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Casimir was doing applied research at Philips in the stability of colloidal suspensions used to deposit films in the manufacture of lamps and cathode ray tubes when he discovered the effect that is now named after him J T G Overbeek and E J W..



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where Ni is the number of molecules of monomer i with functionality fi, and the summations are over all the monomers present in the system. Thus for a system consisting of 2 mol of glycerol (a triol) and 3 mol of phthalic acid (a diacid) there is a total of 12 functional groups per ve monomer molecules, and favg is 12 or 2.4. For a system consisting of a equimolar 5 amounts of glycerol, phthalic anhydride, and a monobasic acid, there is a total of six functional groups per three molecules and favg is 6 or 2. 3 In a system containing equivalent numbers of A and B groups, the number of monomer molecules present initially is N0 and the corresponding total number of functional groups is N0 favg . If N is the number of molecules after reaction has occurred, then 2 N0 N is the number of functional groups that have reacted. The extent of reaction p is the fraction of functional groups lost:





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32 A

p 2 N0 N N0 favg 2-135

reminder about our notation: Operators will have no hats in this section. Hats are reserved to denote unit vectors. 33The same can be said of the blackbody radiation problem, which led to the introduction of Planck's constant. In his original paper, Planck quantizes the material oscillators only, not the radiation field. 34That is not to say that there is no interaction with matter there. The parallel metal plates that the Casimir force attracts toward each other are, after all, matter. But then again, as we shall soon see, for perfectly reflecting plates, the Casimir force turns out to involve only two fundamental constants: Ii, and c. It does not involve the charge of the electron, suggesting that this is an effect that does not involve coupling between matter and radiation.

Xn N0 N 2-136

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Xn 2 2 p favg 2-137

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Verwey at Philips Research had developed a theory of colloidal stability to explain the properties of suspensions of quartz powder. But their experiments indicated that the theory could not be entirely correct. Realizing that a better agreement between theory and experiment could be obtained if the interparticle interaction somehow fell off more rapidly at large distances than had been supposed, Overbeek hinted that this might be related to the finiteness of the speed of light. So Casimir and Polder started to investigate what would happen to the theory of the van der Waals interaction when retardation was included. They found that retardation makes the interaction at large intermolecular separations !::J.R go as 1/!::J.R7 rather than just 1/ !::J.R6 as was usually assumed. Overbeek was right. However, Casimir was still intrigued by the simplicity of the result. Then, as he told Milonni and Shih [455]:

p 2 2 favg X n favg 2-138

Summer or autumn 1947 (but I am not absolutely certain that it [was] not somewhat earlier or later) I mentioned my results to Niels Bohr, during a walk. That is nice, he said, that is something new. I told him that I was puzzled by the extremely simple form of the expressions for the interaction at very large distance and he mumbled something about zero-point energy. That was all, but it put me on a new track. I found that calculating changes of zero-point energy really leads to the same results as the calculations of Polder and myself.... On May 29, 1948 I presented my paper "On the attraction between two perfectly conducting plates" at a meeting of the Royal Netherlands Academy of Arts and Sciences. It was published in the course of the year. . ..

Equation 2-138, often referred to as the Carothers equation, relates the extent of reaction and degree of polymerization to the average functionality of the system. An important consequence of Eq. 2-138 is its limiting form at the gel point where the number-average degree of polymerization becomes in nite. The critical extent of reaction pc at the gel point is given by

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