Hard spheres effective diameter

In this section we restrict ourselves to solvent effects that are due to the first term in the expansion of AG in Eq. (9.4.2). This is equivalent to the assumption that all the particles involved are hard particles, hence only their sizes affect the solvation Gibbs energies. We shall also assume for simplicity that the solvent molecules are hard spheres with diameter a. All other molecules may have any other geometrical shape. 

To extend the usefulness of the model to permit a description of chemical reactions, we must introduce another parameter, the effective duration of a collision. The rectangular well or central force models do this automatically by permitting molecular interaction over a range of distances. However, they are both more complex than the hard sphere model. We can rescue the hard sphere model by specifying a parameter era, the effective diameter for chemical interaction, while keeping hard sphere core diameter. When the centers of two identical molecules are a distance effective reaction volume is 7r([Pg.155]

It should be noted that Equations 5.216 and 5.219 refer to hard spheres of diameter d. In practice, however, the interparticle potential can be soft because of the action of some long-range forces. If such is the case, we can obtain an estimation of the structural force by introducing an effective hard-core diameter " ... 

As a simple example of the additive version of the RG technique we consider a suspension of hard spheres dispersed in a continuous, Newtonian solvent. The effective viscosity of this composite system is a function of the solvent viscosity, rj, the number of hard spheres, N, and a couphng parameter, g = na l() V, defined as the ratio of the volume of a hard sphere of diameter a to the total volume of the suspension. This suspension can be treated as an effectively continuous fluid with a viscosity g = rj N g,rj ) that approaches the value rjo as the volume fraction of spheres, (j) = Ng, tends to zero. 

The meaning of as a volume density is clear for hard spheres of diameter a. For real molecules, this quantity measures the effective volume density of the system. 

Excluded Volume of a Sphere The excluded volume makes the real chains nonideal. The dimension of the real chain is different from that of the ideal chain of the same contour length, for instance. Before considering the excluded volume effect in a chain molecule, we look at the effect in a suspension of hard spheres of diameter d. In Figure 1.33, the center-to-center distance between spheres A and B cannot be less than d. In effect, sphere B is excluded by sphere A. The space not available to the center of sphere B is a sphere of radius d indicated by a dashed line. Thus the excluded volume (Ve) is eight times the volume of the sphere. 

Figure 6. Pore dlfifuslvlty versus pore width. Theory Is for 6-oo LJ fluid with an effective hard sphere diameter cTgff = 0.972. Units of dlfifuslvlty are (3a/8)...

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

AB effective hard sphere diameter for i in a CSTR network... 

We also adopt the above combination rule (Eq. [6]) for the general case of exp-6 mixtures that include polar species. Moreover, in this case, we calculate the polar free energy contribution Afj using the effective hard sphere diameter creff of the variational theory. 

Figure 3.20 The effective hard sphere diameter, r0, calculated from Equation (3.65) for 100 nm radius particles with ( = 50 mV...

The effective hard sphere diameter may be used to estimate the excluded volume of the particles, and hence the low shear limiting viscosity by modifying Equation (3.56). The liquid/solid transition of these charged particles will occur at... 

The term pair potential that contains only the attractive potential, because the repulsion effects have been allowed for by the effective volume fraction and hard sphere diameter. The new potential can be defined as... 

No real system is fully random. Random systems are over-simplified ideal models similar to those of strictly regular structures. Most relevant is the effect of the finite volume of the monomer units which implies that two units can approach each other only up to their diameter. Thus a certain volume is forbidden or excluded for the individual repeating units. For hard sphere monomers this excluded volume is just eight times the monomer volume. This excluded volume... 

Regarding what precedes, it is clear that one of the challenges of the liquid-state theory is to ascribe an effective hard sphere diameter aHS to the real molecule. As stated in the literature [56-59], a number of prescriptions for aHS exist through empirical equations. Among them, Verlet and Weiss [56] proposed... 

Here R = rjd where d is the diameter of the particle and Rm is a reduced effective hard-sphere diameter, chosen such that exp(—u Rm)/kT) is negligible (less than 10-6). The particles cannot come closer than Rm because of the large repulsion at such separations. 

For the potentials for which the repulsive part is not infinitely steep, the reference potential can be reduced to a system of hard spheres by introducing an effective diameter, the Barker—Henderson diameter. This... 

From the concept of a root-mean-square speed we can estimate the collision frequency Z between successive elastic collisions between molecules in a gas and the mean free path X. We assume an effective diameter d of two molecules (assumed to be hard spheres, so that each molecule will collide with another within an area nd1) the collision frequency z is given by... 

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