Localized Excitations and the Morphology of Cooperatively Rearranging Regions in a Colloidal Glass-Forming Liquid

We develop a scheme based on a real space microscopic analysis of particle dynamics to ascertain the relevance of dynamical facilitation as a mechanism of structural relaxation in glass-forming liquids. By analyzing the spatial organization of localized excitations within clusters of mobile particles in a colloidal glass former and examining their partitioning into shell-like and corelike regions, we establish the existence of a crossover from a facilitation-dominated regime at low area fractions to a collective activated hopping-dominated one close to the glass transition. This crossover occurs in the vicinity of the area fraction at which the peak of the mobility transfer function exhibits a maximum and the morphology of cooperatively rearranging regions changes from stringlike to a compact form. Collectively, our findings suggest that dynamical facilitation is dominated by collective hopping close to the glass transition, thereby constituting a crucial step towards identifying the correct theoretical scenario for glass formation.

Figure 1

Representative CRRs containing $N=30$ particles for $\varphi =0.71$ (a), $\varphi =0.75$ (b), and $\varphi =0.76$ (c). In panels (a)–(c), particles belonging to the core are shown in brown and those belonging to the shell are shown in white. Excitations of size $a=0.2\sigma$ that overlap with the CRRs are denoted by “$i$” and those of size $a=0.4\sigma$ are denoted by “$j$.” (d) Procedure for identifying excitations. A representative coarse-grained subtrajectory (blue curve) of a tagged particle and its associated functional $h_i(t,t_a;a)$ (red) over the interval [$-t_a/2,t^{*}+t_a/2$]. The panel on the left side denotes the positions of the tagged particle and its nearest neighbors at the initiation time of the excitation, i.e., the time at which $h_i(t,t_a;a)$ switches from 0 to 1. The panel on the right side denotes the final positions of the particles at the termination of the excitation, i.e., the time at which $h_i(t,t_a;a)$ switches from 1 to 0. The color scheme encodes the magnitude of the displacement, with brown indicating high mobility and blue indicating low mobility. The arrows begin at the initial positions and terminate at the final positions and the lengths are proportional to the magnitude of the displacement.

Figure 2

(a) The fraction of excitations associated with the shell-like ($F_s^a$; hollow circles and triangles) and corelike ($F_c^a$; solid circles and triangles) regions of CRRs of size 25–35 particles for $a=0.2\sigma$ (circles) and $a=0.4\sigma$ (triangles). The fractions of shell-like mobile particles ($F_s^{\text{mob}}$; hollow black squares) and corelike mobile particles ($F_c^{\text{mob}}$; filled black squares) are also shown. (b) The ratio $F_s^a/F_s^{\text{mob}}$ for $a=0.4\sigma$ for CRRs of size 25–35 particles (triangles), 16–24 particles (squares), and 10–15 particles (circles). The horizontal dotted line corresponds to the equipartition of excitations between the shell-like and corelike regions.

Figure 3

(a) The peak $d_m^{\text{peak}}$ of the distributions $P(d_m)$ as a function of $\varphi$ for $a=0.2\sigma$ (filled circle) and $a=0.4\sigma$ (down-pointing triangle) for CRRs containing 25–35 particles. The inset shows the distributions $P(d_m)$ and corresponding fits of the form $A{x}^b\exp (-cx)$ for the three data points circled in the main plot of panel (a). The $d_m^{\text{peak}}$ values in (a) have been extracted using these fits. (b) $⟨d_m/R_g⟩$ vs $\varphi$ for $a=0.4\sigma$ for CRRs of size 25–35 particles (triangles), 16–24 particles (squares), and 10–15 particles (spheres). Here, $R_g$ denotes the radius of gyration of the CRR and $⟨\cdots ⟩$ denotes averaging over the CRRs.