Correlations in Motion of Atoms in Liquid Argon

Contribution: Methodological Validation of MD

This is the archetypal Method paper (dominant classification with secondary Theory contribution). It establishes the architectural validity of Molecular Dynamics (MD) as a scientific tool. Rahman answers the question: “Can a digital computer solving classical difference equations faithfully represent a physical liquid?”

The paper utilizes specific rhetorical indicators of a methodological contribution:

• Algorithmic Explication: A dedicated Appendix details the predictor-corrector difference equations.
• Validation against Ground Truth: Extensive comparison of calculated diffusion constants and pair-correlation functions against experimental neutron and X-ray scattering data.
• Robustness Checks: Ablation studies on the numerical integration stability (one vs. two corrector cycles).

Motivation: Bridging Neutron Scattering and Many-Body Theory

In the early 1960s, neutron scattering data provided insights into the dynamic structure of liquids, but theorists lacked concrete models to explain the observed two-body dynamical correlations. Analytic theories were limited by the difficulty of the many-body problem.

Rahman sought to bypass these analytical bottlenecks by assuming that classical dynamics with a simple 2-body potential (Lennard-Jones) could sufficiently describe the motion of atoms in liquid argon. The goal was to generate “experimental” data via simulation to test theoretical models (like the Vineyard convolution approximation) and provide a microscopic understanding of diffusion.

Core Innovation: System Stability and the Cage Effect

This paper is widely considered the birth of modern molecular dynamics for continuous potentials. Its key novelties include:

1. System Size & Stability: Successfully simulating 864 particles (a significant leap from earlier hard-sphere calculations) with stable energy conservation over long durations ($10^{-11}$ sec).
2. The “Cage Effect”: The discovery that the velocity autocorrelation function becomes negative after a short time: $$ \langle \textbf{v}(0) \cdot \textbf{v}(t) \rangle < 0 \quad \text{for } t > 0.3 \times 10^{-12} \text{ s} $$ This proved that atoms in a liquid “rattle” against the cage of their nearest neighbors.
3. Delayed Convolution: Proposing an improvement to the Vineyard approximation for the distinct Van Hove function $G_d(r,t)$ by introducing a time-delayed convolution to account for the persistence of local structure.

Methodology: Simulating 864 Argon Atoms

Rahman performed a “computer experiment” (simulation) of Liquid Argon:

• System: 864 particles in a cubic box of side $L=10.229\sigma$.
• Conditions: Temperature $94.4^\circ$K, Density $1.374 \text{ g cm}^{-3}$.
• Interaction: Lennard-Jones potential, truncated at $R=2.25\sigma$.
• Duration: Time steps of $\Delta t = 10^{-14}$ s.
• Output Analysis:
  • Radial distribution function $g(r)$.
  • Mean square displacement $\langle r^2 \rangle$.
  • Velocity autocorrelation function $\langle v(0)\cdot v(t) \rangle$.
  • Van Hove space-time correlation functions $G_s(r,t)$ and $G_d(r,t)$.

Results: Validation and Non-Gaussian Diffusion Analysis

• Validation: The calculated pair-distribution function $g(r)$ and self-diffusion constant $D = 2.43 \times 10^{-5} \text{ cm}^2 \text{ sec}^{-1}$ matched experimental values (Eisenstein/Gingrich and Naghizadeh/Rice) with remarkable accuracy.
• Dynamics: The velocity autocorrelation has a negative region, contradicting simple exponential decay models (Langevin). Its frequency spectrum $f(\omega)$ shows a broad maximum at $\omega \approx 0.25 (k_BT/\hbar)$, reminiscent of solid-like behavior.
• Non-Gaussian Behavior: The self-diffusion function $G_s(r,t)$ deviates from a Gaussian shape at intermediate times ($t \approx 3.0 \times 10^{-12}$ s), indicating that Fickian diffusion is an asymptotic limit and does not apply at short times.
• Conclusion: Classical N-body dynamics with a truncated pair potential is a sufficient model to reproduce both the structural and dynamical properties of simple liquids.

Reproducibility Details

Data

The simulation uses physical constants for Argon:

Algorithms

Rahman utilized a Predictor-Corrector scheme for solving the second-order differential equations of motion.

Step Size: $\Delta t = 10^{-14}$ sec.

The Algorithm:

1. Predict positions $\bar{\xi}$ at $t + \Delta t$ based on previous steps: $$\bar{\xi}_i^{(n+1)} = \xi_i^{(n-1)} + 2\Delta u \eta_i^{(n)}$$
2. Calculate Forces (Accelerations $\alpha$) using predicted positions.
3. Correct positions and velocities using the trapezoidal rule: $$ \begin{aligned} \eta_i^{(n+1)} &= \eta_i^{(n)} + \frac{1}{2}\Delta u (\alpha_i^{(n+1)} + \alpha_i^{(n)}) \\ \xi_i^{(n+1)} &= \xi_i^{(n)} + \frac{1}{2}\Delta u (\eta_i^{(n+1)} + \eta_i^{(n)}) \end{aligned} $$

Note: The paper notes that repeating the corrector step (“two repetitions”) improved precision slightly but one pass was generally sufficient.

Models

Interaction Potential: Lennard-Jones 12-6 $$V(r_{ij}) = 4\epsilon \left[ \left(\frac{\sigma}{r_{ij}}\right)^{12} - \left(\frac{\sigma}{r_{ij}}\right)^6 \right]$$

Boundary Conditions: Periodic Boundary Conditions (PBC) in 3 dimensions. When a particle moves out of the box ($x > L$), it re-enters at $x - L$.

Hardware

This is a historical benchmark for computational capability in 1964:

Modern Context: Rahman’s system (864 Argon atoms, LJ-potential) is highly reproducible today and serves as a classic pedagogical exercise. It can be simulated in standard MD frameworks (LAMMPS, OpenMM) in fractions of a second on consumer hardware.

Paper Information

Citation: Rahman, A. (1964). Correlations in the Motion of Atoms in Liquid Argon. Physical Review, 136(2A), A405-A411. https://doi.org/10.1103/PhysRev.136.A405

Publication: Physical Review 1964

@article{rahman1964correlations,
  title={Correlations in the motion of atoms in liquid argon},
  author={Rahman, A.},
  journal={Physical Review},
  volume={136},
  number={2A},
  pages={A405},
  year={1964},
  publisher={APS}
}

Additional Resources:

• Aneesur Rahman - Wikipedia