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Geometric frustration is a concept that has been used to describe some aspects of packing, jamming, and glass formation phenomena. It describes an incompatibility between the local and the global orders in a system. The existence and the strength of frustration are strongly dependent on the properties of the embedding space among which, dimensionality and metrics. For a given space dimension, curvature for instance plays a major role, and one may vary frustration by changing the curvature of space. It is well known that one-component liquids of spherically symmetric particles in flat two dimensional space do not form a glass due to unavoidable ordering in hexatic/hexagonal phases. However, by negatively curving space, ordering is frustrated and glass formation can be observed. This is what we have studied by computer simulation of a monatomic liquid model on the hyperbolic plane. We find that the characteristics of the slowdown of relaxation with decreasing temperature, what is often called the "fragility'' of a glass former, changes with frustration, with the amount of frustration controlled by space curvature~[1].
We have investigated the relation between dynamics and structure, more specifically between the characteristic length scales associated with the growing heterogeneous nature of the dynamics on the one hand and those measuring the amount of local hexatic ordering and the density of topological defects on the other. As temperature is lowered, we identify three different regimes: a first one in which the liquid is insensitive to curvature, an intermediate one which is dominated by collective effects controlled by the proximity to the (avoided) ordering transition in flat space, and a low-temperature one characterized by the dynamics of the few intrinsic topological defects. We have also use this system as a benchmark for testing recently proposed methods to extract characteristic static lengths without a priori knowledge of the local order parameter [2,3]. Finally, we discuss the connection with geometric frustration in three dimensions.
[1] F. Sausset and G. Tarjus. Growing static and dynamic length scales in a glass-forming liquid. Phys. Rev. Lett., (2010). [2] J. Kurchan and D. Levine. Correlation length for amorphous systems. arXiv:0904.4850 (2009). [3] G. Biroli et al. Thermodynamic signature of growing amorphous order in glass-forming liquids. Nature Physics 4, 771 (2008). |
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