Hamaker constant polarizability

A more detailed description of the interaction accounts for the variation of the polarizability of the material with frequency. Then, the Hamaker constant across a vacuum becomes... 

From the above equation, the variation of equilibrium disjoining pressure and the radius of curvature of plateau border with position for a concentrated emulsion can be obtained. If the polarizabilities of the oil, water and the adsorbed protein layer (the effective Hamaker constants), the net charge of protein molecule, ionic strength, protein-solvent interaction and the thickness of the adsorbed protein layer are known, the disjoining pressure II(x/7) can be related to the film thickness using equations 9 -20. The variation of equilitnium film thickness with position in the emulsion can then be calculated. From the knowledge of r and Xp, the variation of cross sectional area of plateau border Qp and the continuous phase liquid holdup e with position can then be calculated using equations 7 and 21 respectively. The results of such calculations for different parameters are presented in the next session. 

Polarizability Attraction. All matter is composed of electrical charges which move in response to (become electrically polarized in) an external field. This field can be created by the distribution and motion of charges in nearby matter. The Hamaker constant for interaction eneigy, A, is a measure of this polarizability. As a first approximation it may be computed from the dielectric permittivity, S, and the refractive index, n, of the material (15), where Vg is the frequency of the principal electronic absorption... 

Though the original work is difficult to understand very good reviews about the van der Waals interaction between macroscopic bodies have appeared [114,120], In the macroscopic treatment the molecular polarizability and the ionization frequency are replaced by the static and frequency dependent dielectric permittivity. The Hamaker constant turns out to be the sum over many frequencies. The sum can be converted into an integral. For a material 1 interacting with material 2 across a medium 3, the non-retarded5 Hamaker constant is... 

Hamaker Constant In a description of the London-van der Waals attractive energy between two dispersed bodies, such as droplets, the Hamaker constant is a proportionality constant characteristic of the droplet composition. It depends on the internal atomic packing and polarizability of the droplets. 

So-8X Aa Ay Ap AA (8o,xi 8ijXi) 2.2.3 local 8(x) slope 2.2.3 difference of segmental polarizabilities of two species 2.2.2 surface tension difference of pure blend components 3.1.2.2 difference of chemical potentials of species (pA-pB) 2.1 difference of Hamaker constant of species 3.1.2.1 ... 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

Hamaker constant in the case of interaction between two different phases in contact is defined by polarizability and density of both phases An (A[A2)112. In three-phase systems in which all three phases have significant concentration of molecules, one has to account for interactions of phases with each other and for those inside an intermediate phase, i.e. three Hamaker constants, A, are needed here the i and j indices are related to the corresponding phases. The decrease in gap thickness results in phases 1 and... 

Different assumptions lead to different expressions for VA (18, 24). The constant h is known as the Hamaker constant and depends on the density and polarizability of atoms in the particles. Typically 10-20 J < H < 10-19 J, or 0.25 kT < H < 25 kT at room temperature (24, 25). When the particles are in a medium other than vacuum, the attraction is reduced. In addition, this can be accounted for by using an effective Hamaker constant (26). The Hamaker constants are usually not well known and must be approximated. 

Since the Hamaker constant is a function of the polarizability of a molecule, LNA, having three double bonds, would have a relatively high value (>8) [5,17]. Plots of the viscosity of alumina suspension as a function of the Hamaker constant of the liquid medium, made by Rives [18], show that the viscosity minimum at a Hamaker constant = 9 for LNA in acidic solvents and = 8 for LNA in basic solvents. The values of Amm in Table 1 show that such solvents as toluene, methylene chloride, chloroform, and THF would be better media for smaller App/m and water the worst. This practice should apply to mixed solvent systems to tailor the Hamaker constant, solubility, electrostatic charge, and so on. This was presented in our recent work elsewhere [19]. 

Adsorption is based on the energetic properties of solid surfaces. At the solid-fluid interface, attractive and repulsive forces are acting on the molecules of the adsorbate (adsorbed molecules). The most important forces are van der Waals or dispersion forces and electrostatic forces. It will be shown later that the Hamaker constant, the electrical charge, the polarizability of the adsorbent molecules, and the dipole and quadrupole moments as well as the polarizability of the adsorptive molecules are the decisive properties for gas-sohd equilibria. These equilibria describe the relationship between the concentration of the adsorptive in the fluid phase and the loading of the adsorbent Principally speaking, two or more components of a fluid can be adsorbed. 

In the original treatment, also called the microscopic approach, the Hamaker constant was calculated from the polarizabilities and number densities of the atoms in the two interacting bodies. Lifshitz presented an alternative, more rigorous approach where each body is treated as a continuum with certain dielectric properties. This approach automatically incorporates many-body effects, which are neglected in the microscopic approach. The Hamaker constants for a number of ceramic materials have been calculated from the Lifshitz theory using optical data of both the material and the media (Table 9.1) (9). Clearly, all ceramic materials are characterized by large unretarded Hamaker constants in air. When the materials interact across a liquid, their Hamaker constants are reduced, but still remain rather high, except for silica. 

So far, the discussion is restricted to particles in vacuum. When the particles are immersed in a fluid, which is generally the case for colloidal systans, the same reasoning applies, but instead of the polarizability of the atoms in the particles, we must now consider the difference between the polarizability a, of the atoms in the particles and the polarizability ot2 of the atoms in the medium. The Hamaker constant for the particles (1) interacting across the medium (2), A,21, varies linearly with (a, - so that... 

The Hamaker constant can be calculated directly from London s theory (Chapter 2, Equation 2.6). The Hamaker constant values range typically between 0.4 and 4 x 10 J. The relatively constant values for different compounds arise because the parameter C (see Equation 2.6) is roughly proportional to the square of the polarizability which is roughly proportional to the square of the volume (or the inverse of number density), as discussed in Chapter 2. Thus the whole product (the Hamaker constant) is roughly constant, which is, of course, a gross oversimplification. 

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