Hard-sphere models approximations

A plot of A versus r, the calibration curve of OTHdC, is shown in Fig. 22.2. The value of constant C depends on whether the solvent/polymer is free draining (totally permeable), a solid sphere (totally nonpermeable), or in between. In the free-draining model by DiMarzio and Guttman (DG model) (3,4), C has a value of approximately 2.7, whereas in the impermeable hard sphere model by Brenner and Gaydos (BG model) (8), its value is approximately 4.89. 

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

The crystal structures observed in ternary fluorides of the transition metals may be explained to a first approximation by reasons of geometry, i. e. by the relative sizes and charges of their constituent ions. The underlying hard sphere model of ions proves to be surprisingly useful. 

R. A. Marcus In Chem. Phys. Lett. 244, 10 (1995), a very rough approximate hard-sphere model used for liquids was mentioned to relate the frictional coefficient to the pair distribution function in the cluster. 

The degree of short-range order in an amorphous material can be characterized by a hard sphere model if the basic structure of an amorphous material is approximated by spheres. The density of packing of atoms around a reference atom is described by the number of atom centers per volume that lie in a spherical shell of thickness, dr, and radius about the reference atom. In a hard sphere model, the number, n, of neighboring spheres with centers between r and dr is measured as a function of r. 

Thus, a hard sphere model is assumed for the ionic crystal, with ions of opposite charge touching one another in the crystal lattice. Such an approximation means that the assignment of individual radii is somewhat arbitrary. The values listed in Table 4.2.2 are a set of data which assume that the radius of 02 is 140 pm and that of F is 133 pm. 

For real fluids the hard-sphere model gives a reasonable approximation to long-time phenomena, such as particle diffusion, but it is not realistic for short-time phenomena, such as collisional excitation of the internal motions of molecules, because of the impulsive nature of hard-sphere impacts. In the next two sections the more realistic case will be studied in which the test particle-bath particle interaction occurs through a continuous central potential. 

Fig. 4.10. Sketch of adsorption of polydisperse particles at (a) low and (b) high salt concentrations. Dotted lines show the effective particle radius (or interaction distance), (c) Surface coverage of the polystyrene particles versus na ( a is a dimensionless screening parameter, where k is the inverse Debye length and a the particle diameter). The more polydisperse particles (41 versus 107nm) have a slightly increased coverage at high na. Solid curves are approximations derived from the effective hard sphere model (see [89] for further details)...

The potential U(r ) is a sum over all intra- and intermolecular interactions in the fluid, and is assumed known. In most applications it is approximated as a sum of binary interactions, 17(r ) = IZ > w(rzj) where ry is the vector distance from particle i to particle j. Some generic models are often used. For atomic fluids the simplest of these is the hard sphere model, in which z/(r) = 0 for r > a and M(r) = c for r < a, where a is the hard sphere radius. A. more sophisticated model is the Lennard Jones potential... 

The polarizability of a species is a measure of the degree to which it may be distorted, e.g. by the electric field due to an adjacent atom or ion. In the hard sphere model of ions in lattices, we assume that there is no polarization of the ions. This is a gross approximation. The polarizability increases rapidly with an increase in atomic size, and large ions (or atoms or molecules) give rise to relatively large induced dipoles and, thus, significant dispersion forces. Values of a can be obtained from measurements of the relative permittivity (dielectric constant, see Section 8.2) or the refractive index of the substance in question. 

At high sample concentrations, the intensities in I Q) are reduced at small Q, and this can lead to artefacts in Guinier plots that exhibit reduced or even negative / o values. The orientations of the particles are correlated with one another, and leads to interparticle interference phenomena. The hard sphere model can be used to calculate /hs(6) as a first approximation to the experimental curves [6,7] ... 

Real fluids are neither ideal gases nor are they composed of hard spheres. But if the density is low, a gas might be nearly ideal, or if the temperature is high, a gas might behave somewhat like a fluid of hard spheres. In such cases the ideal-gas or hard-sphere models may serve as references in expansions that approximate real behavior. In this section we consider Taylor expansions (see Appendix A) of the compressibility factor Z about that for the ideal gas. The expansions may be done using either density or pressure as the independent variable we introduce the density expansion first. 

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