Born-repulsion

In the H2+ ion the bonding is a consequence of the (place) exchange of one electron. In the hydrogen molecule the bonding depends on the zVzterchange of two electrons which form an electron pair with antiparallel spins. In the molecule He2+ with two nuclei and three electrons we have a three electron bond, which as in H2+ depends on (place) exchange or, otherwise stated, on resonance between HeHe+ and He+He. Here an electron pair is exchanged for a separate electron. 

It follows from these figures that the electron pair bond is obviously stronger than the two other types. In fact these latter practically never occur in stable molecules (exception 02, p. 229). 

If we go another step forward and consider a molecule which 

The electron clouds of the two atoms cannot penetrate one another, since this would be in contradiction with the Pauli principle precisely as in the antisymmetrical orbital function of H2, since the spins are always parallel two and two. 

Here one sees the cause of the general repulsion, first, of all ions with closed electron configurations but, in addition, of all atoms which are not bound to each other by an electron pair. Indeed with two hydrogen atoms the probability is a priori 3 for the repulsive state against i for the attractive symmetrical state with antiparallel spins (p. 145). 

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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)]. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

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.

Fumi F. G. and Tosi M. P. (1964). Ionic sizes and Born repulsive parameters in the NaCl-type alkali halides, I The Huggins-Mayer and Pauling forms. J. Phys. Chem. Solids, 25 31 3. 

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)). 

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. 

Another interface that needs to be mentioned in the context of polarized interfaces is the interface between the insulator and the electrolyte. It has been proposed as a means for realization of adsorption-based potentiometric sensors using Teflon, polyethylene, and other hydrophobic polymers of low dielectric constant Z>2, which can serve as the substrates for immobilized charged biomolecules. This type of interface happens also to be the largest area interface on this planet the interface between air (insulator) and sea water (electrolyte). This interface behaves differently from the one found in a typical metal-electrolyte electrode. When an ion approaches such an interface from an aqueous solution (dielectric constant Di) an image charge is formed in the insulator. In other words, the interface acts as an electrostatic mirror. The two charges repel each other, due to the low dielectric constant (Williams, 1975). This repulsion is called the Born repulsion H, and it is given by (5.10). 

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. 

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... 

A model for Brownian coagulation of equal-sized electrically neutral aerosol particles is proposed. The model accounts for the van der Waals attraction and Born repulsion in the calculation of the rate of collisions and subsequent coagulation. In this model, the relative motion between two particles is considered to be free molecular in the neighborhood of the sphere of influence. The thickness of this region is taken to be equal to the correlation length of the relative Brownian motion. The relative motion of the particles outside this region is described... 

The Monte Carlo simulation of Brownian coagulation involves the evaluation of the ensemble average of the coagulation rate over a large number of particle pairs, through the generation of particle trajectories. The inter-particle forces due to the van der Waals attraction and Born repulsion are accounted for in the description of the relative motion [40] two Particles. The relative Brownian motion of two particles is described by the... 

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-... 

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... 

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... 

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. 

Originally one was satisfied to take over a model from macroscopic mechanics, namely that of completely hard spheres (b-correction of Van der Waals) or of almost hard spheres (Born repulsion, p. 36). It did indeed prove possible to obtain interesting results with this simple model (the ionic and Van der Waals radii). The work of Heitler and London on the hydrogen molec ule has laid the foundation for a more correct insight in this case also (p. 147). 

The small influence of the polarization is seen from the quoted calculation of the decrease of separation in the transition from lattice to vapour molecule in which the Coulomb attraction and the Born repulsion only are taken into account. The dipole moments of the molecules of the alkali halides point to a lower value than that corresponding to e-d. It is difficult, both experimentally and theoretically, to distinguish the in-... 

It is not yet quite certain what the explanation of these hindrances is to be. The mutual Born repulsion of the hydrogen atoms attached to different carbon atoms certainly plays an important part, but this probably cannot be made responsible for the whole amount of energy. As another cause one can mention the interaction of dipole (and especially quadrupole) moments of the C—H bonds (Lassettre and Dean, Oosterhoff). 

Since, however, the extra R.E., which stabilizes the coplanarity, is small, it is understandable that the plane structure is lost through steric hindrance (Born repulsion) of groups in the ortho positions (2, 2, 6, 6 ) (optically active compounds). 

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