In 1963, Bob Zwanzig wrote a pair of closely related papers in JCP on dielectric friction, exploring the frictional force on a moving ion1 and the frictional torque on a rotating dipole2 due to dielectric loss in the surrounding medium. These papers—which typify Zwanzig’s elegant, intuitive, incisive, and even didactic style—recount derivations simplifying a complex subject whose importance had been evidenced in, e.g., the ion case by estimates revealing its magnitude to be comparable to the well-known Stokes’s law of friction due to viscosity. His theories in these papers, built on foundations laid by other giants like Born, Debye, Onsager, Landau, and Lifshitz, have proven of ongoing interest, inspiration, and relevance.
Zwanzig’s efforts in the first member of the 1963 JCP pair, focused on ionic motion,1 were inspired by earlier work in JCP by Boyd,3 who—following a suggestion by Fuoss4—first derived an expression for the dielectric friction force for an ion moving through a continuum model dielectric medium. The source of this friction is the dielectric medium’s lack of equilibrium with the moving ion generating an opposing electric field: the solvent’s dipoles require time to adjust to the ion’s changing position. Zwanzig commented dryly: “Boyd’s derivation is based on essentially correct ideas; but it contains certain approximations that are not only unnecessary, but lead to slightly incorrect numerical results. The purpose of this article is to present an exact and more general treatment of the same problem.” He then proceeds to re-derive an expression for the friction, in 2½ printed pages. Zwanzig’s derivation was not restricted to Boyd’s assumption that the dielectric function follows simple Debye relaxation, making his result applicable to more general dielectric behavior [i.e., arbitrary ε(ω)]. Zwanzig extended his own theory seven years later, to include hydrodynamic flow effects due to ion motion relative to the surrounding fluid and an improved description of electrostatic effects.5
During a sabbatical at Columbia University, one of the co-authors (M.B.) contemplated charge carrier motion through hybrid organic–inorganic perovskite materials. These materials, characterized by a crystal–liquid duality, consist of a relatively well-ordered, crystalline inorganic lead halide and a disordered organic cation sub-lattice. The organic cations have a large dipole moment and can reorient relatively freely in their spacious perovskite cages. As such, the motion of a charge carrier can be coupled to the relaxation of surrounding organic dipoles, adding a dielectric drag to the moving charge. In addition to the “normal” electron–phonon coupling that leads to momentum relaxation by electron–phonon scattering, the resulting dielectric friction may also contribute to limiting the mobility of charge carriers in perovskites.6
The simple theory developed by Zwanzig—based on a simple equation of motion—facilitated a semi-quantitative comparison between effects due to electron–phonon coupling and dielectric drag. A theoretical estimate showed that, indeed, the dielectric drag contribution may be non-negligible. Interestingly, more recent mobility measurements of charge carriers in different perovskites are consistent with this conclusion: when the organic, dipolar cation is replaced by an inorganic Cs+ ion without a dipole (for which dielectric friction should be less significant), the electron and hole mobilities increase significantly.7
In the second of his paired 1963 JCP papers,2 Zwanzig extended his exploration of dielectric friction to the case of a rotating dipole, also in a continuum model dielectric medium. Here, the lack of dielectric medium equilibrium leads to a frictional torque on the rotating dipole. This basic idea, with its attendant equations, was extended in 1970 in combination with previous efforts of Onsager, Kirkwood, and other researchers—in the influential Nee–Zwanzig paper8 on dielectric relaxation, with direct connection to and impact on the considerable experimentation in that arena.
The original pair of Zwanzig’s papers1,2 and their familial progeny5,8 have generated extensive and impressive research activity over the years, continuing to the present era.9 The other co-author of the present Reflection (J.T.H.) has participated in this effort in two ways: First, he was able to establish an analytic connection10 of the Nee–Zwanzig dielectric friction with the solvation dynamics probed in time-dependent fluorescence experiments. Here, an initial electronic excitation of a fluorescent solute molecule changes its charge distribution; the surrounding polar medium’s equilibration to this new distribution is then reflected in the fluorescence spectrum’s time evolution. Such experiments provided the first clear characterization of polar solvent dynamical time scales, a critical issue of much previous speculation. Second, the appeal of the Zwanzig papers’ basic dielectric friction idea also motivated the extension to a different electrically responsive environmental situation—chemical reactions in electrolyte solutions. Here, friction associated with an ion atmosphere can influence the rate of chemical reactions in which the reactants’ charge distribution changes in the reaction.11 The frictional effect is directly due to non-equilibrium aspects of the atmosphere on the very short time scale of the reaction barrier crossing, an effect neglected in the standard textbook discussions of the impact of ion atmospheres on reaction rates.
Our brief discussions have given just a glimpse of the ongoing relevance of Zwanzig’s 1960s theories and his subsequent elaborations. This relevance extends to fields ranging, for example, from charge carrier mobility in novel semiconductors,6 through dynamics in normal and supercooled water,9,10 (possibly) even in living cells,12 and to proton transport in fuel cells.13 Such crossroads between classic theories and contemporary applications showcase the enduring impact of his foundational work. This kind of impact is patently characteristic of all of Zwanzig’s numerous and wide-ranging seminal contributions.
We conclude this Reflection with a personal thought (J.T.H.), which we hope captures an essential feature of Zwanzig’s papers, beyond the admirable qualities mentioned above. His writing style possesses a distinctive and almost intimate quality for many of us: reading his papers can easily evoke the sensation of engaging in one of his collegial and enlightening discussions amid the sunny, grassy fields at Gordon Conferences.
