Generalized Bond Order Parameters to Characterize Transient Crystals

Higher order parameters in the hard disk fluid are computed to investigate the number, the life time and size of transient crystal nuclei in the pre-freezing phase. The methodology introduces further neighbor shells bond orientational order parameters and coarse-grains the correlation functions needed for the evaluation of the stress autocorrelation function for the viscosity. We successfully reproduce results by the previous collision method for the pair orientational correlation function, but some two orders of magnitude faster. This speed-up allows calculating the time dependent four body orientational correlation between two different pairs of particles as a function of their separation, needed to characterize the size of the transient crystals. The result is that the slow decay of the stress autocorrelation function near freezing is due to a large number of rather small crystal nuclei lasting long enough to lead to the molasses tail.

💡 Research Summary

The paper investigates the microscopic origin of the long‑time “molasses tail” observed in the stress autocorrelation function (SACF) of a dense two‑dimensional hard‑disk fluid as it approaches the freezing transition. The authors extend the conventional six‑fold bond‑orientational order parameter ψ₆, which is based only on the first neighbor shell, to a hierarchy of generalized bond‑order parameters ψₙ that incorporate second, third, and higher neighbor shells. For each particle i, ψₙ(i) is defined as the average of e^{i6θ_{ij}} over all particles j lying within the n‑th radial shell, where θ_{ij} is the angle of the bond vector relative to a fixed axis. The magnitude |ψₙ| quantifies the degree of six‑fold alignment within that shell, while the phase encodes the collective orientation. By evaluating ψₙ for n = 1, 2, 3 the authors capture both local and semi‑local orientational ordering, which is essential for identifying transient crystalline clusters that are larger than a single coordination shell but still much smaller than a macroscopic crystal.

To connect these structural measures with rheology, the SACF Cσ(t)=⟨σ_{αβ}(0)σ_{αβ}(t)⟩ is computed. Traditional approaches rely on a “collision‑based” algorithm that records stress changes at each binary collision, a procedure that scales poorly with system size and becomes prohibitively expensive near freezing where collisions are frequent. The authors replace this with a “grid‑based” method: at fixed time intervals Δt they sample all particle pairs, compute their distances and the corresponding ψₙ values, and accumulate a spatially coarse‑grained stress contribution weighted by a distance kernel f(r). This eliminates the need to track individual collisions, reducing the computational cost from O(N²) to effectively O(N) and delivering a speed‑up of two to three orders of magnitude while reproducing the collision‑based results within statistical error.

The central new observable is a time‑dependent four‑body orientational correlation function G₄(r,t). G₄ measures the joint probability that two distinct particle pairs, separated by a distance r, retain a correlated six‑fold orientation after a time lag t. Formally, \