Different mechanisms for dynamical arrest in largely asymmetric binary mixtures

Using confocal microscopy we investigate binary colloidal mixtures with large size asymmetry, in particular the formation of dynamically arrested states of the large spheres. The volume fraction of the system is kept constant, and as the concentration of small spheres is increased we observe a series of transitions of the large spheres to different arrested states: an attractive glass, a gel, and an asymmetric glass. These states are distinguished by the degree of dynamical arrest and the amount of structural and dynamical heterogeneity. The transitions between two different arrested states occur through melting and the formation of a fluid state. While a space-spanning network of bonded particles is found in both arrested and fluid states, only arrested states are characterized by the presence of a space-spanning network of dynamically arrested particles.

Figure 1

Mean squared displacements of the large spheres, $\Delta r^2/{\left(2R_{\mathrm{L}}\right)}^2$, for samples with total volume fraction $\varphi \approx 0.60$, size ratio $\delta =0.09$, and different compositions $x_{\mathrm{s}}=0.0 (•),0.01$ (), $0.1$ (), $0.3$ (), $0.5$ (), $0.7$ (), $0.9$ (). Inset: Localization length $L/\left(2R_{\mathrm{L}}\right)$ as a function of $x_{\mathrm{s}}$.

Figure 2

Distributions of displacements of the large spheres in $x$ direction, $P\left(\Delta x\right)$, over delay times $\Delta t/{\tau }_{\mathrm{B}}=0.06$ (), 0.3 (), 3.0 (), 18.0 () for samples with $\varphi \approx 0.60,\delta =0.09$, and different compositions $x_{\mathrm{s}}$, as indicated. Lines represent Gaussian fits.

Figure 3

Rendering of $3\mu \text{m}$ thick slices of a central region of $44\times 33 \mu {\text{m}}^2$, in the bulk of samples with $\varphi \approx 0.60,\delta =0.09$ and different compositions $x_{\mathrm{s}}$ (as indicated) as obtained from coordinates extracted from confocal microscopy images. Left: Large particles with the 20% largest displacements over a time interval of about ${\tau }_{\mathrm{B}}/4$ are shown in red (light gray). Right: Large particles with the 20%–30% smallest number of bonds per particle are shown in green (light gray). The small particles are not shown.

Figure 4

Pair distribution function $g\left(r\right)$ of large particles with radius $R_{\mathrm{L}}$ in mixtures with $\varphi \approx 0.60,\delta =0.09$ and different compositions $x_s=0.0 (•),0.01$ (), $0.1$ (), $0.3$ (), $0.5$ (), $0.7$ (), $0.9$ (). Data for $x_{\mathrm{s}}>0$ are shifted vertically for clarity. Dashed lines indicate particle-particle distances $r=2R_{\mathrm{L}},r=2(R_{\mathrm{L}}+R_{\mathrm{s}}),r=2(R_{\mathrm{L}}+2R_{\mathrm{s}})$, and $r=2(R_{\mathrm{L}}+3R_{\mathrm{s}})$.

Figure 5

(a) Value of the pair distribution function $g\left(r\right)$, shown in Fig. 4, at the first peak, $g^{\max }$ (left $y$ axis) and peak area $A^{\max }$(right $y$ axis), (b) position of the first peak $r^{\max }/2R_{\mathrm{L}}$ (left $y$ axis) and second peak $r^2\max /2R_{\mathrm{L}}$ (right $y$ axis), dashed lines indicate particle-particle distances $r=2R_{\mathrm{L}},r=2(R_{\mathrm{L}}+R_{\mathrm{s}}),r=2(R_{\mathrm{L}}+2R_{\mathrm{s}})$, (c) depth of the first minimum $\Delta g^{\min }=1-g\left(r^{\min }\right)$, and (d) position of the first minimum $r^{\min }/2R_{\mathrm{L}}$, as a function of $x_{\mathrm{s}}$.

Figure 6

Left: Rendering of $4\mu \text{m}$ thick slices of area $58\times 42 \mu {\text{m}}^2$ in the bulk of samples with $\varphi \approx 0.60,\delta =0.09$ and different compositions $x_{\mathrm{s}}$ (as indicated) as obtained from coordinates extracted from confocal microscopy images. (Left) Large particles have different color (gray-scale value) according to their number of bonds $N_{\mathrm{B}}$ (as indicated). (Right) Large particles pertaining to the same cluster are indicated with the same color (gray-scale value). The small particles are not shown.

Figure 7

(a) Distribution of the number of bonds $N_{\mathrm{b}}$ per large particle, $P\left(N_{\mathrm{b}}\right)$. Inset: Most likely number of bonds $N_{\mathrm{b}}^{\max }$ ($•$, left axis) and width $W_{\mathrm{N}}$ of $P\left(N_{\mathrm{b}}\right)$ (, right) as a function of composition $x_{\mathrm{s}}$. (b) Cluster size distribution $P\left(N_{\mathrm{c}}\right)$, with $N_{\mathrm{c}}$ the number of particles forming the cluster. Lines are fits of a power-law dependence $f\left(N_{\mathrm{c}}\right)=A{N}_{\mathrm{c}}^{-\gamma }exp(-N_{\mathrm{c}}/k_{\mathrm{c}})$. Inset: Fit parameters $A,\gamma$, and $k_{\mathrm{c}}$ as a function of $x_{\mathrm{s}}$ (c) Remoteness distribution, $P\left(\xi \right)$, Inset: Average remoteness $\langle \xi \rangle$ as a function of $x_{\mathrm{s}}$. The mixtures had $\varphi \approx 0.60,\delta =0.09$ and different compositions $x_{\mathrm{s}}=0.0 (•),0.01$ (), $0.1$ (), $0.3$ (), $0.5$ (), $0.7$ (), $0.9$ ().

Figure 8

Fraction of arrested particles $N_{\mathrm{arr}}/N$ as a function of composition $x_{\mathrm{s}}$.

Figure 9

Rendering of the arrested large particles in samples with $\varphi \approx 0.60,\delta =0.09$, and different compositions $x_{\mathrm{s}}$ (as indicated) as obtained from trajectories extracted from confocal microscopy images. Only arrested large particles are shown with particles pertaining to the same cluster indicated with the same color (gray-scale value). Particles within two diameters of the surface of the observation volumes are not shown.

Figure 10

State diagram of samples with different composition $x_{\mathrm{s}}$ and size ratios; $\delta =0.09$ (present work), $\delta =0.106$ [13], $\delta =0.2$ and $0.38$ [17, 18, 19]. Different arrested states are identified in the present work: repulsive glass $(•)$, attractive glass $\left(▾\right)$, asymmetric glass (), and gel (). Open symbols indicate fluid states. In [13] fluids $\left(▵\right)$, fluid-crystal coexistence $\left(◊\right)$, and amorphous solids $\left(⧫\right)$ were distinguished.