Microscopic structural relaxation in a sheared supercooled colloidal liquid

The rheology of dense amorphous materials under large shear strain is not fully understood, partly due to the difficulty of directly viewing the microscopic details of such materials. We use a colloidal suspension to simulate amorphous materials and study the shear-induced structural relaxation with fast confocal microscopy. We quantify the plastic rearrangements of the particles in several ways. Each of these measures of plasticity reveals spatially heterogeneous dynamics, with localized regions where many particles are strongly rearranging by these measures. We examine the shapes of these regions and find them to be essentially isotropic, with no alignment in any particular direction. Furthermore, individual particles are equally likely to move in any direction other than the overall bias imposed by the strain.

Figure 1

(Color online) Sketch of the shear cell. A fluorescent sample (gray, green) is put between two parallel glass plates (dark gray, blue) with gap $H$ set by three ball bearings, two of which are shown. The top plate is movable, controlled by a piezomotor driven by a triangular wave. The bottom plate is fixed and the confocal microscope takes images from underneath. Note the definition of the coordinate system, where $x$ is in the velocity direction and $z$ is in the velocity gradient direction.

Figure 2

A $20\times 20\times 20\mu {\text{m}}^3$ image of a supercooled colloidal liquid taken $15\mu \text{m}$ away from the fixed bottom plate by our confocal microscope in less than 1 s. Scale bar represents $10\mu \text{m}$.

Figure 3

(Color online) Shear profile $v_x\left(z\right)$ within a $50\times 50\times 70\mu {\text{m}}^3$ volume, measured at different times during one half period $T/2=84\text{s}$. The total macroscopic strain applied in 84 s is 1.35. The sample has a volume fraction $\varphi \approx 0.50$ and the global applied shear rate is ${\dot{\gamma }}_{\text{macro}}=0.016{\text{s}}^{-1}$ controlled by the piezomotor. If there was not a shear band, one would expect to observe a linear velocity profile as indicated by the lower thick diagonal line. The four curves represent different times during the half period as indicated. Note that these data are obtained using a coarse-grained image velocimetry method [62] rather than particle tracking. Here we take rapid images over the full volume with a spacing of $z=1\mu \text{m}$ in the vertical direction. For each value of $z$, we cross correlate subsequent images to obtain a mean instantaneous velocity $v_x$ with a resolution of $0.05\mu \text{m}/\text{s}$, set by the pixel size and the time between images (see Ref. [62] for further details).

Figure 4

Typical example of a measured local instantaneous shear rate over one period of shearing in a gap of $20–40\mu \text{m}$ away from the fixed bottom plate. The sample has volume fraction $\varphi =0.51$, the applied strain rate is ${\dot{\gamma }}_{\text{macro}}=0.013{\text{s}}^{-1}$, and the period is $T=200\text{s}$.

Figure 5

Trajectories of colloids in an $xz$ slice ($5\mu \text{m}$ thick in the $y$ direction) (see Fig. 1 for the coordinate axes). The sample has $\varphi =0.51$ and the data shown correspond to a locally measured accumulated strain ${\gamma }_{\text{meso}}=0.43$ over 45 s of data, so the effective strain rate is ${\overline{\dot{\gamma }}}_{\text{meso}}={\gamma }_{\text{meso}}/\Delta t=0.0096{\text{s}}^{-1}$. (a) Trajectories in a reference frame comoving with the average velocity $v_x\left(z=20\mu \text{m}\right)$. (b) Displacements corresponding to data from panel (a), where the start point of each particle is marked with a circle and the end point is marked with an arrowhead. (c) The same data, but with the affine motion removed; this is the $\tilde{x}z$ plane. (d) Displacements corresponding to panel (c).

Figure 6

(Color online) (a) Mean-square displacement of the nonaffine motion $\Delta \tilde{r}$ as a function of the time $t$ since the start of shear. (b) The same data plotted as a function of the accumulated strain ${\gamma }_{\text{meso}}={\int }_0^t{\dot{\gamma }}_{\text{meso}}\left(t^{\prime }\right)d{t}^{\prime }$. The five curves represent five data sets with the same volume fraction $\varphi =0.51$ but three different shear rates ${\overline{\dot{\gamma }}}_{\text{meso}}$ as indicated in the figure. The total accumulated strain is the final point reached by each curve in panel (b).

Figure 7

(Color online) Probability distribution function of $\Delta s=\Delta \tilde{r}/{\gamma }_{\text{meso}}^{0.5}$, the nonaffine displacement $\Delta \tilde{r}$ scaled by the local strain ${\gamma }_{\text{meso}}^{0.5}$. Data shown are from one experimental run using portions of the data corresponding to strain increments ${\gamma }_{\text{meso}}=0.01$, 0.02, 0.10, and 0.43. These correspond to time intervals $\Delta t=1.5$, 4.5, 16.5, and 45 s. $P\left(\Delta s\right)$ is normalized so that the integral over all vectors $\Delta \vec{s}$ is equal to 1, similar to Ref. [67]. For small strains, the curves have large exponential tails and for larger strain, the tails become smaller and appear more Gaussian. Dashed line is Gaussian fit for ${\gamma }_{\text{meso}}=0.43$. For this sample, data and parameters are the same as Fig. 5.

Figure 8

(Color online) Probability distribution functions for nonaffine motions in each direction, $\Delta \tilde{x}$ (filled cycle), $\Delta y$ (hollow circle), and $\Delta z$ (hollow triangle). Red curve is a Gaussian fit to the $\Delta \tilde{x}$ data. For this sample, parameters are the same as Fig. 5 $\left(\varphi =0.51,{\gamma }_{\text{meso}}=0.43,{\overline{\dot{\gamma }}}_{\text{meso}}=0.0096{\text{s}}^{-1},\Delta t=45\text{s}\right)$.

Figure 9

(Color online) Spatial correlation functions $S_{\vec{r}}$ and $S_{\delta r}$ characterizing particle motion. Data are the same as Fig. 5 ($\varphi =0.51$, ${\gamma }_{\text{meso}}=0.43$ over 45 s of data, and ${\overline{\dot{\gamma }}}_{\text{meso}}={\gamma }_{\text{meso}}/\Delta t=0.0096{\text{s}}^{-1}$). Solid and dashed lines correspond to $S_{\vec{r}}$ and $S_{\delta r}$ averaged over all the particles in the sample. Dotted lines (red) correspond to exponential fits with length scales ${\xi }_{\vec{r}}=6.4\mu \text{m}$ and ${\xi }_{\delta r}=2.5\mu \text{m}$.

Figure 10

(Color online) Sketch illustrating the definition of the motion of the relative position vector $\Delta {\vec{d}}_{ij}$ of two neighboring particles, red and blue (dark and light gray). ${\vec{r}}_i\left(t\right)$ represents the position of particle $i$ at time $t$, ${\vec{d}}_{ij}\left(t\right)$ is the relative position between particles $i$ and $j$ at time $t$, and $\Delta {\vec{d}}_{ij}\left(t\right)$ is how the vector ${\vec{d}}_{ij}$ changes over the time interval $\left(t,t+\Delta t\right)$.

Figure 11

(Color online) These four sketches show different portrayals of a particle with unusual motion. (a) Particle motion as seen in the laboratory reference frame. Arrows indicate displacement vectors. (b) Similar to (a) except the motion are in the reference frame where the central red (darker) particle is motionless. (c) Large spheres correspond to the expected positions of the particles and the smaller spheres correspond to the actual positions of the particles. The distortion of the box and the expected positions of the particles are calculated based on the measured local strain tensor ${\mathbb{E}}_i$, where red (darker) particle is particle $i$. For this local neighborhood, we have ${\gamma }_{\text{micro}}=0.73$ and a dilation primarily in the $z$ direction (${ϵ}^{xx}=-0.1$, ${ϵ}^{zz}=0.5$, and $\delta e=0.35$). (d) The positions of the same particles after the strain has been applied, where now the box represents the mesoscopic strain ${\gamma }_{\text{meso}}=0.58$. The large spheres represent the expected positions if the motion was affine and the small spheres show the actual positions. For all panels, to show the displacements in three dimensions better, the radii of the large spheres are 0.5 of the real scale. Data correspond to a sample with $\varphi =0.51$, ${\gamma }_{\text{meso}}=0.58$, ${\overline{\dot{\gamma }}}_{\text{meso}}=0.014{\text{s}}^{-1}$, and $\Delta t=40\text{s}$.

Figure 12

(Color online) A $3\text{−}\mu \text{m}$-thin cut through the sample, showing the $xz$ shear plane; axes are as labeled in (a). The particles are drawn in their location at $t=0$, the start of this strain half cycle. (a) Darker particles have larger values of $\Delta x$, thus indicating the applied strain. The top particles (large $z$) are moving left (dark colors) and the bottom particles are moving right (light colors). (b) Darker particles have larger Voronoi volumes at $t=0$. (c) Darker particles have larger values of the nonaffine motion $\Delta {\tilde{r}}^2$. (d) Darker particles have larger values of the plastic deformation $D_{\text{min}}^2$. (e) Darker particles have larger values of the local strain ${\gamma }_{\text{micro}}$. (f) Darker particles have larger values of the dilation $\delta e$. The sample is the same as the data shown in Fig. 5 (see that figure caption for details).

Figure 13

(Color online) 2D histograms of (a) $D_{\text{min}}^2$ and $\Delta {\tilde{r}}^2$, (b) ${\gamma }_{\text{micro}}$ vs dilation $\delta e$, and (c) ${\gamma }_{\text{micro}}$ and $D_{\text{min}}^2$. The sample has the same parameters as Fig. 5 (see that caption for details). In these histograms, darker colors stand for the larger probability. Dotted lines are the average values of the $y$ axis corresponding to each value on the $x$ axis. The correlation coefficients between each pair of variables are (a) $C=0.43\pm 0.11$, (b) $C=0.42\pm 0.09$, and (c) $C=0.17\pm 0.05$ (see text for details).

Figure 14

(Color online) Cluster of neighboring particles with large values of (a) nonaffine motion $\Delta {\tilde{r}}^2$ and (b) plastic deformation $D_{\text{min}}^2$. To avoid biasing the cluster shape, the particles are drawn from within a $15\times 15\times 15\mu {\text{m}}^3$ cube (solid lines). Dashed lines indicate the spatial extent of the two cluster shapes in each direction. The cluster in (a) is $14\times 12\times 11\mu {\text{m}}^3$ and the cluster in (b) is $8\times 14\times 10\mu {\text{m}}^3$. Red particles are common between the two clusters and the black bonds indicate neighboring particles; particle sizes are drawn 0.8 times the actual scale to make the connections more visible. In each case, the particles shown are the top 20% for the parameter chosen. Data corresponds to the same parameters as in Fig. 5 (see that caption for details).

Figure 15

(Color online) Comparison between the cluster extent in $x$, $y$, and $z$ based on different variables. (a) Particles with the largest nonaffine motion $\Delta \tilde{r}$. (b) Particles with the largest plastic deformation $D_{\text{min}}^2$. Symbols represent different volume fractions: $\varphi =0.51$ (circles), 0.56 (triangles), and 0.57 (squares). Colors (or grays from light to dark) represent different accumulative strains ${\gamma }_{\text{meso}}=0.3–0.9$. The clusters are comprised of the top 20% of the particles with the given characteristic.