Hard spheres worked example

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Problems and Worked Examples 6.1 through 6.5, at the end of this chapter, will sharpen your skills in obtaining simple, practical estimations of the viscosity, modulus, and relaxation time of hard-sphere suspensions. 

Problem 6.1 (Worked Example) Estimate the zero-shear viscosity of a suspension of hard spheres 100 nm in diameter at a volume fraction of [Pg.318]

Problem 6.3(a) (Worked Example) At what critical shear rate do you expect the onset of shear thinning in a 40% (by volume) suspension of hard spheres of radius 1 tim in water at room temperature (Hint use the data of Fig, 6-4, plus scaling principles.)... 

In our view, there is a need for much more work in this area. For example, it would be very useful if more extensive studies with the PY approximation were made in order to investigate whether it is possible to obtain results for diatomics of comparable accuracy to those for hard spheres. Also, we do not believe that the influence of the convergence of the spherical harmonic... 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

In a recent work, the translational motion of 4- -hexyl-4 -cyanobiphenyl (6CB) was studied in the isotropic phase by atomistic molecular dynamics simulation [134], The mean-square displacement showed evidence of sub-diffusive dynamics, with a plateau that became very apparent at the lowest temperatures. A three-time self-intermediate scattering function revealed that this plateau was connected with a homogeneous dynamics that, at longer times, became heterogeneous and finally exponential. These features, which are shared by, for example, a high-density system of hard spheres, support the universal character of the translational dynamics of liquids in their supercooled regime. 

Application of the GDI method to the coexistence lines requires establishment of a coexistence datum on each. A point on the vapor-liquid line can be determined by a GE simulation. At high temperature the model behaves as a system of hard spheres, and the liquid-solid coexistence line approaches the fluid-solid transition for hard spheres, which is known [76,77]. Integration of liquid-solid coexistence from the hard-sphere transition proceeds much as described in Section III.C.l for the Lennard-Jones example. The limiting behavior (fi - 0) finds that /IP is well behaved and smoothly approaches the hard-sphere value [76,77] of 11.686 at f = 0 (unlike the LJ case, we need not work with j81/2). Thus the appropriate governing equation for the GDI procedure is... 

The Pratt-Chandler theory has been extended to consider complex molecules. For example, the hard-sphere model of -butane may have an excluded volume Av(f, X), which is a function of the torsion angle (j) and depends on the exclusion radius X of the methylene spheres. Then the part of the PMF (the potential of mean force) arising from the solute-solvent interaction can be related to the reversible work required to create a cavity with the shape and excluded volume Av((/>, X) of the -butane molecule. 

Before elaborating on a specific example involving real solutes, it is instructive to work out a simple case. Consider a solvent of N hard spheres of diameter aw The solute particles are also hard spheres of diameter ag. Clearly, the distance between these two solute particles cannot be smaller than ag. However, if we focus our attention on the function SA(R) and since we know that this function is smooth and finite in the entire region 0 < R < oo, we can inquire about the value of SA(R) for any distance R < a. This point of view is equivalent to replacing the two solute particles by two cavities of appropriate size. While a system of two HS particles in a solvent is certainly different from a system with two cavities, from the point of view of the solvent the two systems are identical. In contrast to the two HS solutes, two cavities can be brought to any distance including R = 0. For this limiting case, expression (4.4.5) reduces to... 

Before elaborating on an example involving real solutes, it is instructive to work out a simple case. Consider a solvent of N hard spheres of diameter The two solute particles are also hard spheres, of diameter o. Clearly, the distance between these two solute particles cannot be less than Og. However, if we focus our attention on the function 5 (R), and since we... 

The simplest case of a solvated 1-D system occurs when the coupling work W(s w) is approximately the same for all configurations s. Two examples are shown in Fig. 8.4. In a, the solvent consists of hard-spheres particles and each unit contributes the same excluded volume with respect to the solvent independently of its state (say up or down spin). In this case 1 (81 ) is simply the work required to create a cavity in the solvent to accommodate the 1-D system. The Helmholtz energy levels (8.6.9) are modified by a constant... 

Faraday Soc. Discussion (1978). Chandler (1978 and 1982) has reviewed some of this work. Specific examples are Lowden and Chandler (197 ), Hsu et al. (1976) and Hsu and Chandler (1978). In its simplest version, that is, when it is applied to single site spherical particles, the RISM theory for hard core molecules reduces to the Percus-Yevick theory for the hard sphere fluid. 

When the dispersed partides are spherical, a hard-sphere fluid can be assumed, for which the interpartide interactions exclude particles from approaching within a center-center distance r equal to the collision diameter for the interaction, that is, <7c = 2tc. For distances r greater than form factors of a particular system can be found in a work from Justice et al. [122]. 

TCF (reversible work, enthalpy, and entropy) are quantities of considerable interest when studying interactions between molecules in fluids. TCF functions in a certain solvent vary with the size and shape of cavity and theories of statistical mechanics are generally employed to evaluate them. For example. SPT has proven a successful method for evaluation of TCF for spherical cavities in hard sphere solvents and Kihara s convex-body theory has proven an adequate method for convex-body solvents. It is interesting to remark that both theories were successful in dealing with real liquids, despite the fact that they are neither hard spheres nor convex bodies. More recent developments in liquid state theory make it possible to treat one solvent molecule as a collection of fused... 

In typical organic crystals, molecular pairs are easily sorted out and ab initio methods that work for gas-phase dimers can be applied to the analysis of molecular dimers in the crystal coordination sphere. The entire lattice energy can then be approximated as a sum of pairwise molecule-molecule interactions examples are crystals of benzene [40], alloxan [41], and of more complex aziridine molecules [42]. This obviously neglects cooperative and, in general, many-body effects, which seem less important in hard closed-shell systems. The positive side of this approach is that molecular coordination spheres in crystals can be dissected and bonding factors can be better analyzed, as examples in the next few sections will show. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

The earliest work was done for hard disks and spheres, with no longer range forces. In that case there is no direct effect due to interactions with distant (but correlated) parts of the periodic system. There is nevertheless some dependence of the results on the size and shape of the periodic sample. This arises in various ways, which of course also persist for systems with other forces. For example, the number of particles within any of the periodic boxes is fixed at N. This is a serious constraint on the density fluctuations for small N, and will lead to errors in the resulting thermodynamic averages, in particular to a diminished entropy. (For mixtures, the concentration fluctuations are similarly constrained, and this could be a still more severe problem.) With small samples the range of structural fluctuations may be similarly constrained. This will clearly be the case if significant interparticle configurational correlations have a... 

---