Born repulsive potential

When the thickness of the film, t, is sufficiently small the Born repulsion potential, Vb, together with other less well understood contributions to the potential, U, become important. In the region of the common black film the Vb and U are negligible, but become significant in the region of the Newton black film. 

The potential energy V of an ionic crystal can be expressed approximately as the sum of the Coulomb interaction and the Born repulsive potential ... 

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.

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)). 

The characteristic repulsive potential falls off very rapidly in value with increase in r, . It was suggested by Born that it be approximated by an inverse power of rt , so that the mutual potential energy of two ions would be written as... 

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. 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

As already mentioned the present treatment attempts to clarify the connection between the sticking probability and the mutual forces of interaction between particles. The van der Waals attraction and Born repulsion forces are included in the calculation of the rate of collisions between two electrically neutral aerosol particles. The overall interaction potential between two particles is calculated through the integration of the inter-molecular potential, modeled as the Lennard-Jones 6-12 potential, under the assumption of pairwise additivity. The expression for the overall interaction potential in terms of the Hamaker constant and the molecular diameter can be found in Appendix 1. The motion of a particle can no longer be assumed to be... 

Bora repulsion is a short-range molecular interaction, resulting from the overlap of electron orbitals. It is the twelfth-order term of the empirical Lennaid-Jones 6-12 potential. To estimate the Born repulsion between a sphere and a plate, the authors of die present paper assumed these molecular interactions also could be linearly superimposed to obtain... 

In order to calculate the potential well, Born repulsion should be included in the calculations. Inslead of using an explicit expression for the Born forces, the interaction potential U is cut at a distance of about 4 A. Considering <5 of the order of 24 A, x-4A, >fp-l5-10 20 J and A — 7.5 10 20 J, one obtains that the optimum At is given by the expression... 

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. 

As mentioned earlier, the shell model is closely related to those based on polarizable point dipoles in the limit of vanishingly small shell displacements, they are electrostatically equivalent. Important differences appear, however, when these electrostatic models are coupled to the nonelectrostatic components of a potential function. In particular, these interactions are the nonelectrostatic repulsion and van der Waals interactions—short-range interactions that are modeled collectively with a variety of functional forms. Point dipole-and EE-based models of molecular systems often use the Lennard-Jones potential. On the other hand, shell-based models frequently use the Buckingham or Born-Mayer potentials, especially when ionic systems are being modeled. 

This equation was first used by Born, Huggins, and Meyer and therefore bears their names. The first two terms represent, respectively, the attractive and repulsive potentials. The last two terms represent dipole-dipole and dipole-quadrupole potentials, respectively. In spite of allowing for the dipole interactions, the calculation is still a hard-sphere one, a mean spherical approximation, because the forces are not allowed to change the shape and the position of the particles. Later on, Saboungi et al. 

D. L. Anderson and O. L. Anderson (1970) showed that, for an ionic crystal with electrostatic attractive forces and a Born power-law repulsive potential, the bulk modulus K is given by... 

The first term is the classical interaction between point charges and the second term is a repulsive Born-Mayer potential. 

Here qi is the effective charge of an atom is a dispersion interaction constant and Ay and bij are parameters of the Born - Mayer atom-atom repulsion potential. To calculate the long range Coulomb term in Eq. (1) one generally has to employ the Ewald summation technique. To obviate this inconvenience, the Coulomb term has been multiplied by the screening factor (and the dispersion term has been neglected) ... 

DFT quantum-chemical codes, the Born repulsive interaction is accurately computed. Expressions (4a) and (4b) are two-body interaction potentials commonly used in codes that predict energies or geometries based on empirical potentials. 

Differentiation of this expression with respect to r, where the repulsive and attractive forces are in equilibrium, produces the Born-Lande potential for one pair of ions. For one mole of ions the expression must be multiplied by Avogadro s number (A ). 

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