The repulsive force

The robustness of the bond valence model derives, therefore, from two factors the independence of the bond valences from the location of the bonding electrons, and the use of fitted values for the bond valence parameters which automatically compensates for systematic changes in the bond length produced by other effects. 

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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).

The van der Waals attraction arises from tlie interaction between instantaneous charge fluctuations m the molecule and surface. The molecule interacts with the surface as a whole. In contrast the repulsive forces are more short-range, localized to just a few surface atoms. The repulsion is, therefore, not homogeneous but depends on the point of impact in the surface plane, that is, the surface is corrugated. 

This term describes the repulsive forces keeping two nonbonded atoms apart at close range and the attractive force drawing them together at long range. 

On August 29,1982, physicists at the Heavy Ion Research Laboratory, Darmstadt, West Germany made and identified element 109 by bombing a target of Bi-209 with accelerated nuclei of Fe-58. If the combined energy of two nuclei is sufficiently high, the repulsive forces between the nuclei can be overcome. 

Van der Waals radius (Section 2 17) A measure of the effec tive size of an atom or a group The repulsive force between two atoms increases rapidly when they approach each other at distances less than the sum of their van der Waals radii Van der Waals strain (Section 3 2) Destabilization that results when two atoms or groups approach each other too closely Also known as van der Waals repulsion Vicinal (Section 6 14) Describing two atoms or groups at tached to adjacent atoms... 

The above potential is referred to as a Lennard-Jones or 6-12 potential and is summed over all nonbonded pairs of atoms ij. The first positive term is the short range repulsion and the second negative term is the long range attraction. The parameters of the interaction are Aj and B... The convenient analytical form of the 6-12 potential means that it is often used, although an exponential repulsion term is usually considered to be a more accurate representation of the repulsive forces (as used in MM-t). 

On a given metallic particle, the repulsive force, E, is dependent on particle mass, AF electrical conductivity. O density, p and shape, s. 

Fig. 18. The repulsion force from adsorbed particles is greater than the van der Waals force between flocculated emulsion droplets under certain...

The molecules of liquids are separated by relatively small distances so the attractive forces between molecules tend to hold firm within a definite volume at fixed temperature. Molecular forces also result in tlie phenomenon of interfacial tension. The repulsive forces between molecules exert a sufficiently powerful influence that volume changes caused by pressure changes can be neglected i.e. liquids are incompressible. 

Whereas the JKR model approached the topic of particle adhesion from a contact mechanics viewpoint, the DMT theory simply assumes that the adhesion-induced contact has the same shape as a Hertzian indentor. The normal pressure distribution Ph(p) for the Hertzian indentor is related to the repulsive force and the distance from the center of the contact circle to the point represented by r according to the relationship [49]... 

Van der Waals radius (Section 2.17) A measure of the effective size of an atom or a group. The repulsive force between two atoms increases rapidly when they approach each other at distances less than the sum of their van der Waals radii. 

Different samples of aqueous solution containing radionuclides of Co and Eu were prepared at different copper sulphate concentrations and constant polymer concentrations (pAM) of 15 mg/1. The addition of salt to the system was done to reduce both the repulsion forces between the radionuclides and the interaction between the polymeric chains [7]. The polymer efficiency for the prepared samples was determined, results are shown in Fig. 15. It is clear that the polymer efficiency for Eu " is higher than for Co. This can be explained by the difference in the tightly bound structured water associated with different cationic species [14,107]. On this basis, we expect that Co is more hydrated than Eu. This is due to the difference in the ionic size. The hydra-... 

A measurement of the density of helium gas shows that it is a monatomic gas. Molecules of He2 do not form. What difference between hydrogen atoms and helium atoms accounts for the absence of bonding for helium The answer to this question also must lie in the attractive and repulsive electrical interactions between two helium atoms when they approach each other. Figure 16-4A shows the attractive forces in one of our hypothetical instantaneous snapshots. There are, of course, four electrons and each is attracted to each nucleus. In Figure 16-4B we see the repulsive forces. Taking score, we find in Figure 16-4A eight attractive interactions, four... 

By comparing these two answers we can see that the repulsive force between two protons in the nucleus is about ten billion times as great as the repulsive force between two protons bound together in a hydrogen molecule. In order to overcome these enormous intranuclear coulomb repulsions and hold the nucleus together there must exist some very strong attractive forces between the nucleons. The nature of these forces is not understood and remains a very important problem in physics. 

The temperature-independent van der Waals parameters a and b are unique for each gas and are determined experimentally (Table 4.5). Parameter a represents the role of attractions so it is relatively large for molecules that attract each other strongly and for large molecules with many electrons. Parameter b represents the role of repulsions it can be thought of as representing the volume of an individual molecule (more precisely, the volume per mole of molecules), because it is the repulsive forces between molecules that prevent one molecule from occupying the space already occupied by another molecule. 

Hitherto it has been generally assumed that the repulsive forces between atoms arise from the interaction of the quadrupole (and higher) moments of the atoms, that is, from their departure from spherical symmetry.7... 

This assumption is no longer valid in its place the wave mechanics provides the simple explanation that the repulsive forces arise from the interpenetration of the atoms. As a simple example, we may consider the hydrogen ion and the chloride ion according to the wave mechanics the potential energy of these two ions at a distance R apart, assuming that no deformation occurs, is6... 

The potential energy of an ionic crystal (ions of valence z) may be written = —a(e2z2)/i + R), the first term representing the Coulomb energy, and the second the potential of the repulsive forces. Equation 6 suggests a simple form for [Pg.260]

The forces between ions have been discussed by Lennard-Jones and his collaborators, who have given tables showing the repulsive forces as a function of the repulsion exponent n [Lennard-Jones and Dent, Proc. Roy. Soc., 112A, 230 (1926)]. In conjunction with Wasastjema s radii, these tables have been used in the theoretical treatment of crystals such as caldte, CaCOj, which, however, we consider not to be composed of monatomic ions. Thus, they assume C+1 and O" to be present in caldte [Lennard-Jones and Dent, Proc. Roy. Soc., 113A, 673, 690 (1927)], although the carbonate ion is generally believed by chemists to contain shared-electron bonds. 

The experimental values for the lithium halides are high. This is due to two different phenomena. In the case of the chloride, bromide and iodide the anions are in mutual contact, that is, the repulsive forces operative are those between the anions, and the anion radius alone determines the inter-atomic distances. The geometry of the sodium chloride structure requires that, for less than 0.414, the anions come into contact... 

In the case of lithium fluoride the ratio R i /Rj>- is 0.44. In this crystal there is double repulsion the repulsive forces between anion and anion and those between anion and cation are simultaneously operative. The inter-atomic distances are determined neither by the sum of the radii for the anion and cation nor by the radius of the anion alone, but are larger than those calculated by either method. Thus the lithium-fluoride distance is 0.05 A. (2.5%) larger than the sum of the radii, and one-half the fluorine-fluorine distance is 0.06 A. larger than the fluoride radius. 

However, the energy of the repulsive forces must be taken into account moreover, the possibility of a difference in R in the two structures must... 

These considerations also explain the occurrence of cases of dimorphism involving the sodium chloride and cesium chloride structures. It would be expected that increase in thermal agitation of the ions would smooth out the repulsive forces, that is, would decrease the value of the exponent n. Hence the cesium chloride structure would be expected to be stable in the low temperature region, and the sodium chloride structure in the high-temperature region. This result may be tested by comparison with the data for the ammonium halides, if we assume the ammonium ion to approximate closely to spherical symmetry. The low-temperature form of all three salts, ammonium chloride, bromide and iodide, has the cesium chloride structure, and the high-temperature form the sodium chloride structure. Cesium chloride and bromide are also dimorphous, changing into another form (presumably with the sociium chloride structure) at temperatures of about 500°. 

The possibility of the existence of two forms of titanium dioxide, rutile and anatase, is evident from the identity of their Coulomb energies taking the energy of the repulsive forces and of possible deformation into account, it is seen that the choice between the two structures would depend on the thermodynamic environment during crystallization. In... 

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