In this chapter, a discussion will first be given of the theory of liquid hydrogen plasmas. Following early work on liquid metals as two-component systems, in which the semiclassical Thomas-Fermi approximation was used (Cowan and Kirkwood, 1958) to find pair-correlation functions for ions and electrons, March and Tosi (1973) described a fully quantal approach to this area which has been described in Chapter 14. This has subsequently been developed in a number of groups, especially through the work of Chihara (1984) and of Dharma-wardana and Perrot (1982). This two-component approach to a pure liquid metal and its correlation functions is, of course, to be contrasted with an approach to structure and forces as in Chapter 5. There the ions were treated as a one-component assembly in which, however, the effective interactions were mediated by the conduction electrons. In the two-component theory, the ions may be assumed, above the melting point of a liquid metal, to be a classical fluid. The electrons, as in Chapters 4 and 7, are usually to be treated as a degenerate quantal fluid.

Following a discussion of integral equation and density functional treatments of metallic liquid hydrogen, some discussion will be given of hydrogen-helium mixtures, their phase diagram, and the relevance to the constitution of Jupiter. Indeed, the properties of these two, the most abundant elements, are important for modeling both the giant planets Jupiter and Saturn. These elements are the major constituents of these planets and are subjected to pressures up to 45 million atmospheres and temperatures up to 20,000 K in Jupiter and about 10 million atmospheres and 14,000 K in Saturn.

Isotope effects are known to exist in liquid metals from a variety of experiments; in particular, those using the light isotopes Li6 and Li7. Some striking—and surprising—regularities exist more generally, especially the effect discovered by Haeffner (1953). Here, in an applied electric field, the light isotope in the isotopic liquid metal mixture is found, invariably, to move toward the anode. No known exceptions to this rule exist. The problem of electromigration is closely related, but presumably the understanding of the Haeffner effect is an essential prerequisite to an understanding of this phenomenon. For a review of electromigration, the article by Huntington (1973; see also Jones, 1980) may be consulted.

The facts, and some basic phenomenology, have been presented by Ginoza and March (1985) for (1) the Haeffner effect just discussed, (2) self- and mutual diffusion, and (3) shear viscosity.

Here, the aim is to present the theory underlying the Haeffner effect, in lowest-order Born approximation, in a form that is directly related to electrical resistivity. Some attempts to transcend this approximation will then be briefly discussed.

Haeffner effect

As mentioned already, it is useful to regard the Haeffner effect as a special case of the more general electromigration problem in liquid metal alloys. This is the effect found in a number of binary systems, where the constituentions drift in opposite directions under the influence of an applied dc electric field (see also Tyrrell and Harris, 1984).

This chapter begins with a relatively brief discussion of the thermodynamics of liquid surfaces. Then the statistical mechanics of inhomogeneous systems, already developed for treating freezing in Chapter 6, are used to obtain some formally exact results for the surface tension of a liquid. These formulae will then, essentially, be developed by gradient expansion methods to yield an interesting relation between bulk and surface properties, related by the “width” of the liquid-vapour interface.

Thermodynamics of liquid surfaces

The atomic density profile, denoted by ρ(z), must vary continuously across the interface from the value ρ1 of the bulk liquid to the value ρv of the bulk vapour. This variation can be expected to take place over a few atomic distances, at least when one is far from the critical point.

The anisotropy of the profile implies a net attraction to the liquid phase of an atom in the transition region: One must perform work to bring an atom from the bulk of the liquid to the surface; i.e., an excess of free energy is associated with the creation of the interface, namely, the surface free energy. It also implies that the tangential pressure, defined as the force per unit area transmitted perpendicularly across an area element in the yz or xz plane is a function pt(z) of position in the transition region. The difference between the components pt(z) and pn = p of the stress tensor in the transition region is negative, i.e., it has the nature of a tension, namely, surface tension.

Let us consider an open region, i.e., one in which particles can come and go freely, drawn in a system of infinite extent. What will now be shown is that the fluctuation in the number of particles in this region is given by the volume integral of g(r) – 1, which is specifically the isothermal compressibility of the liquid. Another interesting example of such a relation between fluctuations and thermodynamic quantities yields the specific heat cv; this is discussed in Appendix A5.4.

One reason for the interest in the above relation between the volume integral of the radial distribution function—or, equivalently, from (2.4), the long wavelength limit of the structure factor S(k)—and the compressibility (first derived by Ornstein and Zernike) is because of the difficulty of extending diffraction experiments to very small scattering angles.

Let us consider a member of the grand canonical ensemble in which the open region, of volume V, contains exactly N particles.

The origin of this volume can be traced to a letter from Dr. P. V. Landshoff (PVL), inviting me to write on liquid metals for his series. By then, my earlier book on the subject, published in 1968, described correctly by Professor N. E. Cusack in his generous review as ‘Bare bones of liquid metals’, was almost 20 years old. Of course, Dr. T. E. Faber's 1972 book was much more extensive, and I immediately recognized that in one important area of the subject: namely, weak scattering theory of electrical transport in liquid metals and alloys (Ziman and Faber-Ziman theory, respectively; see also below), I could not possibly compete with the quality of that.

However, after a long inner debate, I accepted the iniviation and sent PVL a proposed outline which already made clear that it would be a large volume. Back came a reply from PVL and an adviser: could I extend it somewhat?! Perhaps this may have been partly motivated by my long-standing interests in matter under extreme conditions of temperature and pressure, but, nevertheless, these additional proposals led me into areas in which I had not contributed myself for more than a decade. In the end, Chapter 16 became the main response to this challenge, and this could not have been written without the help of the 1985 review by Dr. M. Ross of the Lawrence Livermore Laboratory. While mentioning that, I must also acknowledge the 1987 survey by Dr. R. N. Singh, which was used so extensively in Chapter 13. Dr. Singh worked closely with the late Professor A. B. Bhatia, with whom I was also fortunate enough to collaborate over a decade or so.

The Lezioni Lincee on Topics of Polymer Dynamics were delivered by Professor Pierre Gilles de Gennes at the Politecnico di Milano in December 1986 to an audience of faculty members and graduate students.

The subject of polymer dynamics has been expanding at a very rapid pace in the last few decades and Professor de Gennes has been an active contributor to the field. In particular, several new theoretical ideas have appeared, one of their purposes being to connect different phenomena into a unified frame.

Rather than attempting to give a complete discussion of the whole subject, Professor de Gennes has concentrated on some special topics.

Chapter 1 (which consists of the first and the second lecture as originally delivered in Milan) contains a discussion of some general aspects of polymer chain dynamics. After a brief general illustration of the subject, the motion of isolated chains in a dilute solution is analyzed. Global and local deformation modes are discussed, with reference both to uncharged chains and to flexible polyelectrolytes. Attention is then focused on polymer melts; their viscoelastic behavior is seen in the light of the concept of chain ‘reptation’. Chemical kinetics in entangled media is then taken into consideration, with several examples. Some other aspects of polymer dynamics are also analyzed, including crystallization and the behavior of ramified chains.

The subject of Chapter 2 (originally the fourth lecture) is the conformation of a protein chain around a receptor site, a problem of great importance in molecular biophysics.

Introduction

Flexible polymers in dilute solution can reduce turbulent losses very significantly (Lumley, 1969). The main (tentative) interpretation of this effect is due to Lumley (1973). He emphasized that remarkable viscoelastic effects can occur only when certain hydrodynamic frequencies become higher than the relaxation rate of one coil 1/Tz (the ‘time criterion’). He then proposed a crucial assumption: namely that, in regions of turbulent flow, the solution behaves as a fluid of strongly enhanced viscosity, presumably via regions of elongational flow. On the other hand, Lumley noticed that – for turbulent flow near a wall – the viscosity in the laminar sublayer near the wall should remain low: this last observation does agree with the viscometric data on dilute linear polymers in good solvents, which show shear thinning (Graessley, 1974). Starting from the above assumptions, and performing a careful matching of velocities and stresses beyond the laminar layer, Lumley was able to argue that the overall losses in pipe flow should be reduced.

This explanation has been rather generally accepted. However, it is now open to some question: in recent experiments with polymer injection at the center of a pipe, one finds drag reduction in conditions where wall effects are not involved (McComb and Rabie, 1979; Bewersdorff, 1982, 1984).

This observation prompted Tabor and the present author to try a completely different approach (Tabor and de Gennes, 1986): namely to discuss first the properties of homogeneous, isotropic, three-dimensional turbulence without any wall effect, in the presence of polymer additives.