Multipole forces

Mackrodt86 has used diagrammatic methods to investigate the effect on multipole forces of an external electric field. 

A general solution of Stokes equations can be obtained by analytic continuation of the interstitial velocity and pressure fields into the interior of the regions occupied by the spheres, replacing the particle interiors by singular multipole force distributions located at the sphere centers R (Zuzovsky et al, 1983). Explicitly, (v, p) satisfies the dynamical equation... 

Because of the low kinetic energy available, it is to be expected that centrifugal forces should be of importance, especially so for neutral-neutral reactions, without any charge/multipole force between the reactants. If ions are present, the Langevin approximation/capture theory is more than enough to calculate most chemical rates [83]. On the contrary, for neutral-neutral reactions the situations are vastly different [84]. In particular, some reaction rates diminish rapidly as temperature increases. 

It may be argued that London dispersion forces and multipole forces 51 would add energy to that computed from Kapustins-kiVs equation. This may be so, but this additional term is unlikely to exceed the enthalpy of formation of orthorhombic PtFe crystals from the vapor (i.e., -11 kcal-moIe-J) . Furthermore it is probable that reaction (1) is more exothermic than —25 kcal-mole , since the solid is not sublimable in vacuo below 90 C 2l. Even allowing for additional lattice energy terms, it is probable that E(PtF6) well exceeds —156 kcal-mole i. 

In the last 40 years, techniques to directly measure surface forces and force laws (force vs. separation distance between surfaces) have been developed such as the surface forces apparatus (SFA) [6] and AFM. Surface forces are responsible for the work required when two contacting bodies (such as an AFM tip in contact with a solid surface) are separated from contact to infinite distance. Although the physical origin of all relevant surface forces can be derived from fundamental electromagnetic interactions, it is customary to group these in categories based on characteristic features that dominate the relevant physical behavior. Thus, one speaks of ionic (monopole), dipole—dipole, ion—dipole interactions, electrostatic multipole forces (e.g., quadrupole), induced dipolar forces, van der Waals (London dispersive) interactions, hydrophobic and hydrophilic solvation, structural and hydration forces,... 

It should be noted that despite the similarity between the different estimates, there is still no perfect agreement with all experimental data. The total cross section, the low energy virial coefficient, the elastic constant and the Debye-temperature of the solid are examples of such deviations. Furthermore the question of the three-body forces is not solved completely. The fact that contributions of higher-order non-additive multipole forces were found to be... 

Molecular Rare gas atoms, molecules Dispersion and multipole forces Transparent, and like its molten form Insulators, dissolve in nonionizing solvents Fairly low melting Soft and plastically deformable Argon, paraffins, calomel... 

As pointed out in the introductory section of this chapter, there are two types of electrostatic interaction forces between neutral bodies the multipole-multipole (Keesom-type) forces and the induced multipole forces (Debye-Falkenhagen-type). 

Both far-field and lubrication interaction are computed by a multipole expansion in forces and torques (multipole force expansion—MEE). A low order expansion is used for the far-field interaction (M ), whereas for lubrication, multipoles of high... 

Higher multipoles. Similar interactions occur between higher multipole fields, but these tend to be weaker than typical dipole-dipole interactions. Nevertheless, the higher multipole forces are often significant for interactions between molecules that have no permanent electric dipole moment. These include the homonuclear diatomics, such as N2, and polyatomic molecules of high symmetry, such as BF3 or CH4. 

Eqs (l)-(8) describe the mechanics of multipole force arrays in an elastic solid for both permanent and induced material multipoles. The material multipoles are of course different for different constitutive laws ( see e.g. the micropolar solids [M], nonlocal solids [5], micropolar nonlocal solids [6] ).However, it has been shown that the concept of induced material multipole for any chosen material ( here for elastic solids) can even be applied to coupled materials (see [ ]) ... 

Again, a magnetoelastic composite material can be described by a fictitious equivalent elastic material with material constants C and constrained material multipole forces and... 

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