Hard-sphere bulk systems

For a eonfined system the eritieal temperature is lower than for the bulk system. However, the value of pt, at the eritieal point is nearly the same in all eases, sueh that eonfinement eauses only a slight inerease of the value of the eritieal density. The density funetional theory has also been applied to study the adsorption of assoeiating hard spheres on erystalline surfaees [43]. However, this researeh is in its initial stage at present. 

As in previous theoretical studies of the bulk dispersions of hard spheres we observe in Fig. 1(a) that the PMF exhibits oscillations that develop with increasing solvent density. The phase of the oscillations shifts to smaller intercolloidal separations with augmenting solvent density. Depletion-type attraction is observed close to the contact of two colloids. The structural barrier in the PMF for solvent-separated colloids, at the solvent densities in question, is not at cr /2 but at a larger distance between colloids. These general trends are well known in the theory of colloidal systems and do not require additional comments. 

This is the isothermal bulk modulus. Thus we can use our simulation data in Figure 5.1 and calculate a modulus for a hard sphere system. Equations (5.14) to (5.16) form an interesting hierarchy of equations ... 

For liquids, even simple monatomic liquids such as molten metals, this link between molecular interactions and a bulk property such as viscosity is still there, in principle, but it is difficult to derive a relationship from fundamental interaction energies that is useful for a wide variety of systems. Nonetheless, theoretical expressions for liquid viscosities do exist. Some are based on statistical mechanical arguments, but the model that is most consistent with the discussion so far is that of nonattracting hard spheres... 

Specializing to planar walls for the moment, one has the exact relation [45] that p/kT = where is the local density of the adsorbate in contact with the wall when the pressure of the hard sphere fluid is p. For hard sphere mixtures [46-49] n . is the sum of the individual densities for each of the components in the fluid. Thus, the pressure of the fluid can be obtained from estimates of the intercept of the curve of n z) versus z for example. Fig. 1 indicates that palg/kT is between 8 and 9 for this system. This result, taken together with the calculation of F at a given n from Eq. (10), allows one to construct the isotherm TO). Figure 2 shows the adsorption of a hard sphere fluid on a hard wall as a function of the bulk-phase density [44]. The simulation points compare well with results of two theoretical calculations based on the scaled particle theory. 

FIG. 1 Lower panel Local density in reduced units for the hard sphere-hard wall system discussed in the text. Upper panel Local pressure tensor component perpendicular (constant line) and parallel (oscillating curve) to the surface. These properties are plotted as a function of the distance from the surface in reduced units. The bulk density for this system is 0.90 in reduced units, which is close to the freezing value. (From Ref 44.)... 

FIG. 2 Adsorption (molecules adsorbed per unit area) for the hard sphere-hard wall system plotted in reduced units as a function of the bulk density r) = (7r/6)na j,. The points show the simulations, and the lines show theoretical calculations based on the scaled particle theory. (From Ref. 44.)... 

FIG. 5 Pair correlations plotted as a function of the reduced sphere separation distance for the system of Fig. 6. For these eurves. eaeh sphere is at the same distance from the surface so z = Z2- The number above each curve indicates the value of this distance. The points are simulated, and the solid eurves are the pair correlation functions for the bulk hard sphere fluid at the same seduced density (0.81) as that of the simulation. (From Ref. 49.)... 

The molecular dynamics method was first introduced in the late 1950s by Alder and Wain wright (56,57) to study the interactions of hard spheres. In 1964, Rahman carried out the first simulation using a realistic potential for liquid argon (58). In 1974, the first molecular dynamics simulation of a realistic system was done by Rahman and Stillinger in their simulation of bulk liquid water (59). 

The correspondence between a concentrated dispersion and an assembly of hard spheres has been pursued by several authors. " The Kirkwood-Alder hard-sphere transition is in qualitative agreement with experiment, but the coexistence region is in general too narrow. Introduction of attractive forces, in the Monte Carlo simulations and approximate perturbation-cell theories, leads to iijiprovement at high salt concentrations and large volume fractions. But at low salt concentrations there remains the fundamental problem that the particles are not in proper thermodynamic equilibrium with bulk electrolyte as Ninham and coauthors put it the diffuse double-layers of the particles fill up the entire volume of the system, and there is no place to be regarded as bulk . 

An elementary way of taking into account the granularity of the solvent is the hard sphere-point dipole model, which accounts for the excluded volume effects and part of the polarization effects of the solvent. Unfortunately, even with modern vectorized computers, it is difficult to simulate these systems, even in the bulk phase. 

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