Current and internal phase distribution in an exploding wire until ionization: modeling and experiments

Abstract

An exploding wire discharge begins with a large electrical current that forces the wire metal to melt, vaporize, and later ionize to produce the plasma. Using a combination of modeling and experimental data, in this work it is shown that phase changes at these early stages of the exploding wire process are strongly dependent on metal properties, affecting wire energy dynamics. Simultaneously, current and voltage signals can be very similar regardless of internal phase distribution, which depends strongly on the metal of the wire.

Graphical Abstract

Appendices

Appendix A: Equations and solving methods of the employed simulation

From the previous 3D finite volume, Arbitrary Lagrangian–Eulerian code  [26,27,28,29], a simplified version (one-fluid, one-temperature) was used for the study of the initial stages of the discharge, previous to plasma formation. The equations are

Continuity

$$\begin{aligned} \frac{\partial \rho }{\partial t}+\nabla \cdot \left( \rho \mathbf {u}\right) =0 \end{aligned}$$

Momentum

$$\begin{aligned} \frac{\partial \left( \rho \mathbf {u}\right) }{\partial t}+\mathbf {\nabla }\cdot \left( \rho \mathbf {uu}\right) =-\nabla p+\mathbf {j} \times \mathbf {B}-\nabla \cdot \mathbf {\Pi } \end{aligned}$$

Internal energy

$$\begin{aligned} \frac{\partial \left( \rho \varepsilon \right) }{\partial t}+\mathbf {\nabla }\cdot \left( \rho \varepsilon \mathbf {u}\right) =-p\nabla \cdot \mathbf {u-\nabla }\cdot \mathbf {q}-\mathbf {\Pi :\nabla v} \end{aligned}$$

Faraday’s Law

$$\begin{aligned} \frac{\partial \mathbf {B}}{\partial t}=-\nabla \times \mathbf {E} \end{aligned}$$

Ampere’s Law (neglecting displacement current)

$$\begin{aligned} \nabla \times \mathbf {B}=\mu \ \mathbf {j} \end{aligned}$$

Ohm’s Law

$$\begin{aligned} \mathbf {E}^{\prime }=\eta \mathbf {j} \end{aligned}$$

Lorentz’s transformation

$$\begin{aligned} \mathbf {E}=\mathbf {E}^{\prime }-\mathbf {u}\times \mathbf {B} \end{aligned}$$

Fourier’s law

$$\begin{aligned} \mathbf {q}=-\kappa \nabla T \end{aligned}$$

where $\rho$ is the mass density, t is the time, $\mathbf {u}$ the fluid velocity, p the pressure, $\varepsilon$ the specific internal energy, T the temperature, $\mathbf {q}$ the heat flux, $\mathbf {\Pi }$ the viscosity tensor, $\mathbf {E}$ ($\mathbf {E}^{\prime }$) the electric field in the laboratory (fluid) frame, $\mathbf {B}$ the magnetic field, $\mathbf {j}$ the current density, $\eta$ the electrical resistivity, $\kappa$ the thermal conductivity, and $\mu$ the magnetic permeability. These equations are coupled to an external RLC circuit, which is modeled by

$$\begin{aligned} \left( L_{0}+L_{b}\right) \frac{dI}{dt}+V_{sg}+V_{wire}+\left( R_{0} +\frac{dL_{b}}{dt}\right) I+\frac{q}{C_{0}}=V_{0}\;, \end{aligned}$$

where $L_{0}$, $R_{0}$ and $C_{0}$ are the lumped circuit inductance, resistance and capacity, respectively, $V_{sg}$ the voltage drop at the spark-gap, $V_{wire}$ the “resistive” term of the wire using a path along the boundary, $L_{b}$ the correspondent inductance of that path (see [41] for a discussion on the interplay of resistive and inductive terms along various integration paths in an extensive conductor), $V_{0}$ the initial voltage, I the current, and q the circulating charge defined from (note that the charge of the capacitor is $Q=Q_{0}-q$ being $Q_{0}$ the initial charge)

$$\begin{aligned} I=\frac{dq}{dt}\;. \end{aligned}$$

Therefore, the azimuthal magnetic field at the wire boundary, at a given axial position z, and time t, $B_{b}\left( t,z\right)$, is

$$\begin{aligned} B_{b}\left( t,z\right) =\frac{\mu \; I}{2 \; \pi \; r_{b}\left( t,z\right) }\;, \end{aligned}$$

where $r_{b}\left( t,z\right)$ is the radius of the wire at that given z position and time t.

For the time integration of the coupled system of equations we use the segregated solution method, where each differential equation is solved sequentially in isolation. This means that during the integration of each equation only its dependent variable varies, while the others are frozen. Being nonlinear equations, each equation may require some iterative method. These iterations are called inner iterations. For the treatment of the coupling between the various equations, the complete sequence of all equations is included within a larger iteration cycle, called the outer iterations (Picard iteration). The internal iterations of each equation in this algorithm continue until partial convergence has been reached, typically around three orders of magnitude. It is not worth reducing the residuals to tighter tolerances at this stage because the outer iteration must anyway be repeated until convergence at typically six orders of magnitude.

Actually, rather than dividing the coupled system by equations, we do a separation by physical phenomenon. That is, each time sub-step may correspond to a reduced system of coupled equations, for which well-established integration methods are known. Using Runge–Kutta for the circuit, predictor-corrector method for solving the hydrodynamics, and ADI methods for diffusion, numerical calculation proceeds as follows

• 1. Circuit

• 2. Magnetic diffusion/Joule heating

• 3. Thermal diffusion

• 4. Viscous heating

• 5. Hydrodynamics

• 6. Vertex movement. The volume is obtained from the vertices, and the density is obtained from the volume using the fact that the mass is constant

When the mesh becomes highly distorted a remapping procedure is automatically started. Notice that this was not needed for the present calculations on the initial stage of the exploding wire due to the small deformation attained.

Appendix B: Estimation of time necessary to reach a certain phase

The estimation of the time to reach a certain phase for a wire is performed as follows. Consider the case of an evaporation front propagating in a cylinder, formed by the solid condensed state of the metal with length $l_w$. If this evaporation front is constant along the cylinder axis, and moving at an uniform radial speed v toward the axis center, which following the cylinder symmetry holds, then the moving front evaporates a volume $\mathcal {V}$ at a temporal rate given by

$$\begin{aligned} d\mathcal {V}=2\pi \;r\; v\; l_{w}\; dt\;, \end{aligned}$$

(6)

where r is the front position and $l_w$ wire length. Since v is defined positive toward the axis, it holds $v = -dr/dt$.

Assuming that when the material reaches a gas state, its resistivity grows so steep as to diffuse instantly the electrical current into the inner condensed state surrounded by the gas, Joule heating of the gas phase is then halted. Therefore only the solid, condensed phase, in the central part of the wire is being heating by Joule effect. Calling $e_g$ the specific energy (per unit mass) needed to evaporate the condensed phase, on the one hand, the energy per unit time required to evaporate a volume $\mathcal {V}$ is

$$\begin{aligned} dE_{g}=\rho \; e_{g}\; 2\pi \; r\; v\; l_{w}\; dt\;, \end{aligned}$$

(7)

being $\rho$ the solid mass density. On the other hand, the Joule power delivered to the wire by the electrical current can be written as

$$\begin{aligned} P_{J}=\frac{V^{2}}{R}\;, \end{aligned}$$

(8)

where V is the voltage drop across the wire and R an equivalent resistance. Notice that this equation considers two hypothesis that not always hold in an exploding wire discharge: on one hand, the voltage V must be constant and on the other hand, there must exist an equivalent resistance R to the solid part of the exploding wire. Former assumption can be considered in the dark pause, precisely characterized by the constant values of voltage. Concerning the later, as electrical current is small, magnetic field flux is null or negligible, so the use of a lumped resistance R to represent the solid central part of the exploding wire holds.

Considering now current density uniform in the condensed part that remains in the center of the wire, we have

$$\begin{aligned} R=\frac{\eta \; l_{w}}{\pi \, r^{2}}\;, \end{aligned}$$

(9)

where $\eta$ is the electrical resistivity of the condensed phase (actually it is a mean value for both space and time). Therefore, substituting in Joule power the lumped resistance, Joule energy delivered in dt converts in

$$\begin{aligned} P_J = \frac{dE_J}{dt} = \frac{V^2}{R} \rightarrow dE_{J}=\frac{V^{2}\; \pi \; r^{2}}{\eta \; l_{w}}dt\;, \end{aligned}$$

(10)

which must equate the vaporization energy given in (7), then

$$\begin{aligned} 2\rho \;e_{g}\;l_{w}\;v=\frac{V^{2}r}{\eta \;l_{w}}\;. \end{aligned}$$

(11)

From this equation, it follows that the evaporation front speed is proportional to the radius of the remaining condensed matter, and, therefore, it will slow down as r decreases. Recalling that $v=-dr/dt$ we get for the radius of the remaining condensed matter over time

$$\begin{aligned} r=r_{0}\exp \left( -\frac{t-t_{0}}{\tau }\right) \;; \end{aligned}$$

(12)

being $r_0$ the position of the front at time instant $t_0$, and $\tau$ a characteristic time given by

$$\begin{aligned} \tau =\frac{2\rho \;e_{g}\;\eta \;l_{w}^{2}}{V^{2}}\;. \end{aligned}$$

(13)

Notice that the phase change front velocity is not constant, but exponentially decaying with time, as our calculations predict and the measurements confirm.

Accordingly, resistance (9) becomes

$$\begin{aligned} R=\frac{\eta \; l_{w}}{\pi \;r_{0}^{2}}\exp \left( 2\frac{t-t_{0}}{\tau }\right) \;; \end{aligned}$$

(14)

the current

$$\begin{aligned} I=\frac{\pi \; r_{0}^{2}\;V}{\eta \; l_{w}}\exp \left( -2\frac{t-t_{0}}{\tau }\right) \;. \end{aligned}$$

(15)

Remember that, according to the above simplified model, the current approaches to zero in the dark pause as we have assumed that current circulates only in the condensed phase. This is equivalent to assume an infinite resistivity for the gas phase, which is in principle not physical. But if resistivity of the gas phase was different from zero, the current would instead converge to an asymptotic value. In a similar fashion, the melting front separates the solid phase from the liquid phase, which have different electrical conductivity, making the current exponentially decreases toward an asymptotic value determined by some ratio between conductivity of the different phases.

Still, according to (13), time required for the electrical discharge to reach any phase transition depends mainly of wire geometry and thermophysical properties of the material; also the time until ionization starts into the wire material, experimentally established in this work as explained in the main body of the work. Wire geometry is well measured for these experiments, so the difference between simulated and experimental data is mainly due to the uncertainty on our estimations for the material properties of the metals.

Concerning the resistance, notice that actually it must be calculated over the region in which the current flows, which is not always the whole wire volume due to the already mentioned skin effect. Two limiting cases are considered: current circulating through the entire section of the wire, and current circulating only over an annular section whose width is given by the skin depth, that is:

$$\begin{aligned} R=\left\{ \begin{array}{l} \frac{\eta \; l}{\pi \; r^{2}}\quad \quad \quad \quad \quad \text {if current is uniform in the wire.}\\ \\ \frac{\eta \; l}{2\pi \;r\;\delta }\quad \quad \quad \text {if current circulates in a skin depth }\delta . \end{array} \right. \end{aligned}$$

(16)

Also, during the dark pause, the voltage across the wire is a fraction of the initial voltage at the capacitors bank, $V_{0}$, that can be written as

$$\begin{aligned} V=\frac{Z_{w}}{Z_{w}+Z_{c}}V_{0}\equiv \upalpha V_{0}\,, \end{aligned}$$

(17)

where $Z_{w}$ is the impedance of the wire plus a small part of the circuit (mainly wire holders), the points where the voltage is measured, and $Z_{c}$ impedance from the rest of the circuit. Note that $\upalpha$ is a complicated function of the parameters since, for example, $Z_{w}$ is not constant between shots at different voltages nor in time during a discharge. Also, contact resistance between wire and supporting electrode may vary with voltage, and time, so value of $Z_w$ is altered. Current distribution varies with time too, further modifying $\upalpha$.

Taking all the above into account, an estimation of the characteristic time to reach a given phase, ph, as a function of the initial parameters, for the two limiting cases in resistance of eq. (16), gives

$$\begin{aligned} \Delta t_{ph}\gtrsim \left\{ \begin{array}{l} 2\frac{l_{w}^{2}}{\alpha ^{2}\;V_{0}^{2}}\left( \rho \; e_{ph}\;\eta \right) \quad \text {if current is uniform in the wire.}\\ \\ \frac{r_{b}}{\delta }\frac{l_{w}^{2}}{\alpha ^{2}\;V_{0}^{2}}\left( \rho \;e_{ph}\;\eta \right) \quad \text {if current circulates in a skin depth }\delta . \end{array} \right. \end{aligned}$$

(18)

For a given material the term in parentheses is constant, so this relationship makes it possible to estimate the dependence of time until reaching a given phase (determined by the change in energy) as a function of the geometric parameters of the wire (radius $r_{b}$ and length $l_w$) and the initial voltage $V_{0}$.