Repeated/cyclic shearing can drive amorphous solids to a steady state encoding a memory of the applied strain amplitude. However, recent experiments find that the effect of such memory formation on the mechanical properties of the bulk material is rather weak. Here, we study the memory effect in a yield stress solid formed by a dense suspension of cornstarch particles in paraffin oil. Under cyclic shear, the system evolves toward a steady state showing training-induced strain stiffening and plasticity. A readout reveals that the system encodes a strong memory of the training amplitude (γT) as indicated by a large change in the differential shear modulus. We observe that memory can be encoded for a wide range of γT values both above and below the yielding albeit the strength of the memory decreases with increasing γT. In situ boundary imaging shows strain localization close to the shearing boundaries, while the bulk of the sample moves like a solid plug. In the steady state, the average particle velocity inside the solid-like region slows down with respect to the moving plate as γ approaches γT; however, as the readout strain crosses γT, suddenly increases. We demonstrate that inter-particle adhesive interaction is crucial for such a strong memory effect. Interestingly, our system can also remember more than one input only if the training strain with smaller amplitude is applied last.
Many out-of-equilibrium materials encode memory of past perturbations. The imprint of such perturbation history is stored in the material structure and can be revealed using a suitable readout protocol even long after the perturbation is removed. Cyclic perturbation protocols have been extensively used to encode memories in many of these systems. Such perturbations can be in the form of shear, temperature change, electrical/magnetic fields, etc.1 A few examples include dilute granular suspensions under cyclic shear,2–4 aging and rejuvenation in glasses,5,6 and charge density wave conductors under voltage pulses.7,8
Memory formation in amorphous solids has attracted significant recent interest. Despite the diversity in microscopic details, the presence of long-range correlations, and complex energy landscapes, these materials show very similar localized rearrangements under stress. Each of such rearrangements can be thought of as a transition between two local energy minima of the system.9,10 Interestingly, these systems can also reach a steady state under cyclic deformations encoding a memory of the deformation amplitude. The approach to a steady state and memory formation in amorphous solids under cyclic shear has been demonstrated in numerical simulations of glassy and frictional granular systems.11–17 Experimental studies have explored memory formation in soft glassy systems in both 2D18–21 and 3D,22 colloidal gels,23 and cross-linked biopolymer networks.24,25 Many of these systems also have the ability to remember multiple inputs even in the absence of external noise.
The signature of the memory formed under cyclic shear is reflected as a sudden increase in particle mean square displacement (MSD) as the readout strain crosses the training strain amplitude marking an onset of irreversibility in the system. However, the effect of such reversible–irreversible transition (RIT) on the mechanical properties of the bulk material has rarely been explored. Experiments on dilute non-Brownian suspensions4 and a soft glassy system of 2D bubble raft18 reported that encoding memory induces only a small change in the shear modulus of the system. These observations indicate a limitation of widely tuning the material properties using an imposed training.
In this Communication, we report strong memory formation in an amorphous solid formed by dense granular suspensions of cornstarch (CS) particles in paraffin oil. We find that memory can be encoded for a wide range of strain amplitudes both above and below yielding. Remarkably, such a memory effect is directly reflected as a sharp change in the differential shear modulus of the system. We observe that in the case of consecutive training with the strain amplitudes γ1 ≤ γ2 ≤ ⋯ ≤ γn, only the memory of the largest amplitude γn is retained. However, if the system is trained with a smaller strain amplitude γi(<γn) at the end of the training sequence, then the memory of γi can also be encoded. We show that such strong memory originates from a training-induced non-trivial coupling–decoupling dynamics of the solid-like region inside the bulk of the sample with the shearing plate via high shear rate bands near the boundaries. We also demonstrate the crucial role played by the inter-particle adhesion in forming such strong mechanical memory.
II. MATERIALS AND METHODS
For all our experiments, dense suspensions are prepared by dispersing cornstarch (CS) particles (Sigma-Aldrich) in paraffin oil with a volume fraction (ϕ) of 0.4. The CS particles have a mean diameter of 15 ± 5 µm. This system shows yield stress resulting from a solvent-mediated adhesion.26 We use a MCR-702 stress controlled rheometer (Anton Paar) with a cone and plate geometry (C–P) having rough sand-blasted surfaces for all our measurements. In C–P, the shear rate remains uniform everywhere inside the shearing geometry. The diameter of both the cone and plate is 50 mm, and the cone angle is ≈2°. Our experiments are done in the separate motor transducer mode of the rheometer with the bottom plate moving and the cone remaining stationary at all times. We use an in situ boundary imaging setup with a CCD camera (Lumenera) with a 5× long working distance objective (Mitutoyo) to capture the particle dynamics during the experiments. Images are captured at a rate of 4 Hz with a resolution of 1000 × 2000 pixels2 for all the measurements. For varying the adhesive interaction in the system, we use the non-ionic surfactant Span© 60. For more details about the system and the experimental setup, see the supplementary material and also Ref. 27.
III. RESULTS AND DISCUSSION
To train the system, we apply 300 cycles of a triangular-wave strain deformation with amplitude γT at a constant strain rate of 0.01 s−1, as shown in Fig. 1(a). At this shear rate, the Reynolds number is ≪ 1, indicating that inertial effects can be neglected. We use different colors to indicate the cycles in the beginning (magenta) and in the end (blue). In Figs. 1(b) and 1(c), we show the resulting intra-cycle stress (σ) as a function of the number of cycles (N) for γT = 0.02. We notice that the peak stress (σpeak) shows a systematic drop in the beginning [Fig. 1(b)], whereas near the end, it saturates to a steady state value of ≈15 Pa [Fig. 1(c)]. A similar behavior is also observed for other γT values [Fig. 1(d)]. From Fig. 1(d), we also see that σpeak for any N value decreases with increasing γT. This is related to the strain softening behavior of the system: shear moduli G′ and G″ decrease with increasing strain amplitude as shown in Fig. S2. We note that the nature of the stress waveform also evolves with N. This is more clearly observed from the variation in normalized σ vs γ (Lissajous plot) in Fig. 1(e). Of total 300 cycles, we show the Lissajous plots for only a few discrete N values. We observe from Fig. 1(e) that starting from a quasi-linear visco-elastic response for small values of N, the system gradually shows a highly non-linear response for larger N values, approaching a steady state. In addition, as we go to larger N values, the slope of the Lissajous plots becomes negligible for γ ≪ γT; however, near γT, the slope increases sharply. This indicates the development of plasticity and strain-stiffening behavior under training. A similar behavior has also been observed for colloidal gels and cross-linked biopolymer networks under cyclic shear.24,28
Now, to see if the system encodes memory of the training amplitude, we apply a readout after 300 cycles of training. The readout is a triangular wave pulse having the same strain rate of 0.01 s−1 but with an amplitude γR = 2γT as shown in Fig. 2(a) (top) for γT = 0.02. We plot σ vs γ of the readout cycle for γT = 0.02 in Fig. 2(b) (top, red curve). We observe a sharp change in the slope of σ as we cross γ = γT. For a comparison, we apply the same readout to an untrained sample [Fig. 2(a), bottom] and plot the corresponding σ vs γ in Fig. 2(b) (top, black curve) where we do not observe any such change. This difference is more clearly reflected in the differential shear modulus of the system vs γ as shown in Fig. 2(b) (bottom): the trained sample shows a sharp peak in K with a value Kpeak at γ = γT, indicating that the system encodes a strong memory of the training amplitude that can change the differential shear modulus of the system by a large amount. In our case, Kpeak ≫ Kbaseline (with Kbaseline as the value of K at strain values slightly away from γT, where K varies relatively slowly). However, for earlier studies,4,18 Kpeak ∼ Kbaseline, further highlighting that memory formation is much stronger in our case. Additionally, we find that once the system is trained at a particular γT value, any memory of γ < γT gets erased (Fig. S3). We discuss the possibility of encoding multiple memories in our system later in the manuscript. We also find that the signature of encoded memory stays essentially the same, even when the readout is taken after 1000 s, implying that the relaxation of the structures supporting the memory is very slow (not shown).
To inspect the effect of training amplitude on the strength of memory formation, we encode and readout the memory for a range of γT values. For each γT value, we use a fresh loading of the sample. In Fig. 2(c), we plot average K vs γ for various γT values. We find a steady drop in the strength of the memory (quantified by the peak values of K) as γT increases. In fact, beyond γT = 0.1, Kpeak is significantly lower, albeit still present. This value of γT is close to the fluidization/yielding onset of the system (Fig. S2). Thus, in contrast to the 2D soft glassy system,18 we do not find any enhancement of the memory effect near the yield point.
So far, we have established that the encoded memory gets reflected as a large change in the value of differential shear modulus of the system. In order to understand the mechanism behind this striking behavior, we perform in situ optical imaging [in the flow (X)–gradient (Y) plane]. Details of the setup are given in Ref. 27. A typical image is shown in Fig. 3(a) where the bright speckles correspond to CS particles that protrude out of the air–solvent interface. The internal connectivity of the fractal network giving rise to solid-like yield stress in the system stabilizes such protruding particles against the stress due to surface tension. We map out the velocity profile across the shear-gap using particle imaging velocimetry (PIV) during the training and readout experiments as shown in Fig. S4. We observe that except for narrow regions of high shear rate close to the shearing boundaries, the bulk of the sample moves like a solid plug with a negligible velocity gradient. We use kymographs to map out the space–time variation of the speckle distribution at different locations inside the sample acquired during training and readout experiments (supplementary material). We consider kymographs at three different locations: (1) on the moving plate (Region A), (2) inside the solid-like region close to the moving plate (Region B), and (3) inside the solid-like region close to the static cone (Region C). All these regions contain at least one dominant bright speckle whose trajectory can be tracked during the training and readout. We show in Fig. 3(b) the kymographs corresponding to these regions for the first ten (N = 1–10) and last ten (N = 291–300) training cycles. We find that the trajectory obtained for Region A [Fig. 3(b), bottom] precisely mimics the motion of the shearing plate during training. The self-similarity of the waveform establishes the robustness of the input shear. Interestingly, kymographs for Regions 2 and 3 show more complex displacement waveforms involving a gradual evolution in both amplitude and shape before reaching a steady state during the course of training. However, except for the first cycle (involving start-up transients), the waveform in these regions remains very similar, further confirming the solid plug-like motion of the bulk of the sample.
Interestingly, the displacement waveform inside the solid-like region of the sample indicates a complex coupling–decoupling dynamics between the moving plate and the bulk of the sample through the high shear rate bands near the boundaries. Clearly, such dynamics is developed through training as it is absent in the beginning [left panels in Fig. 3(b)]. In the steady state, particle trajectories in the solid-like region of the sample settle down to a waveform, which is very different from the input, demonstrating a slowing down of the bulk as the intra-cycle input strain (γ) approaches the training amplitude (γT). Now, we take a look at the readout by plotting kymographs corresponding to the last training cycle (N = 300) and the readout cycle in Fig. 3(c). For clarity, we show the data only for Region A (bottom) and Region B (top) with expanded views. We find that the speckle displacement during readout follows the last training cycle up to γ = γT. However, as γ crosses γT, there is a sudden increase in the slope of the trajectory [marked by dashed lines in Fig. 3(c), top]. A similar change in the slope is also observed for other speckle trajectories inside the solid-like region (Fig. S1). This implies that beyond γT, there is a sudden buildup of coupling between the solid-like region and the moving plate. This sudden coupling will lead to an abrupt increase in the velocity of a large number of particles, which can explain the origin of the sharp peak of K around γ = γT. Such an abrupt increase in the velocity of the solid-like region inside the bulk of the sample also reflects in the velocity profile obtained from the PIV analysis (Fig. S5). Difference images constructed through stroboscopic sampling indicate a reversible to irreversible transition (RIT) in the sample beyond γT (Fig. S6) similar to the earlier studies.4,12,18 However, near the yield point, presence of any divergence in time-scale required for reaching the steady state remains to be investigated.
Next, we address the role of inter-particle adhesive interaction on the strong memory formation in our system. Recently, we reported that addition of a small amount of surfactant to the cornstarch–paraffin oil system can reduce the adhesive interaction between the particles as quantified by the systematic change in the jamming volume fraction.27 With a sufficient amount of surfactant, the system completely transforms into a steric-repulsive one. Physically, as we increase the surfactant concentration (c), the large fractal clusters initially stabilized by inter-particle adhesion gradually disintegrate and finally form a well-dispersed suspension as shown in the schematic in Fig. 4(a). We prepare samples with increasing c and subject them to similar training and readout protocols mentioned earlier for different γT values. In Fig. 4(b), we plot the average peak values of K for different γT values obtained from the readouts for a range of c values. We see that the strength of memory (Kpeak) decreases with increasing c and the system loses ability to form strong memory beyond a sufficiently high concentration of surfactant (c ≥ 0.3%). However, even at the highest surfactant concentration, the memory signature is not totally absent, as can be seen from the log–log plot [inset of Fig. 4(b)] and also Fig. S7. We find that for the same range of c values, the training induced strain stiffening also becomes significantly weaker (Fig. S8). Thus, the memory formation in our system is intimately related to the strain stiffening response of the system. From the kymograph analysis, we do not find any discontinuity in the particle trajectory across γT in the readout for sufficiently high surfactant concentrations (Fig. S9).
Finally, we address the possibility of encoding multiple memories in our system. As mentioned earlier, if the system is trained at γT, it erases memories of γ < γT. Interestingly, we find that if the sample is trained with smaller strain amplitude just before the readout, it can retain the memory of the smaller γT value, and more than one memory can be encoded, as has also been reported earlier for other systems.4,19 We use the following protocol: we apply 1 cycle of the larger training strain amplitude (γT2 = 0.04) followed by 20 cycles of the smaller training strain amplitude (γT1 = 0.02). This whole sequence is repeated 15 times, and then, we apply a readout strain of γR = 0.06. The stress (σ) vs strain (γ) plot for the readout shows sharp changes at γ = 0.02 and 0.04 [Fig. 5(a)], giving two prominent peaks for the differential shear modulus K [Fig. 5(b)]. This indicates that the system has the ability to encode memories of more than one strain amplitude under a suitable training protocol.
We report strong memory formation in an adhesive dense particulate suspension under cyclic shear. The differential shear modulus of the bulk system shows a huge enhancement near the training strain amplitude (γT). A possible explanation for such a striking effect is the following: in the presence of adhesion, the fractal particle clusters can form strand-like connected structures. Under cyclic shear deformations, such structures can reorganize and develop a slack up to γT. The adhesive interaction maintains the contact between the particles in these loose strand-like structures for strain values γ < γT. Similar strand plasticity has been reported in colloidal gels under cyclic shear.28 For our case, such reorganization dynamics predominantly takes place in the high shear rate bands close to the shearing boundaries. Due to the slack in the particle strands, the plate can easily move without significantly disturbing the solid-like bulk region inside the sample for γ < γT. This gives rise to an apparent decoupling between the plate and the bulk sample in the steady state. However, when γ crosses γT during the readout, the loose strands suddenly get stretched and the coupling between the moving plate and the solid-like region builds up. This results in a strong stress response. Interestingly, such a picture can also shed some light on the reduction in the strength of memory with increasing γT: for small γT values, a large number of clusters can contribute to strand formation; however, for large γT values, many small clusters can get completely disintegrated and only large connected structures will contribute. Although at present we cannot directly visualize the strand dynamics, the connection between the strand formation/plasticity and the observed memory effect is further supported by the role of adhesive interaction in the system. Once the inter-particle adhesion gets sufficiently weakened by addition of surfactant, the strand formation is no longer possible; as a result, the memory effect gets diminished. This weaker memory, in principle, is similar to that observed earlier in dilute non-Brownian suspensions.4 However, further work is needed to understand the exact correspondence.
The memory effect reported here is reminiscent of similar phenomena in transiently cross-linked biopolymer networks24 and Mullin’s effect in filled-polymeric systems.29 Our system also shows a connection between the memory formation and reversible–irreversible transition like many repulsive particulate systems. Thus, the adhesive particulate system can be thought of as an intermediate between repulsive dense suspensions and polymeric materials. Remarkably, our system presents a distinct advantage: since the particles are very robust and the adhesive particle-bonds are reversible, training and readout for a wide range of strain values do not cause any permanent damage to the system. The sample can be reloaded and reprogrammed arbitrarily. On the other hand, polymeric materials are prone to permanent damage due to the breakage of filaments or chemical bonds. Interestingly, the strand picture together with the reversibility of adhesive bonds can explain the origin of multiple memories in our case. After encoding a memory at a smaller strain amplitude, the application of a larger training strain can destroy the strands supporting the memory at smaller amplitude. However, after this, if the system is again trained at the smaller amplitude, the reversibility of the adhesive bonds ensures that some local connectivity can build up once more (Fig. S10), thus re-encoding the memory of smaller amplitude.
Due to the opaque nature of the particles, we can only probe the particle-dynamics on the sample surface. However, as our system is very dense [Fig. 3(a)], it is hard to directly probe training induced changes in the nodes or connectivity between the particles using boundary imaging alone. Mapping out the system dynamics in 3D using optical/acoustic techniques remains a future challenge. Further theoretical insights into the mechanism of such a strong memory effect in adhesive systems, including the formation of multiple memories, can open up new avenues to explore. Our study can have important implications in designing programmable materials.
See the supplementary material for information about sample preparation, data analysis, and additional measurements.
S.M. acknowledges the SERB (under DST, Government of India) for a Ramanujan Fellowship. We thank Ivo Peters for developing the Matlab codes used for PIV analysis. We thank Daniel Hexner for helpful discussion.
Conflict of Interest
The authors have no conflicts to disclose.
S.C. and S.M. designed the research, S.C. performed the experiments, and S.C. and S.M. analyzed the data and wrote the manuscript.
The data that support the findings of this study are available from the corresponding author upon reasonable request.