Nanosecond time scale, high power electrical wire explosion in water

PHYSICS OF PLASMAS 13, 042701 共2006兲
Nanosecond time scale, high power electrical wire explosion in water
A. Grinenko, Ya. E. Krasik, S. Efimov, A. Fedotov, and V. Tz. Gurovich
Physics Department, Technion, 32000 Haifa, Israel
V. I. Oreshkin
Institute of High Current Electronics, SB RAN, 634055 Tomsk, Russia
共Received 17 December 2005; accepted 24 February 2006; published online 18 April 2006兲
Experimental and magnetohydrodynamic simulation results of nanosecond time scale underwater
electrical explosions of Al, Cu, and W wires are presented. A water forming line generator with
current amplitude up to 100 kA was used. The maximum current rise rate and maximum Joule
heating power achieved during wire explosions were dI / dt 艋 500 A / ns and 6 GW, respectively.
Extremely high energy deposition of up to 60 times the atomization enthalpy was registered
compared to the best reported result of 20 times the atomization enthalpy for energy deposition with
a vacuum wire explosion. Discharge channel evolution and surface temperature were analyzed by
streak shadow imaging and by a fast photodiode with a set of interference filters, respectively. A 1D
magnetohydrodynamic simulation demonstrated good agreement with experimental parameters such
as discharge channel current, voltage, radius, and temperature. Material conductivity was calculated
to produce the best correlation between the simulated and experimentally obtained voltage. It is
shown that material conductivity may significantly vary as a function of energy deposition rate.
© 2006 American Institute of Physics. 关DOI: 10.1063/1.2188085兴
I. INTRODUCTION
Electrical wire explosion has been a subject of interest
since the nineteenth century due to the rich variety of physical phenomena involved in this complicated process and the
exotic material states formed that are similar to those which
occur in stars. Particularly, electrical wire explosions result
in the generation of nonideal, strongly coupled plasmas characterized by a coupling parameter ⌫ 艌 1. The interest in these
plasmas, however, is related not only to the sophisticated
phenomena associated with their formation, but also to their
important applications in inertial confinement fusion, solid
state and plasma-chemical physics, rocket engines, etc.1 A
number of recent experimental studies2–5 have been dedicated to the investigation of the properties of these plasmas
and their transport parameters. New advanced theoretical
models6–8 have been proposed based on the results of this
research.
The main aim in the inertial confinement fusion concept
is to deposit a large amount of energy during a short period
of time into a spatially confined and uniform wire material
by Joule heating in vacuum. However, this aim has not been
realized because of the early termination of energy deposition, and therefore the temperature of the wire core never
exceeded a few eV. The termination of the energy deposition
was caused either by wire disintegration because of magnetohydrodynamic 共MHD兲 instabilities in the case of microsecond 共␮s兲 time scale wire explosions, or by electrical breakdown in the case of nanosecond 共ns兲 time scale wire
explosions. Electrical breakdown, which occurs due to ionization of ablated wire and impurity vapors,9 is accompanied
by the formation of a current carrying low density and fast
expanding plasma shell 共corona兲, the resistance of which is
rapidly falling.
One of the possible methods of increasing energy depo1070-664X/2006/13共4兲/042701/14/$23.00
sition into the wire core was to prevent this breakdown by
exploding wires in breakdown-impeding environments such
as gas, water, or vacuum pump oil9 in order to suppress or
delay the creation of corona. It was shown that an electrical
explosion of Ti wires in air allows the deposition of at least
twice more energy than in vacuum, due to suppression of the
plasma corona expansion by the external air pressure.10 In ␮s
time scale underwater electrical wire explosions 共UEWE兲,3,11
it was shown that high pressures generated in the surrounding water during the discharge suppress generation of a corona plasma. Pressures generated during UEWEs are much
higher than with wire explosions in air. Therefore, the suppression of the plasma corona formation could be more effective with an UEWE. It was shown that in this case the
deposited energy is up to three times higher than the atomization enthalpy.11 Let us note here that with ns time scale
wire explosions in vacuum, a dielectric coating can also slow
down the plasma corona formation, providing an enhancement of the energy deposition of up to three times the atomization enthalpy.12
An increase in the current rise rate dI / dt in the case of ns
wire explosions in vacuum proved to be another method for
significantly improving energy being deposited into a metal
core prior to plasma corona formation.13,14 For instance, in
Ref. 13 it was shown that for nonrefractory metals such as
Ag, Al, Cu, and Au, and for dI / dt ⬵ 150 A / ns, the energy
deposition in the metal core exceeds 1.5–2.9 times the atomization enthalpy. A decrease in dI / dt to 20 A / ns leads to 2–3
times less energy deposited into the metal core. Finally, it
was shown15 that combining a dielectric coating with a
dI / dt ⬵ 150 A / ns allows one to realize corona-free explosion of a polyimide coated W wire in vacuum with an
anomalously high energy deposition of 20 times the atomization enthalpy.
13, 042701-1
© 2006 American Institute of Physics
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Phys. Plasmas 13, 042701 共2006兲
Grinenko et al.
FIG. 1. Experimental setup.
In this paper, a system for UEWEs which combines the
advantages of explosions in breakdown-impeding surroundings and increased dI / dt is studied. Electrical explosions of
Al, Cu, and W wires were produced in deionized water by a
pulsed high voltage 共HV兲 generator. It was shown that the
energy deposited into the wire material by direct Joule heating can surpass the atomization enthalpy by a factor of ⬃60
for Cu and Al wires. The temperature of the discharge channel 共DC兲, the estimate of which was based on the assumption
that the spectrum of radiation emitted during the wire explosion is a blackbody 共BB兲 radiation spectrum, was found to
range from 2 to 9 eV. However, the possibility that a high
temperature spectrum component is emitted from the DC
from a quasiperiodic discharge cannot be ruled out. MHD
simulations of ns time scale UEWEs indicate that the conductivity of the wire material should depend on the current
density and dI / dt, as well as on instantaneous thermodynamic variables such as temperature and density. Simulations
using appropriate conductivity values demonstrate good
quantitative and qualitative agreement with electrical and optical diagnostics of Cu wires in ns explosions.
II. NANOSECOND TIME SCALE EXPLOSION
A. Experimental setup and diagnostics
In the experiments 共see Fig. 1兲 we used a HV generator
consisting of a four-stage Marx generator 共output capacitance Cg = 54 nF, output voltage V0 = 160 kV兲 which charges
a coaxial forming line, 1 m long, filled with deionized water
共line capacitance CL = 20 nF, inductance LL = 50 nH, and line
impedance Z0 = 1.5 ⍀兲. The charging time of the line is
280 ns with maximum amplitude of 230 kV. A selfbreakdown gas switch was used to discharge the line. Typical
resistance and inductance of the switch and the load holder
were estimated as RS = 0.1 ⍀ and LS = 70 nH, respectively.
This generator produces at the matched load a pulse with
voltage and current amplitudes of ⬃110 kV and ⬃70 kA,
respectively, and pulse duration of ⬃80 ns at full width at
half maximum 共FWHM兲. Maximum current densities j obtained during UEWEs were of the order of 109 A / cm2 and
maximum dI / dt ⬵ 500 A / ns. A negative HV pulse was delivered from the generator via an interface insulator to a
metal wire connected between the cathode and grounded anode electrodes. The electrodes with the wire were placed
inside a chamber filled with deionized water that had windows for optical observation. Cu, Al, and W wires of different lengths 共25– 100 mm兲 and diameters 共50– 200 ␮m兲 were
used.
FIG. 2. 共a兲 Typical current and voltage waveforms. 共b兲 Temporal behavior
of the deposited power. 100 mm length and Ø100 ␮m Cu wire.
A self-integrating Rogowsky coil and an active voltage
divider were used for current and voltage measurements at
the wire load, respectively. In order to obtain a resistive voltage drop Vr of the exploding wire, the inductive voltage was
eliminated by simultaneous Ḃ loop measurements that were
calibrated for each wire length. The IdLw / dt component of
the voltage drop was eliminated using inductance estimated
from the streak image of the DC. Here, Lw is the inductance
of the exploding wire. Optical observations of the DC and
the generated shock waves 共SW兲 were carried out by shadow
photography using a VICO-300UV streak camera.
The spectrum of the radiation emitted from the DC was
sampled at eleven wavelengths in the visible and UV light
range using interference filters with central wavelengths of
410 nm, 486 nm, 488 nm, 514 nm, 532 nm, 550 nm, and
656 nm and a bandwidth of 10 nm. Additional filters had
central wavelengths of 255 nm, 303 nm, 336 nm, and
366 nm and a bandwidth of 6 nm. Radiation power was recorded by a photodiode 共OPHIR FPS–10兲 with a risetime of
1 ns. The focusing system, filters and photodetector were
calibrated in situ using a pyroelectric head 共OPHIR
PE50BB-DIF-V2兲 and a BB étalon tungsten halogen lamp
共ORIEL 6318 10 W QTH兲. The recorded light traces were
then analyzed to obtain a time-resolved temperature trace
assuming BB radiation of the DC.
B. Experimental results
1. Electrical and mechanical properties of the DC
Typical current and voltage waveforms obtained for Cu
wire of length lw = 100 mm and diameter Ø100 ␮m are
shown in Fig. 2共a兲. The current reaches its maximum value
of ⬃42 kA in ⬃70 ns and all the energy is deposited into the
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Nanosecond time scale, high power electrical wire...
Phys. Plasmas 13, 042701 共2006兲
FIG. 3. Solid, experimental average discharge resistance with an UEWE.
Dashed, theoretical average discharge resistance for an UEWE. 100 mm
length and Ø100 ␮m Cu wire.
DC at a time interval of 艋100 ns with a maximum value of
deposited power P = IVr = 4 GW 关see Fig. 2共b兲兴. One can see
that an increase in Vr starts with a ⬃50 ns time delay with
respect to the start of the current. In addition, Vr is almost
constant during the first 50 ns, which results in a decrease in
the wire resistance during this period 共see Fig. 3兲. This can
be explained by the skin effect, since at the beginning of the
generator pulse the current flows in a cross-sectional area
that is much smaller than the total cross section of the wire.
Therefore the measured average wire resistance Vr / I is
higher than would be expected if the current were uniformly
distributed in a cross-sectional area of the wire. The skin
time can be estimated as ␶s = 4␲␴Rw2 / c2 ⬇ 150 ns,16 where Rw
is the wire radius and ␴ is the wire conductivity at room
temperature and atmospheric pressure. The estimated skin
time of 150 ns is five times larger than the risetime of the
current I / İ ⬇ 30ns. It is understood that the skin time evaluated using the normal conductivity is overestimated since in
the process of wire heating its conductivity falls. Nevertheless, this estimate shows that skin effect significantly influences this type of discharge. Namely, during this time the
effective conducting area grows and therefore the average
resistance falls. However, the current heats the wire and thus
decreases the specific conductivity and increases the average
wire resistance. These two competing processes are reflected
in the temporal evolution of the average resistance shown in
Fig. 3. This process was reproduced in MHD simulations,
resulting in the average resistance shown as a dashed line in
Fig. 3.
Dependence of the total deposited energy versus wire
length for different wire diameters and materials is shown in
Fig. 4. The value of the deposited energy was obtained by
time integration of the power delivered to the DC by Joule
heating. The limits of the integration were from the start of
the current to the time corresponding to the half amplitude of
the back side of the first voltage pulse. One can see that the
maximum deposited energy of 240± 20 J 共⬃50% of the total
energy stored in the water line兲 is obtained for wires 100 ␮m
in diameter regardless of lengths and materials. The latter
can be explained by the fact that the average value of j is
mainly determined by the wire diameter since dI / dt and the
value of the maximum current are essentially determined by
FIG. 4. Dependency of the total deposited electrical energy 共a兲, average
deposited electrical energy per atom 共b兲 and similarity parameter ⌸ on the
wire diameter for different wire lengths and material.
the parameters of the electrical circuit. Therefore, with thick
wires the electrical energy is not transferred effectively to the
wire material since the average of j is too low to cause an
explosion of the wire. Conversely, thin wires explode too fast
leaving a large amount of energy in the forming line, resulting in the electrical breakdown of exploded wire products
and formation of a high conductivity plasma DC. Between
these extremes lies a region of wire diameters where most of
the electrically stored energy can be deposited in the wire.
Since for wires of medium diameter 共50 ␮m and
100 ␮m兲 the total deposited energies have similar values, the
average energy deposited per unit atom increases with the
decrease in wire diameter 关see Fig. 4共b兲兴. Thus, the maximum estimated value of average deposited energy per atom
for the Cu wire 共lw = 25 mm, Ø50 ␮m兲 is ⬃200 eV/atom.
Taking into account that the enthalpy of atomization of Cu is
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042701-4
Phys. Plasmas 13, 042701 共2006兲
Grinenko et al.
FIG. 6. Total electrically deposited energy per unit wire length vs the similarity parameter ⌸. Designation numbers are the same as in Fig. 6.
FIG. 5. 共a兲 Maximum pressure at the DC boundary vs total electrically
deposited energy; 共b兲 maximum pressure at the DC boundary vs average
electrically deposited energy per atom; 共c兲 maximum pressure at the DC
boundary vs the similarity parameter ⌸; 共d兲 maximum pressure at the DC
boundary vs total electrically deposited energy per unit length; 共1兲 Cu wire,
lw = 50 mm, Ø200 ␮m; 共2兲 Cu wire, lw = 50 mm, Ø50 ␮m; 共3兲 Cu wire,
lw = 100 mm, Ø50 ␮m; 共4兲 Cu wire, lw = 100 mm, Ø100 ␮m; 共5兲 Al wire,
lw = 50 mm, Ø127 ␮m; 共6兲 Cu wire, lw = 50 mm, Ø100 ␮m; 共7兲 Cu wire,
lw = 25 mm, Ø100 ␮m.
3.5 eV/atom, one obtains an overheating value of ⬃60. Here
let us note that the maximum achieved overheating of Cu
wire exploded in vacuum13 is less than 3. For Al wires the
maximum deposited energy obtained in our experiments was
130 eV/atom, which gives an overheating value of 38. Hence
one can see that the water medium plays a crucial role in the
increase in the energy deposition per atom.
The dependency of the parameter ⌸
⌸=
冑冉 冊
␳0 dP
lw dt
共1兲
max
on the wire diameter for different wire lengths is shown in
Fig. 4共c兲. Here ␳0 is the initial density of the wire material,
and P is the electrical power of the discharge. This parameter
is obtained from the combination of the discharge parameters
and it has the dimensionality of pressure. Only the wire
length lw is significant in ⌸ since lw Ⰷ Rw. Therefore lw becomes a critical parameter for similarity analysis. From Fig.
4共c兲 one can see that the parameter ⌸ also has an optimum
value for wires of medium diameter and fixed length, and ⌸
increases with the decrease in length of wires of fixed diameter.
Using a streak image of the expanding DC and the SW,
the value of pmax was estimated by the method described in
Ref. 11. This pressure occurs as a result of water compression resistance to the DC expansion which in turn is a result
of the internal pressure created in the DC due to the electrical
energy input. The dependence of pmax on various discharge
parameters is shown in Fig. 5. One can see that there is a
poor correlation of the value of pmax with total deposited
energy, as well as with the deposited energy per atom. However, a good correlation is obtained between pmax and ⌸ and
between pmax and the deposited energy per unit length. The
last correlation is obvious since lw Ⰷ Rw, so that the wire can
be considered as infinitely thin. In addition, the duration of
the energy deposition is much shorter than the typical duration of the mechanical DC expansion. Following basic dimensional analysis, one can arrive at the conclusion that the
pressure created in such conditions is proportional to the
energy deposited per unit length. By observing the obtained
correlation of pmax and ⌸, and pmax and the deposited energy
per unit length it is, however, easy to see that there should
also be a correlation between ⌸ and the deposited energy per
unit length. This dependence is shown in Fig. 6 in which one
can see that the deposited energy per unit length increases
almost linearly as a function of ⌸.
2. Properties of the white light emission from the DC
It was found that the light radiated from the DC is emitted in two pulses. The first narrow pulse, with a duration of
⬃200 ns, corresponds to the wire explosion and starts with
the onset of wire expansion 关see Fig. 7共a兲兴. Then, after some
delay, whose length depends on the wire material, a secondary light emission pulse starts which continues for tens of
microseconds 关see Fig. 7共b兲兴. A similar temporal behavior of
light emission was observed by Sarkisov et al.15,17 in vacuum
explosion of wires with diameters of tens of microns and
lengths of ⬃2 cm. In experiments14,15,17 with exploding Al
wires in vacuum it was found that the second, long duration
light emission occurs only for small overheating, and this
light emission is related to particle-type radiation.
In plots shown in Fig. 7共a兲 the time scale of the digitizer
was set so that the first narrow emission pulse could be resolved, and in Fig. 7共b兲 the time scale was chosen so that the
second long-duration light emission could be observed. The
intensity is presented in arbitrary units, and is the same for
all the explosions presented in Fig. 7. One can see that in all
the demonstrated UEWEs there exists a late time radiation,
despite the fact that overheating is much higher than in similar experiments in vacuum.14,15,17 Calculations similar to
those in Ref. 17 show that the maximum size of the microparticles is of order of 100 nm, which is comparable with the
size of microparticles obtained in Ref. 17. These microparticles can be created in the process of condensation of wire
explosion products under the high pressures generated during
the discharge. However, the estimated submicron size of the
particles is on the edge of this theory’s applicability, and
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Phys. Plasmas 13, 042701 共2006兲
Nanosecond time scale, high power electrical wire...
TABLE I. Parameters of Cu and Al wire explosions 共␧ is the atomization
enthalpy, l is the wire length, Ø-wire diameter, E is the input energy, P1 is
the first light pulse peak power, P2 is the second light pulse peak power,
dP2 / dt is the second light pulsed peak power derivate兲.
FIG. 7. Intensity of white light emitted by the DC during an UEWE with Cu
wire 共Ø100 ␮m兲 and Al wire 共Ø127 ␮m兲 of 100 mm and 50 mm length. 共a兲
The short time scale emitted light pulse. 共b兲 The later long time scale emitted light pulse.
other explanations of late-time radiation in UEWEs may be
proposed. For example, one can speculate that the DC expansion results in light emission from the entire volume of
the DC instead of only from the DC surface. The long time
scale of this radiation can be attributed to slower expansion
of the DC 共⬍0.1 mm/ ␮s兲 than with vacuum wire explosions
共⬎5 mm/ ␮s兲 and relatively small radiation and mechanical
energy losses compared with total deposited energy. Another
difference between the short time scale of radiation from
UEWEs from the light emission with vacuum wire explosions is that probably no corona plasma takes place in
UEWEs whereas it does with vacuum wire explosions.
Therefore, the short time scale light pulse cannot be attributed to the ionization of the ambient vapor and its fast
expansion.13 The source of this short duration radiation in
UEWE’s is assumed to be the surface of the heated wire
material.
A summary of some of the principle parameters of the
emitted light, such as its maximum intensity in the first shortand second long-duration light pulses, and light power
growth rate of the second light pulse, are shown in Table I.
One can see that the intensity of the first pulse depends on
the energy deposited per atom. For instance, the light intensity increases ⬃2 times with a ⬃2 times increase of input
energy per atom. A similar correlation was obtained for the
second light pulse intensity. Also, it was found that the second light pulse is characterized by light intensity several
times larger than the first light pulse. Let us note that in
explosions of Al wires 共lw = 100 mm兲 the intensity of the sec-
␧
共eV/atom兲
l
共mm兲
Ø
共␮m兲
E
共eV/atom兲
P1
共a.u.兲
P2
共a.u.兲
dP2 / dt
共a.u./ ␮s兲
Cu
3.5
100
50
100
100
22.1
51.0
0.10
0.25
0.37
1.67
2.83
7.27
Al
3.4
100
127
18.3
0.14
1.07
5.57
50
127
40.6
0.25
1.42
9.69
ond light pulse is much larger than the light intensity from a
Cu wire of the same lw. The latter may be attributed to the
combustion of the Al. The difference between Al and Cu
wire explosions becomes even more distinct in the power
growth rate of the secondary light pulse. A similar dependence of the late time emission power on the deposited energy was obtained in vacuum W wire explosions.17
The experimental data obtained for the first short time
scale light emission from exploding 100 mm and 50 mm
long Cu wires 共Ø100 ␮m兲 and Al wires 共Ø127 ␮m兲 are
summarized in Figs. 8–11. In the streak images, both the
light emission from the DC and the shadow image of the
channel after the emission power falls can be seen. In these
experiments a white flash lamp source with lenses was used
to produce a parallel backlighting beam. However, the initial
light pulse emitted from the wire was more intense than the
intensity of the backlighting beam. Since we are interested in
long time scale evolution of the DC, the sensitivity of the
streak camera was set according to the intensity of the backlighting beam. This caused saturation and reduced the resolution quality of the streak image in the first light emitting
stage.
In Figs. 8–11 correlations of the emitted-light power,
current, voltage and electrical power, and channel expansion
can be seen. For instance, in Fig. 8 experimental results of
Cu wire 共Ø100 ␮m, lw = 100 mm兲 explosions are shown. According to Fig. 7 with this UEWE the light emission power
P␥ has the smallest amplitude and shortest duration. Indeed,
the streak image 关see Fig. 8共c兲兴 shows that the light is emitted during a short period of ⬃150 ns at the beginning of wire
expansion. Let us note that the P␥ pulse does not coincide
with the input electrical power PE pulse 关see Fig. 8共b兲兴 but
begins at about ⬃50 ns delay from the peak of the electrical
power. It is reasonable to assume that the start of the light
emission coincides with the fast decrease in the discharge
current at ⬃70 ns 关see Fig. 8共a兲兴. There also seems to be a
correlation between the beginning of wire expansion and the
beginning of light emission. However, this cannot be stated
for sure because of poor spatial resolution during the initial
stage of explosion. The trace with the observed DC boundary
and the SW extracted from the streak image is represented by
the solid line in Fig. 8共b兲. The presumed initial trajectory of
the DC boundary and the SW at t ⬍ 120 ns is represented by
the dashed line. One can conclude that the beginning of light
emission and the beginning of the DC expansion coincide to
within ±15 ns. A similar behavior of the light emission was
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042701-6
Grinenko et al.
FIG. 8. 共a兲 Typical waveforms of the current and voltage; 共b兲 input electrical
power, emitted light power, and trajectories of the DC boundary and SW; 共c兲
streak camera shadow image of the exploding wire and SW. Cu wire
共Ø100 ␮m, lw = 100 mm兲.
observed in vacuum wire explosions where the beginning of
the radiation coincided with the beginning of the DC
expansion.13–15
Experimental results of Al wire 共Ø127 ␮m, lw
= 100 mm兲 explosions are shown in Fig. 9. Here the streak
image 关Fig. 9共c兲兴 has poor spatial resolution during all the
exposure time because of an intense DC light emission
which lasted for ⬃700 ns, causing saturation of the streak
camera. Therefore the DC and SW trajectories could not be
extracted from this image. Similar to the case shown in Fig.
8, the P␥ pulse does not coincide with the PE pulse but
begins at an ⬃50 ns delay from the later 关see Fig. 9共b兲兴. In
addition, in this experiment no pronounced sharp decrease in
the discharge current is observed and the beginning of light
emission is likely to coincide with the maximum of the electrical input power. Also, there seems to be a correlation between the beginning of the DC expansion and the beginning
of the light emission which coincides to within ±20 ns. The
second light-emission pulse starts earlier than with Cu wire
explosions and here also the streak image and the photocathode measurements 关see Fig. 9共b兲兴 do not contradict each
other.
Phys. Plasmas 13, 042701 共2006兲
FIG. 9. 共a兲 Typical waveforms of the current and voltage; 共b兲 input electrical
power, emitted light power, and trajectories of the DC boundary and SW; 共c兲
streak camera shadow image of the exploding wire and SW. Al wire
共Ø127 ␮m, lw = 100 mm兲.
In general, all the conclusions listed above are applicable
for explosions of 50 mm long Al and Cu wires 共see Figs. 10
and 11兲. A difference is that in this case the second light
emission begins earlier than with 100 mm long wires. One
can therefore see the transition from the first light emission
pulse to the second light emission pulse in one 1 ␮s frame of
the streak camera. The dip in the radiation with 50 mm long
Cu wire explosions at 0.25– 0.45 ␮s 关see Fig. 10共b兲兴 can also
be observed in the corresponding streak image 关see Fig.
10共c兲兴. Let us note that with 50 mm long Al wires the delay
between the first and second light pulses is virtually absent
关see Figs. 11共b兲 and 11共c兲兴.
The faster rise and higher peak values of P␥ obtained in
Al as compared to Cu wire explosions with the same amount
of deposited electrical energy per atom can be explained by
an exothermic reaction of Al microparticles and surrounding
water. It is known that with the formation of Al2O3 as a
product of combustion of pure Al with oxidizers since a large
amount of energy is released as a result of an exothermic
reaction:
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042701-7
Nanosecond time scale, high power electrical wire...
FIG. 10. 共a兲 Typical waveforms of the current and voltage; 共b兲 input electrical power, emitted light power, and trajectories of the DC boundary and
SW; 共c兲 streak camera shadow image of the exploding wire and SW. Cu wire
共Ø100 ␮m, lw = 50 mm兲.
4Al + 3O2 Þ 2Al2O3 + ⌬H共62 kJ/g兲.
In order to destroy the protective layer of Al2O3 which always exists and covers pure Al, one has to deliver a significant amount of energy to the Al surface. This could be
achieved by the heating process produced by SWs, by strong
radiation, or by contact with other heated bodies having sufficient temperature for ignition, i.e., close to the Al boiling
point −2520 ° C.18 Thus, the first requirement for Al microparticle ignition is a high temperature. The second requirement is to sustain this temperature at the Al surface for a
time period longer than the ignition time tign. The ignition
time is a function of the surface area of the microparticles
and the temperature of the ambient medium. For instance, for
Al microparticles tign can be approximated as:19 tign = 1.668
⫻ 10−2d2 exp共0.913/ T兲关␮s兴, where d is the dimension of the
microparticles measured in ␮m and T is the temperature of
the ambient medium measured in eV. For example,
tign ⬇ 0.1 ␮s for a 1 ␮m particle at T = 0.5 eV. For the submicron particle size predicted in Ref. 17 the ignition time
will be of the order of a few ns, e.g., for 0.2 ␮m size particles at T = 0.5 eV, tign ⬇ 4 ns, and for 0.5 ␮m size particles
at T = 1 eV, tign ⬇ 10 ns. Thus, in our experiment both requirements for combustion of Al microparticles are satisfied
Phys. Plasmas 13, 042701 共2006兲
FIG. 11. 共a兲 Typical waveforms of the current and voltage; 共b兲 input electrical power, emitted light power, and trajectories of the DC boundary and
SW; 共c兲 streak camera shadow image of the exploding wire and SW. Al wire
共Ø127 ␮m, lw = 50 mm兲.
since the estimated temperature of the DC is of the order of
a few eV, and this temperature is sustained for at least a few
hundreds of nanoseconds. The chemical reaction could give
an additional 210 J energy for the Al wire 共Ø127 ␮m,
lw = 100 mm兲. This energy which is almost equal to the electrically deposited energy, will be generated in the DC in the
case of complete combustion of the Al wire.
3. Spectrally resolved light emission properties
Conventional spectroscopic methods based on spectral
line analysis are not applicable to deducing DC parameters
because of the opacity and intense spectral line broadening
caused by the high density of the DC material. Also, we note
the cylindrical geometry and small dimensions of the DC,
and the nonuniform radial distribution of the DC parameters.
In addition, spectral characteristics of the surrounding water
at the extreme pressures achieved in UEWEs have not been
studied well. Nevertheless, it was speculated that the BB
approximation can be used to estimate the surface temperature of the DC.2,20 For example, it was shown that the plume
ejected from the wire edge in electrical wire explosions inside glass capillaries can be characterized as the BB continuum in the range of 400– 600 nm.2 Below 400 nm, the
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042701-8
Phys. Plasmas 13, 042701 共2006兲
Grinenko et al.
observed continuum spectrum was punctuated by occasional
absorption lines. In an earlier experiment of UEWEs 共Ref.
20兲 it was also indicated that the DC radiates with a BB
spectral distribution. At pressures exceeding 2 ⫻ 107 Pa the
irregularities in the continuum spectrum caused by the spectral lines or by groups of spectral lines become almost irresolvable. Pressures estimated in our experimental conditions
exceed this value by at least two orders of magnitude. A
discussion on conditions crucial for BB radiation can be
found in Ref. 20. The first requirement for BB radiation is
that local thermodynamic equilibrium 共LTE兲 for all the species is realized during a time scale that is significantly less
than the typical duration of the process. The second requirement is that particle density be high enough to render the DC
opaque to the radiation. The equilibration time estimated by
Martin20 for underwater spark plasmas on the basis of elastic
collision time between plasma electrons and ions was
⬃0.1 ns. Thus, one can also assume LTE in our experiments.
The opacity in our experiments can be estimated from the
Rosseland mean free path of photons lR calculated with an
ionization equilibrium model. For an ion density of
1021 cm−3 and temperature of a few eV, lR ⬃ 1 ␮m, and
therefore, for wire diameters 艌50 ␮m the DC can be considered to be optically thick.
In order to measure the time-resolved BB spectral distribution, light emission in the UV to the visible spectral range
has been sampled by interference filters 共see Sec. II B 1兲.
Two methods have been used to infer T from the measured
radiation. The first method consisted of finding two parameters of the Planck equation, i.e., the amplitude and the temperature which should best fit the measured experimental
intensity at several wavelengths at each time step. The BB
radiation intensity at wavelength ␭ is:21
I␭共T,A兲 = A
2␲hc2
␭5
1
,
hc
−1
exp
␭kT
冉 冊
共2兲
where I␭共T , A兲 is the BB intensity and A is an amplitude that
depends on geometric factors. Note that A and T are the
unknown parameters that are common to all the measured
wavelengths. At each time step these parameters are found
from the requirement that the mean error between the measured spectrum intensity and BB intensity be minimal. The
mean relative quadratic error is defined as:
N
Error共Tt,At兲 =
1
兺 关共Iex共␭n,t兲 − I␭n共Tt,At兲兲/I␭n共Tt,At兲兴2 ,
N 1
共3兲
where Iex共␭n , t兲 is the measured intensity of the radiation at
the nth wavelength ␭n at time t, and N is the number filters.
The values of Tt and At at time t were determined as Tt and
At for which the error is minimal. This approach eliminates
the necessity of calibrating the optical system when measuring the absolute value of radiation intensity at a given ␭.
Also, the DC emitting area does not enter into the calculation
since only relative intensities of radiation emitted from the
same surface area at each time step are compared. Therefore,
this approach can yield T independent of other measurements
and can be used as an étalon for other T measurement methods.
A second method to deduce T from the spectrally resolved radiation intensity is to determine the absolute intensity at a single wavelength. The radiation power measured by
a photodiode Iex共⌬␭n , t兲 is translated to the radiation power
Iemitted共␭n , t兲 · ⌬␭n per unit surface of the emitting DC, where
Iemitted共␭n , t兲 is the radiation power per unit area and per unit
wavelength emitted from the DC of radius Rw共t兲, length lw at
time t, and ⌬␭n is the transmission bandwidth of the nth
filter. Assuming a square photodiode with edge length hdet
placed at a distance D from the wire at height Zdet relative to
the wire axis, one obtains:
冉 冊
Iemitted共␭n,t兲 ⬇ 2Rref
D
hdet
2
1
1
⫻ Iex共␭n,t兲,
lwRw共t兲 ⌬␭n
where Rref is the attenuation of the light due to reflection
from the water-quartz and quartz-air boundaries. The value
of T␭n was then obtained by reversing the BB radiation
Planck formula:
T␭n共t兲 =
1.2415 ⫻ 103
␭n
冒冉
ln
3.7435 ⫻ 1029
Iemitted共␭n,t兲␭5n
冊
+1 ,
共4兲
where T␭n共t兲 is the temperature in eV at time t deduced from
the radiation intensity at wavelength ␭n 共nm兲. In this approach, the DC radius enters explicitly into the calculation of
T␭n共t兲, and therefore, an error in T␭n共t兲 associated with the
accuracy of the determination of the DC radius has to be
estimated. A numerical check shows that in the temperature
range of a few eV a 10% error in the value of the DC radius
results in ⬃10% error in the value of temperature.
In the forgoing discussion, data obtained using UV filters
were omitted from consideration because of inaccurate photodiode calibration caused by the weak radiation sources
used in this spectral range. In addition, the photodiode signal
amplitudes in the UV range were an order of magnitude
lower than those measured in the visible range. Therefore,
UV signals were more subject to electrical noise than those
measured in the visible spectral range. Also, the optical properties of the water under extreme pressures of the order of
1010 Pa have not been well studied.
An example of an intensity Iex共␭n , t兲 measured by the
photodiode during the explosion of a Cu wire 共Ø100 ␮m,
lw = 100 mm兲 is shown in Fig. 12. One can see that the
peak of the radiation spectrum occurs at ␭ ⬇ 400 nm,
which, according to Wien’s displacement law21 with
T = 250 共eV nm兲 / ␭共nm兲, gives T ⬇ 0.6 eV. However, the
FWHM of this distribution is ⬃200 nm, whereas the FWHM
of the Planck distribution corresponding to T ⬇ 0.6 eV is
⬃550 nm. Therefore, an attempt to fit the full range of the
sampled spectrum, including the UV range, by a BB curve
results in at least a 50% error.
A plot of the DC temperature during the explosion of a
Cu 共Ø100 ␮m, lw = 100 mm兲 wire evaluated by these two
methods and omitting the UV range is shown in Fig. 13. One
can see that the maximum error resulting from the first
method is 艋20%. The error of the second method is a com-
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042701-9
Nanosecond time scale, high power electrical wire...
Phys. Plasmas 13, 042701 共2006兲
FIG. 14. Typical waveform of the current and temperature of the DC surface
during the explosion of Al wire 共Ø127 ␮m, lw = 100 mm兲. The temperature
was deduced from the absolute intensity at 410 nm, 532 nm, and 550 nm.
FIG. 12. Sampled spectrum of radiation from the Cu wire 共Ø100 ␮m,
lw = 100 mm兲.
bination of two factors. The first is the experimental accuracy
of the determination of the various geometrical parameters of
the optical system, the least accurate of which is the radius of
the DC. The second factor is the scatter of the T␭n共t兲 calculated from the absolute intensity radiated at different wavelengths. In the grey plot 共see Fig. 13兲 the error bars were
determined both from scatter of the T␭n共t兲 calculated from all
of the sampled wavelengths in the visible spectral range, and
the 10% error associated with the accuracy of the DC radius
determination. These were added linearly. One can see good
agreement between the DC temperatures calculated according to the BB fit and according to the absolute intensity
radiated within an isolated wavelength range. This result indicates that the DC radiates here with a BB spectrum in the
visible range in the analyzed explosions of Cu wires.
A lesser agreement between temperatures determined
from the absolute intensities at different wavelengths was
obtained with explosions of Ø127 ␮m, lw = 100 mm Al wire.
Results of temperature calculations based on the absolute
intensities at 410 nm, 532 nm, and 550 nm wavelengths are
shown in Fig. 14. The observed deviation from the BB spec-
trum can be attributed to electrical breakdown which is evident from the current restrike waveform. This current restrike is absent with Ø100 ␮m, lw = 100 mm Cu wire
explosions. Nevertheless, the estimated temperature scatter is
艋20% and therefore, the BB approximation may be considered satisfactory.
Deviation from the BB approximation increases significantly for wires of length lw = 50 mm. In Fig. 15, which
shows data from Cu wire 共Ø100 ␮m, lw = 100 mm兲 explosions, the values of T are clustered into two distinct groups.
The first group corresponds to a maximum DC value of
T ⬃ 4.5 eV calculated from the absolute intensity of radiation
measured at the 410 nm, 486 nm, and 488 nm wavelengths,
and the second group corresponds to a maximum DC value
of T ⬃ 7 eV calculated from the absolute intensity of radiation measured at the 514.5 nm, 532 nm, 550 nm, and
488 nm wavelengths. The scatter of T is relatively small
within each of the groups, i.e., 艋10%, whereas the total
scatter is 艌30%. The BB approximation therefore appears to
hold within each of the indicated spectral regions. This suggests existence of two spatial regions characterized by different T’s each radiating a BB spectrum. If the absorption coefficient is higher on the red end of the spectrum than on the
blue end, then the blue radiation should come from the
FIG. 13. Typical waveform of the current and temperature of the DC surface
during the explosion of Cu wire 共Ø100 ␮m, lw = 100 mm兲. The temperature
estimated by the method of fitting of the observed visible spectrum to the
BB intensity distribution is shown with the solid line with error bars. The
temperature deduced from the absolute intensity at 410 nm, 486 nm,
488 nm, 514.5 nm, 532 nm, 550 nm, and 656 nm is shown in the thick grey
curve.
FIG. 15. Typical waveform of the current and temperature of the DC surface
during the explosion of Cu wire 共Ø100 ␮m, lw = 50 mm兲. The temperature
deduced from the absolute intensity at 410 nm, 486 nm, and 488 nm is
shown in the thick dark-grey curve. The temperature deduced from the
absolute intensity at 514.5 nm, 532 nm, 550 nm, and 656 nm is shown in
the thick light grey curve.
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042701-10
Phys. Plasmas 13, 042701 共2006兲
Grinenko et al.
⳵␳ 1 ⳵共r␳v兲
= 0,
+
⳵t r ⳵r
FIG. 16. Typical waveform of the current and temperature of the DC surface
during the explosion of Al wire 共Ø127 ␮m, lw = 50 mm兲.
deeper regions of the DC and the red radiation from the outer
layers of the DC. Correspondingly, there appears to be a
hotter shell composed of plasma with a temperature of
⬃7 eV surrounding the main DC which is composed of the
hot metal vapor that is generated in the products of the wire
explosion. Currently other experimental diagnostics methods
are being developed to investigate the existence of this
plasma shell. Nevertheless, if plasma is generated in the process of UEWEs it is subject to pressures of the order of
1010 Pa, unlike with vacuum corona discharges formed in
wire ablation products.
The existence of a discharge current restrike may be evidence for the existence of an electrical breakdown leading to
the formation of a plasma channel. One can find a correlation
between the amplitude of the second current restrike I2 and
the deviation of the radiation from a BB spectrum. For Cu
wire 共Ø100 ␮m, lw = 100 mm兲 explosions, the restrike was
virtually absent and there was no deviation from BB radiation. In Al wire 共Ø127 ␮m, lw = 100 mm兲 explosions the restrike current was I2 ⬇ 0.25I1 where I1 is the amplitude of the
first current pulse. Here a deviation from BB radiation was
found with a scatter of the calculated T values 艋20%. With
Cu wire 共Ø100 ␮m, lw = 50 mm兲 explosions I2 ⬇ 0.36I1 and
the scatter of T values is ⬃30%. Similarly, for Al wire
共Ø127 ␮m, lw = 50 mm兲 explosions 共see Fig. 16兲 I2 ⬇ 0.32I1
and the scatter of T values is ⬃30%. An additional possible
reason for the difference between Al and Cu spectra may be
the combustion of Al microparticles.
In addition a correlation can be observed between T and
maximum pressure pmax of the DC calculated in the previous
section. Indeed, the maximum pressure on the DC boundary
for a Cu wire 共lw = 50 mm兲 is pmax ⬇ 10⫻ 109 Pa and for a Cu
wire 共lw = 100 mm兲 pmax ⬇ 5 ⫻ 109 Pa. The estimated temperatures for these two cases are 5 – 9 eV and 2.2 eV, respectively.
III. MHD SIMULATION
A one-dimensional, single temperature approximation of
UEWE is considered. In the case of cylindrical geometry,
MHD equations in Lagrange formulation have the following
form:22
共5a兲
␳
⳵v
⳵v
⳵p 1
+ ␳v = −
− j zB ␸ ,
⳵t
⳵r
⳵r c
␳
1 ⳵共rv兲 jz2 1 ⳵
⳵␧
⳵T
⳵␧
+ ␳v = − p
+ +
r␬
,
⳵t
⳵r
r ⳵r
⳵r
␴ r ⳵r
共5b兲
冉 冊
1 ⳵B␸ ⳵Ez
=
;
c ⳵t
⳵r
jz =
c ⳵共rB␸兲
;
4␲r ⳵r
j z = ␴Ez ,
共5c兲
共5d兲
共5e兲
P = P共␧, ␳兲;
T = T共␧, ␳兲,
共5f兲
␴ = ␴共␧, ␳兲;
␬ = ␬共␧, ␳兲,
共5g兲
where ␳ and T are the density and temperature of the material; p and ␧ are the pressure and energy density of the material; ␴ and ␬ are the coefficients of electrical and heat
conductivities; v is the radial component of the velocity; Ez
is the longitudinal component of the electric field; jz is the
longitudinal component of electric current density; and B␸ is
the azimuthal component of the magnetic field. Equations
共5a兲–共5g兲 were solved numerically using a MHD code in
Lagrange mass coordinates. Here an implicit scheme “cross”
has been used to solve the hydrodynamic Eqs. 共5a兲–共5c兲.23
For the solution of the Maxwell equations 共5d兲 together with
Ohm’s law 共5e兲 and also for the solution of heat conductivity,
an implicit “flow chaser” method was used.23 The calculation
grid consisted of two regions, namely, the DC material and
the water. It was assumed that during the discharge water
conductivity is negligibly small, so that all the current flows
through the wire material and the water exerts only a hydrodynamic effect. The boundary condition used for the Maxwell equations was:
B␸共R兲 =
2Iw
,
cR
共5h兲
where R is the DC channel radius and Iw is the current flowing through the DC. The value of the DC current was determined from the simultaneous solution of the Maxwell and
forming line equations. The implicit scheme “cross”23 was
used for the solution of telegraph equations of the forming
line.
The system of MHD equations is completed with an
equation of state 共EOS兲 关Eq. 共5f兲兴 and related transport parameters 关Eq. 共5g兲兴. The EOS of copper was obtained from
More’s theoretical model.24 More’s quotidian equation of
state 共QEOS兲 is a general purpose EOS model in which electronic properties are obtained from a modified ThomasFermi statistical model, while ion thermal motion is described by a multiphase EOS combining Debye, Grüneisen,
Lindemann, and fluid scaling laws. The EOS of water was
taken from the compiled experimental data of Bridgman.25
Electrical conductivity of the DC material was evaluated
by the adjustable semiempirical method described in Refs.
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042701-11
23 and 26. The two fitting parameters in this model are the
value of conductivity at the critical point ␴cr and a parameter
␣ which determines the dependence of the conductivity as a
function of T at solid density. The conductivity found using
this approach is strongly dependent on the state of the matter.
Namely, in the metallic part of phase space the conductivity
decreases with an increase in T, whereas in the plasma part
of the phase space the conductivity increases. The decrease
in metallic conductivity due to Ohmic heating is the process
that finally causes the current to be rapidly interrupted and
the metallic wire to explode. The corresponding values of
heat transfer coefficients are found from the WiedemannFranz law.27 The conductivity dependence in the metallic and
fluid phases is critical for correct calculation of the initial
stage of the wire explosion.
The semiempirical model described in Ref. 26 allows
the specific conductivity ␴共␦ , T兲 to be calculated, where
␦ = ␳ / ␳0 is the relative density, and ␳0 is the normal density.
The model is applicable in the metallic and fluid phases of
the material for ␦ 艌 ␦cr and T ⬍ 3Tcr, where ␦cr and Tcr are the
density and temperature at the critical point. Following this
model, at the first step the conductivity at the normal density
␦ = 1 is calculated as:
log10关␴共1,T兲兴 = log10关␴0共1,T兲 − ␣兵log10关␴0共1,T兲兴
− log10关␴0共1,T0兲兴其兴,
共6兲
where T0 is the room temperature and ␴0共1 , T兲关1 / s兴 for Cu is
given by:
log10共␴0共1,T兲兲 = 14.525 + 2.03e
−共T+0.1462兲/9.864
+ 1.7e−共T+0.1462兲/0.2
+ 130e
共7兲
−共T+0.1462兲/0.03076
and ␣ is a fitting parameter that is determined from a comparison of the calculation and experimental results. Second,
the conductivity in the metallic and liquid states is calculated
as:
冉
ln
冊
冉
冊
␴共␦,T兲
ln共␦兲
␴cr
,
= ⌽共␹,␧兲ln
␴共1,T兲
␴共1,T兲 ln共␦cr兲
共8兲
where ␦cr is the relative density at the critical point, ␧ is the
internal energy of the material at the given temperature and
density and ␹ is a parameter given by:
␹=
Phys. Plasmas 13, 042701 共2006兲
Nanosecond time scale, high power electrical wire...
ln共␦兲
,
ln关␦bin共T兲兴
共9兲
where ␦bin共T兲 is the relative density of the material at the
binodal boundary for temperature T. Function ⌽共␹ , ␧兲 is calculated according to:
⌽共␹,T兲 =
冦
␹ − 共1 − ␹兲
1
冧
␧
for ␦bin 艋 ␦ 艋 1
␧cr
,
otherwise
共10兲
where ␧cr is the critical energy density. The values of the
extrapolated ␦cr, Tcr, critical pressure pcr, and ␴cr for Cu,
according to Ref. 24 are ␳cr = 3.82 g / cm3, Tcr = 0.96 eV and
pcr = 109 Pa, respectively.
FIG. 17. The conductivity of Cu at the critical point vs the parameter ⌿.
The variation of the parameter ␣ is used to tune the
delay between the calculated voltage and the current pulses
to be similar to the delay that was observed in the experiments 共see Fig. 2兲. This parameter has been varied in the
relatively small range of 0.43–0.48. Once ␣ is determined
the value of ␴cr can be found such that the peak values of
experimental and calculated voltage coincide. Thus ␴cr can
be considered as an implicit function of ␣. The dependence
of ␴cr on the parameter ⌿, which is a combination of the
discharge parameters, is shown in Fig. 17. This parameter is
defined as:
⌿=
lw Imax
jmax
⬀
,
2
Rw Vmax Emax
共11兲
where Imax and jmax are the peak discharge current and discharge current density values, respectively, and Vmax and
Emax are the peak discharge voltage and electric field values,
respectively. The values of ⌿ and corresponding values of
␴cr presented in Fig. 17 were obtained from a set of different
experiments and corresponding MHD calculations. One can
see that a good linear correlation between ␴cr and ⌿ is obtained. The reason for this correlation is not yet clear, and it
is a subject for further research.
This approach to the calculation of conductivity certainly lacks the desired universality and cannot be used in
predicting a large range of experimental results. However,
one can suppose that the conductivity of the material cannot
be determined to be independent of the discharge parameters,
such as, current density, electrical power, and electrical
power deposition rate, when typical times of variation of
these parameters become comparable with relaxation times
characterizing the material. For example, for Cu wire
共Ø100 ␮m, lw = 100 mm兲 explosions, a typical lattice relaxation time can be estimated as Rw / cs ⬇ 14 ns, where
cs = 3.75 mm/ ␮s is the sound velocity in the copper at normal conditions, whereas the typical discharge current risetime I / İ ⬇ 30 ns. Since these typical times are of the same
order of magnitude, one can suppose an influence of the
current time variation on lattice vibration and, therefore, on
the value of ␴. Another comparison worth making is the ratio
of the average velocity of current carrying electrons u and to
the sound velocity. Assuming uniform cross-sectional current
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042701-12
Grinenko et al.
FIG. 18. Experimental 共solid兲 and MHD simulation 共dashed兲 results of the
explosion of Cu wire 共Ø100 ␮m, lw = 100 mm兲. 共a兲 Experimental and simulated current, voltage and DC boundary; 共b兲 surface temperature estimated
from absolute radiation intensity and the simulated mass average
temperature.
density distribution, the maximum value of the average electron current velocity is calculated according to:16
I共t兲
⬇ 4 ⫻ 104 cm/s ⬇ 0.11cs ,
u共t兲 ⬇
␲Rw2 nee
where ne is the density of electrons above the Fermi level,
and e is the electron charge. When taking into account skin
effect, the value of u can assume values of the same order of
magnitude as cs. This again points to the necessity of considering a more complex model of conductivity that depends
on current density.
Despite the absence of universal values of conductivities
that would give a satisfactory agreement with a large set of
experimental results, the MHD calculation can be regarded
as a numerical experiment. When combined with the results
of a real experiment, the MHD computation can be used to
retrieve data inaccessible by experimental diagnostics, or to
confirm independently obtained experimental data. On the
other hand, this approach may be regarded as an indirect way
of computing conductivity, which takes into account parameters such as j and dI / dt.
Comparison of the results of the MHD calculation and
experimental results is shown in Fig. 18 and Fig. 19 for
Ø100 ␮m, lw = 100 mm and Ø100 ␮m, lw = 50 mm Cu wire
explosions, respectively. One can see good agreement between the measured and the calculated values of the current,
voltage and DC radius in both cases 关see Figs. 18共a兲 and
19共a兲兴. In Figs. 18共b兲 and 19共b兲 the calculated and measured
values of T are compared. Since the radial depth that contributes to the radiation measured in the experiments is not
Phys. Plasmas 13, 042701 共2006兲
FIG. 19. Experimental 共solid兲 and MHD simulation 共dashed兲 results of the
explosion of Cu wire 共Ø100 ␮m, lw = 50 mm兲. 共a兲 Experimental and simulated current, voltage and DC boundary; 共b兲 surface temperature estimated
from absolute radiation intensity and the simulated mass average
temperature.
exactly known, the calculated average mass temperature
具T典 = 2␲Lw兰R0 w␳Trdr / M w, where M w is the mass of the wire
material, is compared to the temperature obtained in the experiment assuming BB radiation. A small deviation in these
quantities is observed in both of the discharges considered.
For a 50 mm long wire, the value of T was taken from the
blue part of the spectrum. Satisfactory agreement with this
large number of experimental and measured discharge parameters may suggest that the conductivity values used in the
simulation are correct.
An example of the calculated temporal evolution of temperature, density and pressure of the DC and surrounding
water for in Cu wire 共Ø100 ␮m, lw = 100 mm兲 explosions is
shown in Fig. 20. In this figure a variety of effects can be
observed such as: current skinning 关see Fig. 20共a兲兴; pinch
effect 关see Fig. 20共b兲兴; relaxation wave inside the wire following the drop in the magnetic pressure, and consequent
DC expansion 关see Fig. 20共c兲兴; SW generation in surrounding water; relatively little water heating due to the temperature diffusion and due to the SW 关see Figs. 20共a兲–20共c兲兴; and
nonuniform cross-sectional distribution of density, pressure
and T. One can see that the maximum calculated pressure
inside the DC reaches 4 ⫻ 1010 Pa which is above pcr of Cu.
Therefore, when magnetic pressure drops and an explosion
occurs, the wire material transforms into the gas-plasma state
skipping the quasistable states. This means that no microparticles are formed in the first stage of the discharge, and if
they exist they are created at later stages of the DC expansion.
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042701-13
Phys. Plasmas 13, 042701 共2006兲
Nanosecond time scale, high power electrical wire...
sures generated during the explosion. A pressure of up to
1010 Pa was estimated to develop in the water adjacent to a
DC. Under this extreme pressure the variety of processes that
contribute to ionization of DC surroundings are bound to be
suppressed or at least significantly delayed.
In addition, evidence of combustion of Al microparticles
was found, the initiation of which was shown to be possible
with our experimental conditions. This chemical reaction
could contribute the amount of energy of the same order as
the electrically deposited energy. Thus, even if only part of
the Al material is involved in the combustion process, still a
large additional amount of energy would still be generated in
the DC.
Temporally and spectrally resolved measurements of the
radiation from the DC that were carried out are summarized
as follows:
共a兲 For a ns time scale aperiodical UEWE the DC can be
considered as a source of BB radiation in the visible region.
The temperatures estimated in this case are 2.2± 0.2 eV for
Cu wires and 3 ± 0.5 eV for Al wires.
共b兲 For quasiperiodic UEWEs with discharge current restrike, the radiation spectrum deviates from the BB. In this
case, temperatures estimated from the absolute intensity of
radiation of individual wavelengths are in the range of
5 – 9 eV.
MHD simulations of ns time scale UEWEs indicate that
it is possible that the conductivity of the wire material depends on current density or current rise rate, as well as on
instantaneous thermodynamic variables such as temperature
and density. Simulations using appropriate conductivity values demonstrate good quantitative and qualitative agreement
with electrical and optical diagnostics of fast explosions of
Cu wires.
ACKNOWLEDGMENTS
FIG. 20. Simulated cross-sectional distributions of 共a兲 temperature; 共b兲 density; 共c兲 pressure at different times of Cu wire 共Ø100 ␮m, lw = 100 mm兲
explosion.
IV. CONCLUSIONS
It was demonstrated that in the ns time scale UEWEs
with dI / dt 艋 500 A / ns, energy deposition by Joule heating
significantly surpasses the atomization enthalpy of material
by the extremely high factor of ⬃60. This significantly exceeds the overheating factor of 20 achieved with vacuum
explosion of polyimide-coated wire with dI / dt ⬍ 150 A / ns,15
which is the best reported result obtained for vacuum wire
explosions, and an overheating factor of 11 achieved in the
case of ␮s time scale UEWEs with a dI / dt ⬍ 50 A / ns generator and electrical input power ⬍0.6 GW.11,28,29 These data
show that one can significantly improve the efficiency of
energy deposition for generation of high density plasma by
combining the advantages of wire explosion in breakdownimpeding surroundings and increasing dI / dt.
A possible reason for the increased energy deposition
into the DC is the enhanced impedance of the surrounding
water to electrical breakdown due to the extremely high pres-
We wish to thank Andreas Kemp for generously providing the MPQEOS code for the calculation of the equation of
state based on the QEOS model 共Ref. 24兲.
This research was supported by the Israel Science Foundation Grant No. 1210/04. The work of V. Oreshkin was
supported by the RFBR Grant No. 05-02-16845.
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