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Chapter 3

Solvation Structure in Supercritical Fluid Mixtures Based on Molecular Distribution Functions 1

2

Henry D. Cochran and Lloyd L. Lee

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1

Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831 Department of Chemical Engineering, University of Oklahoma, Norman, OK 73019

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On a molecular level, solubility enhancement and large, negative partial molar volumes near critical points are characterized by buildup of longer-range local solvent density around solute molecules. The magnitude of these clusters has been inferred from partial molar volume data. We have studied such clusters and related solution properties using molecular distribution functions calculated by an accurate integral equation theory. We observe the clustering as long-range correlations in thefluctuationsof solvent molecules about a solute molecule. Partial molar volume is predicted in reasonable agreement with experiment, and solute-solute aggregation is also predicted as suggested by recent excimer fluorescence results. We conclude that integral equation theories are useful in revealing solvation structures of supercritical solutions. The remarkable properties of supercritical fluids have stimulated considerable research interest because of practical applications as solvents for separations processes or reaction media and also because of the considerable challenges supercritical solutions present for modeling and theory. These challenges include the following: 1) fluid properties change drastically near and become nonclassical very near critical points (CPs), 2) conditions of interest and importance span a wide density range, 3) solutions are typically very dilute, 4) solvent and solute are frequently very different in molecular size and interaction energy, and 5) molecular correlations in fluids become long-ranged and fluctuations become large near a CP. In current studies of supercritical mixtures, there is an outstanding hypothesis in the search for a molecular-scale mechanism that underlies all the unusual behavior associated with nearness to the CP. This hypothesis is the clustering of solvent molecules around the solute molecules. A surprising result from a recent fluorescence spectroscopy study shows the possibility of formation of solute-solute aggregates that occur near the CP. We propose in this study to clarify the understanding of solute-solvent clustering and solute-solute aggregation near CPs in supercritical solutions. The fluid microstructure is of central interest in the theoretical study of supercritical solutions for several reasons. Bulk fluid properties can be obtained from knowledge of the fluid structure and the intermolecular potential through distribution function theory. In addition, it has recently become clear, both from experimental evidence and theoretical analyses (1-5) that solvation structure changes rapidly near a CP. Eckert et al. (1) interpreted partial molar volume data for supercritical solutions as indi­ cating the collapse of £g. 100 solvent molecules about a solute molecule. Kim and Johnston (2) interpreted the solvent shift in the U V absorption of phenol blue dissolved in supercritical ethylene to suggest a local solvent density surrounding a solute molecule more than 50% greater 0097-6156V89y0406-0027$06.00/0 ο 1989 American Chemical Society

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SUPERCRITICAL FLUID SCIENCE A N D T E C H N O L O G Y

than the bulk solvent density. Debenedetti (4) has used Kirkwood-Buff (fi) fluctuation theory to infer from experimental partial volume data (1) that a solute molecule is solvated to form clusters of £&. 100 solvent molecules for naphthalene dissolved in supercritical CO2 at CP, for example. Several recent studies provided additional experimental evidence for the clustering phenomenon. Kajimoto et al. (£) interpreted absorption and fluorescence spectra of a polar solute in a polar, supercritical solvent in terms of a simple aggregation model to describe the effects of solvent-solute clustering. Brennecke and Eckert (I and ACS Svmp. Ser. in press) measured the fluorescence spectra of pyrene dissolved in supercritical carbon dioxide and ethylene. Strong enhancement of the symmetry-forbidden first singlet-singlet transition close to the critical point is taken as indicative of very strong solvent-solute interaction (probably only nearest neighbor interactions). Furthermore, the presence of a peak ascribed to the excimer dimer even at bulk pyrene mole fraction 10" suggests very strong solute-solute interactions in the supercritical solution, as well. A n investigation of the disproportionation of toluene over ZSM-5 to form xylene (Collins, N . A . ; Debenedetti, P. G . ; Sundaresan, S. AIChE J . in press) tested the occurrence of solvation. Since toluene acts as both reactant and solvent, near the C P toluene clustered around the para-isomer, thus reducing the chances of secondary isomerization into other isomers.

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In previous work (5.8.9) we have shown that models based on distribution function theory yield a priori predictions of the qualitative behavior of supercritical solutions and yield empirical fits to data which show excellent accuracy for solubility and partial molar volume. An understanding of clustering at a molecular level will have to come from molecular distribution functions, especially the solvent-solute correlation function. These correlation functions could be obtained from neutron scattering experiments, computer simulation, or integral equation theories. No scattering experiments of supercritical solutions have been carried out for this purpose. Computer simulation can shed interesting light and is discussed elsewhere (See Petsche, I. B.; Debenedetti, P. G . J . Chem. Phvs. submitted). For infinitely dilute solute species, the structure (particularly the long-range structure near a CP) is more easily calculated by integral equations. We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxidenaphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (1Û) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M . ; Lee, L . L . ; Cochran, H . D. Int. J . Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. Distribution Function T h e o r y Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type α at a distance between r and r + tfr from a molecule of type β is P 4*r 9afi(r) dr where g fl(r) is the spatial pair correlation function. In principle, g fi(r) may be determined experimentally by scattering experiments; however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. Statistical mechanics gives relationships between the distribution functions and the bulk properties of fluids. The total internal energy of a fluid is given by the energy equation, the pressure is given by the viriai equation, and the isothermal compressibility is given by the compressibility equation, see e. g. . Ref. 11. Through the Kirkwood-Buff formulas (6), 2

a

a

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

a

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COCHRAN AND L E E

Solvation Structure in Supercritical Fluid Mixtures

the compressibility, the partial molar volumes, and the derivatives of the chemical potential with respect to number density are obtained; for simplicity of notation these are presented for binary mixtures. pkTxT

ρΰ

β

= [1 +

P

a

=

Ω/Φ,

(G

- Gafi)] /Φ,

aa

and

where Ω = 1+ pG + Ρββββ + ΡαΡβ (G G - Gl ) , Φ = [Pa + Ρβ + ΡαΡβ (Gaa + Gfifi - 2G )] /p, and G fi is the Kirkwood fluctuation integral,

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a

aa

aa

00

3

a0

a

Jo We have previously shown (&) that the solubility of a pure, incompressible solid, B, in a supercritical solvent, A is given by }

η ±

= [GAB + V

B

+ p v' B

B

(GBB -

G

A B

)\

^

where v' is the solid molecular volume. Useful equations for dilute, supercritical solutions result from appropriate approximations in the above expression; see Ref. 8. Kim and Johnston (2) and Debenedetti (1) have defined the size of the cluster of solvent A molecules about a solute Β molecule in dilute solution by the following: B

ξ

ΑΒ

=

PAG%

B

where the superscript oo signifies the value of the quantity at infinite dilution; CAB expresses the number of solvent A molecules surrounding solute Β molecule in excess of the bulk average.

Calculations In this work we have calculated pair correlation functions and derived properties for systems with interaction potentials representative of supercritical solutions by solution of the OrnsteinZernike (OZ) equation using the HMSA closure (lu) and the efficient algorithm of Labik et al. (12)· Independent and corroborative calculations using the R H N C closure (13) for the same mixtures are omitted for brevity. Although quantitative values of distribution functions from the two theories differed slightly, the general features are the same; the R H N C theory gave somewhat higher peaks, but they occur at the same intermolecular separation. We used the following Lennard-Jones parameters for CC^-CioHg:

Φ (Κ)

σ(Α)

225.3

3.794

CioHg-CioHg

554.4

6.199

CO2—CioHe

353.4

4.997

co -co 2

2

The HMSA closure (10) has been used to solve the O Z equation; it is an interpolation between the soft mean spherical approximation at small r and the H N C closure at large r.

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The interpolating parameter has been chosen to achieve consistency between the isothermal compressibility from the compressibility equation and that from the virial equation. The calculations in this work would have been impractical without the use of an efficient and robust algorithm for solution of the OZ equation. We employed the approach proposed by Labik, et al. (12). For these results we used 2048 grid points in real space, 0.005σ step size, 64 Newton-Raphson terms within 10~ tolerance on Newton-Raphson and 10" tolerance on direct iterations. Because of concern about the long range of correlation functions near the CP, we also explored calculations with 512 and 1024 grid points and with step size 0.010, 0.015, and 0.020 σ^Λ- Calculations with range (number of grid points times step size) 10.24 σ or greater and step size 0.010 CAA or smaller gave essentially the same results. ΑΑ

6

5

ΑΑ

9

The calculations reported in this work were for very dilute, χ Β = 10"~ , solutions. In other work (&) we found the results to be unaffected for solute mole fractions from 10" to 10" . Dilute solutions are characteristic of many experimentally-studied supercritical solutions. The low solute concentration does, however, tend to reduce the accuracy of the solute-solute cor­ relation functions, ÇBB (r); so, we regard them as of qualitative value only.

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9

4

This calculational method has been found to yield internal energy, pressure, and distribution functions in excellent agreement with simulation results for pure and mixed L J fluids as well as fluids obeying other simple force laws (1Q). However, it should be noted that tests of the method have not included the highly compressible, near-critical states such as those studied in the present calculations where both simulations and integral equation calculations would encounter difficulties. Furthermore, we know of no direct way to test the accuracy of the long range structure calculated by the integral method at present.

Results and Discussion

Figure 1 shows the predicted spatial pair correlation functions, ρ β, vs. reduced separation distance, r = r/σΑΑ, for a typical, dilute supercritical solution using the LJ parameters given above and at T * = kT/e = 1.37 ( Γ = 308.4K), ρ* = ρσ = 0.40 (Ρ = 0 . 0 0 7 3 2 Â " ) , and XB = 10~ ; the calculated pressure for this state is 8.7 MPa. The clustering of solvent molecules about a dissolved solute molecule, as remarked earlier, is seen in the solvent-solute pair correlation function with first, second, and even third solvation shells. The long-rangedness of 9ΑΒ(?) is clearly in evidence. Similar behavior is observed for the solute-solute pair correlation function also. α

3

AA

ΑΑ

9

Figures 2a and 2b show how the predicted solvent-solute pair correlation function, QAB, varies with density. At the highest density (p = 0.6) the structure is liquid-like with first, second, third, ... maxima/minima oscillating about 1.0. The size of the solvent-solute cluster for this state was calculated to be about —1 solvent molecule; the presence of one solute molecule at this state excludes about one solvent molecule. At lower density (p* = 0.5) the peak maxima are lower and the peak minima are shallower. Again the long range maxima and minima oscillate about 1.0. The size of the solvent-solute cluster for this state was calculated to be about -f2 solvent molecules; one solute molecule is surrounded by about two excess solvent molecules compared with the bulk average concentration. At density (p* = 0.4) near the C P (the critical point of the pure LJ solvent is at about =1.35 and ρ = 0.35) the oscillations subside into a persistent long-range correlation, d the pair correlation function exhibits a long-range tail that is greater than 1.0. See par­ ticularly Figure 2b which clearly shows this long-range structure. The solvent-solute cluster for this state was calculated to be +14.65 solvent molecules. At states farther from the C P the solute molecule excluded solvent molecules, but near the C P there is a large buildup of longer-range local solvent density around a solute molecule. In Figure 3 we have plotted the number of excess solvent molecules within a sphere of radius L(= r/a ) around a solute molecule, PG B(L), VS. L for the same states as Figure 2. Τ

a n

AA

A

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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3.

Solvation Structure in Supercritical Fluid Mixtures

COCHRAN AND L E E

L

g (r*) M

gAn(r*) 'gAA(r')

M\

"

CD

JL 0.0

V/'V

I , , 2.0 4.0 6.0 dimensionless distance, r*

8.0

Figure 1: Pair correlation functions at T* = 1.37 and p* = 0.40. For the L J fluid the critical point occurs at about Τ =1.35 and p = 0.35.

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SUPERCRITICAL FLUID SCIENCE A N D T E C H N O L O G Y

dimensionless distance, r* 1,10

. . . . 1 . . . . 1 . , , ,,1, ,., . , , ., ,., - b)

1.05 N

S

.p'=0.40 ^ p=0.50 /

/

\'\



Γ

^p=0.60

' .... 1 .... 1 .... 1 ... ,., , , 1 , , . . '

dimensionless distance, r*

*

*

Figure 2: Solute-solvent pair correlation function at Τ = 1.37 and ρ = 0.40, 0.50, and 0.60. a) Short range structure; b) long range structure (note change of axes).

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

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COCHRAN AND L E E

Solvation Structure in Supercritical Fluid Mixtures

1 —

·

• •

.

,

T=1.37

.

.

.

.

I ι

ρ=0.40^^

-

ρ*=0.50\^^

-

3 10 /

6 4 Q. 2 II 0 % -2 Q, -4 u

- V 0.0

/



y? . . ι . . . . ι 5.0 10.0 15.0 dimensionless distance, L

.

20.0

Figuk Î 3: Number of excess solvent molecules within a sphere of radius L around a solute molecule. Same conditions as Figure 2.

PAG (L) AB

= PA

L

Jo

2

f dr4wr [g B(r)-l} A

This graph shows contributions to the fluctuation integral from the successive neighborhoods.

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It is evident that for p* = 0.40, the state close to the critical point, the solvent-solute cluster has grown to almost 15 excess solvent molecules. The contribution from the first solvation shell can be estimated; about 4 excess solvent molecules contribute to the first solvation sphere. It is also evident that the cluster consists mainly of a diffuse, longer-range buildup of local solvent density about the solute which extends 10-14 solvent diameters from the solute. Whether this cluster maintains its identity for a significant period of time or is merely a statistical phenomenon cannot be answered from these equilibrium calculations. Molecular dynamics calculations suggest that the clusters maintain their structural integrity for significant periods although the molecules constituting the cluster change identity more rapidly (See Petsche, I. B.; Debenedetti, P. G . J . Chem. Phys. submitted). Nevertheless, regardless of their lifetime, these clusters are characteristic of solvation structure near the C P and are responsible for the striking behavior of thermodynamic properties near the CP. Figure 4 shows how the size of the solvent-solute cluster varies with density at Τ = 1.37, 1.41, and 1.46 ( 35,45, and 55 ° C ) . As density increases from zero toward the C P the size of the cluster grows rapidly. Because the pair correlation functions become longer and longer in range as the C P is approached, there is a limit beyond which calculations by this technique fail to converge. As a practical matter, the range of the present calculations (< 40.96 σχ>ι) limits convergence at Τ = 1.37 to densities below 0.15 and densities above 0.4. The rapid decline in the cluster size as density is increased away from the C P is also evident. At temperatures higher above the C P the cluster is smaller and the structure is shorter in range. These calculations support the notion of a large solvent-solute cluster near the C P as suggested from spectroscopic results (2.3 7). The estimation (1.4) that the cluster approaches 100 solvent molecules based on the partial molar volume data of Eckert et al. (I) for naphtha­ lene in supercritical carbon dioxide at this temperature also appears entirely consistent with the predictions from the integral equation theory. Figure 5 shows the partial molar volume of naphthalene in supercritical carbon dioxide predicted by theory compared with the data of Eckert et al. (1). Neither naphthalene nor carbon dioxide would be expected to be described quantitatively by the LJ potential. Further­ more, no binary interaction parameter was used with the Lorentz-Berthelot estimates for the CO2-C10H8 interaction. Thus, the degree of agreement between effectively a priori prediction and experiment is judged to be very satisfactory. f

Figures 6a and 6b show the solute-solute pair correlation function at the same conditions as the solvent-solute functions in Figures 2a and 2b. The aggregation of the solute molecules is quantitatively stronger than the solvation structure shown in Figure 2. A quantitative interpretation in terms of the local density of solute molecules surrounding a given solute molecule has not been given for the band attributed to a solute-solute excimer dimer in the fluorescence spectra of Brennecke and Eckert (2); nevertheless, their qualitative interpretation suggesting a significant solute-solute aggregation near the C P appears to be supported by these results. Conclusions The solvation structures which we have determined for the supercritical solutions of LJ mole­ cules appear to be in agreement with the recent theoretical and experimental suggestions of solvent-solute clustering and solute-solute aggregation near the CP. Quantitative testing of the­ oretical models for solvation structure of supercritical solutions (such as that presented here) may become possible if recent efforts at simulation of supercritical solutions (elsewhere in this volume) prove successful. Clearly, the L J model used in our theoretical studies to date does not provide an adequate representation of real molecular interactions. However, the method we have demonstrated is potentially capable of application with much more accurate (and complicated) potential functions, and meaningful quantitative interpretation of experiments may then become possible. In summary, then, we conclude that integral equation theories are useful in revealing

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

COCHRAN AND L E E

Solvation Structure in Supercritical Fluid Mixtures

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3.

dimensionless density, p* Figure 5: Calculated and experimental (1) solute partial molar volume vs. density at Γ * = 1.37.

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SUPERCRITICAL FLUID SCIENCE A N D T E C H N O L O G Y

0.50

1.00

1.50

2.00

2.50

3.00

8.00

9.00

dimensionless distance, r* 2.00

1.50

bo 1.00

0.50 3.00

4.00

5.00

6.00

7.00

dimensionless distance, r* Figure 6: Solute-solute pair correlation function. Same conditions as Figure 2. a) Short range structure; b) long range structure (note change of axes). The curve at Ρ = 0.4 shows a double peak with shallow minimum indicating solute-solute aggregation.

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.

3.

C O C H R A N AND L E E

Solvation Structure in Supercritical Fluid Mixtures

the solvation structures in supercritical solutions. Molecular distribution functions for states representative of typical supercritical solutions exhibit a longer range buildup of local sol­ vent density around solute molecules near the CP. This local structure is consistent with the clustering of solvent molecules about solute molecules which has been hypothesized to occur near a CP. The size of the predicted solvent-solute cluster has been found to be in reasonable agreement with partial molar volume data. The solute-solute distribution functions determined in this work suggest a structure that is consistent with the aggregation of solute molecules suggested by recent excimer fluorescence spectra. Again, the integral equation results cannot reveal whether the solute-solute aggregates persist for significant time.

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Acknowledgment The authors wish to acknowledge their indebtedness to D . M . Pfund for computer program­ ming. Profs. Eckert and Debenedetti kindly shared their results in advance of publication. This work has been supported in part by the Chemical Sciences Division, Office of Basic En­ ergy Science, U . S. Department of Energy, under Contract No. DE-AC05-840R21400 with Martin Marietta Energy Systems, Inc. and subcontract with the University of Oklahoma.

Legend of Symbols A Β G g() k L Ρ r 8 Τ V ν χ

= = = = = = = = = = = = =

component A, solvent, CO2 [-] component B , solute, CioHs [-] Kirkwood fluctuation integral [ À ] pair correlation function [-] Boltzmann constant [ P a A / ° K ] dimensionlees radius of solvation sphere [-] pressure [Pa] separation distance [Â] solid phase [-] temperature [°K] total volume [ À ] volume per molecule [ Â ] mole fraction [-]

or β δ c μ ζ ρ σ Φ XT Ω

= = = = = = = = = = =

arbitrary species index arbitrary species index Kronecker delta [-] LJ energy parameter [ P a  ] chemical potential [PaA ] cluster size [-] number density [A~ ] LJ size parameter [Â] function in Kirkwood-Buff formulas [-] isothermal compressibility [Pa~ ] function in Kirkwood-Buff formulas [-]

3

3

3

3

3

3

s

l

value of quantity at infinite solute dilution partial molar quantity reduced quantity

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SUPERCRITICAL FLUID SCIENCE AND TECHNOLOGY

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Literature Cited 1. Eckert, C. A. ; Ziger, D. H. ; Johnston, K. P. ; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 167 and J. Phys. Chem. 1986,90,2738. 2. Kim, S. ; Johnston, K. P. Ind. Eng. Chem. Res. 1987, 26, 1206 and AIChE J. 1987, 33, 1603. 3. Kajimoto, O. ; Futakami, M. ; Kobayashi, T. ; Yamasaki, K. J. Phys. Chem. 1988, 92, 1347. 4. Debenedetti, P. G. Chem. Eng. Sci. 1987,42, 2203 and Debenedetti, P. G. ; Kumar, S. AIChE J. , 1988, 34, 645. 5. Cochran, H. D. ; Pfund, D. M. ; Lee, L. L. Proc. Int. Symp. on Supercritical Fluids 1988, M. Perrut, ed. , Tom. 1, 245 and Sep. Sci. and Tech. , 1988, 23, 2031. 6. Kirkwood, J. G. ; F. P. Buff, J. Chem. Phys. 1951,19,774. 7. Brennecke, J. F. ; Eckert, C. A. Proc. Int. Symp. on Supercritical Fluids 1988, M. Perrut, ed. , Tom. 1, 263. 8. Cochran, H. D. ; Lee, L. L. ; Pfund, D. Μ. , Fluid Phase Equilib. 1987,34,219 and ibid. 1988, 39, 161. 9. Cochran, H. D. ; Lee, L. L. , AIChE J. 1987, 33, 1391 and ibid. 1988, 34, 170. 10. Zerah, G. ; Hansen, J. -P. , J. Chem. Phys. 1986,84,2336. 11. Lee, L. L. Molecular Thermodynamics of Nonideal Fluids; Butterworths, Stoneham, MA, 1988, Chapter V. 12. Labik, S. ;Malijevsky,A. ; Vonka, P. , Mol. Phys. 1985,56,709. 13. Lado, F. Phys. Rev. A 1973,87,2548. RECEIVED May 1,

1989

Johnston and Penninger; Supercritical Fluid Science and Technology ACS Symposium Series; American Chemical Society: Washington, DC, 1989.