Approximations Using Hard-Sphere Potentials

FIGURE 1.24 The device for evaluation of orientationally averaged projections of polyatomic ions through exposure of wax models rotating around three perpendicular axes. (From Mack, Jr., E., J. Am. Chem. Soc., 47, 2468, 1925.) 

As would be determined 70 years later, orientationally averaged projections are exactly equal to ft (assuming hard-sphere interactions) for and only for contiguous bodies that lack concave surfaces and thus permit neither self-shadowing nor multiple scattering of gas molecules.For other bodies, ft always exceeds the projection. No polyatomic molecule is truly convex because of crevices between the atoms. However, the effect of such small locally concave areas on ft is only a few percent and PA is often passable for largely convex shapes. This is not true for objects 

4 More Sophisticated Treatments of Attractive AND Repulsive Interactions 

Unlike q, the values of Sq, and fr,- could not be determined a priori with sufficient accuracy and have been obtained by fitting K T) for ions of known geometry7 If all atoms of an ion are equivalent such as in o, and r, are equal for all i and the problem reduces to a system of two equations (at different T) with two variables (sq and a) that has a unique solution. For example, K(T) for C, 

FIGURE 1.27 Measured mobility of Ceo in He over T — 80-400 K (circles) are fit (line) using trajectory calculations in the interaction potential of Equation 1.34. (From Mesleh, M.F., Hunter, J.M., Shvartsburg, A.A., Schatz, G.C., Jarrold, M.F.J., J. Phys. Chem. A, 100, 16082, 1996.) 

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Other closures have been used in the solution of the SSOZ equation. We may regard Eqs. (3.2.6) and (3.2.7) as a speeial case of a PY-like approximation for hard sphere potentials, so that more generally we could write... 

In the numerical solution the matrix structure is evaluated from Eqs. (44)-(46). Then Eqs. (47)-(49) with corresponding closure approximations are solved. Details of the solution have been presented in Refs. 32 and 33. Briefly, the numerical algorithm uses an expansion of the two-particle functions into a Fourier-Bessel series. The three-fold integrations are then reduced to sums of one-dimensional integrations. In the case of hard-sphere potentials, the BGY equation contains the delta function due to the derivative of the pair interactions. Therefore, the integrals in Eqs. (48) and (49) are onefold and contain the contact values of the functions... 

The results reported here are illustrative of the information which can be obtained from the hard-sphere potential. Of course, while this approach provides some indication of the stereochemical restrictions in a polypeptide chain, it should be regarded as giving only a first approximation to the most stable structure. Further progress requires the use of more complete energy expressions, as discussed in the next sections. 

When (r) is a hard-sphere potential of diameter d, use of the OZ equation (2.27) plus (2.94) and (2.95), with t>(12) adjusted inside the core region rordered approximation (LOGA). The further approximation that Cq(12) is equal to zero for r>[Pg.216]

The condition [8.81] is exact for hard core potentials since g(12)=0 for r < o, while the condition [8.82] is the approximation, being correct only asymptotically (large r). We thus expect the approximation to be worst near the hard core we also note that the theory is not exact at low density (except for a pure hard sphere potential, where MSA=PY), but this is not too important since good theories exist for low densities, and we are therefore mainly interested in high densities. Once again the solution of the MSA problem is most easily accomplished using the spherical harmonic component of h and c. 

In addition to the electrical interactions, particles interact, at short-range by a repulsive interaction generally taken to be a hard sphere potential. In ER (MR) fluids where the short range repulsions extend over distances much shorter than the size of the particles this is a reasonable approximation. However, in dynamical simulations a discontinuous potential is not convenient and the hard sphere potential is generally replaced by a continuous exponential or inverse power potential with parameters chosen to closely mimic a hard sphere potential. As shown in [173] too soft a potential may lead to im-realistic aggregation of the particles. Potentials of a similar functional form have been used for the wall-particle interaction. 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

One should perhaps mention some other closures that are discussed in the literature. One possibility is to combine the PY approximation for the hard core part of the potential and then use the HNC approximation to compute the corrections due to the attractive forces. Such an approach is called the reference hypernetted chain or RHNC approximation [48,49]. Recently, some new closures for a mixture of hard spheres have been proposed. These include one by Rogers and Young [50] (RY) and the Martynov-Sarkisov [51] (MS) closure as modified by Ballone, Pastore, Galli and Gazzillo [52] (BPGG). The RY and MS/BPGG closure relations take the forms... 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

The value of DFT is evidently dependent on the accessibility and accuracy of the grand potential functional, Si [p(r)]. The usual practice is to treat the molecules as hard spheres and divide the fluid-fluid potential into attractive and repulsive parts. A mean field approximation is used to simplify the former by the elimination of correlation effects. The hard sphere term is further divided into an ideal gas component and an excess component (Lastoskie etal., 1993). The ideal component is considered to be exactly local, since this part of the Helmholtz free energy per molecule depends only on the density at a particular value of r. 

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