Hamaker constant Lifshitz theory

As an example, it has been pointed out that the Hamaker and Lifshitz theories assume (exphcitly and implicitly, respectively) that intensive physical properties of the media involved such as density, and dielectric constant, remain unchanged throughout the phase—that is, right up to the interface between phases. We know, however, that at the atomic or molecular level solids and liquids (and gases under certain circumstances) exhibit short-range periodic fluctuations they are damped oscillating functions. Conceptually, if one visualizes a hquid in contact with a flat solid surface (Fig. 4.8a), one can see that the molecules (assumed to be approximately spherical, in this case) trapped between the surface and the bulk of the liquid will have less translational freedom relative to the bulk and therefore be more structured. That structure will (or may) result in changes in effective intensive properties near the surface. 

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 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 ... 

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 ... 

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. 

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. 

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 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]. 

A fuller theoretical analysis of vdW interactions requires recourse to Lifshitz theory [8[. Lifshitz theory requires a description of the dielectric behavior of materials as a function of frequency, and there are several reviews for the calculation of Hamaker functions using this theory. The method described by Hough and White (H-W) [95], employing the Ninham-Parsegian [96] representation of dielectric data, has proved to be most useful. The nonretarded Hamaker constant (for materials l and 2, separated by material 3) is given by... 

The value of the Hamaker constant can be estimated from the Lifshitz theory of dispersion forces in a continuum in simplified form, one obtains (Israelachvili 1992 p. 184)... 

The major disadvantage of this microscopic approach theory was the fact that Hamaker knowingly neglected the interaction between atoms within the same solid, which is not correct, since the motion of electrons in a solid can be influenced by other electrons in the same solid. So a modification to the Hamaker theory came from Lifshitz in 1956 and is known as the Lifshitz or macroscopic theory." Lifshitz ignored the atoms completely he assumed continuum bodies with specific dielectric properties. Since both van der Waals forces and the dielectric properties are related with the dipoles in the solids, he correlated those two quantities and derived expressions for the Hamaker constant based on the dielectric response of the material. The detailed derivations are beyond the scope of this book and readers are referred to other publications. The final expression that Lifshitz derived is... 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

The description of the van der Waals interaction based on the Lifshitz approach is now sufficiently advanced to provide accurate predictions for the complete interaction energy. For the geometry of two half-spaces, the exact theory is available in a formulation suited for computational purposes. " In parallel with work on planar systems, there has been a focus on the interaction between spheres. " These developed theories have been used as the exact solutions in the validation of the approximate predictions using the Hamaker approach. The significant contribution of the continuum approach to our understanding of the van der Waals interaction lies in the reliable prediction of the Hamaker constant. The interaction energy for two half-spaces and two spheres is summarized below. 

The first improvement is based on the discovery that Eq. (2) of the Hamaker microscopic theory for spheres agrees with Eq. (4) of the continuum macroscopic theory when the Hamaker constant. A, in Eq. (2) is determined from Eq. (3) of the Lifshitz theory for parallel flat plates (Fig. 1). The combined Hamaker-Lifshitz function, A h), can be obtained by comparing the right-hand sides of Eqs. (1) and (3), giving... 

Hough, D.B. White, L.R. The calculation of Hamaker constants from Lifshitz theory with applications to wetting phenomena. Adv. Colloid Interface Sci. 1980, 14 (1), 3-41. 

Since Lifshitz s derivation is too difficult and beyond the scope of this book, the Hamaker constant based on the Lifshitz theory is expressed as... 

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. 

In order to estimate the values of Hamaker constants from Lifshitz s theory, one needs to know optical characteristics of condensed phases. Calculations of this type were carried out for a number of primarily simple systems, including two identical phases separated by vacuum. For instance, such calculations yielded the values of Hamaker constants for water and for quartz of 5.13 x 10"20 and 5.47x 1 O 20 J, respectively. 

The Hamaker constant can be evaluated accmately using the continuum theory, developed by Lifshitz and coworkers [40]. A key property in this theory is the frequency dependence of the dielectric permittivity, e( ). If this spectrum were the same for particles and solvent, then A=0. Since the refractive index n is also related to t ( ), the van der Waals forces tend to be very weak when the particles and solvent have similar refractive indices. A few examples of values for for interactions across vacuum and across water, obtained using the continuum theory, are given in table C2.6.3. 

The objections to the Hamaker theory were overcome by Lifshitz and his coworkers [Lifshitz 1956, Dzyaloshinskii 1961] using the bulk optical properties of the interacting bodies. The approach employed by Lifshitz uses the so-called Lifshitz-van der Waals constant h that depends only on the materials involved provided the separation distance is relatively small. Under some conditions the constant h can be related to the Hamaker constant by... 

In the original treatment, also called the microscopic approach, the Hamaker constant was calculated from the polarizabilities and number densities of the atoms in the two interacting bodies. Lifshitz presented an alternative, more rigorous approach where each body is treated as a continuum with certain dielectric properties. This approach automatically incorporates many-body effects, which are neglected in the microscopic approach. The Hamaker constants for a number of ceramic materials have been calculated from the Lifshitz theory using optical data of both the material and the media (Table 9.1) (9). Clearly, all ceramic materials are characterized by large unretarded Hamaker constants in air. When the materials interact across a liquid, their Hamaker constants are reduced, but still remain rather high, except for silica. 

Lifshitz (1956) and Dzyaloshinskii et al. (1961) developed an approach to the calculation of the Hamaker constant Ah in condensed phases, called the macroscopic theory. The latter is not limited by the assumption for pairwise additivity of the van der Waals interaction. The authors treat each phase as a continuous medium characterized by a given uniform dielectric permittivity, which depends on the frequency, V, of the propagating electromagnetic waves. 

Compare the two expressions for the Hamaker constant according to the van der Waals and Lifshitz theories presented in the chapter and discuss any significant differences one might expect from the application of each theory for the interactions between two spherical particles in a second liquid medium. 

Lifshitz avoided this problem of additivity by developing a continuum theory of van der Waals forces that used quantum field theory. Simple accounts of the theory are given by Israelachvili and Adamson. Being a continuum theory, it does not involve the distinctions associated with the names of London, Debye and Keesom, which follow from considerations of molecular structure. The expressions for interaction between macroscopic bodies (e.g. Eqns. 2-4) remain valid, except that the Hamaker constant has to be calculated in an entirely different way. 

Hamaker constants, which for some materials are given in the literature,can be calculated on the basis of the Lifshitz theory as presented in Eq. 7 below,where tii, ti2, and 3 are the refractive indices of the three media, e, 62, and 3 are the corresponding static dielectric constants, and Ve is the mean value of the absorption frequency of the three media. 

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