Table 4.9 Parameters of negative ions of alkali metals (Gogoleva et al. 1984)

n, cm−3

Fig. 4.9. Charged particle concentration in a plasma of cesium vapors on the 2 MPa isobar

by Khrapak (1979).

among positive ions at T 2200 K, is replaced by an A+2 ion which, in turn, gives way to an A+3 ion. One can expect the emergence of heavier positive ions on further cooling. The A− ion prevails among negatively charged components at T 2000 K. However, the A−2 ion remains inconspicuous.

4.4Droplet model of nonideal plasma of metal vapors. Anomalously high electrical conductivity

4.4.1Droplet model of nonideal plasma

The model was proposed by Iakubov 1979. A vapor of alkali metals close to the saturation line represents a strongly nonideal plasma. Outside the close proximity of the critical point, this plasma can be treated as weakly ionized. In this case, the nonideality is mainly caused by the strong interaction between charged and neutral particles. This interaction promotes the formation of heavy charged complexes – clusters – in the plasma. The plasma of metal vapors may be treated

as a mixture of the electron gas and a gas of clusters of di erent charges and sizes. The e ect of clusters on the degree of plasma ionization was taken into account by Likal’ter (1978), Khrapak (1979), Iakubov (1979), Lagar’kov and Sarychev (1979), and Hernandez et al. (1984) in the framework of di erent models. Inclusion of this e ect enables one to understand the reason why the electrical conductivity of cesium and mercury vapors in the vicinity of the saturation line is anomalously high. For instance, the Saha’sand Lorentz formulas yield the electrical conductivity σ = 3 ·10−5 ohm−1cm−1 at p = 2 MPa and T = 1400 K. The value of σ = 10 ohm−1cm−1 obtained in experiments is five orders of magnitude higher than these (typical for ideal plasmas) estimates.

Likal’ter (1978), Khrapak (1979), Iakubov (1979), Lagar’kov and Sarychev (1979), and Hernandez et al. (1984) share the common assumption of a sharp increase in the concentration of charged polyatomic clusters when approaching the saturation line. Moreover, the concentration of positively charged clusters exceeds considerably that of negatively charged ones. The plasma electroneutrality is ensured by the corresponding increase in the electron number density, which leads to an increase in the electrical conductivity of a plasma. Di erences in theoretical models occur when one tries to specify the cluster properties.

It was proposed by Iakubov (1979) to treat clusters as small droplets. The advantages of this model include the possibility of using the characteristics of metals, such as surface tension and electron work function, to determine the cluster properties.

Changes in the aggregate state of matter, which are caused by electrostriction in the neighborhood of charged particles, are well–known in physics (Frenkel 1955; Jortner and Kestner 1974; Iakubov and Khrapak 1982). Charged droplets can form in vapors, and icicles can form in liquids. On the one hand, in a plasma of metal vapors the conditions for droplet stabilization by a charge are less favorable, because of high temperatures. On the other hand, the very high polarizability enhance electrostriction.

The elementary droplet model by Iakubov (1979) enables one to put forth the basic ideas and make rough estimates. Let us assume that a plasma consists of atoms, positively charged droplets and electrons. The concentration of these particles in a volume V is Na, N +, and Ne, and the total number of all particles

Here ϕL and ϕG are the thermodynamic potentials of liquid and vapor per atom, so that the quantity gkT ln(ps/p) = g(ϕL − ϕG) is the work done to form a neutral droplet of radius R, with g = (4π/3)R3nL being the number of particles in the droplet, nL is the concentration of particles in the liquid, and ps is the saturation pressure. Also, 4πγR2 is the surface energy (γ is the surface tension), and W + e2/(2R) is the electron work function for a droplet. In the last entropy

Table 4.10 Parameters of ion droplets in saturated cesium vapor

term in (4.33), the summation is performed over all sorts of particles. Such a simple approach – when we are not interested in droplets with radii other than the most probable radius R, and do not describe the internal degrees of freedom of the droplets – reduces to Frenkel’s theory of pretransition phenomena (Frenkel 1955).

We minimize, with respect to the radius, the work required to produce a single charged droplet, gkT ln(ps/p) + 4πγR2 + W + e2/(2R), in order to derive the Kelvin equation for determining the radius of the most probable droplet,

kT nL ln(p/ps) = 2γ/R − e2/(8πR4).

In saturated vapor, R is equal to the “electrocapillary” radius

This allows us to estimate the parameters of ion droplets along the saturation line of cesium vapors (Table 4.10). Of course, the emerging droplets are too small for the macroscopic description to be su ciently correct (curvature corrections to γ provide only slight improvement). However, according to the nucleation theory the droplets containing more than 10 particles can be treated as macroscopic.

Below we consider only saturated vapor p = ps. We derive the equation of ionization equilibrium, gA A+g + e, by varying (4.33) and taking into account the stoichiometry, gδna = −δn+ = −δne. Since the plasma is weakly ionized, ne = n+ na, we finally derive

For a number of reasons, the pre–exponential factor in (4.35) has been very poorly determined. However, the major e ect is due to the exponent itself. It contains W , the work function of an electron from the metal, which is known to decrease as the metal temperature rises (as the density decreases). This dependence was investigated by Iakubov et al. (1986). Let us extrapolate the derived

values taking into account that the work function must reach zero at the critical temperature Tc. It turns out that one can write

where W0 = 1.8 eV is the tabular value of the work function for cesium (see Appendix C).

The results of the calculations for ne are listed in Table 4.10 (see above). The interaction e ects led to an increase in ne by orders of magnitude as compared to the ideal–gas approximation. However, such a simple analysis is too crude. The drop model was improved considerably by Pogosov and Khrapak (1988), and Zhukhovitskii (1989). Below we present these results, following mainly Pogosov and Khrapak (1988).

4.4.2Ionization equilibrium

Let us consider the ionization equilibrium in a multicomponent mixture of ideal gases of electrons and charged complexes consisting of g atoms and a charge Z. For such a mixture, the thermodynamic potential Φ is

g,Z

=NgZ (kT ln p + χZg + kT ln(NgZ /F )) + Ne(kT ln p + χe + kT ln(Ne/F )),

gZ

where

F = NgZ + Ne = pV /kT,

In Eqs. (4.37) and (4.38), NgZ and Ne denote the number of particles in volume V , pZg , and pe are the partial pressures, p is the total pressure, and Σ is the partition function over internal and translational degrees of freedom of the species.

Electrons possess only translational degrees of freedom,

where W is the electron work function from a flat surface, λe is the thermal wavelength of the electron, and nes is the equilibrium number density of electrons in the vicinity of the metal surface. We derive the relation between the

concentrations of charged and neutral components by treating the equilibrium AZg AZg +1+ e, which yields

Let us now study another reaction – atom condensation leading to a growth

where λ1 is the thermal wavelength of atom. One usually tries to express fgZ or fg0 in terms of the macroscopic characteristics of the disperse phase. Let us assume that the presence of charge has no e ect on the internal properties of the particle, and that the dependence of fgZ on Z is defined by the electrostatic energy of the excess charge. The latter is only true for large particles, when the fraction of surface atoms is much smaller than the volume fraction. Hence

where nl is the number density of atoms in a liquid and ε is the dielectric permeability of the vapor. The substitution of this expression into (4.40) gives

have maximum concentration. In the vicinity of the saturation line for alkali metals, the charge Z is close to unity.

Let us identify in fg0 the contributions made by the translational, fg,tr, degrees

where λg = g1/2λ1 is the thermal wavelength of a g–atom complex. The quantity Eg0 includes the kinetic energy and the energy of interaction of atoms in the complex. It is evident that, in the limit of high values of g, the energy Eg0 is proportional to g. The dependence of Eg0 on g can be sought for in the form of the expansion over inverse powers of the complex radius,

where µl is the chemical potential of atoms in a liquid. The second term in Eq. (4.47) corresponds to the free surface energy of the complex, and the third

Eg0 /gµL

1.0

0.8

0.6

0.4

0.2

0

100 101 102 103 g

Fig. 4.10. Energy Eg0 as a function of the number of atoms in the complex (Pogosov and Khrapak 1988).

term can be treated as the inclusion of the curvature dependence of the surface tension.

In a large number of studies, Eg0 is calculated from “first principles”. Figure 4.10 gives the values of Eg0 borrowed from papers cited by Pogosov and Khrapak 1988. These values correspond to complexes Ag formed by atoms of di erent elements (Li, Na, K, Fe and Cu), and are marked in Fig. 4.10 using di erent symbols. One can see that, with good accuracy, the quantity Eg0/(gµL) is independent of atoms in the complex. This fact agrees with conclusions by Langmuir and Frenkel (see, Frenkel 1955) about the relation between the sublimation energy and specific surface energy. This justifies the choice of expression (4.47) for Eg0. Indeed, according to Frenkel (1955), at fairly low temperature (when µL −q, where q is the sublimation energy),

Eg0/(gµL) = 1 − α/(qg1/3), α/q (η − η )/η,

where α = 4πγ(3/4πnL)2/3 = 4πγrs2 is the “surface energy” of a single atom, and η and η are the numbers of the nearest neighbors for an atom deep inside the liquid and on the surface, respectively. It is clear that α/q 1/2 < 1. It follows from Table 4.11 that the ratio α/q, taken at triple points of metals, is close to 0.64.

Now let us reconsider Eq. (4.41). We retain the first two terms of the expansion of Eg0 over g1/3. Taking into account that µL − µG = −kT ln(p/ps), we derive the “classical” expression for the concentration of uncharged droplets,