Hamaker pairwise summation

Connection between the Hamaker pairwise-summation picture and the modern theory... 

How does Hamaker pairwise summation emerge from reduction of the Lifshitz theory ... 

In this limit and only in this limit in which each of the materials A, m and B can be considered a dilute gas, does the Hamaker pairwise-summation limit agree rigorously with the modern theory. Compare [Eqs. (L2.123) and (L2.124)]. 

C.3.b. Two parallel cylinders, pairwise-summation approximation, Hamaker-Lifshitz hybrid, retardation screening neglected C.3.b.l. All separations... 

In fact, the graft is exceedingly helpful for geometries in which field equations of the modern theory are too difficult to solve but pairwise summation (actually integration) can be effected. The distance dependence of the interaction is taken from summation whereas the Hamaker coefficient is estimated with modern theory. To see how to connect old and new, consider the formal procedure for summation, then see its equivalence to a much-reduced version of the general theory. 

It became customary to define a "Hamaker constant" AHam (sometimes also written in this text as AH) which in the language of pairwise summation is... 

Van der Waals interactions are noncovalent and nonelectrostatic forces that result from three separate phenomena permanent dipole-dipole (orientation) interactions, dipole-induced dipole (induction) interactions, and induced dipole-induced dipole (dispersion) interactions [46]. The dispersive interactions are universal, occurring between individual atoms and predominant in clay-water systems [23]. The dispersive van der Waals interactions between individual molecules were extended to macroscopic bodies by Hamaker [46]. Hamaker s work showed that the dispersive (or London) van der Waals forces were significant over larger separation distances for macroscopic bodies than they were for singled molecules. Through a pairwise summation of interacting molecules it can be shown that the potential energy of interaction between flat plates is [7, 23]... 

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. 

The earliest quantitative theory to describe van der Waals forces between two colloidal particles, each containing a statistically large number of atoms, was developed by Hamaker, who used pairwise summation of the atom-atom interactions. This approach neglects the multi-body interactions inherent in the interaction of condensed phases. The modem theory for predicting van der Waals forces for continua was developed by Lifshitz who used quantum electrodynamics [19,20] to account for the many-body molecular interactions and retardation within and between materials. Retardation is a reduction of the interaction because of a phase lag in the induced dipole response that increases with distance. 

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. 

The behaviour of the interaction free energy depends strongly on geometry. Sometimes, the geometric factor can be evaluated by taking a "Hamaker function" [3-10] calculated by Lifshitz theory, multiplied by an appropriate geometric distance factor evaluated from pairwise summation. (This assumption fails for the temperature dependent contribution, i.e. in water, the most interesting liquid )... 

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. 

In 1932 Kallmann and Willstatter" and also Bradley recognized that attractive forces between colloidal particles would emerge from a pairwise summation of dispersion forces between atoms. This was further investigated theoretically by Hamaker and de Boer. For a flat geometry, two half-spaces separated by a gap of thickness h, this leads to the form... 

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... 

The two coincide when the pairwise interaction coefficients caCb, cAcm, CBCm, and <4 are evaluated as qc, = (3kT/8it2) J2 Zo i i- The inequality 2 Am/Bm > (EAm/Am + Bm/Bm) of the pure Hamaker form is preserved, but the geometric mean that creates the inequality holds only for the individual terms in the summation over frequencies The total free energy of interaction GAm/Bm(/, T) is not the geometric mean of GAm/Am(7, T) and GBm/Bm( T). 

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