Periodic Potential for Point Defects in a 2D Hexagonal Colloidal Lattice

TL;DR

This study analyzes point defects in a 2D hexagonal colloidal lattice using evolution mechanics to extract drift $\mathbf{f}(\mathbf{q})$ and diffusion $\mathbf{D}(\mathbf{q})$ from trajectories and reconstructs a periodic stochastic potential $\phi(\mathbf{q})$ under an equilibrium approximation. The reconstructed landscape yields energy differences between local minima consistent with prior measurements and reveals nonequilibrium signatures in di-vacancies, indicating potential detailed-balance violations without external driving. By combining time-series analysis with a periodic least-squares fit to a Fourier-expanded $\phi(\mathbf{q})$, the work demonstrates a general approach to infer energy landscapes and assess nonequilibrium dynamics in confined crystalline systems. These insights provide a framework for efficient, large-scale simulations and for exploring defect dynamics in more complex, higher-dimensional settings.

Abstract

We investigate the stochastic dynamics of point defects in a two-dimensional hexagonal colloidal crystal. Using a stochastic dynamical framework (evolution mechanics), we first extract the position-dependent drift vector and diffusion matrix from experimental trajectories, revealing dynamics that go beyond simple diffusion with constant coefficients. We then reconstruct a stochastic potential landscape by applying a periodic constraint and an equilibrium approximation within this framework. The energy differences between local minima in this landscape agree in order of magnitude with prior experimental estimates. Furthermore, analysis of the fit residuals under the equilibrium approximation provides evidence of nonequilibrium dynamics in di-vacancies. This work demonstrates how time-series analysis combined with this theoretical framework can uncover effective energy landscapes and assess detailed-balance violation in defect motion, providing a general approach for studying complex dynamics in confined systems.

Figures (4)

• Figure 1: Trajectories of vacancies and interstitials. Panels (a) to (d) taken from pertsinidis_diffusion_2001kim_dynamical_2020 show trajectories overlaid on the Delaunay triangulation of the hexagonal lattice in the defect-free case, with the black dots representing the center-of-mass position of the disclinations. Panels (e) to (h) present a subset of the trajectories shown in the first set, extracted from the time series data obtained from videos, with the length scale converted to micrometers.
• Figure 2: Reconstructed stochastic potential landscapes for different defect types. The surface plots visualize the stochastic potential $\phi(\mathbf{q})$ (where $2\phi$ is the potential energy in units of $k_{\mathrm{B}}T$) obtained via the equilibrium approximation described in the text. The landscapes for (a) mono-vacancy, (b) di-vacancy, (c) mono-interstitial, and (d) di-interstitial are shown. The color map indicates the value of $\phi$, with cooler (purple/blue) and warmer (green/yellow) colors representing lower and higher potential values, respectively. The parameters are the same as in Table \ref{['table:diffusion_comparison']}. The number of trajectory points $N$ used for the reconstruction in each case is indicated in parentheses.
• Figure 3: Stochastic trajectories simulated from the reconstructed dynamics. The plots show sample trajectories generated by integrating the stochastic differential equation using the continuous drift field $\mathbf{f}(\mathbf{q})=-\mathbf{D}(\mathbf{q})\nabla\phi$ and diffusion matrix $\mathbf{D}(\mathbf{q})$ obtained from fitting a periodic model. Panels correspond to (a) mono-vacancy, (b) di-vacancy, and (c) mono-interstitial, with parameters as in Table \ref{['table:diffusion_comparison']}.
• Figure 4: Quantifying deviations from equilibrium in mono- and di-vacancy dynamics. The plots show the difference in normalized fit residuals $\Delta(\sqrt{W}/N)=(\sqrt{W}/N)_{\mathrm{di\text{-}vac}} - (\sqrt{W}/N)_{\mathrm{mono\text{-}vac}}$ as a function of the time interval $\tau$ for spatial binning radii (a) $r=0.1 \,µm$ and $r=0.2 \,µm$. Different cutoff shells (3, 6, 9) are distinguished by line style.