ISOBARIC JOULE HEATING

47

the equations

R r

T (r) = Ts + F 2	dr(rκR)−1 σr dr ,			(2.7)
r			0
i = (πR2F ) R2		R	σr dr	(2.8)
		0
	2

of the trial functions σ(T ) and κR(T ) followed by the iteration procedure. During the experiments, the maximum pressure of 0.111 GPa was attained and the temperature range of (4–18)·103 K was covered.

It is noted by Dikhter and Zeigarnik (1981) that the main drawback of the technique resides in the impossibility of measuring either the plasma density in the process of its expansion or the plasma temperature. The temperature gradient near the discharge column boundary is very great. Therefore, the calculation of temperature based on its boundary value undoubtedly leads, under such conditions, to great uncertainties. The situation is aggravated by the presence of a quartz wall also having a temperature gradient; there is the further possibility of a reversible loss of transparency by quartz. Moreover, one cannot rule out the possibility of fusion and evaporation of quartz at the boundary with the cesium plasma, as well as partial mixing of the cesium plasma with oxysilicon quartz plasma, that is, a violation of the purity of the material under investigation.

Vorobiov et al. (1981) and Yermokhin et al. (1981) obtained the κR(T ) and σ(T ) values for lithium, sodium, potassium and cesium at pressures of 10–100 MPa and temperatures of (4–14)·103 K. Figure 2.26 shows the isobars of the coe cient of radiative thermal conductivity κR(T ) for cesium.

At low temperatures, that is, at maximum nonidealities, the κR values obtained exceed considerably the values calculated in weak nonideality approximation by Kuznetsova and Lappo (1979). This means that a nonideal alkali plasma proves more transparent than one might assume from ideal–gas concepts. This is especially clear from Fig. 2.27, which shows the ratios between the Rosseland lengths measured by Vorobiov et al. (1981) and those calculated by Kuznetsova and Lappo (1979).

Figure 2.28 gives the isobars and one of the isentropes of the electrical conductivity of cesium plasma.

2.2.2Exploding wire method

The exploding wire method was used for the first time by Lebedev (1957, 1966) (see, also, Lebedev and Savvatimski 1984) for the investigation of the liquid state of metals. Afterwards, this method was employed by Gathers et al. (1974, 1976) to measure the electrical conductivity and the equation of state for liquid uranium in the high–temperature region. Dikhter and Zeigarnik (1975, 1981) investigated the equation of state and electrical conductivity of nonideal cesium and lithium plasma. The apparatus is shown schematically in Fig. 2.29. A cesium

48 ELECTRICAL METHODS OF NONIDEAL PLASMA GENERATION

κR W cm K

Fig. 2.26. Isobars of the coe cient of radiative thermal conductivity of cesium plasma (Vorobiov et al. 1981): 1, 10 MPa; 2, 30 MPa; 3, 100 MPa; broken line, calculation by Kuznetsova and Lappo (1979).

lR/lR0

.

.

.

.

Fig. 2.27. Ratio between the values measured by Vorobiov et al. (1981) and those calculated by Kuznetsova and Lappo (1979) of Rosseland lengths depending on the nonideality parameter γ, T = 6000 K.

(or lithium) wire is placed in a high–pressure chamber 5, and a current with a density of (1–5)·106 A cm−2 is passed through the wire. The plasma formation contained by a high–pressure inert gas expands upon heating. While doing so, the pressure in the column remains constant and equal to the inert gas pressure.

The experiment involves the measurement of pressure in the chamber, the

ISOBARIC JOULE HEATING

49

ohm cm

Fig. 2.28. Coe cient of electrical conductivity of cesium plasma: isobars 1, p = 13 MPa; 2, p = 27.5 MPa; 3, p = 60 MPa; 4, p = 110 MPa (Kulik et al. 1984); 5, p = 50 MPa; 6, p = 26 MPa; 7, p = 126 MPa (Dikhter and Zeigarnik 1981); ∆, isentrope (Sechenov et al.

1977).

Fig. 2.29. Schematic of the apparatus for isobaric heating in an argon atmosphere (Dikhter and Zeigarnik 1981): 1, ballast resistor; 2, capacitor bank; 3, oscillograph; 4, high–speed photorecorder; 5, high–pressure chamber; 6, heated wire; 7, spectrograph.

oscillography of current in the circuit and the voltage drop across the plasma column, and photorecording of the column expansion process with time. The latter enables one to measure the time dependence of the column diameter. Special estimates, as well as the variation of the experimental conditions, demonstrate that the mass of material in the discharge is constant (the mixing of argon and cesium can be ignored) while all the losses of energy, including the radiation losses, are small. Therefore, one can use the measured values to calculate the energy input per unit mass, that is, the enthalpy, density, and electrical conductivity.

According to the estimates by Dikhter and Zeigarnik (1981), the error in measuring the current and voltage does not exceed 7% and for the diameter,

50 ELECTRICAL METHODS OF NONIDEAL PLASMA GENERATION

3%. This leads to the following estimates of errors in measuring the enthalpy, density, and electrical conductivity: ∆H = 15%, ∆ρ = 10% and ∆σ = 20%. The scatter of experimental thermodynamic data does not exceed 22%, that is, it is in good agreement with error estimates. However, the scatter of electrical conductivity data reaches 40%.

In Figure 2.28, the results of electrical conductivity measurements are compared with the data obtained by other authors. While leaving the discussion of the peculiarities in the behavior of electrical conductivity of nonideal plasma for Chapter 7, we shall only draw the reader’s attention to one important detail. The electrical conductivity values on isobars, obtained by the method under discussion, do not increase with heating but, on the contrary, decrease. This can be attributed to the principal shortcoming of the method: the inhomogeneity of the plasma column caused by instability, which leads to superheating.

An interesting phenomenon was observed during the experiments: the plasma column separated into dark and light strata extending across the current. The strata are observed in the opening frames and are clearly visible during the first several hundred microseconds (Fig. 2.30). There is no doubt that, given such fairly slow heating of material, there is enough time for local thermodynamic equilibrium to set in. Therefore, these strata can be referred to as thermal, which is distinct from ionization strata well known in the physics of gas discharges. It was shown by Iakubov (1977) that the observed e ect could be a result of the development of thermal instability of a nonideal plasma with current. We emphasize that the strata revealed by Dikhter and Zeigarnik (1977) still remain the most clearly defined manifestation of the instability of nonideal plasma in external fields.

In a series of experiments (Gathers et al. 1974, 1976, 1979, 1983; Shaner et al. 1986), the exploding wire method was used to study the thermophysical properties of metals in the liquid phase with subsequent evaluation, on this basis, of critical point parameters. As distinct from the techniques used by Dikhter and Zeigarnik, use was made of a slower introduction of energy into the sample under investigation, which had a length of about 25 mm and a diameter of about 1 mm, thereby improving the uniformity of parameters and allowing the avoidance of the instability of the moving metal–gas interface. At the same time, the introduction of energy was su ciently “fast” to preclude the collapse of the plasma column in the gravitational field. During the experiments, the current–voltage characteristics, surface temperature, channel diameter and, in a number of cases, the velocity of sound were recorded. The results of those measurements provided a basis for subsequent determination of the electrical conductivity, density, enthalpy, and heat capacity of about ten metals at pressures of up to 0.5 GPa and temperatures of up to 9·103 K. Whilst doing so, Gathers, Shaner, and others, during their investigations, deliberately confined themselves to the single–phase liquid–metal region of parameters. This was because it had been established that the crossing of the boiling curve in the process of “slow” isobaric expansion causes a sharp development of nonuniformities in heated plasma parameters.

ISOBARIC

JOULE HEATING

51

Fig. 2.30. Structure of plasma column under isobar heating of cesium in a rare gas at-

mosphere (Dikhter and Zeigarnik 1977).

In order to describe the thermodynamics of dense metal plasma, Gathers et al. (1974) and Shaner et al. (1986) employed the soft–sphere model, which helped in the evaluation of the parameters of critical points for copper and lead; these are in good agreement with the data of dynamic measurements of Altshuler et al. (1980) and are listed in Appendix A.

For measuring the sound velocity, Altshuler et al. (1980) subjected a column of matter to the e ect of a focused laser pulse (0.1–0.5 J, 25 ns, 100 µm). An acoustic wave, which was excited in matter, was propagated across the column and caused a compression wave in the gas upon exit from the latter. This enables one to determine the time of the wave exit from the column and, given the column diameter, to calculate the sound velocity. It has been found that the empirical Birch law, which stipulates that the sound velocity depends linearly on the density of matter, is quite valid. However, the reasons for this are not quite clear.

Kloss et al. (1998) and Rakhel et al. (2002) used the exploding wire method for investigation of thermodynamic and electrical properties of liquid tungsten. The experiments were performed with tungsten wires placed in the Teflon covering. The set of measuring parameters was as follows: the current through the sample, the drop voltage, the diameter of the conductor, and the temperature at its surface. A fast framing photo camera was used for control of the sample geometry. For images, with an exposure time 10 ns each, it is important to ensure that no parallel discharges and no change of the wire length occur during the observation time of about 200–300 ns. Distribution of the measured

52 ELECTRICAL METHODS OF NONIDEAL PLASMA GENERATION

quantities throughout the wire cross–section was investigated by means of computational modelling. Heating conditions which ensured inhomogeneity of the conductor down to (3–4)–fold expansion were determined. Estimation of the critical point parameters was the main result of these works: the critical temperature, Tc = (1.5–1.6)·104 K, critical pressure, pc = (1.1–1.3) GPa, and critical density, ρc = (4–5) g cm−3.

The exploding wire method was used for measuring the conductivity of metals in a wide density range (DeSilva and Kunze 1994; DeSilva and Katsouros 1998; Saleem et al. 2001). A rectilineal piece of wire was placed in condensed matter (water, glass capillary, etc.) and heated by the current impulse with density (3– 5)·107 A cm−2. The plasma conductivity was determined under the assumption of inhomogeneous distribution of temperature, pressure, and other quantities in the column. Shadow photographs of the column allowed workers to be sure that it is really inhomogeneous lengthways and axially symmetric. However, radial distribution of measuring quantities could not be controlled. Besides, it was not possible to exclude the influence of evaporation. In these works unexpected results on the conductivity of dense copper, aluminum, and tungsten plasma were obtained: a minimum at the isotherms of conductivity as a function of density at temperatures 1–2 eV was found. This fact testifies to the loss of metallic conductivity only under significant (in 5–7 times) expansion of condensed metals.

In the experiment by Savvatimski (1996) tungsten wires were heated for 10 µs in glass and quartz capillaries in air. The moment of capillary infilling by liquid metal was controlled by conductivity and luminescence. With decreasing density of liquid tungsten from 7.5 to 1.0 g cm−3, the specific resistance increased from 0.5 to 5 ohm cm. The estimation of temperature gave values between 1.0·104 and 1.4·104 K. The radial distribution of the measuring quantities also was not controled by Savvatimski.

Benage et al. (1999) determined the conductivity of aluminum plasma in exploding wire experiments in thick capillaries made from lead glass. The resistance was determined by measured current and voltage, and the wire cross–section was calculated using the one–dimensional hydrodynamic model and the wide–range equation of state SESAME (Lyon and Johnson 1992). The initial phase of the process, when the metal is in a condensed state, was not modeled. Nevertheless, at the initial stage of the process an abrupt increasing of the resistance was observed induced, apparently, by evaporation. It relates to the fact that the increase of the resistance took place at the moment when the scattered in the wire heat did not amount the sublimation heat of aluminum.

The conductivity of aluminum and tungsten plasmas in conditions of isochoric heating were measured by Renaudin et al. (2002). Foil samples wrapped in spiral (around the axis in the current direction) were placed inside thick sapphire cylinders and heated by a current. The ratio of the inner cylinder volume to the initial volume amounts to about 30. At the initial stage the foil evaporated, vapors filled up the inner cylinder volume, and the homogeneous distributions of such values as temperature, density, etc. were established. The authors as-