Hard-sphere solutes

As pointed out earlier, the contributions of the hard cores to the thennodynamic properties of the solution at high concentrations are not negligible. Using the CS equation of state, the osmotic coefficient of an uncharged hard sphere solute (in a continuum solvent) is given by... 

P. Attard. Spherically inhomogeneous fluids. II. Hard-sphere solute in a hard-sphere solvent. J Chem Phys 97 3083-3089, 1989. 

Fig. 6. Excess chemical potential of hard-sphere solutes in SPC water as a function of the exclusion radius d. The symbols are simulation results, compared with the IT prediction using the flat default model (solid line). (Hummer et al., 1998a)...

In this second example, we examine simple systems near the water-hexane interface. Specifically, we calculate the difference in the free energy of hydrating a hard-sphere solute of radius a, considered as the reference state, and a model solute consisting of a point dipole p located at the center of a cavity [11]. We derive the formula for A A assuming that the solute is located at a fixed distance z from the interface, and subsequently we examine the dependence of the free energy on z. The geometry of the system is shown in Fig. 2.3. 

One of the further refinements which seems desirable is to modify Eq. (9) so that it has wiggles (damped oscillatory behavior). Wiggles are expected in any realistic MM-level pair potential as a consequence of the molecular structure of the solvent (2,3,10,11,21,22) they would be found even for two hard sphere solute particles in a hard-sphere liquid or for two H2I80 solute molecules in ordinary liquid HpO, and are found in simulation studies of solutions based on BO-level models. In ionic solutions in a polar solvent another source of wiggles, evidenced in Fig. 2, may be associated with an oscillatory nonlocal dielectric function e(r). ( 36) These various studies may be used to guide the introduction of wiggles into Eq. (9) in a realistic way. 

The theory reflects the solvent properties through the thermody-namic/hydrodynamic input parameters obtained from the accurate equations of state for the two solvents. However, the theory employs a hard sphere solute-solvent direct correlation function (C12), which is a measure of the spatial distribution of the particles. Therefore, the agreement between theory and experiment does not depend on a solute-solvent spatial distribution determined by attractive solute-solvent interactions. In particular, it is not necessary to invoke local density augmentation (solute-solvent clustering) (31,112,113) in the vicinity of the critical point arising from significant attractive solute-solvent interactions to theoretically replicate the data. 

Figure 1.4 Excess chemical potential profiles (kcalmoF ) for a 2.0A hard-sphere solute along the z axis. The tethered chains are on the left. The three curves correspond to pure water (top), a 50/50 mixture (middle), and 90% methanol (bottom) by volume. Energies are in kcalmor Distances along z perpendicular to the interfacial plane are given in A.

Now consider an alternative calculation for a corresponding case of a hard-sphere solute. We start from... 

Figure 8.6 Excess chemical potentials of model hard-sphere solutes of sizes roughly comparable to Ne, Ar, methane (Me), and Xe as a function of temperature. The hard-sphere diameters used were 2.8 A, 3.1 A, 3.3 A, and 3.45 A, respectively. The lines indicate the information theory model results and the symbols are the values computed directly with typical error bars (Garde et al, 1996).

For hard-sphere solutes this entropy convergence point has a nontrivial size dependence that isn t apparent from Fig. 8.7 (Huang and Chandler, 2000 Ashbaugh and Pratt, 2004). Figure 8.9 gives a current estimate of those entropy... 

The quantity Gaa (or G ) is often referred to as representing the solute-solute interaction. In this book, we reserve the term interaction for the direct intermodular interaction operating between two particles. For instance, two hard-sphere solutes of diameter a do not interact with each other at a distance R> a, yet the solute-solute affinity conveyed by Gaa may be different from zero. Therefore, care must be exercised in identifying DI solutions as arising from the absence of solute-solute interactions. 

The second exact integral equation can be derived by considering the addition of a hard sphere solute of diameter J to a large excess of our fluid. It is an immediate consequence of the thermodynamic identity... 

In the Pratt-Chandler theory, the distortion of the liquid structure and the work done when two neutral hard-sphere solutes are brought from infinity to a certain distance gives a measure of the hydrophobic effect. 

This field of force is sometimes referred to as cavities. The term cavity is correct only for hard-sphere solutes. In general, the field of force includes both repulsive and attractive parts. 

Perhaps the simplest and the most convincing argument for justifying the approximation (4.4.12) is to recall that this approximation becomes exact for hard-sphere solutes. An exact statement for hard spheres is likely to be a reasonable approximation for simple non-polar solutes. 

In the following, we elaborate on two generalizations of (4.4.40), one for two real solutes and the second for any number of either real or hard-sphere solutes. Before doing that, we note that relation (4.4.40) means that the HrpO interaction at zero separation is equivalent to the reversal of the solvation process. This is one example of the exact relation between the H[Pg.463]

We now proceed to obtain two generalizations of (4.4.40). For the process of bringing m hard-sphere solutes from infinite separation to zero separation, we have... 

Note that for hard-sphere solutes, 2-s and s are indistinguishable. 

The first of such statistical pore models was proposed by Giddings et al. (31) in 1968. In this landmark study, general expressions were formulated that described the partitioning of hard sphere solutes in a random pore system, described as a porous network. Also unique to this study was an attempt to express SEC partitioning as a function of both complex pore and solute contri-... 

Early theories of hard sphere solute models, in chronological order of appearance, include the random-spheres pore model of Ogston (ref. 19), the... 

We now turn to another useful relation, between the quantity AAcb,y(Rq, pseudo-chemical potential of a hard-sphere solute. 

Note that since the hard-sphere solute is presumed to have no internal structure, its addition to the system does not introduce an internal partition function [compare with Eq. (3.89)]. The quantity U(R /Ro) stands for the total interaction energy between the N molecules in the configuration and the hard-sphere particle at Rq. More specifically. 

Fig. 3.4. A hard-sphere solute with diameter (Ths and a particle with an effective hard-core diameter Oa- The distance of closest approach is a = ((Ths +

For Rij > cTfl, the potential function can have any form. However, the interaction between a particle of the system and the hard-sphere solute is presumed to have the form... 

That is, a particle is not permitted to come closer than a to the center of the hard-sphere solute (see Fig. 3.4). The condition (3.165) can be rewritten in an equivalent form as... 

It is very important to realize that the last statement is valid only when we refer to a fixed position for both processes. The equivalence of the two processes is quite clear on intuitive grounds. Creation of a cavity means imposing a restriction on the centers of all particles, keeping them out of a certain region. This constraint is achieved simply by putting a hard-sphere solute at the position Rq, provided that we have properly chosen its diameter by relation (3.169). In other words, as far as the solvent (i.e., the particles of the system) is concerned, there is no difference if we create a cavity at Rq or put a hard-sphere solute there, provided that (3.169) is fulfilled. 

All of the above reasoning breaks down if we remove the condition of fixed position for both of the processes. We recall that the total work required to add a hard-sphere solute to the system (at T, F, N constant) is equal to the chemical potential of the solute, which can be written as (see Section 3.6)... 

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