Hamaker constant model equation

Fig. 4. Comparison of the dimensionless coagulation coefficient y calculated through Monte Carlo simulation () with the previous models (1) for equal-sized particles of unit density, in air, at 298 K and 1 atm for a Hamaker constant of 10 12 erg. Curve 1 refers to the values of dimensionless coagulation coefficient y calculated from the equation derived previously for large particles (1). Curve 2 refers to the values of the dimensionless coagulation coefficient y calculated from the equation derived previously for sufficiently smalt particles (1) (the lower bound).

Figure 6.8. Variation with separation distance r of the interaction energy due to van der Waals forces as calculated by the pairwise interaction model, between two different atoms (a) and between two surfaces (b). A, the Hamaker constant, equals n2C p,p2 where C is the constant of the atom-atom pair potential in equation (6.11) (case a) and p 1 and p2 are the number of atoms per unit volume in the...

In a number of alternative approaches, equations similar to (5.7.1 and 2) are derived, ignoring the finite thickness of the interfacial region, and/or applying Lifshits theory to obtain (the equivalents of) Hamaker constants (eq. (I.4.7.7) or its variants). In such models our parameter is replaced by the distance of shortest approach between the adjoining phases. However, as is as esoteric as, if only because surfaces are rarely flat on a molecular scale, these alternatives offer little comfort. We recall that our wedge approach (fig. 2.7) obviates the introduction of... 

Values of e, n and ve and Hamaker constants for two identical types of a material in a vacuum, which are calculated from Equation (567) by taking e3 = 1 and 3 = 1, are given in Table 7.1. Unfortunately, the lack of material constants, such as the dielectric constant, as a function of frequency for most of the substances, and also the complexity of the derived formulae have hampered the general use of the Lifshitz model. However, Lifshitz theory made possible the advent of the first theories on the stability of hydrophobic colloids as a balance between London attraction and electrical double-layer repulsion. Later, these theories were further elaborated by Derjaguin and Landau, and independently by Verwey and Overbeek. The general theory of colloidal stability (which is beyond the scope of this book) is based on Lifshitz theory and has become known as the DLVO theory, by combining the initials of these four authors. 

Healy (Chapter 7) and Dumont also prefer the first approach. Healy sets down a model based on the control of coagulation by surface steric barriers of polysilicate plus bound cations. Healy s electrosteric barrier model is designed to stimulate new experimental initiatives in the study of silica sol particles and their surface structure. Dumont believes that many particular aspects of the stability of silica hydrosols could be explained not only by the low value of the Hamaker constant but also by the relative importance of the static term of the Hamaker equation. 

As cells start to adhere to the surface, the uniformity in its characteristics is disturbed. The altered spots on the surface, which consist of adhering cells, present new characteristics concerning the surface potential and Hamaker s constant, and consequently, the rate of cellular deposition is not uniform over the surface. The present model is based on the simplified assumption that the overall rate of deposition is the arithmetic sum of two contributions deposition on the bare surface and deposition on the altered surface. Each of them has its own time constant and depends also on the fraction of the area which is already covered, X. Therefore, by extending equation (19), the overall rate of deposition is given by ... 

In equation (6.5a) and (6.5b), ty is the peld stress, n is ttie power law index, K is the consistency index, C is a positive constant representing ttie total number of nearest neighbours of each sphere, Ah is Hamaker s constant, the maximum volume fraction, Cq tihe dielectric constant of the matrix, ttie thickness of the electrostatic interaction layer, lf the surfece potential of the particulates, D tite particle diameter and y the shear rate. Basically, equation (6.4) along with (6.5b) is identical to the empirical Herschel-Bulkley [91] model given by equation (2.45), other theories which relate yield stress to volume fraction and particle size, and these are available in Rajaiah [93]. 

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