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J. Phys. Chem. B 2005, 109, 7522-7528
Local Solvent Density Augmentation around a Solute in Supercritical Solvent Bath: 1. A Mechanism Explanation and a New Phenomenon Shiqi Zhou† Research Institute of Modern Statistical Mechanics, Zhuzhou Institute of Technology, Wenhua Road, Zhuzhou city, 412008, P. R. China ReceiVed: August 12, 2004; In Final Form: NoVember 24, 2004
A recently proposed partitioned density functional (DF) approximation (Phys. ReV. E 2003, 68, 061201) and an adjustable parameter-free version of a Lagrangian theorem-based DF approximation (LTDFA: Phys. Lett. A 2003, 319, 279) are combined to propose a DF approximation for nonuniform Lennard-Jones (LJ) fluid. Predictions of the present DF approximation for local LJ solvent density inhomogeneity around a large LJ solute particle or hard core Yukawa particle are in good agreement with existing simulation data. An extensive investigation about the effect of solvent bath temperature, solvent-solute interaction range, solvent-solute interaction magnitude, and solute size on the local solvent density inhomogeneity is carried out with the present DF approximation. It is found that a plateau of solvent accumulation number as a function of solvent bath bulk density is due to a coupling between the solvent-solute interaction and solvent correlation whose mathematical expression is a convolution integral appearing in the density profile equation of the DF theory formalism. The coupling becomes stronger as the increasing of the whole solvent-solute interaction strength, solute size relative to solvent size, and the closeness to the critical density and temperature of the solvent bath. When the attractive solvent-solute interaction becomes large enough and the bulk state moves close enough to the critical temperature of the solvent bath, the maximum solvent accumulation number as a function of solvent bath bulk density appears near the solvent bath critical density; the appearance of this maximum is in contrast with a conclusion drawn by a previous investigation based on an inhomogeneous version of Ornstein-Zernike integral equation carried out only for a smaller parameter space than that in the present paper. Advantage of the DFT approach over the integral equation is discussed.
I. Introduction Solvation of solute particle in fluid consisting of solvent particles is a very general phenomena occurring in many industrial processes and scientific practices, some of the examples include extraction under supercritical conditions,1 hydration of biomacromolecules in aqueous environment,2 wetting transition3 induced by large colloidal particles etc. Supercritical solvents in a near-critical conditions have a large compressibility, thus facilitating a change in the fluid number density with moderate changes in pressure, this property makes the supercritical fluid find increasing use in chemical processes.4 The solvents or solvent mixtures under supercritical conditions also represent an attractive area of investigation for statistical mechanics as they cover different fields such as near-critical phenomena, physical chemistry under extreme conditions, or solvation dynamics and kinetics. Studies on the supercritical fluids has been an active field in the last 15 years as described in a recent issue of Chemical ReViews.5 Hydration of biomacromolecules2 is a special solvation phenomenon where the solvent is water molecule, importance of the hydration for conformational stability of proteins, nucleic acids, and their complexes and for specificity and affinity of folding and binding events has been recognized long ago. Generally speaking, an understanding of solute hydration at the microscopic and macroscopic levels is an absolutely necessary initial step toward identifying, resolving, and characterizing individual thermodynamic determinants of the conformational stability and functional activity of biologically relevant molecules.6 For the solvation induced †
E-mail:
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by atomic fluid, the main calculational objectives are the solvent density distribution around a solute particle and the solvation free energy. According to declaration of density functional theory (DFT),7 a macrosystem free energy can be exprssed as a functional of the density distribution; therefore, one can aquire the solvation free energy only if the solvent density distribution is known. Thus the solvent density distribution can be considered as a basic quantity. For the case of hydration, depending on models chosen for water molecule, one may calculate the density distribution of hard sphere fluid with a dipole moment or calculate the density distribution of hydrogen or oxygen atoms. Investigation on the local solvent density inhomogeneity around a solute particle is very important, the local solvent density inhomogeneities are considered as an explanation8 for an observation that 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 Fcri. This feature is reported to become notable as the temperature approaches the solvent critical temperature Tcri. When sufficiently close to Tcri, a region around Fcri even appears where the solute’s properties have a very weak density dependence sometimes called a plateau. The local inhomogeneity observed in vicinity of the solute implies that Ornstein-Zernike (OZ) integral equation theory normally used to describe behavior of the solutes dissolved in dense liquids cannot be applied to the case of the solutes in near-critical fluids because those equations are approximate that
10.1021/jp0463619 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/19/2005
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can only deal with solutes in fluids that are essentially homogeneous.9 A hydrostatic hypernetted chain (HHNC) solution10 of the OZ integral equation for inhomogeneous fluids was used to calculate the local inhomogeneity of the solvent density, also the HHNC with inclusion of a second-order approximation of equation for bridge function was employed for investigation of the local solvent density argumentation.11 It is well-known that the HHNC approximation is applicable to continuous fluids under an external slow-varying field, when the solute’s size is far larger than that of the solvent particle, the external field due to the solute particle cannot be considered a weak and slowvarying field; thus, the validity of the HHNC approximation needs to be tested for this case. Density functional theory (DFT)12 is a powerful tool for studying the local density inhomogeneity. With a good DF approximation for the free energy DF or the first-order direct correlation function (DCF), the DFT approach can predict with almost the same accuracy the local density inhomogeneity induced by arbitrary external fields. Recent investigations13,15b pointed out that global thermodynamic quantities, for example the excess Helmholtz free energy, can be estimated accurately from an approximation for the first-order DCF with the local density inhomogeneity from the latter as input. Therefore, from a DF approximation for the first-order DCF, one can calculate both the local density inhomogeneity of the solvent particles induced by the solute particle and the solvation free energy. A main aim of the present paper is to investigate influence of the solvent bath properties and the solute-solvent interaction properties on the local density inhomogeneity with a DFT approach based on a recently proposed scheme14 for constructing the DF approximation for general interaction potential fluid. In section II, the present DF approximation is presented and its prediction for the local density inhomogeneity is compared with the corresponding simulation data for confirming the theory’s validity. In the same section, the proposed DF approximation is employed for extensive investigation of the local density inhomogeneity under various conditions of solvent bath properties and solute-solvent interaction properties. Finally, in section III, we conclude the paper. II. Presentation of DFT Recipe Local solvent (component 1) density inhomogeneity around a solute particle (component 2) can be considered as generated from an external field due to the solute particle. The external potential is denoted by u21(r), local solvent density inhomogeneity F21(r) around a solute particle can be estimated by a density profile equation of the DFT scheme,
F21(r) ) Fb1 exp{-β u21(r) + c1(1)(r; [F21], T/) c˜ 1(1)(Fb1, T/)} (1) where c1(1)(r; [F21], T/) and c˜ 1(1)(Fb1, T/) are the solvent component nonuniform first-order DCF, and its uniform counterpart, Fb1 is solvent component bulk number density, T/ is reduced temperature of solvent bath. To carry out numerically an estimation of F21(r) by eq 1, we employ a recently proposed partitioned DF approximation14 in which we will apply a recently proposed parameter-free version of a Lagrangian theorem-based density functional approximation (LTDFA)15 for hard sphere fluid to hard core part of the solvent-solvent interaction potential. With these considerations, eq 1 takes the following form:
∫dr1 (F21(r1) Fb1)c11hc(|r - r1|; F˜ hc((r + r1)/2, λ), T/) + ∫dr1 (F21(r1) -
F21(r) ) Fb1 exp{-β u21(r) +
Fb1)C11tail(|r - r1|; Fb1, T/)} (2) Here λ can be a constant 0.5 as shown by the LTDFA,15b the weighted density F˜ hc is defined by
F˜ hc((r + r1)/2, λ) ) r′|;
∫dr′ c11hc(|(r + r1)/2 -
/
T )[Fb1 + λ(F21(r′) - Fb1)]/c˜ 1hc(1)′(Fb1, T/) (3)
Fb1,
As usually, c˜ 1hc(1)′ is density derivative of the bulk first-order DCF for the hard core part and can be estimated by a space integration of the bulk second-order DCF for the hard core part:
c˜ 1hc(1)′(Fb1, T/) )
∫dr c11hc(r; Fb1, T/)
(4)
The solvent second-order DCF c11(r; Fb1, T/) from which the hard core part c11hc(r; Fb1, T/) and tail part c11tail(r; Fb1, T/) are derived, is obtained by solving numerically the OZ equation for the bulk fluid consisting of only the solvent component with the bulk number density Fb1 and the reduced temperature T/. To proceed numerically, a well-known Percus-Yevick (PY) approximation16 is employed to close the OZ equation for the bulk solvent fluid. We choose σ11 as critical separation distance for dividing the solvent second-order DCF c11(r; Fb1, T/) into the hard core part and tail part. i.e.
[
c11hc(r; Fb1, T/) ) c11(r; Fb1, T/) r < σ )0 r>σ
[
c11tail(r; Fb1, T/) ) 0 r < σ ) c11(r; Fb1, T/) r>σ
]
]
(5) (6)
The present paper employs a Lennard-Jones potential as a model of the solvent particle interaction,
[( ) ( ) ]
u11(r) ) 411
r σ11
-12
-
r σ11
-6
(7)
The numerical solution of eq 2 can be obtained by iteration. I.e. at the beginning one assumes the density distribution F21(r) in the region where the fluid particle is present to be Fb1, and the density distribution F21(r) in other region to be zero, then new F21(r) can be produced out in the left-hand side of eq 2 by putting the assumed F21(r) in the right-hand side of the same eq 2. After the first calculation, the new density distribution is mixed with the old density distribution according to the following relation
F21(r)mix ) δF21(r)new + (1 - δ)F21(r)old
(8)
where δ has to be a very small number, for example 0.08, or even 0.01 as the bulk density is high or the reduced temperature is low to avoid divergence. Then puts the F21(r)mix in the righthand side of eq 2, and iterates until convergence. The local density inhomogeneity around a solute particle can be described by the number of solvent molecules that are in the immediate neighborhood of the solute particle. We have determined the number of solvent molecules in the first shell surrounding the solute particle as a function of the reduced solvent density Fb1 σ113 at different reduced temperatures, solvent size relative to solute size, and for different solvent-solute
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Figure 1. Accumulation number N as a function of solvent bath bulk density Fb1σ113 for Xe dissolved in Ne at Tred ) 1.02. Symbols are for simulation data,17 while the line is for the present prediction.
interaction amplitude and interaction range. The first neighbors are defined as those solvent molecules whose centers are in the shell existing between the minimum distance of solvent-solute approach, σ12 ) (σ1 + σ2)/2 and R(1) ) σ12 + σ1/2. Then the total number N of solvent molecules in the first shell is given by
N ) 4π
∫σR
(1)
12
F21(r)r2dr
(9)
Before investigating effect of the temperature, solute-solvent interaction strength, solute-solvent interaction range, solute size on the solvent local density enhancement, one first has to judge the validity of the present DFT approach to account for local inhomogeneity by comparing its predictions with those from somulation study for a given model systems, here a binary Lennard-Jones system with solute component at infinite dilution limit and a Lennard-Jones plus hard core attractive Yukawa system with the Yukawa particle at infinite dilution limit. The solvent bath temperature is denoted by kT/11 or Tred t kT/(1.31911), the solute-solvent size ratio by q ) σ22/σ11, the solute-solvent interaction strength ratio 22/11. A LorentzBerthelot combining rule is used to calculate the molecular parameters characterizing cross-interctions. In Figure 1, the DFT prediction for a set of parameters of the binary Lennard-Jones system is presented with the corresponding Monte Carlo molecular simulation.17 It is found that the theoretical prediction for this global quantity is in very good agreement with accurate simulation data. The parameters’ values producing the curve in Figure 1 correspond roughly to the molecular parameters characterizing Xe dissolved at infinite dilution limit in Ne. It is pointed out in ref 9 that the predictions obtained by the homogeneous PY equation exhibit a larger excess of the solvent molecules for the range of Fb1σ113 that is close to the critical density (for a homogeneous fluid of spherical Lennard-Jones molecules the critical temperature and critical density are determined to be kTc/11 ) 1.319 ( 0.003 and Fc1σ113 ) 0.283 ( 0.003 respectively with the PY calculational procedure18), the results by the homogeneous PY equation even display a maximum that is lager than that at higher fluid density, a physically meaningless result.9 Although the present DFT approach uses the PY approximation-based second-order DCF c11(r; Fb1, T/) for bulk solvent as input, its prediction for density profile is far better than that based on the homogeneous PY equation. We understand this seemingly “strange” observation by noting an inherent structure of the present DFT recipe,
Figure 2. Bulk second-order DCF c11(r; Fb1, T/) for several bulk densities at temperature Tred ) 1.02.
which is based on a partitioned DFT,14 and a recently proposed adjustable parameter-free version of the LTDFA.15b Success of the partitioned DFT depends on two conditions: one is that the tail part c11tail(r; Fb1, T/) is not dependent on the density argument or is only very weakly dependent on the density argument, the other is that a good DF approximation for the hard core part exists. The dependence of c11tail(r; Fb1, T/) on the density argument is plotted in Figure 2 for the case of Tred ) 1.02; it can be seen clearly that for the whole density range, the dependence is very small. As discussed in refs 15b and 19a, the parameter-free version of the LTDFA can perform well for hard sphere fluid or hard core part of the nonhard sphere fluid. As shown in ref 15b, with the PY hard sphere second-order DCF as input, the parameter-free version of LTDFA can work well even for bulk density high up to 0.9135 where the PY hard sphere second-order DCF becomes less accurate. I.e., a less accurate input can preduce a higher accurate output by the parameter-free version of the LTDFA. We think that it is the inherent structure of the LTDFA that underlies the abovementioned favorable phenomena. By incorporating a parameter λ into eqs 2 and 3, a negative feedback mechanism is introduced into the calculational device, which enables the calculational device to hold a mechanism that can restrain the deviation from the actual density distribution. In addition, even for calculation of the density distribution for one state point, information on the second order DCF of many state points is made use of; employment of the global structural information can help the negative feedback mechanism effectively work. Therefore, the final result is that even with a less accurate second order DCF as input, the parameter-free version of the LTDFA can produce out higher accurate result, this is excatly the case of Figure 1. In ref 19b, the negative feedback mechanism of the LTDFA is investigated in detail and confirmed. With an accurate and operational DFT calculational recipe at hand, we will investigate the effect of the solvent bath temperature, solvent-solute interaction range, solvent-solute interaction magnitude, the solute size on the local solvent density inhomogeneity around a solute particle and the curvature of N vs Fb1σ113. From now on, the solvent-solute interaction potential is expressed as follows
[
r < σ12 βu21(r) ) ∝ ) -β12 exp[-κ/(r - σ12)/σ11]/((r - σ12)/σ11 + 1) r > σ12
]
(10)
To furthermore test the accuracy of the present DFT approach for solvent-solute potential of type eq 10. We displayed the
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J. Phys. Chem. B, Vol. 109, No. 15, 2005 7525
Figure 3. Solvent-solute radial distribution function profile for a Yukawa solute particle in Lennard-Jones solvent bath. Symbols are for simulation data,20 while the line is for the present prediction.
calculated solvent-solute radial distribution function g21(r) for near critical state point and corresponding Monte Carlo simulation results20 in Figure 3. One can see that the present DFT approach performs almost the same well as a previous inhomogeneous version of the OZ IE (See Figures 1 and 2 in ref 20). In Figure 4, we compare the effect of solvent-solute interaction range on the curvature of N vs Fb1σ113 for two different solute size q ) 1.44 and 10 respectively, other parameters are shown in the figures. One can see clearly that appearance of the plateau, i.e., the region where the solute’s properties have a very weak density dependence, depends on closely the solute size and the solvent-solute interaction range. The longer the interaction range is, the more obvious the plateau will become, and the large solute size will help the apperance of the plateau. This clearly indicates that the attractive solventsolute interaction is a necessary condition for the appearance of the plateau. When the solvent-solute interaction range becomes longer, the coupling between the solvent-solvent interaction and the solvent-solute interaction also becomes more significant. A mathematical manifestation of the coupling is the two convolution integrals appearing in eq 2
∫dr1 (F21(r1) - Fb1)c11hc(|r - r1|; F˜ hc((r + r1)/2, λ), T*)
(11)
∫dr1 (F21(r1) - Fb1)C11tail(|r - r1|; Fb1, T*)
(12)
When the attractive solvent-solute interaction range becomes longer, βu21(r) will disappears at longer distance away from the solute. If the solvent bulk density is zero, then F21(r) will start to tend to Fb1 at the distance where βu21(r) starts to disappear. However, when the solvent bulk density is not zero, the distance where the F21(r) will start to tend to Fb1 will
Figure 4. Effect of solvent-solute interaction range κ/ on the accumulation number N as a function of solvent bath bulk density Fb1σ113. Other parameters are indicated in the figure.
become longer than the distance where βu21(r) starts to disappear. It is the convolution integrals eqs 11 and 12 that extends the distance where the F21(r) will start to tend to Fb1. The prolonged distance is obviously longer than the range of the second-order DCF. According to the theory of critical phenomena,21 the correlation length is dependent on how the state is close to the critical point. The more close to the critical point the state is, the longer the correlation length is, so when the state is situated at the critical point, the correlation length will become infinitely long. Although the PY approximation cannot accurately describe the above correspondence relation, the tendency that the state that is more close to the critical point has longer correlation length can be captured by the PY approximation. Thus, the appearance of the plateau can easily be explained as follows. For a fixed temperature, when the solvent bulk density approaches the critical density from the lower side, due to the increasing of the correlation length, more particles can be aggregated near the solute, and thus the slope increases. When the solvent bath state deviates from the critical density, and inreases beyond the critical density, the correlation length decreases, and thus the aggregation is not so intense. The resulting consequence is that the slope increases slower (the case of Figure 1 and Figure 4a), or even stops increasing; therefore, the plateau appears as shown in Figure 4b. Obviously, from the mathematical manifestation of the coupling of the solvent-solute interaction and the solvent correlation effect, the coupling can be strengthened by the whole solvent-solute interaction strength, the closeness to the critical point. With this conclusion in mind, the other findings about the affecting factors
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Figure 5. Same as in Figure 4, but for the effect of the solvent-solute size ratio.
Figure 6. Same as in Figure 4, but for the effect of the solventsolute interaction strength.
Figure 7. Same as in Figure 4, but for the effect of the solvent bath temperature Tred.
of the appearance of the plateau can be understood easily. With same solvent-solute potential parameters except for the solute size, the larger solute size corresponds to stronger solventsolute interaction strength, because larger surface gives a stronger confinment to the solvent molecules, this explains the difference between Figure 4, parts a and b, and Figure 5, which shows that the larger size helps the plateau appear more easily. In Figure 6, the effect of the reduced contact potential on the appearance of the plateau is presented, we see that the plateau becomes more obvious when the reduced contact potential becomes more negative. For the case of the most negative example (β12 ) 4), a maximum even appears around the critical density. For the case of weak solvent-solute interaction potential as shown in Figure 1, the appearance of the maximum predicted by the homogeneous PY in eq 9 is regarded as a physically meaningless result resulting from the unsuitability of the theory employed. However, for the case of β12 ) 4 in
Figure 6, where the solvent-solute interaction becomes so strong that the coupling between the solvent-solute interaction and solvent correlation effect also becomes very significant, it is not unlikely for the maximum to appear. In Figure 7, where the effect of the temperture on the appearance of the plateau is presented, one can see that, as the closeness to the critical temperature, the change of N as a function of solvent bulk density Fb1σ113 becomes more moderate. Around the critical density, the slope becomes very near to zero. However, when the temperature deviates from the critical temperature, the plateau becomes very ambiguous. A large deviation from the critical temperature leads to a reduction of the correlation length, thus the coupling between the solvent-solute interaction and the solvent correlation effect is weakened largely, the resultant consequence is the above observation. In ref 20, an investigation of effect of the solvent-solute interaction potential parameters is performed only in a small range, it is concluded that the
Local Solvent Density Augmentation
Figure 8. Effect of the solvent-solute interaction range κ/ and solvent bath temperature Tred on the accumulation number N as a function of solvent bulk density Fb1σ113. Other parameters are indicated in the figure.
appearance of the maximum is spurious, physically meaningless, and only a result due to the incorrectness of the homogeneous OZ-PY theory. In the framework of inhomogeneous version of the homogeneous OZ-PY theory, the maximum never appears. In Figure 8, we plot the dependence of the appearance of the maximum of the function N - Fb1σ113 on the solvent-solute interaction potential and the closeness to the critical temperature, which clearly shows that the maximum can happen only if the negative solvent-solute interaction potential becomes stronger (i.e., small κ/ and large β12), and the solvent bath temperature is near enough to the critical temperature. Therefore, the maximum phenomena is physically meaningful: the conclusion from ref 20 is due to a nonextensive investigation of the parameter’s effects and therefore is incorrect. Although the present DFT approach only achieves the same accuracy as that from the inhomogeneous OZ-PY theory.20 We think that the DFT approach is superior to the inhomogeneous OZ integral equation theory in several aspects. Local solvent density augmentation is actually a nonuniform phenomena originating from an external field due to the solute particle. It is well-known that the theoretical tool appropriate for nonuniform phenomena is the DFT. Homogeneous OZ IE theory is not successful when applied to fluid system under influence of external field. The inhomogeneous version of the OZ IE theory, although successful for the situation investigated,20 where the size asymmetry is modest, has never been tested for high size asymmetry between solute and solvent particle, high solventsolute interaction strength, etc. Since OZ IE theory is not as successful for high size asymmetry as for modest size asymmetrical system, one cannot certainly expect that the inhomogeneous version of the OZ IE theory works the same well for high size asymmetry case as for modest size asymmetry case.
J. Phys. Chem. B, Vol. 109, No. 15, 2005 7527 However a DFT approach will work the same well for different external fields only if it performs well for some external field. This fact is verified by countless DFT calculation experience where the validity of a DFT approach is usually confirmed by comparing its prediction for the density disrtribution profile near a hard wall with that from the corresponding simulation results. It is well-known that when the solute size increases up to some value, for example q ) 20, the external potential due to a solute will become very similar to that due to a wall. Another advantage of the DFT approach over the inhomogeneous version of the OZ IE is that the OZ IE theory is not very suitable for estimation of the global thermodynamic properties. It is exactly the global thermodynamic properties such as the solvation free energy (or the excess chemical potential, cavity formation energy)26 which is of the same importance as the local density argumentation. The DFT approach is especially suitable for estimation of the global thermodynamic properties, this can be seen out clearly from the theoretical structure itself which is exactly based on a functional relationship beween the density distribution and the excess Helmholtz free energy, or the functional relationship beween the density distribution and functional derivative of the excess Helmholtz free energy with respetive to the density distribution. For latter case, as shown in refs 13 and 15b, a functional integration method can be employed to obtain the nonuniform excess Helmholtz free energy from the inhomogeneous density distribution. The third advantage of the DFT approach over the inhomogeneous version of the OZ IE is the complexity of the theoretical structure of inhomogeneous version of the OZ IE theory, which not only makes the numerical solution of the latter equaions computationally intensive, but also enables the physical origin obscure of the observed phenomena. The present DFT recipe expresses mathematically the coupling between the solventsolute interaction and the solvent correlation effect by convolution integrals, from which the observed phenomena can be qualitatively concluded. III. Concluding Remarks The present paper proposed a DFT approach for investigation on the solvent local density augmentation around a solute in supercritical fluid. The validity of the philosophy for construction of the present DFT approach is established in refs 14 and 19a, where the same philosophy is employed for construction of DFT approach for a nonuniform hard core attractive Yukawa (HCAY) fluid.14b,19a The resultant DFT approach can work quitely well for prediction of density distribution of HCAY fluid under various external potentials. Reliability of the present DFT approach for nonuniform LJ fluid was tested by comparing the DFT’s predictions with simulation data for the number of solvent molecules in the first shell surrounding the solute particle and the solvent-solute radial distribution function. The present DFT’s approach is based on a combination of a recently proposed partitioned DF approximation14 and an adjustable parameter-free version of a LTDFA for hard sphere fluid.15 Even with the PY approximation-based second-order DCF as input, the present DFT approach works well for density distribution; the success is thought to be due to an inherent negative feedback mechanism originated from a density field as the argument of the second-order DCF incorporated by the Lagrangian theorem. A detailed investigation about the negative feedback mechanism of the LTDFA will be reported in a forthcoming paper.19b Investigation is carried out for the effect of the solvent-solute interaction detail, solute size, solvent bath properties on the accumulation number of solvent molecules in the first shell
7528 J. Phys. Chem. B, Vol. 109, No. 15, 2005 surrounding the solute particle. It is found that the slope of accumulation number N as a function of solvent bulk density Fb1σ113 will become near zero when the solvent bath bulk density approaches the critical density. i.e., a plateau will appear around the critical density. We point out that existence of the plateau results physically from the coupling between the solvent-solute interaction and solvent correlation effect, as the coupling mathematically originates in the convolution integral appearing in the density profile equation of the DFT formalism. Obviously, the strength of the coupling depends on solvent correlation length, and the whole solvent-solute interaction strength. The longer or stronger the solvent correlation length or the whole solvent-solute interaction strength is, respectively, the more intense the coupling will become. The more intense coupling leads to the more easy appearance of the plateau of the N vs Fb1σ113. The accumulation of the solvent molecules around a solute molecule has implication for potential of mean force between the two solute molecules immersed in a solvent bath. In analogy with the depletion attraction, gathering of the solvent molecules between the two solute molecules will result in repulsive potential of mean force. In recent papers,22 the present author confirmed theoretically the gathering repusion phenomena by proposing a universal framework based on the solvent density distribution around a single solute. The density distribution of the solvent particles around a solute molecule can present some interesting phenomena, such as first- and second-order wetting transition,23 drying transition.24 Such continuous or discontinuous transition of density distribution and the near critical phenomena investigated in the present paper will certainly be reflected in the behavior of repulsive or attractive potential of mean force. Depletion attraction and gathering repulsion may be a main mechanism as an explanation of the hydrophobic attraction and hydration repulsion,25 two important biologically relevant effective interactions responsible for the determination of the three-dimensional configuration of the biomolecules, the formation of biological membrane, and micelle. All of the above phenomena are being investigated theoretically with the universal framework22 and the present partitioned DFT approach. Acknowledgment. Project supported by National Natural Science Foundation of China (Grant No. 20206033) References and Notes (1) Supercritical Fluids. Fundamentals for Applications; Kiran, E. D.; Peters, C. G., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 1990; Vol E366. (2) Golestanian, R.; Liverpool, T. B. Phys. ReV. E 2002, 66, 051802. Bagatella-Flores, N.; Gonzalez-Mozuelos, P. J. Chem. Phys. 2002, 117,
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