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J. Phys. Chem. B 2000, 104, 6248-6257
Origins of Atom-Centered Local Density Enhancements in Compressible Supercritical Fluids Michael W. Maddox, Grant Goodyear, and Susan C. Tucker* Department of Chemistry, UniVersity of California at DaVis, DaVis, California 95616 ReceiVed: January 20, 2000; In Final Form: April 17, 2000
We used molecular dynamics simulation of a neat two-dimensional Lennard-Jones fluid to examine how the distribution of local densities found around an atom varies as the critical point is approached. Through this analysis, we discovered that mean local density enhancements arise as a necessary and direct consequence of the long-range density inhomogeneities present in such fluids, thereby establishing a relationship between the local and long-length-scale phenomena. Additionally, we uncovered a second, “potential-induced” mechanism which generates mean local density enhancements at low bulk densities. The competition between these two mechanisms of local density enhancement formation enables us to explain why these enhancements are not maximized at the critical density, as well as to explain how the location of this maximum will depend on intrinsic experimental parameters such as the size of the local region and the strength of the solutesolvent potential interaction.
I. Introduction In the supercritical regime, which we define here as any state point having a temperature T which exceeds that of the critical temperature Tc, irrespective of the pressure,1 one can access a wide range of fluid densities without the interference of a gasliquid phase transition.2 Moreover, because a fluid’s solvating properties depend strongly on its density, an ability to tune the solvent density translates into an ability to tune solvation energies,3,4 and, of importance to reactivity, an ability to tune the relative solvation along a solute’s reaction path.5 Since the free energy of solvation along this path, i.e., the potential of mean force, is the principal factor in determining the solute’s reaction rate,6 this control of solvent density in supercritical fluids means that reaction outcomes can be manipulated by thermodynamic control of the supercritical solvent density.4,7-10 Of great assistance in harnessing this potential would be the ability to predict such solvent effects across the entire supercritical regime. The weakest link in such predictions occurs over a broad region of intermediate densities surrounding the critical point, and our goal herein is to gain a more in-depth understanding of supercritical fluids in this regime. The difficulty of modeling this regime arises because the already problematic treatment of intermediate densities is complicated here by phenomena associated with the critical point. Specifically, at the critical point itself, thermodynamic susceptibilities, such as the isothermal compressibility, diverge.2 This divergence of the compressibility is a macroscopic manifestation of the fact that the correlation length ξ, which provides a measure of the length scale over which fluctuations in density are correlated, itself diverges. The so-called “compressible regime”, the region of most interest to us here, corresponds to thermodynamic state points in the vicinity of the critical point where the isothermal compressibility, while no longer infinite, is still large relative to that of an ideal gas under the same conditions. Similarly, the correlation length, while not divergent, will be large, such that the associated extended density fluctuations will give rise to distinct, albeit continually fluctuating, regions of high and low
density on a microscopic scale. Such density inhomogeneities can easily be seen in the instantaneous snapshot of a twodimensional Lennard-Jones fluid at a compressible state point shown in Figure 1. The reader should note that the critical scaling region, where universality holds and the fluid’s thermodynamics obey wellknown critical scaling laws,2,11-13 extends over only a very small region around the critical point and is thus contained within a small portion of the compressible regime. Since powerful methods already exist to treat the critical scaling region, 11-13 we focus instead on the supercritical compressible regime existing outside of this scaling region. A second reason for this focus arises because the extended compressible regime is potentially of technological interest, as it is here that one can significantly vary the density with only very modest changes in the pressure while still avoiding many, but as we shall see, not all, of the complications associated with the critical point. Research over the past decade has demonstrated that along with the large macroscopic compressibilities found in this compressible regime come some fairly striking microscopic consequences.8,14,15 In particular, it is now well-known that solute spectroscopy may be significantly altered in the compressible regime.4,14,16-24 As an example, consider a solvatochromic-shift experiment, which is a typical method for assessing solvation effects. The essential idea behind such an experiment is that the presence of a solvent stabilizes, say, both the S0 and the S1 states of a solute, but the fact that the S1 state is generally more polar than the ground-state means that the solvent lowers the energy of the S1 state more than it does the S0 state. As a result, the S1 f S0 transition energy is red-shifted from the gas phase (isolated-molecule) value. With this physical picture it is straightforward to guess the behavior of the solvatochromic shift along an isotherm in a homogeneous fluid: increasing the bulk density increases the number of solvent atoms that are close enough to the solute to assist in stabilizing it, and thus the magnitude of the red shift ought to increase monotonically with increasing bulk density. In liquids, this monotonic behavior is so well established that it forms the
10.1021/jp000379b CCC: $19.00 © 2000 American Chemical Society Published on Web 06/10/2000
Local Densities in Compressible Supercritical Fluids
Figure 1. Representative snapshot of a neat, two-dimensional nearcritical-density supercritical Lennard-Jones fluid at F ) 0.3 σ-2 and T ) 0.53 /kB (F/Fc ) 0.79, T/Tc ) 1.11), with N ) 1152 particles.
basis for standard experimental indicators of solvent polarity.25 Yet, in supercritical fluids sufficiently near the critical point, one sees not a monotonic increase,8,9,14,16,26-32 but rather a threeregime behavior,26,31 in which, after an initial increase in the magnitude of the shift, the shift levels off at a bulk density below, but in the vicinity of, the critical density and becomes relatively invariant to further increases in the density, until, at a density slightly higher than the critical density, the shift again increases with increasing density. Moreover, this three-regime behavior has been found to be a nearly universal signature of the compressible regime,14 showing up not only in static solvatochromic absorbance,16,33-38 fluorescence emission,18,31,33,39,40 Raman,27,41,42 and even EPR17 line-shift and intensity measurements, but also in dynamic measurements18,20,32,43 such as rotational and vibrational line widths and pump-probe vibrational relaxation rate lifetimes.14 It has generally been assumed14,16,26,31,34 that this three-regime behavior arises as a consequence of local solvent density enhancements around the solute which are present in the compressible regime, but not at very low and high densities. Given an experimental observable which is proportional to this local density (around the solute), one sees that this observable will increase with the local density at low densities where the local density tracks the bulk value. In the compressible regime, a local density exceeding the bulk value is thought to build up and remain relatively invariant to further bulk-density changess explaining the invariance of the experimental observables in this regime. Lastly, as the compressible regime is exited, the bulk density catches up with the local density, and further changes in the bulk density again cause equivalent increases in the local density, causing the final rise in the three-regime behavior. Numerous simulations14,15,22,40,44-47 and integralequation theories14,48,49 have confirmed that as long as the solute attracts the solvent at least as strongly as the solvent does itself (as is the case even in a neat fluid), then such local density enhancements will be observed in the compressible regime. Indeed, recent computer simulations have explicitly shown the correlation between local density enhancements and threeregime behavior in both solvatochromism30 and vibrational energy relaxation rates.32
J. Phys. Chem. B, Vol. 104, No. 26, 2000 6249 Despite the considerable amount of work accomplished in this area, many questions still remain. Moreover, the presence of a distinct, attractive solute in the experiments complicates any interpretations about the fluid itself, because in a compressible supercritical fluid an attractive solute induces a significant perturbation.14 Yet the near-universality of local-density enhancement induced behavior suggests that it is the nature of the poorly studied neat fluid itself that is the root cause of the observed anomalies in compressible fluids, and so it is the neat fluid that we need to query to understand this behavior. Thus, we seek to determine the mechanism by which local density enhancements arise in the neat fluid. Toward this end, we ask what, if any, is the relationship between the long-length-scale density inhomogeneities (see, e.g. Figure 1) and the much shorter range local phenomena of local-density enhancements? Or, instead, are local-density enhancements solely a microscopic phenomena, controlled by the intermolecular potential and unrelated to the long-length-scale inhomogeneities (i.e., the critical fluctuations), as has been suggested in the recent literature? What, precisely, is the state-point dependence of these local density enhancements, and, if they are a consequence of the density inhomogeneities associated with critical phenomena, why are they consistently observed to be maximized at a bulk density below the critical density? But if, instead, these local density enhancements are not a product of the critical phenomena, why are they most pronounced along near-critical isotherms? In order to shed light on this global/local relationship, we will examine how the distribution of local densities found around solvent atoms is related to the distribution of local densities found throughout space.50 Additionally, we consider how the local length scale should be chosen, and how the answers to the above questions will change if that length scale is altered. We also examine the ramifications of our findings for experimental observables. Throughout the remainder of this paper, then, we begin to answer these questions by examining the atom-centered local densitiessboth their distributions and meanssin a neat, twodimensional Lennard-Jones fluid as a function of thermodynamic condition. Details about the system and simulation may be found in ref 50. In section II we give precise definitions for the local densities and local-density distributions we shall examine. In section III we tackle the question of how the somewhat arbitrary parameters appearing in the definition of the local density should be chosen, with an eye to maximizing the physical usefulness of the information obtained. In section IV we present the application of these methods to the two-dimensional LennardJones supercritical fluid, and it is here that we provide answers to the questions posed above. II. Local Density Definitions We define here an atom-centered local density (in contrast to the location-based local density defined in a companion paper.50) This atom-centered definition, which characterizes the local density surrounding a “solute” (here a tagged solvent atom), is expected to be of particular relevance to chemically oriented solute properties such as solvation and spectral shifts. A. An Atom-Centered Local Density. We define the (j) instantaneous local density around a tagged particle j, Fl (R), where R ) (r b1, ‚ ‚ ‚, b rN) denotes the instantaneous configuration of all N particles in the system, as the number of solvent particles (excluding the tagged particle) within a microscopic distance, or local radius, rl , of the tagged particle, divided by the
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Maddox et al. to produce the correct homogeneous-fluid result at a high fluid density, 〈Fl 〉 ) F. We discuss the motivation behind our choices for rl and rexcl in detail in section III. B. Local-Environment Distribution. The probability P (Fl ) that at any given instant in time a solvent atom will have a surrounding local solvent density of Fl can be found as
P (Fl ) ) 〈N-1
Figure 2. (a, top) Radial distribution function g(r) for a dense-liquidlike supercritical fluid (F ) 0.711 σ-2, T ) 0.55 /kB). (b, middle) Mean atom-centered local density 〈Fl 〉 as a function of the local radius rl from the central particle at the dense-liquid-like state point of panel a. The different curves correspond to different values of the exclusion radius rexcl (indicated by the legend in panel c), with the exception of the reference curve (ref) which marks the bulk density. (c, bottom) Same as panel b, except the state point is now the compressible-fluid state point of Figure 1a.
t Vl
-1
〈Fl 〉 ) )
)
- rjk)
(1a)
where θ is the Heaviside step function, rjk ) |r bk - b rj|,51 and the local volume is taken to be
Vl )
2πd/2 [(r )d - (rexcl)d] dΓ(d/2) l
N
2
F(j) l (R)
N
∑ θ(rl
〈
1 [F Vl
-rjk)〉
j,k)1 (j F, even in the absence of any intermolecular forces (see ref 68), although clearly the length scale of the density inhomogeneities must exceed the size of the “local” region for local density enhancements to arise. Additionally, this inhomogeneity-induced local density enhancement will be maximal at the critical point, when the inhomogeneities are of the longest range. This fact can be understood by realizing that the larger the size of the high-density domains, the more solvent atoms there are whose surrounding local region falls entirely within a high-density domain, such that the full “high-density-domain density” will be sampled more frequently. Since local regions which extend into both high- and low-density domains will yield a local density closer to the bulk value, higher
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sampling of regions falling entirely within high-density domains will generate a higher mean local density. This increasing enhancement in 〈δFl 〉 with increasing correlation length is apparent in Figure 5, where it is shown that P (Fl ) shifts to higher local densities as T f Tc (for fixed bulk density F). In sum, the density inhomogeneity effect will (i) be maximized as the critical point is approached, and (ii) be observed for any size local region, rl , as long as it is smaller than the extent of the density inhomogeneities. When this condition is met, localdensity enhancements will arise as a direct and necessary consequence of long-length-scale critical fluctuations, although they may also be generated by the structural mechanism discussed below. The “structural” mechanism is primarily the first-peak effect in Fg(r), i.e., the enhancement in the first peak of Fg(r) relative to the bulk density (recall eq 3). At low densities, this effect is due to the leading-order term, F exp[-βu(r)],53 where u(r) is the full potential (including both the repulsive and attractive components) between the central atom and any neighboring particles. At high densities, this “structural” mechanism also reflects the higher-order (many-body) terms in Fg(r) which give rise to the standard oscillatory structure at liquid densities. As discussed in section III, if the local region (i.e., rl ) is defined reasonably, then, at liquid densities, the oscillations beyond the first peak largely cancel in the local density 〈Fl 〉, leaving primarily the first peak enhancement to contribute to 〈δFl 〉 ) 〈Fl 〉 - F. Moreover, since we are not interested in this standard, structural first-peak effect at high densities, we have chosen rexcl to scale out this effect (section III, eq 5), such that 〈Fl 〉 ) F at high densities. We must consider further this “structural” effect, however, because its magnitude is not the same at all densities (or temperatures). At low densities, the structural effect, here due primarily to F exp[-βu(r)], is much larger than it is at liquid densities. Consequently, at low densities we find that 〈Fl 〉 * F, as shown in Figure 6. Indeed, we can show (eqs 5 and 10) that when the density is low enough that g(r) ) exp[-βu(r)], independent of the density, then 〈δFl 〉 will be a linear function of the bulk density having zero intercept:
[
〈δFl 〉 ) F
1 Vl
∫r 0). [Note that this behavior, that 〈δFl 〉 f 0 as F f 0, is often assumed in the creation of reference curves for the extraction of local-density enhancements from experimental data. Yet this same analysis also implies that 〈F˜ l 〉 ≡ 〈Fl 〉/F will not necessarily decay to its reference (zero enhancement) value of unity as F f 0, because, from eqs 9 and 11, 〈F˜ l 〉 |F)0 ) b exp[-βu(r)] > 1, for reasonable (rl , rexcl) pairs.] Vl -1∫r 0, exceeds those at high densities, where 〈δFl 〉 ) 0. Unfortunately, we cannot easily prove that these structural effects are maximized at very low densities, because of the complex density dependence of g(r). Finally, an important feature of this structural mechanism at low densities is that, in this regime, it is due almost entirely to direct potential attractions (exp[-βu(r)]) and can extend only over the range of the potential u(r). Thus, at low densities it is
inherently a “short-ranged” effect, and the result 〈Fl 〉 > F reflects the presence of small physical clusters arising from the nonideal nature of the fluid. On physical grounds, we therefore expect the effects of these nonidealities of the fluid, i.e., the very local density enhancements, to be most important at relatively low densities, well below the critical density. Thus, we see that the structural enhancement effect will (i) be maximized at low (but nonzero) densities and (ii) be observable only for small local regions, i.e., small rl , on the order of the range of u(r). We are now set to address both the location of the maximum in the local-density enhancement 〈δFl 〉 vs F curve (Figure 6c) and the dependence of this location on the size of the local region. At low densities, e.g. 0.05 σ-2, the critical inhomogeneities should be small, such that primarily only the “structural” effectssmall clusters due to the nonideality of the gasswill contribute to the observed local-density enhancements (〈δFl 〉 > 0). Additionally, we know that “structural” local-density enhancements are short-ranged, and will therefore become very small as rl is increased beyond the first solvation shell. This effect is indeed observed in the bottom panel of Figure 6, in which 〈δFl 〉 drops from ∼0.05 to ∼0.02 σ-2 to
