Hamaker microscopic approach

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... 

Microscopic approach of Hamaker between a molecule and a slab surface... 

H.C. Hamaker, in 1937, was the first to treat London dispersion interactions between macroscopic objects. He started with the most basic case, to determine the interaction of a single molecule with a planar solid surface. He considered a molecular pair potential and its relation with the molecules present within the solid surface, to derive the total interaction potential by summing the attractive interaction energies between all pairs of molecules, ignoring multibody perturbations. In this way, he built up the whole from the parts. Thus, Hamaker s method is called the microscopic approach. 

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. 

The classical microscopic approach, developed mainly by Hamaker [51], exploiting the energy additivity principle to derive solutions for more complex geometries from known solutions for atoms and molecules. 

We move from the interaction between two molecules to the interaction between two macroscopic solids. It was recognized soon after London had published his explanation of the dispersion forces that dispersion interaction could be responsible for the attractive forces acting between macroscopic objects. This idea led to the development of a theoretical description of van der Waals forces between macroscopic bodies based on the pairwise summation of the forces between all molecules in the objects. This concept was developed by Hamaker [9] based on earlier work by Bradley [10] and de Boer [11]. This microscopic approach of Hamaker of pairwise summation of the dipole interactions makes the simplifying assumption that the... 

An attractive interaction arises due to the van der Waals forces between molecules of colloidal particles. Depending on the nature of dispersed particles, the Keesom forces (or the dipole-dipole interaction), the Debye forces (or dipole-induced dipole interaction), and the London forces (or induced dipole-induced dipole interaction) may contribute to the van der Waals interaction. First, the van der Waals interaction was theoretically computed using a method of the pairwise summation of interactions between different pairs of molecules of the two macroscopic particles. This method called the microscopic approximation neglects collective effects, and, as a consequence, misrepresents the Hamaker constant. For many problems of a practical use, however, specific features of the total interaction are determined by a repulsive part, and such an effective, gross description of the van der Waals interaction may often be accepted [3]. The collective effects in the van der Waals interaction have been taken into account in the calculations of Lifshitz et al. [4], and their method is known in the literature as the macroscopic approach. 

Depending on the geometrical model (Figure 17) used and on the theoretical approach taken, different relationships exist for the approximation of adhesion by van der Waals forces. The best known equations are those developed by Hamaker based on the microscopic theory of London-Heitler. For the model sphere/plane. Figure 17(a), a distance a< 100 nm, and the particle diameter x, the adhesion force A, i is... 

Qualitatively the same result may be obtained if one utilizes more strict treatment of molecular interactions in disperse systems. This approach is based on the so-called macroscopic theory of van der Waals forces developed by E.M. Lifshitz, I.E. Dzyaloshinski and L.P. Pitaevski [14], In contrast to Hamaker s microscopic theory, the macroscopic theory does not use a simplified assumption of additivity of interactions between molecules, on which their summation is based (see Chapter I, 2). Mutual influence of molecules in condensed phases on each other may alter polarizabilities and ionization energies, making them different from those established for isolated molecules, which results in molecular interactions being not fully additive. 

The simple additive approach of Hamaker has been criticised as being inaccurate [Kitchener 1973] and Oliveira [1992] states that it is not valid for condensed media interactions. Visser [1988] has explained the reason why the original Hamaker approach was inaccurate being due to the false assumption that the molecular forces are additive. It cannot be so, since the interactions between molecules with larger separations than near or direct contact, are screened by any molecules at a closer distance. As a consequence, layers of adsorbed materials for instance can influence the interaction between the two interacting macroscopic bodies. Langbein [1969] has demonstrated that when the thicknesses of the surface layers on microscopic bodies are larger than the separation distance, their interaction is... 

The microscopic method, credited to Hamaker, came first and is based on pair-wise summation of the individual dispersion interaction between molecules. Casmir and Polder later supplemented this approach by including the correction for electromagnetic retardation. The molecular interaction potential used is typically represented by the following expression ... 

The key property in these calculations is the so-called Hamaker constant (A), which is directly linked to the C parameter of Equations 2.3 and 2.4 via the so-called microscopic (London) approach ... 

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