The Triangular Lattice Gas (TLG)

The Model

The triangular lattice gas (TLG) of Jäckle and Krönig is a simple model used to investigate glassy behaviour. Subject to kinetic constraints particles are allowed to make nearest neighbour moves on a two-dimensional lattice of triangular geometry. Although not intended to represent a physical system the kinetic constraint may be interpreted as a hard core repulsion between particles. For the (2)-TLG model a particle may only move if both mutual neighbours of the initial and target site are empty.

Glassy Dynamics

As particle density is increased the (2)-TLG shows a rapid slowdown in dynamical behaviour.

The movement of a single particle within the system requires the cooperative rearrangement of many particles.

Shown below are particle trajectories for the (2)-TLG model at various densities. Click on an image to open a movie in a new window. Blue particles are those which have not moved during the course of the simulation. Movies correspond to trajectories of 10⁶ Monte Carlo sweeps for a lattice of 10² sites.

Particle density = 0.77	Particle density = 0.78
Particle density = 0.79	Particle density = 0.80

At high densities particles become trapped in cages formed by their neighbours. Large scale motion requires cooperative behaviour resulting in the opening of the cage, thus allowing a particle to escape.

The following movie shows the motion of several "probe" particles within a (2)-TLG lattice at a particle density of 0.80. The particles have been overlaid on the mobility field of the lattice where regions of the lattice are shaded according to how much motion there has been within a time window Δt. The colour scheme is chosen such that the most mobile areas are white whilst the most immobile are black. Note that the probes have been artificially enlarged to aid visualisation.

Probe movie

For more information on the TLG models read Albert Pan's paper 'Heterogeneity and growing lengthscales in the dynamics of kinetically constrained lattice gases in two dimensions', Phys. Rev. E 72, 041106 (2005). , or alternative visit Albert's site here.

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