Chapter 12
Chemical Potentials in Ternary Supercritical Fluid Mixtures 1
2
David M . Pfund and Henry D. Cochran 1
Chemical Sciences Department, Pacific Northwest Laboratory, Richland, WA 99352 Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831
2
The solubility of a solute in a supercritical fluid can be greatly enhanced by adding an appropriate co-solvent or entrainer. We have explored this solubility enhancement at a molecular level using integral equation theory. With a method we developed previously, based on scaled particle theory (SPT), we are able to estimate the chemical potential of a solute from molecular distribution functions calculated from the hybrid mean spherical approximation. A multistep charging process permits a separate determination of the effects of short-ranged repulsive and long-ranged attractive interactions on the chemical potential. The SPT-based method provides a means of explaining how a small concentration of a third component can greatly alter the chemical potential, and therefore the solubility, of a solute in a supercritical solvent.
Theoretical studies of supercritical fluid extraction have focused their attention on the structure and properties of binary mixtures (1-7). Attention needs to be turned to ternary mixtures since the presence of a third component can significantly affect the solubility of a solute in a supercritical solvent (8-11). The third component can also affect the selectivity of a supercritical solvent for one solute or another (12,13). The Peng-Robinson equation of state can sometimes provide a correlation of these effects, however the values of the binary interaction parameters obtained in such correlations can not be interpreted in terms of the properties of the molecules (12). In previous work (14) we presented an interpretation of these effects in terms of fluctuation theory. The authors developed expressions relating the solubility to the Kirkwood-Buff affinities of the solute for the solvent and of the solute for the third component. In other work(75,76) we developed and tested a technique for estimating chemical potentials in supercritical fluid mixtures. The technique described how large solvent-solute attractions act to reduce the chemical potential, and therefore increase the solubility of a solute. In the present paper we extend the chemical potential prediction technique to ternary mixtures. Only recently have theoretical methods been devised for and applied to the calculation of chemical potentials. Methods based on molecular simulation techniques have been
0097-6156/93/0514-0149$06.00/0 © 1993 American Chemical Society
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used to estimate chemical potentials in fluids using realistic fluid models such as the Lennard-Jones model (17-25). These techniques can provide rigorous estimates of chemical potentials in the chosen model for mixtures of gases or liquids that are not too strongly non-ideal. Simulations are less useful in the study of supercritical solutions (which tend to be dilute mixtures of unlike species and to have very long-ranged correlation functions). Lotfi and Fischer (7 9) have found that large system sizes and long run times are required in order to obtain reproducible results from the Widom test particle method (77) for large solutes in mixtures at infinite dilution. Approximate theories can provide a fast way of estimating chemical potentials in such asymmetric mixtures. Simulations provide the means for verifying the accuracy of theories. Theory Integral equation theories can be used to estimate potentially long-ranged correlation functions in dilute mixtures of unlike species (2). The hybrid mean spherical approximation (HMSA) integral equation theory is a theory for the correlation functions in fluids modeled with spherical particles. It provides accurate estimates of the pressures and internal energies of such fluids (26). In this theory the Ornstein - Zernike (OZ) equation,
h (r)-C (r) = ij
ij
+
p J C (r , 1
v
il
i
l
lj
rp àr
v
x
c
P 2 j i 2 ( r i » r ) h ( r , rp d r v
+
r )h (r 2
c
P J i3(iV 3
v
2 j
2
r ) h ( r , rp d r 3
3 j
3
( 1 ) 2
3
(written for a ternary mixture) is solved simultaneously with the closure equation (26), 0 =Η -(Ι)ψ Μ
υ
8 ί
.ββ»5-( -1)]-β^ ΐ
(2) 0
to yield the direct and total correlation functions, Ç. and h.. respectively, in the fluid, u^ and u.} are the repulsive and attractive parts of the intermolecular pair potential according to the division of Weeks, Chandler, and Anderson (27). The adjustable switching function s is varied to achieve thermodynamic consistency between the bulk moduli obtained from the virial and compressibility equations. The reliability of integral equation theories (including the HMS A) for states very near to the liquid-vapor critical point has not been extensively studied. Numerical schemes for solving Equations (1) and (2) which require h (r)= 0 for r > are unable to approach the critical point closely and the results will deviate from known analytic solutions for simple models and theories (28). The conditions for which the H M S A theory will be applied in the present work will be sufficiently removed from the critical point so that the theory will supply accurate estimates of structures, pressures, and internal energies (76). The estimated correlation functions for the near critical states of Reference (5) are consistent with simulation results s
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151
but lack the statistical noise characteristic of such simulations. In this work we investigate the chemical potentials calculated from the H M S A theory using a Kirkwood charging process based on scaled particle theory. In the Kirkwood charging process, the residual chemical potential of a solute species j equals the change in Helmholtz free energy (or work) at constant temperature and volume which occurs on moving a molecule of j (referred to as a test particle) from a fixed position in an ideal gas to a fixed position in the fluid of interest (29). This free energy change will be referred to as the work of insertion. The new (or test) particle may be inserted in an arbitrary number of steps over an arbitrary path. The insertion process is equivalent to switching on a field source at that position; for a given path the switching on is accomplished by varying a coupling parameter λ \ The resulting free energy change is (30), (3)
A(1)-A(0) =
uit(r
l t
^)]dr
l t
dX
Γ
λ
δ2ι< 2Ρ )
Pi J
A u 2 t ( r A )] dr 2t dX 2 t
P μ (Γ ;λ)Λ U 3 t ( r 3
3ι
J
3ι
3t
^ )j dr
3t
dX
where the pair correlation function of molecules of species i surrounding the test particle is denoted by g and the pair potential between the test particle and molecules of species i is u . As is shown in Equation (3), the work of insertion is a function of the correlation functions for fluid particles surrounding a test particle. These functions will be determined from the H M S A theory. The O Z equations which are used are those for a mixture consisting of a test particle species which is at infinite dilution in the fluid of interest (which may contain the solute at finite concentrations). The components of the test particle pair potentials u and xx which appear in Equations (2) and (3) vary with the coupling parameter for each step in the charging process. k
tt
0
h
l
k
In applying Kirkwood charging one must first specify a charging path by which the test particle is inserted into the fluid. If Kirkwood charging is used with integral equation theories (which are inherently inexact) then different charging paths can yield different chemical potentials (31). It is possible that results obtained from different charging paths differ in their accuracy. In this work a multi-step charging process is used to divide the
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work of insertion into positive repulsive and negative attractive contributions. These contributions result from the short-ranged W C A repulsive potentials u^ and long-ranged W C A attractive potentials u.. , respectively. The free energy change for each step is determined from Equation (3) and the results are summed to give the residual chemical potential. 0
1
The charging path followed in this work is a traditional one used in solvation theory (52). This traditional approach views interphase transfers as occuring in two steps, the first being the formation of a cavity big enough to hold the repulsive core of a test solute molecule (which defines the molecular size) and the second being the placement of such a molecule within the cavity and allowing it to interact with the surrounding molecules. Inserting a hard sphere test particle into the fluid is a process equivalent to forming a cavity. The hard sphere is inserted into the fluid using the particle scaling technique originally developed for scaled particle theory (SPT) (33-35). Associated with this step is a free energy change or work of cavity formation. Effective hard sphere diameters must be defined for the softly repulsive species in the fluid. These diameters determine the cavity radii at the end of the hard sphere insertion process. They are obtained during the process of switching-on the softly repulsive interactions between the test particle and the surrounding particles. The hard sphere test particle of species j is softened until it becomes the W C A repulsive particle. The W C A repulsive test particle of species j interacts with surrounding particles of species i according to the pair potential u^ . Associated with this step is a free energy change or work of softening. The total repulsive contribution to the residual chemical potential equals the work of cavity formation plus the work of softening. The effective hard sphere diameters for interactions with the hard sphere test particle are chosen so that the work of softening is approximately zero. The work of softening is estimated using a technique adapted from liquid state perturbation theory (36). In the last step the attractive components u^ of the pair potentials for interactions of species i with the test particle of species j are switched-on; the repulsive components u^ being fully charged after the hard sphere insertion and softening steps. Associated with this step is a free energy change or work of binding. After completing this step the test particle is equivalent to an ordinary particle of species j , interacting with species i according to the potential Uj. = u^ + u.. . 0
1
0
0
1
In a previous work (75), we described the development of the SPT-based method and its application to the H M S A theory. Extension of the method to ternary mixtures of Lennard-Jones particles or to a similar model of fluids is straightforward. The complete charging process is illustrated in Figure 1. In the first step (proceding counter clockwise from the upper left) a free mass-point is replaced by a hard point test particle. The hard point can approach the center of a particle of species j to within a distance dJ2, where d~ is the effective diameter of species j . In the second particle scaling step the hard sphere test particle of species i grows to a diameter of d . In the third softening step the hard sphere is replaced by a soft W C A repulsive test particle. The effective diameters are chosen so that the free energy change for the third step is zero. In the fourth step the W C A attractive potentials are added to the test particle and it becomes a new molecule of species i. The work of cavity formation is the sum of free energy changes for the first two steps. The work of solvent-solute binding equals the free energy change for the fourth. u
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153
The assumptions made in the development of the method were presented previously (75). These assumptions were found to introduce a negligible error in predicted chemical potentials. The principle source of error was determined to be the predicted test particle correlation functions obtained from the HMS A theory (76). The SPT-based charging path minimizes this error under certain conditions. The error was found to be small at low pressures, in compressed liquids, and at supercritical temperatures. It was found that the method presented here is useful in the study of chemical potentials in those supercritical fluids that can be described with spherically symmetric potential models.
Potential Parameters Ternary mixtures were constructed of the following Lennard-Jones species:
Species
Type
σ.Α
e /k. Kelvin
1
Solvent
3.753
246.1
2 3
Co-solvent Solute
3.626 5.883
481.8 674.1
The solvent parameters (taken from Reference 37) were developed from virial coefficient data for carbon dioxide. The co-solvent parameters (taken from Reference 38) were developed from dilute gas viscosity data of methanol. The solute parameters were calculated from an equation in Reference 38 using the critical temperature, critical pressure, and acentric factor of benzoic acid. The parameters are not intended to provide a quantitative description of these components. They are merely intended to be a set of Lennard-Jones parameters likely to yield co-solvent effects.
Results Solute chemical potentials are a function of both the bulk density of the fluid and the local environment of the solute. Solute residual chemical potentials at infinite dilution were calculated in the following mixtures at a temperature of 344.54 Κ (kT/e = 1.4): n
Mole Fraction
Density gmoL/cm
3
Co-solvent
Pressure,
Contributions
arm.
Repulsive
Attractive
(μ3-μ *)^τ
4.01 4.08 1.86 7.92 6.72
-12.10
-8.09 -7.80 -5.22
0.01099
0.035
92.4
0.01099 0.00599 0.01571 0.01443
0.0 0.0 0.035 0.0
108.02 92.4 134.1 134.1
-11.88 -7.08 -16.92 -15.34
3
-9.05 -8.62
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SUPERCRITICAL FLUID ENGINEERING SCIENCE
where the pressures have been calculated using the H M S A theory. Since the critical temperature of a pure Lennard-Jones fluid occurs at approximately kT/e = 1.35 (or 332.24 K), the mixtures are at a reduced temperature T of approximately 1.037. The n
R
3
critical density of a pure Lennard-Jones fluid occurs at approximately ρ σ = 0.35 (or 0.01099 gmol./cm ). Since the mixtures were relatively close to the critical point at a presssure of 92.4 atmospheres, the addition of a small amount of co-solvent resulted in nearly a doubling of the density. The increase in density was the principle cause of the reduction in the residual chemical potential from -5.22 to -8.09. To obtain lower residual chemical potentials for the solute in a binary mixture requires increasing the pressure. At the higher pressure of 134.1 atmospheres the density of the mixture containing the cosolvent was more nearly equal to the density of pure carbon dioxide at the same pressure. Under such conditions the effect of co-solvent addition was to change the local environ ment of the solute resulting in a reduction of the residual chemical potential from -8.62 to -9.05. η
3
Figure 2 displays the pair correlation functions of solvent and co-solvent molecules with the solute in the first three mixtures listed above. The corresponding aggregation numbers for species with the solute are: Density gmol./cm 0.01099 0.01099 0.00599
3
Mole Fraction Co-solvent 0.035 0.0 0.0
Pressure, aim. 92.4 108.02 92.4
Number of Solvent Molecules 60.8 27.7 44.5
Number of Cosolvent Molecules 6.0
-
The aggregation numbers were obtainedfromthe Kirkwood-Buff fluctuation integrals as discussed in a previous work (2). The aggregation numbers are sensitive to long-ranged correlations. The solute/solvent correlation functions are similar in the binary and ternary mixtures when examined at equal densities. The co-solvent has a very small effect on the first and second maxima of the solute/solvent pair correlation function at constant density. The co-solvent appears to enhance the long-range correlation of solvent molecules around solute molecules. The co-solvent exhibits very strong short-range and long-range corre lations with solute molecules. The predicted effect of co-solvent addition on the solute chemical potential yields solubility enhancements in qualitative agreement with experiment. The solubility of a solid solute in a supercritical fluid depends inversely on the bulk density and inversely on the exponential of the residual chemical potential of the solute:
1
(18)
The left hand side of Equation 18 is plotted versus pressure in Figure 3. The solubility was enhanced by a factor of 9.6 at 92.4 atmospheres and by a factor of 1.4 at 134.1 atmospheres by the addition of co-solvent. Experimental enhancements of benzoic acid solubility due to methanol addition into carbon dioxide range from approximately 6.0 at low pressures to approximately 3.5 at high pressures at a reduced temperature of 1.013 (11).
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Chemical Potentials in Supercritical Fluid Mixtures 155
PFUND & COCHRAN
Softly Repulsive,. Repulsxve^i^
v
v
Species j
Cores
/ /
Attractive Tails / / ΔΑ^μ,-μ.; New particle ^ ^ / j ' O f species i
Species
*
Weal Gas Point
Softly Repulsive Particle Hard Sphere
Figure 1: The residual chemical potential of species i equals the sum of Helmholtz free energy changes for four steps.
Reduced separation distance r ΛΤ33
Figure 2: Pair correlation functions for fluid species interacting with solute mole cules. Conditions are given in the text.
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SUPERCRITICAL FLUID ENGINEERING SCIENCE 6000001
0H 90
1
1
100
1
1
110
•
1
•
120
1
130
•
1
140
Pressure, atmospheres
Figure 3: Solute solubility is proportional to the quantity plotted on the ordinate. Conditions are given in the text.
Conclusions In summary, the residual chemical potential of a solute in a pairwise additive, spherically symmetric, ternary fluid model has been expressed as the sum of an approximately determined work of inserting a hard point plus formally exact contributions due to particle scaling, softening the hard sphere, and adding attractive potentials. The method has been used qualitatively to predict the solubility enhancement of a solid in a supercritical fluid due to addition of a co-solvent. Near the critical point of the solvent the solubility was greatly enhanced because of an increase in the density of the fluid upon co-solvent addition. At higher pressures the bulk density remained essentually constant upon co-solvent addition. The observed solubility enhancement was therefore due to an alteration of the local environment of the solute molecule. This work shows that simple fluids (spherically symmetric, non-polar fluids) are capable of exhibiting enhanced supercritical solubility upon addition of a co-solvent and that appeals to "special" or "chemical" interactions (though such interactions may exist in a given system) are not necessary to explain the phenomenon. Acknowledgments This research was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy (DOE) under contract DE-AC06-76-RLO1830 with Pacific Northwest Laboratory and under contract DE-AC05-84OR21400 with Martin Marietta Energy Systems, Inc. Pacific Northwest Laboratory is operated for D O E by Battelle Memorial Institute. Literature Cited 1. Debenedetti, P.G. Chem. Eng. Sci. 1987, 42, 2203. 2. Cochran, H.D.; Pfund, D.M.; Lee, L.L. In Proc. Int. Symp. on Supercritical Fluids (Nice, France); Perrut, M. Ed.; Institut National Polytechnique de Lorraine: 1988; p. 245.
12. PFUND & COCHRAN
Chemical Potentials in Supercritical Fluid Mixtures 157
3. Debenedetti, P.G.; Mohamed, R.S. J. Chem. Phys. 1989, 90, 4528. 4. Cochran, H.D.; Pfund, D.M.; Lee, L.L. Sep. Sci. Technol. 1988, 23, 2031. 5. Petsche,I.B.;Debenedetti, P.G. J. Chem. Phys. 1989, 91, 7075. 6. Cochran, H.D.; Pfund, D.M.; Lee, L.L. Fluid Phase Equil. 1988, 39, 161. 7. Peter, S.; Brunner, G.; Rhia R. In Zur Trennung Schwerfluchtiger Stoffe mit Helfe Fluider Phasen; Monograph 73; DECHEMA: 1974. 8. Yonker, C.R.; Frye, S.L.; Kalkwarf, D.R.; Smith, R.D. J. Phys. Chem 1986, 90, 3022. 9. Schmitt, W.J.; Reid, R.C. Fluid Phase Equil. 1986, 32, 77. 10. van Alsten,J.D.PhD. Dissertation, University of Illinois, Urbana-Champaign: 1986. 11. Dobbs, J.M.; Wong, J.M.; Lahiere, R.J.; Johnston, K.P. I&EC Res. 1987, 26, 56. 12. Kurnik, R.T.; Reid, R.C. Fluid Phase Equil. 1982, 8, 93. 13. Kwiatkowski, J.; Lisicki, Z.; Majewski, W. Ber. Berbunsenges. Phys. Chem. 1984, 88, 865. 14. Cochran, H.D.; Johnson, E.; Lee, L.L. J. Supercrit. Fluids 1990, 3, 157. 15. Cochran, H.D.; Pfund, D.M.; Lee, L.L.J. Chem. Phys. 1991, 94, 3107. 16. Cochran, H.D.; Pfund, D.M.; Lee, L.L. J. Chem. Phys. 1991, 94, 3114. 17. Widom, B. J. Chem. Phys. 1963, 39, 2808. 18. Heinbuch, U.; Fischer, J. Mol. Sim. 1987, 1, 109. 19. Lotfi, Α.; Fischer, J. Mol. Phys. 1989, 66, 199. 20. Shing, K.S.; Gubbins, K.E.; Lucas, K. Mol. Phys. 1988, 65, 1235. 21. Shing, K.S.; Gubbins, K.E. Mol. Phys. 1982, 46, 1109. 22. Powles, J.G.; Evans, W.A.; Quirke, N. Mol. Phys. 1982, 46, 1347. 23. Panagiotopoulos, Α.; Suter, U.W.; Reid, R.C. Ind. Eng. Chem. Fundam.. 1986, 25, 525. 24. Haile, J.M. Fluid Phase Eq. 1986, 26, 103. 25. Shukla, K.P.; Haile, J.M. Mol. Phys. 1987, 62, 617. 26. Zerah, G.; Hansen, J. J. Chem. Phys. 1986, 84, 2336. 27. Weeks, J.D.; Chandler, D.; Andersen, H.C. J. Chem. Phys. 1971, 54, 5237. 28. Cummings, P.T.; Monson, P.A. J. Chem. Phys. 1985, 82, 4303. 29. Ben-Naim, A. Solvation Thermodynamics ; Plenum Press: New York, 1987. 30. Kirkwood, J.G. J. Chem. Phys. 1935, 3, 300. 31. Kjellander, R.; Sarmen, S. J. Chem. Phys. 1989, 90, 2768. 32. Eley, D.D. Trans. Faraday Soc. 1938, 35, 1281. 33. Reiss, H.; Frisch, H.L.; Lebowitz, J.L. J. Chem. Phys. 1959,31,369. 34. Lebowitz, J.L.; Helfand, E.; Praestgaard, E. J. Chem. Phys. 1963, 43, 774. 35. Reiss, H.; Frisch, H.L.; Helfand, E.; Lebowitz, J.L. J. Chem. Phys. 1960, 32, 119. 36. Lado, F. Phys. Rev. A 1973, 8, 2548. 37. Maitland, G.C.; Rigby, M.; Smith, E.B.; Wakeham,W.A. Intermolecular Forces, Their Origin and Determination; Clarendon Press: Oxford, 1981. 38. Reid, R.C.; Prausnitz, J.M.; Sherwood,T.K. The Properties of Gases and Liquids; third edition; McGraw-Hill: New York, 1977. RECEIVED April 27, 1992
