**2.1 Metallic foil electrical explosion**

Here we introduce the model of metallic foil electrically exploding driving highvelocity flyers to describe the physical process of electrical explosion of metallic foil shown in Fig.1. A large pulsed current is released to the metallic foil of the circuit, which is produced by a typically pulsed power generator. The circuit can be described by *R*-*C*-*L* electrical circuit equations[15]. During the circuit, the metallic foil is with larger resistance than that of other part, so the energy is mainly absorbed by the metallic foil, and then the physical states of metallic foil change with time. Fig.2 shows the typical current and voltage histories between metallic aluminum foil during the discharging process of pulsed power generator.

Fig. 1. The model of metallic foil electrically exploding driving highvelocity flyers.

compression was firstly presented by J.R. Asay at Sandia National Laboratory[10]. In last decade, this planar loading technique had being developed fastly and accepted by many researchers in the world, such as France[11], United Kingdom[12],and China[13]. As J.R. Asay said, it will be a new experimental technique widely used in shock dynamics, astrophysics, high energy density physics, material science and so on. The process of magnetically driven quasi-isentropic compression is typical magnetodynamics[14], which refers to dynamic

As described above, the electrical explosion of metallic foil and magnetically driven quasiisentropic compression is typically magnetohydrodynamic problem. Although it develops fastly and maybe many difficulties and problems exist in our work, we present our important and summary understanding and results to everyone in experiments and simulations of electrical explosion of metallic foil and magnetically driven quasi-isentropic

In the following discussions, more attentions are paid to the physical process, the experimental techniques and simulation of electrical explosion of metallic foil and

Here we introduce the model of metallic foil electrically exploding driving highvelocity flyers to describe the physical process of electrical explosion of metallic foil shown in Fig.1. A large pulsed current is released to the metallic foil of the circuit, which is produced by a typically pulsed power generator. The circuit can be described by *R*-*C*-*L* electrical circuit equations[15]. During the circuit, the metallic foil is with larger resistance than that of other part, so the energy is mainly absorbed by the metallic foil, and then the physical states of metallic foil change with time. Fig.2 shows the typical current and voltage histories between

**2. Physical process of metallic foil electrical explosion and magnetically** 

metallic aluminum foil during the discharging process of pulsed power generator.

Fig. 1. The model of metallic foil electrically exploding driving highvelocity flyers.

compression, magnetic field diffusion, heat conduction and so on.

magnetically driven quasi-isentropic compression.

**driven quasi-isentropic compression** 

**2.1 Metallic foil electrical explosion** 

compression in last decade.

Fig. 2. The typically discharging current and voltage histories between bridge Aluminum foil.

According to the density changing extent of metallic foil when the first pulsed current flows through it, the whole process of electrical explosion of metallic foil can be classified to two stages. The initial stage includes the heating stage , the melting stage and the heating stage of liquid metal before vaporizing. During this process, the density of metallic foil changes relatively slow. The second stage includes the vaporizing stage and the following plasma forming. The typical feature of electrical explosion of metallic foil is that the foil expands rapidly and violently, and that the resistance increases to be two or more orders than that of initial time (*R*/*R*0~100). The resistance increases to be maximum when the state of metallic foil is at the vaporizing stage. During this stage, the voltage of between foil also increases to be maximum, and then the breakdown occurs and the plamas is forming. The inflection point of the discharging current shown in Fig.2 exhibits the feature.

At the initial satge, the expansion of metallic foil is not obvious, and the change of physical states can be described with one thermodynamic variable *T* (temperature) or specific enthalpy. The energy loss of the interaction between the foil and the ambient medium can be neglected when there is no surface voltaic arcs. Therefore, some assumptions can be used to simplify the problem. We can think that the heating of the metallic foil is uniform and the instability, heat conduction and skin effect can not be considered at initial stage. For this stage, the physical states of metallic foil vary from solid to liquid, and the model of melting phase transition can be used to described it well[16].

For the second stage, the physical states varies from liquid to gas, and then from gas to plasma. There are several vaporizing mechanisms to describe this transition, such as surface evaporation and whole boil[16]. The rapid vaporizing of liquid metal make its resistance increases violently, and the current decreases correspondingly. At this time, the induction voltage between bridge foil increases fastly. If the induction voltage can make the metallic vapor breakdown and the plasma is formed, the circuit is conducted again. Of course, the

Magnetohydrodynamics of Metallic Foil Electrical

Explosion and Magnetically Driven Quasi-Isentropic Compression 351

1

( )0 <sup>2</sup>

;v ( )

*xu x u q*

v

1

*ppT T T*

(v, ); (v, ); (v, )

Where, -symmetric exponent(for metallic wire or cylindrical foil =2,and for planar

magnetic field;*E*-axial component of electrical field; *j*-current density; *Q*V-specific power of Joule heating; *p*-artificial viscosity coefficient;*u*-transverse moving

For this apparatus, the discharging ciruit is a typical *RCL* circuit, which can be expressed by

0 0

*<sup>d</sup> L L I U RI U*

() ;

*foil foil c*

'

0 is the vacuum magnetic permeability, *k* is a coefficient related with the length *l*

*foil* (3)

In the equation (2), when the time *t*=0, the primary current and voltage *I*(0)=0 and *U*c(0)= *U*0, *C*0 and *U*0 are the capacitance and charging voltage of capacitor or capacitor bank, *L*0 and *R*0 are the inductance and efficient resistance of circuit, *U*foil is the voltage between the ends of metallic foil, which is related with the length *l*foil of metallic foil and the magetic field of the space around the foil. the dynamic inductance *L*foil can be obtained by equation (3).

0 00 () ( / ) *L t kl b x X foil*

The concept of magnetically driven quasi-isentropic compression is illustrated in Fig.4. A direct short between the anode and cathode produces a planar magnetic field between the conductors when a pulsed current flows through the electrodes over a time scale of 300~ 800ns. The interaction between the current (density *J*) and the induction magnetic field

and width *b* of metallic foil. *x* is the expanding displacement of metallic foil.

0

*C*

*dU I dt C*

*dt*

 

 

**2.2 Magnetically driven quasi-isentropic compression** 

*foil foil*

*U l EtXt*

;

~ , () ,

 


v/x; q-Lagrange mass coordinate;*B*-transverse component of

1

*ux pp q pp Q d E x B dt q*

( )v

( v)

1

 

 

foil =1); /q=x1-

velocity;*p*-pressure;

equation(2)below.

Where  0

 

v

v ; v

*E xB q j E Q jE*

<sup>1</sup> ( )

2

*B*

 

0

(1)

(2)

breakdown of metallic vapor needs some time, which is called relaxation time as shown in Fig.3. For different charging voltages, the relaxation time varies, which can be seen from the experimental current hostories in Fig.3.

Fig. 3. The breakdown relaxation time shown in the discharging current histories at different charging voltage for the pulsed power generator.

One important application of the electrical explosion of metallic foil is to launch highvelocity flyers with the rapid expansion of tha gas and plasma from electrical explosion of metallic foil. Some metallic materials are with good conductivity and explosion property, such as gold, silver, copper, aluminum and so on. The experimental results[17] show that the aluminum foil is the best material for the application of metallic foil electrically exploding driven highvelocity flyers. There are many models used to describe the process, such as eletrical Gurney model[18], Schmidt model[19] and one dimensional magnetohydrodynamic model[20]. The electrical Gurney model and Schmidt model are two empirical models which are derived from energy conservation equation based on some assumptions. For a specific electrical parameters of the circuit of some apparatus, the electrical Gurney model can be used to predict the final velocity of the flyers when the Gurney parameters are determined based on some experimental results. And the Schmidt model can be used to predict the velocity history of the flyers because the Gurney energy part is substituted with an energy part with the function of time, which is depended on the measured current and voltage histories between bridge foil to correct the specific power coefficient. These two models can't reflect other physical variables of electrical explosion of metallic foil except the velocity of the flyer. Therefore, a more complex model is put forward based on magnetohydrodynamics, which considers heat conduction, magnetic pressure and electrical power. The magnetohydrodynamic model can well reflect the physical process of electrical explosion of metallic foil. The equations are given below[16,20].

$$\begin{cases} \dot{\mathbf{x}} = \mathbf{u}; \ \dot{\mathbf{v}} = \frac{\mathcal{\mathcal{E}}}{\mathcal{\mathcal{E}}\eta} (\mathbf{x}^{\mathcal{I}^{-1}}\mathbf{u}) \\\\ \dot{\mathbf{u}} + \mathbf{x}^{\mathcal{I}^{-1}}\frac{\mathcal{\mathcal{E}}}{\mathcal{\mathcal{E}}\eta} (p + p\_{o\bullet} + \frac{\mathbf{B}^{2}}{2\mu\_{0}}) = \mathbf{0} \\ \dot{\mathbf{e}} + (p + p\_{o\bullet})\dot{\mathbf{v}} = Q\_{\mathbf{v}} \\ \frac{d}{dt}(\mathbf{x}^{1-\mathcal{I}} \mathbf{v}\ \mathbf{B}) = \frac{\mathcal{E}\mathcal{E}}{\mathcal{\mathcal{E}}\eta} \\ E = \frac{1}{\mu\_{0}\sigma\mathbf{v}} \frac{\mathcal{E}}{\mathcal{\mathcal{E}}q} (\mathbf{x}^{\mathcal{I}^{-1}}\mathbf{B}) \\ j = \sigma E; \ Q\_{\mathbf{v}} = \mathbf{v}\ \mathbf{j}\mathbf{E} \\ p = p(\mathbf{v}, T); \ \mathbf{e} = \mathbf{e} \ (\mathbf{v}, T); \quad \sigma = \sigma(\mathbf{v}, T) \end{cases} \tag{1}$$

Where, -symmetric exponent(for metallic wire or cylindrical foil =2,and for planar foil =1); /q=x1 v/x; q-Lagrange mass coordinate;*B*-transverse component of magnetic field;*E*-axial component of electrical field; *j*-current density; *Q*V-specific power of Joule heating; *p*-artificial viscosity coefficient;*u*-transverse moving velocity;*p*-pressure;-internal energy; *v*-unit volume; -conductivity. For this apparatus, the discharging ciruit is a typical *RCL* circuit, which can be expressed by

equation(2)below.

350 Hydrodynamics – Advanced Topics

breakdown of metallic vapor needs some time, which is called relaxation time as shown in Fig.3. For different charging voltages, the relaxation time varies, which can be seen from the

Fig. 3. The breakdown relaxation time shown in the discharging current histories at different

One important application of the electrical explosion of metallic foil is to launch highvelocity flyers with the rapid expansion of tha gas and plasma from electrical explosion of metallic foil. Some metallic materials are with good conductivity and explosion property, such as gold, silver, copper, aluminum and so on. The experimental results[17] show that the aluminum foil is the best material for the application of metallic foil electrically exploding driven highvelocity flyers. There are many models used to describe the process, such as eletrical Gurney model[18], Schmidt model[19] and one dimensional magnetohydrodynamic model[20]. The electrical Gurney model and Schmidt model are two empirical models which are derived from energy conservation equation based on some assumptions. For a specific electrical parameters of the circuit of some apparatus, the electrical Gurney model can be used to predict the final velocity of the flyers when the Gurney parameters are determined based on some experimental results. And the Schmidt model can be used to predict the velocity history of the flyers because the Gurney energy part is substituted with an energy part with the function of time, which is depended on the measured current and voltage histories between bridge foil to correct the specific power coefficient. These two models can't reflect other physical variables of electrical explosion of metallic foil except the velocity of the flyer. Therefore, a more complex model is put forward based on magnetohydrodynamics, which considers heat conduction, magnetic pressure and electrical power. The magnetohydrodynamic model can well reflect the physical process of electrical explosion of metallic foil. The

experimental current hostories in Fig.3.

charging voltage for the pulsed power generator.

equations are given below[16,20].

$$\begin{cases} \frac{d}{dt} \left[ (L\_0 + L\_{foli})I \right] + \mathcal{U}\_{foli} + R\_0 I = \mathcal{U}\_c; \\ \frac{d\mathcal{U}\_C}{dt} = -\frac{I}{C\_0}; \\ \mathcal{U}\_{foli} \supseteq I\_{foli} \mathcal{E} \left[ t, \mathcal{X}(t) \right] \end{cases} \tag{2}$$

In the equation (2), when the time *t*=0, the primary current and voltage *I*(0)=0 and *U*c(0)= *U*0, *C*0 and *U*0 are the capacitance and charging voltage of capacitor or capacitor bank, *L*0 and *R*0 are the inductance and efficient resistance of circuit, *U*foil is the voltage between the ends of metallic foil, which is related with the length *l*foil of metallic foil and the magetic field of the space around the foil. the dynamic inductance *L*foil can be obtained by equation (3).

$$L\_{foil}(t) = \mu\_0 k (l\_{foil} \,/\, b) \Big[ \, \mathbf{x}\_0^{\prime} - \mathbf{X}\_0 \, \Big] \tag{3}$$

Where 0 is the vacuum magnetic permeability, *k* is a coefficient related with the length *l* and width *b* of metallic foil. *x* is the expanding displacement of metallic foil.

#### **2.2 Magnetically driven quasi-isentropic compression**

The concept of magnetically driven quasi-isentropic compression is illustrated in Fig.4. A direct short between the anode and cathode produces a planar magnetic field between the conductors when a pulsed current flows through the electrodes over a time scale of 300~ 800ns. The interaction between the current (density *J*) and the induction magnetic field

Magnetohydrodynamics of Metallic Foil Electrical

conductivity of electrodes and

developed by ourselves.

Where 

Explosion and Magnetically Driven Quasi-Isentropic Compression 353

 is thermal conducitivity. Similar to the technique of electrical explosion of metallic foil, the large current is also produced by some pulsed power generators, for example, the ZR facility at Sandia National Laboratory can produce a pulsed current with peak value from 16 MA to 26 MA and rising time from 600 ns to 100 ns[23]. In the following part, we will introduce the techniques of magnetically driven quasi-isentropic compression based on the pulsed power generators

**3. Techniques of metallic foil electrically exploding driving highvelocity flyers** 

The techniques of metallic foil electrically exploding driving highvelocity flyers and magnetically driven quasi-isentropic compression have been widely used to research the dynamic properties of materials and highvelocity impact phenomena in the conditions of shock and shockless(quasi-isentropic or ramp wave) loading. By means of these two techniques, we can know the physical, mechnical and thermodynamic properties of materials over different state area (phase space), such as Hugoniot and off-Hugoniot states.

As descibed above, the high pressure gas and plasma are used to launch highvelovity flyer plates, which are produced from the electrical explosion of metallic foil. The working principle diagram of the metallic foil electrically exploding driving highvelocity flyers is presented in Fig.5. Usually we choose the pure aluminum foil as the explosion material because of its good electrical conductivity and explosion property. The flyers may be polyester films, such as Mylar or Kapton, or complex ones consisted of polyester film and metallic foil. The material of barrel for accelerating the flyers may be metals or non-polyester films, such as Mylar or Kapton, or complex ones consisted of polyester film and metallic foil. The material of barrel for accelerating the flyers may be metals or non-metals, such as

Fig. 5. The diagram of working principle of metallic foil electrically exploding driving flyer.

*p* is pressure, *q* is artificial viscosity pressure, *e* is specific internal energy,

**and magnetically driven quasi-isentropic compression** 

**3.1 Metallic foil electrically exploding driving highvelocity flyers[24,25,26]** 

m is mass density of electrodes, *u* is velocity, J is current density, *B* is magnetic field,

is electrical

Fig. 4. The principle diagram of magnetically driven quasi-isentropic compression.

*B* produces the magnetic pressure ( *J B* ) proportional to the square of the field. The force is loaded to the internal surface that the current flows through. The loading pressure wave is a ramp wave, which is a continuous wave. Compared with the shock wave, the increment of temperature and entropy is very lower. However, because of the effects of viscosity and plastic work, the sample can't turn back to the original state after the loading wave. That is to say, in solids the longitudinal stress differs from the hydrostatic pressure because of resolved shear stresses that produce an entropy increase from the irreversible work done by deviator[21, 22]. For this reason, the ramp wave loading process is usually assumed to be quasi-isentropic compression. Besides the loading force is magnetic pressure, it is called magnetically driven quasi-isentropic compression.

In order to produce high pressure, the amplitude of the current is ususally up to several megamperes or tens of megamperes. Because of the effects of Joule heating and magnetic field diffusion, the physical states of the loading surface will change from solid to liquid, and to gas and plasma. And these changes will propagate along the thickness direction of the electrodes originated from the loading surface. These phenomena are typically magnetohydrodynamic problems. In order to describe the physical process, the equation of magnetic field diffusion is considered besides the equations of mass, momentum and energy. The magnetohydrodynamic equations are presented below.

$$\begin{cases} \frac{\partial \rho\_m}{\partial t} + \nabla \cdot \left(\rho\_m \vec{u}\right) = 0\\ \rho\_m \frac{d\vec{u}}{dt} + \nabla \left(p + q\right) - \vec{f} \times \vec{B} = 0\\ \frac{de}{dt} + \left(p + q\right) \frac{d\left(1 \mid \rho\_m\right)}{dt} - \dot{e}\_D = 0\\ \rho\_m \frac{d}{dt} \left(\frac{\vec{B}}{\rho\_m}\right) - \left(\vec{B} \cdot \nabla\right)\vec{\mu} = -\nabla \times \left[\frac{\eta}{\mu\_0} \left(\nabla \times \vec{B}\right)\right] \\\\ \vec{f} = \sigma \vec{E} = \frac{1}{\mu\_0} \nabla \times \vec{B} \\ \vec{\mu} = \frac{d\vec{\chi}}{dt}, \dot{e}\_D = \kappa \nabla T \end{cases} \tag{4}$$

Fig. 4. The principle diagram of magnetically driven quasi-isentropic compression.

loaded to the internal surface that the current flows through. The loading pressure wave is a ramp wave, which is a continuous wave. Compared with the shock wave, the increment of temperature and entropy is very lower. However, because of the effects of viscosity and plastic work, the sample can't turn back to the original state after the loading wave. That is to say, in solids the longitudinal stress differs from the hydrostatic pressure because of resolved shear stresses that produce an entropy increase from the irreversible work done by deviator[21, 22]. For this reason, the ramp wave loading process is usually assumed to be quasi-isentropic compression. Besides the loading force is magnetic pressure, it is called

P

P

In order to produce high pressure, the amplitude of the current is ususally up to several megamperes or tens of megamperes. Because of the effects of Joule heating and magnetic field diffusion, the physical states of the loading surface will change from solid to liquid, and to gas and plasma. And these changes will propagate along the thickness direction of the electrodes originated from the loading surface. These phenomena are typically magnetohydrodynamic problems. In order to describe the physical process, the equation of magnetic field diffusion is considered besides the equations of mass, momentum and

) proportional to the square of the field. The force is

*B* produces the magnetic pressure ( *J B*

magnetically driven quasi-isentropic compression.

energy. The magnetohydrodynamic equations are presented below.

*<sup>m</sup> <sup>m</sup>*

*de d*

*m*

*t*

*m*

*dt*

0

,

*D*

*dx ue T dt*

1

*m*

*JE B*

*u*

0

(1 / ) <sup>0</sup>

*<sup>m</sup> <sup>D</sup>*

 

*d B B u <sup>B</sup>*

*pq e dt dt*

*du pq JB dt*

0

0

(4)

  Where m is mass density of electrodes, *u* is velocity, J is current density, *B* is magnetic field, *p* is pressure, *q* is artificial viscosity pressure, *e* is specific internal energy, is electrical conductivity of electrodes and is thermal conducitivity.

Similar to the technique of electrical explosion of metallic foil, the large current is also produced by some pulsed power generators, for example, the ZR facility at Sandia National Laboratory can produce a pulsed current with peak value from 16 MA to 26 MA and rising time from 600 ns to 100 ns[23]. In the following part, we will introduce the techniques of magnetically driven quasi-isentropic compression based on the pulsed power generators developed by ourselves.
