Hard spheres interaction energy

The hard sphere interaction energy is an accurate approximation for short-range interactions between particles. This occurs when we have steric stabilization [33,34] due to polymer adsorption and electrostatic stabilization with a thin double layer [35,36] (i.e., high ionic... 

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

The interaction V"f /f) depends only on the unperturbed surface potential ipoi of sphere i (/ = 1, 2) and can be interpreted as the interaction between sphere i and its image with respect to sphere j ( j= 1, 2 j i). When both spheres are hard, the interaction energy is given by... 

The different terms represent segment-segment hard spheres interactions, a, mean field contribution, permanent bond energy between segments in the chain, a "", and free energy of specific hydrogen bond interactions between associating sites, a ° , if any. 

In the PHSC EoS, the Helmholtz free energy is expressed as the sum of two different terms, one is a reference term accounting for chain connectivity and hard sphere interactions, and the other is a perturbation term, which represents the contributions of mean-field forces ... 

As discussed above, solvation free energy is t3q)ically divided into two contributions polar and nonpolar components. In one popular description, polar portion refers to electrostatic contributions while the nonpolar component includes all other effects. Scaled particle theory (SPT) is often used to describe the hard-sphere interactions between the solute and the solvent by including the surface free energy and mechanical work of creating a cavity of the solute size in the solvent [148,149]. 

Here we have introduced the normalized Helmholtz energy F = Fvo/kTV. The first term on the right-hand side of (3.8) is the ideal contribution, while the second hard-sphere interaction term is the Camahan-Starling equation of state [12]. 

Modern synthetic methods allow preparation of highly monodisperse spherical particles that at least approach closely the behavior of hard-spheres, in that interactions other than volume exclusion have only small influences on the thermodynamic properties of the system. These particles provide simple model systems for comparison with theories of colloidal dynamics. Because the hard-sphere potential energy is 0 or 00, the thermodynamic and static structural properties of a hard-sphere system are determined by the volume fraction of the spheres but are not affected by the temperature. Solutions of hard spheres are not simple hard-sphere systems. At very small separations, the molecular granularity of the solvent modifies the direct and hydrodynamic interactions between suspended particles. 

The hard-sphere interaction has a clearly defined and energy-independent range, namely d, and so the colhsion cross-section has the exact value nd. Even when the interaction is not that of hard spheres, it is the case that the cross-section can be interpreted in such a form, where now is an effective range of the interaction. To do so we need to know more about realistic forces between molecules. 

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. 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

Hard-sphere models lack a characteristic energy scale and, hence, only entropic packing effects can be investigated. A more realistic modelling has to take hard-core-like repulsion at small distances and an attractive interaction at intennediate distances into account. In non-polar liquids the attraction is of the van der Waals type and decays with the sixth power of the interparticle distance r. It can be modelled in the fonn of a Leimard-Jones potential Fj j(r) between segments... 

P. Attard, D. R. Berard, C. P. Ursenbach, G. N. Patey. Interaction free energy between planar walls in dense fluids an Omstein-Zernike approach for hard-sphere, Lennard-Jones, and dipolar systems. Phys Rev A 44 8224-8234, 1991. 

Let us thus consider a model in which the association energy depth changes when two reacting particles are approaching the surface see Refs. 86,90. If in the vicinity of the surface the binding energy is lower than it is far from the surface, the probability of the chemical reaction to occur in the surface zone decreases. Similarly to the previous case, we consider an equimolar mixture of associating hard spheres of equal diameters. The interaction between the species a and (3 is assumed in the form... 

The hard sphere (HS) interaction is an excellent approximation for sterically stabilized colloids. However, there are other interactions present in colloidal systems that may replace or extend the pure HS interaction. As an example let us consider soft spheres given by an inverse power law (0 = The energy scale Vq and the length scale cr can be com-... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

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). 

In order to utilise our colloids as near hard spheres in terms of the thermodynamics we need to account for the presence of the medium and the species it contains. If the ions and molecules intervening between a pair of colloidal particles are small relative to the colloidal species we can treat the medium as a continuum. The role of the molecules and ions can be allowed for by the use of pair potentials between particles. These can be determined so as to include the role of the solution species as an energy of interaction with distance. The limit of the medium forms the boundary of the system and so determines its volume. We can consider the thermodynamic properties of the colloidal system as those in excess of the solvent. The pressure exerted by the colloidal species is now that in excess of the solvent, and is the osmotic pressure II of the colloid. These ideas form the basis of pseudo one-component thermodynamics. This allows us to calculate an elastic rheological property. Let us consider some important thermodynamic quantities for the system. We may apply the first law of thermodynamics to the system. The work done in an osmotic pressure and volume experiment on the colloidal system is related to the excess heat adsorbed d Q and the internal energy change d E ... 

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