Pairwise summation

PAIRWISE SUMMATION, LESSONS LEARNED FROM GASES... 

Pairwise summation, lessons learned from gases applied to solids and liquids... 

Not so fast Those formulae don t look so different to me from what people have learned by using old-fashioned pairwise summation... 

What s different from pairwise summation Simple You let nature do the volume average for you and unashamedly take the electrical and magnetic behavior of the entire material. You don t try to take the properties of constituent atoms and weave them into the properties of the liquid or solid. 

In these simplified formulae, the distance dependence is what would result from pairwise summation. The huge difference is in the coefficients AHam that are now computed from whole-material properties rather than from the polarizabilities of constituent atoms or molecules. Even in formal correspondence with the old way of summing incremental contributions, the resemblance is in the distance dependence but not in the coefficient. Only in another limit, in which the media are all gases so dilute that their atoms interact two at a time as though no other particles were present, is there rigorous correspondence between old and new theories. 

In this picture, we are seeing the p-responding particles on the right interact with the a-responding particles on the left with a strength that is given in the summation by aP- It should be obvious that this kind of pairwise summation of a/p interactions is permissible only when the suspensions are so dilute that their dielectric response is linear in particle density (see Fig. LI.43).7... 

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

C.8.a. Circular disk or rod of finite length, with axis parallel to infinitely long cylinder, pairwise-summation form... 

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... 

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)]. 

The causative part of GAm/Bm(Z, T) is merely the sum of products (eA — em)(eB — em) formally (but only formally ) like the product of differences (NAcA — Nmcm)(NBcB — Nmcm) in pairwise summation. It is as though the electromagnetic waves that constitute the electrodynamic force were waves in a medium m that suffered only small perturbations because of the small difference between sA, sB and em. It is not because the atoms in the different media see each other individually, as imagined in pairwise summation the i s are not proportional to the respective number density N,. 

II.C. van der Waals Attraction. The van der Waals attraction between the hollow shells of the ferritin can be easily computed in the pairwise summation approximation. Let us consider a large sphere B made up of a small sphere b and a spherical shell S. The interaction free energy Fbb between two large spheres can be written (assuming pairwise interactions) as... 

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. 

This formidable formula is not so forbidding as it seems, and has been analysed in great detail, reduced to tractable forms that make sense in [3]. Pairwise summation emerges as a very special case, valid for gases only, and even then is a bad approximation to the full many-body interaction. Calculation of the interaction free energy for particular cases is not difficult [3-10] and requires a knowledge of measured dielectric properties and adsorption frequencies in the infrared, visible and ultraviolet, all known, in principle. 

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. 

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