Graded density impactors (GDIs) are multi-material composite impactors used in gas gun experiments to tailor the drive conditions imparted to a sample test material. Previous graded density impactors generally rely on thin, but discrete, layers of different materials. The thinner and the greater number of layers will result in smoother compression. Taken to the limit of very thin layers would be pure material 1 at one surface, such as the front surface of an impactor, smoothly transitioning at the atomic scale to pure material 2 on the back surface. Such an impactor can initially shock, then smoothly compresses a material during a dynamic experiment. This type of experiment can serve to explore a larger region of thermodynamic space than a single or even multi-shock experiments. An overview of how graded density impactors are made is reviewed and sample results are given. A strategy for modeling these kinds of impactors is presented. The length scales of constituent mixing are given from the experimental build through electrochemical-deposition. Equation of state models for pure constituents and their subsequent mixtures are presented. It is demonstrated that the time scales for pressure and temperature equilibration, for atomically mixed GDIs, are short enough to be a justifiable closure for the resulting multiphase flow. Furthermore, we present simulation results of dynamic shock followed by a ramp compression, utilizing a silver/gold graded density impactor, onto a tantalum sample.

## I. INTRODUCTION AND BACKGROUND

Understanding the behavior of materials under extreme conditions (i.e., megabar pressures and thousands of degrees of temperature), both experimentally and computationally, remains a long-standing challenge. Manipulating the high pressure response of materials is critical to advancing multiple commercial and academic fields including geology, high explosive and planetary science, condensed matter physics, materials science, and aerospace applications.

A variety of experimental techniques have been developed to investigate material properties under extreme pressures, $P$, and temperatures, $T$. These methods have evolved from quasistatic measurements using a diamond anvil cell to dynamic compression experiments driven by gas guns, lasers, and pulse power.^{1} Conventional dynamic compression experiments are generally restricted to the generation of shock-compressed states, which results in a rapid increase in temperature.^{2} This thermal energy can not only melt or vaporize the sample but also limits the maximum compression. In contrast, a controlled, continuous increase in the dynamic pressure (ramp compression) can enable compression rapid enough to make heat flow negligible but slow enough to avoid a shock. These experiments, termed isentropic/quasi-isentropic compression experiments (ICEs), place less constraints on the maximum density, $\rho $, achieved and can open up previously inaccessible regions in the $P$–$T$ space through single shock Hugoniots. Furthermore, ICE experiments yield a continuous curve of pressure–density states along the load curve compared to single or discrete points from shock or multi-shock experiments.

However, ramp compression experiments are restricted to a few facilities worldwide with costly lasers or pulsed power facilities. One such instrument is Sandia National Laboratory’s Z machine which is limited to 150–200 experiments a year at tremendous experimental cost.^{3} Recently, Lawrence Livermore National Laboratory’s National Ignition Facility has been utilized to obtain ramp compression of several materials.^{4} In contrast, light-gas guns are ubiquitous^{5–7} in high energy density physics laboratories and are routinely used to explore materials under extreme conditions. In a normal gas gun operation, a single material impact causes a single shock; to achieve ramp compression after this initial shock, a graded density impactor (GDI) is required. GDIs enable a tailored approach to custom design the shock impedance/density profile through the thickness of the impactor. In this manner, GDIs can be designed to contain combinations of shock, quasi-isentropic compression, and release in a single experiment.^{8} Thereby, GDI experiments can aid in the calibration of equation of state (EOS) models.

Figure 1 displays a model equation of state (EOS) of tantalum, with details presented in Sec. III C. Single shock experiments lie along the principal Hugoniot and they effectively query one point in EOS space. Measurements of release waves from single shock experiments can inform the region “to the left” of the principal Hugoniot (these release waves are not shown in the figure). GDI impacts can not only query the EOS along the principal Hugoniot, but can also serve to explore higher compression states at cooler temperatures relative to Hugoniot, as indicated by the shock and ramp compression curves. Like single shock experiments, release waves can be utilized to measure expansions from this compressed state. Importantly, a series of GDI impact experiments can be employed to measure the EOS surface in a continuous manner on both sides of the principal Hugoniot. This methodology can be utilized as a means to achieve high compression in reactive materials such as polymers and high explosives, where single shock measurements will induce chemical decomposition.

Various methods to fabricate GDIs for light gas gun experiments have been presented in the literature including tape casting,^{8} hot-pressed powder compacts,^{9} and bilayer sputter deposition.^{10} These techniques rely on the layer-by-layer assembly of thin slices with different compositions that are fused together to produce the targeted density/impendance gradient, which can be thought of as the limit of many layered impactors.^{11} Outside of experimental limits, these structures present challenges performing hydrodynamic simulations due to (1) need to fully resolve the various individual layers, which can become burdensome and computationally costly or (2) utilizing questionable thermodynamic closure models between the various phases of materials; such as assuming the layers/phases of materials are in pressure and temperature equilibration, when those equilibration time scales may take longer than the impact phenomena. A variety of thermodynamic closure models are explored in detail elsewhere.^{12}

Electrochemical deposition^{13} is an alternative approach to fabricate GDIs with a fully continuous composition variation throughout the structure. Elements remain atomically mixed in the required proportion to manipulate the impedance profile.

The objective of this work is to present a methodology for modeling atomically mixed GDIs. The structure of this manuscript is described next. Section II examines the length scales of GDIs as they pertain to hydrodynamic modeling. Section III presents thermodynamically complete Davis EOS and calibration for the specific materials examined in this GDI study. Section IV describes the modeling strategy for treating atomically mixed GDIs. Predictions of a prototypical GDI impact experiment are given in Sec. V. Section VI summarizes the conclusions and possible avenues for future work.

## II. LENGTH AND TIME SCALES

Considering all GDIs involve the use of multiple materials, one must consider the length scales of the resulting composite material heterogeneity. In particular, if there are O(100–1000) layers of materials, one may consider modeling the constituents individually in computational hydrodynamic modeling. Recently, Brown *et al*.^{10} have directly modeled 100 bilayers, each 0.34 $\mu m$ in thickness with such a methodology. For GDIs that are atomically mixed, such a strategy of modeling the individual materials would of course be infeasible, as it would require computationally bridging atomic length scales with engineering length scales millions of times larger. The cross section of an atomically mixed GDI is given in Fig. 2. It shows scanning electron microscope images at several locations throughout the GDI, as well as a schematic showing the atomically mixed nature. These GDI materials were recently fabricated in our laboratory, and detailed presentation of the methods of production and characterization are the subject of an upcoming paper. Some key points of the experimental process and results are summarized here: Experimental graded density impactors were electroformed from electrolytes containing silver cyanide and potassium gold cyanide. The samples were cross-sectioned and polished with silicon carbide paper with grits at 400, 600, 800, and 1200. The cross-sectional composition was collected via three distinct line scans using energy-dispersive spectroscopy. These line scans were then averaged together and smoothed via a Savitzky–Golay filter of polynomial order 2 and window frame length of 51. The smoothed compositional profiles shown in Fig. 3 indicates three distinct segments: (1) a segment of pure gold, (2) a gradual transition from gold to silver, and (3) a pure silver segment. Note the 10% discrepancy between the displayed compositional curve and the expected compositional profile likely arises due to smearing of the soft silver and gold during polishing. The details of our experimental methodology will be presented in an upcoming manuscript.

Computational modeling of mixtures of two constituents at the continuum level stems from examining full two-phase flow modeling.^{14} Full two-phase flow modeling allows independent velocity, pressure, and temperature fields to evolve in the different phases, with source terms, such as drag and thermal diffusion to drive equilibrium between the phases. Considering the the GDIs are solid materials, it is presumed that each phase will be in velocity equilibrium.

The time scale, $\tau p$, for pressure equilibration between phases scales will scale proportionally with the length scale, $L$, of the material and inversely with the acoustic speed, $c$, as

The time scale for thermal diffusion, $\tau T$, is proportional to the length scale squared, $L2$, and inversely proportional to the thermal diffusivity, $\alpha $, as

Generally speaking for condensed phase materials, an upper bound on the time scales can be made by the examination of the above time scales at the quiescent state. This is due to the fact that the length scales will become shorter under compression, and both acoustic speed and thermal diffusivity generally become larger under compression.

For both silver and gold, the sound speeds are approximately 3 mm/$\mu $s and the thermal diffusivities are approximately $10\u22124$ mm$2$/$\mu $s. For an atomically mixed GDI, the appropriate length scale is the atomic spacing, which for both silver and gold are approximately $10\u22127$ mm. Thus, both the pressure and thermal time scales are very short: $\tau p\u223c10\u22127$ $\mu $s and $\tau T\u223c10\u221210$ $\mu $s. One may argue that the above time scale estimates are not applicable at the atomic length scale, but even if one examined mixing at the $10\u22123$ mm scale, one finds the equilibrium time scales to be $10\u22123$ $\mu $s, which is sufficient for most GDI experiments as modern photon Doppler velocimetry is of that same time scale. From this scale analysis, it is likely that the experiments by Brown *et al*.^{10} reach local thermal equilibrium within their bilayered GDI. This demonstrates the justification of pressure–temperature equilibration between phases within an atomically mixed GDI. We note that a variety of empirical mixing rules for multi-material flow have been utilized in the past.^{15–19} Many of these methodologies only give resulting mixture Hugoniots or are not thermodynamically complete, which does not allow naturally changing the initial temperature of the GDI, for example.

## III. EQUATION OF STATE OF CONSTITUENTS

Here, a general methodology is presented for the calibration of single phase constituents applicable to the metals of interest for GDI simulations. A complete and thermodynamically consistent EOS calibration procedure is outlined and parameters are given for the three metals of interest, namely, silver, gold, and tantalum. Constructing a thermodynamically complete EOS is required when performing pressure and temperature equilibration between phases. A complete EOS can naturally allow GDIs to be used at a variety of initial temperatures, if so desired. This could be important in situations where one needs to account for thermal expansion/contraction of the GDI, where temperature can vary, such as the experiments of Wise *et al*.^{20}

The specific EOS form studied here is the Davis EOS for solids (sometimes referred to as the Davis reactants EOS when used in explosives modeling). The Davis EOS is of the Mie–Grüneisen form, where the reference curve, $Pr(\rho )$ is given along the principal, room-temperature, isentrope. Based on the work of Enig,^{21} Davis^{22,23} derived a reference curve using the Walsh mirror-image approximation^{24} and the Riemann relation for the particle velocity on an isentrope. To allow for greater flexibility at high pressures, Stewart *et al.*^{25} changed the fitting form by using a truncated expansion of the original exponential form. The final form of the reference curve is written as

where $y=1\u2212\rho 0/\rho $, $P^=\rho 0A2/4B$ and $\rho 0$ is the density at room temperature and zero pressure. Both branches in Eq. (3) are continuous at $\rho =\rho 0$ up to O($y$), so the derivatives are also continuous. It is important to keep both branches as the compression side has more parameters to allow greater flexibility in fitting experimental data, while the simple exponential in the expansion branch allows for robust behavior in expansion. For positive $A$, $B$, and $C$, it can be shown that the Davis reference curve has a positive bulk modulus for all positive densities.^{26} Equation (3) contains three fitting parameters $A$, $B$, and $C$. The parameters $A$ and $B$ can be related to the isentropic bulk modulus, $K0$, and its derivative with respect to pressure at the zero pressure state, $K0\u2032$, and room temperature:^{27} $K0=\rho 0A2$, $K0\u2032=4B+1/2B\u22121$. As such, $A$ represents the adiabatic bulk sound speed at the reference state of $P=0$ and $T=T0$ and $B$ is related to the slope, $s$, of the Hugoniot curve in the shock particle, $Up$, and shock speed, $Us$, plane: $s=14(K0\u2032+1)=B+18B$. The $C$ parameter accounts for changes to the reference curve at high compression and is related to the third derivative off of the isentropic bulk modulus at the zero presure state, $K0\u2034$.

In addition to the reference isentrope, one needs a model for the Grüneisen parameter to model the EOS of the reference curve

where $e$ is the specific internal energy and $er$ is the specific internal energy on the reference curve

For the Davis model, the Grüneisen parameter is given by

where $Z$ is a constant used to describe the changes to $\Gamma $ with respect to density. The above description gives the hydrodynamic response of the material, namely, the $e\u2212P\u2212\rho $ relations. Additionally, to describe the thermal portion of the EOS, one needs a model for the specific heat at constant volume, $Cv$. It can be shown that the Davis model for $Cv$ at the reference density is given by^{27} a power law

For the Davis EOS, the following parameters are required to define the model: $\rho 0$, $A$, $B$, $C$, $\Gamma 0$, $Z$, $Cv0$, $\alpha st$, and $T0$. Here, the reference temperature is taken to be $T0=297$ K. Also, $Cv0$ is the specific heat at constant volume at the reference density, $\rho =\rho 0$, and reference temperature, $T=T0$. The Davis EOS has been successfully utilized in a variety of models including the Wescott-Stewart–Davis (WSD)^{28} and Arrhenius–Wescott–Stewart–Davis (AWSD)^{29,30} reactive burn models. Further EOS details can be found in those references. Nothing precludes one from utilizing this EOS form for metals. Following those references, the calibration procedure can be described in a few, relatively straightforward steps. First, the specific heat model is calibrated to experimental or theoretical models, such as a Debye model, up to the melt temperature, $Tm$, which sets $Cv0$ and $\alpha st$. The Grüneisen model is then fit to match thermal expansion behavior at the reference state $\Gamma 0=\beta A2Cp0$, where $A$ is the sound speed, $Cp0$ is the specific heat at constant pressure at the reference state and $\beta $ is the volumetric thermal expansion at the reference state $P=0$ and $T=T0$. For metals, the parameters $A$, $\beta $, and $Cp0$ are found in standard reference texts. Here, we take $Z=1/2\u2212\Gamma 0$ to yield a Grüenisen parameter of $1/2$ at infinite density.^{31} Finally, experimental shock Hugoniot data are utilized to determine $B$ and $C$.

### A. Silver

For silver, if we take a Debye temperature of 215 K with a Dulong–Petit specific heat limit, $CvDP=0.000231$ kJ/g K, and a melt temperature of 1235 K, the Davis specific heat parameters that best match are determined to be $Cv0=0.000224$ kJ/g K and $\alpha st=0.0289$. Measured room temperature specific heat at constant pressure, $Cp$, volumetric thermal expansion, $\beta $, and sound speed measurements^{32} lead to a Grüneisen parameter at the reference state of $\Gamma 0=2.5$. With these thermal and Grüneisen parameters, the other reference isentrope parameters, $B$ and $C$, can be determined by fitting to experimental Hugoniot data.^{32–36} Figure 4 displays the match to data with the final parameter set given in Table I.

### B. Gold

With a Debye temperature of 170 K with a Dulong–Petit specific heat limit, $CvDP=0.000127$ kJ/g K, and a melt temperature of 1337 K, the Davis specific heat parameters for gold that best match are given by $Cv0=0.000124$ kJ/g K and $\alpha st=0.0226$. Measured room temperature specific heat at constant pressure, volumetric thermal expansion, and sound speed measurements^{32} lead to a Grüneisen parameter for gold at the reference state of $\Gamma 0=2.9$. With these thermal and Grüneisen parameters, the reference isentrope parameters can be determined by fitting to experimental Hugoniot data.^{32–37} Table II gives the final parameter set and Fig. 5 displays the match to Hugoniot data.

### C. Tantalum

For tantalum, with a Debye temperature of 240 K with a Dulong–Petit specific heat limit, $CvDP=0.000138$ kJ/g K, and a relatively high melt temperature of 3290 K, the Davis specific heat parameters that best match are determined to be $Cv0=0.000134$ kJ/g K and $\alpha st=0.0142$. Measured room temperature specific heat at constant pressure, volumetric thermal expansion, and sound speed measurements^{32} lead to a Grüneisen parameter for tantalum at the reference state of $\Gamma 0=1.6$. With these thermal and Grüneisen parameters, the reference isentrope parameters can be determined by fitting to experimental Hugoniot data.^{32,33,36,38–44} Figure 6 displays the match to data with the final parameter set given in Table III. This complete EOS is shown in Fig. 1.

Parameter . | Value . | Unit . |
---|---|---|

ρ_{0} | 16.656 | g/cm^{3} |

A | 3.4 | mm/μs |

B | 1.0 | |

C | 2.0 | |

Γ^{0} | 1.6 | |

Z | −1.1 | |

$Cv0$ | 0.000 134 | kJ/g K |

α_{st} | 0.0142 | |

T_{0} | 297 | K |

Parameter . | Value . | Unit . |
---|---|---|

ρ_{0} | 16.656 | g/cm^{3} |

A | 3.4 | mm/μs |

B | 1.0 | |

C | 2.0 | |

Γ^{0} | 1.6 | |

Z | −1.1 | |

$Cv0$ | 0.000 134 | kJ/g K |

α_{st} | 0.0142 | |

T_{0} | 297 | K |

It is noted that the Davis EOS model represents the behavior of a single-phase material. As such, at temperatures above the melt point, the model will lose accuracy. The EOS at that point is likely to have larger thermal errors, but the overall fits in the $Us\u2212Up$ plane are apparently not overly sensitive, as the highest shock compressions reached in Figs. 4–6 assuredly are molten.^{45}

## IV. THERMODYNAMIC CLOSURE AND HYDRODYNAMICS OF MIXTURE

As indicated in Sec. II, it is appropriate to presume the two-phase flow is under limits of local pressure and temperature equilibrium with a single-particle velocity. As such, the Euler equations of conservation of mass momentum and energy are appropriate,

where $u\u2192$ is the velocity vector. Additionally, one needs to keep track of the mass fraction of constituents for the GDI. Here, $\lambda $ is defined as the mass fraction of gold, and thus by construction the mass fraction of silver is simply $1\u2212\lambda $. The evolution equation of $\lambda $ is simply given by

Here, $DDt$ is the material, Lagrangian or total derivative defined by $DDt=\u2202\u2202t+u\u2192\u22c5\u2207\u2192$.

From the above, we have the primitive variables $\rho $, $u\u2192$, $e$, and $\lambda $. To close the system, one needs $P(\rho ,e,\lambda )$. For $P\u2212T$ equilibration between the two-phases, with a fixed mass fraction of silver and gold, the underlying algebraic systems of equations to be solved is effectively determined by how the volumes and energies are distributed between the phases. For Mie–Grüneisen EOS models, this can be boiled down to a single nonlinear algebraic equation in the volume fraction, $\varphi $, of gold.^{46} Other thermodynamic quantities necessary for hydrodynamic simulation can be found in the literature.^{46–48}

## V. SIMULATION OF SILVER–GOLD GDI IMPACT INTO TANTALUM

Here, a prototypical computational demonstration of a silver–gold GDI impacting tantalum is investigated. One of the quantities of interest, when devising GDIs, is knowing how to distribute the mass fraction profile and how that distribution affects the shock and ramp compression in test materials. This is a prime example where modeling and simulation can play an important role.

From Fig. 3, we observe a sigmoidal spatial profile for the mass fraction of constituents, with roughly $\u223c1$ mm of transition from silver to gold. For demonstration purposes, the assumed mass fraction of gold, $\lambda $, throughout the GDI at the initial state is be given by

The mixture density is given by the ideal mixing relation

Note that it is important that during the fabrication of GDIs that the modeled density is achieved. More often than not, mixtures of materials or tape fabrication of GDI^{50} can potentially introduce porosity inadvertently.

For the demonstration problem, we assume the GDI extends from $\u22124.75mm<x<0.75mm$. We presume an initial GDI velocity of $2$ mm/$\mu $s. The quiescent tantalum target is presumed to be 2 mm in thickness and initially is located $0.75mm<x<2.75mm$. The resulting density, zoomed in near the GDI and tantalum, at the time of impact, is shown in Fig. 7. The smooth transition from silver density of 10.49 g/cm$3$ to a gold density of 19.24 g/cm$3$ is observed.

The computation is carried out in a Lagrangian framework, with an initial grid spacing of 0.005 mm. The simulation is integrated until $t=1$ $\mu $s. Figure 8 shows the predicted free surface velocity of the tantalum, initially located at $x=2.75$ mm. Figure 9 shows the simulated pressure at the interface between the GDI and the tantalum. The initial jump at the free surface is primarily due to the shock drive from the silver side of the GDI at $\u223c62$ GPa. As time progresses, the ramp compression from the gold side of the GDI at $\u223c82$ GPa accelerates the tantalum even further, in a smooth fashion. Near $t=0.8\mu s$, one observes the rapid decrease of pressure at the interface from the reflected rarefaction from the shock breaking out from the free surface.

A complete $x\u2212t$ diagram showing the entire simulated density field is shown in Fig. 10. Here, one can see the initial gradient in density, material interfaces and the resulting shocks from the impact and release wave from the tantalum free surface.

## VI. CONCLUSIONS

A physically realistic modeling framework for atomically mixed graded density impactors has been presented. For atomically mixed two-phase flow, a model based on local pressure and temperature equilibration is fully justifiable by a simple scaling analysis. A sample prototypical simulation using a representative experimental GDI was performed and showed what EOS states can be achieved with an impact of a silver–gold GDI into tantalum. Simulations, of the sort demonstrated here, can be used to help guide GDI gradients and design of experiments to further understand materials at extreme conditions.

## ACKNOWLEDGMENTS

The authors thank Eduardo Lozano for examining properties of the Davis EOS. This work was supported by the U.S. Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). This research was supported by the Advanced Simulation and Computing Program (ASC) and the Dynamic Materials Properties Campaign (C2) under DOE-NNSA. Work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Research presented in Sec. II of this article was supported by the Laboratory Directed Research and Development program of Los Alamos National Laboratory under Project No. 20190658PRD4.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Tariq D. Aslam:** Conceptualization (equal); Formal analysis (lead); Investigation (equal); Methodology (equal); Software (lead); Visualization (equal); Writing – original draft (lead); Writing – review & editing (lead). **Michael A. McBride:** Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Validation (equal); Visualization (equal); Writing – original draft (equal). **Nirmal Rai:** Formal analysis (equal); Investigation (equal); Software (supporting); Writing – original draft (equal). **Daniel E. Hooks:** Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal). **Jamie A. Stull:** Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). **Brian J. Jensen:** Conceptualization (equal); Investigation (equal); Writing – original draft (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in references as well as the Shock Wave Database: http://www.ihed.ras.ru/rusbank/.

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