1 I. INTRODUCTION

The Advanced Liner Technology (ALT)-3 device (Fig. 1) has been designed^1–4 to test the efficiency of the magnetic implosion of a cylindrical Al liner as an impactor driven up to 20 km/s by an azimuthal magnetic field B_φ ∼ 6 MG (current ∼70 MA). In the ALT-1,2 experiments⁵ with a similar device and a 10-module disk explosive magnetic generator (DEMG), the same Al liner, having a thickness of 2 mm, was accelerated to 12 km/s (impact radius R_imp = 1 cm, field B_φ ∼ 2 MG, current ∼30 MA).

Fig. 1. Layout and basic parameters of the ALT-3 device (projected). 1 and 2: Ø0.4 m helical and 15-module disk explosive magnetic generators (HEMGs and DEMGs); 3 and 4: explosive closing switches (ECS)—crowbar 3 disconnects the HEMG at initial DEMG current I₀ = 7 MA (t = t₀); ECS 4, having low resistivity R_kl, connects a load of inductance L₀ = 6 nH at a given time t_0l, at fuse opening switch (FOS) voltage U_0l < 10 kV; 5: electrically exploded FOS with Cu foil of thickness Δ_f = 0.12–15 mm and height ∼90 cm; 6 and 7: coaxial-radial transmission line (TL); 8: ponderomotive unit (PU) with an Al liner of outer and inner radius R_l = 4 cm and R_in0 = 3.7 cm (Δ_Al = 3 mm) and height H_l ∼ 1.2R_l; 9: PU end walls; 10 and 11: current probes; 12: measuring unit of radius R_imp = 1 cm (implosion depth R_in0/R_imp = 3.7) with photon doppler velocimetry (PDV) probes and test samples.

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One-layer (Al, Cu) and two-layer liners are used.^1–9 The latter consist of an inner layer (impactor or tested material) and an adjacent highly conducting layer (Al, Cu) on the outside (Al, Cu pusher⁶), in which the current I(t) is flowing. The outer surface of the liner R_out(t) is exposed to a magnetic field (1), which diffuses into the skin layer and produces the magnetic pressure (2):

Devices such as the ALT-3 can be used to generate ramp pressures above 10 Mbar (Ref. 8) in materials by reducing the radii of the liner and the measuring units as a result of higher magnetic fields and, in particular, deeper implosion of the liners, such as the two-layer Al/Cu and Al/W liners with the parameters shown in (3). An increase in the relative thickness of such liners (with the mass being the same, ∼20 g/cm) is expected to control the growth of their basic instabilities,⁹ and with the height given above, the influence of the end walls (glide planes) on their implosion can be weakened.¹⁰ Previous reports^7,8 have proposed studying ramp compression of such liners during their implosion using precision PDV measurements of the velocity v_in(t) of the inner liner surface—this is analogous to the liner implosion experiments,⁶ with the parameters shown in (4):

In this paper, we present revised simulations of the ALT-1–3 devices (Sec. II) and possible designs for the ALT-3 with different liner assemblies and measuring units (Sec. III). The 1D(MHD)_n code that we used¹¹ was developed by Buyko, Ivanova, and Sofronov based on the UP-OK technique¹³ and has been verified by a number of experiments.^5,12 Within the 1D-MHD simulation framework, we demonstrate the capabilities of the ALT-3-like devices as applied to isentropic compression of materials to pressures of 20 Mbar by magnetically driven liner implosion. To test these capabilities, one needs to perform similar^4,10 2D-MHD simulations of the proposed liners taking into account the development of their basic instabilities and glide-plane effects. In such simulations, it is necessary to provide efficient implosion, which should be considerably deeper than that of the liners in (4) or the ALT-3 liner. Of even greater importance is experimental verification of the efficiency of the required deep liner implosion.

2 II. MODELING OF THE ALT-1–3 DEVICES

In the 1D(MHD)_n code,¹¹ an arbitrary number (n) of 1D-MHD problems modeling major device components (Fig. 1) and coupled by boundary conditions of types (1) and (5) are solved in parallel. In the computational scheme of the devices under consideration, n = 13, and two interconnected current circuits are used: a DEMG-FOS with current I_g(t) and an FOS-PU with current I(t). Each circuit is composed of building blocks taken from “libraries” of energy sources (DEMG, FOS, etc.), transmission lines (coaxial, radial, etc.), and liner PUs (cylindrical, quasi-spherical, etc.). The currents I_g(t) at t > t₀ and I(t) at t > t_0l are found from the equations of magnetic flux balance in the respective circuits (at t < t₀, the simulated HEMG current has a required maximum, I₀, which is important for providing the correct state of the Cu foil by the DEMG actuation time t₀). For the current I(t), the circuit equation is given by (μs, nH, MA, kV)

Here, L(t) is the load inductance; U_f(t) and U₋(t) are the FOS foil voltage and the total rate of magnetic flux losses in the load: U_tl(t) and U_pu(t) on the TL and PU walls, U_l(t) on the liner, and U_kl(t) = R_kl(t)I(t) on the ECS [R_kl(t) is taken from experiments]. To simulate the devices without ECS,^1,2 we assume that t_0l = 0 (U_0l = 0). These voltages and the change in the load inductance are calculated based on the outputs of respective 1D-MHD simulations: U_i(t) is based on the electric fields E_i(t) generated as a result of magnetic diffusion on the surface of the FOS Cu foil, on the TL and PU walls, and on the liner; ΔL(t) is based on the displacements of these walls δ_i(t) and the liner boundary R_l − R_out(t). In the 1D-MHD simulations, the conductors are described by wide-range combinations of equation of state and conductivity—from their solid to their vaporized (Cu)¹⁴ or plasma (Al, Cu)^15,16 state.

The revised simulations employ finer Lagrangian meshes with a mesh size of ∼1 μm, which provide nearly converging simulation results. It is also important that they enable more consistent modeling of the load from the FOS to the liner (including magnetic flux losses and changes in its inductance). For example, eight 1D-MHD computations with the current I(t) are used (previously six), including two to simulate the liner, the return conductor, and the insulators between them. The liner implosion simulations take into account radiation transfer in the “back and forth” approximation¹⁷ along with electron heat conduction; the radiation flux across the outer liner surface is taken into account, but it has no effect on the insulator (an H-released discharge and current branch-off into it are possible, but, according to estimates, they have a minor effect on implosion). As in the previous simulations, five 1D-MHD problems simulate the DEMG-FOS system. Two problems with current I_g(t) are used to model magnetic flux losses by diffusion into DEMG cavity walls and current conductors from there to the FOS; the essentially 2D motion of these walls is modeled by two functions, the minimum wall radius R_min(t) (to calculate the losses mentioned above) and cavity inductance L_g(t), which are taken from the DEMG “library” (they have been obtained in special 2D computations of cavity compression by the products of explosion of the DEMG). Three problems are used to model the FOS as a multi-layer system that has three current-carrying inner layers with insulators between them (Fig. 1): the DEMG current conductor with current I_g(t), the Cu foil with back current I_g(t) and current I(t), and the load current conductor with back current I(t). The boundary conditions of these problems are interrelated and enable “through” 1D-MHD modeling of the whole system.

The ALT-1 and ALT-2 experiments demonstrated stable operation of the devices (with load currents reaching 31.5 ± 1.5 MA and 30.0 ± 1.3 MA)⁵ and close agreement between the basic calculated characteristics and the results of previous simulations ^{[Fig. 2(a)]}. The results of the revised and previous simulations are nearly identical ^{[Fig. 2(b)]}.

Fig. 2. (a) Load current derivative dI/dt in the ALT-2 experiment and in the previous simulation (heavy and fine lines, dashed line represents the calculated FOS voltage U_f); (b) dI/dt and inner liner surface velocity v_in(t) in the revised and previous simulations (heavy and fine lines).

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The revised and previous⁴ simulations of the ALT-3 device differ notably ^{[Figs. 3(a)–3(c)]}. For example, the revised simulation gives the same DEMG current peak as before, 71 MA, but distinct differences in the FOS voltage and load current derivative, and a significant increase in the rate of magnetic flux losses in the load U₋(t) along with an 11% decrease in its inductance at peak current. The peak current decreases weakly (62.7 MA at 7.5 nH instead of 64.1 MA at 8.4 nH), but the ramp pressure in the liner by the end of its implosion decreases by 14% (down to 0.97 Mbar), with nearly the same maximum liner velocity, ∼21 km/s ^{[Figs. 3(a), 3(b), and 3(d)]}. Simulation 1 ^{[Figs. 3(a)–3(d), dashed line], as our analysis shows, understated the inductance, and, in particular, the magnetic flux losses in the load, which resulted in overstated values of the peak current and magnetic pressure in the liner and its ramp pressure and velocity.}

Fig. 3. Results of revised simulation of the ALT-3 device (heavy lines), previous simulation (fine lines), and simulation 1 (dashed line): (a) currents I_g(t) and I(t) and inductance L(t) in the load; (b) current derivative dI/dt and velocity of the inner liner surface v_in(t); (c) FOS voltage U_f(t) and rate of magnetic flux losses in the load U₋(t); (d) profiles of ramp P(r) and magnetic P_B(r) pressures in the liner at the end of implosion (R_in = R_imp = 1 cm).

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3 III. REVISED SIMULATIONS OF ALT-3-LIKE DEVICES

Results of such simulations of the devices with two-layer liners (3) are presented in Table I (simulations 1–4). They differ from the results of the same simulations reported from a previous study⁸ more considerably than the similar computations of the ALT-3 device presented at the end of Sec. I. For example, at Cu foil thicknesses of 0.12–15 mm, the maximum current in the liners decreased by 12%–14% (previously 65–70 MA), which resulted in a decrease in the ramp pressures reached by the end of their implosion by 26%–29% in the W layer (previously 17.2–18.0 Mbar) and by 25%–27% in the Cu layer (previously 13.1–13.7 Mbar), respectively.

It is expected that this device will be able to use both Cu and Cu/W liners with the parameters in (6): “small copper PU” in the load with the previous load inductance (6 nH). The deepest implosion of such liners is achieved at an impact radius of R_imp = 0.7 mm, which seems to be feasible: it is twice the radius of the experimental measuring unit (4),

The results of simulations 5–10 of such a device are presented in Table I and Figs. 4(a)–4(d). The DEMG currents and the FOS and load wall peak voltages in simulations 5–8 are close to those of simulations 1–4, and the liner currents have decreased by 5%–6%. The values of magnetic pressure in the copper skin layer, however, are nearly twice as high, which has increased attainable ramp pressures by a factor of 1.4–1.5.

Fig. 4. Results of device simulations with Cu/W and Cu liners (6) at FOS Cu foil thickness of 0.15 mm [nos. 6(4w), 8(4cu), and 10(4w₀) in Table I]. (a) Currents in DEMG I_g(t) and liner I(t), load inductance L(t); (b) voltages on foil U_f(t) and load walls U₋(t); (c) pressure P_ex(t) on the outer boundary R_ex(t) of the tested liner layer and velocity v_in(t) of the inner boundary R_in(t) of this layer; (d) profiles of ramp P(r) and magnetic P_B(r) pressures in the liner at the end of implosion (R_in = R_imp = 0.7 mm).

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At a Cu foil thickness of 0.15 mm [simulations 6(4w) and 8(4cu), Figs. 4(a)–4(d)], the values of attainable pressure and liner velocity are the highest, 19.7 Mbar and 14.5 Mbar, 29 km/s and 32 km/s, in the Cu/W and Cu liners at R_in0/R_imp = 26. The pressures reached at shallower implosion are lower, 16.1 Mbar and 12.3 Mbar at R_in0/R_imp = 18, but they are substantially higher than the pressures in the Al/W and Al/Cu liners (3) at their deeper implosion (12.8 Mbar and 9.9 Mbar, R_in0/R_imp ∼ 27). Note that these pressures in the Cu/W liner are reached near the boundary of the W layer: in copper at R_imp = 1.0 mm and in tungsten at R_imp = 0.7 mm. Also note that the first electrical-explosion peak of FOS voltage is the highest in these simulations, at ∼350 kV (the second peak is ∼70 kV lower).

Decreasing the Cu foil thickness from 0.15 mm to 0.12 mm [simulations 5(2w) and 7(2cu) in Table I] enables preserving the attainable pressures at nearly the same level and significantly reducing the first peak of FOS voltage—by about 90 kV (so that it becomes 35–40 kV lower than the second peak of this voltage).

Excluding the ECS—which would simplify the design significantly—would not result in any significant decrease in attainable pressures, but the first FOS voltage peaks would grow substantially, by 100–120 kV [simulations 9(2w₀) and 10(4w₀), Figs. 4(a), 4(b), and,4(d)].

Note that the pressures P_ex(t) on the outer surface of the liner’s tested W layer and the velocities v_in(t) of the inner boundary of this layer—at early implosion times, at P_ex(t) < 0.3 Mbar—have a two-wave structure ^{[Fig. 4(c)]}. This is created by MHD processes in the liner’s skin layer and is particularly visible in the ECS-free devices. These pressures P_ex and velocities v_in grow with the decreasing radius of the corresponding boundaries during implosion (Fig. 5), approximately as P_ex ∼ (R_ex)^−2.0 and v_in ∼ (R_in)^−0.2 at R_in < 0.17 mm (cumulation). This can decrease the accuracy of computational analysis of the liner velocities v_in(t),^6,7 which is necessary for studying the isentropes of the tested materials, especially for the highest attainable pressures.

Fig. 5. (P_ex vs R_ex) and (v_in vs R_in) plots of the outer and inner boundaries of the liner’s tested W layer from simulations 6(4w) and 10(4w₀), see Table I and Fig. 4.

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4 IV. CONCLUSION

Section I provides a summary of the numerical code used to simulate the devices of interest and to revise the simulations of their liner load. The revised and previous simulations of the experimental ALT-1,2 devices give practically the same results. The revised simulation of the ALT-3 device (design project) gives similar values of the load current, ∼63 MA, and Al liner velocity, ∼21 km/s (R_l = 40 mm, Δ_Al = 3 mm, R_imp = 10 mm), as the previous simulation, but with a 14% decrease in ramp pressure to ∼1 Mbar, which is achieved at an implosion depth of R_in0/R_imp ∼ 4.

Section II reports the revised simulations of the ALT-3-like devices—first with the parameters of the two-layer Al/W and Al/Cu liners of R_l = 30 mm, Δ_Al = 3.0 mm, Δ_Cu(W) = 0.34(0.16) mm, R_imp = 1 mm (R_in0/R_imp ∼ 27). Their results differ substantially from the similar previous computations. For example, for a Cu foil thickness of 15 mm, the maximum liner current decreased by 14%, which resulted in a drop of ramp pressures reached in the W layer by 29% (to 12.8 Mbar). Next, we considered similar devices with Cu and Cu/W liners of R_l = 20 mm, Δ_Cu = 2.0 mm, Δ_Cu+W = 1.75 + 0.25 mm, R_imp = 1.0–0.7 mm, R_in0/R_imp = 18–26. Their current decreased to 55 MA (by 5 MA), but the magnetic pressures grew to 4.5 Mbar (nearly doubling), and the attainable ramp pressures increased by approximately half. The maximum ramp pressure of 19.7 Mbar was reached in the Cu/W liner at Δ_f = 0.15 mm and an implosion depth of R_in0/R_imp = 26 (v_imp = 29 km/s). At R_in0/R_imp = 18, the pressure was lower, 16.1 Mbar, but this substantially exceeds the pressure in the Al/W liner at its deep implosion (see above). Reducing the thickness of the Cu foil from 0.15 mm to 0.12 mm, one can deliver nearly the same attainable pressures and bring the FOS peak voltage down to 90 kV. In the absence of the ECS, which would simplify the design significantly, there was no prominent decrease in the liner currents and pressures, but the FOS voltage peak grew substantially, from 350 kV to 450 kV.

The pressures P_ex(t) on the outer surface of the liner’s tested layer and the velocities v_in(t) of this layer—at early implosion times, at P_ex(t) < 0.3 Mbar—have a two-wave structure, which becomes particularly clear in the ECS-free devices. These pressures P_ex and velocities v_in grow with the decreasing radius of the corresponding boundaries during implosion, approximately as P_ex ∼ (R_ex)^−2.0 and v_in ∼ (R_in)^−0.2 at R_in < 0.17 mm (cumulation). This can decrease the accuracy of computational analysis of the liner velocities v_in(t), which is necessary for studying the isentropes of the tested materials, especially at the highest attainable pressures.

Our results can be applied to the design of experiments on isentropic compression of materials up to pressures ∼20 Mbar.