Born repulsion energy

The value of m is usually 10-12, while that of n depends on the nature of attractive forces. A steep increase in the Born repulsion energy is observed as the molecules closely approach each other (Fig. 1-8). As a result, the potential well depth, u, for small n values (corresponding to the Coulombic interaction of ions) is mainly determined by the attractive energy of molecules, corresponding to the equilibrium separation distance. 

If we neglect Born repulsion between the two anions and between the two cations, i.e. if we assume that only the nearest neighbors touch one another, the Born repulsion energy is... 

Eq. (16) gives the total attraction energy at the molecular level. It indicates that the attraction energy becomes more negative as the separation distance decreases. When the separation distance becomes so small, to the extent that the electron clouds of two units start to overlap, a repulsive force named as the Born repulsion energy is generated and can be expressed as ... 

Particles with anchored surfactant layers cannot come into direct contact. The distance of closest apprrwch can be, at most, two time the adsorption layer thickness. Using nm particles the F<,-energy of particle core no longer acts at closest approach, therefore the Van der Waals attraction energy of the adsorbed layer (F,) [15], the Born repulsive energy (Fb) [16], and the of the external ionic adsorption layer determine the colloidal stability of the dispersed particles. 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

The DLVO theory, with the addition of hydration forces, may be used as a first approximation to explain the preceding experimental results. The potential energy of interaction between spherical particles and a plane surface may be plotted as a function of particle-surface separation distance. The total potential energy, Vt, includes contributions from Van der Waals energy of interaction, the Born repulsion, the electrostatic potential, and the hydration force potential. [Israelachvili (13)]. 

Schematic forms of the curves of interaction energies (electrostatic repulsion Vr, van der Waals attraction Va, and total (net) interaction Vj) as a function of the distance of surface separation. Summing up repulsive (conventionally considered positive) and attractive energies (considered negative) gives the total energy of interaction. Electrolyte concentration cs is smaller than cj. At very small distances a repulsion between the electronic clouds (Born repulsion) becomes effective. Thus, at the distance of closest approach, a deep potential energy minimum reflecting particle aggregation occurs. A shallow so-called secondary minimum may cause a kind of aggregation that is easily counteracted by stirring.

Table 5.37 Lattice energy terms for C2/c pyroxenes. Values in kJ/mole. bhf = energy of Born-Haber-Fayans thermochemical cycle U- = lattice energy Ec = coulombic energy = repulsive energy Edd = dipole-dipole interactions E q = dipole quadrupole interactions =...

A second major contributor or the lattice energy of an ionic crystal is the repulsive energy. Following Born and Huang (1954), the repulsive energy per mole may be written as... 

To prevent misunderstanding (94), we emphasize that neither experimental hydration energies nor experimental coordination numbers are necessary for these calculations. Moreover, the coordination numbers obtained are generally not comparable to empirical hydration numbers. The only experimental quantities that enter the calculations are a) cationic radius and charge b) van der Waals radius of water c) dipole and quadrupole moment of water d) polarizabilities e) ionization potentials and f) Born repulsion exponents as well as fundamental constants (see Ref. (92)). 

We expect methane to be formed by the combination of an H-atom with the remaining unpaired pz orbital of CH3. If the principal configuration is still that involving the C 5 state and its nondirectional character predominates, we expect methane to be tetrahedral, thereby minimizing the repulsion energy between pairs of H atoms. This is borne out by the calculations as we see in Table 13.7. 

For the linear model cos cp = 0, sin (p = 1, and it will only be stable when d2E/d(p2 > 1 so that (1 — 8oc/r3) > 0 and (8oc/r3) < 1. Therefore, if we could allow the value 8a/r3 of a molecule to increase continuously, when it reaches unity the molecule would begin to have lower symmetry. If we still wish to correct the energy for the change of r, then we must differentiate the expression for E with respect to r and eliminate B. At this point, however, the exact calculation can be taken no further because in hydrogen compounds no normal Born repulsion exists. The calculation can be carried through successfully for all molecules not containing hydrogen ions. 

Curve P represents the physical interaction energy between M and X2. It inevitably includes a short-range negative (attractive) contribution arising from London-van der Waals dispersion forces and an even shorter-range positive contribution (Born repulsion) due to an overlapping of electron clouds. It will also include a further van der Waals attractive contribution if permanent dipoles are involved. The nature of van der Waals forces is discussed on page 215. 

Repulsion due to overlapping of electron clouds (Born repulsion) predominates at very small distances when the particles come into contact, and so there is a deep minimum in the potential energy curve which is not shown in Figures 8.2-8.4. 

Several repulsive and attractive forces operate between colloidal species and determine their stability [12,13,15,26,152,194], In the simplest example of colloid stability, dispersed species would be stabilized entirely by the repulsive forces created when two charged surfaces approach each other and their electric double layers overlap. The overlap causes a coulombic repulsive force acting against each surface, which will act in opposition to any attempt to decrease the separation distance (see Figure 5.2). One can express the coulombic repulsive force between plates as a potential energy of repulsion. There is another important repulsive force causing a strong repulsion at very small separation distances where the atomic electron clouds overlap, called Born repulsion. 

Physical adsorption arises bom physical interactions between the suspended particles and die collector, such as van der Waals attraction, double-layer repulsion, and Born repulsion. The total interaction energy, as a function of the particle-collector gap width, displays either one minimum and no maximum or two minima and one maximum. Several mechanisms for chromatographic sep-... 

By including Born repulsion in the calculation of the interaction energy profile, the primaty minimum is finite and depends on ionic strength. Allowing the primary to be above the secondary minimum one is... 

Since this approach does not account for long-range electrostatic potentials present in the extended material, the second approach chosen was the rigid-ion lattice energy minimization technique, widely used in solid-state chemistry. This method is based on the use of electrostatic potentials, as well as Born repulsion and bond-bending potentials parametrized such that computed atom—atom distances and angles and other material properties, such as, for instance, elastic constants, are well reproduced for related materials. In our case, parameters were chosen to fit a-quartz. 

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