Shock-Compression Model Overview

• Shock-Compression Model is a predictive framework linking atomic-level mechanisms to macroscopic state changes under rapid, high-strain-rate conditions.
• It leverages deep neural-network potentials, ab initio molecular dynamics, and large-scale MD simulations to capture phase transitions and structural disorder in materials like gold.
• The model quantitatively aligns simulation results with experiments by resolving FCC to BCC transition pressures and incorporating disorder-induced entropic effects.

A shock-compression model provides a rigorous, predictive description of a material’s response to rapid, high-strain-rate loading, relating microscopic atomic or mesoscale mechanisms to the macroscopic evolution of state variables (pressure, density, internal energy) under propagating shock waves. Such models are foundational to interpreting dynamic-compression platforms, extracting equations of state (EOS), and understanding phase transitions and transformations under extreme conditions. In contemporary research, the construction and validation of shock-compression models leverage advanced simulation techniques (e.g., deep neural-network potentials), ab initio molecular dynamics (DFT), and non-equilibrium thermodynamic integration. Below, the state-of-the-art framework for shock-compressed gold, as established by deep potential-driven atomistic simulation and free-energy modeling, is detailed systematically (Chen et al., 2020).

1. Construction of the Deep Potential for Gold

The interatomic interactions underpinning the model are encoded by a neural network-based potential, accurately trained to the ab initio (DFT-PBE-GGA) reference. The training set comprises atomic configurations sampled via the concurrent-learning DP-GEN workflow across all major phases and relevant defected/liquid environments, spanning 0–15,500 K and 0–500 GPa:

• The local atomic environment of atom $i$ is represented by descriptors $\{D_{i\alpha}\}$, determined from neighbors within a cutoff $r_c \sim 8$ Å.
• Two subnetworks: an embedding net, mapping descriptors to “atomic fingerprints” $\mathbf{G}_i$, and a fitting net, mapping $\mathbf{G}_i$ to atomic energies $\epsilon_i$; the total energy $E = \sum_i \epsilon_i$.
• Loss function:

$$L = w_E |E^{\rm DP} - E^{\rm DFT}|^2 + w_F \sum_i |F_i^{\rm DP} - F_i^{\rm DFT}|^2 + w_V ||V^{\rm DP} - V^{\rm DFT}||^2$$

with weights $w_E$, $w_F$, $w_V$ ramped during training. The DP-GEN scheme iteratively adds high-uncertainty molecular dynamics snapshots—identified by model uncertainty—to the DFT dataset until force convergence $<0.05$ eV/Å is achieved.

2. Large-Scale Molecular Dynamics Shock Simulations

Molecular dynamics simulations are performed on single-crystal FCC gold cells of up to $N \sim 2 \times 10^5$ atoms:

• Boundaries: periodic in directions transverse to shock; a modified Lagrangian multi-scale shock technique (MSST) restrains motion along the shock axis, enforcing the Hugoniot locus (i.e., energy conservation consistent with shock passage) without explicit flyer plates.
• Shock loading: MSST implemented in LAMMPS with shock velocity $u_s$ swept from 4.0 to 8.0 km/s (corresponding to $P \lesssim 400$ GPa). The shock direction is varied:
  • $\langle 100 \rangle$—$u_s = 4.0$–$7.0$ km/s ($P \sim 0$–325 GPa)
  • $\langle 110 \rangle$—$u_s = 4.5$–$8.0$ km/s ($P \sim 0$–260 GPa)
• System size and run duration are converged to robustly resolve the steady-state shock front and post-shock averages.

3. Phase Transformation and Structural Diagnosis

The FCC $\to$ BCC phase transition under shock is observed via multiple diagnostics:

• Thresholds for the onset of BCC:
  • $\langle 100 \rangle$ shock: $P_t \approx 159$ GPa
  • $\langle 110 \rangle$ shock: $P_t \approx 219$ GPa
• Structural transitions are identified by:
  1. Simulated powder XRD: the emergence of BCC peaks (structure factor $S(Q)$ at $\lambda=0.5266$ Å, evaluated by Exp–Gauss lineshape fitting);
  2. Adaptive Common-Neighbor Analysis (a-CNA): quantifies fractions of FCC, HCP, BCC, and “Other” (disordered) atoms;
  3. Effective Coordination Number (ECN): discriminates between instantaneous disorder and time-averaged lattice occupation;
  4. Radial distribution function $g(r)$: reveals medium-range and short-range order via peak shape and emergence of additional features.

4. Atomistic Nature and Quantification of Shock-Induced Disorders

Shock-compressed BCC gold exhibits significant disorder absent in equilibrium BCC:

• “Disorders” are defined as atoms whose neighbor-shell configuration departs from the perfect lattice (labeled “Other” by a-CNA) but which, upon time-averaging, occupy the BCC lattice sites (ECN ∼8).

• Quantitative features:

  • Short-range order (SRO): Broadened first-shell $g(r)$ peak and emergent shoulder at $r \approx 2.5$ Å.
  • Medium-range order (MRO): Weak third-shell peak at $r \approx 4.5$ Å, bond-angle histogram $P(\theta)$ broadens about BCC angles (70.5°, 109.5°).
• Disordered atom fraction: up to 30–40% at transition pressure, decreasing with increasing $P$ and thermal activation.

5. Thermodynamic Free-Energy Model with Disorder Contributions

Phase stability is determined by Gibbs free energies on the post-shock Hugoniot:

• $G(P,T) = U + PV - TS$ is computed via non-equilibrium thermodynamic integration and reversible scaling.
• The total BCC-with-disorder free energy is

$$G_{\text{total}} = G_\text{BCC}^\text{perf} + \Delta G_\text{disorders}$$

where disorder corrections are

$$\Delta G_{\text{disorders}}(η) = N E_{\text{exc}}\,η - N k_B T [η\lnη + (1-η)\ln(1-η)] + \cdots$$

with $E_{\text{exc}}$ the excess enthalpy per disordered atom and $η$ the disordered fraction.

• The transition pressure at given orientation and temperature is found by

$$ΔG(P,T,η) = G_\text{FCC}(P,T) - G_\text{BCC+dis}(P,T,η) = 0$$

evaluated on the shock locus $T(P)$.

6. Quantitative Model Predictions and Physical Mechanism

The central quantitative result is the dramatic shift in FCC–BCC transition pressure when disorder is included:

Structure	$\langle 100 \rangle$ $P_{t}$ (GPa)	$\langle 110 \rangle$ $P_{t}$ (GPa)
FCC $\to$ BCC (perfect)	$\approx$ 280–350	$\approx$ 280–350
FCC $\to$ BCC+disorder	140–160	215–220
MD-XRD threshold	159	219
Recent experiment	150–176	223

• Entropic stabilization from disorder (–$TS_{\text{dis}}$) and modest enthalpy cost $E_{\text{exc}}$ lower the BCC transition pressure by more than 100 GPa relative to the perfect-crystal thermodynamic prediction—quantitatively resolving the long-standing discrepancy between static-compression and shock (dynamic) results.
• The model captures orientation dependence: $\langle 100 \rangle$ transition occurs much earlier than $\langle 110 \rangle$, in precise agreement with simulation and experiment.

This integrated atomistic–thermodynamic model, incorporating neural-network potentials, large-scale MD, explicit structural disorder quantification, and free-energy modeling, provides a definitive, predictive framework for shock-induced phase transitions in gold (Chen et al., 2020). The methodology is extendable to other metals, alloys, and complex materials subjected to dynamic compression on experimentally relevant time and length scales.