---
title: Shock-Compression Model Overview
url: https://www.emergentmind.com/topics/shock-compression-model
type: topic
---

# Shock-Compression Model Overview

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 [2006.13136].

## 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 [2006.13136]. The methodology is extendable to other metals, alloys, and complex materials subjected to dynamic compression on experimentally relevant time and length scales.

Source: https://www.emergentmind.com/topics/shock-compression-model