Forces between macroscopic bodies

The interaction of two macroscopic bodies can sometimes be obtained by summing the dispersion energy between all pairs of molecules or unit cells in the two bodies. There is no electrostatic or induction contribution when the material is uncharged and isotropic. If the separation of the units is large compared to the reduced wavelength associated with the strong electronic transitions, the dispersion interaction is retarded and therefore weakened it varies as R rather than [34]. If the dispersion energy between the units is 

---

Generally speaking, intermolecular forces act over a short range. Were this not the case, the specific energy of a portion of matter would depend on its size quantities such as molar enthalpies of formation would be extensive variables On the other hand, the cumulative effects of these forces between macroscopic bodies extend over a rather long range and the discussion of such situations constitutes the chief subject of this chapter. 

This chapter and the two that follow are introduced at this time to illustrate some of the many extensive areas in which there are important applications of surface chemistry. Friction and lubrication as topics properly deserve mention in a textbook on surface chemistiy, partly because these subjects do involve surfaces directly and partly because many aspects of lubrication depend on the properties of surface films. The subject of adhesion is treated briefly in this chapter mainly because it, too, depends greatly on the behavior of surface films at a solid interface and also because friction and adhesion have some interrelations. Studies of the interaction between two solid surfaces, with or without an intervening liquid phase, have been stimulated in recent years by the development of equipment capable of the direct measurement of the forces between macroscopic bodies. 

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

Our objectives in this chapter are to look into the origin of van der Waals forces, see how they affect macroscopic behavior and properties of materials, and establish relations for scaling up the molecular-level forces to forces between macroscopic bodies. 

CALCULATING VAN DER WAALS FORCES BETWEEN MACROSCOPIC BODIES... 

It is extremely difficult to measure the Hamaker constant directly, although this has been the object of considerable research efforts. Direct evaluation, however, is complicated either by experimental difficulties or by uncertainties in the values of other variables that affect the observations. The direct measurement of van der Waals forces has been undertaken by literally measuring the force between macroscopic bodies as a function of their separation. The distances, of course, must be very small, so optical interference methods may be used to evaluate the separation. The force has been measured from the displacement of a sensitive spring (or from capacitance-type measurements). 

One such approximation was proposed by Good and Girilalco assuming that mainly van der Waals forces act between all molecules 11081. Based on the theory of van der Waals forces between macroscopic bodies they suggested... 

Figure 15.1 Arrangement of crossed cylinders as used for the direct determination of the force between macroscopic bodies.

J. N. Israelachvili, The calculation of van der Waals dispersion forces between macroscopic bodies, Proc. Royal Soc. (London) 331A 39 (1972). See also Chap. 7 in J. Mahanty and B. W. Ninham, Dispersion Forces. Academic Press, London, 1976. 

The above short description of the theory of van der Waals forces is based upon interparticle interactions. The extension of this approach to many-body systems is a difficult task, as discussed in refs. 55 and 56. Since the van der Waals force between macroscopic bodies is measurable and since it originates in the fluctuations of electrons, Lifschitz sought to develop a theory based upon macroscopic electromagnetic concepts. The derivations are rather lengthy and the reader is referred to Lifschitz and coworkers articles and to a very readable review by Grimley. A somewhat simplified result is given in Eq. (93),... 

Attractive surface forces, generally referred to as van der Waals forces, exist between all atoms and molecules regardless of what other forces may be involved. They have been discussed in detail by Isrealachvili (4). We shall first examine the origins of the van der Waals forces between atoms and molecules and later consider the attractive forces between macroscopic bodies such as particles. 

To determine the van der Waals forces between macroscopic bodies (e.g., two particles), we assume that the interaction between one molecule and a macroscopic body is simply the sum of the interactions with all the molecules in the body. We are therefore assuming simple additivity of the van da- Waals forces and ignore how the induced fields are affected by the intervening molecules. 

Figure 4.2 Additivity of the van der Waals force between macroscopic bodies. The total interaction is taken as the sum of the interactions between infinitesimally small elements in the two bodies.

Hamaker [9] calculated the attractive forces between macroscopic bodies using a simple additivity principle. For two-semi infinite flat plates separated by a distance d, the attractive force F is given by... 

Note the difference between Hq.(20) and (25). The attraction van der Waals force between macroscopic bodies is clearly dependent on the geometries of two units. 

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... 

---