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J. Phys. Chem. B 2000, 104, 6266-6270
Effect of Critical Slowing Down on Local-Density Dynamics 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
Through molecular dynamics simulation we demonstrate that the time scale for local solvent reorganizations, which may control solute dynamics, can become very long for low- and near-critical-density supercritical fluids. We show that this can be ascribed to two processes, the first being due to direct, interparticle potential interactions, and the second, which dominates on all but the shortest length scales, being due to a coupling between the dynamics of the local solvent environment and those of the long-length-scale density fluctuations which are present in compressible supercritical fluids. Specifically, as the critical point is approached and the correlation length of these density inhomogeneities increases, we find that the associated slowing of these extended collective fluctuations, known as “critical slowing down”, generates a concomitant slowing of the local reorganization time.
I. Introduction The compressible supercritical regime encompasses a range of state points in the general vicinity of the critical point, and the high compressibilities found near a solvent’s critical point are known to be a macroscopic manifestation of spatially extended correlations in the microscopic density fluctuations.1,2 These extended correlations generate a fluid whose density is highly inhomogeneous on micro- or mesoscopic length scales,3-7 as can be seen in the molecular dynamics snapshot of a twodimensional Lennard-Jones supercritical fluid at a compressible state point (Figure 1a). Not surprisingly, these inhomogeneities may significantly affect the local solvent environments supported by the neat fluid. In fact, it is well-known that in such compressible fluids the mean local solvent density 〈Fl 〉 around an attractive solute,3,8-12 and even around a tagged solvent molecule,3,13 will exceed the thermodynamic bulk density, i.e., 〈Fl 〉 > F. Additionally, the distribution of environments experienced by a tagged solvent molecule (and by some solute molecules4) becomes significantly broadened relative to the distribution expected in homogeneous fluids.13-16 Of course, whether a particular dynamic solute process will be governed by the mean solvent environment, 〈Fl 〉, or by the distribution of instantaneous local environments, P(Fl ), will depend on the relative time scales of the solute process and solvent reorganization. Hence, if the local solvent environments persist for only very short times relative to the dynamic solute process, the solute will experience many local environments over the duration of its dynamic event, and the resultant rate will be controlled by the mean local environment 〈Fl 〉. On the other hand, if the local environments persist for times long relative to that of the solute process, each solute will feel only one of the distribution of possible solvent environments over the entire duration of its dynamic event. This latter scenario would give rise to a phenomenon known as inhomogeneous broadening,17-20 in which the dynamics of an ensemble of solutes will be controlled by the full distribution of local environments P(Fl ), rather than by the mean 〈Fl 〉. Thus, in order to determine whether the distribution P(Fl ) will be relevant to any solute process, i.e., whether inhomogeneous broadening effects will be present, we must first determine the lifetime of
the local solvent environmentssa quantity which, in supercritical fluids, is likely to depend on the thermodynamic state point. We therefore seek to answer the question, “What are the lifetimes of the local solvent environments found in neat supercritical fluids as a function of thermodynamic condition?” The astute reader may be wondering, however, why we concern ourselves with the question of inhomogeneous broadening in supercritical fluids which are, by definition, at high temperatures, given that inhomogeneous broadening effects are most typically observed in low-temperature amorphous materials.21 Indeed, such inhomogeneous effects become less and less important as local solvent environments interconvert more rapidly, as they are generally expected to do with increasing temperature. However, as the critical point is approached, the collective dynamics associated with fluctuations of the highand low-density regions become progressively slower as these regions grow in size, a phenomenon known as “critical slowing down”.22-24 What effect, if any, critical slowing down will have on dynamic solute processes is not yet understood. Interest in this question was generated by early critical-scaling arguments suggesting that chemical reaction rates should slow dramatically with the onset of critical slowing down.25,26 Since that time, these arguments have been shown to be incorrect,16,27,28 and little further attention has been directed toward this question. Indeed, the currently prevailing attitude appears to be the idea that long-length-scale phenomena such as critical slowing down are not relevant to local phenomena, and will thus have no affect on solute dynamics. While a separation of local and long-range effects is justified in the compressible regime, the oversimplified assumption that these effects are uncoupled from one another is not. That such coupling is present was recently demonstrated through the introduction of a local-density-specific radial distribution function g(r|Fl ).29 Application to a compressible, and therefore inhomogeneous, supercritical fluid showed a definite correlation between the local density around an atom and the extended surrounding solvent environment, reflecting the fact that very high local densities are most probably found embedded far within high-density domains while very low local densities are most probably found far within low-density
10.1021/jp0003813 CCC: $19.00 © 2000 American Chemical Society Published on Web 06/10/2000
Critical Slowing Down
J. Phys. Chem. B, Vol. 104, No. 26, 2000 6267 solute lifetimes in compressible supercritical fluids, although how these studies should be interpreted is not yet clear.3 Our goal in the present work is to examine whether local environment lifetimes do indeed grow large with the onset of critical slowing down as the critical point is approached. We therefore identify a local-environment lifetime and evaluate this quantity, via computer simulation, over a range of thermodynamic conditions for a supercritical two-dimensional LennardJones fluid. Combined with our earlier analysis of the local density distributions, P(Fl ),13 and local-density-specific correlation functions, {g(r|Fl )},30 for this same supercritical fluid, these results enable us to ascertain the physical origins of the observed local lifetimes. In section II we introduce the local density and the local environment lifetimes, while in section III we present the system and simulation details. Results and Discussion comprise section IV, and conclusions follow in section V. II. Methods We characterize the local environment around an atom at time t by the surrounding local density Fl (t). This instantaneous local density is defined as the number of atoms found within a distance rl of the central atom, divided by the appropriate local volume; see ref 13 for details. The dynamics of the local-density environment around an arbitrary solvent particle can now be monitored through a localdensity autocorrelation function, CF(t), defined as4
CF(t) )
〈δFl(0) δFl(t)〉 〈[δFl(0)]2〉
(2a)
where
δFl(t) ) Fl(t) - 〈Fl〉
Figure 1. Representative snapshots of a neat supercritical LennardJones fluid in two dimensions (62 σ × 62 σ) at T ) 0.55 /kB (Tr ≡ T/Tc ) 1.15) and at (a, top) an intermediate-density (F ) 0.3 σ-2) compressible fluid, and (b, bottom) a low-density (F ) 0.1 σ-2) state point. Panels a and b contain 1152 and 384 atoms, respectively.
domains and intermediate local densities are generally found at the edges of domains.13,29-31 This correlation between the local density and the extended inhomogeneous structure provides a possible mechanism by which the local solvent dynamics may be coupled to the dynamics of the extended structures. Thus, we hypothesize that the lengthening of this latter time scale, i.e., critical slowing down, may, in fact, generate a concomitant slowing of the local solvent environment dynamics, exactly the behavior which would generate inhomogeneous broadening of the solute dynamics. The presence of such inhomogeneous behavior has also been detected in supercritical fluids. First, the spectral dephasing of N2 in neat supercritical N2 was observed, experimentally,32 to broaden as the critical point is approached, and this broadening was attributed to inhomogeneous, rather than to homogeneous, origins. Similarly, recent computer simulations of vibrational relaxation in a highly compressible supercritical Lennard-Jones fluid have demonstrated that local environment lifetimes in such compressible fluids can be very long4 relative to some dynamic solute processes,16 indirectly supporting the inhomogeneous interpretation of the earlier dephasing study. Two other experimental studies33,34 have uncovered apparent distributions of
(2b)
and Fl (t) is the local density around an arbitrary particle in the system, and here the angle brackets indicate both an ensemble average and an average over all of the particles in the system. (Note that a similar function was earlier proposed by Randolph, O’Brien, and co-workers.)35 The desired quantity, the time scale on which CF(t) decays, is then determined by the standard formulation, i.e., by picking out the lowest frequency component of the relaxation via the zero-frequency Fourier transform:
τF )
∫0∞ dt CF(t)
(3)
The lifetime τF corresponds to the time required for the local environment, or density, around a tagged particle to change substantiallyswhat one might term the local-density reorganization time. III. Simulation Details Details about the system and the simulation may be found in ref 6. To calculate the local-environment time autocorrelation function, CF(t), data was collected every 40 time steps for the entire duration of each simulation run. Because of computer disk-space constraints, not all of the atoms in every simulation were used in the autocorrelation analysis. Instead, a randomly chosen set of 300 atoms (or all of the atoms in the system when the total number was less than 300)6 was found to give sufficiently good convergence of this function. IV. Results and Discussion Recall that our goal is to determine how long it takes for the solvent environment around a solute molecule to change
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Figure 3. Solvent reorganization times (in τLJ) determined from the data in Figure 2c. The legend indicates the various rl values used (in units of σ).
Figure 2. Atom-centered local-density autocorrelation functions, CF(t) versus time (eq 2). (a, top) Temperature dependence (F ) 0.3 σ-2, T given by the legend, rl ) 3.09 σ, rexcl ) 0.78 σ). (b, middle) Bulkdensity dependence (F given by the legend, T ) 0.55 e/kB, rl ) 3.09 σ, rexcl ) 0.78 σ). (c, bottom) Local-radius dependence (F ) 0.3 σ-2, T ) 0.55 /kB, rl given by the legend, rexcl ) 0.56, 0.73, 0.78, and 0.73 σ for rl ) 4.72 to 1.78 σ, respectively).
substantially, in order to determine whether inhomogeneous broadening should be expected for any given dynamic solute process. Toward this end, we show in Figure 2 the local-density autocorrelation functions CF(t) for the two-dimensional LennardJones fluid. Panel a presents CF(t) at a variety of temperatures (T ) 0.55, 0.75, and 5.0 /kB) along the F ) 0.3 σ-2 nearcritical-density isochore; the critical temperature and density for this fluid are Tc ) 0.477 ( 0.003 kB/ and Fc ) 0.38 ( 0.02 σ-2.36 We see that decreasing the temperature dramatically increases the time it takes for CF(t) to decay, such that, for example, a decay of 50% is attained at 0.5 τLJ (∼1 ps in Ar) when T ) 5.0 /kB, but not until 3.0 τLJ (∼6.5 ps) when T ) 0.75 /kB, or until 8.5 τLJ (∼18 ps) when T ) 0.55 /k.B. The factor of 3 increase in the relaxation time observed between T ) 0.75 and 0.55 /kB, which is only a 36% decrease in the temperature, suggests that the slow relaxation of CF(t) at the compressible state point (T ) 0.55 /kB) arises because of density inhomogeneities, not just because the individual particles are moving more slowly than at the higher temperature (T ) 0.75 /kB). To further examine this conjecture, we show CF(t) as a function of bulk density (F ) 0.1, 0.3, 0.55, 0.63, and 0.711 σ-2) on the T ) 0.55 /kB near-critical isotherm, panel b. Indeed, we observe the decay time of CF(t) to increase substantially as the bulk density is decreased from the liquidlike density of 0.711 σ-2 to the near-critical density of 0.30 σ-2, suggesting that the local-density relaxation time increases as the critical point is approached and the extended density inhomogeneities grow in. However, at the lower bulk density of 0.10 σ-2, CF(t) decays only slightly faster than it does at the more near-critical density of 0.30 σ-2, even though the latter state point is significantly more inhomogeneous,30 as can be observed in the corresponding snapshots for these state points, Figure 1, b and a, respectively. Confirmation of our hypothesis that extended density inhomo-
geneities cause an increased local-density relaxation time will thus require a more quantitative measure of the time scale. The local-density reorganization times computed from CF(t) (which cannot be well fit with single-exponential decays), eq 3, are shown as a function of the bulk density along this same near-critical isotherm, T ) 0.55 /kB. When the local regions are defined so as to contain approximately two solvation shells at liquid densities (rl ) 3.09 σ), as it was in Figure 2a,b, the local reorganization time τr is indeed observed (solid circles) to attain its maximum at the most near-critical density considered, F ) 0.35 σ-2. Thus, we observe τF to be maximized as the critical fluctuations are maximized. Yet, this relaxation time remains much larger at low densities than it does at high densities, and this large asymmetry implies that critical fluctuations are probably not the only source of the slow local-density reorganization times observed. To gain additional insight into the nature of the physical mechanisms contributing to the observed reorganization times, we examine the dependence of CF(t) and τF on the size of the local region (rl ) 1.78, 3.09, 3.99, and 4.72 σ). Figure 2c shows that, under the compressible-fluid condition of T ) 0.55 /kB, F ) 0.30 σ-2, increasing the size of the local region increases the time required for the local-density autocorrelation function to decay. This trend results because the collective fluctuation required to generate a specified percentage change in the local density of a larger local region will involve a greater number of particles than would be required to generate this same percentage change in a smaller local region, thus making the former slower than the latter. Of more importance, Figure 3 illustrates that the bulk-density dependence of the local-density reorganization time is sensitive to the size of the local region. For example, for the largest local region, rl ) 4.72 σ (∼4 solvent shells), τF’s maximum at the near-critical density F ) 0.35 σ-2 is 39.3 τLJ, 1.8 times its value at the lowest bulk density of F ) 0.10 σ-2 and 7.4 times its value at the highest bulk density of F ) 0.711 σ-2. As rl , and thus the size of the local region, is decreased, we find that τF becomes more asymmetrical around F ≈ Fc. Thus, by the smallest local radius, rl ) 1.78 σ (∼1 solvent shell), no clear maximum is observed, and its value of 13.5 τLJ at F ) 0.35 σ-2 is approximately equal to its value at F ) 0.10 σ-2. On the other hand, this value of 13.5 τLJ is now 13.1 times that at the high density (F ) 0.711 σ-2). Thus, we see that as the local region becomes smaller, the mechanism generating slow local reorganization times at low bulk densities
Critical Slowing Down becomes increasingly more important relative to the mechanism generating slow local reorganization times at near-critical densities. In previous work,13 we uncovered two mechanistic sources of density inhomogeneities in neat supercritical fluids, one which acts on a short length scale and is maximally important at low bulk densities (and “low” supercritical temperatures) and one which acts over an extended length scale and is maximized at the critical density (and temperature). The former mechanism we termed “potential-induced”, or “structural”, as it results from direct interparticle potential interactions which, at low bulk densities and near-critical temperatures, compete effectively with entropy to create small, metastable clusters of just a few atoms. The latter mechanism is, of course, the critical fluctuations, which result from indirect interparticle interactions and which create extended regions of high and low densities; we termed this the “inhomogeneity-induced” mechanism. On near-critical isotherms, the interplay of these two mechanisms should cause the maximum of any inhomogeneity-driven quantity to be observed at the critical density when the local region considered is large but to shift toward lower bulk densities when smaller local regions are considered. As these trends are exactly what were observed for the local-density reorganization time, we suggest that there is both a potential-induced slowing of τF which arises when small metastable clusters are present, and inhomogeneity-induced slowing which arises when critical fluctuations are present. To support this interpretation, we provide explanations of how each of these mechanisms would generate slowed local-density reorganization times, τF. The potential-induced mechanism for slowing τF is simple to understand. At very low bulk densities (Figure 1b) a solvent atom should have no other particles inside its local region a significant fraction of the time, although because of the shortranged potential attraction it will also have nearest neighbors for a small fraction of the time. Yet, these potential interactions also ensure that the process of losing a near neighbor is a metastable one; i.e., it is an activated process; consequently, this step is both rate-limiting and slow, generating a long τF. Note, however, that the exact origin of the slow reorganization time observed at F ) 0.10 σ2 is not entirely clear, because, as shown by the local-density-specific radial distribution functions g(r|Fl ),30 the system still exhibits noticeable remnants of spatially extended density inhomogeneities at this state point. Consequently, both “mechanisms” are still operative and expected to contribute to the lengthening of τF under these conditions. To understand how the “inhomogeneity-induced” mechanism will slow the local-density reorganization time τF, consider first a homogeneous system. In homogeneous systems, only a small number of different solvent environments are easily accessible to the solute (P(Fl ) is narrow, see ref 13), and, because these all correspond to local densities near the mean, the system can quickly flicker from one environment to the next with only one or two atoms actually changing position, causing CF(t) to decorrelate rapidly, e.g., Figure 2b, F ) 0.711 σ-2. When spatially extended density fluctuations are present, however, the correlation between local densities and the extended fluid structure, which we uncovered via the local-density-specific radial distribution functions g(r|Fl ) in refs 29 and 30, implies that the short-ranged dynamics of the local region will be entangled with the long-range dynamics of the extended inhomogeneities. Since this latter, long-range process involves the collective motion of many particles, it is necessarily governed by a long relaxation time. The coupling between the dynamics of the local and extended structures therefore ensures
J. Phys. Chem. B, Vol. 104, No. 26, 2000 6269 that this slow relaxation time of the extended structure will be reflected in the local-density reorganization time. Physically, this coupling can be understood as follows. First, recall that the correlations observed in g(r|Fl ) indicate that very high (low) local densities are found embedded within extended high (low)density domains. And, within a given extended domain the tagged solvent particle can easily sample a restricted range of local densities, e.g., in a high-density domain a range of fairly high local densities will be easily accessible, such that the local density can flicker rapidly between these high local densities. Yet, for the local-density autocorrelation function to decay fully, such a high-density solvent particle must also move to lowdensity regions, and this transition appears to be dependent upon the invariably slow breakup of the high-density region. Indeed, examination of molecular dynamics snapshots (for the state point shown in Figure 1a) over increments of ∼10 000 time steps suggests that the extended domain structure fluctuates via highdensity-domain fracture and fusion, rather than by single particle evaporative or diffusive mechanisms. We therefore propose that these collective domain fluctuations, fracture and fusion, are responsible for the escape of solvent atoms (“solutes”) from high- and low-density domains, respectively, and hence that the local-density reorganization time is controlled by the time scale of such collective processes. Clearly, the longer-ranged the density inhomogeneities, the further embedded in a high (or low)-density domain the tagged solvent particle can become, and thus the longer it can take for the tagged particle to “escape” to a low (or high)-density environment. Thus, we see that the well-known process of “critical slowing down”,22-24 which arises as a result of the long-range critical fluctuations, makes itself felt in the dynamics of the local regions. Note that although we present only solute-equal-to-solvent results here, long localdensity relaxation times have been observed for attractive solutes as well; see refs 4, 37, and 38). Returning to Figure 3, it is interesting to compare the bulkdensity dependence of the local-density reorganization time τF with that of the mean local-density enhancement, 〈δFl 〉 ) 〈Fl 〉 - F. On near-critical isotherms this latter quantity is generally found to be maximized at densities well below the critical density,3 and, additionally, it is very sensitive to the size of the local regions, with the maximum moving away from the critical point toward lower densities as the size is decreased.3,13 The local-size dependence of the mean local-density enhancement, 〈δFl 〉, parallels that which we observed for the local-density reorganization time, τF, in Figure 3, consistent with the supposition that in both cases there are two contributing mechanismssa “potential-induced” mechanism, which acts only over very short ranges and becomes important at low densities and which reflects the presence of small metastable clusters, and an “inhomogeneity-induced” mechanism, which becomes maximally important as spatially extended density inhomogeneities grow in as the critical point is approached.39 However, the location of the observed maxima in 〈δFl 〉 and τF differ (compare Figure 3 with Figure 6 of ref 13); 〈δFl 〉 exhibits maxima at densities well below critical for all size local regions, while τF exhibits a maximum at the critical density for all but the smallest local region. This difference indicates that the inhomogeneity-induced mechanism tends to dominate the localdensity reorganization time τF, while the local density enhancement 〈δFl 〉 is more sensitive to the potential-induced mechanism. V. Concluding Remarks Our recent work has shown local solvent environments in compressible supercritical fluids to have interesting and varied
6270 J. Phys. Chem. B, Vol. 104, No. 26, 2000 structures which change dramatically with state point. Yet, the relevance of these local solvent structures to solute processes cannot be ascertained without knowledge of the relative time scales for solvent reorganization and solute dynamics. We have therefore examined a local-density reorganization time τF, representing the time required for the density within a specified local radius of a tagged solvent particle to become uncorrelated with its initial value, as a function of state point for a simple two-dimensional Lennard-Jones supercritical fluid. We found that, for near-critical temperatures, the local-density reorganization time at low and near-critical bulk densities may exceed the reorganization time observed at liquidlike bulk densities by more than a factor of 10. This result supports earlier studies suggesting that inhomogeneous broadening may be observed in compressible supercritical fluids. Additionally, the observed density dependencies suggest that two different mechanisms contribute to the lengthening of the local-density lifetime. First, at low bulk densities, local-density lifetimes are lengthened by the presence of small, metastable clusters which dissociate only rarely. This effect, which arises from direct, attractive interparticle potential interactions, is a short-range effect and decreases in importance as larger local regions are considered. Perhaps more surprisingly, the presence of spatially extended domains of high and low density associated with the critical region are also found to contribute to the lengthening of the local-density reorganization time. This coupling of local and extended phenomena arises because, in such inhomogeneous fluids, very high (low) local densities are most probably found embedded far within extended high (low)-density domains. Consequently, the escape of a particle from a high-density domain, for example, would likely be controlled by a collective fluctuations of the extended domain structure.29,30 Thus, as the critical point is approached and such collective fluctuations become progressively slower, a phenomenon known as critical slowing down, the local-density reorganization time also becomes slower, confirming that critical slowing down can affect local dynamics. Acknowledgment. S.C.T. gratefully acknowledges an NSF Young Investigator award and a Camille Dreyfus TeacherScholar award. This work was supported by NSF grant CHE9727361. References and Notes (1) Fisher, M. E. In Critical Phenomena, Proceeding, Stellenbosch, South Africa; Lecture Notes in Physics 186; Hahne, F. J. W., Ed.; SpringerVerlag: Stellenbosch, South Africa, 1983; p 1. (2) Domb, C. The Critical Point; Taylor & Francis: London, 1996.
Maddox et al. (3) Tucker, S. C. Chem. ReV. 1999, 99, 391-418. (4) Tucker, S. C.; Maddox, M. W. J. Phys. Chem. B 1998, 102, 24372453. (5) Maddox, M. W.; Goodyear, G.; Tucker, S. C. J. Phys. Chem. 2000, 104, 6258. (6) Goodyear, G.; Maddox, M. W.; Tucker, S. C. J. Phys. Chem. 2000, 104, 6240. (7) Rovere, M.; Heermann, D. W.; Binder, K. J. Phys. Condens. Matter 1990, 2, 7009. (8) Petsche, I. B.; Debenedetti, P. G. J. Chem. Phys. 1989, 91, 7075. (9) Brennecke, J. F.; Chateauneuf, J. E. Chem. ReV. 1999, 99, 433452. (10) Kajimoto, O. Chem. ReV. 1999, 99, 355-390. (11) Sun, Y.-P.; Bennet, G.; Johnston, K. P.; Fox, M. A. J. Phys. Chem. 1992, 96, 10001. (12) O’Brien, J. A.; Randolph, T. W.; Carlier, C.; Ganapathy, S. aiche 1993, 39, 876. (13) Maddox, M. W.; Goodyear, G.; Tucker, S. C. J. Phys. Chem. 2000, 104, 6248. (14) Tucker, S. C.; Goodyear, G. In Supercritical Fluids II; Kiran, E.; Debenedetti, P. G., Peters, C. J., Eds.; Kluwer: Dordrecht, in press. (15) Kajimoto, O.; Sekiguchi, a. K.; Nayuki, T.; Kobayashi, T. Ber Bunsen-Ges. Phys. Chem. 1997, 101, 600. (16) Goodyear, G.; Tucker, S. C. J. Chem. Phys. 1999, 111, 96739677. (17) Kubo, R. In Fluctuations, Relaxation, and Resonance in Magnetic Systems; Haar, D. T., Ed.; Oliver & Boyd: Edinburgh, UK, 1962. (18) Kubo, R. AdV. Chem. Phys. 1963, 13, 101. (19) Stratt, R. M.; Maroncelli, M. J. Phys. Chem. 1996, 100, 12981. (20) Geva, E.; Reilly, P. D.; Skinner, J. L. Acc. Chem. Res. 1996, 29, 579-584. (21) Tokmakoff, A.; Fayer, M. D. J. Chem. Phys. 1995, 103, 28102826. (22) Binney, J. J.; Dowrick, N. J.; Fisher, A. J.; Newman, M. E. J. The Theory of Critical Phenomena: An Introduction to the Renormalization Group; Clarendon Press: Oxford, UK, 1993. (23) Chang, R. F.; Keyes, P. H.; Sengers, J. V.; Alley, C. O. prl 1971, 27, 1706-1709. (24) Burstyn, H. C.; Sengers, J. V. Phys. ReV. A 1983, 27, 1071-1085. (25) Procaccia, I.; Gitterman, M. Phys. ReV. Lett. 1981, 46, 1163. (26) Procaccia, I.; Gitterman, M. Phys. ReV. A 1983, 27, 555. (27) Milner, S. T.; Martin, P. C. Phys. ReV. A 1986, 33, 1996. (28) Greer, S. C. Phys. ReV. A 1985, 31, 3240. (29) Goodyear, G.; Maddox, M. W.; Tucker, S. C. J. Chem. Phys., in press. (30) Goodyear, G.; Maddox, M. W.; Tucker, S. C. J. Phys. Chem. 2000, 104, 6258. (31) Martinez, H. L.; Ravi, R.; Tucker, S. C. J. Chem. Phys. 1996, 104, 1067-1080. (32) Clouter, M. J.; Kiefte, H.; Deacon, C. G. Phys. ReV. A 1986, 33, 2749. (33) Betts, T. A.; Zagrobelny, J.; Bright, F. V. J. Supercrit. Fluids 1992, 5, 48. (34) Rhodes, T. A.; Fox, M. A. J. Phys. Chem. 1996, 100, 17931. (35) Randolph, T. W.; O’Brien, J. A.; Ganapathy, S. J. Phys. Chem. 1994, 98, 4173. (36) Panagiotopoulos, A. Z. Int. J. Thermophys. 1994, 15, 1057. (37) Graf, P.; Nitzan, A. Chem. Phys. 1998, 235, 297-312. (38) Re, M.; Laria, D. J. Phys. Chem. B 1997, 101, 10494. (39) Or, more rigorously, at the point of maximum compressibility along the (possibly noncritical) isotherm.
