Hard-sphere exclusion volume

The first term in Eq. [429] is the hard-sphere exclusion volume term, which decreases the counterion concentration at the surface the second and third terms respectively represent the fluctuation potential and increase the surface concentration of ions. These terms can be included as an activity coefficient in a general-purpose algorithm through the exponential term in Eq. [378]. For ion distances Ar closer to the surface than three ion radii, Bratko and Vlachy multiply the ion concentrations in Eqs. [430] by an excluded volume correction factor B r) = (Ar- - a)j4a for distances Ar < a, ion concentrations are, of course, zero because of hard-sphere exclusion. 

The total exclusion chromatogram provides the means to obtain the e values and this was found to be 0.423. It is interesting to compare this value with that reported ( ) for the interstitial volume of randomly packed rigid spheres which is 0.364. We assume that our value deviates from the hard sphere value primarily because of the inefficient packing of particles in the case of the column used in this work varied substantially in size (35 -75 p). 

The volume exclusion effect has received recently renewed attention.5 9 Paunov et al.5 argued that the excluded volume should be taken as eight times the real volume of the ions (this corresponds to the first-order correction in the virial expansion for a hard-sphere fluid)... 

When the role of hard spheres, like those depicted in Figure 5.26, is played by the molecules of solvent, the resulting volume exclusion force is called the oscillatory solvation force, or sometimes when the solvent is water, the oscillatory hydration force. The latter should be distinguished from the monotonic hydration force, which has a different physical origin and is considered separately in Section 5.4.5.4 below. 

Exact results derived for the three-dimensional hard sphere system can often be immediately transcribed to the two- or the one-dimensional hard sphere system from purely dimensional considerations. To do this we must correctly identify in the formulas in which they occur the volume V of the container, the volume element dr, and the density p = N/V in three dimensions with the area A of the container, the areal element dr, and the density p = N/A in two dimensions and the total lineal length L of the container, twice the lineal element 2dr, and the density p = NjL in one dimension. Thus, e.g., the analogs of (26) are found as soon as the three-dimensional mutual exclusion volume fn-fl is transcribed, leading to... 

The Pratt-Chandler theory has been extended to consider complex molecules. For example, the hard-sphere model of -butane may have an excluded volume Av(f, X), which is a function of the torsion angle (j) and depends on the exclusion radius X of the methylene spheres. Then the part of the PMF (the potential of mean force) arising from the solute-solvent interaction can be related to the reversible work required to create a cavity with the shape and excluded volume Av((/>, X) of the -butane molecule. 

The HNC is the most accurate theory for bulk electrolytes. One would expect that this fact would remain true in the plane electrode limit. However, because of the inaccuracy of the HNC for uncharged hard sphere fluids the HNCl does no do well in representing the exclusion volume of the ions, and is not on the whole, such a good approximation for the electric double layer. The bulk direct correlation function... 

The mechanical properties of suspensions containing a narrow size distribution of particles have been studied extensively because they offer the best chances for testing models for flow behavior. The most detailed studies can be found for hard spheres where particles experience only volume exclusion, thermal and hydrodynamic interactions. Based on the models developed for these systems, a great deal can be learned about the behavior of suspensions experiencing longer range repulsions and attractions. 

Repulsive interactions are important when molecules are close to each other. They result from the overlap of electrons when atoms approach one another. As molecules move very close to each other the potential energy rises steeply, due partly to repulsive interactions between electrons, but also due to forces with a quantum mechanical origin in the Pauli exclusion principle. Repulsive interactions effectively correspond to steric or excluded volume interactions. Because a molecule cannot come into contact with other molecules, it effectively excludes volume to these other molecules. The simplest model for an excluded volume interaction is the hard sphere model. The hard sphere model has direct application to one class of soft materials, namely sterically stabilized colloidal dispersions. These are described in Section 3.6. It is also used as a reference system for modelling the behaviour of simple fluids. The hard sphere potential, V(r), has a particularly simple form ... 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

Figure 2.3 Left panel collision of hard spheres. Upper hard spheres of equal radius d/2 centered at A and B. The excluded volume (dashed), drawn centered at B, is of radius d. Lower hard spheres of unequal diameters, di and dj. The "equivalent" exclusion sphere is of radius d = (di +d2)/2. The hard-sphere potential as a function of the center-to-center separation is shown in the right panel.

Armed with the hard-sphere model we can make the definition of a cross-section more transparent. Imagine that an exclusion sphere is centered around each beam molecule. Corresponding to this sphere is a circle of radius in the plane perpendicular to the beam velocity. Thus the beam molecule sweeps out a cylinder of volume nd Ax as it moves a distance Ax through the target gas. If the center of a target molecule lies within that volume a collision will occur and the beam molecule will be deflected off the x axis and therefore lost to the detector. From the volume of the cylinder swept we see that for the hard-sphere model... 

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