Hard spheres approximation

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

Fig. 9. Derivation diagram showing which atomic collisions (using a hard-sphere approximation) produce the restrictions on main chain dihedral angles 0 and ifi. The crosshatched regions are allowed for all residues, and each boundary of a prohibited region is labeled with the atoms which collide in that conformation. Atom names are the same as in Fig. 5. Adapted from Mandel et al. (1977), with permission.

In a gas or a mixture of two gases 1 and 2, the mean free path must be obtained as an average over the velocity distribution. We quote here a simple formula based on the (crude) hard sphere approximation,... 

Frequency of collisions. The mean frequency of collisions is similarly expressed in the hard spheres approximation as... 

For rough estimates, the collisional cross section may be assumed to be velocity-independent, Qn(vn) = Qo = constant (hard-sphere approximation), so that the mean time between collisions becomes... 

The diffusional rate constant kD is calculated on the basis of the Debye-Hiickel theory (Equation 6.107), where the distance tr is the sum of A and B radii in the hard-sphere approximation. 

We also confine attention here to Hnbr, that portion of IIcaused by the repulsive part of unb, and the hard-sphere approximation to Eq. (25) leads to... 

In the case of neutral species at thermal equilibrium ((ca) (Cb))> the hard sphere approximation leads to... 

In the molecular dynamics approaches to such systems, one of the principal novelties compared with the work on simpler systems such as KCI (Section 5.3.3) is the taking into account of anion polarizability, i.e., elimination of the hard-sphere approximation. Anions are by and large more polarizable than cations. The polarizability ce is proportional to rf. The radii of cations, formed by losing an outer electron, are smaller than those of the anions, which are formed by adding an electron. Hence, anions are mainly the ions affected when polarizability is to be accounted for. 

Calculations taking into account the anion polarizability in AICI4 reduce the approximation associated with the simple additivity of pairwise potentiais in computational modeling (hard-sphere approximation) of molten salts. They predict new entities (e.g., AljClg) and in this respect have an advantage over earlier calculations. 

The equilibrium distribution p (uj) = [Pg.320]

The volume restriction effect as discussed in this paper was proposed several years ago by Asakura and Oosawa (12,13). Their theory accounted for the instability observed in mixtures of colloidal particles and free polymer molecules. Such mixed systems have been investigated experimentally for decades (14-16). However, the work of Asakura and Oosawa did not receive much attention until recently (17,18). A few years ago, Vrij (19) treated the volume restriction effect independently, and also observed phase separation in a microemulsion with added polymer. Recently, DeHek and Vrij (20) have reported phase separation in non-aqueous systems containing hydrophilic silica particles and polymer molecules. The results have been treated quite well in terms of a "hard-sphere-cavity" model. Sperry (21) has also used a hard-sphere approximation in a quantitative model for the volume restriction flocculation of latex by water-soluble polymers. 

In the model, the agreement between theoretical and experimental curves Is satisfactory. It may be possible to Improve the agreement by removing some of the assumptions In the model. Also, one may use a hard-sphere approximation to compute the free energy of dispersion. But the overall behavior predicted would roughly be the same. 

Fig. 19. The general significance of the conformational map for the alanyl residue. Allowed limits of the hard sphere approximation...

The picture postulated by Van Kranendonk (7) leads to a molecular cross-section, corresponding to the hard spheres approximation. 

The application of this approach to spin waves was called by Dyson naive and criticized as incorrect and leading to results different from those obtained by him (see (6), the end of 3). We shall show in what follows, however, on the basis of an exact representation of Pauli operators in terms of Bose operators, that the picture described above does take place for Frenkel excitons. This takes place only because the excitation energy A for Frenkel excitons is large compared with the width of the exciton band. As for the spin waves, where the inequality indicated above is not satisfied, the cross-section for the scattering of long-wavelength spin waves by each other can indeed, in agreement with Dyson, differ substantially from a value that follows from the hard sphere approximation (7). 

THERMAL CONDUCTIVITY OF REAL DENSE FLUIDS. HARD SPHERE APPROXIMATION. 

The development and application of molecular shape descriptors is an active area in computational chemistry and biology. The main goal of our work is to develop mathematical descriptors that can determine whether two molecules have comparable shapes. In this chapter we present a series of molecular shape descriptors developed oti the basis of molecular vdW space. The molecules are treated in the hard sphere approximation, as a body composed from a collection of atomic fused spheres. Each sphere is centered in the corresponding nucleus and it is characterized by its Cartesian coordinates and by its vdW radius, r. These molecular vdW shape descriptors depend only on the internal structure of the molecule, being invariants to any translation and rotation movement. Consequently, they may inform us that two molecules have comparable shapes, but since they carry no information about the absolute orientation or position of the molecule, they are not useful for computing molecular superposition. 

The space occupied by molecules can be conveniently described in the approximation of hard spheres each atom of the molecule M is represented by an isotropic sphere having the center in the equilibrium position (Xj, Yj, Zj) of the atomic nucleus i, and the radius equal with its van der Waals radius, r-. A molecular van der Waals envelope, F, can be defined in the hard-spheres approximation as the external surface resulted from the interaction of all vdW spheres corresponding to the atoms of molecule M. The points (x, y, z) inside the envelope satisfy at least one of the following inequalities ... 

The molecules are treated within the hard sphere approximation (see previous section). Points (x, y, z) residing on vdW envelope F of a molecule M satisfy one of the equations (15.6) where (Xj, Yj, Zj) are the Cartesian coordinates of the m atoms belonging to molecule M, and is the vdW radius of atom i. 

To allow a better understanding of the condensed phase, the volume of a sample can be divided into two parts the van der Waals volume, V , which represents the actual volume of the molecules or ions in the hard-sphere approximation, taken from Fig. 4.23, and the total, experimental volume V. It must be remembered that the van der Waals radius depends somewhat on the forces that determine the approach of the... 

Figure 10.20. (a) Particle with strongly bonded soluble polymer on Its surface, (b) Interaction energy between two surfaces in solvent with bonded soluble polymer, (c) Hard sphere approximation to interaction energy. 

FIGURE 1.7. Variation of with redox site concentration c. From top to bottom To == 1.3 nm, 0.6 nm, and 0 nm, respectively. Curves calculated using Eqn. 25 (hard-sphere approximation) and Eqn. 27 (point molecule approximation). Point A indicates (r > and for closest packing conditions. 

The hard-sphere approximation of Section 8.4 is the simplest model for reaction. The molecules A and B are treated as spheres which must approach within a distance g to react. Only the translational motion of A and B is considered, and from (9.13) and (9.A6) of Appendix A... 

At least in the case of liquid simple metals, a knowledge of the effective pan-potentials describing the interaction between the ions in the liquid metal can also be utilized to calculate g(r) and A K). The most common such method involves the assumption of a hard-sphere potential in the Percus-Yevick (PY) equation its solution provides the hard-sphere structure factor, /4hs( C). (See Ashcroft and Lekner 1966.) The two parameters that must be provided for a calculation of Ahs( ) are the hard-sphere diameter, a, and the packing fraction, x. It is found that j = 0.45 for most liquid metals at temperatures just above their melting points. A hard-sphere solution of the PY equation has also been obtained for binary liquid metal alloys, and provides estimates of the three partial structure factors describing the alloy structure (Ashcroft and Langreth 1967). To the extent that the hard-sphere approximation appears to be valid for the liquid R s, pair potentials should dominate these metals also, at least at short distances. 

In each case, the first peak in A K) is very nearly symmetric the oscillations in A K) for K are also consistent with a hard sphere lic[uid model. See fi . 2.) It does appear, however, that the hard-sphere approximation improves as one moves across the lanthanide series from La to Lu. The more rapid damping of the oscillations in A K) for the light R s impUes a somewhat softer core (Waseda and Tamaki 1977b, Waseda and Miller 1978) (fig. 2). Rao and Sitpathy (1981) see this directly in their calculations of the effective interionic potentials, (r), where they use the A K) data of Waseda and co-workers. Fits of the experimental A K) to Hs( ) imply that the packing fraction, x, lies in the range of 0-42-0.43 for most R s the range extends to 0.40-0.44 for some of the heavy R s, however. 

The conclusion is that there is a minimum in , when there is exactly one monolayer of adsorbate between two plates. This, however is on average exactly 1/2 a monolayer for one plate. Using the symbol Fj for a mono-layer surface excess, then = 1/2 Fj within the first approximation assumed with the hard sphere approximation. Using this together with Eq. (79), (80) and (82) one arrives at... 

To relate Eq. (97), or C, to the surface area, a value for the excluded area, needs to be determined. First, the hard sphere approximation to an adsorbed molecule will be determined. The area one would expect an average liquid molecule to cover is given by the molar area. This physical quantity, designated as a here, is given by the equation... 

This is not the entire picture, however. First, according to Eq. (91) and the approximation thereafter, half the time an adsorbate molecule will exclude another adsorbate molecule from its area and half the time it will not. Therefore, with the hard sphere approximation the excluded area is half of the van der Waal area or twice the hquid area. Second, the hard sphere approximation assumes that the energy profile as a molecule rolls over ... 

Using only the hard sphere approximation it is possible to provide the relationship between and A. The hard sphere approximation for the x equation becomes... 

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