J. Phys. Chem. B 1998, 102, 7455-7461
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Solvatochromism in a Near-Critical Solution: A Direct Correlation with Local Solution Structure John E. Adams Department of Chemistry, UniVersity of MissourisColumbia, Columbia, Missouri 65211 ReceiVed: March 24, 1998; In Final Form: July 2, 1998
Although many studies of supercritical fluid solutions have been interpreted in terms of solute-solvent clustering, no work directly relating microscopic solution structure to solvatochromism measurements has been described previously. Here we report a simulation of a model supercritical fluid solution that directly yields the information required for the calculation of solvatochromic shifts, the density dependence of which mirrors the general behavior (that the most rapid change in the spectral shift with increasing solution density is found in the subcritical regime) reported in numerous experiments. Our calculations also confirm that it is the extent of population of the first solvent shell that most strongly influences the solute’s spectroscopy (i.e., “direct” solvation effects dominate the spectral shift) and that even a rather small simulation yields an adequate description of the local solvation environment. However, the wide distribution of local environments does require that special attention be paid to adequate configuration averaging. We discuss the implications of this work with regard to the popular physical cluster picture of solvation in a near-critical fluid.
I. Introduction The commercial importance of supercritical fluids (SCFs) as extraction media1-6 has spurred numerous attempts to elucidate the unique solvating properties of these systems. Although many studies have yielded evidence for a microscopic model of the solution involving solvent-solute “clustering” (more properly, local density enhancement) in the cybotactic region,7-14 that evidence, while indeed suggestive, is only indirect. In particular, no theory has been presented that permits a direct linkage between observed solvatochromism and microscopic solution structure. We recount here a theoretical study that affords just such a linkage, reproducing in a model system (“infinitely dilute” benzene in Ar) the generally observed density dependence of solvatochromic shifts, identifying the solution structural features responsible for those shifts, and clarifying the nature of solute-solvent clustering in a SCF near its critical point. While the report of large solute partial molal volumes in dilute systems by Eckert and co-workers7 stimulated considerable interest in the extended structure of near-critical fluids, many investigators have focused on the local solution structural information that may be extracted from a spectroscopic probe of the solute. Notable in this regard are rotational correlation time studies, such as those by Bright and co-workers8 and by Anderton and Kauffman,9 involving a characterization of the local solvent friction as a function of solution density. More frequently, however, the quantity measured directly is the change in the position of the spectral absorption maximum, the solvatochromic shift.10-12 Recently there have been indications that shifts measured in disparate systems exhibit remarkably similar solution density dependences, a result indicative of common underlying solution structural features. In particular, both Johnston and co-workers,10 who investigated (dimethylamino)benzonitrile and ethyl (dimethylamino)benzoate in mixed solvents (CHF3 and CO2), and Fayer and co-workers,11 who probed the asymmetric carbonyl stretch of W(CO)6 in SCF CO2,
have found the bathochromic shift to be essentially independent of solvent density in the near-critical regime but monotonically increasing outside this region of the phase diagram. We thus are drawn to a consideration of how these spectral measurements directly reflect the microscopic structure of the cybotactic region and, since previous simulations have predicted solute-solvent clustering in near-critical systems, whether clustering in some fashion is directly manifested in the spectroscopic observations. In stating that the present work is novel insofar as it makes a direct connection between microscopic solution structure and spectroscopic measurements, we do not wish to imply that no one has considered this general problem or that no progress has been made in previous work. Indeed, Knutson et al.15 obtained good agreement between local density enhancements in a pyrene-CO2 solution calculated using molecular dynamics simulations and those deduced from fluorescence measurements on the same system. Our goal, however, is to predict a quantity (the solvatochromic shift) that can itself be measured rather than to base the comparison on a quantity that must be derived from the raw experimental data, even if the analysis of that data appears to be reasonably straightforward. Many of the basic features of solvation in SCF systems have been summarized recently by Tucker and Maddox,16 who have reviewed the important distinction between local density enhancement stemming from (nondivergent) “direct” solvation effects17-19 (those that extend over a distance comparable to the range of the solute-solvent interaction potential) and the distinctive (divergent) bulk “indirect” effects20,21 (those that arise as a consequence of the long-range correlations appearing near a solution’s critical point22). Of particular practical relevance to the present work is their observation that spectral influences are expected to fall into the category of direct effects, for which only an accurate modeling of the local solvation environment is required. Necessarily finite simulations of these systems thus may yield quite useful information about the spectral properties of solutes dissolved in SCFs even though they fail to reproduce the macroscopic correlation lengths appropriate to the bulk SCF
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7456 J. Phys. Chem. B, Vol. 102, No. 38, 1998
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in the near-critical regime. But even when only direct density enhancements need to be considered, careful averaging over many typical system configurations still may be required if one is to account for potentially large fluctuations in the local density. II. Calculation of Spectral Shifts A. Theory. We adopt the same formalism for calculating shifts of the 610 spectral line in the benzene-Ar system that we used previously in studies of both bulk and gas-phase cluster systems.23-25 The physical picture motivating this approach to the problem is quite simple: the nonpolar but polarizable Ar solvent atoms interact with one another to yield collective polarization modes26 of the fluid, the interactions of which with the (effective) transition dipole of the benzene molecule shift the benzene’s absorption frequency toward the red. (In benzene, which has no permanent dipole moment in either its ground or first excited electronic state, this effect is small, being on the order of a few tens of wavenumbers. One thus might anticipate a perturbation theory approach23 to be appropriate here, and indeed that has proved to be the case.23-25) This dielectric effect cannot, however, be identified with the experimentally observed total shift. One must also include (at least empirically) a contribution from short-ranged residual forces.23,27,28 This repulsive contribution to the total shift is toward the blue and thus partially offsets the red-shifted dielectric component. The explicit expressions for these two contributions are respectively
cR2
3N
δ(hV)dielectric ) -d
2
∑
R)1 2m ω v R
2ωu 2
2
ωR - ωu
z
(r0j)-2 ∑ j)1
δ(hV)repulsive ) b
(1)
In these equations d denotes the magnitude of the effective transition dipole moment of benzene, ωu the benzene transition frequency in the absence of solvent, mv an effective oscillator mass determined from the solvent’s polarizability, ωR ) ωR(R) the frequency of the Rth collective solvent polarization mode determined at a particular system geometry R, cR ) cR(R) the strength of the coupling between this mode and the chromophore’s effective transition dipole, b an empirical constant scaling the short-range repulsive forces, and r0j the distance between the centers of mass of benzene and the jth Ar atom. (Only the z nearest-neighbor Ar atoms contribute to the shortrange component. As in previous work,23-25 we somewhat arbitrarily consider a solvent atom to be a “nearest neighbor” if it lies within a sphere having a radius equal to the equilibrium benzene-Ar binding distance in benzene‚(Ar)2 clusters29 plus one Ar van der Waals radius, a total distance of 0.54 nm.) Note that the direct link between microscopic solution structure and the calculated solvatochromic shift appears in the explicit dependence of ωR, cR, and r0j on the system configuration. The quantities appearing in eq 1 above must be averaged over an appropriate ensemble of characteristic system configurations in order to obtain results that can be compared with experiments. B. Simulations. In the present study the configurationaveraged shifts have been calculated using structures generated in an isothermal-isobaric (NPT) Monte Carlo random walk,30,31 with all potential interactions taken to be of the Lennard-Jones (LJ) form. While values of the system parameters and other computational details are available elsewhere,23-25,32,33 a few
specifics are worth mentioning. The majority of the results reported here derive from simulations utilizing 256 Ar atoms and a single benzene molecule (with periodic boundary conditions) and at a system temperature of 168 K, equal to a LennardJones reduced temperature T* of 1.39. Unfortunately, though, the appropriate values of the critical constants for the present system are somewhat uncertain. The conventionally cited values of Tc* ) 1.35 and Fc* ) 0.35 determined by Verlet34 strictly apply to a finite NVT simulation and are slightly higher than those derived by Smit et al.35 on the basis of a calculation utilizing the Gibbs ensemble. As an added complication, there is evidence that critical constants tend to be overestimated in a finite simulation.36 If we adopt the values cited by Verlet,34 then our isotherm lies within 3% of the critical isotherm, while if we accept the Tc* estimate of Smit et al.35 to be appropriate to our calculation, then the simulations correspond instead to a temperature that is 6% above Tc. The choice of N ) 256 stems from practical considerations: calculating the dielectric component of the spectral shift is the computationally “costly” step because it requires the diagonalization of a 3N × 3N matrix (the elements of which are induceddipole-induced-dipole couplings) at each system configuration. As noted in the Introduction, spectral shifts are expected to depend most sensitively on the local solution environment,16-18 which should be adequately modeled in a reasonably small simulation. One obvious consideration, however, is whether a system of 256 solvent atoms is large enough that the correlation length of the pure solvent in the near-critical regime is less than the simulation box dimensions. Using the scaling relationship and the scaling amplitude for Ar given by Sengers and Levelt Sengers,37 we find the solvent correlation length appropriate to our isotherm to be 1.3 nm, a value that should be compared with our average (cubic) simulation box edge length (at F* ) 0.35) of approximately 3.1 nm. Thus, we have some confidence that our simulations are not yielding results that are dominated by spurious edge effects. Nonetheless, since dependence of the results on the simulation size is a point of obvious concern, simulations involving 864 Ar atoms were also performed. Finally, we note that interactions between Ar atoms were described using the LJ parameters given by Rahman33 (Ar-Ar ) 121 K, FAr-Ar ) 0.34 nm), while the benzene-Ar interactions were taken to be those suggested by Ondrechen et al.,32 consisting of sums of pairwise-additive carbon-argon and carbon-hydrogen LJ potentials (C-Ar ) 58.1 K, σC-Ar ) 0.342 nm; H-Ar ) 64.7 K, σH-Ar ) 0.321 nm). In this potential model the total benzene-Ar interaction is not spherically symmetric. Throughout our simulations the single benzene molecule was treated as a rigid body, with bond lengths fixed at the experimental values. III. Results A. Solvatochromic Shifts. In Figure 1 are shown the net spectral shifts (relative to the absorption frequency of an isolated benzene molecule) calculated for a range of (Lennard-Jones reduced) solvent densities and determined on the basis of 500 equilibrated configurations with N ) 256. (Each Monte Carlo cycle consisted of successive attempted moves of the solvent atoms and an attempted volume change. After equilibration at the given temperature and pressure, configurations for analysis were stored every 500 complete cycles. In simulations involving 256 solvent atoms, therefore, 128 000 single-particle moves were attempted between stored geometries.) Of primary importance is the overall trend that the magnitude of the shift is a monotonically increasing function of the density, increasing
Solvatochromism in a Near-Critical Solution
Figure 1. Density dependence of the solvatochromic shift of the 610 spectral line of benzene dissolved in Ar (“infinite dilution”) at 168 K determined from simulations involving 256 Ar atoms. Densities are reported in Lennard-Jones reduced units (ref 31). The error bars represent two standard deviations of the mean shift (as opposed to standard deviations of the distribution of shifts, which are larger by a factor of x500).
Figure 2. Density dependence of the solvatochromic shift as in Figure 1, but for simulations involving 864 Ar atoms.
rapidly in the subcritical region of the phase diagram but less so as the system enters the near-critical regime. (As we noted above, the critical density for the neat fluid in Lennard-Jones reduced units is roughly 0.35.34) This behavior is in general accord with experimental results such as those obtained by the Johnston10 and Fayer11 groups, although the plateau seen in the data from those groups (reflecting a near independence of the shift on density) is not identifiable here. (On the other hand, recent calculations by Larsen and Stratt38 of the vibrational relaxation of I2 in fluid Xe at an analogous temperature and over the same solvent density range have not yielded a clear plateau behavior either.) A similar dependence is found in the shifts derived from the larger simulations (N ) 864) displayed in Figure 2. Only 200 equilibrated configurations were used in the calculation of each of these shifts, and accordingly the error bars shown in this figure are larger. In fact, they are sufficiently large that the more abrupt change in the density dependence upon entering the near-critical regime suggested by these results cannot with any confidence be judged significant. Although in a number of other contexts quite reliable spectral shifts have been extracted from 200 configurations,23-25 the near-critical fluid environment here is such that proper averaging requires a greater number of configurations.
J. Phys. Chem. B, Vol. 102, No. 38, 1998 7457
Figure 3. Spectrum of couplings J(ω) for a simulation involving 256 Ar atoms at Lennard-Jones reduced densities of 0.19 (open circles), 0.35 (closed circles), and 0.46 (triangles). Frequencies on the horizontal scale are given as fractions of the pure solvent frequency.
It is important to note that an NPT simulation yields the average system density directly; the particle number is fixed, and the mean system volume is obtained in a simple average over the steps of the Markov chain generated in the Metropolis Monte Carlo simulation.31 Experimental densities are not as easily obtained, however. One conventionally measures the pressure of the system and, knowing the mass of the fluid, calculates the density from an appropriate equation of state. But, of course, in an SCF’s compressible regime large density changes derive from relatively small changes in the system pressure.22 Any comparison of the results from simulations with those from experiments must therefore recognize that the two approaches have their unique potential difficulties: the simulations can be subject to large uncertainties unless careful averaging is performed over a large number of characteristic configurations, while the corresponding experimental measurements can suffer from large uncertainties in the derived densities unless the pressure measurements are very precise. Before passing to the characterization of the local solvent structure in the vicinity of a solute molecule, one might reasonably ask whether the formalism for calculating dielectric spectral shift contributions23 used in the present work is unnecessarily complicated. In particular, is the polarization of the solvent really a collective phenomenon such that a sum of independent solvent atom-solute molecule dipolar couplings is an insufficient model of the dielectric response of the solvent? In Figure 3 one finds plots of the spectrum of couplings J(ω), defined via the equation 3N
J(ω) ) π
∑ cR2δ(mvω2 - mvωR2)
R)1
and representing the distribution of polarization modes weighted by cR2, i.e., by the ability of each mode to couple to the solute’s effective transition dipole moment. (J(ω) reduces to a delta function at ω/ωv ) 1 in the limit of zero solvent-solvent coupling, where ωv is the frequency associated with the polarization of an isolated solvent atom.) The width of the spectra here is on the order of 10% or so of the Ar frequency, with the spectra being strongly peaked toward the highfrequency side. (This peaking toward higher frequencies simply reflects the preferential coupling of the effective solute transition dipole to the radially directed longitudinal polarization modes rather than to the circulatory transverse modes.) Thus, even though some computational expense is involved in the inclusion of collective solvent polarizations in our model, they indeed
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Figure 4. Average number of Ar nearest neighbors about a benzene molecule derived from the same simulations from which the spectral shifts of Figure 1 were obtained. The line is the equivalent result expected for a uniform fluid.
are required if one desires an accurate representation of the dielectric response at the critical density. (Previous work has shown that their inclusion becomes even more important as the solution density is increased.23) It is also significant that so little difference is observed in the plot of J(ω) when passing from the near-critical regime (F* ) 0.35) to a higher density (F* ) 0.46), a result that is consistent with a relatively slow variation in local density over this range of bulk densities. This observation is corroborated by more direct probes of local density, a subject to which we now turn our attention. B. Microscopic Solution Structure. Our focus here is on the characterization of the relationship between the calculated solvatochromic shift and the microscopic solution structure and, in particular, on the question of the nature of clustering in these systems. When one thinks of a physical cluster of solvent atoms surrounding a solute molecule, one tends to envision a filled or partially filled solvation shell similar to what one finds in the analogous small gas-phase clusters.24,25,39,40 That picture is inconsistent, however, with the numbers of nearest neighbors found in the present calculations. Figure 4 is a plot of the average number of nearest neighbors z as a function of the (LJ) reduced density, where z is determined in this case by counting the number of argon atoms within 0.54 nm of the center of the benzene ring. (This distance is identical to the cutoff radius used in determining the repulsive component of the spectral shift in eq 1 above and accounts only for the innermost solvation shell.) Also shown is a line indicating the analogous density dependence of 〈z〉 that would be expected if the solvent were instead a uniform fluid at the bulk density. (This line is determined by multiplying the bulk density by the volume of a spherical shell15 having inner and outer radii of 0.31 and 0.54 nm, respectively. The former value accounts for the “size” of the benzene molecule itself and is admittedly somewhat arbitrary; our choice corresponds to a LJ reduced distance of 0.9, which is roughly the point at which the benzene-Ar radial distribution function discussed below becomes nonzero. Since the benzene-Ar potential is not spherically symmetric in our model, using a spherical shell volume here is not strictly correct, but it nonetheless provides a useful approximation to the volume of the nearest-neighbor solvation shell.) Thus, the distance between the data points and this line reflects any density change effected by attractive solute-solvent interactions. One sees immediately the signature of local density augmentation in the subcritical regime and a diminution of the effect at densities beyond the critical density. This relationship between local and
Adams
Figure 5. Net spectral shift plotted versus the average number of Ar nearest neighbors, the latter being a measure of the local solvent density. (The simulations are the same as those giving rise to the data appearing in Figure 1.)
bulk solvent densities is qualitatively similar to the result found in an analogous investigation of pyrene-CO2, another system in which the local solvation environment is largely determined by attractive solute-solvent interactions, reported by Knutson et al.15 But, of course, one wishes for more than just a qualitative indication that local density augmentation may be occurring in a fluid solution. It therefore is useful to note the relationship between net spectral shift and local density, a relationship that is conventionally assumed to be linear. A plot of these quantities, shown in Figure 5, indeed reveals the expected linearity; a least-squares fit to the calculated points yields a squared correlation coefficient (r2) of 0.98. Even though the data displayed in Figure 4 indicate that the number of nearest neighbors increases with increasing bulk solvent density, that number remains less than half of the value found in simulations of large benzene-argon clusters.25 Thus, in the supercritical regime, the conception of a solute-solvent “cluster” must differ from the straightforward one pertinent to cryogenic gas-phase systems. (This observation is hardly novel; the distinction between the environments was clearly noted by Kajimoto and co-workers12,41 a decade ago.) Extracting this structural difference from the experimental shift values alone, however, need not be straightforward. The spectral shifts calculated near the critical point (equal, roughly, to -45 cm-1) in the fluid system are found to be of the same magnitude as those calculated for cryogenic clusters containing far more (4050) atoms. One might thus be tempted to conclude that the observed shifts are indicative of quite substantial density enhancements and of the formation of well-defined physical clusters. It must be remembered, though, that the net spectral shift results from dielectric (red-shifted) and repulsive (blueshifted) components that largely offset one another, and so there need not be a unique spatial configuration corresponding to a particular observed solvatochromic shift. An important question one must ask concerns the range of the structural information present in the spectral shifts. As suggested above, spectral shifts are expected to reflect direct solvation effects, i.e., those acting over a distance commensurate with the solute-solvent potential interaction length. This expectation is borne out in the results presented in Figure 6, which is a plot of the fraction of the total dielectric coupling resulting from interactions with atoms lying within a given distance from the benzene molecule’s center of mass. (Distances are measured in units of the Ar-Ar LJ σ parameter, which is equal to 0.34 nm.) For both the lowest and highest densities considered, the coupling ratio rises sharply in the range
Solvatochromism in a Near-Critical Solution
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Figure 6. Radial dependence of the polarization mode coupling strength (as defined in the text) at two fluid densities. The values plotted are normalized by the total coupling strength, i.e., by the sum over contributions from all the Ar atoms in the simulation.
r* ) 1.0-1.7 and then begins a slow increase to the asymptotic value of unity. This distance over which one finds a rapidly increasing total coupling ratio is exactly where one expects to find the nearest-neighbor atoms of the first solvation shell. The conspicuous message of this figure, therefore, is that the spectroscopy is dominated by the local structure of the solution and has at most only a very weak dependence on the disposition of those solvent species lying beyond the innermost shell. For the present work this result immediately suggests that that relevant structural information can be derived from relatively small simulations of SCF systems as long as the probe is sensitive primarily to direct solvation effects. While we cannot always hope to model accurately the macroscopic correlations that appear in the bulk SCF, especially when the system is very near the critical point, doing so may not be crucial in simulations of attractive systems in which the local structure is dominated by the solvent-solute interactions. Our finding that the spectroscopy depends most sensitively on the population of the first solvation shell is consistent with the results obtained in modeling studies of local solvation environments in similar near-critical solutions by Munoz and Chimowitz,42 who carried out integral equation calculations on a LJ system and in so doing obtained a quantitative estimate of the range of the “local neighborhood” in which density augmentation affected the solute’s calculated chemical potential. They reported the radius of this neighborhood to be between 3 and 5 times the diameter of a solvent molecule. Subsequently, a study of naphthalene dissolved in SCF CO2 by Tom and Debenedetti43 suggested that solvation depends only on the solvent structure out to a distance of 2.0 nm from the naphthalene molecule; a converged value of the solute’s fugacity coefficient could be obtained over that distance. Their results clearly show, however, a very steep rise in the fugacity coefficient over a quite narrow interval of r* ) 1.4-1.7, thus indicating that nearest-neighbor interactions are dominating the solvation. More information concerning the basic structure of the solution in the cybotactic region is revealed in the radial distribution functions given in Figure 7 for three different densities, including the critical density of the Ar solvent. For each density gAr-Ar(r*) (the distribution of Ar atoms around another Ar atom) and gBz-Ar(r*) (the distribution of Ar atoms around the benzene molecule) are shown. The local density enhancement around the benzene molecule is especially evident at the two lower densities, at which the areas under the innermost gBz-Ar(r*) peaks clearly exceed those of the corresponding gAr-Ar(r*) peaks. Note also that the density enhancement
Figure 7. Radial distribution functions obtained from simulations involving 256 Ar atoms at the indicated Lennard-Jones reduced densities. The closed circles denote the radial distribution of Ar atoms around the benzene molecule, while the open circles denote the radial distribution of Ar atoms around Ar.
extends well beyond the first solvation shell at r* = 1.5 (=0.51 nm), especially at the lowest density shown, for which the tail is more pronounced over the range r* ) 2-3 than it is in the near-critical case. (This result is also obtained in N ) 864 simulations.) That the enhancement effect is most apparent at the lowest density is not unexpectedsthe net interactions between the solute and solvent are attractive here, and the local structure is dominated by these relatively short-range interactions rather than the bulk solution structure. In contrast, indirect solvation effects leading to local density augmentation would be most evident at near-critical densities and would be accompanied by a broad distribution of structural environments.13,16,19,44 Another way of analyzing these same radial distribution functions, one that has been described by Debenedetti,45-47 is in terms of the excess number of solvent atoms surrounding a solute species relative to the number that is expected based on a uniformly distributed bulk solvent at the given density. (This approach is similar to the calculation of pure solvent coordination numbers as carried out by Kim and Johnston.48,49) Specif-
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Figure 8. Radial dependence of the excess solvent atom number for the densities indicated.
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Figure 10. Radial dependence of the excess solvent atom number at the critical solvent density for simulations involving different numbers of solvent atoms.
not affected appreciably. Similarly, if one compares the radial dependence of FG(r) for the same two systems, results that are displayed in Figure 10, one obtains good agreement only for r* e 2.5; beyond this value, the larger simulation reveals a longerranged density enhancement. (It is, of course, hardly surprising that the larger simulation does a better job of modeling the longrange solute-solvent correlations near the critical point.) It certainly may be true that even the N ) 864 simulation is insufficient for the accurate determination of the solvent excess ξusa yet larger simulation would be needed to ensure that FG(r) is indeed approaching an asymptotic value that is not an artifact of the necessarily finite system size. Figure 9. Radial distribution function for a solvent density equal to the critical density as in the central panel of Figure 7, but for a simulation involving 864 solvent atoms.
ically, this solvent excess ξu as a function of distance from the infinitely dilute solute molecule can be written
ξu(r) ) FG(r)
∫0rdr′[gBz-Ar(r′) - 1]4πr′2
)F
where F is the average bulk solvent density and G (in the limit r f ∞) is the Kirkwood-Buff integral.50 A plot of FG (shown as a function of the reduced distance r*) is given in Figure 8 for the same three densities indicated above. One sees immediately that, in accordance with the observed bulk dependence of the radial distribution function, it is the lowest-density solution that is characterized by the largest relative cybotactic density enhancement. But, it is also important to note that, at the critical density (F* ) 0.35), the local density enhancement at the radius corresponding to the first solvation shell consists on the average of only about two additional solvent atoms. This result is similar to the estimate of density enhancements deduced by Kajimoto and co-workers41 and is quite different from the solvation shell that is observed in cryogenic gas-phase clusters. This is an appropriate point at which to comment further on the sacrifices involved in using a relatively small simulation in the calculation of solvatochromic shifts. If one examines the radial distribution functions derived from a larger (N ) 864) simulation at a density equal to the critical density, results which are shown in Figure 9, one finds there to be very little difference between the first peak of gBz-Ar(r*) here and the corresponding result seen in the smaller simulation (Figure 7, center panel). The differences are significantly greater beyond this peak representing the nearest neighbors, but this first solvent shell is
IV. Summary and Conclusions We herein have described the application of a formalism that makes possible a direct correlation between the microscopic structure of a fluid and the solvatochromic shift of a dilute chromophore. Unique to this analysis is the inclusion of the collective dielectric response of the solvent as well as an empirical correction for the short-range repulsive interactions unaccounted for in the dielectric theory. Of central importance is the result that the spectroscopy depends sensitively only on the disposition of atoms within the first solvation shell, the structure of which is determined by the short-range intermolecular interactions. Accordingly, only a small simulation is required for modeling the spectroscopy, even though such a simulation does not yield an accurate model of longer-range density fluctuations and certainly not the macroscopic correlations characteristic of the bulk solvent near its critical point. Our calculations also suggest that mapping the solvatochromic shift in the immediate vicinity of the critical point can be fraught with problems unless one exercises some control over the errors (insufficient statistical averaging on the computational side and on the experimental side, the extremely sensitive pressure dependence of the system density) that arise inherently in this regime. In fairness we should note, however, that whereas our “best” calculations do yield a monotonically increasing spectral shift toward the red as the solution density is increased, they do not reproduce the distinct density-independent plateau behavior observed in experiments of the Johnston10 and Fayer11 groups. The origin of this difference is as yet not clear, but a couple of points warrant mention. First, it is possible that near the critical point our empirical models of the benzene-Ar and Ar-Ar potential interactions32,33 are simply inadequate. Even though we have obtained considerable success in describing the gasphase cluster24,25 and liquid23 environments in previous studies,
Solvatochromism in a Near-Critical Solution there is no guarantee that our pairwise-additive Lennard-Jones potential models will be sufficiently accurate in all situations. (This question concerning the utility of empirical potential functions always haunts those doing computational studies, of course; our case is by no means unusual in that regard.) Second, there remains a question concerning configurational averaging that need not necessarily be resolved merely by doubling or tripling the number of configurations. There is ample evidence that a fluid near its critical point is characterized by multiple local environments.16,19,44 It is therefore conceivable that in any finite simulation one might only sample a fraction of these local environments even though that sampling of particular environments is done very well indeed. In the limit of strong solutesolvent potential interactions, this concern is mitigated by virtue of the fact that local solvation environments will be determined primarily by these short-ranged interactions. In the present case, though, we cannot eliminate the possibility that bulk density fluctuations exert an influence, albeit weak, on the local solvation environment probed by spectroscopy. (Ultimately, this second consideration is similar to the first one in that it involves questions regarding the details of the potential interactions.) What is quite clear in the present work is that the nature of solvent clustering in a simple fluid such as the one considered here differs significantly from what one finds in cryogenic gasphase clusters, where one observes the formation of densely packed, closed or partially closed solvation shells with increasing cluster size. In contrast, physical clustering in a fluid is not as easily characterized. The “clusters” that one finds in an SCF (or even in the subcritical system) are by no means composed of close-packed solvent atoms, but rather consist of only a few additional nearest-neighbor atoms beyond the number expected from the average density of the bulk solvent. Furthermore, larger simulations indicate that regions of enhanced local density can extend a considerable distance into the bulk solvent. The caveat, however, is that information about these longer-range density enhancements may be difficult to extract from spectroscopic measurements in the absence of insight derived from other sources. Large-scale simulations will have an important role to play in understanding this intermediate region beyond the nearest neighbors but still close enough to the solute species that the solvent structure cannot be characterized solely on the basis of our understanding of bulk fluids. Examples of SCFs for which modeling calculations such as those pursued in the present work may provide very useful information are the cosolvent-modified systems.49,51 These fluids afford the opportunity to achieve an unparalleled “tunability” of the chemical potential (and hence the solvating properties) of the solvent. Since usually the addition of only a small amount of cosolvent is required (5-10% of the fluid mixture), one suspects that preferential clustering of the cosolvent species with the solute is occurring. A direct solvation effect such as this should be reflected in the spectroscopy (or any other approach that depends primarily on the local solvation environment) of the solution, and so simulations of these twocomponent solvent systems should yield a means of confirming this structural supposition. Our future work will be directed toward an elucidation of just such systems. Acknowledgment. The author is pleased to acknowledge several very helpful discussions with R. M. Stratt and J. F. Kauffman concerning this work and is grateful for a number of helpful suggestions contributed by the referees. Support for this work was provided by the University of Missouri Research Board.
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