Pair potential models hard-sphere

In the theory of the liquid state, the hard-sphere model plays an important role. For hard spheres, the pair interaction potential V r) = qo for r < J, where d is the particle diameter, whereas V(r) = 0 for r s d. The stmcture of a simple fluid, such as argon, is very similar to that of a hard-sphere fluid. Hard-sphere atoms do, of course, not exist. Certain model colloids, however, come very close to hard-sphere behaviour. These systems have been studied in much detail and some results will be quoted below. 

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... 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

Instead of the hard-sphere model, the Lennard-Jones (LJ) interaction pair potential can be used to describe soft-core repulsion and dispersion forces. The LJ interaction potential is... 

The second generalization is the reinterpretation of the excluded volume per particle V(). Realizing that only binary collisions are likely in a low-density gas, van der Waals suggested the value Ina /I for hard spheres of diameter a and for particles which were modeled as hard spheres with attractive tails. Thus, for the Lennard-Jones fluid where the pair potential actually is... 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

Equation 8 was also applied by Sperry (12), although the underlying assumptions are different in his model. There is also a close analogy between Equation 8 and the pair potential used by De Hek and Vrij. Indeed, Equation 4 of Ref. 6 reduces to our Equation 8 for H = 0, provided that 2A is interpreted as the hard sphere diameter of the polymer molecule. Hence, in dilute solutions (where A a rg) the two approaches are very similar. However, in our model A is a function of the polymer concentration. Because most experimental depletion studies are carried out at values for that are comparable in magnitude to <)>, our model... 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

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,... 

Until very recently there was no information about how an MM pair potential should look, based upon calculations from the deeper BO level. In the simplest BO level model for an ionic solution the solvent molecules are represented as hard spheres with centered point dipoles and the ions as hard spheres with centered charges. Now there are two sets of calculations, (16,17) by very different approximation methods, for this model where all of the spheres are 3A in diameter, where the dipole moments are near 1 Debye, and where the ions are singly charged. The temperature is 25° and the solvent concentration is about 50M, corresponding to a liquid state. The dielectric constant of the model solvent is believed to be near 9 6. 

One of the further refinements which seems desirable is to modify Eq. (9) so that it has wiggles (damped oscillatory behavior). Wiggles are expected in any realistic MM-level pair potential as a consequence of the molecular structure of the solvent (2,3,10,11,21,22) they would be found even for two hard sphere solute particles in a hard-sphere liquid or for two H2I80 solute molecules in ordinary liquid HpO, and are found in simulation studies of solutions based on BO-level models. In ionic solutions in a polar solvent another source of wiggles, evidenced in Fig. 2, may be associated with an oscillatory nonlocal dielectric function e(r). ( 36) These various studies may be used to guide the introduction of wiggles into Eq. (9) in a realistic way. 

For the case of hard spheres Eq. (6.3) collapses to the result of Eq. (6.2). The only information required for the model is an estimate of the form of the atom-molecule potential F(l ) within several kT of the binding energy of the pair. The collision integral is... 

In this section we consider the possibility of applying the ion association concept to the description of the properties of electrolyte solutions in the ion-molecular or Born-Oppenheimer level approach. The simplest ion-molecular model for electrolyte solution can be represented by the mixture of charged hard spheres and hard spheres with embedded dipoles, the so-called ion-dipolar model. For simplification we consider that ions and solvent molecules are characterized by diameters R and Rs, correspondingly. The model is given by the pair potentials,... 

In the absence of a reliable theory, computer simulations have become the most important means of tackling the question of ionic criticality. In the last ten years there have been numerous attempts to identify the universality class of what is perhaps the most basic model of ionic fluids, the restricted primitive model (RPM). The RPM consists of an equimolar mixture of positively and negatively charged hard spheres with diameter a, immersed in a dielectric continuum with dielectric constant D. The pair potential is,... 

Henceforth, we consider a Lennard-Jones(12,6) (LJ) fluid between two plane parallel solid substrates. The ba.sic. setup of our model is schematically depicted in Fig. 4.3. We as.sume the fluid-substrate interaction to be a pair-wise additive sum of L.1 potentials. As the reference system we take a hard-sphere fluid (diameter hard-sphere substrates (diameter (Tg). Moreover, we smear the hard spheres of the surface layer of each... 

The conventional van der Waals approach where model parameters d and a are the constants cannot describe more than one first order phase transition and one critical point. Therefore a key question is a formulation of temperature -density dependency for EoS parameters generating more than one critical point in the mono-component matter. There are several approaches of the effective hard sphere determination from spherical interaction potential models that have a region of negative curvature in their repulsive core (the so-called core softened potentials). To avoid the sophistication of EoS and study a qualitative picture of phase behavior we adopt an approach Skibinsky et al. ° for one-dimensional system of particles interacting via pair potential... 

A ternary collision may be conveniently pictured as a very rapid succession of two binary collisions one to form the unstable product, and the second, occurring within a period of about 10 sec or less, to stabilize the product. It is immediately obvious that it is not possible to use the elastic-hard-sphere molecular model to represent ternary collisions since two such spheres would be in collision contact for zero time, the probability of a third molecule making contact with the colliding pair would be strictly zero. It is therefore necessary to assume a potential model involving forces which are exerted over an extended range. One such model is that of point centers having either inverse-power repulsive or inverse-power attractive central forces. This potential, shown in Fig. 2-If, is represented by U r) = K/r. For the sake of convenience, we shall make several additional assumptions first, at the interaction distances of interest the intermolecular forces are weak, that is, U(r) < kT second, when the reactants A and B approach each other, they form an unstable product molecule A B when their internuclear separations are in the range b third, the unstable product is in essential... 

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