Dispersion molecular theories

In several cases, the experimental plots were indeed linear. According to a molecular theory,11 39 40 the slope Hq and the intercept Sq are related to the chiral dispersion forces and to the entropic gain obtained by a twisted relationship between the polymer molecules (Figure 7.4), respectively. The value of Sq reflects the fact that the helices take up less space when packed in a helical arrangement and should be related to the hard-core steric contribution to the twist. Positive or negative values of Sq should be related to a right- or left-handed twist between the polymer molecules, respectively, and should be determined solely by steric effects. 

This form for reduced compliance is suggested by the dilute solution molecular theories, according to which JeR is governed by the dispersity of molecular relaxation times [from Eq. (4.12)] ... 

A general formula for calculation of the dispersion molecular interactions in any type of condensed phases has been proposed in [148], The attraction between bodies results from the existence of fluctuational electromagnetic field of the substance. If this field is known in a thin film, then it is possible to determine the disjoining pressure in it. The more strict macroscopic theory avoids the approximations assumed in the microscopic theory, i.e. additivity of forces integration extrapolation of interactions of individual molecules in the gas to interactions in condensed phase. The following function for IIvw was derived in [148] for thick free films... 

Abstract We are definitely witnessing an ever-increasing need to study dispersion molecular interactions that govern a weakly bound molecular world within the density functional theory. This chapter outlines the basic approaches currently undertaken to resolve this density functional paradigm. 

Thin liquid films can be formed between two coUiding emulsion droplets or between the bubbles in foam. Formation of thin films accompanies the particle-particle and particle-wall interactions in colloids. From a mathematical viewpoint, a film is thin when its thickness is much smaller than its lateral dimension. From a physical viewpoint, a liquid film formed between two macroscopic phases is thin when the energy of interaction between the two phases across the film is not negligible. The specific forces causing the interactions in a thin liquid film are called surface forces. Repulsive surface forces stabilize thin films and dispersions, whereas attractive surface forces cause film rupture and coagulation. This section is devoted to the macroscopic (hydrostatic and thermodynamic) theory of thin films, while the molecular theory of surface forces is reviewed in Section 4.4. 

In this section, we review the molecular theory of surface forces with special attention to the effect of surfactant adsorption and surfactant micelles on the interactions in the thin liquid films and between the particles in dispersions. 

The Hamaker approach of pairwise addition of London dispersion forces is approximate because multi-body intermolecular interactions are neglected. In addition, it is implicitly assumed in the London equation that induced dipole-induced dipole interactions are not retarded by the finite time taken for one dipole to reorient in response to instantaneous fluctuations in the other. Because of these approximations an alternative approach was introduced by Lifshitz. This method assumes that the interacting particles and the dispersion medium are all continuous i.e. it is not a molecular theory. The theory involves quantum mechanical calculations of the dielectric permittivity of the continuous media. These calculations are complex, and are not detailed further here. 

Hard-core repulsion between anisotropic molecules, discussed in the previous subsection, can be the driving force of the I-N transition in lyotropic systems. In contrast, in thermotropic liquid crystals the transition occurs at some particular temperature and therefore some attraction interaction must be involved. The corresponding molecular theory, based on anisotropic dispersion interactions, was proposed by Maier and Saupe[ll, 12]. 

We note that the full dispersion interaction energy (Eq. 24) can be approximated to by the dipole-dipole term (Eq. 25) only if the molecules are sufficiently far apart. However, Eq. (25) is often used in molecular theories of liquid crystals to draw qualitative conclusions. For example, the potential (Eq. 25) has been used in the Maier-Saupe theory. 

The specific value includes both polar and hydrograi bonding effects. Noticeably, the dispersion value indicates that approximately 30% of the surface tension value of water is due to dispersion forces, which agrees well with theoretical considerations and the molecular theories mentioned in Chapter 2. (The contribution of dispersion to water s potential energy is around 24% at 298 K.) This illustrates the importance of the van der Waals forces even for strongly hydrogen bonding compounds such as water. The dispersion surface tension value of water from the Fowkes theory can be considCTed to be widely accepted and is therefore used in further calculations. 

It represents a reasonable percentage of dispersion forces in agreement with molecular theories. 

In the case of the induction and dispersion energies the treatment of the shape divergence requires a partitioning of molecular polarizability into molecular fragments. The theory of distributed multipole polarizabilities was initiated by Stone and further developed by Le Sueur and Stone. Recently Angyan and collaborators used the partitioning of molecular volume into disjoint domains (basins) provided by Bader s atoms in molecular theory to define distributed moments and dynamic polarizabilities in a rigorous, basis set independent way ... 

The starting point for the development of the contribution of the rods to the extra stress is Doi s molecular theory for mono-disperse rod-like molecules suspended in a Newtonian fluid. The theory begins with the dilute solution case where a rod is free to rotate and translate without interacting with other rods. It was then extended to concentrated systems which spontaneously become anisotropic after a critical concentration without the presence of any external fields due to excluded volume effects [11]. 

The dispersion coefficient is orders of magnitude larger than the molecular diffusion coefficient. Some rough correlations of the Peclet number are proposed by Wen (in Petho and Noble, eds.. Residence Time Distribution Theory in Chemical Tngineeiing, Verlag Chemie, 1982), including some for flmdized beds. Those for axial dispersion are ... 

The theory of molecular interactions can become extremely involved and the mathematical manipulations very unwieldy. To facilitate the discussion, certain simplifying assumptions will be made. These assumptions will be inexact and the expressions given for both dispersive and polar forces will not be precise. However, they will be reasonably accurate and sufficiently so, to reveal those variables that control the different types of interaction. At a first approximation, the interaction energy, (Ud), involved with dispersive forces has been calculated to be... 

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