Non-Gaussian Rotational Diffusion and Swing Motion of Dumbbell Probes in Two Dimensional Colloids

Two dimensional (2D) colloids exhibit intriguing phase behaviors distinct from those in three dimensions, as well as dynamic heterogeneity reminiscent of glass-forming liquids. Here, using discontinuous molecular dynamics simulations, we investigate the reporting dynamics of dicolloidal dumbbell probes in 2D colloids across the liquid-hexatic phase transition, where hexagonal bond-orientational order (HBOO) extends to quasi-long-ranged one. The rotational dynamics of dumbbell probes faithfully capture the structural and dynamical features of the host: Brownian in the isotropic liquid, and non-Gaussian in the hexatic and solid phases, reflecting both HBOO and dynamic heterogeneity of the medium. In the 2D hexatic and solid phases, probe rotation reflects heterogeneity as the dumbbells sample multiple dynamical domains of the host system: in mobile domains, they undergo rotational jumps of $π/3$ in accordance with HBOO, whereas in immobile domains they librate within cages formed by surrounding discs. Such non-Gaussianity disappears upon reentrant melting of the host medium driven by size polydispersity, highlighting a close connection between HBOO and probe dynamics. Furthermore, probe dynamics reveal both coupling (at a single particle level) and decoupling (at an ensemble-averaged level) between translation and rotation: swing motion emerges as their primary diffusion mode, while the Debye-Stokes-Einstein relation breaks down regardless of how the rotational diffusion coefficient is defined.

💡 Research Summary

In this work the authors employ discontinuous molecular dynamics (DMD) simulations to study how a dilute population of dicolloidal dumbbell probes reports on the structural and dynamical state of a two‑dimensional (2D) hard‑disk colloidal suspension across the liquid‑hexatic‑solid transition. The host system consists of hard disks of diameter σ and mass m at area fractions ϕ ranging from 0.5 to 0.74, thereby spanning an isotropic liquid, a hexatic phase with quasi‑long‑range six‑fold bond‑orientational order (HBOO), and a crystalline solid. Ten dumbbells (each a rigid dimer of two disks with a fluctuating bond length of 0.95–1.05 σ) are introduced at an extremely low probe area fraction ϕ_d≈0.006, ensuring that they act as passive reporters rather than perturbing the host.

Translational dynamics – The mean‑squared displacement (MSD) of the dumbbell centers of mass mirrors that of the host disks at all ϕ, confirming the probes’ fidelity. Near ϕ≈0.7, where the liquid‑hexatic transition occurs, the MSD exhibits a pronounced sub‑diffusive regime (⟨Δr²⟩∝t^b, b<1) and a long plateau at ϕ=0.72, indicative of transient caging. At longer times the MSD recovers linear diffusion, yet the self‑part of the van Hove function remains non‑Gaussian, reflecting persistent spatial heterogeneity.

Rotational dynamics – The unbounded orientation angle φ(t) of each dumbbell is tracked to compute the mean‑squared angular displacement (MSAD) ⟨Δφ²(t)⟩, the angular displacement distribution G(φ,t), and the rotational non‑Gaussian parameter α₂,R(t). In the isotropic liquid (ϕ=0.65) the MSAD grows linearly, G(φ,t) is Gaussian, and α₂,R≈0, i.e., the probes behave as ordinary Brownian rotors. In the hexatic and solid phases (ϕ≥0.71) the MSAD shows an initial sub‑diffusive segment followed by a linear regime, but α₂,R becomes large and persists even when ⟨Δφ²⟩ has reached the diffusive limit. G(φ,t) displays a striking oscillatory pattern with sharp peaks spaced by π/3 (≈60°). This pattern originates from two distinct dynamical domains: (i) mobile regions where the six‑fold HBOO allows the dumbbell to execute sudden π/3 reorientations that align with the underlying bond network, and (ii) immobile regions where the probe is trapped in a cage and only librates around a fixed orientation. The coexistence of these behaviors produces a highly non‑Gaussian angular displacement distribution that serves as a sensitive reporter of both structural order and dynamic heterogeneity.

Effect of polydispersity and re‑entrant melting – By introducing a Gaussian size polydispersity Δ (standard deviation of particle diameters) of 0.05, 0.09, and 0.13 at fixed ϕ=0.71, the authors induce a re‑entrant melting transition driven by disorder. In the polydisperse, melted state the HBOO disappears, the MSAD regains a purely diffusive, Gaussian character, and α₂,R collapses to zero. Hence the non‑Gaussian rotational signatures are directly linked to the presence of quasi‑long‑range bond‑orientational order.

Translation‑rotation coupling and swing motion – In the ordered phases the translational and rotational motions of individual dumbbells become strongly correlated. The authors identify a “swing motion” in which a probe simultaneously translates (small cage‑rattling) and rotates, reminiscent of a pendulum swing rather than a pure glide. This coupled motion is reflected in the orientational autocorrelation function U(t)=⟨ê(t)·ê(0)⟩, which deviates from a simple exponential decay and is better described by a stretched Kohlrausch‑Williams‑Watts (KWW) form, U(t)≈exp