From Glass Formation to Icosahedral Ordering by Curving Three-Dimensional Space

Geometric frustration describes the inability of a local molecular arrangement, such as icosahedra found in metallic glasses and in model atomic glass formers, to tile space. Local icosahedral order, however, is strongly frustrated in Euclidean space, which obscures any causal relationship with the observed dynamical slowdown. Here we relieve frustration in a model glass-forming liquid by curving three-dimensional space onto the surface of a 4-dimensional hypersphere. For sufficient curvature, frustration vanishes and the liquid “freezes” in a fully icosahedral structure via a sharp “transition.” Frustration increases upon reducing the curvature, and the transition to the icosahedral state smoothens while glassy dynamics emerge. Decreasing the curvature leads to decoupling between dynamical and structural length scales and the decrease of kinetic fragility. This sheds light on the observed glass-forming behavior in Euclidean space.

Figure 1

(a) Concentration $n$ of particles found in domains of icosahedra (see ball-and-stick model) as a function of the inverse of the reduced temperature $T$ for several curvatures characterized by the system size $N$ [see Eq. (1)]. The lines are hyperbolic-tangent fits, from which the maximum of the derivative ${\chi }_n=dn/dT$ can be estimated: see the case $N=120$ in the inset. (b) Same data, as a function of the curvature $1/R$ at $T=0.6$, 0.7, 0.8, 2, 3. Lines are guides to the eye.

Figure 2

(a) Logarithm of the relaxation time (a) and Effective activation energy $\mathrm{\Delta }{E}_{\mathrm{eff}}=T\log \tau /{\tau }_{\infty }$ (b) versus $1/T$ for several curvatures. The vertical dashed lines approximately indicate the temperatures at which the $N=120$ and $N=140$ systems freeze into a solid icosahedral phase.

Figure 3

(a) Rescaled dynamic length versus $1/T$ for different curvatures. (b) Radial distribution functions $g_{\text{slow}}(r;\tau )$ (continuous) and $g_{\mathrm{icos}}(r)$ (dotted) for $N=160$, 720 and $T=0.8$ scaled by the first peak height and shifted for clarity. Fits to extract length scales (see Supplemental Material [34]) in gray. (c) Rescaled dynamic length versus the rescaled structural length for different curvatures, down to the Euclidean limit.