Fig. 1. MD simulation of ramp-compressed liquid can be well described by the forward and backward hydrodynamic calculations. (a) Illustration of MD simulation. (b) Velocity contours extracted from MD simulation (red) and forward hydrodynamic model (blue) for a 108 nm-thick sample of liquid iron. Dashed lines mark the trajectories for the first rarefaction wave and the last compression wave given by subsequent back-calculations. (c) Free-surface velocity profiles for 108, 120, 132, and 144 nm-thick samples subjected to the same ramp drive. Solid lines are the MD results and dashed lines are those given by the forward hydrodynamic model. (d) c_L(u_p) relation obtained by Lagrangian analysis (red) and that corresponding to the reference “wave-free” stress–density curve. (e) Stress–density relations: red, back-calculated; dashed black, wave-free compression. The real longitudinal stress and density values directly measured from the MD trajectories are shown in blue; the central line, darker, and lighter bands correspond to the mean, standard deviation, and minimum/maximum values, respectively. The fluctuations are too small to be visible in this plot. (f) The wave-free P(ρ) relation is subtracted for better visualization of the errors. Panels (e) and (f) share the same horizontal axis labels.

Fig. 2. Designing the ramp loading piston acceleration history for materials showing nonmonotonic dependence of sound velocity on density. (a) Preliminary wave-free MD simulation of solid iron at ε̇=2×1010 s⁻¹. The sample starts as a single-crystal bcc solid at 300 K. Compression is applied along the [111] crystalline direction. Elastic–plastic transition and phase transition begin at ∼15 GPa. Note that P stands for the longitudinal stress in the direction of uniaxial compression throughout this paper. After smoothing, this P(ρ) relation is taken as the reference for the forward designs (dashed curve). The corresponding Lagrangian sound velocity is shown in (b). (c) and (d) Characteristic lines for the basic and optimized designs in the Lagrangian coordinate. (c) For this case with nonmonotonic c_L(ρ), the basic design in which all compression waves converge to a single point (black) failed to avoid shock in the presence of rarefaction waves reflected from the free surface (red). (d) Self-intersection of characteristic lines can be effectively eliminated by Monte Carlo optimization of the piston acceleration history. Panels (c) and (d) share the same horizontal axis labels. (e) This optimized piston v(t) trajectory differs remarkably from the basic design.

Fig. 3. Back-calculation shows errors in the case of ramp-compressed solid iron. (a) Free-surface velocity profiles given by MD simulation. Dashed black lines are those from the forward hydrodynamic model. (b) c_L(u_p) relation given by Lagrangian analysis of the data in (a). The dashed line is the c_L(u_p) derived from the reference P(ρ) [dashed curve in Fig. 2(a)]. The place where deviations begin to appear is marked by the arrowhead. (c) and (d) Negative control No. 1: for simulation data generated by the forward hydrodynamic model, although the underlying c_L(u_p) relation is nonmonotonic, Lagrangian analysis still yields the correct result. Panels (c) and (d) have the same vertical axis legends as panels (a) and (b), respectively. (e) and (f) Negative control No. 2: measuring the c_L(u_p) relation of simple compression waves directly from the MD results. A 240 nm-thick case is shown. Before being affected by the rarefaction waves (within the dashed magenta polygon), the slope of each u_p contour represents the corresponding Lagrangian sound velocity c_L. This directly measured c_L(u_p) is very close to the reference curve. (g) Stress–density relations. The darker and lighter bands represent the statistics of the real stress–density data in the MD trajectories: the darker bands show the mean plus/minus standard deviation, and the lighter bands the maximum and minimum longitudinal stress recorded for each density point. The result of Lagrangian analysis is shown as the solid red curve. The magenta curve is derived from the directly measured c_L [i.e., (f)]. The reference P(ρ) used for forward design is shown as the dashed black curve, which is quite close to the measurements.

Fig. 4. The MD flow field of ramp-compressed solid iron is not consistent with the isentropic hydrodynamic model. (a) Lagrangian flow field given by the forward hydrodynamic model. The 208 nm-thick case is used as an example. The red–blue heat map is of ∂P_zz/∂t at all Lagrangian positions at all time points, with red standing for compression and blue for rarefaction. The red–blue color scales for panels (a)–(c) are identical, ranging from −14 to +14 GPa/ps. Black curves are the back-calculated characteristic lines. Only the first rarefaction wave and the last compression wave are shown. (b) Lagrangian flow field extracted from the MD simulation. The vertical axis range and legends are identical to those in panel (a). The first rarefaction wave given by back-calculation (solid black line) does not match the flow field. Here, the dashed curve is the true rarefaction wave trajectory traced manually from the MD flow field. The rarefaction wave in MD is visibly faster. (c) Magnification of the MD flow field where the disagreement appears. The mirrored compression wave trajectories do not align with the tangent directions of the rarefaction wave. Compression and rarefaction waves seem to travel with two different sound velocities. (d) Dashed black lines are phenomenological characteristic lines directly constructed from the contours of velocity (blue) and stress (red); see Appendix. A for details. Solid black lines are characteristic lines given by back-calculation. Differences between the two constructions exist beyond the elastic limit (red dashed polygon). (e) and (f) Stress–time and stress–density curves for seven equally spaced Lagrangian positions 78–118 nm away from the left boundary. Unloading due to the rarefaction waves can be seen (black arrows), during which the stress–density relation deviates from the original loading curve. Curves for adjacent positions have been shifted vertically by 5 GPa for clarity. The vertical axes in panels (e) and (f) are the same. (g) Wave-free loading–unloading–reloading simulations of a homogeneous solid system of iron (2 × 10⁵ atoms) at fixed strain rate ±10¹⁰ s⁻¹, along the [111] crystal direction. Unloading is applied at ρ/ρ₀ = 1.1, 1.15, and 1.2 (blue, green, and red curves, respectively), by 2% in volume. As a reference the solid black curve represents the stress–density relation without unloading, on which the elastic limit (E.L.) and the region of structural phase transition are marked. Adjacent curves are shifted by 10 GPa along the vertical axis for better visualization.

Fig. 5. The case of ramp-compressed nanocrystalline iron. (a) Illustration of MD simulation. Wave fronts relating to plasticity and phase transition are shown. (b) Free-surface velocity profiles given by MD (red) for four different sample thickness ranging from 144 to 180 nm. Dashed lines represent the corresponding hydrodynamic model. (c) and (d) Errors exist in the back-calculation results, for both c_L(u_p) and P(ρ). In particular, the stress–density relation extracted from MD shows two “steps” corresponding to plasticity and phase transition, respectively. Without these features being revealed, the back-calculation result could be misleading. The P(ρ) curve of wave-free simulation is also shown as a reference (dashed black curve), and the c_L(u_p) curve derived from it is shown in (c).

Fig. 6. Hysteresis of phase transition in the uniaxial loading–unloading–reloading simulation. Data shown here correspond to the blue curve in Fig. 4(g). Results of two different methods implemented in OVITO,³⁵ namely, CNA and PTM, are presented.