Theories repulsion

Hertzian Theory (Repulsion between Elastic Bodies)... 

Debye-Hiickel theory The activity coefficient of an electrolyte depends markedly upon concentration. Jn dilute solutions, due to the Coulombic forces of attraction and repulsion, the ions tend to surround themselves with an atmosphere of oppositely charged ions. Debye and Hiickel showed that it was possible to explain the abnormal activity coefficients at least for very dilute solutions of electrolytes. 

Fig. V-5. The repulsive force between crossed cylinders of radius R (1 cm) covered with mica and immersed in propylene carbonate solutions of tetraethylammonium bromide at the indicated concentrations. The dotted lines are from double-layer theory (From Ref. 51).

For example, van den Tempel [35] reports the results shown in Fig. XIV-9 on the effect of electrolyte concentration on flocculation rates of an O/W emulsion. Note that d ln)ldt (equal to k in the simple theory) increases rapidly with ionic strength, presumably due to the decrease in double-layer half-thickness and perhaps also due to some Stem layer adsorption of positive ions. The preexponential factor in Eq. XIV-7, ko = (8kr/3 ), should have the value of about 10 " cm, but at low electrolyte concentration, the values in the figure are smaller by tenfold or a hundredfold. This reduction may be qualitatively ascribed to charged repulsion. 

The first reliable energy band theories were based on a powerfiil approximation, call the pseudopotential approximation. Within this approximation, the all-electron potential corresponding to interaction of a valence electron with the iimer, core electrons and the nucleus is replaced by a pseudopotential. The pseudopotential reproduces only the properties of the outer electrons. There are rigorous theorems such as the Phillips-Kleinman cancellation theorem that can be used to justify the pseudopotential model [2, 3, 26]. The Phillips-Kleimnan cancellation theorem states that the orthogonality requirement of the valence states to the core states can be described by an effective repulsive... 

The parameters a and b are characteristic of the substance, and represent corrections to the ideal gas law dne to the attractive (dispersion) interactions between the atoms and the volnme they occupy dne to their repulsive cores. We will discnss van der Waals equation in some detail as a typical example of a mean-field theory. 

The interaction between ions of the same sign is assumed to be a pure hard sphere repulsion for r < a. It follows from simple steric considerations that an exact solution will predict dimerization only if i < a/2, but polymerization may occur for o/2 < L = o. However, an approximate solution may not reveal the fiill extent of polymerization that occurs in a more accurate or exact theory. Cummings and Stell [ ] used the model to study chemical association of uncharged atoms. It is closely related to the model for adliesive hard spheres studied by Baxter [70]. 

Weeks J D, Katsov K and Vollmayr K 1998 Roles of repulsive and attractive forces in determining the structure of non uniform liquids generalized mean field theory Phys. Rev. Lett. 81 4400... 

R), i.e. there is no effect due to caging of the encounter complex in the common solvation shell. There exist numerous modifications and extensions of this basic theory that not only involve different initial and boundary conditions, but also the inclusion of microscopic structural aspects [31]. Among these are hydrodynamic repulsion at short distances that may be modelled, for example, by a distance-dependent diffiision coefficient... 

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

In PPP-SCF calculations, we make the Bom-Oppenheimer, a-rr separation, and single-electron approximations just as we did in Huckel theor y (see section on approximate solutions in Chapter 6) but we take into account mutual electrostatic repulsion of n electrons, which was not done in Huckel theory. We write the modified Schroedinger equation in a form similar to Eq. 6.2.6... 

In the Huckel theory of simple hydrocarbons, one assumes that the election density on a carbon atom and the order of bonds connected to it (which is an election density between atoms) are uninfluenced by election densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to election repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method. 

One of the limitations of HF calculations is that they do not include electron correlation. This means that HF takes into account the average affect of electron repulsion, but not the explicit electron-electron interaction. Within HF theory the probability of finding an electron at some location around an atom is determined by the distance from the nucleus but not the distance to the other electrons as shown in Figure 3.1. This is not physically true, but it is the consequence of the central field approximation, which defines the HF method. 

---