Experimental and magnetohydrodynamic numerical simulation results and analysis of a *μ*s- and sub-*μ*s-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.

## I. INTRODUCTION

The study of electrical explosion of wires has been of interest for over two centuries^{1} with several review papers^{2–4} and text books^{5–11} dedicated to the subject. Research has been mostly concentrated on the study of wire explosions in vacuum and in gas, on ns and *μ*s timescales with current densities in the range of 10^{7}–10^{9} A/cm^{2}. The main interest in wire explosions in vacuum was in x-pinch^{12} and cylindrical wire array explosion^{13–15} 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 energy^{16–18} 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 × 10^{5} cm/s) radial expansion of the wire^{4,23–25} and surface flashover^{4,26} 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 (10^{7} 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 ∼10^{5} 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 $Rwcr\u22652Lt/C$, 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 *μ*s-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}

Here, $W0$ and $\phi 0$ are the stored energy and charging voltage of the capacitor, respectively, $\rho 0$ is the density, $\sigma 0$ is the conductivity at normal condition of the material, $\lambda $ 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, $\omega th,$ and current density, $jth,$ were derived as follows:

Using this approach for copper, $\omega th=20.1\u2009kJ/g\u2009$. One should note that dependencies (1) and (2) were obtained for relatively slow explosions (rise time of the current ≥3 *μ*s) 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, *μ*s- or sub-*μ*s-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 (∼10^{3} 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) simulations^{52,53} 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 *μ*s- and sub-*μ*s-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.

## II. EXPERIMENTAL SETUP AND DIAGNOSTICS

Experiments were carried out using a *μ*s generator^{55} (*C _{μ}* = 10

*μ*F, 25 kV charging voltage, ∼250 kA current amplitude, 1.2

*μ*s rise time on a short circuit load with total inductance of the discharge circuit

*L*∼ 75 nH, stored energy of ∼3.2 kJ) and a sub-

_{μt}*μ*s generator

^{56}(

*C*= 0.32

_{sμ}*μ*F, 58 kV charging voltage, 170 kA current amplitude, 0.22

*μ*s rise time on a short circuit load with total inductance of the discharge circuit

*L*∼ 55 nH, stored energy of ∼0.54 kJ). To keep the discharge critically damped, the total resistance $Rdc\u22652L/C1/2$ of the discharge circuit should be ≥0.17 Ω for the

_{sμt}*μ*s generator and ≥0.8 Ω for the sub-

*μ*s 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 *μ*s and sub-*μ*s experimental setups were similar and are shown in Fig. 1(a). In experiments with the *μ*s 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-*μ*s 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 *μ*s-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.

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 $S=1h\u222b0\tau \u2009exp\u2009I2tdt0.5$ was estimated using the action current integral for copper *h *=* *4.1 × 10^{9} A^{2} s/cm^{4}.^{21,22} Here, $\tau \u2009exp\u2009$ 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 $\tau \u2009exp\u2009$. Preliminary experiments showed that for 40-mm-long wires, for the *μ*s generator, the discharges were overdamped for 600-*μ*m- and 500-*μ*m-diameter wires, and for 250-*μ*m- and 200-*μ*m-diameter wires for the sub-*μ*s 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 $\tau \u2009exp\u2009,$ 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 $\tau \u2009exp\u2009$, can be the reason for this discrepancy.

μs generator . | Sub-μs generator . | ||||
---|---|---|---|---|---|

Length (cm) . | Cross-sectional area (cm^{2})
. | Weight (g) . | Length (cm) . | Cross-sectional area (cm^{2})
. | Weight (g) . |

Single wire 600 μm diameter | Single wire 250 μm diameter | ||||

4 | 2.79 × 10^{–3} | 0.1 | 4 | 4.91 × 10^{–4} | 0.017 |

3 | 0.075 | 3 | 0.013 | ||

2.3 | 0.058 | 2.3 | 0.01 | ||

Single wire 500 μm diameter | Single wire 200 μm diameter | ||||

4 | 1.96 × 10^{–3} | 0.07 | 4 | 3.14 × 10^{–4} | 0.011 |

3 | 0.053 | ||||

2.3 | 0.04 | ||||

Wire array^{a}: 27 wires each of 130 μm diameter | Wire array^{a}: 5 wires each of 130 μm diameter | ||||

4 | 3.58 × 10^{–3}/1.33 × 10^{–4} | 0.128/0.0048 | 4 | 6.64 × 10^{–4}/1.33 × 10^{–4} | 0.024/0.0048 |

3 | 0.096/0.0036 | 3 | 0.018/0.0036 | ||

2.3 | 0.073/0.0028 | 2.3 | 0.014/0.0027 | ||

Wire array^{a}: 56 wires each of 90 μm diameter | Wire array^{a}: 10 wires each of 90 μm diameter | ||||

4 | 3.56 × 10^{–3}/6.36 × 10^{–5} | 0.127/0.0022 | 4 | 6.36 × 10^{–4}/6.36 × 10^{–5} | 0.023/0.0023 |

3 | 0.095/0.0017 | 3 | 0.017/0.0017 | ||

2.3 | 0.073/0.0013 | 2.3 | 0.013/0.0013 |

μs generator . | Sub-μs generator . | ||||
---|---|---|---|---|---|

Length (cm) . | Cross-sectional area (cm^{2})
. | Weight (g) . | Length (cm) . | Cross-sectional area (cm^{2})
. | Weight (g) . |

Single wire 600 μm diameter | Single wire 250 μm diameter | ||||

4 | 2.79 × 10^{–3} | 0.1 | 4 | 4.91 × 10^{–4} | 0.017 |

3 | 0.075 | 3 | 0.013 | ||

2.3 | 0.058 | 2.3 | 0.01 | ||

Single wire 500 μm diameter | Single wire 200 μm diameter | ||||

4 | 1.96 × 10^{–3} | 0.07 | 4 | 3.14 × 10^{–4} | 0.011 |

3 | 0.053 | ||||

2.3 | 0.04 | ||||

Wire array^{a}: 27 wires each of 130 μm diameter | Wire array^{a}: 5 wires each of 130 μm diameter | ||||

4 | 3.58 × 10^{–3}/1.33 × 10^{–4} | 0.128/0.0048 | 4 | 6.64 × 10^{–4}/1.33 × 10^{–4} | 0.024/0.0048 |

3 | 0.096/0.0036 | 3 | 0.018/0.0036 | ||

2.3 | 0.073/0.0028 | 2.3 | 0.014/0.0027 | ||

Wire array^{a}: 56 wires each of 90 μm diameter | Wire array^{a}: 10 wires each of 90 μm diameter | ||||

4 | 3.56 × 10^{–3}/6.36 × 10^{–5} | 0.127/0.0022 | 4 | 6.36 × 10^{–4}/6.36 × 10^{–5} | 0.023/0.0023 |

3 | 0.095/0.0017 | 3 | 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 *L _{l}dI/dt* was subtracted from the measured value of

*V*to obtain the resistive voltage

*V*. Here,

_{res}*L*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 $\xb16%$. 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 $\mu m$ 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 $\xb1200$ m/s. The error in wire length is of $\xb10.1\u2009mm$. 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.

_{l}## III. EXPERIMENTAL AND 1D MHD SIMULATION RESULTS

### A. Results of experiments with the *μ*s-timescale generator

In Fig. 2, we present waveforms of the current and resistive voltage, obtained in explosions for 600- and 90-*μ*m-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-*μ*m-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.

Wire diameter ($\mu m)$ . | 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 ($\mu m)$ . | 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 $Wd=\u222b0t*I\phi rdt,\u2009$ where $t*$ is the time when the discharge current drops to 20% of its maximal amplitude. The efficiency of the stored energy deposition *η* =$Wd/Wst\xd7100%$ increases from ∼70% to ∼80% with decreasing wire length from 40 to 23 mm. Here, $Wst\u2009$ is the initially stored energy in the pulse generator. For all the studied single wires and wire array explosions, approximately half of *W _{st}* 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.

L (mm)
. _{wire} | Rise time to I (ns)
. _{max} | Time of max. power (ns) . | Max. resistive voltage (kV) . | Energy density (kJ/g) . | Energy density rate (kJ/g s) × 10^{6}
. | R at max. current (Ω)
. _{1} | R at max. power (Ω)
. _{2} |
---|---|---|---|---|---|---|---|

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 |

L (mm)
. _{wire} | Rise time to I (ns)
. _{max} | Time of max. power (ns) . | Max. resistive voltage (kV) . | Energy density (kJ/g) . | Energy density rate (kJ/g s) × 10^{6}
. | R at max. current (Ω)
. _{1} | R at max. power (Ω)
. _{2} |
---|---|---|---|---|---|---|---|

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-*μ*m-diameter wire as compared with 600-*μ*m-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-*μ*m-diameter single copper wires and 90-*μ*m-diameter wires in a cylindrical array and the generated shocks in water are shown in Fig. 3. At the beginning of the explosion at $t=\tau m$ (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 *μ*s 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-*μ*m-diameter wires since for wires of smaller diameter, determining the inclination points was difficult.

In Table IV, the shock velocity of the 600-*μ*m-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.

### B. Results of 1D MHD simulations of *μ*s-timescale wire explosions

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 water^{54} 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 *σ _{cr}* 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

*σ*= (2.3 ± 0.5) × 10

_{cr}^{15}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

*I*

_{i}= (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 *μ*s-timescale explosions of single 600-*μ*m-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 $\sigma =1/\rho $ at the time corresponding to maximal current and power for 600-*μ*m-diameter, 40-mm-long and 90-*μ*m-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 $\sigma $ for the 600-*μ*m-diameter wire at peak current, whereas at the time when the maximal deposited power is reached, it is less pronounced (∼10%).

In Table V, the main parameters characterizing explosions of 40-, 30-, and 23-mm-long and 600-, 500-, 130-, and 90-*μ*m 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 (*r _{p}*/

*r*), the current density, temperature, mass density, and pressure decrease. Here,

_{o}*r*and

_{p}*r*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.

_{o}Wire length (mm) . | R (Ω)
. | ρ (Ω × m) × 10^{–6}
. | Max. wire velocity (m/s) . | Wire radius r [_{p}μm]
. | n (kg/m^{3})
. | P (Pa) × 10^{9}
. | T (kK)
. | j (A/cm^{2}) × 10^{7}
. | $\omega d\u2009max$ (J/kg) × 10^{6}
. | $\epsilon t$ (J/kg s) × 10^{12}
. | $d\omega d\u2009/dSmax$ (J/kg⋅cm^{2}) × 10^{6}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

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^{–6}
. | Max. wire velocity (m/s) . | Wire radius r [_{p}μm]
. | n (kg/m^{3})
. | P (Pa) × 10^{9}
. | T (kK)
. | j (A/cm^{2}) × 10^{7}
. | $\omega d\u2009max$ (J/kg) × 10^{6}
. | $\epsilon t$ (J/kg s) × 10^{12}
. | $d\omega d\u2009/dSmax$ (J/kg⋅cm^{2}) × 10^{6}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

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 $,\u2009\omega d\u2009max$, energy density deposition rate $\epsilon t=d\omega d\u2009/dtmax,$ and deposited energy density $d\omega d\u2009/dSmax$ 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 (

*r*_{p}/*r*_{c}), significant increase in the deposited energy density rate and energy density per unit area.Decreasing wire expansion velocity, density, pressure, temperature, and current density.

### C. Results of experiments with the sub-*μ*s-timescale generator

In Fig. 5, waveforms of the current and resistive voltage for explosions of 40- and 23-mm-long, 250-*μ*m single wires and 90-*μ*m diameter wire arrays are presented together with the deposited power and individual wire resistance. Similar to *μ*s-timescale experiments, decreasing the length of the wires to 23 mm keeps the discharge overdamped. Also, at the maximum deposited power, the 90-*μ*m-diameter wire resistance increases ∼10-fold compared to the 250-*μ*m wire. However, in contrast to the *μ*s-timescale explosions, for sub-*μ*s-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-*μ*m-diameter wire [see Fig. 6(d)]. These experiments show that similar to *μ*s-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 *μ*s-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 *μ*s-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.

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

L (mm)
. _{wire} | Rise time to I (ns)
. _{max} | Time of max. power (ns) . | Max. resistive voltage (kV) . | Energy density (kJ/g) . | Energy density rate (kJ/g s) × 10^{6}
. | R at max. current (Ω)
. _{1} | R at max. power (Ω)
. _{2} |
---|---|---|---|---|---|---|---|

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 |

L (mm)
. _{wire} | Rise time to I (ns)
. _{max} | Time of max. power (ns) . | Max. resistive voltage (kV) . | Energy density (kJ/g) . | Energy density rate (kJ/g s) × 10^{6}
. | R at max. current (Ω)
. _{1} | R at max. power (Ω)
. _{2} |
---|---|---|---|---|---|---|---|

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 *μ*s-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 *μ*s-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-

*μ*s-timescale explosions is almost double compared with*μ*s-timescale explosions.

Shadow streak images of explosions of single copper wires, 40 and 23 mm long and 250 *μ*m diameter and the generated shocks in water are shown in Fig. 7. Similar to the *μ*s-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 *μ*s-timescale wire explosions (see Table VII).

### D. Results of 1D MHD simulations of sub-*μ*s timescale wire explosions

1D MHD simulations were also carried out for sub-*μ*s-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-*μ*m-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.

Examples of radial distributions of *T*, *n*, *P*, and $\sigma $, at the current and power maxima for 250- and 90-*μ*m-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 *μ*s-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 *μ*s-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.

In Table VIII, the main parameters characterizing explosions of 40-, 30-, and 23-mm-long, 250-, 200-, 130-, and 90-*μ*m-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 (*μ*s or sub-*μ*s) 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 (*r _{p}*/

*r*), energy density rate, and energy density deposition per unit area. However, the wire expansion velocity, density, pressure, temperature, and current density decrease.

_{c}Wire length (mm) . | R Ω
. | ρ (Ω × m) × 10^{–6}
. | Wire velocity (m/s) . | Wire radius r (_{p}μm)
. | n (kg/m^{3})
. | P (Pa) × 10^{9}
. | T (kK)
. | J (A/cm^{2}) × 10^{7}
. | $(\omega d)max$ (J/kg) × 10^{6}
. | $\epsilon t$ (J/kg s) × 10^{12}
. | $d\omega d\u2009/dSmax$ (J/kg cm^{2}) × 10^{6}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

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 Ω
. | ρ (Ω × m) × 10^{–6}
. | Wire velocity (m/s) . | Wire radius r (_{p}μm)
. | n (kg/m^{3})
. | P (Pa) × 10^{9}
. | T (kK)
. | J (A/cm^{2}) × 10^{7}
. | $(\omega d)max$ (J/kg) × 10^{6}
. | $\epsilon t$ (J/kg s) × 10^{12}
. | $d\omega d\u2009/dSmax$ (J/kg cm^{2}) × 10^{6}
. |
---|---|---|---|---|---|---|---|---|---|---|---|

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-*μ*s-timescale explosions compared with *μ*s-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-

*μ*s explosions.

## IV. DISCUSSION

The main purpose of the present research was (a) to obtain maximal values of energy density *ω _{d}*, energy density rate

*dω*, and energy density per unit area

_{d}/dt*dω*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?

_{d}/dSFirst let us note that analysis of the data presented in Tables V and VIII showed that in both *μ*s- and sub-*μ*s-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 $\omega d\u2009,\u2009d\omega d\u2009/dt,$ and $d\omega d\u2009/dS,$ 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).

. | Energy density (kJ/g) . | Energy density deposition rate (kJ/(g s)) × 10^{6}
. | Energy density per unit area at maximal power [kJ/(g cm^{2})] × 10^{9}
. |
---|---|---|---|

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)) × 10^{6}
. | Energy density per unit area at maximal power [kJ/(g cm^{2})] × 10^{9}
. |
---|---|---|---|

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 $\omega d\u2009$ was obtained on *μ*s timescale, but the values of $d\omega d\u2009/dt$ and $d\omega d\u2009/dS$ were largest for sub-*μ*s-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 *μ*s, using estimate of cross section of a 15-mm-diameter cylindrical copper foil (see Sec. II), the corresponding foil thickness is 127 *μ*m. 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 *μ*s-timescale and in Tables VI and VIII for sub-*μ*s-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, $\pi r2$, $[$($Rl1/Rl2=k\rho l1/\rho l2rl2/rl12$, 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 $k=1.74\u2009$(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.

. | $Rl1/Rl2\u2009exp\u2009Rl1/Rl2sim$ . | $\rho l1/\rho l2$ . | $rl2/rl12$ . | Temperature $Tl1/Tl2$ . | Density $nl1/nl2$ . |
---|---|---|---|---|---|

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 |

. | $Rl1/Rl2\u2009exp\u2009Rl1/Rl2sim$ . | $\rho l1/\rho l2$ . | $rl2/rl12$ . | Temperature $Tl1/Tl2$ . | Density $nl1/nl2$ . |
---|---|---|---|---|---|

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 as^{61} $\rho e\u2212i(\Omega \u2009cm)=5.26\xd710\u22123T\u22121.5(eV)ln\Lambda $, where $ln\Lambda $ 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:^{61} $\rho e\u2212nn\Omega \u2009\u2009cm=3.52\xd7103ne\u22121(cm\u22123)\xb7\nu e\u2212n(s\u22121)$. Here, *n _{e}* is the plasma electron density and $\nu e\u2212ns\u22121=6.21\xd7107nn0\u2009rw0/rt2\xd7\pi ra2Te(t)(K)$ is the electron–atom collision frequency.

*T*(

_{e}*t*) and

*m*are the temperature and electron mass, respectively; $nn0\u2009$ is the density of atoms at normal conditions; $rw0$ is the initial radius of the wire; and $ra$ is the atom radius. Thus, $\nu e\u2212n\u221dr\u22122tTe.$ Here, the Maxwellian energy distribution of electrons is assumed. The density of electrons can be estimated as $ne=I(t)/\pi r2teVdr,$ where

_{e}*I*(

*t*) is the current, $Vdr=1.2Vth(t)me/Ma0.5$ is the drift velocity of current carrying electrons, and $Vth=8kBTe/\pi me0.5$ is the electron mean thermal velocity. Thus, one obtains that $ne\u221dr\u22122tTe\u22120.5$ that results in $\rho e\u2212nn\u221d\u2009Te$. For our experimental conditions, when the density of neutrals is in the range 10

^{21}–10

^{22}cm

^{−3}, the value of $\nu e\u2212n$ is in the range 10

^{13}–10

^{14}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 *μ*s- and sub-*μ*s-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 *μ*s-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 $\u221dTe\u22121.5.$ For sub-*μ*s-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 *μ*s-timescale explosions. Thus, for sub-*μ*s-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-*μ*m-diameter, 40-mm-long wires exploded on the *μ*s and sub-*μ*s timescales. Figure 10 shows good agreement between the experimental and calculated conductivities for *μ*s-timescale explosion assuming a lowly ionized plasma. For sub-*μ*s-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)].

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 $\sigma t=I(t)lw/\phi R\u2009(t)\pi rw\u20092(t),$ where $I(t)$ and $\phi R\u2009(t)$ are the current and resistive voltage, respectively; $lw\u2009$ is the length of the wire; and $rw\u2009(t)$ 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 *μ*m, 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 earlier^{62} and present research, the coupling coefficient is $\Gamma \u2009\u2248\u20091$ 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.

$\mu s$ generator . | $\Gamma $ . | Sub-$\mu s$ generator . | $\Gamma $ . |
---|---|---|---|

∅600 $\mu m$, $lw\u2009$=40 mm | 0.57 | ∅250 $\mu m$, $lw\u2009$=40 mm | 0.71 |

∅600 $\mu m$, $lw\u2009$=23 mm | 0.46 | ∅250 $\mu m$, $lw\u2009$=23 mm | 0.47 |

∅90 $\mu m$, $lw\u2009$=40 mm | 0.92 | ∅90 $\mu m$, $lw\u2009$=40 mm | 0.61 |

∅90 $\mu m$, $lw\u2009$=23 mm | 0.48 | ∅90 $\mu m$, $lw\u2009$=23 mm | 0.41 |

$\mu s$ generator . | $\Gamma $ . | Sub-$\mu s$ generator . | $\Gamma $ . |
---|---|---|---|

∅600 $\mu m$, $lw\u2009$=40 mm | 0.57 | ∅250 $\mu m$, $lw\u2009$=40 mm | 0.71 |

∅600 $\mu m$, $lw\u2009$=23 mm | 0.46 | ∅250 $\mu m$, $lw\u2009$=23 mm | 0.47 |

∅90 $\mu m$, $lw\u2009$=40 mm | 0.92 | ∅90 $\mu m$, $lw\u2009$=40 mm | 0.61 |

∅90 $\mu m$, $lw\u2009$=23 mm | 0.48 | ∅90 $\mu m$, $lw\u2009$=23 mm | 0.41 |

## V. SUMMARY

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.

## ACKNOWLEDGMENTS

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.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**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).

## DATA AVAILABILITY

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