Read an Excerpt

Cambridge University Press
0521839068 - Van der Waals Forces - A Handbook for Biologists, Chemists, Engineers, and Physicists - by V. Adrian Parsegian
Excerpt

Prelude

PR.1. The dance of the charges, 4
Early insight, 5 ● Point-particle interactions in gases, 5 ● Pairwise summation, lessons learned from gases applied to solids and liquids, 7 ● Derjaguin-Landau-Verwey-Overbeek theory, colloids, 8 ● The modern view, 9 ● Is there a way to connect interaction energies and the shapes of interacting objects?, 13 ● What are the powers of shapes and sizes?, 14 ● Units, 14 ● Not so fast! Those formulae don't look so different to me from what people have learned by using old-fashioned pairwise summation, 17 ● How strong are long-range van der Waals interactions?, 18 ● How deeply do van der Waals forces "see"?, 22 ● The subject is "force" but the formulae are "energies." How to connect them? 23

PR.2. How do we convert absorption spectra to charge-fluctuation forces? 24 ● Sampling frequencies?, 25 ● If I take that formula seriously, then two bubbles or even two pockets of vacuum will attract across a material body. How can two nothings do anything?, 26 ● Does charge-fluctuation resonance translate into specificity of interaction?, 26 ● How does retardation come in?, 27 ● Can there ever be a negative Hamaker coefficient and a positive charge-fluctuation energy? 28

PR.3. How good are measurements? Do they really confirm the theory? 30 ● Atomic beams, 31 ● Nanoparticles, 31 ● Force microsopy on coated surfaces, 31 ● Glass surfaces, 31 ● Mica, 32 ● Bilayers as thin films and as vesicles, 32 ● Cells and colloids, 33 ● Aerosols, 34 ● Bright stuff. Sonoluminescence, 34 ● Fun stuff, 34 ● Slippery stuff, ice and water, 34 ● What about interfacial energies and energies of cohesion? Aren't van der Waals forces important there too, not just between bodies at a distance? 35

PR.4. What can I expect to get from this book? 37

"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 der Waals 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. His work followed that of Robert Boyle on dilute gases of infinitesimal noninteracting particles. In high school, or even in junior high, we learn Boyle's law: pV is constant. In today's modern form, we write it as pV = NkT, where N is a given number of particles and k is the universal Boltzmann constant. In 1660 Boyle called his relation the "spring of air." Hold the temperature fixed; squeeze the volume; and the pressure goes up. Hold the volume fixed; heat; and the pressure goes up too.

This "spring" is still the ideal example of today's "entropic" forces. Any particle or set of particles will try to realize all allowed possibilities. Particles of the ideal gas fill all the available volume V. The confining pressure depends on the volume V/N available to each particle and on kT, the vigor of thermal motion.

Boyle and Boltzmann describe most of the spring, but to describe dense-gas pressures, van der Waals found he had to replace pV = NkT with [p + (a/V²)](V - b) = NkT:

The full volume V became (V - b). A positive constant b accounts for that small part of the space V occupied by the gas particles themselves. Today we speak of "steric" interactions in which bodies bump into each other and thereby increase the pressure needed to confine them.

The earlier role of p_Boyle is now played by p_vdW + aV². With constant positive a, p_vdW=p_Boyle - a/V² is less than the ideally expected p_Boyle by an amount a/V². This difference gives evidence that the particles are attracted to each other and thus exert less outward pressure on the walls of the container. When the volume V is allowed to go to infinity, the difference disappears. The form a/V² tells us that this attractive correction varies with the average distance between particles. The smaller the volume V allowed to the N particles, the stronger the pressure-lowering attraction between the particles.

It is this attraction between molecules in a gas that most people first think of as the van der Waals interaction. These attractions in gases are so weak that they are small compared with thermal energy kT. Nevertheless, nonideal gases taught us a general truth: Electrically neutral bodies attract.

PR.1. The dance of the charges

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.

Why? Because the frequencies at which charges spontaneously fluctuate are the same as those at which they naturally move, or resonate, to absorb external electromagnetic waves. This is the essence of the "fluctuation-dissipation theorem." It states that the spectrum (frequency distribution) over which charges in a material spontaneously fluctuate directly connects with the spectrum of their ability to dissipate (absorb) electromagnetic waves imposed on them. Computation of charge-fluctuation forces is essentially a conversion of observed absorption spectra. By its very nature, the measured absorption spectrum of a liquid or solid automatically includes all the interactions and couplings among constituent atoms or molecules.

Early insight

Contemporary with the 1870s formulation of the van der Waals gas equation were huge steps in electromagnetic theory. Between 1864 and 1873 James Clerk Maxwell distilled all of electricity and magnetism into four equations. In 1888 Heinrich Hertz showed how electric-charge oscillations could create and absorb (and thus detect) electromagnetic waves. In his 1894 doctoral dissertation, P. N. Lebedev saw that atoms and molecules must behave as microscopic antennae that both send and receive electrical signals. He recognized that these actions and reactions create a physical force. He saw too that in condensed materials these sendings and receivings, with their consequent forces, would have to involve many atoms or molecules at the same time. Although widely quoted, Lebedev's prescient words are worth repeating to guide us here:

Hidden in Hertz's research, in the interpretation of light oscillations as electromagnetic processes, is still another as yet undealt with question, that of the source of light emission of the processes which take place in the molecular vibrator at the time when it gives up light energy to the surrounding space; such a problem leads us, on the one hand, into the region of spectral analysis [absorption spectra], and on the other hand, quite unexpectedly as it were, to one of the most complicated problems of modern physics—the study of molecular forces.

The latter circumstance follows from the following considerations: Adopting the point of view of the electromagnetic theory of light, we must state that between two radiating molecules, just as between two vibrators in which electromagnetic oscillations are excited, there exist ponderomotive [physical body] forces: They are due to the electromagnetic interaction between the alternating electric current in the molecules…or the alternating charges in them…; we must therefore state that there exist between the molecules in such a case molecular forces whose cause is inseparably linked with the radiation processes.

Of greatest interest and of greatest difficulty is the case of a physical body in which many molecules act simultaneously on one another, the vibrations of the latter not being independent owing to their close proximity.

Lebedev's program for learning was not carried out for decades. Through the 1930s, there was rapid progress in formulating interactions between particles in gases; but there was no corresponding success in formulating these forces within liquids and solids "in which many molecules act simultaneously on one another…owing to their close proximity." From 1894, when it was first realized that there must be a connection between absorption spectra and charge-fluctuation forces, to the 1950s when the connection was actually made, to the 1970s when it became practical to convert measured spectra to predicted forces, people still thought in terms of van der Waals forces between condensed materials as though these forces acted the same as in gases.

Point-particle interactions in gases

By the end of the 1930s, the catechism of neutral-molecule interactions, at separations large compared with their size, included three kinds of dipole-dipole interactions. Their free energies—the work needed to bring them from infinite separation to finite separation r—all vary as the inverse-sixth power of distance, -(C/r⁶), with different positive coefficients C:

• Keesom interactions of permanent dipoles whose mutual angles are, on average, in attractive orientations:

  Image not available in HTML version

  Dipole-dipole electrostatic interaction perturbs the randomness of orientation. If the dipole on the left is pointing "up," then there is a slightly greater chance that the dipole on the right will point "down" (or vice versa; both particles are equivalent in mutual perturbation). By increasing the chances of an attractive mutual orientation, the perturbation creates a net-attractive interaction energy.
• Debye interactions in which a permanent dipole induces a dipole in another nonpolar molecule, with the induction necessarily in an attractive direction:

  Image not available in HTML version

  The relatively sluggish permanent dipole polarizes the relatively frisky electrons on the nonpolar molecule and induces a dipole of opposite orientation. The direction of the induced dipole is such as to create attraction.
• London dispersion interactions between transient dipoles of nonpolar but polarizable bodies:

  Image not available in HTML version

  Here, the electrons on each molecule create transient dipoles. They couple the directions of their dipoles to lower mutual energy. "Dispersion" recognizes that natural frequencies of resonance, necessary for the dipoles to dance in step, have the same physical cause as that of the absorption spectrum—the wavelength-dependent drag on light that underlies the dispersion of white light into the spectrum of a rainbow.

There is an easy way to remember why these interaction free energies go as an inverse-sixth power. Think of the interaction between a "first" dipole pointing in a particular direction and a "second" dipole that has been oriented or induced by the 1/r³ electric field of the first. The degree of its orientation or induction, favorable for attraction, is proportional to the strength of the orienting or inducing electric field. The oriented or induced part of the second dipole then interacts back with the first. Because the interaction energy of two dipoles goes as 1/r³, we have

1/r³ (for induction or orientation force) x 1/r³ (for interaction between the two dipoles) = 1/r⁶.

This is not an explanation of the inverse-sixth-power energy in gases; it is only a mnemonic.

In quantum mechanics, we think of each atom or molecule as having its own wave functions that describe the distribution of its electrons. The expected basis of interaction is that two atoms or molecules react to each other as dipoles, each atom's or molecule's dipolar electric field shining out as 1/r³ with distance r from the center. This dipole interaction averages to zero when taken over the set of electron positions predicted for the isolated atoms. However, when the isolated-atom electron distributions are themselves perturbed by each other's dipolar fields, "second-order perturbation" in the parlance, the resulting position-averaged mutual perturbation makes the extra energy go as 1/r⁶ at separations r much greater than dipole size.

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

For practical and fundamental reasons, there was a need to learn about the interactions of bodies much larger than the atoms and small molecules in gases. What interested people were systems we now call mesoscopic, with particles whose finite size Wilhelm Ostwald famously termed "the neglected dimension": 100-nm to 100-μm colloids suspended in solutions, submicrometer aerosols sprayed into air, surfaces and interfaces between condensed phases, films of nanometer to millimeter thickness. What to do?

In 1937 H. C. Hamaker,^, following the work of Bradley, DeBoer, and others in the Dutch school, published an influential paper investigating the properties of van der Waals interactions between large bodies, as distinct from the small-molecule interactions that had been considered previously. Hamaker used the pairwise-summation approximation. The idea of this approximation was to imagine that incremental parts of large bodies could interact by -C/r⁶ energies as though the remaining material were absent. The influence of the intermediate material was included as the electromagnetic equivalent of Archimedean buoyancy.

De Boer had shown that, summed over the volumes of two parallel planar blocks whose separation l was smaller than their depth and lateral extent, the -C/r⁶ energy became an energy per area that varied as the inverse square of the separation l (1/l² for small l):

In the Hamaker summation over the volumes of two spheres, the interaction energy approaches inverse-first-power variation near contact (1/l when l ≪ R) and reverts to the expected inverse-sixth-power dependence of point particles when the spheres are widely separated compared with their size (1/r⁶ when r ≫ R):

The newly recognized possibility that van der Waals forces could be of much longer range than the 1/r⁶ reach previously expected from Keesom, Debye, and London forces easily explains the well-earned influence of Hamaker's paper. The coefficient of the interaction between large objects came to be termed the "Hamaker constant" A_Hamaker, (abbreviated in this book as A_Ham).

Derjaguin-Landau-Verwey-Overbeek theory, colloids

There was still the problem of how to evaluate A_Ham. When the model was applied to colloidal interactions, this coefficient was usually fitted to data rather than estimated from any independent information. Papers written in the 1940s and early 1950s allowed "constants" that could span many orders of magnitude. This span revealed ambiguity. Nevertheless, even with such quantitative problems, van der Waals forces became recognized as the dominant long-range interaction governing the stability of colloids and often the dominant energy of interfaces. At least to first approximation, van der Waals forces followed a power law, whereas electrostatic forces, screened by the mobile ions of a salt solution, dropped off exponentially. At long enough distance a power law will always win over exponential decay. (It will also win at very short distances, but by then there are many other factors to consider.)

Neglecting all but electrostatic and van der Waals forces, and treating these two forces as though they were separable, the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory created a framework (and soon a dogma) for describing the stability of colloidal suspensions. In their inspiring book, published in 1948, Verwey and Overbeek make this remark: "This stability problem has been placed on a firmer physical basis by the introduction of the concept of van der Waals London dispersion forces together with the theory of the electrolytic or electro-chemical double layer." Within this framework, exponentially varying electrostatic repulsion (indicated by es in the following figure) competed with power-law van der Waals attraction (indicated by vdW) to create energy versus separation curves of a form:

The idea has been applied to the interactions of bodies of many shapes. At long distance, the net interaction between two colloidal particles is dominated by the van der Waals attraction. This longer-range attraction creates a weak "secondary" (2^°) energy minimum at a point of force balance against shorter-range electrostatic repulsion. The depth of this minimum compared with thermal energy kT allows the possibility of loose association. The height of the energy maximum opposing closer approach, again in kT units, determines the rate at which the particles might coalesce into the deep energy well of "primary" (1^°) contact at which, mathematically, the inverse-power van der Waals energy takes on large negative values. Kinetic measurements, usually  difficult to interpret, have been fitted to this DLVO scheme. Although more quantitative examination would require equilibrium or static measurements, and although the assumed separability of electrodynamic and electrostatic forces is incorrect, nevertheless the DLVO theory established van der Waals forces as the essential feature in the aggregation of nanometer-to-micrometer-sized particles.

The modern view

The first steps to remove the ambiguity about the magnitude of van der Waals interactions were taken almost at the same time as the development of DLVO theory, when H. B. G. Casimir formulated the interaction between two parallel metal plates. The antecedent of Casimir's view was the analysis by Max Planck in 1900 to explain the heat capacity of an empty "black box." The problem at that time was to explain the amount of heat needed to warm or cool a hollow cavity. In such a space, electromagnetic fields can exist as long as these fields satisfy the polarization properties of the vacuum in the cavity and of the bounding walls at finite temperature. In particular, the fields of this "blackbody" radiation can be expressed as oscillatory standing waves within the cavity; the energy of each wave of frequency ν changes with the emission or absorption of energy by the walls. It was Planck's famous postulate that these exchanges of energy occur as discrete units, hν, soon to be called photons, with a finite value of h, now known as Planck's constant. This proposed discreteness of energy transfer was the birth of the "quantum" theory. By summing the free energies of all allowed discrete oscillations within ideally conducting walls, and then by differentiating this total free energy with respect to temperature, Planck was able to account for the previously puzzling measured heat capacity, the energy absorbed or radiated by the space when it was heated or cooled.

In 1948 Casimir used the same principle, focusing on the free energy of electromagnetic modes, to derive the force between ideally conducting walls of a box. He considered a "box" as having two of its six faces infinitely bigger than the remaining four; his box was really two large facing plates. Summing the energies of the electromagnetic modes allowed in such a box and then differentiating with respect to the distance between the two large walls, he obtained an expression for the electromagnetic pressure between the two plates. Defining the energy per area relative to a zero value when the plates are infinitely far apart then yielded the electromagnetic energy of interaction and pressures between two metal plates. Casimir's shift in perspective broke a spell. Suddenly the human race was able to step away from microscopic thinking about atoms and to survey the macroscopic whole. As Casimir told it, 50 years later:

Here is what happened. During a visit I paid to Copenhagen, it must have been in 1946 or 1947, [Niels] Bohr asked me what I had been doing and I explained our work on van der Waals forces. "That is nice," he said, "that is something new." I then explained I should like to find a simple and elegant derivation of my results. Bohr thought this over, then mumbled something like "must have something to do with the zero-point energy." That was all, but in retrospect I have to admit that I owe much to this remark.

For this reason, the Casimir formulation had another far-reaching effect. It made us recognize that "zero-point" electromagnetic-field fluctuations in a vacuum are as valid as fluctuations viewed in terms of charge motions. As clearly predicted by the Heisenberg uncertainty principle, even in a vacuum, the change ΔE in the energy of a quantum mechanically evolving system depends inversely on the duration of time Δt between observations: ΔE Δt ≥ h/2π. By the inclusion of the very shortest intervals Δt, which correspond to periods of the highest frequencies or the shortest-lived fluctuations, the energy change of all possible fluctuations formally diverges to a physically impossible infinity. We are all bathed in this "vacuum infinity" of virtual electromagnetic waves, although we would never guess it from listening to a weather report.

We now recognize that "empty space" is a turmoil of electromagnetic waves of all frequencies and wavelengths. They wash through and past us in ways familiar from watching the two-dimensional version, a buoy or boat bobbing in rough water. We can turn the dancing-charges idea around. From the vacuum point of view, imagine two bodies, such as two boats in rough water or a single boat near a dock, pushed by waves from all directions except that of a wave-quelling neighbor. The net result is that the bodies are pushed together. You get close to a dock, you can stop rowing. The waves will push you in.

We can think of electromagnetic modes between the two bodies as the fluctuations that remain as tiny deviations from the outer turmoil. The extent of the quelling is, obviously, in proportion to the material-absorption spectra. So we can think of absorption frequencies in two ways: those at which the charges naturally dance; those at which charge polarization quells the vacuum fluctuations and stills the space between the surfaces.

A careful examination of forces between point particles at finite temperature and asymptotically infinite separations illustrates the value of focusing on vacuum fluctuations. These forces can be seen as driven by thermally excited, infinitely long-wavelength electromagnetic-field fluctuations that come from the surrounding vacuum. Particle polarizations are a passive response to these external fields. It is as though the particles bob on infinitely extended waves and feel each other only to the extent that their passive polarization modifies the incoming fields. Again think of the two boats on rough (read "thermally excited") water, but this time think of them as so far away as to make only the slightest perturbation on the waves.

In 1948 Casimir and Polder published a second conceptual leap, showing how "retardation" changed the charge-fluctuation force between point particles. If the distance between fluctuating charges is large enough, it takes a finite amount of time for the electromagnetic field from one charge to fly across to the other. By the time the second charge responds to the field, the momentary charge configuration at the first position will have changed, and the charge fluctuations will fall out of step with each other. The strength of interaction is always weakened; its distance dependence is changed. For example, 1/r⁶ energy between small particles becomes 1/r⁷. In fact, retardation was implicitly included in the Planck and Casimir formulations; finite light velocity c means finite wavelength λ = c/ν for a wave of frequency ν. In fact there is no "first" sender and "second" recipient, only the coordinated fluctuation of charges at different places interacting through waves of finite wavelength. The Casimir-Polder tour de force ends with the observation that the simple form of the results "suggest that it might be possible to derive these expressions, perhaps apart from the numerical factors, by more elementary considerations. This would be desirable since it would also give a more physical background to our result, a result which in our opinion is rather remarkable. So far we have not been able to find such a simple argument." A few years later, the Lifshitz formulation accomplished just that by deriving the general theory, built on Casimir's earlier insight on the connection between the electromagnetic energy in a perfect-conductor Planck black box and the work of moving the walls of that box. Physicists now refer to this connection as the "Casimir effect." Some "effect"!