Solid hard sphere

The statistical mechanical approach, density functional theory, allows description of the solid-liquid interface based on knowledge of the liquid properties [60, 61], This approach has been applied to the solid-liquid interface for hard spheres where experimental data on colloidal suspensions and theory [62] both indicate 0.6 this... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

We discuss classical non-ideal liquids before treating solids. The strongly interacting fluid systems of interest are hard spheres characterized by their harsh repulsions, atoms and molecules with dispersion interactions responsible for the liquid-vapour transitions of the rare gases, ionic systems including strong and weak electrolytes, simple and not quite so simple polar fluids like water. The solid phase systems discussed are ferroniagnets and alloys. 

Statistical mechanical theory and computer simulations provide a link between the equation of state and the interatomic potential energy functions. A fluid-solid transition at high density has been inferred from computer simulations of hard spheres. A vapour-liquid phase transition also appears when an attractive component is present hr the interatomic potential (e.g. atoms interacting tlirough a Leimard-Jones potential) provided the temperature lies below T, the critical temperature for this transition. This is illustrated in figure A2.3.2 where the critical point is a point of inflexion of tire critical isothemr in the P - Vplane. 

The CS pressures are close to the machine calculations in the fluid phase, and are bracketed by the pressures from the virial and compressibility equations using the PY approximation. Computer simulations show a fluid-solid phase transition tiiat is not reproduced by any of these equations of state. The theory has been extended to mixtures of hard spheres with additive diameters by Lebowitz [35], Lebowitz and Rowlinson [35], and Baxter [36]. 

Reiss H and Hammerich ADS 1986 Hard spheres scaled particle theory and exact relations on the existence and structure of the fluid/solid phase transition J. Phys. Chem. 90 6252... 

There are two classes of solids that are not crystalline, that is, p(r) is not periodic. The more familiar one is a glass, for which there are again two models, which may be called the random network and tlie random packing of hard spheres. An example of the first is silica glass or fiised quartz. It consists of tetrahedral SiO groups that are linked at their vertices by Si-O-Si bonds, but, unlike the various crystalline phases of Si02, there is no systematic relation between... 

Figure B3.3.9. Phase diagram for polydisperse hard spheres, in the volume fraction ((]))-polydispersity (s) plane. Some tie-lines are shown connecting coexistmg fluid and solid phases. Thanks are due to D A Kofke and P G Bolhuis for this figure. For frirther details see [181. 182].

In practice, tliere are various ways by which ( ) can be detennined for a given sample, and tire results may be (slightly) different. In particular, for sterically stabilized particles, tire effective hard-sphere volume fraction will be different from tire value obtained from tire total solid content. 

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Figure 8.12 Schematic illustration showing with the solid line how the probability of placement varies with the distance of separation between the centers of the coils. The broken line is the equivalent result for hard spheres.

FIG. 2 The equation of state for hard spheres, obtained from the HNC equation (part a) and the PY equation (part b). The dot-dashed and dotted curves and the circles have the same meaning as in Fig. 1. The solid curves give the results of the CS equation. 

Although Eqs. (33), (34), and especially (35), are useful they have a problem. They all predict that the hard sphere system is a fluid until = 1. This is beyond close packing and quite impossible. In fact, hard spheres undergo a first order phase transition to a solid phase at around pd 0.9. This has been estabhshed by simulations [3-5]. To a point, the BGY approximation has the advantage here. As is seen in Fig. 1, the BGY equation does predict that dp dp)j = 0 at high densities. However, the location of the transition is quite wrong. Another problem with the PY theory is that it can lead to negative values of g(r). This is a result of the linearization of y(r) - 1 that... 

FIG. 6 Density profile of a hard sphere fiuid near a hard wall. The bulk density is pd = 0.6. The curve gives the results of the lOZ equation with the PY closure and the circles give the simulation results. The results obtained using the HAB equation together with the MV closure are very close to the solid curve whereas the results obtained from the HAB equation with the HNC and PY closures are too large and too small, respectively. 

FIG. 7 Values of the density profile at eontaet for hard spheres in a sht of width H as a funetion of H. The density of the hard sphere fluid that is in equilibrium with the fluid in the slit is pd = 0.6. The solid eurve gives the lOZ equation results obtained using the PY elosure. The broken and dotted eurves give the results of the HAB equation obtained using the HNC and PY elosures, respeetively. The results obtained from the HAB equation with the MV elosure are very similar to the solid eurve. The eireles give the simulation results. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

FIG. 5 Adsorption isotherms for a hard sphere fluid from the ROZ-PY and ROZ-HNC theory (solid and dashed lines, respectively) and GCMC simulations (symbols). Three pairs of curves from top to bottom correspond to matrix packing fraction = 0.052, 0.126, and 0.25, respectively. The matrix in simulations has been made of four beads (m = M = 4). 

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 BBM gas consists of an arbitrary number of hard spheres (or balls) of finite diameter that collide elastically both among themselves and with any solid walls (or mirrors) that they may encounter during their motion. Starting out on some site of a two-dimensional Euclidean lattice, each ball is allowed to move only in one of four directions (see figure 6.10). The lattice spacing, d = l/ /2 (in arbitrary units), is chosen so that balls collide while occupying adjacent sites. Unit time is... 

Back reflection of translational and rotational velocity is rather reasonable, but the extremum in the free-path time distribution was never found when collisional statistics were checked by computer simulation. Even in the hard-sphere solid the statistics only deviate slightly from Pois-sonian at the highest free-paths [74] in contrast to the prediction of free volume theories. The collisional statistics have recently been investigated by MD simulation of 108 hard spheres at reduced density n/ o = 0.65 (where no is the density of closest packing) [75], The obtained ratio t2/l2 = 2.07 was very close to 2, which is indirect evidence for uniform... 

This is an indication of the collective nature of the effect. Although collisions between hard spheres are instantaneous the model itself is not binary. Very careful analysis of the free-path distribution has been undertaken in an excellent old work [74], It showed quite definite although small deviations from Poissonian statistics not only in solids, but also in a liquid hard-sphere system. The mean free-path X is used as a scaling length to make a dimensionless free-path distribution, Xp, as a function of a free-path length r/X. In the zero-density limit this is an ideal exponential function (Ap)o- In a one-dimensional system this is an exact result, i.e., Xp/(Xp)0 = 1 at any density. In two dimensions the dense-fluid scaled free-path distributions agree quite well with each other, but not so well with the zero-density scaled distribution, which is represented by a horizontal line (Fig. 1.21(a)). The maximum deviation is about... 

Krypton crystallizes with a face-centered cubic unit cell of edge 559 pm. (a) What is the density of solid krypton (b) What is the atomic radius of krypton (c) What is the volume of one krypton atom (d) What percentage of the unit cell is empty space if each atom is treated as a hard sphere ... 

Fig. 6. Excess chemical potential of hard-sphere solutes in SPC water as a function of the exclusion radius d. The symbols are simulation results, compared with the IT prediction using the flat default model (solid line). (Hummer et al., 1998a)...

Frenkel, D. Ladd, A. J. C., New Monte Carlo method to compute the free energy of arbitrary solids. Application to the fee and hep phases of hard spheres, J. Chem. Phys. 1984, Si, 3188-3193... 

Runge, K. J. Chester, G. V., Solid-fluid phase transition of quantum hard spheres at finite temperature, Phys. Rev. B 1988, 38, 135-162... 

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