Ind. Eng. Chem. Res. 1992, 31, 1391-1397
r = radius S = area T = temperature, inside tank diameter U = overall heat-transport coefficient u = velocity v = PI? W = width of impeller Z = depth of fluid in tank Greek Letters = density = viscosity
p p
Subscripts f = fluid i = inside o = outside w = wall z = axial direction
1391
Chapman, F. S.; Holland, F. A. Heat Transfer Correlations for Agitated Liquids in Process Vessels. Chem. Eng. 1965a, 72 (2), 155-162. Chapman, F. S.; Holland, F. A. Heat-Transfer Correlations in Jacketed Vessels. Chem. Eng. 1965b,72 (4), 175-184. Chemineer Co. Liquid Agitation; McGraw-Hill: New York, 1985. FOB,I.; Placek, J.; Strek, F.; Jaworski, Z.; Karcz, J. Heat and Momentum Transfer in the Wall Region of a Cylindrical Vessel Mixed by a Turbine Impeller. Collect. Czech Chem. Commun. 1979,44,684-698. Kupcik, F. Heat transfer at the bottom and at the walls of agitated vessels, Parts I & 11. Znt. Chem. Eng. 1975,15(4), 658-664; 1976, I6 (l),91-96. Oldshue, J. Y. Fluid Mixing in 1989. Chem. Eng. B o g . 1989,s(5), 33-42. Poggeemann, R.;Steiff, A.; Weinspach, P.-M. Heat transfer in agitated vessels with single-phase liquids. Ger. Chem. Eng. 1980,3, 162-1 - - - - 74. . -.
Literature Cited Ackley, R. J. Film Coefficients of Heat Transfer for Agitated Process Vessels. Chem. Eng. 1960,69(17), 133-140. Brodkey, R. S.; Hershey, H. C. Transport Phenomena, A Unified Approach; McGraw-Hill: New York, 1988. Brooks, G.; Su, G.-J. Heat Transfer in Agitated Kettles. Chern.Eng. R o g . 1959,55 (lo), 54-57.
Strek, F.; Masiuk, S. Heat transfer in liquid mixers. Znt. Chem. Eng. 1967,7 (4). 693-702. Uhl,V. W. Heat Transfer to Viscous Materials in Jacketed Agitated Kettles. Chem. Eng. Prog. Symp. Ser. 1952, 51 (17), 93-107. Uhl, V. W. Mechanically Aided Heat Transfer. In Mixing; Theory and Practice; Uhl, V. W., Grey, J. B.,Eds.; Academic Press: New York, 1966; Vol. 1, Chapter 5.
Received for review September 17, 1991 Revised manuscript received January 17, 1992 Accepted January 30, 1992
Molecular Dynamics Study of Solute-Solute Microstructure in Attractive and Repulsive Supercritical Mixtures Ariel A. Chialvot and P a b l o G. Debenedetti* Department of Chemical Engineering, Princeton University, Princeton, New Jersey 08544-5263
Molecular dynamics simulations show enhancements in the density of solute molecules around each other in model attractive and repulsive dilute supercritical Lennard-Jones mixtures. The effect is pronounced in the former case and moderate in the latter. For the attractive mixture, the quantification of the density enhancement in terms of excess number of solute molecules suggests an increased collision frequency rather than the formation of a compact solute-solute aggregate. The simulations confiim the resulta of recent integral equation calculations for the same model mixtures, and suggest that solute-solute interactions in dilute supercritical mixtures could play an important role in situations involving a reactive solute. Introduction Fluid behavior near critical points is governed by longranged correlations in density fluctuations (pure substances) or in concentration fluctuations (mixtures). These order parameter fluctuations are correlated over a characteristic distance, the correlation length, that diverges at the critical point. The practical applications of supercritical mixtures, on the other hand, occur over the approximate range 1IT,I1.1; 1 IP,I2 (where subscript r denotes the value of a property divided by its value at the solvent’s critical point). In this region, the short-ranged structure, that is, deviations from a random distribution of molecules occurring over distances of the order of 2-3 molecular diameters around a given species, appears also *To whom correspondence should be addressed. Current ~ ~University i of~ address: Department of Chemical ~ California, Berkeley, CA 94720-9989. t Current address: Department of Chemical Engineering, University of Virginia, Charlottesville, VA 22903-2442.
to be verv imwrtant. Long-ranged fluctuations cause the solute’s mecdanical partiaimolir properties in a mixture to increase in magnitude, but ita chemical potential is determined solely by ita local solvation environment; hence the importance of short-ranged structure, even in the presence of long-ranged fluctuations. We shall henceforth refer to such short-ranged features as microstructure, Much of what makes supercritical solutions of scientific interest originates from the coexistence of long-ranged correlations (whose length scale, though finite, is still large compared to molecular dimensions) and a microstructure that generally involves large deviations with respect to bulk conditions. For example, fluorescence spectroscopy (Brennecke and Eckert, 1988, 1989a; Brennecke et al., 1990a,b), solvatochromic probe experiments (Kim and Johnston, 1987a,b), and computer simulations (Petsche and Debenedetti, 1989; Knutson et al., 1992) indicate a density around ~substantial ~ ~increase i in ~the local ~ solvent , -~ Fursolute molecules with respect to bulk conditions. thermore, solvatochromic probe experiments (Kim and Johnston, 1987a,b) suggest that the remarkable enhance-
0888-5885/92/2631-1391$03.00/00 1992 American Chemical Society
1392 Ind. Eng. Chem. Res., Vol. 31, No. 5,1992
ment in solvent power that can result when small amounts of certain cosolventa are added to a supercritical fluid (Wong and Johnston, 1986) occurs as a result of a pronounced increase in the concentration of cosolvent around solute molecules. Understanding the microstructure, that is, being able to predict by how much and over what distance the neighborhood of a given solute is enriched in solvent (or cosolvent or solute) with respect to bulk conditions, offers the interesting possibility of designing a specific solvent for a particular reaction or separation application (Brennecke and Eckert, 1989b). There are therefore fundamental and practical reasons for studying the microstructure of supercritical mixtures. Because supercritical mixtures of practical interest are dilute (solute mole fractions rarely exceed 1%)most theoretical, computational, and experimental work on molecular interactions in such systems has focused on solute-solvent interactions (Kim and Johnston, 1987a,b; Brennecke and Eckert, 1988, 1989a; Brennecke et al., 1990a,b; Yonker and Smith, 1988; Petsche and Debenedetti, 1989,1991; Cochran et al., 1988; Cochran and Lee, 1989; Wu et al., 1990; McGuigan and Monson, 1990; Knutson et al., 1992). Recent experimental and theoretical investigations, however, suggest that solute-solute interactions can be important in supercritical systems, even at very low solute mole fractions. The evidence includes one calculation of solute-solute molecular distribution functions in model supercritical mixtures that shows remarkably high first peaks (Wu et al., 1990), and the observation of cholesterol aggregation at very low mole fractions in supercritical carbon dioxide (Randolph et al., 1988a,b). Rate and selectivity enhancements in a bimolecular photochemical reaction in a supercritical solvent (Combes et al., 1992) have been interpreted in terms of enhanced solute-solute interactions at high dilution. Solubility enhancements in a mixed solute supercritical system (Kurnik and Reid, 1982) have also been interpreted in terms of solute-solute interactions at high dilution (Brennecke et al., 1990b). Excimer fluorescence experiments (Brennecke et al., 1990a,b) on highly dilute pyrene in supercritical ethylene appear to indicate enhanced solute-solute interactions at high dilution. In the latter case, however, the mechanistic interpretation of the spectroscopic measurements is still a matter of debate (Bright et al., 1991). In this work, we investigate the solute-solute microstructure (in addition to solute-solvent and solvent-solvent microstructure) in two model dilute supercritical mixtures using molecular dynamics computer simulations. One mixture is attractive; the other, repulsive (Debenedetti and Mohamed, 1989; Petsche and Debenedetti, 1991). In the former type (which corresponds to most supercritical mixtures of practical interest), there is local and longranged solvent enrichment around the solute. In the latter, there is solvent depletion around solute molecules. This comparative study of solute-solute microstructure in attractive and repulsive systems was undertaken with several objectives in mind. In the first place, all of the above-mentioned experiments that are consistent with the notion of enhanced solute-solute interactions involve attractive mixtures. It is thus natural to inquire whether the observed (and inferred) solute-solute enrichment is a general feature, or whether it is specific to attractive mixtures. It has been shown recently (Debenedetti and Chialvo, 1992) that, in the limit of infinite dilution and close to criticality, the solute-solute distribution function (gll) always decays to ita bulk value from above (gll > l ) , irrespective of the type of mixture (attractive, weakly at-
tractive, or repulsive). That theoretical derivation, however, did not address the corresponding microstructure (local effects). Finally, there is only one theoretical calculation to date on solute-solute microstructure in highly dilute supercritical mixtures, namely, the recent fundamental study of Wu et al. (1990). These authors performed integral equation calculations of radial distribution functions in attractive and repulsive supercritical mixtures at infinite dilution. Their calculations show both short and long-ranged enrichment in the solute-solute distribution function for attractive and repulsive mixtures, the effect being more pronounced in the former case. The effects under investigation, however, involve precisely the type of situation (pronounced size asymmetry between species) for which integral equation calculations can give incorrect results [see, for example, the recently reported discrepancy between simulation results and integral equation predictions for the size asymmetry dependence of a solute’s residual chemical potential at infinite dilution in a Lennard-Jones system (Pfund et d,1991)l. It is therefore very important to verify integral equation predictions, especially since no other theoretical or computational study of solute-solute interactions in dilute supercritical mixtures exists to date. Accordingly, in this study we use an exact technique to investigate the nature of solute-solute microstructure in dilute supercritical mixtures, with emphasis placed on comparing attractive and repulsive behavior. Ultimately, we seek to understand how important are solute-solute interactions in dilute supercritical mixtures. There are considerable computational difficulties associated with studying solute-solute interactions at high dilution via molecular dynamics. Foremost among these is the need to use very large systems, 80 as to accommodate a sufficient number of solute molecules (good solute-solute statistics) while maintaining a low solute mole fraction. The efficient simulation of large systems calls for the implementation of special algorithms (Chialvo and Debenedetti, 1991; see Molecular Dynamics Simulations below). We purposefully chose molecular dynamics because its results are exact (an essential requirement when testing both theoretical predictions and integral equations on the same microscopic model) and because it provides timedependent information (to be discussed in a separate paper) on such quantities as lifetimes of solute-solute aggregates.
Molecular Dynamics Simulations Two model Lennard-Jones mixtures were studied. At infinite dilution, one is attractive and the other repulsive (Petsche and Debenedetti, 1991). In an attractive mixture, there is long-ranged solvent enrichment around solute molecules in the vicinity of the solvent’s critical point. The solute-solvent pair correlation function decays to unity from above, and the statistical excess number of solvent molecules surroundinga given elute molecule with respect to a uniform distribution at bulk density is large and positive. In a repulsive mixture, there is long-ranged solvent depletion around solute molecules in the vicinity of the solvent’s critical point. The solute-solvent pair correlation function decays to unity from below, and the statistical excess number of solvent molecules surrounding a given solute molecule with respect to a uniform distribution at bulk density is large and negative. Criteria for the classification of a given pair of substances as attractive or repulsive in terms of differences in size and characteristic energies for spherical and chain molecules have been published recently (Petache and Debenedetti, 1991). In general, attractive behavior (which characterizes most supercritical mixtures of practical interest) occurs when
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1393 Table I. Lennard-Jones Parameters for Studied ~ _ _ _ interaction Attractive Mixture pyrene-pyrene (1-1) carbon dioxide-carbon dioxide (2-2) pyrene-carbon dioxide (1-2)
the Two Mixtures
elk, K
u, A
12.0 -
662.8 225.3 386.4
7.14 3.794 5.467
10.0
32.8 231.0 87.0
2.82 4.047 3.434
-
8.0 -
Repulsive Mixture neon-neon (1-1) xenon-xenon (2-2) neon-xenon (1-2)
-
the solute is larger than the solvent and the characteristic energy for solute-solute (and hence solute-solvent) interactions is larger than the corresponding solvent-solvent value. In recent simulation (Knutson et al., 1992; Petsche and Debenedetti, 1989) and integral equation calculations (Wu et al., 19901, it has been found that, in addition to these long-ranged features, the pair distribution functions exhibit interesting microstructure around the solute probe, with pronounced local solvent density enhancements in attractive mixtures and local solvent density depletion (cavity formation) in the repulsive case. Fluorescence spectroscopy measurements (Brennecke and Eckert, 1988, 1989a; Brennecke et al., 1990a,b) also show evidence of pronounced local solvent density enhancements in attractive mixtures at supercritical conditions. These local effects occur over a broad density range including (but not limited to) the highly compressible region near the solvent's critical point and, at fixed bulk density, become less pronounced the higher the reduced temperature. Contrary to long-ranged features, which occur over a single length scale (the correlation length) the details of the microstructure (e.g., the number of solvation shells over which it occurs, the magnitude of the local density enhancement or depletion) are system-specific,although the qualitative trends appear to be quite general. The Lennard-Jones intermolecular potential parameters for the attractive and repulsive mixtures studied in this work are listed in Table I (1 = solute; 2 = solvent). These potentials have been used to represent pyrene-carbon dioxide and neon-xenon mixtures at supercritical conditions (Petache and Debenedetti, 1989; Wu et al., 1990; Knutaon et al., 1992). The mixtures consisted of 10 solute molecules and 3990 solvent molecules (solute mole fraction 2.5 X In general, it is not possible to simulate efficiently such large system sizes (N) with standard molecular dynamics codes. We used the automated Verlet neighbor list algorithm described in detail elsewhere (Chialvo and Debenedetti, 1991). With N = 4000, it consumes less than 0.2 CPU s per time step on a Cray Y-MP machine (Chialvo and Debenedetti, 1991). Simulations were performed in the canonical ensemble, keeping constant the volume, number of molecules, and temperature. Thermostating was done using a momentum scaling procedure (Haile and Gupta, 1983). Interactions were truncated at 3a2. The equations of motion were integrated using Gear's fifth-order predidor-corrector algorithm (Gear, 1971), with a time step of 4 X [in units of ~ ~ ( r n ~ / e ~ ) ' /where ~, m2is the sols for the vent's mass]. This corresponds to 7.355 X attractive mixture and 1.338 X s for the repulsive mixture. Every run was preceded by an equilibration period to allow for melting of the initial face-centered-cubic structure (with random distribution of solute molecules); the equilibration stage was allowed to proceed until the solvent's mean squared displacement reached approximately 20a2 (roughly 2000 time steps). The number of
6.04.0
-
2.0
1
0.0
b
DISTANCE
Figure 1. Solute-solute pair correlation function (gI1)for the attractive mixture (pyrene in carbon dioxide). p* = 0.25; Tz = 1.40. Solute mole fraction = 2.5 X Distance is in units of u2.
steps per simulation was lo5 (attractive mixture; 736 ps of simulated time) and 1.8 X lo5 (repulsive mixture; 2408 ps of simulated time). Beyond this time, the first peak of the solute-solute correlation function did not change. In order to verify that the stimulated systems were in the single-phase region, we performed analytical and numerical checks for every state point studied. The former consisted of equation-of-state calculations for the Lennard-Jones fluid (Nicolas et al., 1979), using van der Waals type 1 mixing rules. The latter consisted of isobaric-isothermal molecular dynamics simulations with the imposed pressure equal to the canonical equilibrium pressure. The equation of state calculations showed that the mixtures were always in the one-phase region. The isobaric simulations did not show a drift in density toward a different (binodal) value (see, e.g., Abraham, 1986).
Results and Discussion Figure 1 shows the solute-solute pair correlation function for the attractive mixture at p* = 0.25 and T* = 1.4 (throughout this paper, p* = N C T ~ ~ VT*- ' = ; k!!'e2-l). The best estimate of the critical point of the pure LennardJones fluid (Smit et al., 1989) is p*c = 0.31 and PC= 1.31. Thus, in units of the solvent's critical parameters, the density and temperature in Figure 1 are 0.806 and 1.07. Statistics for pair correlation functions were collected every time step. A recent study of local density enhancements of the solvent around the solute in the same attractive mixture but a t infinite dilution (Knutson et al., 1992) showed that pronounced deviations between local and bulk conditions occurred over a broad density range encompassing the highly compressible region, and were most pronounced around p* = 0.15 (for T* close to 1.4). The first peak shows an enhancement in the local solute density of more than an order of magnitude with respect to bulk conditions. The maximum, 11.73, occurs at a separation of 2.1 (throughout this paper, all distances are in units of u2). The minimum between the two first peaks is much higher than unity (gll = 5.1 at a distance of 2.6), and the second peak reaches 7.3 at a separation of 3.08. Finally, the distribution function exhibits a rather broad shoulder, with a local extremum, 4.1, at a distance of 4.25. The first peak corresponds to the well of the solutesolute pair potential, which occurs at a separation of 2.11. The location of the second peak suggests a contribution from
1394 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1
I:="-
4
.
12.010.0-
z
8.0
~
g11
-
P I3 mU +
6.0 -
0
4.0
i!?
0
-
d
9 0 a
2.0 -
U
0.0
J
0.0
I 1 .o 2.0 3.0 4.0 5.0 DISTANCE
0.5
1 .o
1.5
2.0
2.5
DISTANCE
Figure 2. Solute-solute (sll),solute-solvent (g12),and solventsolvent (gZ2)pair correlation functions for the attractive mixture (pyrene in carbon dioxide). p* = 0.25; P = 1.40. Solute mole fraction = 2.5 X Distance is in units of u p
v)
0.0
Figure 4. Solutm3olute pair correlation function (glJ for the repulsive mixture (neon in xenon). p* = 0.35; P = 1.40. Solute mole fraction = 2.5 X Distance is in units of up
Figure 3 shows the average, ideal, and excess number of solute molecules around a central solute molecule, as a function of radial distance, R (same units as Figures 1 and 2). They are given, respectively, by the quantities
0.8
W
-I
3
o
Y0
0.6
Nli = (4?r/3)R3pxl
a
W I-
3 B
R
(N1)- Nli = 4rpxlX (gll - l)r2 dr
0.4
U
0
0.2
K W
m
4z
0.0
-0.2 0.0
1 .o
2.0
3.0
4.0
5.0
DISTANCE
Figure 3. Average (@TI)), ideal (Ni), and excess ((Nl)- Nli)number of solute molecules around a central solute molecule, as a function of separation. Attractive mixture (pyrene in carbon dioxide). p* = 0.25; P = 1.40. Solute mole fraction = 2.5 X Distance is in units of up
solute-solvent-solute triplets (the solute-solvent potential well occurs a t a distance of 1.62). The location of the shoulder suggests a contribution from solute-solutesolute lineartriplets. These are the same qualitative features first reported by Wu et al. (1990) in their integral equation calculations of this model system at infinite dilution. Note, however, that because the present simulations were conducted at a lower density [p* = 0.25, close to where we observed the maximum deviation between local and bulk conditions for the unlike pair distribution function (Knutson et al., 1992)], the observed solute-solute local density enhancement is much more pronounced than in the integral equation calculations [first peak 3.6 a t p* = 0.4248, T* = 1.37 (Wu et al., 1990)l. Figure 2 shows all three radial distribution functions for the attractive mixture a t the same conditions as in Figure 1. Note the attractive behavior of g12,which is always greater than unity beyond the first peak.
(1)
where x1 is the solute's mole fraction. The average number of solute molecules found within a sphere of radius 16 A around a solute molecule is 1. If molecules were randomly distributed, this number would be 0.2. Since this radius equals twice the distance to the potential well, this represents a 5-fold density enhancement averaged over two solvation shells. However, at the conditions investigated here, the picture is not one of a compact solute-solute aggregate. Rather, it seems plausible to think in terms of loosely bound pairs, interacting much more strongly than if solute molecules were randomly distributed. Work is currently in progress on the structure and lifetime of solutesolute aggregates (pairs, triplets, etc.) in this system. The above picture is not inconsistent with recent observations of enhancements in the rate of photodimerization of dilute cyclohexenone, and in the corresponding regioselectivity toward the more polar head-tohead dimer (versus the lees polar head-to-tail dimer), upon decreasing the preasure of the supercritical solvent (ethane) toward the critical region. The rate increase has been attributed to enhanced solute-solute interactions (increased collision frequency), and the increased selectivity, to the resulting enhancement in local polarity (Combes et al., 1992). Recent excimer fluorescence experiments (Brennecke et al., 1990a,b)on highly dilute pyrene in supercritical ethylene are also not inconsistent with the above interpretation of Figure 3, insofar as the measurements appear to indicate enhanced solute-solute interactions even at high dilution. In the latter case,however, the connection with the present simultations is less straightforward, since the mechanistic interpretation of the spectroscopic measurements is still a matter of debate (Bright et al., 1991). Figure 4 shows the solute-solute pair correlation function for the dilute repulsive mixture (neon in supercritical
Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992 1396 2.4
UIU
2.0 -0.2
-
1.6
1.2
\
A
-0.4
0.8 -0.6
v
0.4
0.0 0.0
1 .o
2.0
-c
-0.8 0.0
3.0
I 1 .o
DISTANCE
Figure 5. Solutesolute &), solute-solvent k12), and solventsolvent &**) pair correlation functions for the repulsive mixture (neon in xenon). p* = 0.35; T* = 1.40. Solute mole fraction = 2.5 X Distance is in units of u p
xenon) in the relative vicinity of the solvent’s critical point (p* = 0.35; P = 1.4;reduced density and temperature 1.13 and 1.07, respectively). I t is consistent with the recent theoretical prediction (Debenedetti and Chialvo, 1992) that this function always decays to unity from above (gll > 1) in the limit of high dilution and near the solvent’s critical point, regardless of the type of mixture. However, the magnitude of the first peak and of the resulting local density enhancement is much smaller than in the attractive case. For the repulsive mixture shown in Figure 4, the first peak (separation 0.75; potential well at 0.78) is only 1.92. The valley between the first and second peaks, as in the attractive case,is significantly higher than unity (gll = 1.57 at a distance of 0.88). The fact that the solute-solvent potential well occurs at a separation of 0.95 suggests that this feature is primarily due to a local depletion of solute molecules caused by the first solvent shell around the solute. Figure 5 shows all three radial distribution functions for the repulsive mixture at the same conditions as in Figure 4. The solute-solvent curve displays typical repulsive behavior. This can best be seen from the excess number of solvent molecules surrounding each solute molecule with respect to a uniform distribution at bulk conditions W2l)
- N21i = 4m-41 - ~
~ ) J ~ -r 1)~ dr ( b ~(2)~
This quantity is negative for the entire range of separations shown in Figure 5 (0 < R < 3). Its behavior as a function of separation, R, is shown in Figure 6. Figure 7 shows the average, ideal, and excess number of solute molecules around a central solute molecule, as a function of radial distance (see eq l ) , for the repulsive mixture (same conditions as in Figures 4-6). The average number of solute molecules found within a sphere of radius 6.3 A (two solvation shells) around a solute molecule is 0.0175. If molecules were randomly distributed, this number would be 0.0136. Thus,the density (averaged over two solvation shells) is now only 29% higher than in the bulk. Evidently, the effect is much milder than in the attractive case. This follows from the fact that, in typical repulsive mixtures, the characteristic energy (well depth) for soluteaolvent interactions is smaller in magnitude than the corresponding solvent-solvent value. Hence, both
2.0
I
3.0
DISTANCE
Figure 6. Average excess number of solvent moleculea ((Nlz)- Nl$) surrounding each solute molecule with respect to a uniform distribution at bulk conditions for the repulsive mixture (neon in xenon). p* = 0.35; T* = 1.40. Solute mole fraction = 2.5 X lo-*. Distance in units of up 0.100
3A 3
E
0.075
..I
0
a W
I-
3
0.050
B LL
0 U
0.025
2z 0.000 0.0
1 .o
2.0
3.0
DISTANCE
Figure 7. Average ((Nl)),ideal ( N j ) ,and excess ( ( N , )- Nj) number of solute molecules around a central solute molecule, as a function of separation. Repulsive mixture (neon in xenon). p* = 0.35; T* = 1.40. Solute mole fraction = 2.5 X Distance is in units of 62.
soluteaolvent and (even more so) solute-solute correlations tend to be much weaker than in attractive mixtures, and the pair correlation integrals are correspondingly smaller in magnitude. Figure 8 shows the three pair correlation functions for the repulsive mixture (p* = 0.6; T* = 1.4;reduced density and temperature 1.94 and 1.07, respectively). It can be seen from their oscillatory character that the curves are now liquidlike. Interestingly, upon compressing the mixture isothermally from p* = 0.35 to 0.6, the nearestneighbor peak decreases (from 1.92 to 1.65) for the solutesolute distribution, but it increases (from 1.42 to 1.76) for the solute-solvent distribution. Both features are consistent with the fact that, as the mixture is compressed, the microstructures associated with the highly compressible region (moderate solute enrichment and solvent de-
1396 Ind. Eng. Chem. Res., Vol. 31, No. 5, 1992
v)
z 0
2.4
tackled if full use is made of the complementary insights provided by experiments, theory, and simulations.
2.0
Acknowledgment
c
0
f
1.6
U
z 1.2 2
ma I-
E! n
0.8
-I
9
2
0.4
a
0.0
0.0
1 .o
2.0
3
DISTANCE
Figure 8. Solutesolute (gI1), solute-solvent &), and solventsolvent (gZ2)pair correlation functions for the repulsive mixture (neon in xenon). p* = 0.60; !P = 1.40. Solute mole fraction = 2.5 X lo+. Distance is in units of u p
pletion around solute molecules) disappear. Conclusion The average concentration of solute molecules is always higher in the vicinity of other solute molecules than it is in the bulk, when the mixture is dilute and the solvent is highly compressible. This phenomenon occurs in both attractive and repulsive mixtures, but tends to be more pronounced in the former case, where the force constants for solute-solute interactions tend to be larger. Our calculations appear to suggest an enhancement in the rate of solute-solute encounters rather than the formation of stable aggregates. This question is currently under study. It must be borne in mind, however, that microstructural details, as well as the kinetia and statistics of solutesolute interactions, are system-specific. The simulations confirm recent theoretical predictions and integral equation calculations as to solute-solute enrichment in both attractive and repulsive supercritical mixtures. Integral equations should therefore be the natural choice for the study of both short- and long-ranged equilibrium features in dilute supercritical mixtures. Currently unanswered questions on the temporal stability of soluteaolute aggregates, however, must be investigated via molecular dynamics. The picture of a supercritical mixture consisting of noninteracting solute molecules appears to be incorrect. However, this does not necessarily mean that contributions to bulk properties due to solute-solute interactions are large (since such contributions scale with the square of the solute concentration). Rather, enhancements in the rate of solutesolute encounters should be important primarily in situations involving chemical reactions, an example of which appears to be the recently reported rate and selectivity enhancement (the latter toward the more polar dimer) in the photodimerization of dilute cyclohexenone in supercritical ethane upon approaching the critical region (Combes et al., 1992). Solute-solute interactions in dilute supercritical mixtures are only now being investigated. Our present knowledge of the topic is very limited. Attention needs to be given to the formation kinetics and stability of solute-solute aggregates, and to solute-solute interactions in the presence of cosolventa. Such questions can be best
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