Hard Sphere Diameter Effects

FIGURE 4.8. The structuring of liquid molecules near a surface (a) leads to an ordering of the trapped molecules to produce a regular, damped molecular density distribution as a function of molecular diameter as one moves away from the surface. The structuring near a surface may result in changes in the properties of the liquid near the surface (b). 

Interaction energy (as a function of liquid molecular diameter) 

FIGURE 4.9. A liquid trapped between two closely spaced surfaces can produce a highly ordered structure leading to possibly interesting changes in the characteristics of the liquid. 

Although the impact of this molecular-level effect is quite small (negligible in most cases), it can, under some circumstances, produce an appreciable energetic effect—repulsive in the case of two approaching surfaces—that is, it will be difficult to displace the last molecular layers separating the surfaces—or attractive in the separation of two contacting (adhesive) surfaces. These topics will appear again in Chapters 10 (on colloids and colloidal stability), 15 on association colloids micelles, vesicles and membranes), and 19 on adhesion). 

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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... 

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. 

The direct calculation of the collective contribution DJDs to the self-diffusion coefficient is complicated by the inadequate temperature dependence of the shear viscosity in ref. [ ]. Indeed, it is easy to verify that the ratio r / r g for the model argon increases with temperature on isochors. From the physical viewpoint, this result is inadequate. It is worth noting that for ( ) < 0.4 the values of r from ref. f ] and those determined on the basis of the Enskog theory for hard spheres diameter of which coincides with the effective diameter... 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

A general method of predicting the effective molecular diameters and the thermodynamic properties for fluid mix-tures based on the hard-sphere expansion conformal solution theory is developed. The method of Verlet and Weis produces effective hard-sphere diameters for use with this method for those fluids whose intermolecular potentials are known. For fluids with unknown potentials, a new method has been developed for obtaining the effective diameters from isochoric behavior of pure fluids. These methods have been extended to polar fluids by adding a new polar excess function, to account for polar contributions in a mixture. A new set of pseudo parameters has been developed for this purpose. The calculation of thermodynamic properties for several fluid mixtures including CH —C02 has been carried out successfully. 

A theoretical basis for the computation of effective hard-sphere diameters is now well developed for the perturbation theory. Its numerical evaluation requires exact knowledge of the intermolecular potential so that the method is not immediately applicable to real molecules for which this potential is usually unknown. Despite this difficulty, the principles involved are important in designing a procedure for real molecules with the HSE theory. 

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