Experimental and magnetohydrodynamic numerical simulation results and analysis of a μs- and sub-μs-timescale overdamped underwater electrical explosion of copper wires having different lengths and diameters are presented. For these explosions, ∼80% of the energy stored in the pulse generator is deposited into the wire during a time comparable or shorter than a quarter period of the underdamped discharge. It was found that the threshold values of the deposited energy density, energy density rate, and energy density per unit area, which satisfy overdamped discharge, depend on the wire parameters and on the timescale of the explosion. It was shown that the mechanism responsible for this is the process during which the wire experiences phase transitions to a low-ionized plasma, the resistivity of which is determined by the electron–neutral collision rate, which, in turn, depends on the wire radial expansion velocity, current density, and temperature.

The study of electrical explosion of wires has been of interest for over two centuries1 with several review papers2–4 and text books5–11 dedicated to the subject. Research has been mostly concentrated on the study of wire explosions in vacuum and in gas, on ns and μs timescales with current densities in the range of 107–109 A/cm2. The main interest in wire explosions in vacuum was in x-pinch12 and cylindrical wire array explosion13–15 experiments due to their important application as extremely bright, localized nanosecond timescale duration sources of soft x rays and neutron fluxes. Wire explosions in gaseous environments were studied for various other important applications, such as fast opening switches, which are a key element in high-voltage (HV) generators based on inductively stored energy16–18 and for the production of micrometer-scale powders.19,20

In general, a wire explosion is characterized by rapid solid–liquid–vapor–plasma phase transitions. To achieve explosion of a wire at the maximal amplitude of the discharge current, the action integral is commonly used to estimate the value of the required wire cross section for a given rise time and maximal amplitude of the discharge current.21,22 In the present article, the beginning of the wire explosion will be chosen to be the time when the wire resistance starts to increase rapidly, while the corresponding discharge current decreases. This time corresponds to the beginning of the transition of the wire to the low-ionized vapor phase. Wire explosions in gas and vacuum are similar, manifested in fast (≥5 × 105 cm/s) radial expansion of the wire4,23–25 and surface flashover4,26 due to the HV induced by the fast increase in the current and the finite wire inductance. The fast wire expansion is accompanied by a decrease in the wire density and the current density resulting in the decrease in the energy density deposition. The surface plasma expands rapidly (107 cm/s)27 and captures part of the discharge current, decreasing the current through the wire and, consequently, the energy deposition into the wire. There is, though, a major difference between wire explosion in gas and vacuum. When a sufficiently long wire explodes in a gas, the current can stop (dwell)28,29 for several tens of ns which does not occur in the case of wire explosion in vacuum. This dwell is due to almost complete loss of conductivity in the wire vapor, which is followed by a current re-strike. This is a result of an electron avalanche in the vapor accompanied by the formation of a high-conductivity plasma channel leading to an underdamped discharge.

The disadvantages of wire explosion in gas or vacuum, such as fast wire expansion, surface flashover, and current re-strike, do not exist in underwater electrical wire explosions (UEWE).30,31 Water suppresses wire expansion velocities of ∼105 cm/s (Refs. 32 and 33) and prevents wire surface flashover because of its high electrical strength.16 These advantages make UEWE interesting in warm dense matter and high energy density physics research.11,34,35 In addition, an UEWE is accompanied by highly efficient generation of strong shocks,36–38 which can be utilized for different applications.39–41 Moreover, explosions of either cylindrical or spherical wire arrays generate strong converging shocks which can be used to study extreme water states.42–44 For all these reasons, in our laboratory, we concentrate on UEWE.

In UEWEs by optimizing the length and cross-sectional area of a single wire, a critically damped discharge can be obtained for which a major part of the energy stored in the pulse generator will be deposited into the wire during a time shorter than a quarter period of the underdamped discharge. Thus, one obtains the maximal energy density and energy density rate deposition into the wire and, consequently, the most extreme parameters of the material together with the generation of the strongest possible shock wave (SW).45 Despite the large volume of existing research,5–10,46–49 the threshold values for maximal energy density and energy density rate deposition into the wire, while keeping the wire resistance sufficiently high to satisfy critically damped discharge condition Rwcr2Lt/C, were not studied in detail. Here Lt and C are the total inductance of the discharge circuit and capacitance of the pulse generator, respectively. In Ref. 8, different μs-timescale generators were used for underwater electrical explosion of copper and aluminum wires of different length and diameters. Based on the data obtained in experiments, empirical dependencies were suggested for the wire length lth and diameter dth for which critically damped discharge can be obtained,8 

dth=2W0(Lt/C)ρ0σ0(λ+r)0.25;lth=2×103φ0(LC)0.25.
(1)

Here, W0 and φ0 are the stored energy and charging voltage of the capacitor, respectively, ρ0 is the density, σ0 is the conductivity at normal condition of the material, λ is the specific heat for fusion, and r is the specific heat for vaporization. Using these empirical dependencies, in Ref. 46, the threshold values for maximal energy density, ωth, and current density, jth, were derived as follows:

ωth=2σ0(λ+r)8π2ρ00.5;jth=103σ02(L/C)0.25.
(2)

Using this approach for copper, ωth=20.1kJ/g. One should note that dependencies (1) and (2) were obtained for relatively slow explosions (rise time of the current ≥3 μs) and are not to be used for explosions with faster current rise time.

Resistance of the wire is a crucial parameter which determines the type of the discharge. Apart from the initial length and diameter, to determine the wire resistance Rwcr, the time-dependent resistivity and radial expansion velocity of the wire need to be determined. The latter determines the increase in the wire cross-sectional area accompanied by a decrease in wire density n and current density j. The expansion velocity, in turn, depends on pressure inside the wire. This is determined by the temperature which is a function of the energy density deposition rate. The resistivity of the exploding wire, which during the explosion can be characterized as a weakly nonideal plasma,11 also depends on the time-dependent density and temperature T. In general, the wire explosion occurs close to the critical point of the phase diagram where the liquid and vapor phases coexist. Several models are used to evaluate the conductivity (see, for instance, Refs. 4, 11, 47, 50, and 51). The initially stored energy in the pulse generator is also an important parameter which determines the amount of energy left in the generator at the time when the wire reaches its maximal resistance. When too much energy remains in the generator, continuous large energy density deposition could lead to a fast decrease in the wire resistance, and the discharge becomes underdamped. In addition, μs- or sub-μs-timescale discharges can be characterized by different threshold values of the energy density and energy density deposition rates due to different wire expansion velocities resulting in different wire cross-sectional areas. Finally, let us note that “pure” critically damped discharge with fastest energy density deposition can be considered only for a constant resistor. During wire explosion, the resistance changes in time, especially during the last phase of the explosion when the resistance increases drastically (∼103 times). Thus, it becomes important to obtain a criterion for critically damped discharge for an exploding wire underwater.

In this paper, we present the results of experiments on underwater electrical explosion of a single copper wire and cylindrical copper wire arrays using two high-current pulse generators differing in their stored energy and timescale. The energy stored in the pulse generators was kept constant, whereas the wire length and the number of wires of the same radius in the arrays were varied, keeping the total cross-sectional area of array's wires closely equal to that of a single wire. This allows to keep the same initial current density for a single wire and each wire of the array at the beginning of the discharge. The results of these experiments were compared with one-dimensional magnetohydrodynamic (1D MHD) simulations52,53 coupled with the equations of state (EOS)54 for copper and water, BKL conductivity model,51 Ohm's law, and the discharge electrical circuit. From the simulation results, the threshold values of the energy density deposition and the rate of the energy density deposition were determined, as well as the critical values of the temperature and pressure at the time when maximal power of the explosion was realized. In Sec. II, we describe the experimental setups and the diagnostic used. In Secs. III A and III C, the experimental results for single and wire array explosions using μs- and sub-μs-timescale high-current pulse generators will be presented. In Secs. III B and III D, the experimental data obtained are compared with 1D MHD simulations, while Sec. IV summarizes the main conclusions.

Experiments were carried out using a μs generator55 (Cμ = 10 μF, 25 kV charging voltage, ∼250 kA current amplitude, 1.2 μs rise time on a short circuit load with total inductance of the discharge circuit Lμt ∼ 75 nH, stored energy of ∼3.2 kJ) and a sub-μs generator56 (C = 0.32 μF, 58 kV charging voltage, 170 kA current amplitude, 0.22 μs rise time on a short circuit load with total inductance of the discharge circuit Lsμt ∼ 55 nH, stored energy of ∼0.54 kJ). To keep the discharge critically damped, the total resistance Rdc2L/C1/2 of the discharge circuit should be ≥0.17 Ω for the μs generator and ≥0.8 Ω for the sub-μs generator. The internal resistance of the pulse generators, obtained for short-circuit shots, was of several mΩ; thus, the time-dependent resistance of the exploding wire determines the value of the total resistance.

The μs and sub-μs experimental setups were similar and are shown in Fig. 1(a). In experiments with the μs generator, the load (single wire or wire array) was placed between the grounded cathode and the HV anode electrodes inside an experimental chamber filled with deionized water [see Fig. 1(b)]. For the sub-μs generator, we used a 60-mm-inner diameter cylindrical Perspex tube, filled with deionized water where the load was placed inside the tube between HV and grounded electrodes [see Fig. 1(c)]. The tube had two 15-mm-diameter windows located 180° relative to each other and was hermetically sealed by 0.5-mm-thick glass plates. It was thereafter connected to the HV electrode and placed inside the experimental chamber having the same diameter as the chamber used in μs-timescale explosions but not filled with water. This design significantly decreases dynamic load on the pulse generator during the explosion of wires. Both experimental chambers had two 20-mm-diameter windows placed 180° with respect to each other. The windows were hermitically sealed by 25-mm-thick Perspex plates.

FIG. 1.

Experimental setup for underwater electrical explosions of single wire and wire arrays (a), μs timescale (b), and sub-μs timescale (c).

FIG. 1.

Experimental setup for underwater electrical explosions of single wire and wire arrays (a), μs timescale (b), and sub-μs timescale (c).

Close modal

In experiments, we studied explosions of single copper wires and copper wire arrays of different cross sections and lengths. The initial cross-sectional area of the wire S=1h0τexpI2tdt0.5 was estimated using the action current integral for copper h =4.1 × 109 A2 s/cm4.21,22 Here, τexp is the time when the discharge current approaches it is maximal amplitude and starts to decrease because of the sharp increase in wire resistivity. It is understood that this estimate is rough, due to the assumption that the current waveform is close to that obtained in short-circuit shots and that the wire cross section remains constant until τexp. Preliminary experiments showed that for 40-mm-long wires, for the μs generator, the discharges were overdamped for 600-μm- and 500-μm-diameter wires, and for 250-μm- and 200-μm-diameter wires for the sub-μs generator. The length of the wires was varied in the range of 40–19 mm. For 20-mm-diameter arrays of the same length as the corresponding single wire, the diameter and number of wires were varied so that the total cross-sectional area of wires is almost equal to that of the single wire. Exploratory experiments showed that to obtain approximately same τexp, the total cross section of the array's wires should be larger than that of the corresponding single wire. Moreover, it was found that the difference in total cross sections increases with decreasing wire diameter in the array (see Table I). One can consider that this difference is related to a slightly different amplitude of the current which is larger for an array due to smaller load inductance. Also, the observed different expansion velocities of wires of different diameters, which lead to different evolution of the wire cross section prior to τexp, can be the reason for this discrepancy.

TABLE I.

Parameters of the exploding wires.

μs generatorSub-μs generator
Length (cm)Cross-sectional area (cm2)Weight (g)Length (cm)Cross-sectional area (cm2)Weight (g)
Single wire 600 μm diameter Single wire 250 μm diameter 
2.79 × 10–3 0.1 4.91 × 10–4 0.017 
0.075  0.013 
2.3 0.058 2.3  0.01 
Single wire 500 μm diameter Single wire 200 μm diameter 
1.96 × 10–3 0.07 3.14 × 10–4 0.011 
0.053 
2.3 0.04 
Wire arraya: 27 wires each of 130 μm diameter Wire arraya: 5 wires each of 130 μm diameter 
3.58 × 10–3/1.33 × 10–4 0.128/0.0048 6.64 × 10–4/1.33 × 10–4 0.024/0.0048 
0.096/0.0036 0.018/0.0036 
2.3 0.073/0.0028 2.3 0.014/0.0027 
Wire arraya: 56 wires each of 90 μm diameter Wire arraya: 10 wires each of 90 μm diameter 
3.56 × 10–3/6.36 × 10–5 0.127/0.0022 6.36 × 10–4/6.36 × 10–5 0.023/0.0023 
0.095/0.0017 0.017/0.0017 
2.3 0.073/0.0013 2.3 0.013/0.0013 
μs generatorSub-μs generator
Length (cm)Cross-sectional area (cm2)Weight (g)Length (cm)Cross-sectional area (cm2)Weight (g)
Single wire 600 μm diameter Single wire 250 μm diameter 
2.79 × 10–3 0.1 4.91 × 10–4 0.017 
0.075  0.013 
2.3 0.058 2.3  0.01 
Single wire 500 μm diameter Single wire 200 μm diameter 
1.96 × 10–3 0.07 3.14 × 10–4 0.011 
0.053 
2.3 0.04 
Wire arraya: 27 wires each of 130 μm diameter Wire arraya: 5 wires each of 130 μm diameter 
3.58 × 10–3/1.33 × 10–4 0.128/0.0048 6.64 × 10–4/1.33 × 10–4 0.024/0.0048 
0.096/0.0036 0.018/0.0036 
2.3 0.073/0.0028 2.3 0.014/0.0027 
Wire arraya: 56 wires each of 90 μm diameter Wire arraya: 10 wires each of 90 μm diameter 
3.56 × 10–3/6.36 × 10–5 0.127/0.0022 6.36 × 10–4/6.36 × 10–5 0.023/0.0023 
0.095/0.0017 0.017/0.0017 
2.3 0.073/0.0013 2.3 0.013/0.0013 
a

The slash symbol “/” in the bottom of the four quadrants is related to the total weight of the array and the weight of an individual wire. In all explosions, the wire array diameter was 20 mm.

The waveforms of the discharge current (I) and voltage (V) were measured using a self-integrated Rogowski coil (measurement error of ±5%) and a Tektronix P6015A voltage divider (measurement error of ±3%). The inductive voltage LldI/dt was subtracted from the measured value of V to obtain the resistive voltage Vres. Here, Ll is the inductance of the wire or wire array determined in short-circuit shots with non-exploding loads imitating the single wire or wire array. Namely, in these calibration shots, we used a 1-mm-diameter copper wire to imitate a single wire explosion and a copper rod having the same diameter as the wire array. The experimentally calculated error in resistance, power, and deposited energy is of ±6%. A diode-pumped continuous wave single-mode laser (∼1.5 W, λ = 532 nm) was used to backlight the exploding wires and the shock generated in water. The shadow images were obtained using a streak Optronis Optoscope SC-10 camera, with an entrance slit of 100 μm in width. To suppress self-emission of the exploding wire, neutral density filters and a 532-nm interference filter (bandwidth of 1 nm) were used in front of the streak camera. The measured shock velocity using the streak image gives an error of ±200 m/s. The error in wire length is of ±0.1mm. For both timescale explosions of arrays with multiple wires, the observed wire in streak images kept a ∼2.5-mm gap from its nearest neighbors.

In Fig. 2, we present waveforms of the current and resistive voltage, obtained in explosions for 600- and 90-μm-diameter single wires and, respectively, 40- and 23-mm-long array wires, together with the corresponding deposited power and wire resistance. The decrease in the length of wires from 40 to 23 mm does not result in a fast-decaying oscillatory discharge, even though the obtained maximal value of the resistance decreases. This decreased resistance is sufficiently large to maintain an overdamped discharge, though longer current fall time is observed. The most important result is the drastic increase in the resistance of the exploded wire with a decreasing wire diameter. Resistance of the exploding 90-μm-diameter wire, for example, reaches tens of Ohms at the time of the maximum deposited power. Moreover, for 40-mm-long wires, independent of the wire diameter, the resistance continues to increase after the maximum of the deposited power is reached. As the wire length is decreased, the time delay between the maxima of the deposited power and the wire resistance shortens (see Table II). We should point out that for explosions with 19-mm-long wires, the discharge becomes underdamped. The latter corresponds to the formation of a high-conductivity plasma whose resistance is significantly lower than 0.17 Ω following higher energy density deposition into the wire with decreasing weight. Thus, the deposited energy density and energy density rate obtained with a wire length of 23 mm can be considered as the maximal threshold values for an almost critically damped discharge.

FIG. 2.

Current, resistive voltage, deposited power, and resistance waveforms, obtained in shots with a single copper wire of 600 μm diameter (a) and (b) and a 20-mm-diameter cylindrical array of 56 copper wires each of 90 μm diameter (c) and (d). The results are 40-mm-long (solid lines) and 23-mm-long (dashed lines) wires. (The colors of the curves correspond to color of the axes titles.)

FIG. 2.

Current, resistive voltage, deposited power, and resistance waveforms, obtained in shots with a single copper wire of 600 μm diameter (a) and (b) and a 20-mm-diameter cylindrical array of 56 copper wires each of 90 μm diameter (c) and (d). The results are 40-mm-long (solid lines) and 23-mm-long (dashed lines) wires. (The colors of the curves correspond to color of the axes titles.)

Close modal
TABLE II.

Time delays between the maxima of the deposited power and wire resistance for various wire explosions.

Wire diameter (μm)Wire length (mm)Time delay (ns)Deposited energy until maximum power (kJ)Remaining energy in maximum resistance (kJ)
600 40 ⋯ 1.24 ⋯ 
30 190 1.4 1.22 
23 120 1.17 1.54 
500 40 210 1.01 1.48 
23 70 0.94 1.92 
130 40 ⋯ 1.41 ⋯ 
30 110 1.29 1.2 
23 80 1.34 1.4 
90 40 ⋯ 1.38 ⋯ 
23 70 1.25 1.33 
Wire diameter (μm)Wire length (mm)Time delay (ns)Deposited energy until maximum power (kJ)Remaining energy in maximum resistance (kJ)
600 40 ⋯ 1.24 ⋯ 
30 190 1.4 1.22 
23 120 1.17 1.54 
500 40 210 1.01 1.48 
23 70 0.94 1.92 
130 40 ⋯ 1.41 ⋯ 
30 110 1.29 1.2 
23 80 1.34 1.4 
90 40 ⋯ 1.38 ⋯ 
23 70 1.25 1.33 

These experiments show that for a fixed wire diameter, decreasing the length from 40 to 23 mm of the single wire or wire array increases the maximal amplitude of the discharge current in the range of 165–210 and 220–250 kA, respectively. This dependence relates to the decrease in the load inductance, as shown in Table III, and in the decrease in the load resistance. The total deposited energy was calculated as Wd=0t*Iφrdt, where t* is the time when the discharge current drops to 20% of its maximal amplitude. The efficiency of the stored energy deposition η =Wd/Wst×100% increases from ∼70% to ∼80% with decreasing wire length from 40 to 23 mm. Here, Wst is the initially stored energy in the pulse generator. For all the studied single wires and wire array explosions, approximately half of Wst was deposited into the wires up to the time corresponding to the maximal power. In Table III, we present the main parameters characterizing these explosions.

TABLE III.

The main parameters characterizing μs-timescale wire explosions.

Lwire (mm)Rise time to Imax (ns)Time of max. power (ns)Max. resistive voltage (kV)Energy density (kJ/g)Energy density rate (kJ/g s) × 106R1 at max. current (Ω)R2 at max. power (Ω)
Single 600-μm-diameter copper wire 
40 1170 1510 29 21.4 27 0.061 0.21 
30 1100 1410 24.7 31.8 36 0.058 0.16 
23 1070 1340 20.8 44.3 51.5 0.053 0.11 
Single 500-μm-diameter copper wire 
40 990 1250 37 34.8 41.7 0.11 0.32 
23 970 1150 27 65.2 75.2 0.09 0.18 
Array of 27 copper wires of 130 μm diameter 
40 980 1170 53 17.32 47.16 1.29 7.45 
30 930 1010 45.9 25.33 65.55 1.21 5.34 
23 930 1110 41.5 39.19 68.6 1.15 4.67 
Array of 56 copper wires of 90 μm diameter 
40 950 1180 52.9 17.3 48.8 2.93 16.74 
23 930 1110 43.2 35.3 91.2 2.67 11.17 
Lwire (mm)Rise time to Imax (ns)Time of max. power (ns)Max. resistive voltage (kV)Energy density (kJ/g)Energy density rate (kJ/g s) × 106R1 at max. current (Ω)R2 at max. power (Ω)
Single 600-μm-diameter copper wire 
40 1170 1510 29 21.4 27 0.061 0.21 
30 1100 1410 24.7 31.8 36 0.058 0.16 
23 1070 1340 20.8 44.3 51.5 0.053 0.11 
Single 500-μm-diameter copper wire 
40 990 1250 37 34.8 41.7 0.11 0.32 
23 970 1150 27 65.2 75.2 0.09 0.18 
Array of 27 copper wires of 130 μm diameter 
40 980 1170 53 17.32 47.16 1.29 7.45 
30 930 1010 45.9 25.33 65.55 1.21 5.34 
23 930 1110 41.5 39.19 68.6 1.15 4.67 
Array of 56 copper wires of 90 μm diameter 
40 950 1180 52.9 17.3 48.8 2.93 16.74 
23 930 1110 43.2 35.3 91.2 2.67 11.17 

The following conclusions can be drawn from these data.

  • The threshold values of the deposited energy density and energy density rate for fixed 23-mm-long wire/wire array depend on the wire diameter. Namely, the deposited energy density slightly decreases for smaller wire diameter, and the energy density rate has opposite dependence, i.e., it increases ∼1.8 times for 90-μm-diameter wire as compared with 600-μm-diameter wire.

  • For fixed wire diameter, the deposited energy density and the energy density deposition rate increase with decreasing wire length.

  • There is a large difference (up to a factor of 100) between the resistance of a single wire and an individual array wire. Moreover, smaller diameter wires reach higher values of resistance upon explosion.

Typical shadow streak images of exploding 40- and 23-mm-long 600-μm-diameter single copper wires and 90-μm-diameter wires in a cylindrical array and the generated shocks in water are shown in Fig. 3. At the beginning of the explosion at t=τm (i.e., when the current starts to decrease), the discharge channel's rapid expansion is screened by the generated strong shock. Only at t >1 μs relative to the maximal amplitude of the current, can one distinguish between the discharge channel and the shock. Thus, only radial expansion of the strong shock was used to calculate the dependence of its velocity as a function of the exploding wire parameters. The shock velocity was calculated after the inclination point, i.e., when this shock overlaps the weak shock generated at earlier time.33 This calculation was possible only for the explosion of 600-μm-diameter wires since for wires of smaller diameter, determining the inclination points was difficult.

FIG. 3.

Shadow streak images of exploding copper wires, 600-μm-diameter single wire (a) and (b) and 90-μm-diameter wires in an array of 56 wires (c) and (d) at 40- (a) and (c) and 23-mm (b) and (d) in length, respectively, and generated shock in water.

FIG. 3.

Shadow streak images of exploding copper wires, 600-μm-diameter single wire (a) and (b) and 90-μm-diameter wires in an array of 56 wires (c) and (d) at 40- (a) and (c) and 23-mm (b) and (d) in length, respectively, and generated shock in water.

Close modal

In Table IV, the shock velocity of the 600-μm-diameter wire for a few values of wire lengths are listed. Decreasing the wire length results in the increase in the shock velocity. This dependence agrees qualitatively with the data presented in Table II which shows increase in the energy density rate deposition with decreasing wire length. The data for the channel expansion at later times were used to validate the results of the 1D MHD simulation presented in Sec. III B.

TABLE IV.

Shock velocity (km/s) for 600-μm-diameter wire explosions.

Length (mm) 40 30 23 
Velocity (km/s) 2.7 3.3 
Length (mm) 40 30 23 
Velocity (km/s) 2.7 3.3 

To obtain the evolution of the temperature, density, pressure, and conductivity of the wire, we carried out 1D MHD simulations using the model described in detail in Refs. 52 and 53. In these simulations, hydro-dynamic equations are solved numerically with Maxwell equations in Lagrange mass coordinates together with Ohm's law. The calculation grid consists of two regions, namely, the wire material and the water. The system of MHD equations is coupled with the equation of state (EOS) of copper and water54 and the electrical discharge circuit. The electrical conductivity of the wire is evaluated using the semiempirical model described in Refs. 4, 9, 51, and 52. The fitting parameters in this model are the value of the conductivity at the critical point σcr and a parameter α which determines the temperature dependence of the conductivity at solid density. The corresponding values of the heat transfer coefficients are found using the Wiedemann–Franz law.57 The initial conditions in these simulations are the charging voltage of the capacitor together with parameters such as the capacitance and inductance of the discharge circuit. Small variations in α = 1.2 ± 0.1 and σcr = (2.3 ± 0.5) × 1015 s−1 are used to obtain satisfactory agreement between the calculated and experimental waveforms of the voltage and current, as well as between the calculated and experimentally measured shock and wire expansion. For wire array explosions, the current through an individual wire is calculated as Ii = (ϕ/m)[(C/L)]1/2, where m is the number of wires and assuming uniform current distribution among the wires.45 

The experimental and simulated current and voltage waveforms for μs-timescale explosions of single 600-μm-diameter wires of 40 and 23 mm length are presented in Fig. 4. Additionally, the simulated trajectories of the wire radial expansion are also shown in Fig. 4. There is satisfactory agreement between the experimental and simulated waveforms of the current and resistive voltage. These also result in good agreement between MHD simulated and experimentally calculated energy density and energy density rate depositions into the wires. Similar agreement was obtained for other types of wire loads tested in these experiments, which allows to obtain the temporal dependence of the wire radial expansion velocity, temperature (T), number density (n), pressure (P), and resistivity (ρ). Examples of radial distributions of T, n, P, and σ=1/ρ at the time corresponding to maximal current and power for 600-μm-diameter, 40-mm-long and 90-μm-diameter, 23-mm-long wires are shown in Fig. 5 together with calculated and experimental resistances. There is a nonuniformity in the radial distributions of T, n, P, and σ for the 600-μm-diameter wire at peak current, whereas at the time when the maximal deposited power is reached, it is less pronounced (∼10%).

FIG. 4.

The experimental (solid) and simulated (dashed) current, voltage waveforms, and wire radius vs time for a μs-timescale explosion of a 600-μm-diameter, 40-mm-long (a) and (b) and 23-mm-long (c) and (d) single wire.

FIG. 4.

The experimental (solid) and simulated (dashed) current, voltage waveforms, and wire radius vs time for a μs-timescale explosion of a 600-μm-diameter, 40-mm-long (a) and (b) and 23-mm-long (c) and (d) single wire.

Close modal
FIG. 5.

Radial distributions of T, P, n, and σ=1/ρ at maximum current (solid) and power (dashed); calculated and experimental resistance of 600-μm-diameter, 40-mm-long (a)–(c) and 56 wire array of 90-μm-diameter, 23-mm-long (d)–(f) wires.

FIG. 5.

Radial distributions of T, P, n, and σ=1/ρ at maximum current (solid) and power (dashed); calculated and experimental resistance of 600-μm-diameter, 40-mm-long (a)–(c) and 56 wire array of 90-μm-diameter, 23-mm-long (d)–(f) wires.

Close modal

In Table V, the main parameters characterizing explosions of 40-, 30-, and 23-mm-long and 600-, 500-, 130-, and 90-μm diameter wires are shown at peak power. One can see that overdamped discharge can be realized for quite a broad range of wire parameters. This is the result of different co-depending processes. Namely, the time evolution of the wire's resistance, which determines the type of the discharge, is governed by the evolution of the material density and temperature. These parameters depend on the deposited energy density and energy density deposition rate. The latter also determine the pressure and, which then determines the wire expansion velocity, density and current density through the wire. For instance, for the same wire length, a decrease in diameter increases the energy density deposition. However, due to the increase in the relative wire radius (rp/ro), the current density, temperature, mass density, and pressure decrease. Here, rp and ro are the wire radius at maximal deposited power and initial radii, respectively. Thus, depending on the purpose of the study (WDM—for which the density and the pressure are the dominant parameters or shock generation—for which the rate of wire expansion is important), an appropriate choice of the wire parameters can be made.

TABLE V.

Results of MHD simulations of 40-, 30-, and 23-mm-long, 600-, 500-, 130-, and 90-μm-diameter wire explosions.

Wire length (mm)R (Ω)ρ (Ω × m) × 10–6Max. wire velocity (m/s)Wire radius rp [μm]n (kg/m3)P (Pa) × 109T (kK)j (A/cm2) × 107ωdmax (J/kg) × 106εt (J/kg s) × 1012dωd/dSmax (J/kg⋅cm2) × 106
Copper wire of 600 μm diameter 
40 0.199 6.23 1586 631 1993 6.4 21.8 1.12 24.3 29.8 1183 
30 0.159 6.84 1738 642 1953 7.5 24.9 1.21 32.0 39.6 1306 
23 0.119 7.22 1844 658 1842 8.3 28 1.27 40.7 48.0 1395 
Copper wire of 500 μm diameter 
40 0.317 7.2 1835 538 1869 7.5 26.8 1.34 33.7 49.9 1886 
23 0.184 7.88 2045 560 1746 9.9 34.2 1.55 64.3 77.6 2263 
Copper wire of 130 μm diameter 
40 7.905 21.5 1165 185 1088 2.56 19.3 0.67 22.2 67.6 16434 
30 5.771 23.0 1242 195 989 2.85 22.1 0.67 29.9 79.4 16823 
23 5.188 27.5 1395 197 957 3.23 24.7 0.68 39.4 105.7 17285 
Copper wire of 90 μm diameter 
40 13.41 15.87 1017 123 1215 1.656 13.2 0.81 22.6 63.5 29623 
23 10.41 29.24 1200 143 862 2.33 20.9 0. 62 39.0 105.1 33561 
Wire length (mm)R (Ω)ρ (Ω × m) × 10–6Max. wire velocity (m/s)Wire radius rp [μm]n (kg/m3)P (Pa) × 109T (kK)j (A/cm2) × 107ωdmax (J/kg) × 106εt (J/kg s) × 1012dωd/dSmax (J/kg⋅cm2) × 106
Copper wire of 600 μm diameter 
40 0.199 6.23 1586 631 1993 6.4 21.8 1.12 24.3 29.8 1183 
30 0.159 6.84 1738 642 1953 7.5 24.9 1.21 32.0 39.6 1306 
23 0.119 7.22 1844 658 1842 8.3 28 1.27 40.7 48.0 1395 
Copper wire of 500 μm diameter 
40 0.317 7.2 1835 538 1869 7.5 26.8 1.34 33.7 49.9 1886 
23 0.184 7.88 2045 560 1746 9.9 34.2 1.55 64.3 77.6 2263 
Copper wire of 130 μm diameter 
40 7.905 21.5 1165 185 1088 2.56 19.3 0.67 22.2 67.6 16434 
30 5.771 23.0 1242 195 989 2.85 22.1 0.67 29.9 79.4 16823 
23 5.188 27.5 1395 197 957 3.23 24.7 0.68 39.4 105.7 17285 
Copper wire of 90 μm diameter 
40 13.41 15.87 1017 123 1215 1.656 13.2 0.81 22.6 63.5 29623 
23 10.41 29.24 1200 143 862 2.33 20.9 0. 62 39.0 105.1 33561 

Based on the simulated MHD data in Table V, the following conclusions can be drawn:

  • For the same wire diameter, decreasing the wire length is accompanied with:

    • Increasing resistivity, velocity, wire radius, pressure, temperature, maximum of deposited energy density ,ωdmax, energy density deposition rate εt=dωd/dtmax, and deposited energy density dωd/dSmax per unit area.

    • Decreasing current density, wire resistance, and density.

  • For the same wire length, decreasing the wire diameter is accompanied by:

    • Increasing resistivity, total resistance, relative increase of the wire radius (rp/rc), significant increase in the deposited energy density rate and energy density per unit area.

    • Decreasing wire expansion velocity, density, pressure, temperature, and current density.

In Fig. 5, waveforms of the current and resistive voltage for explosions of 40- and 23-mm-long, 250-μm single wires and 90-μm diameter wire arrays are presented together with the deposited power and individual wire resistance. Similar to μs-timescale experiments, decreasing the length of the wires to 23 mm keeps the discharge overdamped. Also, at the maximum deposited power, the 90-μm-diameter wire resistance increases ∼10-fold compared to the 250-μm wire. However, in contrast to the μs-timescale explosions, for sub-μs-timescale explosions, the wire resistance does not decrease after the maximum of the deposited power. Quite the opposite, independently of wire diameter, the resistance continues to increase, except for the 23-mm-long, 90-μm-diameter wire [see Fig. 6(d)]. These experiments show that similar to μs-timescale explosions, for fixed wire diameter, decreasing the wire length from 40 to 23 mm results in an increase in the discharge current's maximum amplitude. However, this increase is ≤10%, and it does not change the efficiency of the stored energy deposition, η ≈ 68 ± 3%. In comparison with μs-timescale explosions, we see that here more than half of the stored energy was deposited into the wires until the time which corresponds to maximum power. Also, similar to μs-timescale experiments, decreasing the wire length to 19 mm led to an underdamped discharge. Thus, the parameters of explosions obtained in experiments with 23-mm-long wires can be considered as the maximal values for overdamped discharge.

FIG. 6.

Current, resistive voltage, deposited power, and resistance waveforms, obtained in shots with single copper wire of 250 μm diameter (a) and (b) and 20-mm-diameter cylindrical array of 10 copper wires with diameter of 90 μm (c) and (d). Solid and dashed lines are for 40- and 23-mm-long wires, respectively.

FIG. 6.

Current, resistive voltage, deposited power, and resistance waveforms, obtained in shots with single copper wire of 250 μm diameter (a) and (b) and 20-mm-diameter cylindrical array of 10 copper wires with diameter of 90 μm (c) and (d). Solid and dashed lines are for 40- and 23-mm-long wires, respectively.

Close modal

In Table VI, we present the main parameters characterizing these explosions with notation similar to that in Table III.

TABLE VI.

Main parameters which characterize sub-μs-timescale explosions of wires.

Lwire (mm)Rise time to Imax (ns)Time of max. power (ns)Max. resistive voltage (kV)Energy density (kJ/g)Energy density rate (kJ/g s) × 106R1 at max. current (Ω)R2 at max. power (Ω)
Single 250-μm-diameter copper wire 
40 150 320 39.9 20.9 94.1 0.26 0.59 
30 141 320 37.2 27.4 131.1 0.25 0.58 
23 146 310 41 34.8 173.7 0.23 0.56 
Single 200-μm-diameter copper wire 
40 115 260 57.4 21.8 186.4 0.53 1.19 
Array of 5 copper wires of 130 μm wire diameter 
40 133 300 40.8 14.1 81.2 0.90 2.5 
30 132 300 37.9 19.3 118.5 0.80 2.45 
23 139 300 32.1 23.9 143.8 0.95 1.85 
Array of 10 copper wires of 90 μm wire diameter 
40 131 290 48.5 15.8 108.5 1.8 5.8 
30 130 280 44.8 22.1 158.2 1.5 5.2 
23 131 280 41 25.2 174.8 1.4 5.1 
Lwire (mm)Rise time to Imax (ns)Time of max. power (ns)Max. resistive voltage (kV)Energy density (kJ/g)Energy density rate (kJ/g s) × 106R1 at max. current (Ω)R2 at max. power (Ω)
Single 250-μm-diameter copper wire 
40 150 320 39.9 20.9 94.1 0.26 0.59 
30 141 320 37.2 27.4 131.1 0.25 0.58 
23 146 310 41 34.8 173.7 0.23 0.56 
Single 200-μm-diameter copper wire 
40 115 260 57.4 21.8 186.4 0.53 1.19 
Array of 5 copper wires of 130 μm wire diameter 
40 133 300 40.8 14.1 81.2 0.90 2.5 
30 132 300 37.9 19.3 118.5 0.80 2.45 
23 139 300 32.1 23.9 143.8 0.95 1.85 
Array of 10 copper wires of 90 μm wire diameter 
40 131 290 48.5 15.8 108.5 1.8 5.8 
30 130 280 44.8 22.1 158.2 1.5 5.2 
23 131 280 41 25.2 174.8 1.4 5.1 

Comparison with Table III shows that similar to μs-timescale explosions:

  • For the same wire diameter, decreasing the wire length leads to an increase in the deposited energy density and the energy density deposition rate.

  • The resistance of one array wire is ∼19 times higher than that of a single wire. Moreover, smaller wire diameter leads to a higher resistance value upon explosion.

However, there are also differences compared with μs-timescale explosions, namely:

  • The deposited energy density and energy density rate are approximately equal for wires with different diameters but similar lengths.

  • The deposited energy density and energy density rate in sub-μs-timescale explosions is almost double compared with μs-timescale explosions.

Shadow streak images of explosions of single copper wires, 40 and 23 mm long and 250 μm diameter and the generated shocks in water are shown in Fig. 7. Similar to the μs-timescale case (see Fig. 3), at the beginning of the explosion, the discharge channel expansion is screened by the strong shock generated by the rapid expansion of the wire. The velocity of this shock, calculated after the inclination point, increases slightly with decreasing wire length and is similar to that obtained for μs-timescale wire explosions (see Table VII).

FIG. 7.

Shadow streak images of the exploding single 250-μm-diameter, 40-mm-long (a) and 23-mm-long (b) copper wires and the generated shock waves (SW) in water.

FIG. 7.

Shadow streak images of the exploding single 250-μm-diameter, 40-mm-long (a) and 23-mm-long (b) copper wires and the generated shock waves (SW) in water.

Close modal
TABLE VII.

Velocity of the strong shock generated by 250-μm-diameter wire explosions.

Length (mm) 40 30 23 
Velocity (km/s) 2.8 2.9 3.1 
Length (mm) 40 30 23 
Velocity (km/s) 2.8 2.9 3.1 

1D MHD simulations were also carried out for sub-μs-timescale wire explosions, to obtain the evolution of the temperature, density, pressure, and resistivity of the exploding wire. The experimental and simulated current and voltage waveforms for explosion of 40- and 23-mm-long, 250-μm-diameter single copper wires are presented in Fig. 8. One can see a reasonable fit between experimental and simulated waveforms of the current and resistive voltage for the presented wire parameters. Some discrepancy seen in simulated and experimentally obtained waveforms can be related to the necessity of more accurate corrections of EOS and conductivity model which becomes essential especially for sub-microsecond timescale explosions. This subject was discussed in our earlier research in Refs. 58–60.

FIG. 8.

The experimental (solid) and simulated (dashed) current and voltage waveforms (a) and (c) and the wire radius (b) and (d) for sub-μs-timescale explosions of 250-μm-diameter single wire of 40 mm (a) and (b) and 23 mm (c) and (d) length.

FIG. 8.

The experimental (solid) and simulated (dashed) current and voltage waveforms (a) and (c) and the wire radius (b) and (d) for sub-μs-timescale explosions of 250-μm-diameter single wire of 40 mm (a) and (b) and 23 mm (c) and (d) length.

Close modal

Examples of radial distributions of T, n, P, and σ, at the current and power maxima for 250- and 90-μm-diameter wires are shown in Fig. 9, together with the calculated time-dependent resistance of the wire. Nonuniformities in these radial distributions are obtained only until the maximum current is reached. These nonuniformities are significantly less pronounced when the maximal power of the explosion is realized. Similar to the μs-timescale explosions simulations (see Fig. 5), here, at the maximal power deposition, for decreasing wire length and fixed wire diameter, the temperature and pressure increase. However, in contrast to μs-timescale explosions, there is no significant change in the density and conductivity for decreasing wire length and fixed initial wire diameter. Also, decreasing the initial wire diameter, for constant wire length, does not lead to a significant decrease in the temperature and pressure, but the conductivity and density decrease significantly.

FIG. 9.

Radial distributions of the T, P, n, and σ at maximal current (solid) and resistive voltage (power) (dashed) and calculated and experimental resistance for 250-μm-diameter, 40-mm-long (a)–(c) and 90-μm-diameter, 23-mm-long (d)–(f) wires.

FIG. 9.

Radial distributions of the T, P, n, and σ at maximal current (solid) and resistive voltage (power) (dashed) and calculated and experimental resistance for 250-μm-diameter, 40-mm-long (a)–(c) and 90-μm-diameter, 23-mm-long (d)–(f) wires.

Close modal

In Table VIII, the main parameters characterizing explosions of 40-, 30-, and 23-mm-long, 250-, 200-, 130-, and 90-μm-diameter wires are listed at the time corresponding to maximal deposited power. Comparing Table VIII with Table V shows that various dependencies on wire parameters for the two timescales (μs or sub-μs) are similar. Namely, decreasing the wire length for fixed wire diameter leads to an increase in the resistivity, expansion velocity, radius of the wire, pressure, temperature, energy density, energy density rate, and energy density deposition per unit area. Furthermore, the wire resistance, density, and current density decrease. Decreasing the wire diameter for fixed wire length leads to an increase in the resistivity, resistance, relative increase in the wire radius (rp/rc), energy density rate, and energy density deposition per unit area. However, the wire expansion velocity, density, pressure, temperature, and current density decrease.

TABLE VIII.

Results of MHD simulations which characterize explosions of 250-, 200-, 130-, and 90-μm-diameter wires having 40 and 23 mm lengths.

Wire length (mm)R Ωρ (Ω × m) × 10–6Wire velocity (m/s)Wire radius rp (μm)n (kg/m3)P (Pa) × 109T (kK)J (A/cm2) × 107(ωd)max (J/kg) × 106εt (J/kg s) × 1012dωd/dSmax (J/kg cm2) × 106
Copper wire of 250 μm diameter 
40 0.58 1.7 1820 194 3707 10.37 16.4 5.78 25.83 117 8537 
23 0.53 3.42 2250 210 3164 13.98 24.6 5.32 40.81 201 10 363 
Copper wire of 200 μm diameter 
40 1.228 3.74 2078 197 2301 10.38 28.1 3.79 38 191 14 567 
Copper wire of 130 μm diameter 
40 2.23 2.05 1428 108 3235 4.57 12.4 4.54 19.46 103 21 960 
23 1.88 4.25 1612 134 2101 5.71 20.2 3.01 26.77 146 23 152 
Copper wire of 90 μm diameter 
40 5.85 3.75 1350 89 2214 3.85 15.1 3.18 21.61 131 44 373 
23 4.9 0.12 1580 110 1430 4.83 24.5 2.23 32.8 208 47 392 
Wire length (mm)R Ωρ (Ω × m) × 10–6Wire velocity (m/s)Wire radius rp (μm)n (kg/m3)P (Pa) × 109T (kK)J (A/cm2) × 107(ωd)max (J/kg) × 106εt (J/kg s) × 1012dωd/dSmax (J/kg cm2) × 106
Copper wire of 250 μm diameter 
40 0.58 1.7 1820 194 3707 10.37 16.4 5.78 25.83 117 8537 
23 0.53 3.42 2250 210 3164 13.98 24.6 5.32 40.81 201 10 363 
Copper wire of 200 μm diameter 
40 1.228 3.74 2078 197 2301 10.38 28.1 3.79 38 191 14 567 
Copper wire of 130 μm diameter 
40 2.23 2.05 1428 108 3235 4.57 12.4 4.54 19.46 103 21 960 
23 1.88 4.25 1612 134 2101 5.71 20.2 3.01 26.77 146 23 152 
Copper wire of 90 μm diameter 
40 5.85 3.75 1350 89 2214 3.85 15.1 3.18 21.61 131 44 373 
23 4.9 0.12 1580 110 1430 4.83 24.5 2.23 32.8 208 47 392 

Nonetheless, the simulation results show that for sub-μs-timescale explosions compared with μs-timescale explosions, one obtains:

  • Significantly larger current densities (∼4 times) and slightly larger density and pressure when the deposited power reaches its maximum value.

  • Larger wire expansion velocity.

  • Significantly smaller resistivity.

  • For similar energy density deposition, the energy density rate and energy density per area are considerably higher for sub-μs explosions.

The main purpose of the present research was (a) to obtain maximal values of energy density ωd, energy density rate d/dt, and energy density per unit area d/dS deposition into a copper wire keeping the discharge overdamped. The latter depends on wire and pulse generator parameters which allow one to realize maximal values of these parameters as compared with underdamped discharge, thus achieving maximal available pressure, density, and temperature of exploding wire and generation of strongest shock wave in water. (b) What are the phenomena which limit these parameters from keeping the discharge overdamped? (c) What would be the optimal wire parameters suited for studies of a warm dense plasma and shock wave generation in water where the shock velocity is maximized?

First let us note that analysis of the data presented in Tables V and VIII showed that in both μs- and sub-μs-timescale explosions, the wire maximal values of temperature, pressure, density, current density, and expansion velocity are obtained for explosion of single wires with the shortest length of 23 mm. Thus, for studies related to a warm dense plasma and generation of cylindrically divergent strong shocks, this load geometry is preferable.

Experimental and MHD simulation results showed that an overdamped wire explosion can be realized for different parameter choices of a wire characterized by different values of ωd,dωd/dt, and dωd/dS, which also depend on the timescale of the explosion. In the present research, the maximal values of these parameters were obtained in explosions with the shortest 23-mm wires (see Table IX).

TABLE IX.

Maximal threshold parameters for overdamped explosions.

Energy density (kJ/g)Energy density deposition rate (kJ/(g s)) × 106Energy density per unit area at maximal power [kJ/(g cm2)] × 109
μs timescale 65 75 2.3 
Copper wire: ∅500 μm, 23 mm length 
sub-μs timescale 40 201 10.3 
Copper wire: ∅250 μm, 23 mm length 
Energy density (kJ/g)Energy density deposition rate (kJ/(g s)) × 106Energy density per unit area at maximal power [kJ/(g cm2)] × 109
μs timescale 65 75 2.3 
Copper wire: ∅500 μm, 23 mm length 
sub-μs timescale 40 201 10.3 
Copper wire: ∅250 μm, 23 mm length 

The highest value of ωd was obtained on μs timescale, but the values of dωd/dt and dωd/dS were largest for sub-μs-timescale explosions. These findings can be used in experiments with current pulses of several MA by appropriate scaling. For instance, given a current amplitude of 5 MA with rise time of ∼1 μs, using estimate of cross section of a 15-mm-diameter cylindrical copper foil (see Sec. II), the corresponding foil thickness is 127 μm. For a 6-cm-long foil, the total weight of the load is ∼3.2 g which results in ∼140 kJ (here ∼44 kJ/g energy density was used). The latter indicates that a 175-kJ stored energy pulse generator (here we used efficiency of the stored in the pulse generator energy transfer to the load as ∼75%) can be used for such an overdamped explosion.

Tables III and V for μs-timescale and in Tables VI and VIII for sub-μs-timescale explosions showed that for similar wire diameter, the resistivity increases but the total resistance decreases with decreasing wire length. We consider different values of the ratio resistances of wires of different lengths (l), resistivities (ρ), and cross sections, πr2, [(Rl1/Rl2=kρl1/ρl2rl2/rl12, where k=l1/l2=Constant], and assume axial uniformity of the exploding wire parameters. In Table X, using the data in Tables III, V, VI, and VIII, we compare the experimentally measured and simulated ratios of resistances, obtained for k=1.74(40 and 23 mm wire lengths) at the maximal deposited power. The agreement between experiment and simulations is reasonably good. Decrease in the total resistance is mainly due to decreasing length and diameter increase, which is only partially compensated by the resistivity increase.

TABLE X.

Comparison between experimental and simulation results of different wire parameters.

Rl1/Rl2expRl1/Rl2simρl1/ρl2rl2/rl12Temperature Tl1/Tl2Density nl1/nl2
μs-timescale explosion. Copper wires: ∅600 μm with lengths 40 and 23 mm 1.9/1.63 0.863 1.087 0.78 1.08 
μs-timescale explosion. Copper wires: ∅90 μm with lengths 40 and 23 mm 1.49/1.27 0.542 1.351 0.63 1.41 
Sub-μs-timescale explosion. Copper wires: ∅250 μm with lengths 40 and 23 mm 1.06/1.01 0.49 1.171 0.67 1.17 
Sub-μs-timescale explosion. Copper wires: ∅130 μm with lengths 40 and 23 mm 1.35/1.29 0.48 1.54 0.61 1.54 
Rl1/Rl2expRl1/Rl2simρl1/ρl2rl2/rl12Temperature Tl1/Tl2Density nl1/nl2
μs-timescale explosion. Copper wires: ∅600 μm with lengths 40 and 23 mm 1.9/1.63 0.863 1.087 0.78 1.08 
μs-timescale explosion. Copper wires: ∅90 μm with lengths 40 and 23 mm 1.49/1.27 0.542 1.351 0.63 1.41 
Sub-μs-timescale explosion. Copper wires: ∅250 μm with lengths 40 and 23 mm 1.06/1.01 0.49 1.171 0.67 1.17 
Sub-μs-timescale explosion. Copper wires: ∅130 μm with lengths 40 and 23 mm 1.35/1.29 0.48 1.54 0.61 1.54 

Also, in Table X, the ratios of simulated temperatures and densities at peak deposited power for these lengths of wires are presented. In spite of the increase in temperature with the decreasing wire length, the resistivity increases. Considering the case when Coulomb collisions prevail, the resistivity reads as61ρei(Ωcm)=5.26×103T1.5(eV)lnΛ, where lnΛ is the Coulomb logarithm which depends only little on the temperature and the electron density. Thus, the data obtained indicate that the plasma is lowly ionized, and its resistivity is governed by electron–neutral collisions:61ρennΩcm=3.52×103ne1(cm3)·νen(s1). Here, ne is the plasma electron density and νens1=6.21×107nn0rw0/rt2×πra2Te(t)(K) is the electron–atom collision frequency. Te(t) and me are the temperature and electron mass, respectively; nn0 is the density of atoms at normal conditions; rw0 is the initial radius of the wire; and ra is the atom radius. Thus, νenr2tTe. Here, the Maxwellian energy distribution of electrons is assumed. The density of electrons can be estimated as ne=I(t)/πr2teVdr, where I(t) is the current, Vdr=1.2Vth(t)me/Ma0.5 is the drift velocity of current carrying electrons, and Vth=8kBTe/πme0.5 is the electron mean thermal velocity. Thus, one obtains that ner2tTe0.5 that results in ρennTe. For our experimental conditions, when the density of neutrals is in the range 1021–1022 cm−3, the value of νen is in the range 1013–1014 s−1 which corresponds to ∼10−8 s equilibration time between electrons and neutrals. Then, the ratio of temperatures obtained in the MHD simulations for wires of the same initial diameter but different lengths should be equal to the ratio of the corresponding resistivities. In Table X, these ratios are roughly equal to each other within ∼20%. This confirms that our suggestion that the plasma is in a lowly ionized state is correct until the maximal deposited power (and respectively, a rather large resistance to satisfy overdamped discharge conditions) is reached.

Experimental results, for both μs- and sub-μs-timescale explosions, show that 55 ± 10% of the energy deposited into the wire is reached prior to achieving maximal power. In Table II and Fig. 5(c), for μs-timescale explosions, for 40-mm-long wires and all tested diameters, the resistance continues to increase after the deposited power reaches its maximum. This indicates that the wire remains in its low-ionized plasma state. Decreasing the wire length led to the appearance of a maximal wire resistance after the deposited power peak. In Table II, the time delay between the power and resistance maxima decreases for smaller wire's length which corresponds to a larger part of the initially stored energy remaining in the generator bank. This explains qualitatively the faster decrease in the resistance for shorter wire length as the result of the larger energy deposition after the maximum of the deposited power. The latter results in the formation of a sufficiently highly ionized plasma the resistance of which is determined by electron–ion collisions and Te1.5. For sub-μs-timescale explosions, a significantly smaller part of the energy remains in the pulse generator after the time when the deposited power has reached its maximum value, than for μs-timescale explosions. Thus, for sub-μs-timescale explosions, even for 23-mm-long wires, the resistance continues to increase or it remains almost unchanged.

Applying the same approach as that described in detail in Ref. 62, we calculate the evolution of the exploding wire conductivity using only electron–neutral or electron–ion collisions. The results of these calculations are compared with experimentally obtained conductivities in Fig. 10 for 90-μm-diameter, 40-mm-long wires exploded on the μs and sub-μs timescales. Figure 10 shows good agreement between the experimental and calculated conductivities for μs-timescale explosion assuming a lowly ionized plasma. For sub-μs-timescale explosions, the low-ionized state of the wire prevails until the maximum of the deposited power is realized [see Figs. 6(c) and 6(d)].

FIG. 10.

Time evolution of plasma conductivities obtained in explosions of 90-μm-diameter, 40-mm-long wires in (a) μs- and (b) sub-μs-timescale explosions (black) and calculated under the assumption of electron–neutral (red) and electron–ion (blue) collisions approximations.

FIG. 10.

Time evolution of plasma conductivities obtained in explosions of 90-μm-diameter, 40-mm-long wires in (a) μs- and (b) sub-μs-timescale explosions (black) and calculated under the assumption of electron–neutral (red) and electron–ion (blue) collisions approximations.

Close modal

For larger wire diameters, assuming a low-ionized plasma does not reproduce the experimental conductivity well enough. We think that this discrepancy is due to the assumption that the conductivity across the cross-sectional area of the wire is uniform. Here, the conductivity is calculated as σt=I(t)lw/φR(t)πrw2(t), where I(t) and φR(t) are the current and resistive voltage, respectively; lw is the length of the wire; and rw(t) is the wire radius, the evolution of which was calculated by MHD simulations. However, the results of the MHD simulations show [see Figs. 5(b) and 9(b)] that the conductivity of large diameter wires is not radially uniform because of a skin effect, even at times when the maximal power deposition is realized. Only for the smallest diameters of 90 μm, the assumption of radially uniform conductivity to be considered is correct. Finally, it is understood that the results of these simplified calculations should be corrected for a weakly nonideal plasma which affects the plasma conductivity. Indeed, in our earlier62 and present research, the coupling coefficient is Γ1 at the time when the current reaches its maximum. Although it decreases gradually during the main discharge due to a combination of temperature increase and density decrease in the plasma, it still remains close to 1 (see Table XI). This changes electron collision cross sections, because the collective electric field should be accounted for, and, respectively, the electron temperature, the ionization level of the plasma, and the plasma conductivity would change.

TABLE XI.

Coupling parameter at time when the maximal deposited power is realized.

μs generatorΓSub-μs generatorΓ
∅600 μm, lw=40 mm 0.57 ∅250 μm, lw=40 mm 0.71 
∅600μm, lw=23 mm 0.46 ∅250 μm, lw=23 mm 0.47 
∅90 μm, lw=40 mm 0.92 ∅90 μm, lw=40 mm 0.61 
∅90 μm, lw=23 mm 0.48 ∅90 μm, lw=23 mm 0.41 
μs generatorΓSub-μs generatorΓ
∅600 μm, lw=40 mm 0.57 ∅250 μm, lw=40 mm 0.71 
∅600μm, lw=23 mm 0.46 ∅250 μm, lw=23 mm 0.47 
∅90 μm, lw=40 mm 0.92 ∅90 μm, lw=40 mm 0.61 
∅90 μm, lw=23 mm 0.48 ∅90 μm, lw=23 mm 0.41 

In this research, we found that the maximal values of the energy density, energy density rate, and energy density per unit area deposited into a single wire or a wire in an array exploding under water, which satisfy overdamped discharge, depend on the wire length and diameter. Moreover, these parameters are strongly dependent on the timescale of the explosion and the remaining energy in the pulse generator after the maximum of the deposited power. It was shown that prior to realizing maximal deposited power, the exploded wire experiences a transition to a low-ionized plasma. In this state, the wire resistivity is determined by the electron–neutral collision rate, which, in turn, depends on the radial expansion velocity of the wire, current density, and temperature. To keep this state of plasma, it is important that at least half of the total energy deposited into the wire is delivered before the power maximum. Also, it was found that for a warm dense plasma and strong shocks generation research purposes, where the highest values of density, temperature, and pressure are required, the smallest possible wire lengths should be chosen. The latter, for a given load cross section, stored energy in the pulse generator and its timescale of operation, will produce the total weight of the load for which the energy density and energy density rate deposition would not exceed corresponding values obtained in this research. The data regarding the maximal values of energy density and energy density rate deposition in the exploding wire, while keeping an overdamped discharge, obtained in the present research can be scaled to applications where significantly powerful pulse generators are available.

We are grateful to Dr. J. Leopold and Dr. S. Bland for fruitful discussions and L. Merzlikin and E. Flyat for technical assistance. This research was supported by the Israel Science Foundation Grant No. 492/18. The financial support of Vetenskapsrådet (Swedish Research Council) and Wenner-Gren Foundations are gratefully acknowledged.

The authors have no conflicts to disclose.

Daniel Maler: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Writing – original draft (equal); Writing – review & editing (equal). Michael Liverts: Data curation (supporting); Formal analysis (supporting); Writing – review & editing (equal). Sergey Efimov: Conceptualization (equal); Data curation (supporting); Writing – review & editing (supporting). Alexander Virozub: Formal analysis (supporting); Software (equal). Yakov E. Krasik: Conceptualization (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon request.

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