Hamaker constants

These simplified expressions depend on the assumption of a constant Hamaker coefficient with dielectric responses e1, e2, and em of similar magnitudes. 

The van der Waals interaction between different materials depends on their differences in optical properties and can be characterized by the so-called Ha-maker constant. Hamaker constants either can be calculated from dielectric and spectroscopic information of the materials using the Lifshitz theory or can be... 

Fig. 4 illustrates the time-dependence of the length of top s water column in conical capillary of the dimensions R = 15 pm and lo =310 pm at temperature T = 22°C. Experimental data for the top s column are approximated by the formula (11). The value of A is selected under the requirement to ensure optimum correlation between experimental and theoretical data. It gives Ae =3,810 J. One can see that there is satisfactory correlation between experimental and theoretical dependencies. Moreover, the value Ae has the same order of magnitude as Hamaker constant Ah. But just Ah describes one of the main components of disjoining pressure IT [13]. It confirms the rightness of our physical arguments, described above, to explain the mechanism of two-side liquid penetration into dead-end capillaries. 

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... 

A common approach to treating retardation in dispersion forces is to define an effective Hamaker constant that is not constant but depends on separation distance. Lxioking back at Eq. VI-22, this defines the effective Hamaker constant... 

The Hamaker constant can also be related to physical properties. The dispersion... 

An assortment of values of the Hamaker constant A is collected in Table VI-4. These are a mixture of theoretical and experimental values there is reasonable agreement between theory and experiment in the cases of silica, mica, and polystyrene. 

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.)...

In an extensive SFA study of protein receptor-ligand interactions, Leckband and co-workers [114] showed the importance of electrostatic, dispersion, steric, and hydrophobic forces in mediating the strong streptavidin-biotin interaction. Israelachvili and co-workers [66, 115] have measured the Hamaker constant for the dispersion interaction between phospholipid bilayers and find A = 7.5 1.5 X 10 erg in water. 

The Hamaker constant for the case of a substance having two kinds of atoms is... 

Using Eqs. VI-30-VI-32 and data from the General References or handbooks, plot the retarded Hamaker constant for quartz interacting through water and through n-decane. Comment on the relative importance of the zero frequency contribution and that from the vuv peak. 

A thin film of hydrocarbon spread on a horizontal surface of quartz will experience a negative dispersion interaction. Treating these as 1 = quartz, 2 = n-decane, 3 = vacuum, determine the Hamaker constant A123 for the interaction. Balance the negative dispersion force (nonretarded) against the gravitational force to find the equilibrium film thickness. 

Determine the net DLVO interaction (electrostatic plus dispersion forces) for two large colloidal spheres having a surface potential 0 = 51.4 mV and a Hamaker constant of 3 x 10 erg in a 0.002Af solution of 1 1 electrolyte at 25°C. Plot U(x) as a function of x for the individual electrostatic and dispersion interactions as well as the net interaction. 

The interfacial free energy per unit area is given by the adhesion force Fo/lrR", estimate the Hamaker constant responsible for the adhesion force in the crossed-cylinder geometry illustrated in the inset to Fig. VI-6. 

A related approach carries out lattice sums using a suitable interatomic potential, much as has been done for rare gas crystals [82]. One may also obtain the dispersion component to E by estimating the Hamaker constant A by means of the Lifshitz theory (Eq. VI-30), but again using lattice sums [83]. Thus for a FCC crystal the dispersion contributions are... 

Calculate the Hamaker constant for Ar crystal, using Eq. VII-18. Compare your value with the one that you can estimate from the data and equations of Chapter VI. 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

More generally, approximate relations can be used to estimate tlie Hamaker constant for particles 1 and 2, suspended in a medium 3, such as... 

Surface Coating. A dense surface coating (encapsulation) that contains no occluded solvent decreases interparticle attraction provided that the coating has a Hamaker constant intermediate between the particle and the Hquid. This is called semisteric stabilization (ST). The energy of interaction between coated spheres is as follows (26) ... 

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