Time resolved x-ray diffraction in shock compressed systems

In this paper, we give a perspective on the topic of time resolved x-ray diffraction in shock compressed systems. Our goal is to highlight results in this rapidly growing field and place them in the context of what can be learned from these techniques. We caution that this is by no means a comprehensive review of work in this area. In addition, while the primary focus is on x-ray diffraction, other time resolved x-ray techniques have been included as well. We have chosen to primarily highlight the work that has been published in the past 2 years (2019 and 2020). Even here, we have chosen a subset of material representing different aspects of techniques and the scientific questions being addressed by these techniques. These examples are drawn in part from recent work by the authors, as well as examples from many groups working in this field. We have attempted to give some balance between experimental results and those of simulation/modeling.

We begin with a brief overview of the physics of shock compression (Sec. I), followed by a discussion of current shock platforms next to x-ray sources, discussing both the x-ray sources and the shock drivers (Sec. II). We then discuss recent experimental results, including the topics of melting, solid–solid phase transformations, equation of state, and chemical phenomena (Sec. III). This is followed by a discussion of simulation and modeling work often needed to fully utilize the experimental x-ray results, including techniques such as Kohn–Sham Density Functional Theory (DFT) as well as new molecular dynamics simulation approaches (Sec. IV). Finally, we conclude with a perspective on the future of the field (Sec. V).

Historically, dynamic compression experiments have been employed to obtain the equation of state of materials via measurements of the compression wave and material speeds within the shocked material. The Hugoniot relations¹ that represent the conservation of mass, momentum, and energy across the shock front allow the determination of the pressure, density, and specific energy of the final equilibrium state. Dynamic compression experiments are usually performed in two limits: (1) shock wave compression, where shock rise times can be as short as picoseconds,² and (2) quasi-isentropic compression, where compression (while still faster than thermal diffusion, i.e., adiabatic) is, in the ideal case, slow enough to be thermodynamically reversible.³ 

In an ideal shock wave experiment, homogeneous fully equilibrated initial and final (compressed) states are separated by a spatially infinitesimal, temporally instantaneous planar shock front. Actual shock wave experiments are confounded by a number of practical complications, including non-planar shock fronts,¹⁰ strength effects such as elastic precursors,^11,12 and structure due to sample inhomogeneity and anisotropy.¹³ Often, materials undergo kinetically limited transitions prior to reaching the final state, typically in the order: 1D elastic compression, plastic relaxation of the initial phase, followed by phase transitions (if the final state is a different phase).

Although characterization of shocked materials via pulsed x rays can be used to determine quasi-static equations of state, often material properties are significantly impacted by the path-dependent compression conditions imposed by shock compression. The strongly non-equilibrium states induced by shock compression play a fundamental and as yet poorly understood role in fundamental issues such as dissipation generation via strength,^11,14 chemistry,¹⁵ and phase transitions.^16–19 

Quasi-isentropic compression experiments are typically more complex in both execution and analysis, involving sophisticated methods to generate relatively slow,^3,20,21 but well characterized, compression waves, in some cases over hundreds of nanoseconds or longer. Because ideal isentropic compression does not generate energy dissipation, quasi-isentropic compression experiments generally obtain thermodynamic states at lower temperature and pressure than shock compression to a given final density and are often used to explore the phase diagram and transformation kinetics^3,22 under conditions that cannot be obtained by shock compression (see Fig. 2). (The density of shock compressed states is limited by dissipation generation. As shock strength increases, density approaches a limiting value as pressure and temperature increase without limit.²³ For monatomic gases, this limit is 4× the initial density.) For the sake of brevity, here we limit discussion to shock wave compression.

Shock waves compress the sample at a faster rate than quasi-static (or isentropic) equilibration can occur. Thus, the sample accesses far-from-equilibrium mechanical, phase, or chemical states whose relaxation generates dissipative energy. A common example of this is shock-induced elastic–plastic relaxation. In this case, the rapid application of longitudinal stress induces transient, extreme states of elastic compression in solids. These stress states relax via plastic deformation under sustained longitudinal stress. In ultrafast compression experiments, elastic stresses (due to reversible compression in the absence of dissipation) can be been found to be as high as 100× larger than the yield strength (pressure above which a solid system begins to deform irreversibly or plastically).^2,12,24 This can occur subsequent to a picosecond-scale rise in the longitudinal stress. (In shock compression experiments, longitudinal compression is enforced by employing a sample with a much larger transverse scale than the expected longitudinal scale of the experiment, i.e., 1D compression. Under these conditions, transverse rarefaction waves from the uncompressed edges of the experiment do not have sufficient time to reach most of the compressed region of the sample. This configuration effectively isolates the transverse-center region of the sample, maintaining 1D compression over this region.)

In conventional materials, sound speed increases with increasing compression. Thus, regions of higher compression downstream continually catch up to regions of lower compression upstream, termed “wave steepening” in the dynamic compression literature. Wave steepening is ultimately limited by the formation of shock waves, where, as the rise in stress becomes faster (ultimately becoming faster than the time to equilibration), the sample in the dynamically compressed region accesses further-from-equilibrium states, generating more dissipative energy at shorter compression times. Ultimately, the generation of dissipative energy upon relaxation of far-from-equilibrium states limits further wave steepening, resulting in steady shock compression.^1,6

One interesting feature of shock experiments is that they can differ in results on different timescales. For example, gas gun experiments tend to be on nanosecond timescales or longer, usually sufficient to result in equilibrium phases or chemical products.^25,26 In contrast, ultrafast table top experiments can probe picosecond timescales, which can result in the observation of transient species. This is not a problem with either technique, but rather exemplifies that the physics occurring can be different in the different time regimes. This is particularly true for phase transitions, and we discuss specific examples in Sec. III.

In contrast to phase transitions, other measurements such as the Hugoniot equation of state measurements generally yield equivalent results between long time gas gun data (the gold standard for equation of state data) and those obtained from 100 ps experiments.²⁷ This implies that kinetics does not play a large role in reaching the thermodynamic limit for the compressed material and remains principally unchanged once established. Phase transitions, on the other hand, can be dominated by kinetic effects and look incredibly different on the different timescales.

X-ray diffraction is widely used in static and dynamic experiments to measure the density and crystal structure of compressed states, most often under quasi-equilibrium conditions (i.e., the final shock compressed state). Although the transition region between the uncompressed and compressed regions (i.e., the shock front) has been addressed via simulations in a variety of contexts,^4,11,28–30 obtaining actual measurements over the shock rise is challenging because this region is typically extremely thin (in the sub-micrometer range for many materials compressed to >10 GPa pressure). This naturally limits the observable signal and yields a physical state that is both far from equilibrium and heterogeneous at the nanometer scale. For instance, MD simulations of dislocation generation/migration under the extreme anisotropic compression found in a shock wave rise^11,31 suggest a highly inhomogeneous mixture of elastic/plastic compressed states in the presence of a rapidly evolving high density distribution of dislocations. Further, it is critical to connect the mechanical state (macroscopic stresses and strains) of the sample to microscopic variables such as dislocation density. The principal conventional diagnostic used to characterize materials under dynamic compression is surface velocimetry, which obtains only aggregate continuum behavior (e.g., average deviatoric stress/strain)^13,32,33 and does not provide microscopic information about fundamental inelastic or chemical transformation mechanisms. As such, a comprehensive connection from micro- to macro-scale requires an admixture of molecular dynamics (MD) simulations, hydrodynamic simulation, continuum hydrodynamic data (i.e., velocimetry), and atomic to dislocation scale data. If fundamental phenomena are thought to play a significant role in the final state of the material or continuum modeled dynamic behavior such as kinetically limited transformation rates, it is likely that x-ray methods (and possibly other short wavelength probes such as neutrons) are the only available option to empirically calibrate MD simulations of nonequilibrium shock conditions.

Facility-scale scientific short pulse x-ray generators currently include synchrotron³⁴ (third generation) sources, x-ray free electron laser³⁵ (XFEL, or fourth generation, which also employs synchrotron radiation) sources, and thermal x-ray emission from laser-driven plasmas.^18,36 A principal difference between these sources is the total interaction length—the distance over which accelerating electrons emit x rays. Longer interaction lengths increase both the total photon flux and the coherence of the emitted radiation. While both types of synchrotron radiation-based sources employ undulators to generate x-ray radiation from high energy (5–10 GeV) electron beams, synchrotrons typically employ at most a few undulators with undulator lengths at most ∼10 m along a given beam line. In contrast, XFELs employ several 10 s of undulator modules and can reach total lengths greater than 100 m. Analogous to optical laser gain, the x-ray pulse energy grows nonlinearly with interaction length, so these very long interaction lengths allow XFELs to obtain approximately 1000× higher x-ray pulse energy than a synchrotron, and 10⁹ higher brightness.³⁵ 

Synchrotrons are storage ring accelerators³⁴ typically with many experimental beamlines, possibly employing multiple undulator options per beamline, and typically have multiple hutches/experimental stations (with different instrumentation) for a given facility (35 beamlines at APL, 14 beamlines at PETRA III). Because synchrotrons usually have several options for undulators and accelerator operational modes, the range of operation (particularly x-ray photon energy) is large—photon energy at synchrotrons can range from the mid-IR optical range to >100 keV (but not with the same undulator or beamline usually). Here, we will mostly limit discussion to the most commonly used x-ray energy and accelerator modes for compression experiments. Laser plasma-driven sources emit thermal x-ray radiation from an ablation-generated plasma and are limited by Stimulated Raman Scattering (SRS) to the brightness of thermal emission under illumination at ∼10¹⁵ W/cm².³⁷ 

For dynamic compression experiments (which are typically single shot), the number of photons in a single x-ray pulse is more relevant than average power since experiments will typically interrogate samples with a single pulse (for a single detected frame per shot) or a sequence of time-delayed pulses, enabling a “movie mode.” Generally, the throughput of such experiments is strongly tied to the scale of the experiment. For laser-driven experiments, the required pulse energy scales as the third power of the duration of the experiment.³⁹ In other words, a nanosecond long experiment will require 10⁻³ of the power of a 10 ns study. This implies that (for constant power and variable repetition rate) the x-ray throughput varies as the inverse square of the desired experimental timescale—a 10× smaller experiment can obtain 100× higher throughput⁴⁰ (see Fig. 3 for an example of a high throughput sample substrate). For the sake of experimental throughput, laser-driven experiments should be the smallest scale necessary to observe the phenomena of interest. However, in many cases, experimental scales must be in the 100s μm–mm range. This includes inter-grain strength effects and high explosive phenomena, where time and length scales to equilibrium span these larger domains.

Peak brightness is a commonly used measurement to determine the quality of an x-ray pulse. This is defined as a density of the pulse energy per unit of a number of different experimental variables, including pulse duration, area, angular divergence, and spectral width. XFELs have substantially higher brightness than synchrotrons (typically around 10⁹ higher), roughly half of this increase is contained in the time-bandwidth product, which results in ∼1000× shorter pulses (∼100 fs vs ∼100 ps for synchrotrons). However, often for dynamic compression experiments, at least some of this extra brightness is not needed because the required time resolution is usually at least 10 s of ps or greater. Ultrafast compression experiments^2,41,42 (where it is desired to observe phenomena at 10 ps timescales or less) are an exception. In part because of the time resolution and in part because the samples are substantially thinner (some a few micrometers for ultrafast vs tens of millimeters for larger scale compression) such experiments can only be performed at XFELs. Similarly, for imaging experiments, often sufficiently small beam divergence can be obtained at the desired spot size (i.e., a large spot size that does not require a tight focus), even at synchrotrons, but where an ∼μm size spot with small beam divergence is needed (also required for ultrafast experiments), XFELs are preferred.

Various x-ray sources are displayed in Table I. Ablation-based sources are typically limited by stimulated Raman scattering and have similar properties regardless of facility. Since the output of undulator-based sources can vary over a wide range in many parameters, we cite typical values for techniques/facilities. Accelerator-based x-ray generators may also have several operational modes, which impact the distribution of total current (and x-ray energy per pulse) among pulses within a pulse train. Here, we have focused on experimental stations whose capabilities best align with dynamically compressed materials [e.g., the Dynamic Compression Sector (DCS) at APS] and parameters for common methods. Generally, experimental capabilities depend strongly on context, so potential experiments should be discussed with a beamline scientist well in advance of submitting a proposal.

Laser emission sources have substantially lower photon flux and coherence than synchrotron or XFEL sources, but are generally the only short pulse x-ray sources available at high flux laser-driven sources such as the National Ignition Facility at Lawrence Livermore National Laboratory, or the Omega laser at the University of Rochester.⁴³ It is difficult to summarize the capabilities of all facilities. It is possible to obtain estimates of the likely data quality of a given method by considering prototypical previous work and scaling arguments considering bandwidth, beam quality, and the number of photons per pulse.

Shock drivers can be categorized by timescale of the driving event, ranging from very long (up to some μs) in gas gun and pulsed power-based drivers and down to 100 s of ps in ultrafast laser drives (see Table II). Typically, long timescale drivers are best suited to shock and quasi-isentropic EOS measurements. Lasers can also obtain EOS measurements, but with caveats for material kinetics, particularly for sub-ns compression, where material relaxation (e.g., inelastic deformation in brittle materials such as SiO₂) and long timescale phenomena (such as deflagration to detonation) can be much longer than the compression time. For all experiments intending to obtain quasi-static parameters (such as the EOS), the timescale of the driver must be longer than the relaxation kinetics.

Shock drivers generally include explosive,⁴⁴ impact,⁴⁵ and pulsed lasers.⁴⁶ Since laser energy scales nonlinearly with experimental duration,³⁹ laser compression to pressures greater than some 10 s of GPa for more than ∼10 ns requires facility-scale compression (i.e., NIF/Omega), which is typically not available at x-ray source facilities. A typical trade-off between the need for performing experiments at a large-scale facility vs an in-house setup is a balance between compression needs vs throughput (either directly through repetition rate or via limited access to large facilities). For instance, liquids (which relax quickly) can be easily investigated in the table top experiments.^42,47,48 However, materials with longer timescale dynamics (e.g., fused silica or detonating carbon-rich explosives) require large-scale facilities for compression studies in order to obtain enough energy output for the required duration. X-ray beam quality is also exchanged for long-timescale/very high-pressure compression since very high-quality x-ray facilities do not exist at very high energy laser facilities.

Experimental observables of interest in dynamic compression experiments historically includes equation of state (EOS), dynamic material strength, phase transitions, and fundamental material transformation mechanisms such as the dislocation dynamics in plasticity or molecular intermediates in shock induced chemistry (and the resulting kinetics). One of the great challenges of characterizing material transformation properties is the extremely broad range of spatial and temporal scales required to fully understand these complex phenomena. In liquids where chemistry does not occur (shocked liquid argon, for instance), equilibration subsequent to shock compression is thought to occur over ps or faster timescales. In some solids (such as aluminum) where phase transitions do not occur at low (<100 GPa) pressure, relaxation of shock compression-induced deviatoric stress to an equilibrated final state can occur over timescales ranging from tens of ps (at around 40 GPa pressure) to many hundreds of nanoseconds at single GPa pressures^2,14—a variation over four orders of magnitude in timescale for a one order of magnitude change in pressure.

Comparable nonlinearities are found throughout shock compression physics. The Swegle–Grady law—the relation between final shock pressure (or stress) and the shock strain rate—is a fourth power law. Aluminum and glass vary by only a factor of two in quasi-static compression strength, yet the timescale of relaxation of these materials under comparable shock pressures varies over orders of magnitude. Fast chemistry in explosives without carbon can occur over tens of picoseconds,^26,47 but with the addition of carbon, full chemical equilibration requires hundreds of nanoseconds or longer.⁴⁹ These types of considerations and the desired outputs are important for the choice of an x-ray source for a given experiment.

In this section, we describe results from a variety of different experiments, all published within the past 2 years. These descriptions are by no means comprehensive and represent a sampling from the authors’ own work as well as those from a variety of groups worldwide. Our goal is to give the reader a reasonable broad brush of the work in this rapidly expanding area of science. We focus on three areas: melting, solid–solid phase transitions, and equation of state/chemical transformations. While a majority of the experiments discussed involve time resolved x-ray diffraction, we also include some that deploy other time resolved x-ray techniques such as imaging, spectrally resolved x-ray scattering,⁵⁰ and small angle x-ray scattering (SAXS). In experiments using ns drives, the pressure conditions are normally determined from velocimetry, while for those using ps drives the pressure conditions are determined from the x-ray diffraction data since ps velocimetry is not yet available at facilities such as Linac Coherent Light Source (LCLS).

Melting and freezing are commonly observed phase changes,⁵¹ yet can be difficult to characterize since the fundamental mechanisms occur on near-atomic and picosecond timescales.^52–55 A recent paper⁴¹ demonstrates melting and recrystallization in zirconium, observed using ultrafast compression experiments with femtosecond x-ray diffraction at the matter in extreme conditions (MEC) sector of the Linac Coherent Light Source. Figures 4 and 5 show that it is possible to observe the melting of metals on the timescale of a few hundred picoseconds. This work indicates the possibility of following recrystallization paths into different phases from the time of compression out to the very long time of 50 ns. In this case, recrystallization yields the bcc β phase, as opposed to returning to the original α hcp phase. To our knowledge, this observation at a few hundred picoseconds is the shortest time observation of mechanical melting, as differentiated from non-equilibrium short time melting observed at the femtosecond timescale.

Briggs et al.⁵⁶ have recently looked at atomic nearest neighbor coordination changes in the liquid tin under shock compression using femtosecond x-ray diffraction at LCLS (Fig. 6). They report on liquid structure factor and radial distribution function measurements. A 15 ns laser pulse (527 nm Nd:glass laser) was focused on a 250–350 μm spot on a polyimide plastic ablator to generate a shock within the sample. Sn foils of 20 μm thickness were attached to the ablator and a 150 μm LiF window using thin glue bonds of ∼1 μm thickness. The Sn samples were shock compressed along the principal Hugoniot into the liquid phase, where the evolution of the liquid structure was determined up to ∼87 GPa. They find that the liquid structure evolves from a complex structure, with a low coordination number, to a simple liquid structure with a coordination number of ∼12 with increasing pressure.

In an example of shock melting of simple metals, Sharma et al. conducted x-ray diffraction measurements on shock-compressed gold using the 100 Joule laser drive system at DCS (see Table II), spanning pressures up to 350 GPa and temperatures up to 4000 K. They observed a set of liquid-bcc coexistence points between 220 and 302 GPa, with complete melting found by 355 GPa.⁵⁸ In contrast, Millot et al.⁵⁹ have used ns laser-driven shockwaves to simultaneously compress and heat liquid water samples to 100–400 GPa and 2000–3000 K. They show that under these conditions water solidifies within a few nanoseconds into nanometer-sized ice grains that exhibit unambiguous evidence for the crystalline oxygen lattice of superionic water ice. These studies are indicative of how shock compression and x-ray sources can be applied to commonly studied systems such as gold and water and are expected to be applied to increasingly complex systems.

As an example of a nanosecond driven shock experiment, Tracy et al.⁶⁰ have used the MEC at LCLS to laser-shock SiC up to 206 GPa. Single crystals and polycrystals of different polytypes exhibit a transformation from a low-pressure tetrahedral phase to the high-pressure rocksalt-type (B1) structure. These experiments demonstrate the coexistence of the low- and high-pressure phases in a mixed-phase region and complete transformation to the B1 phase above 200 GPa. These data are shown in Figs. 7 and 8 and focus on the coexistence upon release.

Moving to short-pulsed laser experiments, Hwang et al.⁶¹ have used a 140 ps laser driver at the Pohang Accelerator Laboratory XFEL to look at the structural dynamics of iron from high-quality x-ray diffraction data, measured at 50-ps intervals up to 2500 ps. They identify a three-wave structure during the initial compression and a two-wave structure during the decaying shock, involving three phases of iron (α-, γ-, and ε-phases). The shock conditions are determined only from hydro simulations, without any benchmarks to calibrate the calculations. They compare the calculated results with the expected density measured by x-ray diffraction and find reasonable agreement. Finally, we discuss work on solid zirconium where the system was driven with a short-pulse 150 ps laser drive at the MEC/LCLS³⁸ and pressures were determined directly from x-ray diffraction data.³⁸ These short pulses create unsteady waves that required specific physical characterization, as opposed to standard longer pulse studies that achieve steady (flat top) compression waves but require long time periods and larger samples. However, short pulses allow for observation with a much greater time resolution of the phase transitions which tend to occur on ps times. In addition, there was an observation for the first time of an intermediate beta phase, which had been previously predicted. Also observed was the transition into the equilibrium bcc phase over sub-ns timescales. Figure 9 shows the x-ray diffraction pattern for three different pressures, along with the hydro simulations of the waves propagating at those conditions.

There have been a number of works looking at phase transformations/chemistry in carbon-containing species^63,64—we highlight here several examples from the last 2 years. In many cases, interesting results were found that did not obtain x-ray measurements (e.g., CO), and we expect such studies to be performed with x rays at some point in the near future.⁴² 

Bagge-Hansen et al.,⁴⁴ in measurements at DCS, reported dynamic measurements of liquid carbon condensation and solidification into nano-onions over approximately 200 ns by using time resolved, small-angle x-ray scattering data acquired during detonation of a hydrogen-free explosive, DNTF [3,4-bis(3-nitrofurazan-4-yl)furoxan]. Their SAXS results (see Fig. 10) suggest a potential pathway to the efficient production of carbon nano-onions and offer a view of the phase transformation kinetics of liquid carbon under extreme pressures and temperatures.

Schuster et al.⁶⁵ determined the nucleation rate into a diamond lattice in dynamically compressed polystyrene at MEC/LCLS. For a single shock reaching 70 GPa and 3000 K, they observed no diamond formation, while using a double shock to drive the polystyrene to pressures around 150 GPa and temperatures around 5000 K, nucleation rates between 10²⁹ and 10³⁴ m⁻³ s⁻¹ were recorded. Helfrich et al.⁵⁰ shocked dense carbon to pressures between 100 GPa and 200 GPa and temperatures between 5000 K and 15 000 K. The temperatures were determined using spectrally resolved x-ray scattering to determine ion–ion structure factors. MacDonald et al.⁶⁶ determined the strength of laser shock-compressed polycrystalline diamond at stresses above the Hugoniot elastic limit from x-ray diffraction and velocimetry at LCLS. They were able to use diffraction to measure lattice strains and combine this with calculations to determine the material strength of diamond above the Hugoniot elastic limit.

Brown et al.⁶⁷ performed simultaneously in situ x-ray diffraction and x-ray phase-contrast imaging using LCLS of shock-compressed silicon (see Fig. 11). They were able to capture images of multiple transient elastic regimes, characterize the crystalline structure of each progressive shock wave, and extract the compression across both elastic features in situ.

Finally, in an example from large laser facilities, Fratanduono et al.⁶⁸ were able to probe the equation of state for copper at terapascal conditions. For these experiments, x rays are generated by laser-ablation driven x-ray fluorescence. Copper samples were ramp compressed to peak pressures of 2.30 TPa and densities of nearly 30 g/cc. They conclude that the ambient fcc structure is stable to pressures up to 1.15 TPa—the maximum pressure at which XRD measurements were performed.

In attempting to understand data from time-dependent x-ray experiments, atomic-level simulations can elucidate phase changes and chemical rearrangements that are difficult to discern from experimental data alone. Frequently, insufficient data exist for the equation of state and chemical reactivity of these materials under extreme conditions, and experiments require theoretical studies to elucidate measured physical and chemical properties. For example, small angle x-ray scattering (SAXS) experiments frequently yield a statistical sampling of scattering curves that cannot be assigned to a given ensemble of solid clusters or shapes without some form of molecular dynamics or quantum calculation.⁶⁹ In addition, these types of simulations generally require some form of experimental validation to assess their accuracy.

Furthermore, reported experimental temperatures can contain large uncertainties, making it difficult to adequately constrain the equation of state measured within an experiment. The use of pyrometry to determine shock Hugoniot temperatures of many non-transparent systems remains an unresolved issue.⁷⁰ As a result, many x-ray studies do not include any measurement of temperature. Velocity interferometer system for any reflector (VISAR) data can yield reliable equation of state (EOS) data, though temperatures must then be determined through an additional method since EOS data give the total energy, rather than the partition in terms of the kinetic and potential contributions. Consequently, experiments tend to rely on parameterized EOS models for these data, which can yield a wide range of results depending on a given parameterization.⁷¹ 

Molecular dynamics simulations provide an independent route to temperature determination, where material properties such as the shock Hugoniot states are readily computed.⁷² MD simulations can also help determine other physical properties such as phase changes, which have occurred during shock compression. In this case, the objective is to model nature/experiments/reality as observed in x-ray measurements as closely as possible. These approaches can provide simple chemical pictures of ionized intermediates and reaction mechanisms and can be helpful in identifying atomic-scale properties that govern the observed macroscopic kinetics (e.g., descriptors). MD modeling results can thus be extremely useful in making experiments more tractable by aiding in their interpretation and helping to decrease the vast physico-chemical phase space that can be accessed in shock compression x-ray experiments.

Accurate modeling of the breaking and forming of bonds in condensed phases frequently requires the use of quantum theories such as Density Functional Theory (DFT). DFT remains one of the most popular and widely used modeling methods in condensed matter physics, computational chemistry, and materials science for the prediction of material properties under extreme conditions. DFT has been shown to accurately reproduce the phase boundaries and thermal decomposition of many materials,⁷³ particularly at the extreme pressures and temperatures of shock loading, where the choice of a specific basis set or exchange-correlation functional is less important.⁷⁴ These efforts include martensitic phase transformations under shock loading^75,76 as well as exotic high-pressure phases.^77–79 

Quantum simulations, though, require immense computational effort per simulation time step and consequently are usually limited to picosecond timescales and nanometer system sizes. In contrast, many chemical events and phase changes occur over nanosecond timescales or longer, and experiments can probe micrometer length scales or beyond. Regardless, quantum calculations can supply necessary constraints on experimental observables, even if the calculations themselves are limited to significantly smaller time and length scales. Recently, Kroonblawd and Goldman used semi-empirical quantum simulations (Density Functional Tight Binding, DFTB⁸⁰) to study graphite seeded with varying degrees of defects under rapid compressive strain rates to determine the material properties of the diamond end state⁷⁶ (Fig. 12). The DFTB method combines approximate quantum mechanical interactions with fitted functions to allow for computationally efficient, high-throughput simulations while retaining the accuracy of higher order quantum methods. These results indicated that allowing for multiple vacancy sites within the graphitic basal planes would seed a number of defects such as twin boundaries and amorphous regions with significantly lower bandgaps than found in bulk diamond. This type of calculation could inform x-ray studies.⁸¹ 

The need to overcome the time and length scale gap between quantum calculations and experiments necessitates the development of new simulation methods that can capture their accuracy while providing many orders of magnitude improvement in computational efficiency. The high computational costs can frequently be surmounted through the creation of reactive molecular dynamics force fields. In this case, a system's forces, energies, and stresses are projected onto a given parameterized functional form (through linear and/or non-linear optimization) that allows for the bond breaking and forming. These approximations can yield improved computational scaling alongside orders of magnitude improved efficiency, affording a more one-to-one correspondence with experiments. Recently, Lindsey et al. used the ChIMES (Chebyshev Interaction Model for Efficient Simulation) machine learned force field to conduct million-atom reactive MD simulations of shock compressed liquid CO.^42,82 Their results yielded a distribution of liquid carbon droplets with a shell of oxygen atoms bound on the surface that formed over picosecond timescales (Fig. 13). The million-atom simulations allowed for accurate determination of the nanoparticle distribution as a function of time. Such studies can yield nucleation and growth rates for the droplet formation as well as determine possible SAXS signals for future experimentation.

Reactive force fields themselves frequently have computationally intensive functional forms with large numbers of parameters that can preclude simulating beyond scales beyond 10s of nanoseconds. Classical MD force field calculations based on exceedingly simple functional forms can be used in these cases to achieve micrometer length scales and microseconds or larger timescales. This can allow for the simulation of more realistic chemical or point defects concentrations, grain sizes and boundaries, and lattice defects (i.e., twinning). In a recent joint experimental–theoretical study, Sliwa et al. used a combination of femtosecond x-ray diffraction and one such simple MD model (an embedded atom potential) to describe the elimination of defects and associated rotation on release in polycrystalline tantalum targets shock compressed to 37–253 GPa.⁸³ In agreement with the MD simulation, lattice rotation and the twins that formed under shock loading were observed to be almost fully eliminated during the rarefaction process (Fig. 14). This direct observation of the reversal of microstructural changes upon release emphasizes the importance of these types of joint studies in elucidating the physics of shock deformation at the lattice level.

Continued advances in MD force field method development and supercomputing approaching the exascale (i.e., 10¹⁸ calculations per second) allow for some bridging of the time and length scale gap between experiments and atomistic simulations. However, even state of the art methods can frequently approach experimental scales only within a few orders of magnitude and cannot actually achieve them. In addition, many calculations focus on perfect solids and have more difficulty modeling actual defect and impurity concentrations. Most importantly, computational results can require weeks of computing time, using models that are generally developed specifically for a given material and experiment (which can be a labor intensive process in itself). As a result, these methods cannot provide real-time feedback for x-ray experiments. For example, the million-atom liquid CO simulations⁴² required approximately 1 week to determine less than 1 ns of simulation time. Still, the ability of these approaches to provide a direct simulation of experiments is crucial, in particular, in situations where an experimental measurement can have large errors (e.g., shock Hugoniot temperature) or is somewhat unconstrained and difficult to attribute to a specific chemical state (e.g., SAXS signals of carbon cluster formation). In these cases, accurate atomistic simulations can be an essential component in the interpretation and design of current and future x-ray studies of shocked materials.

In this section, we give our perspective on several aspects of this growing field. Two of the main drivers advancing the field are the availability of new platforms and new diagnostics. A current limiting factor for the field, however, is the scarcity of beam time on many of the current facilities. This is particularly true of XFEL facilities that have shock drivers available. The role of limited beam time certainly drives the science which is proposed for beam time. With many facilities coming online with these capabilities, this will be a major driver of new opportunities and discoveries.

One of the new diagnostics expected to come online in the near future is combining x-ray diffraction and velocimetry at ps timescales. This is particularly important for allowing EOS and transformation kinetics measurements at the ps timescale that match what is currently available at the ns timescale, where VISAR measurements are typically available as a standard diagnostic at the beam line. Having a similar velocimetry diagnostic available at the ps drive times will vastly improve the opportunities for experiments at these short times and the ability of researchers to utilize the techniques of unsteady shocks.

Hot topics on the experimental side from the last few years include high repetition rate XFELs and laser drivers, including the LCLS II upgrade, the APS-U upgrade, and HAPLS. Such capabilities have the potential to (1) increase the throughput of XFELs and (2) streamline the collection of dynamic material data, similar to the increase in biological x-ray structural characterization at x-ray facilities over the last few decades. It is recognized that the development of high throughput data production will require parallel development of x-ray detectors, and handling and analysis methods for large datasets, possibly spanning a wide range of drive characteristics and variations in samples (such as microstructure). Automated computational model design and machine learning approaches may prove crucial in analyzing the expected massive increase in shocked material x-ray data expected in the next few years. In addition, developments in exa-scale computing will help further the ability of atomistic simulation methods to make more rapid and direct predictions for experiments, over a broad range of materials and conditions.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. This material is based in part upon work supported by the Defense Threat Reduction Agency under Award No. HDTRA12020001.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.