Smooth hard-sphere theory

Smooth hard-sphere theory for monatomic fluids... 

The smooth hard-sphere theory discussed above has been remarkably successful for monatomic fluids, as exemplified by xenon (see Chapter 10). For application to polyatomic fluids, it is necessary to take into account additional considerations ... 

In view of the success of the methods based on hard-sphere theories for the accurate correlation and prediction of transport properties of single-component dense fluids, it is worthwhile to consider the application of the hard-sphere model to dense fluid mixtures. The methods of Enskog were extended to mixtures by Thome (see Chapman Cowling 1952). The binary diffusion coefficient >12 for a smooth hard-sphere system is given by... 

Dense fluid transport property data are successfully correlated by a scheme which is based on a consideration of smooth hard-sphere transport theory. For monatomic fluids, only one adjustable parameter, the close-packed volume, is required for a simultaneous fit of isothermal self-diffusion, viscosity and thermal conductivity data. This parameter decreases in value smoothly as the temperature is raised, as expected for real fluids. Diffusion and viscosity data for methane, a typical pseudo-spherical molecular fluid, are satisfactorily reproduced with one additional temperamre-independent parameter, the translational-rotational coupling factor, for each property. On the assumption that transport properties for dense nonspherical molecular fluids are also directly proportional to smooth hard-sphere values, self-diffusion, viscosity and thermal conductivity data for unbranched alkanes, aromatic hydrocarbons, alkan-l-ols, certain refrigerants and other simple fluids are very satisfactorily fitted. From the temperature and carbon number dependency of the characteristic volume and the carbon number dependency of the proportionality (roughness) factors, transport properties can be accurately predicted for other members of these homologous series, and for other conditions of temperature and density. Furthermore, by incorporating the modified Tait equation for density into... 

The wall-PRISM equation has been implemented for a number of hard-chain models including freely jointed [94] and semiflexible [96] tangent hard-sphere chains, freely rotating fused-hard-sphere chains [97], and united atom models of alkanes, isotactic polypropylene, polyisobutylene, and polydimethyl siloxane [95]. In all implementations to date, to my knowledge, the theory has been used exclusively for the stmcture of hard-sphere chains at smooth structureless hard walls. 

The choice of the weighting function w depends on the version of density functional theory used. For highly inhomogeneous confined fluids, a smoothed or nonlocal density approximation is introduced, in which the weighting function is chosen to give a good description of the hard sphere direct pair correlation function for the uniform fluid over a... 

The electrical conductivity of hard-sphere-like microemulsions increases smoothly as the volume fraction < > is increased. On the contrary, the conductivity of microemulsions with attractive interaction between droplets increases steeply around pM).08-0.14 (Figure 3). The behavior of the conductivity may be accounted for by percolation theories and < >p is identified to the percolation threshold. However, in such systems one must distinguish between geometrical percolation and conductivity percolation. 

Doll64 has applied classical S-matrix theory to the collinear A + BC collision where atoms A and B interact via a hard sphere collision this is the model studied quantum mechanically by Shuler and Zwanzig.65 Doll treats classically allowed and forbidden processes and finds good agreement between semiclassical and quantum mechanical transition probabilities. This is a remarkable achievement for the semiclassical theory, for the hard sphere interaction is far from the smooth potential that one normally assumes to be necessary for the dynamics to be classical-like. 

Equations of motion and the pertinent constitutive equations for the flow of granular materials have been developed by Lun et al. (1984) using the hard sphere kinetic theory of dense gas approach. In this derivation a fixed control volume was considered in which a discrete number of smooth but inelastic particles are undergoing deformation. The resulting system of equations was given as follows ... 

In the density functional theory of Tarazona, the excess hard-sphere free energy is calculated in a nonlocal manner by employing the concept of smoothed density [261-263] ... 

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

The form of the function efr ( ) is different in different versions of the smoothed-density approximation proposed by Somo-za and Tarazona [71, 72] and by Poniwier-ski and Sluckin [69, 73]. The density functional model of Somoza and Tarazona is based on the reference system of parallel hard ellipsoids that can be mapped into hard spheres. In the Poniwierski and Sluckin theory the effective weight function is determined by the Maier function for hard sphe-rocylinders and the expression for Ayr (p) is obtained from the Carnahan-Starling ex-... 

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