^{1}, David R. Reichman

^{1,a)}and Giulio Biroli

^{2}

### Abstract

In this paper, we consider in detail the properties of dynamical heterogeneity in lattice glass models (LGMs). LGMs are lattice models whose dynamical rules are based on thermodynamic, as opposed to purely kinetic, considerations. We devise a LGM that is not prone to crystallization and displays properties of a fragile glass-forming liquid. Particle motion in this model tends to be locally anisotropic on intermediate time scales even though the rules governing the model are isotropic. The model demonstrates violations of the Stokes–Einstein relation and the growth of various length scales associated with dynamical heterogeneity. We discuss future avenues of research comparing the predictions of LGMs and kinetically constrained models to atomistic systems.

R.K.D. would like to thank the John and Fannie Hertz Foundation for research support via a Hertz Foundation Graduate Fellowship. R.K.D. and D.R.R. would like to thank the NSF, Grant No. CHE-0719089 for financial support. We would like to thank Ludovic Berthier, Joel Eaves, Peter Harrowell, Robert Jack, Peter Mayer, and Marco Tarzia for useful discussions.

I. INTRODUCTION

II. MODEL

III. DYNAMICAL BEHAVIOR

A. Simple bulk dynamics

B. Motion on the atomic scale

C. Stokes–Einstein violation

D. van Hove function

E. Fickian length

F. and fluctuation measures

IV. CONCLUSION

### Key Topics

- Crystallization
- 13.0
- Entropy
- 10.0
- Relaxation times
- 10.0
- Glass transitions
- 9.0
- Diffusion
- 8.0

## Figures

Comparison and distinction of a caricature of a KCM with a LGM. In the KCM, any configuration is allowed, but move may only be made if a particle has at least one missing neighbor before and after the move. In the LGM, the global configuration is defined such that all particles must have at least one missing neighbor, and all dynamical moves must respect this rule. Note that the local environment around the moving particle is identical in this example, while the global configurations are distinct. Periodic boundary conditions are assumed for both panels.

Comparison and distinction of a caricature of a KCM with a LGM. In the KCM, any configuration is allowed, but move may only be made if a particle has at least one missing neighbor before and after the move. In the LGM, the global configuration is defined such that all particles must have at least one missing neighbor, and all dynamical moves must respect this rule. Note that the local environment around the moving particle is identical in this example, while the global configurations are distinct. Periodic boundary conditions are assumed for both panels.

Crystallization thermodynamics in LGM. Top: The t154 model. refers to the chemical potential of the type 1 particles. The maximum density observed for the lattice is 0.5479 (exactly 1849 out of 3375 lattice sites occupied). The three plotted quenching rates vary between a 0.01 and 0.05 increase of per 10000 cycles. Bottom: A close up of the equivalent plot for the BM model. Note the clear discontinuity upon crystallization. Slower -increase rates produce a sharper discontinuity.

Crystallization thermodynamics in LGM. Top: The t154 model. refers to the chemical potential of the type 1 particles. The maximum density observed for the lattice is 0.5479 (exactly 1849 out of 3375 lattice sites occupied). The three plotted quenching rates vary between a 0.01 and 0.05 increase of per 10000 cycles. Bottom: A close up of the equivalent plot for the BM model. Note the clear discontinuity upon crystallization. Slower -increase rates produce a sharper discontinuity.

Decay of the self-intermediate scattering function for . Densities are 0.3, 0.4, 0.45, 0.48, 0.50, 0.51, 0.52, 0.53, 0.535, 0.5375, 0.5400, 0.5425 from fastest relaxation to slowest relaxation. These densities are used in all plots in this paper unless otherwise indicated. Top: Plotted on a linear-log scale. Bottom: Same data as upper panel plotted on a vs scale. Lowest density curves are at the top left.

Decay of the self-intermediate scattering function for . Densities are 0.3, 0.4, 0.45, 0.48, 0.50, 0.51, 0.52, 0.53, 0.535, 0.5375, 0.5400, 0.5425 from fastest relaxation to slowest relaxation. These densities are used in all plots in this paper unless otherwise indicated. Top: Plotted on a linear-log scale. Bottom: Same data as upper panel plotted on a vs scale. Lowest density curves are at the top left.

Top: (time at which ) as a function of density, . Plotted for , with lowest at the top. Center: Beta stretching exponent of [from terminal fits ]. Lowest curve is at the top of the plot. Bottom: Plot of log scale against chemical potential of type 2 particles. The behavior is consistent with .

Top: (time at which ) as a function of density, . Plotted for , with lowest at the top. Center: Beta stretching exponent of [from terminal fits ]. Lowest curve is at the top of the plot. Bottom: Plot of log scale against chemical potential of type 2 particles. The behavior is consistent with .

Examples of stringlike motion apparent in the t154 model. (a) An example of a string with all neighboring particles removed. (b) A similar string in the context of other particles. Note that the string here is truly isolated in space, away from other mobile particles. In these figures, type 1 particles are white, type 2 particles are blue, and type 3 are green. Sites occupied at the initial time but vacated at the final time are shown in red. These pictures show only the differences in position of particles between the origin of time and the final time, not the path the particles took to achieve that displacement. All figures are at a density of 0.5400, with times in (a) 251, (b) 199 526. The -relaxation time for at this density is about .

Examples of stringlike motion apparent in the t154 model. (a) An example of a string with all neighboring particles removed. (b) A similar string in the context of other particles. Note that the string here is truly isolated in space, away from other mobile particles. In these figures, type 1 particles are white, type 2 particles are blue, and type 3 are green. Sites occupied at the initial time but vacated at the final time are shown in red. These pictures show only the differences in position of particles between the origin of time and the final time, not the path the particles took to achieve that displacement. All figures are at a density of 0.5400, with times in (a) 251, (b) 199 526. The -relaxation time for at this density is about .

Examples cluster shapes in the (a) the t154, model, density and (b) the KA model, density . Arrows indicate motion between initial and final times. Time separation is 1/10th of the -relaxation time. In the t154 model, we see more fractal and disconnected clusters, while in the KA model, mobile domains tend to be smoother clusters.

Examples cluster shapes in the (a) the t154, model, density and (b) the KA model, density . Arrows indicate motion between initial and final times. Time separation is 1/10th of the -relaxation time. In the t154 model, we see more fractal and disconnected clusters, while in the KA model, mobile domains tend to be smoother clusters.

Violation of the Stokes–Einstein relation, , using at . Data has been normalized to at the lowest density.

Violation of the Stokes–Einstein relation, , using at . Data has been normalized to at the lowest density.

van Hove function for and various times. Distances are measured independently along each coordinate axis. The times plotted, from left to right, are , 316227 (approximately the -relaxation time), and . An exponential fit to the tail of the case is shown by a dotted line.

van Hove function for and various times. Distances are measured independently along each coordinate axis. The times plotted, from left to right, are , 316227 (approximately the -relaxation time), and . An exponential fit to the tail of the case is shown by a dotted line.

-dependent diffusion . Densities of 0.3000 (upper) and 0.5425 (lower). The higher density curve is multiplied by a scale factor of for ease of comparison. A dotted flat line is included for reference of behavior expected in the purely Fickian case.

-dependent diffusion . Densities of 0.3000 (upper) and 0.5425 (lower). The higher density curve is multiplied by a scale factor of for ease of comparison. A dotted flat line is included for reference of behavior expected in the purely Fickian case.

Top: Plot of at for densities 0.51, 0.52, 0.53, and 0.54. Bottom: Plot of for the same densities. Peak values correspond to lower bounds for of the value of in the upper panel at .

Top: Plot of at for densities 0.51, 0.52, 0.53, and 0.54. Bottom: Plot of for the same densities. Peak values correspond to lower bounds for of the value of in the upper panel at .

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