Effective Hamaker constant equation

The effective Hamaker constant equation shows that the attraction between particles is weakest when the particles and medium are most chemically similar (Ai As). The Hamaker constants are usually not well known and must be approximated. 

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. 

It is understood that A in Equation (1) is the effective Hamaker constant A2I2 for the system. Of the variable parameters in this equation, it is the one over which we have least control its value is determined by the chemical nature of the dispersed and continuous phases. The presence of small amounts of solute in the continuous phase leads to a negligible alteration of the value of A for the solvent. 

The presence of a liquid dispersion medium, rather than a vacuum (or air), between the particles (as considered so far) notably lowers the van der Waals interaction energy. The constant A in equations (8.8)-(8.10) must be replaced by an effective Hamaker constant. Consider the interaction between two particles, 1 and 2, in a dispersion medium, 3. When the particles are far apart (Figure 8.1a),... 

Equations (2)-(4) show that the total potential energy of interaction between two colloidal spherical particles depends on the surface potential of the particles, the effective Hamaker constant, and the ionic strength of the suspending medium. It is known that the addition of an indifferent electrolyte can cause a colloid to undergo aggregation. Furthermore, for a particular salt, a fairly sharply defined concentration, called critical aggregation concentration (CAC), is needed to induce aggregation. 

From the preceding discussion of the DLVO theory and equations 9.1-9.5 and 9.8, it is apparent that the stability of a lyophobic dispersion is a function of the particle radius and surface potential, the ionic strength and dielectric constant of the dispersing medium, the value of the Hamaker constant, and the temperature. Stability is increased by increase in the particle radius or surface potential or in the dielectric constant of the medium and by decrease in the effective Hamaker constant, the ionic strength of the dispersing liquid, or the temperature. 

The expression in brackets in this equation is the effective Hamaker constant for two particles of A and B separated by a medium C. 

In this case, the effective Hamaker constant is always positive (i.e., the interaction between like particles is an attractive one). But the strength of the attraction is usually less than if the dispersing medium C were absent, and effective Hamaker constants not much greater than 1(E " erg (Kh i J) have been reported. We note that effective Hamaker constants of the forms indicated in Equations 3.14 and 3.15 may be used in any of the expressions of Equations 3.5 through 3.10 to account for the continuous phase C of a colloidal dispersion. 

It is important to note that Eqs. (3.1) and (3.2) apply to colloidal particles in a vacuum. If there is instead a liquid medium between the particles, the van der Waals potential is substantially reduced. The Hamaker constant in these equations is then replaced by an effective value. Consider the interaction between two colloidal particles 1 and 2 in a medium 3. If the particles are far apart, then effectively each interacts with medium 3 independently and the total Hamaker constant is the sum of two particle-medium terms. However, if particle 2 is brought close to particle 1, then particle 2 displaces a particle of type 3. Then particle 1 is interacting with a similar body (particle 2), the only difference being that molecules of particle 2 have been replaced by those of medium 3. Thus the potential energy change associated with bringing particle 2 close to particle 1 in the presence of medium 3 is less than it would be in vacuo. The effective Hamaker constant is thus a sum of particle-particle plus medium-medium contributions. 

For octane particles in water we can estimate the effective Hamaker constant using Equation 2.7b and obtain ... 

Figure 10.3 DLVO theory gives the potential energy as the sum of the electrostatic (repulsive) and the van der Waals (attractive) forces. The equation shown here is for equal-sized spherical particles (H R). If the particles Interact over a (liquid) medium (as opposed to vacuum), the Hamaker constant (A) becomes an effective Hamaker constant. The secondary shallow minimum (of a few kaTj at rather high separations indicates reversible flocculation. The aggregates are rather loose (weak) and are called floes. This secondary minimum has been confirmed experimentally and often can disappear with a small energy input, e.g. gentle stirring...

It has been shown (IsraelachvUi, 2011) that for diameters beyond 0.5 nm a molecule must be considered a small particle and the equations of Table 2.2 should be used for the van der Waals forces. The effective Hamaker constant, which depends on the particles (chemical composition) and the medium, determines the effective strength of the vdW interactions. 

Equation 10.6 is true when we have the same type of particles in a medium, e.g. polystyrene particles in water. However, one simple equation for the effective Hamaker constant in the case of two different types of particles (1 and 3) in a medium 2 is ... 

Thus, in the (often encountered) case of particles of the same type or when air/vacuum are the medium (zero Hamaker constant for vacuum and almost zero for air). Equations 10.6 and 10.7 lead always to a positive Hamaker constant and attractive van der Waals forces, which is the most usual case. However, in the case of unlike particles and when the Hamaker constant. A, of the medium has a value in between that of two different types of particles, the effective Hamaker constant can be negative, which implies repulsive van der Waals forces. In this case one material interacts more strongly with the medium than with the second body. 

A more rigorous estimation of the effective Hamaker constant for mixtures avoids simplified combining rules and uses the Lifshitz theory (based on relative permittivities and refractive indices see Israelachvili (2011) and Chapter 2, Equation 2.8). The Lifshitz theory is particularly useful for calculating the van der Waals force for any surface and in any medium, also because it relates the Hamaker constant with the material properties (relative permittivity and refractive index). Thus, the theory shows how the van der Waals forces can be changed via changing the Hamaker constant. The Lifshitz theory is a continuum theory, i.e. the dispersion medium, typically water, is... 

In this case, the Hamaker constant, A, has been replaced by an effective Hamaker constant, A123, which is valid for two different materials in a liquid medium. The value of A123 can be calculated from Equation 10.7 ... 

This is the effective value of the Hamaker constant to be used in evaluating the attraction between (like) particles embedded in a medium. Equation (79) leads to three important generalizations about the value of A2l2 ... 

The coagulation coefficient is a function of the radius of the particle Rp, its mass m , the effective particle Knudsen number k, the temperature of the medium T, and the depth of the interaction potential well between two particles. Using the expression for the overall interaction potential given in Appendix I, the depth of the interaction potential well can be calculated from the knowledge of the Hamaker constant for the particle. The friction coefficient f is related to the diffusion coefficient of the particle, D, through the Einstein equation... 

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