattering in the low-q region increases until it reaches a straight line with a gradient of −2, while the intensity in the high-q region stays constant (see inset). This indicates that the total volume of scattering particles in solution remains constant in this temperature range whereas the attractive interactions between them increase. At 319
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the critical temperature (279.6 K, green wave) a strong q−2 dependence is shown. When cooling continues below this temperature (279.4−275 K, red waves), a decrease in intensity is seen in both the high and low q-regions as the reverse micelle-type scattering particles coalesce into the third phase. After establishing the critical temperature, the next stage was to probe the I(0) behavior, which was conducted by heating a fresh aliquot of the same solution and taking SAXS measurements in situ. The SAXS waves are shown in Figure 6b with temperature increasing from 286 K (dark blue wave) to 350 K (dark red wave). Using eq 1, the temperature-dependent I(0) values were estimated by fitting the data using GIFT. In the same way as with the H3PO4 extraction study, the p(r) functions were used to identify the temperature range at which the morphology changes significantly. Such changes occurred only at temperatures below 313 K (40 °C), and these data were thus omitted from the critical exponent plot. By plotting t against I(0) on the log−log scale for temperatures above 313 K (Figure 7), we can see that the points follow a power law expected for a critical field variable according to eq 5.
phase. The lower critical point has been suggested56,58−60 to arise from where short-ranged directional interactions dominate between unlike constituents (such as H-bonds between water and amphiphile) and therefore favor miscibility at low temperature. At higher temperatures, entropy makes these directional interactions break apart so that the nondirectional van der Waals-type interactions between like constituents (water−water and amphiphile−amphiphile) dominate and favor separate phases. Solvent extraction organic phases are far more complex than the systems studied by Wheeler in that they contain reverse micelles that interact over different length scales and contain molecules that form H-bonding interactions as well as ionizable solutes that can form charged centers. Therefore, a complex hierarchical interplay between entropic effects (that would favor the mixing of reverse micelles in the oil solvent) and enthalpic effects (attractive interactions between reverse micelles that drive toward third phase formation) exist in a microemulsion that is constructed from H-bonding, dipole−dipole, and electrostatic interactions. Although solvent extraction systems are too complex to explain easily by referring to a simple interaction-based model, the critical-like behavior elegantly reveals the universal nature of third phase formation and the doubled exponent suggests that a critical double point may lay at its heart. In this way, the interactions between reverse micellar aggregates in this solvent extraction system were shown to respond in a predictable manner to field variables whether it is solute extraction or temperature variationso that the critical point of third phase formation was related to system conditions via a simple exponent value. Thus, by recasting solvent extraction as a critical system, a more predictive understanding of third phase phenomena is achieved.
Figure 7. Plot of log I(0) versus log t for the SAXS waves in Figure 6b show the linear behavior expected from a critical field variable with exponent γ = 2.55 calculated from the least-squares fit to the data (R = 0.997, dotted line). This is close to the doubled Ising value of 2.48 shown by the gray line. The SAXS waves from solutions close to the critical limit do not tend toward a plateau within the measured q-range and therefore do not give good estimates for I(0) and are thus not included in the plot.
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CONCLUSIONS
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AUTHOR INFORMATION
Small-angle X-ray scattering was used to explore the critical characteristics of third phase formation in the H3PO4−TRUEX system. I(0) gave a linear dependence on ε (where ε = (XC − X)/XC; XC and X are the critical and precritical concentrations of H3PO4) on the log−log plot in regions where little change in particle morphology was observed to give an exponent γ approximately double the universal Ising value. To corroborate this result, temperature variations were used to probe the point of third phase formation and a similar critical exponent was derived. These results represent the first critical exponents derived for a solvent extraction system and suggest that third phase formation is a manifestation of a critical double point that underpins demixing or a vanishing miscibility gap. This is the first study that has sought critical exponents corresponding to the extraction of ionizable solutes between phases. Future researchers might use this methodology to seek critical exponents in other solvent extraction systems.
A least-squares fitting to the data (dotted line in Figure 7) gives a value for critical exponent γ = 2.55. Like in the H3PO4 concentration variation, this is close to double the universal Ising value for correlation length shown by the offset gray line in Figure 7. Both the H3PO4 extraction and temperature variation results gave similar critical exponents, providing strong evidence that H3PO4 extraction is a critical field variable and that third phase formation has a critical exponent approximately double the universal Ising value. Doubled exponents have been found near the vanishing miscibility gap (when liquid mixtures of two or more components separate into different phases) in oil−water systems where upper and lower critical points are said to coalesce into a “critical double point”.16,23,56,57 Critical double points arise in solution mixtures that have two critical points (an upper point and a lower point) that exist in the same phase space due to complex combinations of different interactions and entropic effects.55 According to the model suggested by Wheeler,56,58−60 the upper critical point arises from simple energy considerations where enthalpic interactions between like particles or molecules (say oil−oil or water−water) dominate at low temperature and favor separate phases; whereas at high temperatures, the entropic effects dominate that favor miscibility so that the oil and water phases collapse into one
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work and the use of the Advanced Photon Source are supported by the U.S. Department of Energy, Office of Basic 320
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