Femtosecond Visualization of Lattice Dynamics in Shock-Compressed Matter

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Science  11 Oct 2013:
Vol. 342, Issue 6155, pp. 220-223
DOI: 10.1126/science.1239566

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Elastic to Plastic

When a crystal is mechanically compressed, it first reacts elastically (reversibly), and then enters the plastic regime, in which the structure of the material is irreversibly changed. This process can be studied with molecular dynamics (MD) simulations on very fine temporal and spatial scales, but experimental analysis has lagged behind. Milathianaki et al. (p. 220) shocked polycrystalline copper with a laser beam, and then took successive snapshots of the crystal structure at 10-picosecond intervals. The results were compared directly with atomistic simulations and revealed that the yield stress—the point of transition from plastic to elastic response—agreed well with MD predictions.


The ultrafast evolution of microstructure is key to understanding high-pressure and strain-rate phenomena. However, the visualization of lattice dynamics at scales commensurate with those of atomistic simulations has been challenging. Here, we report femtosecond x-ray diffraction measurements unveiling the response of copper to laser shock-compression at peak normal elastic stresses of ~73 gigapascals (GPa) and strain rates of 109 per second. We capture the evolution of the lattice from a one-dimensional (1D) elastic to a 3D plastically relaxed state within a few tens of picoseconds, after reaching shear stresses of 18 GPa. Our in situ high-precision measurement of material strength at spatial (<1 micrometer) and temporal (<50 picoseconds) scales provides a direct comparison with multimillion-atom molecular dynamics simulations.

The distinct properties of materials at high-pressure and/or strain-rate conditions lead to a broad range of phenomena in fields such as high-energy-density physics (1), Earth and planetary sciences (2, 3), aerospace engineering (4), and materials science (5, 6). For the latter, a predictive understanding and control of mechanical properties, enabled by the direct comparison of experiments with large-scale atomistic simulations, is the ultimate goal. Whereas the bulk material behavior can be inferred by macroscopic measurements (7, 8), key information on the mechanical properties requires knowledge of the physics embedded at the lattice level. Such knowledge has traditionally been obtained via nanosecond-resolution x-ray diffraction measurements (914) from dynamically compressed samples that are tens of micrometers in thickness.

In recent years, molecular dynamics (MD) simulations (1518) on massively parallel computers have elucidated the orientation-dependent response of single-crystal samples to shock and ramp compression (19, 20). Because of computational complexities, such simulations have interrogated spatial scales of <1 μm and temporal scales of <1 ns. Specifically, MD simulations have predicted that for picosecond compression time scales, the response of a face-centered cubic crystal, such as Cu, should initially be elastic up to very high strains (12 to 20%), depending on crystallographic orientation and strain rate (15, 18). Similar behavior has been suggested by dislocation dynamics (DD) simulations (21). However, these results have remained largely unverified, either because in situ strain measurements have been indirect (22, 23) or because they had limited temporal and/or angular resolution (9, 11, 12). We present high-precision x-ray diffraction measurements of the lattice evolution in laser-shocked polycrystalline Cu (24). A sequence of femtosecond snapshots at pump-probe intervals of 10 ps (corresponding to an incremental shock propagation distance of ~55 nm) produced a lattice-level movie of the strain state within the material. As the x-ray pulse length was less than even the shortest phonon period, diffraction captured strain profiles in the absence of temporal smearing. Furthermore, the diffraction geometry was designed to differentiate between a purely elastic response and plastic relaxation resulting from the generation and motion of dislocations. The sample thickness (1 μm), stress rise time (<80 ps), and propagation time (~180 ps) were directly comparable with the spatial and temporal dimensions of MD simulations.

The experiment was performed at the Coherent X-ray Imaging Instrument (25) of the Linac Coherent Light Source (LCLS). An ablation-driven compression front was launched parallel to the sample normal and across a Gaussian 260-μm (1/e2 diameter, where e is equal to 2.718281828) focus using the 800 nm, ≤20 mJ, ~170 ps (full width at half-maximum) output of a Ti:sapphire laser system (Fig. 1). The texture of the polycrystalline Cu samples (~400-nm average grain size) was such that crystallites were preferentially shocked along the <111> direction. Quasi-monochromatic (ΔE/E = 0.2 to 0.5%, where E is energy and ΔE is the change in energy) 8.8-keV x-ray pulses with 48-fs duration and an average of ~1012 photons per pulse were incident over a 30-by-30–μm2 spot. Diffraction rings were recorded from the Embedded Image reflections on an in-vacuum, 2.3-megapixel array detector [the Cornell Stanford Pixel Array Detector (CSPAD)] (26) in a Debye-Scherrer geometry (27). Pump-probe delay scans with incremental 10-ps intervals allowed us to obtain a time series of x-ray diffraction patterns from the shock-compressed lattice (28).

Fig. 1 Experimental configuration of the pump (optical laser) and lattice probe (LCLS) scheme.

The lattice response of the shock-compressed 1-μm polycrystalline Cu films deposited on 85-μm <100> Si substrates was captured in a Debye-Scherrer geometry by a series of 48-fs snapshots. A preferential orientation of the Cu crystallites constrained the axis of the applied stress along the [111] direction. FEL, free electron laser; θB, Bragg angle.

A full diffraction ring from the ambient sample is shown in Fig. 2A, where the uniformity in signal intensity indicates rotational symmetry of the crystallites around the x-ray and shock axis. As the sample is uniaxially compressed, the strain profile across the sample depth results in an angular distribution of the diffracted x-rays that depends on the instantaneous elastic strains both parallel and perpendicular to the shock propagation direction (28) (Fig. 2B and movies S1 and S2). Our Debye-Scherrer geometry allows a clear demarcation of elastic and hydrostatic responses. Specifically, the low ambient Bragg angle (19.8°) leads to a substantial difference in the angular shift of the x-rays for a given elastic strain along the shock propagation direction; if the sample response is purely elastic (i.e., Embedded Image, the elastic strain along the shock direction, is finite, but Embedded Image, where Embedded Image is the transverse elastic strain, and Embedded Image the transverse plastic strain) the angular shift is almost nine times less than that of a nearly hydrostatic response (where Embedded Image) (28).

Fig. 2 Raw data illustrating the Debye-Scherrer cone projections captured on the CSPAD detector.

(A) Projections from the Cu Embedded Image and Cu (200) planes are displayed. The intensity of the Cu Embedded Image plane is higher than expected from the calculated structure factors for a powder specimen because of texture. (B) A section of the diffraction ring is magnified, and its evolution is shown at 20-ps intervals. Movies S1 and S2 demonstrate the angular shift in the x-ray signal as a function of the lattice strain state, recorded at 10-ps intervals.

Azimuthal integration of the diffracted intensities as a function of x-ray scattering angle 2θ and for different delay times (Fig. 3A) shows (i) a reduction in intensity of the ambient Embedded Image diffraction peak at 2θ0 = 39.6°; (ii) the subsequent emergence of a diffraction peak at 2θ = 40.4°, an angle that remains constant with laser irradiance (fig. S1); and (iii) a broad feature extending to scattering angle 2θ ~ 43° at later delays. An understanding of these features in the time-dependent diffraction profiles can be gained via comparison with simulated strain and diffraction profiles (Fig. 3, B to D) calculated from a hydrocode incorporating a simple plasticity model (28).

Fig. 3 Elastic and plastic strain profiles in shocked Cu and corresponding diffraction signal, as calculated from the captured x-ray data.

(A) Experimental diffraction data resulting from the azimuthal integration around 2π of the Cu Embedded Image x-ray signal. The diffraction profiles are divided into three regions to illustrate the characteristic lattice response: region I, the unstrained lattice; region II, the elastically compressed lattice; and region III, the lattice exhibiting three-dimensional relaxation. a.u., arbitrary units. (B) Simulated diffraction data resulting from the calculated strain profiles, showing good agreement with experiment. Discrepancies in the signal amplitude are mainly due to artifacts introduced in the azimuthal integration of the diffraction peaks by the area detector tiling, as well as the Δt = 0 synchronization uncertainty (~20 ps) in the time of arrival of the optical and x-ray pulse. (C) The modeled time-dependent normal elastic strain and (D) transverse plastic strain profiles versus sample depth.

To distinguish the diffraction features representative of the shock-induced strain state in the lattice, we divide the range of scattering angles into regions I, II, and III (Fig. 3A). At any instant in time, the diffraction patterns are a superposition of the depth-dependent strain states along the entire sample thickness, as the x-ray probe depth at 8.8 keV in Cu is >> 1 μm. The prominent feature in region I is the diffraction peak from the unstrained Embedded Image lattice plane at 2θ0 = 39.6°. As the compression front traverses the sample, the ambient diffraction peak intensity decreases because of the reduction in thickness of unstrained material. Within the first 40 ps of shock propagation, the lattice exhibits a considerable elastic response that is evident in region II. Such response can be attributed to an initial homogeneous nucleation regime, as predicted by MD and DD simulations (15, 21), with duration depending on the rise time of the compression front. A large elastic strain Embedded Image of 18% is induced along the shock propagation axis, as manifested by an angular shift of the ambient diffraction peak to 2θ = 40.4°. Note that in this experimental geometry, such normal elastic strain value corresponds to a relatively low angular shift (0.8°) of the ambient diffraction peak as Embedded Image in the absence of plastic flow. Other notable features of this elastic diffraction peak are (i) amplitude that increases with time, as the thickness of the elastically compressed region also increases, and (ii) narrow angular distribution because of the low elastic strain gradient present (Fig. 3C). Our calculations, based on the experimental shock parameters and captured diffraction profiles, indicate that this purely elastic response persists up to a peak normal stress of ~73 GPa and shear stress of 18 GPa (Fig. 4A). The latter, representing the yield stress of the material, is in excellent agreement with MD simulations in single crystal Cu (18) at a strain rate of (109 s−1) and for uniaxial compression along the [111] direction, thus confirming the considerably higher yield stress values predicted by simulations compared with those extracted from nanosecond shock experiments on samples of >>1-μm thickness (12).

Fig. 4 Snapshot of the normal and shear stress in the sample and strain history.

(A) Calculated normal and shear stress from the experimentally determined angular shift in the Cu Embedded Image reflection at t = 140 ps. (B) The elastic and plastic strain history at a sample depth of 200 nm. Note that plastic relaxation initiates when the normal elastic strain reaches a peak value of ~18%.

Plastic deformation is induced when the normal elastic stress exceeds the material elastic limit, in this case ~73 GPa, causing multiplication and motion of existing dislocations. Region III encompasses diffraction from regions of the lattice evolving from a purely elastic to a nearly hydrostatic state after compression to the elastic limit; here, whereas the magnitude of Embedded Image decreases with time, the magnitude of Embedded Image increases (Fig. 4B). Despite Embedded Image being much lower than for the elastic regime of region II, the change in the scattering angle is considerably larger, as in the hydrostatic limit Embedded Image. Scattering extends out to 2θ ~43°, consistent with a plastic strain state of Embedded Image. The plastic strain rate Embedded Image can be estimated at ~109 s−1, considering that Embedded Image is reached over a time period of ~60 ps. Taking a magnitude b = 2.6 Å for the Burgers vector, we deduce a dislocation density-velocity product of 4.0 × 1018 m−1 s−1 via Orowan’s equation (29) Embedded Image, where constant A is of order unity, ρm is the mobile dislocation density, and v is the average dislocation velocity. We note that MD (15) simulations predict average dislocation velocities of ≤1300 m s−1 at similar strain rates, implying a minimum dislocation density in our experiment of 3.0 × 1015 m−2. This value, although lower than the MD simulations (1017 m−2) (15), is in close agreement with that calculated in DD simulations (1016 m−2) (21). This discrepancy could be a consequence of the methods used to extract dislocation densities or of dislocation tangling and locking, which result in the total number of dislocations produced in MD to be artificially higher than the number of mobile dislocations responsible for plastic flow. We note that, in the future, small-angle scattering techniques exploiting the high spectral brightness of LCLS could provide direct information on the transient dislocation density itself.

Our results highlight the necessity for experiments designed with length scales equal to these of atomistic simulations; such fundamental measurements of the material response could be used to evaluate and improve time-dependent deformation models, thus extending our knowledge on the ultrafast elastic-plastic behavior of different crystal structures. Beyond simple crystal structures, the mechanical properties of complex engineered materials (30) could also be studied and optimized.

Supplementary Materials

Materials and Methods

Supplementary Text

Fig. S1

References (3135)

Movies S1 and S2

References and Notes

  1. See supplementary materials on Science Online.
  2. Acknowledgments: We thank M. Bionta, A. Fry, S. Edstrom, J. Koglin, S. Guillet, I. Ofte, and G. Stewart for assisting with our experimental requirements, data processing, and illustrations. Raw x-ray data are available upon request. This work was funded as part of the in-house research effort of LCLS, a National User Facility operated by Stanford University on behalf of the U.S. Department of Energy, Office of Basic Energy Sciences. A.H. acknowledges support from the UK Atomic Weapons Establishment and J.S.W. from the UK Engineering and Physical Sciences Research Council under grant EP/J017256/1.
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