Correlations in Motion of Atoms in Liquid Argon

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

Additional Resources:

• Aneesur Rahman - Wikipedia

What kind of paper is this?

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. Rather than deriving liquid properties solely from first-principles statistical mechanics, 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).

What is the motivation?

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.

What is the novelty here?

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 $\langle v(0)\cdot v(t)\rangle$ becomes negative after a short time ($t \approx 0.3 \times 10^{-12}$ s). This proved that atoms in a liquid do not diffuse via simple Langevin Brownian motion but “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.

What experiments were performed?

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 $G_d(r,t)$.

What outcomes/conclusions?

• 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, not a short-time reality.
• 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: $$\eta_i^{(n+1)} = \eta_i^{(n)} + \frac{1}{2}\Delta u (\alpha_i^{(n+1)} + \alpha_i^{(n)})$$ $$\xi_i^{(n+1)} = \xi_i^{(n)} + \frac{1}{2}\Delta u (\eta_i^{(n+1)} + \eta_i^{(n)})$$

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:

Citation

@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}
}