Heterogeneous Activation, Local Structure, and Softness in Supercooled Colloidal Liquids

We experimentally characterize heterogeneous nonexponential relaxation in bidisperse supercooled colloidal liquids utilizing a recent concept called “softness” [Phys. Rev. Lett. 114, 108001 (2015)]. Particle trajectory and structure data enable classification of particles into subgroups with different local environments and propensities to hop. We determine residence times $t_R$ between particle hops and show that $t_R$ derived from particles in the same softness subgroup are exponentially distributed. Using the mean residence time ${\overline{t}}_R$ for each softness subgroup, and a Kramers’ reaction rate model, we estimate the activation energy barriers $E_b$ for particle hops, and show that both ${\overline{t}}_R$ and $E_b$ are monotonic functions of softness. Finally, we derive information about the combinations of large and small particle neighbors that determine particle softness, and we explicitly show that multiple exponential relaxation channels in the supercooled liquid give rise to its nonexponential behavior.

Figure 1

$P(t_R)$ measured for large particles. Error bars indicate standard deviations. The red dashed line is the stretched exponential fit $P(t_R)\sim \exp [-(t_R/220{)}^{0.85}]$. The black solid line is calculated from Eq. (3). The blue dash-dot line is the exponential fit $P(t_R)\sim \exp [-(t_R/300)]$. Inset (left): the relative residuals of both fits. The residual plot is cut off at 2000 sec; points at longer times exhibit even greater disagreement with the exponential fit, but obscure early-time data which have excellent signal-to-noise. Residuals are fit with quadratic functions as guides for the eye. Inset (right) shows a trajectory (black) of particle displacement, $\mathrm{\Delta }{r}_i(t)\equiv |{\overrightarrow{r}}_i(t)-{\overrightarrow{r}}_i(0)|$, relative to an arbitrary reference time $t=0$. Here, displacement is scaled by large particle diameter, $\sigma$. $p_{\mathrm{hop},i}$ is calculated from the same trajectory.

Figure 2

Two unnormalized conditional PDFs, $P(t_R|{⟨S⟩}_{t_R})$, for $t_R$ with similar $⟨S{⟩}_{t_R}$. The red solid and blue dashed lines are exponential fits, $P(t_R|⟨S{⟩}_{t_R})\sim (1/{\overline{t}}_R)e^{-t_R/{\overline{t}}_R}$, with ${\overline{t}}_R=400$ and 270 sec for ${⟨S⟩}_{t_R}=-0.8$ and 0.8, respectively; fitting is to the tails ($t_R>200\sec$) of the conditional PDFs. Inset shows the overall PDF, $P(⟨S{⟩}_{t_R})$. The solid line is a Gaussian fit. The subgroups that constitute the conditional PDFs are shaded with the same colors.

Figure 3

(a) Measured $g_l^l(r)$ of large particles around a central large particle. (b) Measured $g_l^s(r)$ of small particles around a central large particle. (c) Numbers of large (blue circle) and small (green triangle) neighbors within $r<1.2\sigma$ as a function of $⟨S{⟩}_{t_R}$ (see main text).

Figure 4

(a) Measured ${\overline{t}}_R$ as a function of $⟨S{⟩}_{t_R}$, for large (black squares) and small (red circles) particles. Solid and dashed lines are exponential fits (see main text). The error bars indicate the standard deviations.

Figure 5

(a) Estimated $E_b$ as a function of $⟨S{⟩}_{t_R}$ for both species. Inset: Schematic of the barrier-crossing activation process. (b) $P(E_b)$ of activation energy barriers.