We report on a new technique for obtaining off-Hugoniot pressure vs. density data for solid metals compressed to extreme pressure by a magnetically driven liner implosion on the Z-machine (Z) at Sandia National Laboratories. In our experiments, the liner comprises inner and outer metal tubes. The inner tube is composed of a sample material (e.g., Ta and Cu) whose compressed state is to be inferred. The outer tube is composed of Al and serves as the current carrying cathode. Another aluminum liner at much larger radius serves as the anode. A shaped current pulse quasi-isentropically compresses the sample as it implodes. The iterative method used to infer pressure vs. density requires two velocity measurements. Photonic Doppler velocimetry probes measure the implosion velocity of the free (inner) surface of the sample material and the explosion velocity of the anode free (outer) surface. These two velocities are used in conjunction with magnetohydrodynamic simulation and mathematical optimization to obtain the current driving the liner implosion, and to infer pressure and density in the sample through maximum compression. This new equation of state calibration technique is illustrated using a simulated experiment with a Cu sample. Monte Carlo uncertainty quantification of synthetic data establishes convergence criteria for experiments. Results are presented from experiments with Al/Ta, Al/Cu, and Al liners. Symmetric liner implosion with quasi-isentropic compression to peak pressure ∼1000 GPa is achieved in all cases. These experiments exhibit unexpectedly softer behavior above 200 GPa, which we conjecture is related to differences in the actual and modeled properties of aluminum.

For over a decade, the Z-machine (Z) at Sandia National Laboratories (SNL) has been the premier current source in the world for precision, magnetically driven, high-pressure dynamic materials experiments. Flyer plate shock impact and ramp (quasi-isentropic) loading experiments on Z have produced state-of-the art results relevant to planetary science,1 formation of the earth-moon system,2 material strength,3 and equation-of-state (EOS) research in general4–6 with a wide range of applications. These experiments employed planar platforms exclusively; that is, the current carrying electrodes are coplanar, perpendicular to which is the magnetic force that quasi-isentropically accelerates a flyer plate7–10 or directly loads a material sample.11 While the experiments produce state-of-the-art results for material science, with limited current it is difficult to produce magnetic drive pressures exceeding 600 GPa because the planar platform is divergent. Magnetic pressure explodes the platform, electrodes fly apart resulting in extreme deformation of conductors.7–9 The ratio of magnetic field to current (B/I) decreases monotonically during an experiment, so the magnetic field produced by a constant current decreases with time. In contrast, a cylindrical liner implosion is convergent, B/I is inversely proportional to liner radius (R) and magnetic pressure PB is proportional to I2/R2. Hence, as a liner implodes the magnetic pressure driving it can increase monotonically, even when the applied current is decreasing.

Through convergence, it is possible to achieve a magnetic drive pressure approaching 5000 GPa in a cylindrical experiment with the available current on Z (≈20 MA). Furthermore, solid metals can be quasi-isentropically compressed to stresses in the range 1000–2000 GPa via cylindrical liner implosion, 2.5–5 times larger than the peak stress that can be inferred from velocimetry data using the planar platform, and second only to the capabilities of the National Ignition Facility12,13 (NIF). This provides motivation to use a cylindrical platform for dynamic materials experiments on Z. However, diagnosing the state of the liner material in cylindrical geometry is challenging due to the small dimensions of the liner. A radiography technique was used to successfully infer the principal isentrope of Be to 550 GPa in cylindrical liner implosion experiments on Z.14 While exceeding the maximum pressure that can be inferred from planar experiments, the radiography technique is limited by the energy of the x-ray source. A velocimetry-based technique is preferred because it is more accurate and provides wider applicability. However, the standard analysis technique used to obtain stress vs. density from velocimetry data in ramp compression experiments, Lagrangian analysis,5,15–17 cannot be used similarly in cylindrical experiments due to small sample dimensions and high wave speeds that result in multiple reverberations in the liner on short time scales. In Secs. II–IV, we discuss a new velocimetry based experimental technique for obtaining off-Hugoniot pressure vs. density data for solid metals compressed to extreme pressure by a magnetically driven liner implosion.

Figure 1 illustrates a cylindrical, liner implosion experiment. Figure 1(a) is a cross-section in the rθ-plane (not to scale). Figure 1(b) is a cross-section in the rz-plane (r horizontal, z vertical) of an experimental load (with Cu sample) with correct relative scaling. J and B denote current density and magnetic field, respectively, with direction indicated by arrows. The liner comprises inner and outer metal tubes. The inner tube is composed of a sample material (e.g., Ta and Cu), of unknown EOS, whose compressed state is to be inferred. The outer tube (the pusher) is composed of Al (6061-T6) and serves as the current carrying cathode; it is designed to prevent current from reaching the inner liner for the duration of the implosion, which precludes having to know the electrical properties of the sample material. For a Cu experiment, the sample, indicated by an arrow in Fig. 1(b), is 530 μm thick with inner radius (IR) 1.9 mm, while the Al pusher is 1 mm thick with outer radius (OR) 3.43 mm. Also shown in Fig. 1(b) are upper and lower end caps composed of Ta, and upper and lower electrodes composed of Cu and stainless steel (SS), respectively. Vertical distance between top surface of the SS cathode and bottom surface of the Cu anode is 10 mm.

FIG. 1.

(a) Cross-section of cylindrical quasi-isentropic compression experiment in the rθ-plane (not to scale). Internal PDV fibers are housed in a tube composed of Pt (red); they measure free surface velocity of sample liner (blue). Straight black (red) arrows show direction of motion of liner (anode) and indicate azimuthal location of velocimetry probes. (b) Schematic of experimental load with Cu sample. Parts labeled Ta are end caps designed to prevent destruction of the internal PDV probes by edge waves.

FIG. 1.

(a) Cross-section of cylindrical quasi-isentropic compression experiment in the rθ-plane (not to scale). Internal PDV fibers are housed in a tube composed of Pt (red); they measure free surface velocity of sample liner (blue). Straight black (red) arrows show direction of motion of liner (anode) and indicate azimuthal location of velocimetry probes. (b) Schematic of experimental load with Cu sample. Parts labeled Ta are end caps designed to prevent destruction of the internal PDV probes by edge waves.

Close modal

We have developed a multi-point, radial, PDV (Photonic Doppler Velocimetry18–20) diagnostic specifically for measuring the velocity of an imploding liner in cylindrical experiments.21 Internal to the composite liner is a Pt (platinum) tube (colored red in Fig. 1(a)), IR = 200 μm and OR = 350 μm, that houses six, 125 μm diameter PDV probes spaced at 60° intervals (colored blue dots inside Pt tube, Fig. 1(a)). Black arrows originating at the sample free surface, indicating its direction of motion, terminate on the probes. The green dot at the center of the PDV ensemble represents a wire composed of high strength, carbon steel alloy that holds the probes in place, and helps to align them with a conical mirror. In Fig. 1(b), the Pt tube and PDV probes (which enter the load at the top) terminate just above a conical Al mirror that reflects laser light emanating from the fibers onto the sample free surface. Each probe serves as both send and receive fiber and is powered by a laser with nominal wavelength of 1550.95 nm.21 Data are recorded on a set of digitizers having a bandwidth of 25 GHz at sampling rate 80 GS/s. The use of six probes reduces the risk of not collecting any liner velocity data in an experiment; also, collectively these data are indicative of implosion symmetry.

External to the composite liner at much larger radius is a return current anode; composed of Al (6061-T6) it is 450 μm thick with IR = 13 mm in all experiments. The outward radial velocity of the anode free surface is measured by an array of external VISAR (Velocity Interferometry System for Any Reflector22) and PDV probes; red arrows in Fig. 1(a) indicate probe locations in azimuth and direction of motion. There is one external probe for each internal probe at the same azimuthal location. The arrays of velocimetry probes are located precisely at the liner mid-height, 5 mm above the surface of the SS lower electrode (Fig. 1(b)).

Two velocity measurements are required for this iterative, EOS calibration technique: (1) The free surface velocity of the Al anode and (2) the free surface velocity of the inner surface of the sample material (i.e., the composite liner). Velocity data from a given probe pair (i.e., sample liner and anode at the same azimuth) can be used in conjunction with resistive, magnetohydrodynamic (MHD) simulation and mathematical optimization to infer the pressure and density in the sample material under the following assumptions: (1) the EOS and electrical properties of Al are known; a reasonable assumption because we have partially validated our models of Al through flyer plate and ramp loading experiments on Z,6–10 (2) there is an accurate strength model for the sample/pusher and that strength has a small effect (relative to EOS) on the free surface velocity, (3) the time varying magnetic field driving the liner implosion can be determined accurately, (4) current driving the imploding liner is identical to current on the anode (follows from Ampere's Law), (5) current is confined to the Al pusher for the duration of the velocity measurement, (6) edge waves do not affect either the sample or anode velocities during the time of the measurement, and (7) the magnetic Rayleigh-Taylor (MRT) instability that develops at the Al pusher drive surface does not feed through to the inner sample. With these assumptions (the validity of which will be addressed subsequently), the EOS of the sample material is the one unknown that determines the response of the sample free surface velocity to the magnetic drive. Consequently, the unknown EOS of the sample material can be determined by solving two independent inverse problems.

First, MHD and optimization codes are employed in conjunction with the measured (PDV or VISAR) free surface velocity of the anode to solve an inverse problem that yields the current I(t) driving the liner implosion.7,8 Then I(t), the measured liner PDV velocity (at the same azimuthal location I(t) was determined), and MHD and optimization codes are used to solve a second inverse problem that yields pressure vs. density on approximately the principal isentrope of the sample material. We refer to this process as EOS calibration; it is illustrated in more detail in Sec. II B using a simulated experiment in which the sample is composed of Cu.

Two Sandia codes are used for MHD modeling and optimizations. ALEGRA23 (Arbitrary Lagrangian Eulerian General Research Application) is a 2D/3D, radiation, resistive MHD code with demonstrated predictive capability for planar flyer plate and ramp loading experiments, and for solid liner z-pinch experiments.14 A suite of verification problems ensures that ALEGRA algorithms accurately produce known solutions. Also, a MHD model for Al flyer plates has been validated using velocimetry data from shock impact experiments,7–10 so we have confidence that ALEGRA can accurately solve the liner problem. DAKOTA24 (Design Analysis Kit for Optimization and Terascale Applications) optimization capability is used extensively to design shaped currents for dynamic material experiments on Z, and to obtain the actual current that drove the experiment.7,8,11,14 It is also used for parameter estimation, sensitivity analysis, and uncertainty quantification (UQ).

The goal of this simulated experiment is to quantify the accuracy with which pressure vs. density can be determined from the EOS calibration technique using the measured velocities. To this end, we use an Al/Cu (pusher/sample) liner with dimensions given in Sec. II A. Assumptions (6) and (7) above imply there are no 2D effects. Therefore, we can accurately model the problem with Lagrangian MHD including only the liner in 1D, cylindrical coordinates with azimuthal symmetry. The initial liner density vs. radius is plotted in Fig. 2(a). Material models include EOS and strength for Cu and Al, and electrical conductivity for Al (see the  Appendix).

FIG. 2.

(a) Initial density of Al/Cu liner vs. radius in 1D, Lagrangian, MHD simulation. (b) Current vs. time that drives the simulation. (c) Simulated liner free surface velocity vs. time; the synthetic data used for EOS calibration.

FIG. 2.

(a) Initial density of Al/Cu liner vs. radius in 1D, Lagrangian, MHD simulation. (b) Current vs. time that drives the simulation. (c) Simulated liner free surface velocity vs. time; the synthetic data used for EOS calibration.

Close modal

The shaped current plotted in Fig. 2(b) drives a forward simulation of the liner implosion using what we consider our best models of Al and Cu (e.g., wide range tabular EOSs, see the  Appendix) to produce synthetic PDV data; that is, the free surface velocity of the Cu sample plotted in Fig. 2(c) versus time. (Note that since we are given a current, it is not necessary to perform the first step described in Sec. II A in which the drive current is obtained by solving an inverse problem using the anode velocity.) The Cu sample is quasi-isentropically compressed to peak pressure 1250 GPa. Although the simulated PDV velocity increases from 0 starting at 2.55 μs, it is truncated for time <2.915 μs for use in the EOS calibration because it is not sensitive to the Cu EOS in this temporal range. The plotted velocity spans the pressure range 200–1250 GPa at the Al/Cu interface. In the Lagrangian simulation, current is applied at the outer surface of the Al pusher (the outer radial boundary). The Cu inner (free) surface is a zero-pressure boundary. The radial cell size Δr = 1.0 μm was determined by a numerical convergence study.

We apply the DAKOTA-ALEGRA EOS calibration method (described above) to the synthetic PDV data in Fig. 2(c). In this exercise, the EOS of Cu is parameterized using the extended form of Vinet25 that includes the effect of temperature26 (see the  Appendix) with four free parameters d2–d5. Starting from initial guess values d2–d5 = (7.0, 10.0, 1.0, 10.0), DAKOTA iterates on the Vinet parameters using a non-linear least squares solver, running ALEGRA for each parameter per iteration, until the simulated velocity converges to the synthetic data; that is, until the velocity residual norm given by Vres=i(Vsim,iVdata,i)2 is minimized, where the sum is over the number of time steps used to fit the synthetic data Vdata, and Vsim is the simulated result produced by a DAKOTA iteration.

The initial and converged PDV velocities are compared with the synthetic data in Fig. 3(a). The initial and final residual norms are 1885 m/s and 47 m/s, respectively. Velocity residuals of the converged solution are in the range −8 to 14 m/s, indicating convergence to the synthetic data with high accuracy. Converged values of the Vinet parameters are d2–d5 = (6.019, −0.079, −41.749, 79.999) with standard deviations σ2–σ5 = (0.203, 0.262, 12.668, 45.983); the latter are consistent with a 95% confidence interval (given approximately by dn ± 2σn with n = 2–5) under the assumption that the dn are random variables with normal distribution. If this was an actual experiment, the 95% confidence interval indicates that out of 100 experiments only 5 outcomes (i.e., the converged EOS) would not be included in the confidence interval (which may be different for each experiment).

FIG. 3.

Results of EOS calibration for Al/Cu liner. (a) Simulated Cu free surface velocity vs. time for initial and converged Vinet EOS compared with the synthetic (tabular EOS) data (Fig. 2(c)). (b) Pressure vs. density in Cu at the Al/Cu interface for the initial and converged Vinet EOS compared to the simulated result obtained using a tabular EOS.

FIG. 3.

Results of EOS calibration for Al/Cu liner. (a) Simulated Cu free surface velocity vs. time for initial and converged Vinet EOS compared with the synthetic (tabular EOS) data (Fig. 2(c)). (b) Pressure vs. density in Cu at the Al/Cu interface for the initial and converged Vinet EOS compared to the simulated result obtained using a tabular EOS.

Close modal

Peak pressure vs. peak density in Cu for the initial and converged Vinet EOSs is compared to synthetic data (obtained using a tabular EOS for Cu, see the  Appendix) in Fig. 3(b). It is clear that the EOS calibration converged to values of Vinet parameters that produce the synthetic data with high accuracy; at density 22.4 g/cm3, the difference in pressure is 0.8%, at pressure 1250 GPa the difference in density is 0.2%.

Using the previous values for the σn, a Monte Carlo UQ calculation is performed to quantify the uncertainty in the converged pressure vs. density. The resulting average pressure vs. density and ±1σ variation about the average are plotted in Fig. 4. The maximum uncertainty in pressure (density) is 5.7% (1.8%) at 1170 GPa for a residual norm of 47 m/s.

FIG. 4.

Average pressure vs. density at the Al/Cu interface with ±1σ (standard deviation) variation about the mean resulting from a Monte Carlo UQ analysis of the synthetic experiment.

FIG. 4.

Average pressure vs. density at the Al/Cu interface with ±1σ (standard deviation) variation about the mean resulting from a Monte Carlo UQ analysis of the synthetic experiment.

Close modal

From this simulated experiment with EOS calibration, we can conclude that when the simulated PDV velocity converges to the data with high accuracy we can expect that the converged Vinet EOS accurately reproduces the pressure vs. density in the sample to maximum compression. Figure 3 shows that the EOS is sensitive to ∼1% (∼100 m/s) differences in the sample free surface velocity. Our radial PDV system has a measurement uncertainty ∼10 m/s over the entire range of velocity,19 which is within the range of velocity residuals of the converged solution. Hence, in principle, this EOS calibration technique can yield off-Hugoniot pressure vs. density to ∼1000 GPa with uncertainty 5.7% (1.8%) in pressure (density) or less.

1. 2D MHD simulation

Two-dimensional ALEGRA simulation is used to illustrate the dynamics in a liner implosion, and to address the validity of some of the assumptions given in Sec. II A. The experimental configuration in Fig. 1(b) is modeled in a cylindrically symmetric (r-z) coordinate system using Eulerian, resistive MHD. The 2D simulation model is shown in Fig. 5(a), which plots filled contours of log density at the initial time 2.2 μs. Void is colored black. Material models for Al and Cu are identical to those used in the 1D, simulated experiment above; models of EOS and strength for Pt (PDV tube near axis), SS, and Ta are added (see the  Appendix). The axial height is 2.0 cm with uniform cell size Δz = 10.0 μm. The radial cell size Δr = 10.0 μm from the axis to the outer radius of the Al pusher; from here it increases nonuniformly to 25.0 μm at r = 1.05 cm where it is constant to 1.15 cm; it decreases to 10.0 μm again at the IR of the anode at 1.3 cm and is constant for larger r.

FIG. 5.

Snapshots of log density from 2D, Eulerian, MHD simulation of Al/Cu liner, quasi-isentropic compression experiment at (a) the initial time 2.20 μs (compare with Fig. 1(b)), (b) 2.975 μs, and (c) 3.016 μs, 4 ns before the Cu liner free surface impacts the Pt tube.

FIG. 5.

Snapshots of log density from 2D, Eulerian, MHD simulation of Al/Cu liner, quasi-isentropic compression experiment at (a) the initial time 2.20 μs (compare with Fig. 1(b)), (b) 2.975 μs, and (c) 3.016 μs, 4 ns before the Cu liner free surface impacts the Pt tube.

Close modal

The shaped current from the experiment (to be discussed subsequently) drives the simulation; it is plotted vs. time in Fig. 6(a). Current flows into the simulation model via a magnetic boundary condition across the void-gap at the bottom of Fig. 5(a). The axial current density (Jz) produces an azimuthal component of magnetic field (Bϑ) that implodes (explodes) the liner (anode) via the J × B force. Snapshots of log density are plotted in Figs. 5(b) and 5(c) at times 2.975 μs and 3.016 μs, respectively; the latter is about 4 ns before the liner impacts the PDV (Pt) tube.

FIG. 6.

(a) Current vs. time that drives the simulation is the actual experimental current. (b) Cu free surface velocity vs. time from 1D and 2D, MHD simulations. (c) Results from 2D, MHD simulation; liner density, magnetic pressure, and hydrodynamic pressure vs. radius at time 3.016 μs (same time as Fig. 5(c)).

FIG. 6.

(a) Current vs. time that drives the simulation is the actual experimental current. (b) Cu free surface velocity vs. time from 1D and 2D, MHD simulations. (c) Results from 2D, MHD simulation; liner density, magnetic pressure, and hydrodynamic pressure vs. radius at time 3.016 μs (same time as Fig. 5(c)).

Close modal

Current density is enhanced at the corners formed at the junction of the Al pusher and top and bottom electrodes (Fig. 5(a)). Rapid burn-through of the magnetic field due to enhanced Joule heating produces irregularly shaped, magnetic bubbles in these regions (roughly at z = 0.4 cm and 1.6 cm, r = 0.25 cm in Fig. 5(b)) and produces radially converging shocks in the Ta end caps. When the end caps are composed of Al, as in early experiments, the shock in the upper electrode destroys the internal PDV probes before full convergence of the liner (at about the time of Fig. 5(b)), thereby reducing the maximum possible measured velocity and inferred pressure and density. The Ta end caps are designed specifically to protect the PDV probes from edge effects. The conical termination prevents material at the edges from advancing ahead of the liner.

Hydrodynamic waves move into the liner from the top and bottom edges. If they reach the axial center of the liner (where the PDV measurement is made) before impact with the Pt tube, then the EOS calibration cannot be performed using 1D simulation (2D would be required) rendering this technique impractical. Figure 6(b) shows that 2D and 1D velocities are in excellent agreement, indicating that edge waves are not an issue in this configuration. If they were, the 2D and 1D velocities would begin to diverge at some time; thereafter, the 2D velocity would be less than the 1D. This justifies assumption (6) in Sec. II A, and the same conclusion also applies to the Al anode.

Small amplitude modulations on the liner due to MRT (initiated by noise in the simulation) are evident in Figs. 5(b) and 5(c). However, the MRT does not feed through to the Cu (verified by plotting hydrodynamic variables as a function of z), justifying assumption (7) in Sec. II A. Significant MRT modulation would cause the 2D and 1D velocities in Fig. 6(b) to diverge at some time; thereafter, the 2D velocity would be larger than the 1D. This would preclude using a 1D simulation model for the EOS calibration (as with edge waves). In fact, due to the combination of the relatively thick and massive composite liner (1140 mg/cm) in combination with the long current pulse, it is unlikely that the MRT has a significant effect on the measured liner velocity.

Figure 6(c) plots liner density, hydrodynamic pressure, and magnetic pressure vs. r at the axial center of the liner (where the PDV measurement is made) at the same time as Fig. 5(c). Solid Cu, spanning the range 0.04 cm < r < 0.14 cm, has been quasi-isentropically compressed to peak pressure and density 1300 GPa and 22.5 g/cm3, respectively, at temperature 1720 K. Approximately 300 μm of the Al pusher remains in a solid state spanning the range 0.11 cm < r< 0.14 cm. It is clear from the magnetic pressure curve that the magnetic field (current) is confined to the pusher, justifying assumption (5) in Sec. II A.

The extended Vinet form with temperature effects was chosen to represent the unknown sample EOS because it has sufficient adjustable parameters to match the curvature in isotherms across the entire pressure range expected in experiments. Different analytic models of the EOS could be used. Simulation shows that the sample density is more sensitive to the EOS than pressure. Thus, the EOS calibration will adjust sample density more than pressure in order to converge to the measured liner free surface velocity.

We present results from cylindrical liner, quasi-isentropic compression experiments with Cu, Ta, and Al samples. In the latter, the liner is composed entirely of 6061-T6 Al. Figure 1(b) depicts the basic design of each experiment; only the liner dimensions change, which depend on sample material and available current.

Composite liners (Al/Cu and Al/Ta) are fabricated via a heat-shrink technique in which the sample and pusher tubes are heated to 573 K. Due to the difference in thermal expansion coefficients, the Al pusher expands more than the sample. As it cools the pusher shrinks onto the sample tube. The final IR and OR of the composite liner are accurate to within ±5 μm. It is possible to have void gaps ∼1 μm thick between the sample and pusher; simulations show that this has no effect on the sample free surface velocity. The inner (free) surface of the sample tube and the outer (free) surface of the anode are pressure blasted with 44 μm diameter glass beads to a surface roughness of ∼300 nm. This roughness improves return stability in the PDV measurement and reduces multiple reflections that lead to harmonics.21 

The current used to drive each experiment is plotted vs. time in Fig. 7. They are obtained by solving the inverse problem using the measured free surface velocity of the aluminum anode (as described in Sec. II A) for each experiment. The current for Cu is the same as in Fig. 6(a). Initially, the current desired for a given sample is designed using a combination of 1D and 2D MHD simulation.7,11 The latter provides a load inductance for a transmission line calculation that models the Z circuit to determine the timing of 36 laser triggered gas switches that produces the desired current. However, the current that drives the liner implosion is less than the design current (particularly around the maximum) because of current loss in the machine. Consequently, the actual current driving the experiment must be obtained by solving an inverse problem using the measured velocity of the anode free surface (described above).

FIG. 7.

Current vs. time from cylindrical quasi-isentropic compression experiments with Al/Cu, Al/Ta, and Al liners, as determined by solving the inverse problem using the measured (PDV), Al anode free surface velocity for each case.

FIG. 7.

Current vs. time from cylindrical quasi-isentropic compression experiments with Al/Cu, Al/Ta, and Al liners, as determined by solving the inverse problem using the measured (PDV), Al anode free surface velocity for each case.

Close modal

The currents in Fig. 7 are very similar, despite the fact that they depend on both the EOS of the sample and dimensions of the composite liner. This is a consequence (and an advantage) of using an analysis technique not based on Lagrangian (wave profile) analysis5 to infer pressure vs. density from the data, which eliminates considerations pertaining to reverberations thereby reducing the sensitivity to sample material and dimensions. The same current shape can be used to quasi-isentropically compress different sample materials, though not to the same pressure.

Details of the Al/Cu experiment were provided in Sec. II. Four of 6 internal PDV probes returned data (azimuthal locations 0°, 120°, 180°, and 300°). The spectrum of the Cu free surface velocity measured at the 300° location is plotted vs. time in Fig. 8(a). The latter is obtained by fast Fourier transforms of beat frequency data over short (∼1 ns) time intervals of duration τ. For fixed laser wavelength, the spread in velocity about the peak is inversely proportional to τ. Velocity vs. time is extracted from the spectrum using a local centroid algorithm. The digitizer bandwidth limits the peak measurable velocity to about 19.4 km/s. The experiment was designed so that peak velocity at impact with the Pt tube is less than this; about 18 km/s. More details on PDV can be found in Refs. 18–21. Velocities at the four azimuthal locations are plotted vs. time in Fig. 8(b); noise due to intensity variations in the spectrum is removed using a low-pass filter with frequency 50 MHz. They overlay to within the uncertainty in the relative timing of the probes (∼200 ps), thereby indicating a symmetric implosion. A significant departure from symmetry would result in a 3D implosion, thereby making EOS calibration impractical.

FIG. 8.

(a) Spectrum of Cu free surface velocity vs. time from PDV probe at 300° location. Colors indicate relative intensity in dB (decibels). (b) Measured Cu free surface velocity vs. time extracted from the spectra of all probes that returned data; azimuthal locations 0°, 120°, 180°, and 300°.

FIG. 8.

(a) Spectrum of Cu free surface velocity vs. time from PDV probe at 300° location. Colors indicate relative intensity in dB (decibels). (b) Measured Cu free surface velocity vs. time extracted from the spectra of all probes that returned data; azimuthal locations 0°, 120°, 180°, and 300°.

Close modal

The free surface velocity of the Al anode was measured using VISAR at 0°, 60°, 120°, 180°, and 240°, and PDV at 300°. The PDV velocity spectrum and extracted velocity are plotted in Figs. 9(a) and 9(b), respectively. The change in the signal beginning at about 3.08 μs is indicative of a change in the surface properties of aluminum (e.g., electrical conductivity). Simulation of the anode produces a bump (or knee) in free surface velocity at this same time that is correlated with arrival of the melt-front (caused by Joule heating). Any one of the anode velocity measurements can be used to obtain an accurate drive current by solving the inverse problem described in Sec. II A. However, the VISAR and PDV data are on different time bases; consequently, they must be mapped to the same time base for accurate EOS calibration, which we have not yet been able to accomplish with sufficient accuracy (<1 ns). The relative timing of the various PDV probes is accurate to ∼200 ps. Thus, data from pairs of PDV probes (internal and external) at the same azimuth are used to obtain drive current, and in the EOS calibration. Using the anode PDV velocity (Fig. 9(b)) over the range 2.47–3.08 μs, ALEGRA-DAKOTA optimization produced the drive current I(t) plotted in Figs. 6(a) and 7 with high accuracy; Vres = 12.0 m/s, 0.23% relative to peak velocity.

FIG. 9.

(a) Spectrum of Al anode free surface velocity vs. time from external PDV probe at 300° location. Colors indicate relative intensity in dB (decibels). (b) Al anode free surface velocity vs. time extracted from the spectrum.

FIG. 9.

(a) Spectrum of Al anode free surface velocity vs. time from external PDV probe at 300° location. Colors indicate relative intensity in dB (decibels). (b) Al anode free surface velocity vs. time extracted from the spectrum.

Close modal

Using the drive current I(t) obtained from the anode free surface velocity measurement at 300°, and the liner free surface velocity measured at 300°, the EOS calibration did not converge to an accurate solution. An ALEGRA, 1D Lagrangian, MHD simulation of the experiment using I(t) and our best material models for Cu and Al ( Appendix) provides insight. Simulated and measured PDV velocities of the Cu free surface vs. time (300° location, Fig. 8) are compared in Fig. 10(a), which includes the measured free surface location. PDV tracked the liner to within 50 μm of the Pt tube. The simulated and measured velocities are in good agreement through 2.87 μs; thereafter, they diverge with the measurement lower by 2%–3%, an unexpectedly large difference that, given the assumptions in Sec. II A, could indicate the actual EOS of Cu is softer (more compressible) than our model EOS. Figure 10(b) plots the simulated pressure vs. density attained in Cu (at the Al/Cu interface) through free surface velocity 15.8 km/s, indicating quasi-isentropic compression to ≈1200 GPa. This is indicative of the actual pressure produced in Cu because simulated pressure is not sensitive to the EOS.

FIG. 10.

(a) Measured (300° location) and simulated Cu free surface velocity vs. time, and measured location of Cu free surface obtained by integrating the velocity. (b) Simulated pressure vs. density in Cu at Al/Cu interface through free surface velocity15.8 km/s.

FIG. 10.

(a) Measured (300° location) and simulated Cu free surface velocity vs. time, and measured location of Cu free surface obtained by integrating the velocity. (b) Simulated pressure vs. density in Cu at Al/Cu interface through free surface velocity15.8 km/s.

Close modal

In the ALEGRA-DAKOTA EOS calibration with extended Vinet parameterization of the Cu EOS, the solution to the inverse problem tended toward a softer principal isentrope for pressure >500 GPa, but failed to converge to an accurate solution in general. Manually adjusting the values of the Vinet parameterization shows that the actual EOS of Cu would have to be unrealistically softer than our tabular EOS to produce the measured velocity. Thus, we rule out the Cu EOS as the cause of the difference in measured and simulated PDV velocity.

A DAKOTA-ALEGRA Monte Carlo UQ calculation of the simulated experiment explores the possibility that the difference in measured and simulated liner velocity is due to uncertainties in the experiment. The calculation includes: (1) ±5 μm uncertainty in the IR and OR of the composite liner, (2) ±2.5 μm uncertainty in the thickness of the Al anode, (3) 0.4% uncertainty in the drive current obtained via solving the inverse problem using the measured anode can velocity, and (4) ±200 ps uncertainty in the relative timing of PDV probes. Under the assumption of a normal distribution, DAKOTA runs ALEGRA with random variations in the aforementioned quantities; thousands of runs are performed to get good statistics. Figure 11 plots mean simulated liner (PDV) velocity, ±1σ variations about the mean, and measured PDV velocity vs. time. It is clear that these uncertainties in the experiment cannot account for the difference in simulated and measured liner PDV velocity for time later than 2.87 μs.

FIG. 11.

Results from Monte Carlo UQ analysis of Al/Cu cylindrical, quasi-isentropic compression experiment; measured (300° location) and average Cu free surface velocity vs. time, and ±1σ (standard deviation) variation about the mean.

FIG. 11.

Results from Monte Carlo UQ analysis of Al/Cu cylindrical, quasi-isentropic compression experiment; measured (300° location) and average Cu free surface velocity vs. time, and ±1σ (standard deviation) variation about the mean.

Close modal

In this experiment, the Al/Ta composite liner comprises an inner tube composed of Ta with IR = 1.7 mm and thickness 300 μm, and an outer tube composed of Al (the pusher) with IR = 2.0 mm and thickness 1200 μm. The velocimetry diagnostics are the same as for Cu. Four of 6 internal PDV probes (at 0°, 120°, 240°, and 300°) returned excellent data on the Ta free surface velocity indicating a symmetric implosion. The external PDV at 300° returned excellent data on the anode free surface velocity enabling an accurate determination of the drive current I(t) plotted in Fig. 7.

A forward simulation (1D, Lagrangian MHD) of the experiment using I(t) as the drive current, and what we think is an accurate EOS for Ta (see the  Appendix), produces results qualitatively similar to Cu. The measured velocity of the Ta free surface at the 300° location, the simulated velocity, and the simulated pressure in Ta at the Al/Ta interface are plotted vs. time in Fig. 12(a). The simulated pressure in solid Ta reaches 1000 GPa, which is indicative of the actual pressure. The measured and simulated velocities are in good agreement until 2.947 μs where they diverge; thereafter, the simulated result is greater than the measurement.

FIG. 12.

(a) Measured (300° location) and simulated Ta free surface velocity vs. time, and simulated pressure vs. time in Ta at Al/Ta interface. (b) Measured velocity and acceleration (time derivative velocity) vs. time Ta free surface. Unit of acceleration is giga-g, where g = 9.81 m/s is the gravitational acceleration of the earth.

FIG. 12.

(a) Measured (300° location) and simulated Ta free surface velocity vs. time, and simulated pressure vs. time in Ta at Al/Ta interface. (b) Measured velocity and acceleration (time derivative velocity) vs. time Ta free surface. Unit of acceleration is giga-g, where g = 9.81 m/s is the gravitational acceleration of the earth.

Close modal

The measured liner velocity and acceleration vs. time are plotted in Fig. 12(b). The liner abruptly decelerates at 2.947 μs, when the peak pressure in Ta is 478 GPa and the simulated and measured velocities diverge. This could be caused by a solid-solid phase transition in Ta, but there is no published evidence for this (to our knowledge). First principles, DFT (density functional theory) calculations indicate that the ambient bcc phase of Ta is stable through 1000 GPa.27 Furthermore, a ramp experiment at NIF produced pressure vs. density in Ta to 600 GPa (close to the principal isentrope of a model EOS) with no indication of a solid-solid phase transition.12 Thus, our results for Ta and Cu suggest that the softer behavior exhibited by the liner velocity might be due to the aluminum pusher, motivating an experiment with a liner composed entirely of Al.

The load configuration is similar to the aforementioned experiments, but the liner is composed entirely of 6061-T6 Al with IR = 2.0 mm and OR = 3.9 mm; velocimetry diagnostics are similar. Four of 6 internal PDV probes (at 0°, 120°, 180°, and 300°) returned excellent data on the Al free surface velocity indicating a symmetric implosion. The external PDV at 300° returned excellent data on the anode free surface velocity enabling an accurate determination of the drive current I(t) plotted in Fig. 7.

As for Cu and Ta, the I(t) is used to drive a forward simulation of the experiment. The measured velocity of the Al free surface at the 300° location, the simulated velocity, and the simulated pressure in Al (at the outer-most radius location that remains solid, ≈1300 μm from the free surface) are plotted vs. time in Fig. 13(a). Solid aluminum is quasi-isentropically compressed to peak pressure of ≈850 GPa. The simulated and measured free surface velocities are in good agreement until 2.92 μs where they diverge; thereafter, the simulated result is greater than the measurement indicating the actual properties of Al may be different than the model. This would account for the systematic difference in measured and free surface velocities in the Cu and Ta experiments.

FIG. 13.

(a) Measured (300° location) and simulated Al free surface velocity vs. time, and simulated pressure vs. time at Lagrangian tracer location of outer-most radius that remains solid, ≈1300 μm from free surface. (b) Measured Al free surface velocity and acceleration vs. time (300° location). Unit of acceleration is giga-g, where g = 9.81 m/s is the gravitational acceleration of the earth.

FIG. 13.

(a) Measured (300° location) and simulated Al free surface velocity vs. time, and simulated pressure vs. time at Lagrangian tracer location of outer-most radius that remains solid, ≈1300 μm from free surface. (b) Measured Al free surface velocity and acceleration vs. time (300° location). Unit of acceleration is giga-g, where g = 9.81 m/s is the gravitational acceleration of the earth.

Close modal

Figure 13(b) plots the measured velocity (300° location) and acceleration of the Al free surface vs. time. Despite the monotonically increasing current (Fig. 7), the liner decelerates in the temporal range 2.83–2.90 μs, and then accelerates on a different trajectory. This behavior is consistent with a solid-solid transition to a softer phase of Al. There is theoretical and experimental evidence to support this conjecture.

Kudasov et al.28 used ab initio calculations to produce a phase diagram for Al that shows an fcc-hcp phase boundary in the range 185–235 GPa for T < 2000 K, an hcp-bcc phase boundary in the range 235–360 GPa for T < 2000 K, and an fcc-bcc phase boundary in the range 160–235 GPa for T > 2000 K. The Kudasov phase boundaries are reproduced in Fig. 14(a) with simulated temperature vs. pressure curves from five Lagrangian tracer locations in the Al liner. Pressure vs. time at these same locations is plotted in Fig. 14(b). The tracer positions span approximately the middle third of the liner. The tracer with the lowest temperature follows approximately the principle isentrope of the Al EOS used in the simulation. Prior to Kudasov's work, the fcc-hcp transition had been detected by Akahama et al.29 in the range 189–215 GPa in DAC (Diamond Anvil Cell) experiments.

FIG. 14.

(a) Phase diagram of Al as determined through ab initio calculations by Kudasov et al.28 (solid lines) with simulated results at select Lagrangian locations in Al liner superimposed. (b) Simulated pressure vs. time at select Lagrangian locations in Al liner.

FIG. 14.

(a) Phase diagram of Al as determined through ab initio calculations by Kudasov et al.28 (solid lines) with simulated results at select Lagrangian locations in Al liner superimposed. (b) Simulated pressure vs. time at select Lagrangian locations in Al liner.

Close modal

It is evident from Fig. 14(a) that if the Kudasov phase diagram is correct, then fcc-hcp and hcp-bcc phase transitions are possible in the Al liner for T < 2000 K. The time at which these would begin to occur can be estimated using the results in Fig. 14(b). The pressure at tracer locations 3.30 mm, 3.18 mm, and 3.00 mm reaches 200 GPa at times 2.921 μs, 2.930 μs, and 2.943 μs, respectively, and 300 GPa at 2.953 μs, 2.960 μs, and 2.970 μs, respectively. Thus, the onset of fcc-hcp transitions in this segment of liner would occur approximately at 2.921 μs. This time is not inconsistent with the time at which the measured and simulated free surface velocities begin to diverge in Fig. 13(a). Moreover, the observed deceleration would be expected from a phase transition that results in a softer isentrope, as Kudasov et al. predict occurs. At 2.953 μs, when pressure in this segment of liner reaches 300 GPa, hcp-bcc transitions could occur. The fcc-bcc transition is possible at larger radius (than 3.30 mm) where the temperature can exceed 2000 K.

The choice of Al for the pusher material was based on, in part, experimental evidence from over a decade of Al flyer plate experiments on Z, and analysis of these experiments. In contrast to the liner implosion, flyer plates are quasi-isentropically accelerated to ballistic velocity. In most cases, the release wave from the free surface keeps the stress below the threshold for solid-solid transitions to occur. Although peak pressure in solid Al can reach 300–400 GPa in the highest velocity experiments, there is no obvious signature of solid-solid phase transitions in the velocimetry data. Hence, the apparently softer behavior exhibited by Al in cylindrical experiments was unexpected.

The goal of the experiments with Ta and Cu is to infer off-Hugoniot pressure vs. density in these materials to ∼1000 GPa using a new EOS calibration technique discussed in Sec. II. A simulated experiment with a Cu sample demonstrated that it is possible to infer pressure vs. density to an accuracy of 5.7% (1.8%) or better in pressure (density) at ∼1000 GPa using velocimetry measurements in conjunction with ALEGRA-DAKOTA optimization. This iterative technique requires two velocity measurements: (1) the free surface velocity of an Al anode that is used to obtain the drive current and (2) the free surface velocity of the imploding liner. To this end, we developed a multi-point, radial PDV diagnostic for measuring the free surface velocity of an imploding liner. Measured liner velocities from different azimuthal locations overlay to within the timing accuracy of the diagnostic, thereby demonstrating a symmetric implosion (required for the analysis technique). An external PDV probe returned an accurate measurement of the Al anode free surface velocity from which the current driving the liner was obtained by solving an inverse problem.

ALEGRA 1D, Lagrangian simulation reveals quasi-isentropic compression to ∼1000 GPa is achieved for Cu, Ta, and Al samples, over a factor of 2 greater than what is possible in planar ramp loading experiments on Z. However, comparison of simulated and measured liner velocities reveals that the actual mechanical response of Al to the drive is softer than expected. The apparently softer behavior of Al precluded obtaining accurate off-Hugoniot pressure vs. density with the proposed technique. While there is evidence for solid-solid transitions in Al, as discussed above, there are alternative explanations for the apparently softer behavior exhibited in these experiments.

We make two independent measurements of liner velocity; a direct measurement from the internal PDV probes, and an indirect measurement using the drive current obtained from the anode free surface velocity (via solving an inverse problem) to drive a forward simulation of the liner implosion. These two agree in simulations of all experiments until some time, after which the direct measurement is less than the simulated velocity. This might indicate that after some time the currents are different in the cathode and anode, which would violate Ampere's Law. However, it is possible that after some time the current distribution in the cathode (i.e., the Al pusher) is different than in ALEGRA MHD simulations. For example, as inductance increases during the liner implosion, current could at some time be shunted to larger radius, thereby reducing the liner velocity giving the appearance of softer behavior. This could be due to differences in the actual and model electrical conductivity of Al that develop as pressure approaches 10 Mbar, or related to an unknown parallel current path in the liner region.

The efficacy of the proposed EOS calibration technique depends in part on having an accurate model for the properties (EOS and electrical conductivity) of the pusher material. These cylindrical ramp loading experiments indicate our models of Al may not be accurate enough for this application. Future experiments using Cu for the pusher material will provide insight on whether the systematic, late-time difference in simulated and measured liner velocity observed in these experiments is related to either current distribution or solid-solid phase transitions in Al.

The authors thank Dawn Flicker for supporting this research; Charles Meyer, Jeff Gluth, Dustin Heinz-Romero, Anthony Romero, and Sheri Payne for their work on velocimetry diagnostics; Jerry Taylor, Randy Holt, and Chris Wilson (all with General Atomics) for their contributions to target fabrication; Jean-Paul Davis for helpful discussions on EOS; Laura Swiler and Bert Debusschere for helpful discussions on DAKOTA and UQ, and the entire Z team for their contributions to this work. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

1. Tabular EOS

The tabular EOS used in this work is considered to be our best available material models as determined by validation over limited ranges of pressure (P) and density (ρ) via flyer plate shock impact and quasi-isentropic compression experiments on Z. Generally, the table for a given material includes internal energy and pressure as functions of temperature (T) and density. ALEGRA calculates density and internal energy, uses these in conjunction with the energy table to obtain temperature, then the latter and density are used to obtain pressure.

Sesame 3700 is used for the EOS of aluminum.30 It is valid for P ≤ 53.7 TPa, ρ ≤ 50.0 g/cm3, and T ≤ 100 000 K. It includes a solid-liquid phase transition. The principal isentrope is in good agreement with quasi-isentropic loading experiments to 240 GPa.6 Also, it has proven to be an accurate model for simulating shocklessly accelerated flyer plates through P ≈ 300 GPa.7–10 

The copper table 3325 is a solid-fluid model that incorporates melting and vaporization (developed by J. H. Carpenter). It is built using semi-empirical models based upon the standard separation of the Helmholtz free energy into cold and thermal terms. These models were calibrated to experimental and first-principle calculation data available in regions along the cold curve, Hugoniot, isobaric expansion, melt, and vaporization. Tabulation was performed on a fine grid and includes Maxwell constructions in the melt and vaporization transitions. It is valid for P ≤ 3560 TPa, ρ ≤ 80.0 g/cm3, and T ≤ 9.986 eV.

The tabular EOS for Ta was developed by Greeff et al.;31 it is valid for P < 880 GPa, ρ ≤ 38.0 g/cm3, and T ≤ 5000 K. The principal isentrope is in good agreement with data to 330 GPa from planar, quasi-isentropic compression experiments on Z.5 

Sesame 3700 is generally used for the EOS of Al in all calculations; the inverse problem that solves for the drive current using measured anode velocity, EOS calibration, and forward simulations of the liner implosion. Tabular EOS for Cu and Ta (inner liners) is used in forward simulations of the liner implosion only to assess overall performance of the experiment. For EOS calibration, a form that can be parameterized is used to model the sample EOS.

2. Extended Vinet EOS

The EOS calibration technique requires an analytic form for the unknown sample EOS that can be parameterized for use in the ALEGRA MHD model. To this end, we use an extended form of the Vinet universal EOS for solids25 that includes temperature effects.26 The temperaturedependence of the pressure is given by

P(ρ,T)=PVinet(ρ)+α0B0[TT0],
(A1)

where T0 is a reference temperature on an isotherm, B0 denotes the isothermal bulk modulus, α0 denotes thermal volume expansion coefficient, and ρ denotes density

PVinet(X)=3B0X2Zeη0Z(1+n=2NdnZn),
(A2)

with X=(ρ0/ρ)1/3, Z=1X, and η0=(3/2)[(B/P)01].

Choosing this form for the extended Vinet EOS, we can derive a thermodynamically consistent energy equation (A. E. Mattsson); it is given by

E(X,T)=EVinet(X)+CV0(TT0),
(A3)

where

EVinet(X)=9B0V0η02{f0eη0Z[f0η0Z(f0+n=1NfnZn)]}α0B0V0(1X3)T0,
(A4)

with fN=dN, fn=dn(n+2)fn+1/η0, and V0 denotes volume at the reference temperature. The product α0B0is held constant, as is CV0. ALEGRA calculates density and energy, then temperature is obtained using Eq. (A3), and Eq. (A1) is used to calculate pressure. This thermodynamically consistent form of the extended Vinet EOS is developed through a simple temperature addition to an isotherm and is best in the neighborhood of an isotherm.

3. Viscoplastic and electrical conductivity

The Steinberg-Guinan-Lund viscoplastic model32 is used for all materials.The yield strength and shear modulus depend linearly on the pressure. A melt model is included.

In 1D, Lagrangian MHD simulations of either the liner or anode, a model of electrical conductivity is required only for aluminum. To this end, the improved model of Desjarlais et al.10,33 is used. Similar models are used for other materials when required (as in 2D, MHD simulations of the experiment, Fig. 1(b)). Thermal conduction has no effect on simulated results.

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