Between the ecclesiastic (definition 1) and the operative (definition 4), between origin (Level 3) and result (Level 2 tables), is exegesis. Different minds will find different ways to work from Level 3 Foundations to Level 2 tabulations. Each of the following sections describes not simply one particular set of steps but, more important, the kinds of steps to take:

• Examining the full expression for the interaction between half-spaces to see which features are revealed in its specialized limiting forms (Section L2.3.A.);

• Generalizing the original half-space geometry to layered and inhomogeneous planar bodies (Section L2.3.B);

• Converting to cases of curved structures (Section L2.3.C);

• Reducing to Hamaker theory (Section L2.3.D) for gases and dilute suspensions, but now

• Incorporating fluctuations and screening in ionic solutions (Section L2.3.E), and then

• Extending to include interactions between small particles and substrates (Section L2.3.F) as well as between one-dimensional linear bodies (Section L2.3.G).

In what follows, the numbers in parentheses next to or below the equations are the actual equation numbers.

In all matter there are continuous jostlings of positive and negative charges; at every point in a material body or in a vacuum, transient electric and magnetic fields arise spontaneously. These fluctuations in charge and in field occur not only because of thermal agitation but also because of inescapable quantum-mechanical uncertainties in the positions and momenta of particles and in the strengths of electromagnetic fields. The momentary positions and electric currents of moving charges act on, and react to, other charges and their fields. It is the collective coordinated interactions of moving electric charges and currents and fields, averaged over time, that create the van der Waals or “charge-fluctuation” force.

It turns out that such charge-fluctuation or “electrodynamic” forces are far more powerful within and between condensed phases—liquids and solids—than they are in gases. In fact, these forces frequently create condensed phases out of gases. The electric fields that billow out from moving charges act on many other atoms or molecules at the same time. The particles in a dilute gas are so sparsely distributed that we can safely compute the total interaction energy as the sum of interactions between the molecules considered two at a time. Rather than think in terms of a gas of pairwise particles, the modern and practical way to look at van der Waals forces is in terms of the electromagnetic properties of fully formed condensed materials. These properties can be determined from the electromagnetic absorption spectrum, i.e., from the response to externally applied electric fields.

It is curious how people working in different disciplines seek and see validation in different kinds of experiments. Only the liquid helium up-the-walls measurement, described in the preceding subsection, seems to satisfy most people that the Lifshitz theory quantitatively accounts for measured forces (see note in the preceding subsection). As with the historical comments, this review of measurements is not intended to be exhaustive.

Force-balance measurements and atomic-beam measurements give qualitatively or semiquantitatively satisfying results, but they present worries about the smoothness of the surfaces and about the effective “zero” of separation at contact. For example, with no fitting parameters, the atomic-beam measurements already described give the predicted 1/l3 power law for the energy of interaction between atom and surface, but then there is a stubborn 60–75% overestimate of the coefficient computed by Lifshitz theory fed with good spectral data versus the experiments that admit little adjustment (see note 34 in the subsection titled “Measurably strong”). This level of disagreement might be resolved by better spectra or by better modeling of the surface structure. Surface roughness is often the fingered culprit, but specific attempts to formulate and compute its consequences fail. Formulations that include roughness are often based on pairwise summation and must themselves be validated by systematic measurement or by comparison with exact solutions.

That's the worst. Several kinds of measurements reassure us that the Lifshitz strategy of converting spectra into forces works reliably enough to capture the key features of van der Waals forces.

“How much does a ladybug weigh?” A straightforward question from a straightforward six-year old. I made a guess that seemed to satisfy. Thirty-five years later, I finally weighed a ladybug: 21 mg. I never doubted that she could walk on a ceiling or on the window where I caught her, but could she be holding on by the van derWaals forces about which I was writing? That 21 mg, plus a quick calculation, reassured me that these bugs might have learned some good physics a very long time ago. If so, what about other creatures? We are told that geckos might use these forces; their feet have hairy bottoms with contact areas like those of insects. They might put to good use forces whose appearance to us humans emerged from details in the pressures of gases, whose formulation resides in difficult theories, whose practicality is seen in paints and aerosols, and whose measurement requires delicate equipment.Were these same forces showing themselves to us during childhood summers?

The first clear evidence of forces between what were soon to be called molecules came from Johannes Diderik van der Waals' 1873 Ph.D. thesis formulation of the pressure p, volume V, and temperature T of dense gases.

Attitude

It is remarkable how many people think it difficult to compute van der Waals forces directly by using the Lifshitz theory. Invariably, after a few minutes' instruction, there is the reaction “I didn't know how easy it was.” Essentially it is a matter of introducing tabulated experimental information for ε's and numerically summing or integrating for the interaction energy.

Thanks to rapid progress in spectroscopy we can soon expect to compute van der Waals forces by direct conversion of material responses to applied electromagnetic fields. The future of precise computation relies on measuring these responses on the same materials as those used to measure forces. This is because small changes—in composition of materials, dopants that confer conductance, solutes that modify spectra, even in atomic arrangement that create anisotropies—all have quantitative consequences. Whether the unintended result of handling materials or artful modifications to create forces, spectral details merit respect.

Combining these data with the theory of dielectrics helps us to think about the connection between specific features of absorption spectra and forces in order to create strategies for designing materials or to find reasons to explain measured forces. Outside the purview of this text, the actual practice of assembling data into usable ε(ω)'s and ε(iε)'s has been well described in the physics and the engineering literature. To connect readers with that literature, this chapter first introduces the essential physical features and language used to create ε's of computation.

Because they have so many unexpected features and because they are kind of a hybrid with electrostatic double-layer forces, ionic-charge-fluctuation forces deserve separate consideration.

In the language of dielectric response, how is one to regard the movement of mobile ions?

First, through conductivity. An applied electric field creates an electric current in a salt solution. Formally, a conductivity σ appears in the form that varies with frequency as ∼ {iσ /[ω (1 − iωτ)]} in the dielectric permittivity ɛ(ω). In the limit of low frequency, ωτ → 0, this diverges as ∼ (iσ/ω). In that limit, a conducting material begins to appear as an infinitely polarizable medium, its mobile charges able to move indefinitely long distances.

In real life, we know this is not necessarily so. An electric field applied to a conducting medium can be maintained only as long as reactions or transfers of charges occur at the walls. The electrical outlet delivers and removes electrons; the electrodes react to remove or to produce ions. In real life we must recognize what goes on at the walls bounding a conducting medium. In the ideal “bad-electrode” limit there is no removing or producing a reaction at the walls. In that limit, under the action of a constant applied electric field, the charges pile up to create electrostatic double layers. When oscillating fields settle down at ω → 0 there is a spatially varying electric field across the space between the bounding walls.

As described in earlier sections, any two material bodies will interact across an intermediate substance or space. This interaction is rooted in the electromagnetic fluctuations—spontaneous, transient electric and magnetic fields—that occur in material bodies as well as in vacuum cavities. The frequency spectrum of these fluctuations is uniquely related to the electromagnetic absorption spectrum, the natural resonance frequencies of the particular material. In principle, electrodynamic forces can be calculated from absorption spectra.

Lifshitz's original formulation in 1954 (see Prelude, note 17) used a method, due to Rytov, to consider the correlation in electromagnetic fluctuations between two bodies separated by a vacuum gap. The force between the bodies is derived from the Maxwell stress tensor corresponding to the spontaneous electromagnetic fields that arise in the gap between boundary surfaces—the walls of the Planck–Casimir box. His result, for the case of two semi-infinite media separated by a planar slab gap, reduces in special limits to all previous valid results, specifically those of Casimir and of Casimir and Polder for, respectively, the interaction between two metal plates or between two point particles. In 1959, Dzyaloshinskii, Lifshitz, and Pitaevskii (DLP) published a derivation that used diagram techniques of quantum field theory to allow the gap between the two bodies to be filled with a nonvacuous material.

The DLP result can be derived as well through an intuitive and heuristic method wherein the energy of the electromagnetic interaction is viewed as the energy of electromagnetic waves that fit between the dielectric boundaries of the planar gap.

“What is this about entropy really decreasing?” I didn't know how to answer my family, worried by some preposterous news report. My best try was, “I don't know the words that you and I can use in the same way. I tell you what. Let me give you examples of where you see entropy changing, as when you put cream and sugar in coffee. You think a while about these examples. Then we can answer your question together.”

That was part of the dream to which I woke the morning I was to write this welcome to readers. I connected the dream with the way my friend David Gingell came to learn about van der Waals forces 30 years ago. He began immediately by computing with previously written programs, then improved these programs to ask better questions, and finally worked back to foundations otherwise inaccessible to a zoologist.

Written using the “Gingell method,” this book is an experiment in what another friend called “quantum electrodynamics for the people.” First the main ideas and the general picture (Level 1); after that, practice (Level 2); then, finally, the bedrock science (Level 3), culled and rephrased from abstruse sources. This is a strategy intended to defeat the fear that stops many who need to use the theory of van der Waals forces from taking advantage of progress over the past 50 or 60 years.

Many excellent physically sophisticated texts already exist, but they remain inaccessible to too many potential users.