Multiple Transient Memories in Experiments on Sheared Non-Brownian Suspensions

A system with multiple transient memories can remember a set of inputs but subsequently forgets almost all of them, even as they are continually applied. If noise is added, the system can store all memories indefinitely. The phenomenon has recently been predicted for cyclically sheared non-Brownian suspensions. Here we present experiments on such suspensions, finding behavior consistent with multiple transient memories and showing how memories can be stabilized by noise.

Figure 1

Experimental setup. (a) Top view of the circular Couette cell containing a viscous suspension between two concentric cylinders. A two-dimensional slice of the suspension is imaged by shining a laser sheet into the fluorescently dyed fluid. The emitted light is imaged through a long-pass filter. (b) A small region of the imaged slice. (c),(d) Visual readout of a memory formed at ${\gamma }_1=1.2$. Each image is the difference of pictures taken before and after a single back-and-forth rotation of amplitude (c) $\gamma =1.2$ and (d) $\gamma =1.4$. The subtractions show that the particle trajectories are reversible at $\gamma =1.2$ but irreversible at $\gamma =1.4$.

Figure 2

Single memories. (a) Inset: ${\sigma }_x^2/d^2$ versus cycle number for ${\gamma }_1=1.6$. The system relaxes to a reversible steady state in $\sim 30$ cycles. Closed circles: $\mathrm{Re}=0.007$. Open circles: 20 cycles with $\mathrm{Re}=0.001$. Main: memory readout. ${\sigma }_x^2/d^2$ versus readout strain, $\gamma$. Circles: readout after training with 100 cycles of ${\gamma }_1=1.6$. The suspension is reversible up to ${\gamma }_1$. Triangles: readout for a randomized suspension shows no memory. (b) $S$ [defined by Eq. (1)] versus readout strain, for systems trained for 100 cycles at ${\gamma }_1=0.4$, 0.8, 1.2, and 1.6. The peaks identify the memory values. (c) Rheology of a single memory showing the stress versus strain during a readout shear. After training with 10 cycles of ${\gamma }_1=1.44$, the stress ($\tau$, left axis) on the inner cylinder is measured during a unidirectional constant strain-rate shear ($\dot{\gamma }=0.018{\mathrm{s}}^{-1}$). The stress sharply increases at ${\gamma }_1=1.44$ (dashed line), where there is a peak in the slope of the data ($d\tau /d\gamma$, right axis), indicating the memory. Here, $\mathrm{Re}=0.0002$, $d=90$ to $106\mu \mathrm{m}$, and the inner cylinder $\text{radius}=13.3\mathrm{mm}$. (d) Two sides to a single memory: ${\sigma }_x^2/d^2$ versus readout strain for single memories, showing readouts in both the $+$ (clockwise) and $-$ (anticlockwise) directions (with $\varphi =0.45$). Suspensions were trained between $\gamma =0$ and ${\gamma }_1=0.5$ (circles), and between ${\gamma }_{1-}=-0.3$ and ${\gamma }_{1+}=0.5$ (triangles).

Figure 3

Multiple memories. (a) Readout of 48 independent experiments with the training sequence $\gamma =2.0,1.2,1.2,1.2,1.2$ repeated 4 times. Points: individual runs. Line: average values. The system shows a sharp increase in irreversibility at the larger training amplitude, ${\gamma }_2=2.0$. Inset: A memory of the smaller training value, ${\gamma }_1=1.2$, is visible on an expanded $y$ axis. (b) $S$ of the averaged data in (a). The two memories are signified by the peaks. (c) Area under the peaks in $S$, versus the number of times the training sequence ($\gamma =2.0,1.2,1.2,1.2,1.2$) is applied. As the peak at ${\gamma }_2=2.0$ (closed symbols) becomes stronger, the peak at ${\gamma }_1=1.2$ (open symbols) gradually disappears until it cannot be resolved from the background (indicated by the error bars), and is effectively forgotten. Panel (d) shows similar results for memories at ${\gamma }_1=0.8$ (open symbols) and ${\gamma }_2=1.6$ (closed symbols), using the training sequence $\gamma =1.6,0.8,0.8,0.8,0.8,0.8,0.8$.

Figure 4

Stabilization by noise. Area under the peaks in $S$ signifying the memories at ${\gamma }_1=1.2$ (open symbols) and ${\gamma }_2=2.0$ (closed symbols), versus the duration of the pause between each shear cycle. The memory at ${\gamma }_2=2.0$ is weaker for longer pauses. The memory at ${\gamma }_1=1.2$ cannot be distinguished from the background (indicated by the error bars) for short pauses, but is stabilized by longer pauses. The insets show $S$ versus readout strain for 0 and 8 min pause durations.