Point-to-set lengths, local structure, and glassiness

The growing sluggishness of glass-forming liquids is thought to be accompanied by growing structural order. The nature of such order, however, remains hotly debated. A decade ago, point-to-set (PTS) correlation lengths were proposed as measures of amorphous order in glass formers, but recent results raise doubts as to their generality. Here, we extend the definition of PTS correlations to agnostically capture any type of growing order in liquids, be it local or amorphous. This advance enables the formulation of a clear distinction between slowing down due to conventional critical ordering and that due to glassiness, and provides a unified framework to assess the relative importance of specific local order and generic amorphous order in glass formation.

Figure 1

Overlaps for monodisperse hard disks at packing fraction $\varphi =0.695<{\varphi }_{\mathrm{hexatic}}$ and cavity radius $R=6.25\approx {\xi }_{\ell =6}^{\mathrm{PTS}}\approx 2{\xi }_{\mathrm{pos}}^{\mathrm{PTS}}$. The positional (left) and bond-orientational overlaps for $\ell =6$ (middle) and 8 (right) between a specific equilibrated configuration $\mathbf{Y}$ with a given reference configuration $\mathbf{X}$ are evaluated at the particle centers of the former (top). Averaging over reference and equilibrated configurations yields smooth overlap profiles (bottom). To compare different types of overlaps, their average values for two identical configurations are rescaled to unity. Cavity overlap profiles clearly confirm that the $\ell =6$ overlap detects the incipient hexatic order while positional and other bond-orientational overlaps are blind to it.

Figure 2

Decay of the PTS correlations with cavity radius $R$ for monodisperse hard disks: (a) positional $g_{\mathrm{pos}}^{\mathrm{PTS}}\left(R\right)$ and [(b) and (c)] bond-orientational $g_{\ell }^{\mathrm{PTS}}\left(R\right)$ for $\ell =6,8$ at packing fractions $\varphi =0.600$ (red cross), 0.650 (green circle), 0.680 (cyan square), 0.690 (blue diamond), and 0.695 (black plus). Solid lines are exponential fits. Note that the $y$-axis range is an order of magnitude larger for $\ell =6$ than for $\ell =8$. (b) The growth of positional (red square) and hexatic (blue diamond) PTS correlation lengths extracted through the exponential fits track the correlation lengths (dashed lines) extracted from the radial and coarse-grained $\ell =6$ correlation functions, respectively. Lengths are relative to the results for ${\xi }_0\equiv \xi ({\varphi }_0=0.650)$.

Figure 3

PTS correlations [(a)–(c)], susceptibilities [(d)–(f)], and lengths (g) for the $d=3$ KABLJ model as a function of cavity radius $R$: positional [(a) and (d)] and bond-orientational results for $\ell =6$ [(b) and (e)] and 8 [(c) and (f)] at $T=1.00$ (red cross), 0.80 (green circle), 0.60 (cyan square), 0.51 (blue diamond), and 0.45 (black-plus). For PTS correlations, solid lines are compressed exponential fits $g^{\mathrm{PTS}}=Aexp[-{(R/{\xi }_{\mathrm{fit}}^{\mathrm{PTS}})}^{\eta }]$ with $\eta =3$; for susceptibilities, solid lines are guides for the eye, and dashed lines for $T=0.45$ are extrapolated from $R=3.2$ to the extrapolated peak (star) at $R\approx 3.8$ [36]. (g) The bond-orientational PTS lengths ${\xi }_{\ell }^{\mathrm{PTS}}$ for $\ell =6$ (blue diamond), 7 (green cross), 8 (black plus), and 12 (cyan circle) follow the positional PTS length ${\xi }_{\mathrm{pos}}^{\mathrm{PTS}}$ (red-square). Dotted lines depict the standard pair positional length ${\xi }_{\mathrm{pos}}$ and bond-orientational lengths ${\xi }_{\ell }$ for $\ell =6,7,8,12$, all of which increase at a slower pace with decreasing temperature, in agreement with Ref. [35]. Each length is relative to ${\xi }_0\equiv \xi (T_0=0.800)$.

Figure 4

Growth of the structural relaxation time, ${\tau }_{\alpha }$, with the dominant length, normalized with ${\xi }_0\equiv {\xi }_{\mathrm{pos}}^{\mathrm{PTS}}\left(T_0\right)$ at high temperature $T_0=0.80$ for the KABLJ model (blue triangles) and with ${\xi }_0\equiv {\xi }_{\ell =6}\left({\varphi }_0\right)$ at low packing fraction ${\varphi }_0=0.650$ for monodisperse (orange squares) and ${\varphi }_0=0.700$ for $11\%$ polydisperse hard disks (purple stars). For the KABLJ model, ${\tau }_{\alpha }$ increases exponentially while the PTS length grows modestly. By contrast, for hard disks, ${\tau }_{\alpha ;\ell =6}$ as a function of ${\xi }_{\ell =6}$ (see Appendix pp3) increases much more slowly, compatible with the power-law scaling of Eq. (2) (see the inset for the log-log plot).

Figure 5

Evolution of weakly first-order transition line (magenta) with polydispersity from a semigrandcanonical ensemble calculation [50]. Dots indicate packing fractions at which PTS measurements are performed for each polydispersity, $\varphi ={\varphi }_1$ (red cross), ${\varphi }_2$ (green circle), ${\varphi }_3$ (cyan square), ${\varphi }_4$ (blue diamond), and ${\varphi }_5$ (black plus).

Figure 6

Radial decay of the bond-orientational PTS correlations for monodisperse hard disks, $g_{\ell }^{\mathrm{PTS}}\left(R\right)$, for $\ell =1,...,16$ at $\varphi =0.600$ (red cross), 0.650 (green circle), 0.680 (cyan square), 0.690 (blue diamond), and 0.695 (black plus). Note that the vertical axes have the same range for all $\ell$ except for $\ell =6$. Solid lines are exponential fits.

Figure 7

Positional ${\chi }_{\mathrm{pos}}^{\mathrm{PTS}}$ (top row) and $\ell =6$ bond-orientational ${\chi }_6^{\mathrm{PTS}}$ (middle row) PTS susceptibilities as a function of cavity radius $R$ for hard disks with polydispersity $\mathrm{\Delta }=0\%,3\%,6\%,9\%,11\%$ (from left to right) at $\varphi ={\varphi }_1$ (red cross), ${\varphi }_2$ (green circle), ${\varphi }_3$ (cyan square), ${\varphi }_4$ (blue diamond), and ${\varphi }_5$ (black plus), where the values of ${\varphi }_i$ at each polydispersity are the same as in Fig. 5. The bottom row, for the same range of polydispersity, records the growth of positional (red square) and hexatic (blue diamond) PTS lengths extracted through the exponential fits to PTS correlations. Dashed lines are positional and sixfold pair correlation lengths. Lengths are relative to the results for ${\xi }_0\equiv \xi ({\varphi }_0={\varphi }_2)$.

Figure 8

Radial decay of the bond-orientational PTS correlations for the KABLJ model, $g_{\ell }^{\mathrm{PTS}}\left(R\right)$, for $\ell =1,...,16$ at $T=1.00$ (red cross), 0.80 (green circle), 0.60 (cyan square), 0.51 (blue diamond), and 0.45 (black plus). Solid lines are compressed exponential fits $g^{\mathrm{PTS}}=Aexp[-{(R/{\xi }_{\mathrm{fit}}^{\mathrm{PTS}})}^{\eta }]$ with $\eta =3$.