Hard sphere fluids calculation

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

Figure 8 Compressibility factor P/fiksT versus density p = pa3 of the hard-sphere system as calculated from both free-volume information (Eq. [8]) and the collision rate measured in molecular dynamics simulations. The empirically successful Camahan-Starling84 equation of state for the hard-sphere fluid is also shown for comparison. (Adapted from Ref. 71).

The program of calculating the BO-level potentials from Schroedinger level cannot often be carried through with the accuracy required for the intermolecular forces in solution theory. (9.) Fortunately a great deal can be learned through the study of BO-level models in which the N-body potential is pairwise additive (as in Eq. (3)) and in which the pair potentials have very simple forms. (2, 3, 6) Thus for the hard sphere fluid we have, with a=sphere diameter,... 

The 6-12 potential is only qualitatively like the realistic potentials that can be derived by calculations at Schroedinger level for, say, Ar-Ar interactions. But it requires careful and detailed study to see how real simple fluids (i.e. one component fluids with monatomic particles) deviate from the behavior calculated from the 6-12 model. Moreover the principal structural features of simple fluids are already quite realistically given by the hard sphere fluid. 

In order to use the above expressions for calculating the thermodynamic properties, appropriate expressions for the radial distribution function and for the equation of state for the hard-sphere reference system are required which are given in Appendix A. Fortunately, accurate information for the hard-sphere fluid as well as for the hard-sphere solid is available and this enables the determination of the properties of the coexisting dilute and concentrated phases of colloidal dispersions. 

In eq 3.1, the activity coefficients appear as a result of the hard-sphere repulsions among the droplets. Since the calculations focus on the most populous aggregates, the hard-sphere repulsions will be expressed in terms of a single droplet size corresponding to the most populous aggregates. One can derive expressions for the activity coefficients y ko of a component k in the continuous phase O starting from an equation for the osmotic pressure of a hard-sphere fluid,3-4 such as that based on the Carnahan—Starling equation of state (see Appendix B for the derivation) ... 

The excess chemical potential of a hard-sphere fluid can be calculated on the basis of an equation of state using the expressions17... 

HLC have suggested that the solvent dipoles near the colloidal particles are preferentially aligned. This effect is well known in theories of the electrical double layer. One simple way of accounting for this effect is through the use of a Stern layer of low dielectric constant near the colloidal particles. It is difficult to calculate this correction for spherical particles. As a result, HLC considered a hard sphere fluid between two hard walls and with a region of low dielectric constant near the walls. They found that Eq. (62) should be generalized to... 

The purpose of calculating Henry s Law constants is usually to determine the parameters of the adsorption potential. This was the approach in Ref. [17], where the Henry s Law constant was calculated for a spherically symmetric model of CH4 molecules in a model microporous (specific surface area ca. 800 m /g) silica gel. The porous structure of this silica was taken to be the interstitial space between spherical particles (diameter ca. 2.7 nm ) arranged in two different ways as an equilibrium system that had the structure of a hard sphere fluid, and as a cluster consisting of spheres in contact. The atomic structure of the silica spheres was also modeled in two ways as a continuous medium (CM) and as an amorphous oxide (AO). The CM model considered each microsphere of silica gel to be a continuous density of oxide ions. The interaction of an adsorbed atom with such a sphere was then calculated by integration over the volume of the sphere. The CM model was also employed in Refs. [36] where an analytic expression for the atom - microsphere potential was obtained. In Ref. [37], the Henry s Law constants for spherically symmetric atoms in the CM model of silica gel were calculated for different temperatures and compared with the experimental data for Ar and CH4. This made it possible to determine the well-depth parameter of the LJ-potential e for the adsorbed atom - oxygen ion. This proved to be 339 K for CH4 and 305 K for Ar [37]. On the other hand, the summation over ions in the more realistic AO model yielded efk = 184A" for the CH4 - oxide ion LJ-potential [17]. Thus, the value of e for the CH4 - oxide ion interaction for a continuous model of the adsorbent is 1.8 times larger than for the atomic model. 

Here p(r) is the smoothed density and A is the thermal de Broglie wavelength. The repulsive part of the Helmholtz free energy is usually calculated by the Carnahan-Starling equation derived for the hard sphere fluid [80] ... 

In Equation 1 XnSM is the value of X for a hard-sphere mixture of diameters dn, d22. . . etc., and XIIS is the value of X for a pure hard-sphere fluid with diameter d. The value of XHs is calculated from the Camahan-Starling (CS) equation (4). XREf represents the value of X as obtained from a reduced equation of state for the pure reference fluid evaluated at T and p made dimensionless by the pseudo parameters c and d3. 

The procedure for finding the approximate equation of state of the two-dimensional hard sphere fluid follows almost step by step that outlined above for the three-dimensional fluid except that the curvature term in Eq. (67) can be neglected. Without entering into further details of the calculations we merely quote the result... 

We have already in this and the previous sections made a number of comparisons between the various theories of fluids and the machine computations for the hard sphere system. Unfortunately, many recent developments in theory have been evaluated numerically only to the extent that the fourth and fifth virial coefficients can be compared. The table below lists the values of the fourth and fifth virial coefficients for the three-dimensional hard sphere fluid in units of the second virial coefficient b [cf. Eq. (33)]. The bases of calculation have been identified already in Section III except for the older "netted-chain approximation of Rushbrooke and Scoins. ... 

Smith and Triska have performed NVT and NpT RCMC simulations for several different reactions involving hard-sphere fluids [10]. They compared their results with theoretical calculations for hard-sphere mixtures. They considered an isomerization reaction of the form... 

High-temperature-approximation calculations have been performed primarily for the Lennard-Jones fluid and the square-well fluid. Higher order pei turbation theory results for the Lennard-Jones fluid, the square-well fluid,and the dipolar hard-sphere fluid have also been obtained. The Fade approximant method has also been used to extend the results. ... 

It should be noted that the SPT is not a pure molecular theory in the following sense. A molecular theory is supposed to provide, say, the Gibbs free energy as a function of T, P, N as well as of the molecular parameters of the system. Once this function is available, the density of the system can be computed from the relation p = (9/x/9 )t (with pi = G/N). The SPT utilizes the effective diameter of the solvent molecules as the only molecular parameter (which is the case for a hard-sphere fluid) and, in addition to the specification of T and P, the solvent density Pw is also used as input in the theory. The latter being a measurable quantity carries with it implicitly any other molecular properties of the system. The first application of the SPT to calculate the thermodynamics of solvation in liquids was carried out by Pierotti (1963, 1965). 

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. 

Yoon and Ohr derived an expression for the compressibility of hard-sphere fluids in terms of the radial free space distribution function, (r), which is the probability of acceptance of a (radial) displacement, r, in a Monte Carlo simulation.Alemany et al. performed MD simulations of liquid 50 and calculated its diffusion coefficient and shear viscosity. They simulated a system of 1372 Ceo molecules interacting through Girifalco s potential. 

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as... 

---