Solute-Boundary Interactions

When a solute diffuses through small pores, its speed may be affected by the size and the chemistry of the pores. For example, a solute will diffuse faster through a large straight pore than through a small crooked one. It may diffuse differently if it adsorbs on the pore s wall and then scoots along the wall at a faster rate than it moves in the bulk. 

I have tried to force an organization on these examples as follows. In Section 6.4.1, I have discussed the simplest empirical methods of organizing experimental results. In Section 6.4.2, I have reviewed theories for solute diffusion in a solvent trapped within cylindrical pores in an impermeable solid. In this case, solute-solvent interactions still control diffusion and the solid only imposes boundary conditions. Cases where the interactions are between the diffusion solute and the pores boundaries are covered in Section 6.4.3. Finally, cases not of cylindrical pores but of other composite structures are described in Section 6.4.4. 

The effects of longer pores and smaller areas are often lumped together in the definition of a new, effective diffusion coefficient efr 

We next turn to diffusion of a solution held within an array of cylindrical pores. Normally, these large pores are assumed to span a thin film, and to all be perpendicular to the surfaces of the film. By large pores, we imply that the solvent acts as a continuum and that the solute diameter is much smaller than the pore diameter. Not surprisingly, this idealized geometry has been the focus of considerable theoretical effort. In spite of its idealizations, it does provide physical insight. 

We first consider results for a gas. For large pores, gas transport through the pores will be described by the Hagen-Poiseville law 

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In the last section of this chapter, we summarize diffusion affected by solute-boundary interactions, which is the third important group of interactions. Solute-boundary interactions occur in porous solids with fluid-filled pores. They include such diverse phenomena as Knudsen diffusion, capillary condensation, and molecular sieving. Because these phenomena promise high selectivity for separations, they are an active area for research. They and the other interactions illustrate the chemical factors that can be hidden in the diffusion coefficients which are determined by experiment. 

Continuum models remove the difficulties associated with the statistical sampling of phase space, but they do so at the cost of losing molecular-level detail. In most continuum models, dynamical properties associated with the solvent and with solute-solvent interactions are replaced by equilibrium averages. Furthermore, the choice of where the primary subsystem ends and the dielectric continuum begins , i.e., the boundary and the shape of the cavity containing the primary subsystem, is ambiguous (since such a boundary is intrinsically nonphysical). Typically this boundary is placed on some sort of van der Waals envelope of either the solute or the solute plus a few key solvent molecules. 

Much like the RISM method, the LD approach is intermediate between a continuum model and an explicit model. In the limit of an infinite dipole density, the uniform continuum model is recovered, but with a density equivalent to, say, the density of water molecules in liquid water, some character of the explicit solvent is present as well, since the magnitude of the dipoles and their polarizability are chosen to mimic the particular solvent (Papazyan and Warshel 1997). Since the QM/MM interaction in this case is purely electrostatic, other non-bonded interaction terms must be included in order to compute, say, solvation free energies. When the same surface-tension approach as that used in many continuum models is adopted (Section 11.3.2), the resulting solvation free energies are as accurate as those from pure continuum models (Florian and Warshel 1997). Unlike atomistic models, however, the use of a fixed grid does not permit any real information about solvent structure to be obtained, and indeed the fixed grid introduces issues of how best to place the solute into the grid, where to draw the solute boundary, etc. These latter limitations have curtailed the application of the LD model. 

There are currently three different approaches for carrying out ASC-PCM calculations [1,3]. In the original method, called dielectric D-PCM [18], the magnitude of the point charges is determined on the basis of the dielectric constant of the solvent. The second approach is C-PCM by Cossi and Barone [24], in which the surrounding medium is modelled as a conductor instead of a dielectric. The third, IEF-PCM method (Integral Equation Formalism) by Cances et al the most recently developed [16], uses a molecular-shaped cavity to define the boundary between solute and dielectric solvent. We have to mention also the COSMO method (COnductorlike Screening MOdel), a modification of the C-PCM method by Klamt and coworkers [26-28], In the latter part of the review we will restrict our discussion to the methods that actually are used to model solute-solvent interactions in NMR spectroscopy. 

Clearly, the present approach precludes the detailed study of the dynamics of explicit solute-solvent interactions because the solvent (bath) degrees of freedom have been eliminated. Also, as in the stochastic boundary model,... 

Picosecond laser spectroscopy offers direct access to molecular vibrational (T,) and phase (Tj) relaxation as well as orientational dynamics of molecules, t,< . In contrast to experiments on vibrational relaxation in liquids, all those on the reorientational process have been confined to large polyatomic molecules, particularly dye molecules in probe solvents for pragmatic reasons. Since the slip boundary conditions are also a sensitive function of the shape of the molecules and solute-solvent interactions, there is some uncertainty in deciding whether or not it is the local interaction in terms of the solvation volume, or the boundary condition, or both, that varies for a given molecule in a range of liquids such as the... 

To obtain Eqs. 5-10, it was assumed that the concentration of solute within the adsorption boundary layer is related to the solute-surface interaction energy by a Boltzmann distribution. The essence of the thin-layer polarization approach is that a thin diffuse layer can still transport a significant amoimt of solute molecules so as to affect the solute transport outside the diffuse layer. For a strongly adsorbing solute (e.g., a surfactant), the dimensionless relaxation parameter fila (or Kid) can be much greater than imity. If all the adsorbed solute were stuck to the surface of the particle (the diffuseness of the adsorption layer disappears), then L = 0 and there would be no diffusiophoretic migration of the particle. In the limit of [l/a 0 (very weak adsorption), the polarization of the diffuse solute in the interfacial layer vanishes and Eq. 5 reduces to Eq. 1. 

But, is a continuum representation of solvent always sufficient May we neglect specific solute-solvent interactions such as H-bonds And what about reactions in which the solvent plays the role of a reactant Furthermore, do species in solution interact, e.g., a charged catalyst with its counterion, and how would this affect the catalyst s reactivity To address these and similar questions, we first have to extend the model systems considered to include an explicit representation of solvent and of those molecular species which may have an impact on reactivity. A first step in this direction is the use of cluster-continuum models in which a reduced number of explicit solvent molecules are introduced in the model. This approach has been successfully applied in computational studies of the organometallic reactivity [12-14], yet it suffers from some limitations [15]. How many solvent molecules should be explicitly included Is the first solvation sphere enough In order to mimic bulk conditicHis, models have to include enough solvent molecules to fully solvate the solute. These models, which are built to reproduce experimental densities, are generally treated as periodically repeating units in order to remove the explicit/ continuum (or vacuum) boundary. 

In the PCM the solvent is described as a homogeneous dielectric which is polarized by the solute. The latter is placed within a cavity in the solvent medium (built as the envelope of spheres centered on the solute atoms) and the proper electrostatic problem at the cavity surface is solved using a boundary element approach [78]. In the PCM framework, the solvent loses its molecularity and, especially in hydrogen-bonding solvents, the explicit inclusion of solute-solvent interactions is very important for getting accurate results. In these cases, as discussed in detail in the next section, mixed discrete/continuum models, where a limited number of solvent molecules are included in the computational model, usually provide accurate results. 

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