Proximal radial distribution functions

Figure 1.9 Carbon- and water-oxygen interfacial densities as a function of z. The dashed and solid lines indicate the observed carbon and oxygen densities, respectively, at 300 K determined from molecular simulation. The disks plot the water-oxygen densities reconstmcted from the proximal radial distribution function for carbon-oxygen (see Fig. 1.2), averaged over alkyl chain conformations sampled by the molecular simulation. The interfacial mid-point (z = 0) is set at the point where the alkyl carbon- and water-oxygen densities are equal. See Figs. 1.1 and 1.2, p. 7.

Consider now the mean oxygen density conditional on a specific alkyl configuration. Since that conditional mean oxygen density is less traditionally analyzed than the density profile shown in Fig. 1.9, we exploit another characterization tool, the proximal radial distribution (Ashbaugh and Paulaitis, 2001). Consider the volume that is the union of the volumes of spheres of radius r centered on each carbon atom see Fig. 1.10. The surface of that volume that is closer to atom i than to any other carbon atom has area fl, (r) with 0 < ff, (r) < 4tt. The proximal radial distribution function ( ) is defined as... 

The proximal radial distribution functions for carbon-oxygen and carbon-(water)hydrogen in the example are shown in Fig. 1.11. The proximal radial distribution function for carbon-oxygen is significantly more structured than the interfacial profile (Fig. 1.9), showing a maximum value of 2. This proximal radial distribution function agrees closely with the carbon-oxygen radial distribution function for methane in water, determined from simulation of a solitary methane molecule in water. While more structured than expected from the... 

Figure 1.10 Geometrical quantities in defining the proximal radial distribution function gp j(r) of Eq. (1.14). The surface proximal to the outermost carbon (carbon /), with area fi, (r) r, permits definition of the mean oxygen density in the surface volume element, conditional on the chain configuration p gprox ( )-...

The relationship between (r) for water oxygen atoms (Fig. 1.11) and the oxygen atom interfacial density profile (Fig. 1.9) can be established by superposing these proximal radial distribution functions to model the conditional densities as... 

We conclude that the proximal radial distribution function (Fig. 1.11) provides an effective deblurring of this interfacial profile (Fig. 1.9), and the deblurred structure is similar to that structure known from small molecule hydration results. The subtle differences of the ( ) for carbon-(water)hydrogen exhibited in Fig. 1.11 suggest how the thermodynamic properties of this interface, fully addressed, can differ from those obtained by simple analogy from a small molecular solute like methane such distinctions should be kept in mind together to form a correct physical understanding of these systems. 

Figure 1.11 Carbon-water proximal and radial distribution functions at 300 K. The solid and dashed lines indicate the alkyl chain carbon-(water)oxygen and -(water)hydrogen proximal correlation functions, respectively, evaluated from simulations of grafted alkyl chains in contact with water. The dots indicate the methane-(water)oxygen and -(water)hydrogen radial distribution functions, respectively, evaluated from simulations of a single methane molecule in water.

This effect must not be confused with the cybotactic effects we have mentioned, nor with the hole in the solute-solvent correlation function gMs(t) (see Figure 8.5). The hole in the radial correlation function is a consequence of its definition, corresponding to a conditional property, namely that it gives the radial probability distribution of the solvent S, when the solute M is kept at the origin of the coordinate system. Cybotactic effects are related to changes in the correlation function gMs(t) (or better gMs(r> )) with respect to a reference situation. Surface proximity effects can be derived by the analysis of the gMs(r,fi) functions, or directly computed with continuum solvation methods. It must be remarked that the obtention of gMs(r) functions near the surface is more difficult than for bulk homogeneous liquids. Reliable descriptions of gMs(ri re even harder to reach. 

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