Short-range force fields

Incomplete Dissociation into Free Ions. As is well known, there are many substances which behave as a strong electrolyte when dissolved in one solvent, but as a weak electrolyte when dissolved in another solvent. In any solvent the Debye-IIiickel-Onsager theory predicts how the ions of a solute should behave in an applied electric field, if the solute is completely dissociated into free ions. When we wish to survey the electrical conductivity of those solutes which (in certain solvents) behave as weak electrolytes, we have to ask, in each case, the question posed in Sec. 20 in this solution is it true that, at any moment, every ion responds to the applied electric field in the way predicted by the Debye-Hiickel theory, or does a certain fraction of the solute fail to respond to the field in this way In cases where it is true that, at any moment, a certain fraction of the solute fails to contribute to the conductivity, we have to ask the further question is this failure due to the presence of short-range forces of attraction, or can it be due merely to the presence of strong electrostatic forces ... 

The mobility of ions in melts (ionic liquids) has not been clearly elucidated. A very strong, constant electric field results in the ionic motion being affected primarily by short-range forces between ions. It would seem that the ionic motion is affected most strongly either by fluctuations in the liquid density (on a molecular level) as a result of the thermal motion of ions or directly by the formation of cavities in the liquid. Both of these possibilities would allow ion transport in a melt. 

The remainder of Section I is devoted to a rather brief review of earlier work in the field in order to gain a little perspective. In Sections II to IV the basic results of the cluster method are derived. In Section V a very brief account of the application of the formal equations to some systems with short-range forces is given. Section VI is devoted to a review of the application to systems with Coulomb forces between defects, where the cluster formalism is particularly advantageous for bringing the discussion to the level of modern ionic-solution theory.86 Finally, in Section VII a brief account is given of Mayer s formalism for lattice defects69 since it is in certain respects complementary to that principally discussed here. We would like to emphasize that the material in Sections V and VI is illustrative of the method. This is not meant to be an exhaustive review of results obtainable. 

The major forms of van der Waals forces between molecules that are not bonded together are the permanent dipole-dipole interaction, the dispersion-induced temporary dipole interaction, and the hydrogen bond. They are short-range forces that operate only when two atoms or molecules are in close proximity. The Lennard-Jones potential of 6-12 is a model of this potential field ... 

The ionic model divides the forces acting on atoms into an electrostatic component that can be calculated using classical electrostatic theory and a short-range component that is determined empirically. The previous chapter explored the properties of the classical electrostatic field. This chapter explores the properties of the empirically determined short-range force which is represented in the electrostatic model by the bond capacitance, C,y, defined in eqn (2.8). Rather than try to determine the values of Cy directly, it is better to step back and look at the way in which the bond valence model developed historically. Its connection with the electrostatic model of Chapter 2 will then become apparent. 

The material properties of solids are affected by a number of complex factors. In a gas-solid flow, the particles are subjected to adsorption, electrification, various types of deformation (elastic, plastic, elastoplastic, or fracture), thermal conduction and radiation, and stresses induced by gas-solid interactions and solid-solid collisions. In addition, the particles may also be subjected to various field forces such as magnetic, electrostatic, and gravitational forces, as well as short-range forces such as van der Waals forces, which may affect the motion of particles. 

Hereby, the branches with E - and / -symmetry are twofold degenerated. Both A - and / d-modes are polar, and split into transverse optical (TO) and longitudinal optical (LO) phonons with different frequencies wto and wlo, respectively, because of the macroscopic electric fields associated with the LO phonons. The short-range interatomic forces cause anisotropy, and A - and / d-modcs possess, therefore, different frequencies. The electrostatic forces dominate the anisotropy in the short-range forces in ZnO, such that the TO-LO splitting is larger than the A -E splitting. For the lattice vibrations with Ai- and F -symmetry, the atoms move parallel and perpendicular to the c-axis, respectively (Fig. 3.2). 

In the field of intermolecular forces a book has been published by Kaplan241 which provides a coverage of the theory from long-range forces (including retardation effects) to short-range forces and nonadditivity. The determination of molecular potentials from experimental data is also considered in one chapter of this book. 

In order to account for the attraction between non-polar molecules on the basis of the electrostatic theory, it is necessary to regard the molecule as a quadrupole, which may be pictured as a molecule with two identical dipoles directed in opposite directions. The quadrupole may be either linear or rectangular as depicted in Figure 39. Such a treatment may be applied not only to molecules such as COs, which consist of two identical polar bonds pointing in opposite directions, but also to homopolar molecules such as H2. It is possible to calculate the field due to the quadrupole and to consider, in a way similar to that described above, two types of interaction, mutual orientation of the quadrupoles and polarization of one molecule in the field of another. The calculation shows that the field due to the quadrupole decreases more rapidly with the distance from the molecule than does the field due to the dipole. The attractive force is thus a short range force which rapidly falls to zero as the distance from the molecule is increased. 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

A Monte Carlo simulation [102] of a system with short-range forces confirmed these notions. The correlation function clearly exhibited exponentially damped oscillations. From the ratio of the wavelength and correlation length, the value of y characterizing the system could be obtained from Eq. (35), and it was found that 1 > y > 0, indicating that the microemulsion was structured but weakly so. Within the mean-field calculation, however, this is still strong enough that the middle phase should not wet the oil/water interface. However, measurement of all three interfacial tensions within the simulation revealed that Antonow s rule was obeyed, so that the interface was indeed wetted by the middle phase, an effect clearly attributable to the fluctuations included in the simulation. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

The commonest additional constraint has been a short-ranged attractive field near the bottom of the cell (z 0) which serves to anchor the liquid in the lower half, lliis field is chosen to mimic simply the attractive forces that would emanate from a uniform liquid phase below the bottom... 

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