Estimation of effective Hamaker constants

As discussed in Chapter 2, the effective Hamaker constants can be obtained either by empirical combining 

The Hamaker constant is also related to the surface tension but this estimation method (not shown here) depends on the value chosen for the distance H. Once reliable values of surface tensions and Hamaker constants arc available from other techniques or measurements, then the relevant equations can be used to estimate the interparticle distance. Such calculations yield for many compounds like hydrocarbons, water and polystyrene values aroimd 0.2-0.3 tun, which are physically reasonable. 

If the properties required in the Lifshitz theory are not available, effective Hamaker constants can be estimated via combining rules. The effective Hamaker constant of particles (1) in a medium (2) is given, as shown in Chapter 2, as  

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  

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. 

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I -9. Estimation of effective Hamaker constant from water adsorption behavior in soils under study using Eq. [18a]. The top (fine) curve with Asvl = -5 x 10-19 (J) fitted to the kaolinite-dominated Salkum soil, and the bottom (bold) curve with Asyl - -6 x 10-2° (J) was fitted to adsorption data from all other soils in this study " hich have mixed montmorillonite-kaolinite clay mineralogy). 

From a plot of log W versus log c determine the CCC value and T0 [by means of Eq. (53)]. Use the approximation for T0 given in Problem 6 to estimate for this colloid. Use the values of the CCC and T0 determined in Eqs. (5) and (6) to estimate the effective Hamaker constant Ain for polystyrene dispersed in water. Describe how A might be estimated using a more realistic model than that used in the derivation of Eqs. (5) and (6). 

Estimation of the Effective Hamaker Constant for Solid-Vapor Interactions for Different Soils. The Hamaker constant represents interactions between macro-objects such as mineral surfaces and liquid due to short-range (< 100A) van der Waals forces (Ackler et al., 1996 Bergstrom, 1997). The presence of van der Waals surfacial interactions in most liquid adsorption processes in nalural porous media renders the proper estimation of the Hamaker constant an important task. The Hamaker constant for liquid-liquid interactions through the intervening vapor is a... 

We proceed with illustrative examples for application of the proposed up-scaling scheme to seven soil types with properties listed in Table 1-2. The closed-form solution for degree of saturation (Eq. [23]) was fitted to measured data by optimizing parameters p, go, X, and the chemical potential pd at air entry point (that defines Lmax). Note that the Hamaker constant was estimated beforehand, as described in Estimation of the Effective Hamaker Constant for Solid-Vapor Interactions for Different Soils above. The estimated parameters were then used to calculate the liquid-vapor interfacial area for each soil (Eq. [28]). We used square shaped central pores for all soil types except the artificial sand mixture, where triangular pores were applied to emphasize capillaiy processes over adsorption in sand. I lowcver, the closed-form solutions for retention and interfacial area were derived lo accommodate any regular polygon-shaped central pore. Constants for various shapes are described in Table I-1. The values of primary physical constants employed in (he calculations and (heir units are shown in Table 1-3. 

Table 5.1. Effective Hamaker constants (in lO J) for CNTs in air and polycarbonate melt at 300 °C. The values have been estimated by means of the surface free energies of the polycarbonate [35] and the CNTs (specified references) [13]...

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... 

Estimate the Debye length at the conditions of the problem. How are the Debye length and the stability expected to change if the NaCl solution is replaced by an AICI3 solution of the same concentration Draw qualitatively the V-H curve in the cases that (i) the effective Hamaker constant is changed to 0.4 x 10 ° J (all other parameters have the same values) and (ii) the zeta potential is changed to -20 mV (aU other parameters have the same values). 

Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

For aq.KOH on mica, the dependence of the effective contact angle on droplet height is mnch weaker than that for the aq.KOH-graphite system. The estimated Hamaker constant of the van der Waals interaction for this system is -1.9 X 10 ° J (repnlsive), and the fitting gives ... 

The Hamaker constant A can, in principle, be determined from the C6 coefficient characterizing the strength of the van der Waals interaction between two molecules in vacuum. In practice, however, the value for A is also influenced by the dielectric properties of the interstitial medium, as well as the roughness of the surface of the spheres. Reliable estimates from theory are therefore difficult to make, and unfortunately it also proves difficult to directly determine A from experiment. So, establishing a value for A remains the main difficulty in the numerical studies of the effect of cohesive forces, where the value for glass particles is assumed to be somewhere in the range of 10 21 joule. 

The effective value of A123 is related to Hamaker constants of individual materials An, A22, and A33 and can be estimated as [14] ... 

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