J. Phys. Chem. B 2002, 106, 3217-3225
3217
Study of Local Density Enhancement in Near-Critical Solutions of Attractive Solutes Using Hydrostatic Hypernetted Chain Theory Roberto Ferna´ ndez-Prini* Unidad ActiVidad Quı´mica, Comisio´ n Nacional de Energı´a Ato´ mica, AV. Libertador 8250, 1429-Capital Federal, Argentina, and INQUIMAE, Facultad Ciencias Exactas y Naturales, UniVersidad de Buenos Aires, Ciudad UniVersitaria, Pabello´ n II, 1428-Capital Federal, Argentina ReceiVed: August 7, 2001; In Final Form: NoVember 26, 2001
The properties of infinitely dilute solutes in near-critical solvents often show a strong change in curvature when plotted against the bulk fluid density, in some cases even a plateau region is observed. In this article, it is shown that this phenomenon, due to a local solvent density enhancement, can be construed as a consequence of the linear response of fluids having large compressibilities when the solutes interact strongly with the solvent molecules. The hydrostatic hypernetted chain (HHNC) solution of the Ornstein-Zernike (OZ) equation for inhomogeneous fluids was used to calculate the solvent-solute radial distribution function and the number of solvent molecules that surround the solute particle for different solute-solvent interaction energies and reduced temperatures. The use of HHNC eliminates physically unsound features of the calculations, which employ theories for homogeneous fluids, and it also limits the extent and range of the local density enhancement. It is shown that the performance of the relatively simple HHNC theory is as satisfactory as that of more sophisticated theories and that it is able to describe adequately experimental spectroscopic and kinetic results confirming that the phenomenon involves the coupling of intermolecular interactions with the high susceptibility of the near-critical fluids. It is noted, however, that the magnitude of the solute-solvent interactions required to fit the experimental results are larger than expected from known values of the solutes’ intermolecular energy (2/k).
Introduction The knowledge of the behavior of infinitely dilute solutions in near-critical solvents is important on two accounts. In the first place, the use of these solvents allows the exploration of the effect of the solvent density, F1, upon solvation phenomena; this is necessary information to complete the description of solvation. Second, these fluids play a central role in the increasing use of supercritical fluids in chemical processes. To accomplish both objectives, it is necessary to have the supercritical solvents in a near-critical condition because then they have a large compressibility, thus facilitating the change in the fluid number density with moderate changes in pressure. The study of these systems has been pursued intensely in the last fifteen years as described in a recent issue of Chemical ReViews dedicated to supercritical fluids.1 However, there are still some basic aspects of near-critical mixtures that require a deeper understanding, namely, the observed density dependence of some solute properties. The curves representing the isothermal density dependence of many properties of dilute solutes in near-critical solvents, notably equilibrium, structural, and spectroscopic properties, often exhibit a strong change in the curvature at densities somewhat lower than the solvent’s critical density, Fc1. This feature is reported to be more notable as the temperature approaches the solvent critical temperature, Tc1; when sufficiently close to Tc1, it even leads to a region around Fc1 where the solute’s properties have a very weak density dependence and which is sometimes denoted as a plateau. The most * To whom correspondence should be addressed. Member of Carrera del Investigador (CONICET). E-mail address:
[email protected].
frequently invoked quantitative explanation for this feature is that the number density of solvent molecules in the neihgborhood of a solute particle is different from the bulk average density of the fluid. The remaining question is what causes the local density inhomogeneity. The origin of this phenomenon has been attributed variously to the solvent near-critical state, to intermolecular interactions, and to an interplay between both causes. The first evidence of this type of behavior was construed as an indication that, close to its critical state, solvent molecules crowd around an attractive solute particle; thus, the term critical clustering was coined. Huge solvation numbers were sometimes reported, which were calculated with models applicable to triplepoint liquids; however, trivial extensions of these models to nearcritical behavior is not guaranteed.2 Early in the study of solutions in supercritical fluids, Eckert and co-workers3 suggested the convenience of separating contributions due to the enhanced solvent susceptibility, in our case the isothermal compressibility, from those due to intermolecular interactions. The coupling of pure solvent compressibility, which is affected by criticality and produces long-range effects, with short-range intermolecular interactions, which are independent of the vicinity of a critical point, is what causes the divergence of many partial molar quantitities of the solute at the solvent critical point. The existence of a change in the intermolecular interaction regime as F1 changes, that is, predominantly repulsive at high solvent densities and attractive in the lower density range, was invoked2,4 as the cause for the observed increase in the local density surrounding the solute particle. The fact that many properties exhibit a region of weak bulk density dependence either determined experimentally,5,6
10.1021/jp013034h CCC: $22.00 © 2002 American Chemical Society Published on Web 02/28/2002
3218 J. Phys. Chem. B, Vol. 106, No. 12, 2002 or predicted by theory2 or simulation7 centered at fluid densities somewhat lower than the critical density seemed to support this contentionswhen the fluid’s packing fraction is smaller than that of close-packed liquids (triple-point densities) attractive interactions prevail, generating an increased local density. Invoking a change in the interaction regimes with F1 as the only cause for the observed near-critical behavior of solutes implies that a similar type of density dependence should be observed as the temperature increases above its critical value. However, recent studies of spectroscopic properties of molecules dissolved in supercritical fluids reported that these properties exhibit a much stronger change of curvature in their bulk density dependence as the temperature was closer Tc1, leading even to a plateau region.8,9 Such a strong temperature dependence of the studied properties would not be expected according to the hypothesis that increased number of solvent molecules around the solute is only caused by the attractive forces in a medium of low packing fraction. Last but not least, near-critical systems close to Tc1 have a high compressibility, and consequently, special care is required to obtain meaningful experimental results. To calculate with enough precision the solvent density from measurements of p and T, it is necessary to have a good control of these two thermodynamic variables and also to assess carefully the effect of the solute in changing the system’s equation of state. Considering this background, it has been suggested recently10 that although the cause of the abrupt change in curvature of the bulk density dependence of many properties operates at short range, where intermolecular interactions prevail, the observed overall effect is promoted by the vicinity of the solvent critical point. This contention may appear paradoxical11 on account of the different length scales that are characteristic of critical phenomena and that are characteristic of intermolecular interactions; however, this is not necessarily so, and this idea has recently received support from new experimental and thoretical studies.12,13 Very recently, Egorov12,14 showed that the local inhomogeneity observed in the vicinity of the solute implies that the integral equations normally used to describe the behavior of solutes dissolved in dense liquids cannot be applied to the case of solutes in near-critical fluids because those equations are approximations that can only deal with solutes in fluids that are essentially homogeneous. In view of this picture, in this work, we explore first the interpretation of the phenomenon of enhanced local density in the frame of linear response to describe local density enhancement in near-critical systems. This description underlies the interplay between solvent-solute interactions and solvent criticality. Second, the structure of the fluid surrounding attractive solutes is analyzed using a relatively simple integral equation and its inhomogeneous counterpart. Due to the existence of different interpretations and some conflicting evidence, the present study privileges obtaining a general picture for the observed phenomena rather than a detailed quantitative explanation of each individual property. For this reason, we used the hydrostatic hypernetted chain (HHNC),15 which employs the Percus-Yevick (PY) solution of the Ornstein-Zernike (OZ) integral equation for the solvent-solvent interactions and is of simple implementation. This allows a very flexible and extensive exploration of behavior as the solvent’s critical point is approached. It seems important to know what a simple, albeit well-founded, theoretical description predicts for the behavior of solutes dissolved at infinite dilution in near-critical fluids. Although this work will concentrate on structural and spectro-
Ferna´ndez-Prini scopic properties (in a future study thermodynamic properties will be analyzed), the ideas that will be discussed might be extended to the dynamic solvation behavior of solutes, taking into account that linear response theory is applicable to them16,17 even when the solvent is in a near-critical state.18,19 Theoretical Framework In this work, we study the case of nonionic attractive solutes dissolved in solvents at infinite dilution. The term attractiVe solutes denotes solutes interacting with the solvent molecules more strongly than two solvent molecules do. Let us consider a system consisting of N solvent molecules and a single solute particle, that is, at infinite dilution. It is instructive to analyze the behavior of this system considering that the introduction of the solute molecule in a homogeneous fluid brings in solutesolvent interactions, which perturb the solvent molecules in the same manner as an external field applied to the system would. This situation was analyzed in detail by Hansen and McDonald.20 Assuming that the potential energy in the system is pairwise additive, then for an isotropic system of spherical particles having N solvent molecules and one solute particle (subindex 0), we have N
UN+1 )
∑
N
uij +
i,j)1;j>i
∑ i)1
N
u0i ) UN +
Ψ(ri) ∑ i)1
(1)
where uij is the interaction energy between solvent molecules i and j and u0i represents the interaction energy between the solute and solvent molecule i. This equation shows that the potential energy of the system may be described also in an alternative equivalent waysthe solvent molecules at distance ri may be considered subjected to a local external field, Ψ(ri). When an external field exists, the n-particle density, F(n)(r1,...,rn) in the grand canonical ensemble is given by
F(n)(r1,...,rn) )
1
∞
∑
N
1
ΞN)n(N - n)!
∫ ∫∑ ...
N
z*(ri)
i)1
∑
i
