Hamaker constant theory

These quartz measurements, together with several other less-successful attempts by others, had been fiercely contested.20 Theories had been fitted to faulty measurements there had been no adequate theory yet available for good measurements. "Measurement" drove theory. Hamaker constants (coefficients of interaction energy) were so uncertain that they were allowed to vary by factors of 100 or 1000 in order to fit the data. The Lifshitz theory put an end to all that. Disagreement meant that either there was a bad measurement or there was something acting besides a charge-fluctuation force. 

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. 

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... 

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). 

SFA has been traditionally used to measure the forces between modified mica surfaces. Before the JKR theory was developed, Israelachvili and Tabor [57] measured the force versus distance (F vs. d) profile and pull-off force (Pf) between steric acid monolayers assembled on mica surfaces. The authors calculated the surface energy of these monolayers from the Hamaker constant determined from the F versus d data. In a later paper on the measurement of forces between surfaces immersed in a variety of electrolytic solutions, Israelachvili [93] reported that the interfacial energies in aqueous electrolytes varies over a wide range (0.01-10 mJ/m-). In this work Israelachvili found that the adhesion energies depended on pH, type of cation, and the crystallographic orientation of mica. 

The first term is related to the van der Waals interaction, with A being the Hamaker constant. The second term includes other forces that decay exponentially with distance. As discussed, these may include double-layer, solvation, and hydration forces. In our data analysis, B and C were used as fitting variables the Hamaker constant A was calculated using Lifshitz theory [6]. 

For the aq.KOH-graphite system, the van der Waals interaction should be repulsive, because Lifshitz theory predicts a negative Hamaker constant A, which we calculated to be approximately -7.7 X 10 ° J. Using this value, the fit 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. 

Historical deveiopment of van der Waais forces. The Lennard-Jones potentiai. intermoiecuiar forces. Van der Waais forces between surfaces and coiioids. The Hamaker constant. The DLVO theory of coi-loidal stability. 

Use schematic diagrams to describe the influence of electrolyte concentration, type of electrolyte, magnitude of surface electrostatic potential and strength of the Hamaker constant on the interaction energy between two colloidal-sized spherical particles in aqueous solution. What theory did you use to obtain your description Briefly describe the main features of this theory. 

The van der Waals forces scale up from atomic distances to colloidal distances undiminished. How the molecular forces scale up in the case of large objects, expressions for such forces, definition of the Hamaker constant, and theories based on bulk material properties follow in Sections 10.5-10.7. 

The Hamaker constants of nonpolar fluids and polymeric liquids can be obtained using an expression similar to Equation (67) in combination with the corresponding state theory of thermodynamics and an expression for interfacial energy based on statistical thermodynamics (Croucher 1981). This leads to a simple, but reasonably accurate and useful, relation for Hamaker constants for nonpolar fluids and polymeric liquids. We present in this section the basic details and an illustration of the use of the equation derived by Croucher. 

Throughout most of this chapter the emphasis has been on the evaluation of zeta potentials from electrokinetic measurements. This emphasis is entirely fitting in view of the important role played by the potential in the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloidal stability. From a theoretical point of view, a fairly complete picture of the stability of dilute dispersions can be built up from a knowledge of potential, electrolyte content, Hamaker constants, and particle geometry, as we discuss in Chapter 13. From this perspective the fundamental importance of the f potential is evident. Below we present a brief list of some of the applications of electrokinetic measurements. 

Figure 6.11 Gibbs free interaction energy (in units of ki>T) versus distance for two identical spherical particles of R = 100 nm radius in water, containing different concentrations of monovalent salt. The calculation is based on DLVO theory using Eqs. (6.57) and (6.32). The Hamaker constant was Ah = 7 x 10 21 J, the surface potential was set to )/>o = 30 mV. The insert shows the weak attractive interaction (secondary energy minimum) at very large distances.

In an aqueous electrolyte we have spherical silicon oxide particles. The dispersion is assumed to be monodisperse with a particle radius of 1 /rm. Please estimate the concentration of monovalent salt at which aggregation sets in. Use the DLVO theory and assume that aggregation starts, when the energy barrier decreases below 0ksT. The surface potential is assumed to be independent of the salt concentration at -20 mV. Use a Hamaker constant of 0.4 x 10-20 J. 

Additional complications can arise when two bodies, i.e. the tip and the sample, interact in liquid (Fig. 3c). The interaction energy of two macroscopic phases across a dielectric medium can be calculated based on the Lifshitz continuum theory. In contact, when the distance between the phases corresponds to the nonretarded regime, the Hamaker constant in Eq. (2) is approximated by ... 

For a typical experimental hydrosol critical coagulation concentration at 25°C of 0.1 mol dm-3 for z = 1, and, again, taking if/d = 75 mV, the effective Hamaker constant, A, is calculated to be equal to 8 X 10 20 J. This is consistent with the order of magnitude of A which is predicted from the theory of London-van der Waals forces (see Table 8.3). 

Prieve, D.C. and Russel, W.B., (1988), Simplified predictions of Hamaker constants from Lifshitz theory , J. Colloid and Interface Science, 125 (1), 1-13. 

Hough, D. B. and White, L. R. (1980). The Calculation of Hamaker Constants from Lifshitz Theory with Applications to Wetting Phenomena. Adv. Colloid Interface Sci., 14, 3. 

The role of electrostatic repulsion in the stability of suspensions of particles in non-aqueous media is not yet clear. In order to attempt to apply theories such as the DLVO theory (to be introduced in Section 5.2) one must know the electrical potential at the surface, the Hamaker constant, and the ionic strength to be used for the non-aqueous medium these are difficult to estimate. The ionic strength will be low so the electric double layer will be thick, the electric potential will vary slowly with separation distance, and so will the net electric potential as the double layers overlap. For this reason the repulsion between particles can be expected to be weak. A summary of work on the applicability or lack of applicability of DLVO theory to non-aqueous media has been given by Morrison [268],... 

The Lifshitz theory of dispersion forces, which does not imply pairwise additivity and takes into account retardation effects, shows that the Hamaker constant AH is actually a function of the separation distance. However, for the stability calculations that follow, only the values of the attraction potential at distances less than a few nanometers are relevant, and in this range one can consider that AH is constant. 

The second issue is the extent of the decrease of the van der Waals interaction. Experiment and calculation of the van der Waals interactions between polystyrene latex beads and either a bare glass plane or a polystyrene coated glass plane [17] revealed that the Hamaker constant decreases only by about 25% at complete screening, while the experiments of Petrache et al. for neutral lipid bilayers require a decrease of about 75% (from 1.2kT to OAkT). Such a strong decrease of the van der Waals interaction upon addition of salt would be expected to have strong consequences in the general theory of colloid stability, and not only in the stability of lipid bilayers. 

The expression van der Waals attraction is widely used and is here defined as the sum of dispersion forces [9], Debye forces [17] and the Keesom forces [18]. Debye forces are Boltzmann-averaged dipole-induced dipole forces, while Keesom forces are Boltzmann-averaged dipole-dipole forces. The interaction for all three terms decays as 1 /r6, where r is the separation between the interacting particles, and they are combined into one term with the proportionality constant denoted the Hamaker constant. In order to determine the van der Waals force there are at least two approaches, either to calculate the force between two particles assuming that the interaction is additive, (this is usually called the Hamaker approach) or to use a variant of Lifshitz theory. 

Dispersion. Dispersion or London-van der Waals forces are ubiquitous. The most rigorous calculations of such forces are based on an analysis of the macroscopic electrodynamic properties of the interacting media. However, such a full description is exceptionally demanding both computationally and in terms of the physical property data required. For engineering applications there is a need to adopt a procedure for calculation which accurately represents the results of modem theory yet has more modest computational and data needs. An efficient approach is to use an effective Lifshitz-Hamaker constant for flat plates with a Hamaker geometric factor [9]. Effective Lifshitz-Hamaker constants may be calculated from readily available optical and dielectric data [10]. 

FIGURE 1.12 Comparison of (a) experimentally observed and (b) calculated r/-values using DLVO theory, with an adjustable parameter (proportional to the Hamaker constant) chosen to fit the data at c = 0.01 M... 

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