Hamaker summation method

The two system-specific parameters in the LJ equation encompass a and s. If their values, the number density of species within the interacting bodies, and the form/shape of the bodies are known, the mesoscopic/macroscopic interaction forces between two bodies can be calculated. The usual treatment of calculating net forces between objects includes a pairwise summation of the interaction forces between the species. Here, we neglect multibody interactions, which can also be considered at the expense of mathematical simplicity. Additivity of forces is assumed during summation of the pairwise interactions, and retardation effects are neglected. The corresponding so-called Hamaker summation method is well described in standard texts and references [5,6]. Below we summarize a few results relevant for AFM. 

Hamaker (1932) and de Boer (1936) calculated van der Waals forces between macroscopic bodies using the summation method. 

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. 

In 1937 Hamaker had the idea of expanding the concept of the van der Waals forces from atoms and molecules to solid bodies. He assumed that each atom in body 1 interacts with all atoms in body 2, and with a method known as pairwise summation (Figure 11.3), found an expression for the interaction between two spheres of radius f , and R2. 

Fig. 1-10. Summation of dispersion interactions according to the method established by Hamaker and De Boer...

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 ... 

Dispersion forces are universal because they attract all molecules together, regardless of their specific chemical nature. The potential energy of dispersion attraction between two isolated molecules decays with the sixth power of the separation distance. Based on the so-called Hamaker theory (i.e., the method of pair-wise summation of intermolecular forces) or the more modern Lifshitz macroscopic treatment of strictly additive London forces, it is possible to develop the so-called Lifshitz-Van der Waals expression for the macroscopic interactions between macroscopic-in-size objects (i.e., macrobodies) [19, 21], Such an expression strongly depends on the shapes of the interacting macrobodies as well as on the separation distance (non-retarded or retarded interaction). For two portions of the same phase of infinite extent bounded by parallel flat surfaces, at a distance h apart, the potential energy of macroscopic attraction is ... 

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