Multiple Glass Singularities and Isodynamics in a Core-Softened Model for Glass-Forming Systems

We investigate the slow dynamics of a simple glass former whose interaction potential is the sum of a hard core and a square shoulder repulsion. According to mode coupling theory, the competition between the two repulsive length scales gives rise to a complex dynamic scenario: besides the fluid-glass line, the theory predicts a glass-glass line in the temperature-packing fraction plane with two end points. Interestingly, for critical values of the square-shoulder parameters, such end points can be accessed from the liquid phase. We verify, via extensive numerical simulations, the existence of both points through the observation of an unconventional subdiffusive/logarithmic dynamical behavior. Unexpectedly, we also discover that the simultaneous presence of two end points generates special loci in the state diagram along which the dynamics is identical at all length and time scales.

Figure 1

Schematic evolution of the MCT dynamic state diagram of the SS system for increasing values of the shoulder width $\mathrm{\Delta }$. (a) For small $\mathrm{\Delta }$, beside the fluid-glass line (solid line), a disconnected glass-glass line is predicted (dashed line), ending in two $A_3$ singularities (stars). (b) On increasing $\mathrm{\Delta }$, the glass-glass line merges with the fluid-glass line and an $A_4$ singularity appears when one of the two $A_3$ points meets the fluid-glass line (filled triangle). (c) At even larger $\mathrm{\Delta }$ also the second $A_3$ point eventually intersects the fluid-glass line generating a distinct $A_4$ singularity (filled triangle). The arrows in (b) and (c) indicate the paths followed to locate the $A_4$ singularities in the simulations. The vertical dashed lines indicate the glass transitions at high and low densities: while the former happens at the well-known hard sphere glass transition packing fraction ${\varphi }_g^{(1{)}_{\mathrm{HS}}}$, the latter takes place at ${\varphi }_g^{(2{)}_{\mathrm{HS}}}={\varphi }_g^{(1{)}_{\mathrm{HS}}}/(1+\mathrm{\Delta }{)}^3$.

Figure 2

Dynamical quantities close to the two singularities $A_4^{(1)}\equiv ({\varphi }^{*(1)}\simeq 0.6,T^{*(1)}\simeq 0.55,{\mathrm{\Delta }}^{*(1)}\simeq 0.21)$ and $A_4^{(2)}\equiv ({\varphi }^{*(2)}\simeq 0.4,T^{*(2)}\simeq 0.15,{\mathrm{\Delta }}^{*(2)}\simeq 0.24)$, along paths shown in Figs. 1 and 1, respectively. (a) $A$-particles MSD as a function of the scaled time $t_0$ for different $\varphi$ at $T^{*(1)}$, ${\mathrm{\Delta }}^{*(1)}$. The dashed line is a power law ($\propto t^{0.3}$) to highlight the subdiffusive regime. The inset shows the MSD for the $B$ species. (b) Collective density correlators of the $A$ species ${\mathrm{\Phi }}_q^{AA}(t)$ evaluated at $A_4^{(1)}$ for different wave vectors $q{\sigma }_{AA}$. The dotted vertical lines delimit the time window in which ${\mathrm{\Phi }}_q^{AA}(t)$ display a logarithmic behavior. Dashed lines are fits obtained by a second degree polynomial in $\ln (t)$. (c) ${\mathrm{\Phi }}_q^{AA}(t)$ at $T^{*(1)}$, ${\mathrm{\Delta }}^{*(1)}$ as a function of $\varphi$. (d) MSD for $A$ particles at ${\varphi }^{*(2)}$, ${\mathrm{\Delta }}^{*(2)}$ as function of the scaled time $t_0$ for different $T$. The subdiffusive regime is characterized by a power-law $\propto t^{0.37}$. Inset: MSD of $B$ particles. (e) The same as (b) but evaluated at $A_4^{(2)}$. Despite the interference of the fluid-glass line on the dynamics (see text), a long-time logarithmic behavior can be identified. (f) ${\mathrm{\Phi }}_{q^{*(2)}}^{AA}(t)$ at ${\varphi }^{*(2)}$, ${\mathrm{\Delta }}^{*(2)}$ as a function of $T$.

Figure 3

Dynamical properties for $\mathrm{\Delta }=0.17$ along three isodynamics lines with rescaled diffusivity $D/D_0=3.5\times 10^{-4}$ (orange symbols), $D/D_0=5.5\times 10^{-5}$ (red symbols), and $D/D_0=9.8\times 10^{-6}$ (magenta symbols). (a) MSD of the $A$ species along the isodynamics lines as a function of $t/t_0$. (b) Density correlator for the same sets of isodynamics points in (a) as a function of $t/t_0$. The inset shows the position of the iso-$D/D_0$ lines and of the expected fluid-glass line [43].

Figure 4

${\mathrm{\Phi }}_q^{AA}(t)$ along the iso-$D/D_0=5.5\times 10^{-5}$ line for $\mathrm{\Delta }=0.17$ at different wave vectors $q{\sigma }_{AA}$ for (a) isodynamics and (b) nonisodynamics state points. The inset shows the location of such points in the $T-\varphi$ diagram. Squares (stars) indicate isodynamics (nonisodynamics) points.