Memory formation in matter

Memory formation in matter is a theme of broad intellectual relevance; it sits at the interdisciplinary crossroads of physics, biology, chemistry, and computer science. Memory connotes the ability to encode, access, and erase signatures of past history in the state of a system. Once the system has completely relaxed to thermal equilibrium, it is no longer able to recall aspects of its evolution. The memory of initial conditions or previous training protocols will be lost. Thus many forms of memory are intrinsically tied to far-from-equilibrium behavior and to transient response to a perturbation. This general behavior arises in diverse contexts in condensed-matter physics and materials, including phase change memory, shape memory, echoes, memory effects in glasses, return-point memory in disordered magnets, as well as related contexts in computer science. Yet, as opposed to the situation in biology, there is currently no common categorization and description of the memory behavior that appears to be prevalent throughout condensed-matter systems. Here the focus is on material memories. The basic phenomenology of a few of the known behaviors that can be understood as constituting a memory will be described. The hope is that this will be a guide toward developing the unifying conceptual underpinnings for a broad understanding of memory effects that appear in materials.

Figure 1

Memory of a direction. (a) Two concentric cylinders that are rotated with respect to one another may be used to measure the shear stresses in a suspension of neutrally buoyant particles in a viscous liquid that fills the gap between the cylinders. The top graph shows a series of rotations of the outer cylinder (counterclockwise, clockwise, clockwise, counterclockwise), with pauses between each rotation. The bottom graph shows a schematic curve of the torque felt on the stationary inner cylinder; the presence (or absence) of a transient may be viewed as a memory of the previous shear direction. From [50]. (b) Pulse-sign memory in the traveling charge-density wave conductor ${\mathrm{K}}_{0.30}{\mathrm{MoO}}_3$. If an applied current pulse has the same sign as the previous one, the voltage response lacks a strong transient. From [45]. (c) Backlash between two gears as a memory of a direction. If the right gear is rotated clockwise, the left gear responds immediately. If the right gear is rotated counterclockwise, there is a small lag. Adapted from [142].

Figure 2

Kaiser effect. The height of a piston compressing a Mylar sheet is shown, following the protocol described in the text. The inset shows the Mylar sheet compressed within a cylindrical tube by a piston of mass $m$. The piston height $h(m)$ is reproducible (shown by blue circles for diminishing mass and by red triangles for increasing mass) as long as the maximum mass does not exceed the previously applied maximum value of $m=M_1=2.6\mathrm{kg}$ indicated by the dashed line. When a larger mass (red triangles above 2.6 kg) is placed on the piston, the curve changes; green square data points show the reproducible curve after training with $m=M_2=5\mathrm{kg}$. Adapted from [107]; inset: Carin Cain (APS).

Figure 3

Mullins effect. Schematic stress-strain curve for the Mullins effect in rubber. The pristine loading curve (black full line) is smooth. After partial loading, when the stress $\sigma$ is removed the response changes and the unloading curves (red dashed and blue dash-dotted curves) do not follow the original loading. On reloading, the response is reversible up to where the previous largest stress had been applied. At that point there is a kink where the curve rejoins the original pristine loading curve. Adapted from [23].

Figure 4

Return-point memory. (a) Magnetization of a simulated ferromagnet model ([139]) as the applied magnetic field $H$ is varied. The sequence $a\to B\to b$ creates a memory at $H=H_B$. As long as $H_a\le H\le H_B$, returning to $H_B$ will restore the system to the same state $B$ every time, regardless of intervening events (such as the excursion to $b$). The sequence $b\to C\to c$ creates a new subloop and encodes a second memory at $H=H_C$. (b) Detail of trajectory from (a), showing signatures of the two nested memories as $H$ is swept from $H_c$ to $H_A$. Each time a subloop is exited (points $C$ and $B$), its history is erased and the slope of the curve changes. The inset shows the slope $dM/dH$ during readout. The jumps at $H_C$ and $H_B$ indicate memories. For this figure $5\times {10}^4$ hysterons were simulated, with a Gaussian distribution of $H_i^{+}$ and $H_i^{-}$ with a standard deviation of 1.8, and selected so that $H_i^{+}>H_i^{-}$.

Figure 5

Memories from cyclic driving. (a) To read out memory in a sheared suspension, cycles are applied with successively larger strain amplitude (horizontal axis). After each cycle, each particle’s position along the direction of shear is compared with its position at the start of the cycle (vertical axis). An untrained system (black triangles) shows no memory, while a system trained for many cycles with amplitude ${\gamma }_0=1.6$ (red circles) shows a memory. Adapted from [129]. (b) After ten cycles with ${\gamma }_0=1.44$, measuring shear stress $\tau$ vs strain $\gamma$, and its derivative $d\tau /d\gamma$, shows a clear memory in the mechanical response. Adapted from [129]. (c) Multiple transient memories: In a simulation of the sheared suspension, memories are read out by monitoring collisions as strain amplitude is increased, analogous to (a). Curves are labeled with the number of training cycles applied. Training with amplitudes 2.0 and 3.0 is evident after just 100 cycles, but eventually a steady state is reached with only one memory. The shaded curve shows the result after many cycles when noise is added. Adapted from [80].

Figure 6

Multiple transient memories in a row of benches on a lawn. The schematic shows the park viewed from above. The park entrance is at the left, and lighter colors indicate worn grass. Even though there is only one path, the grass heights encode the memory that benches 2 and 3 were visited. If this pattern of visits continues without the grass regrowing, eventually only the largest memory (bench 3) will remain. From [128].

Figure 7

Multiple memories in a jammed solid. Shown is a readout of memories of two strain amplitudes in a simulation of a jammed, amorphous solid. The system was trained by alternating between strain amplitudes ${\gamma }_1=0.06$ and ${\gamma }_2=0.04$, for 30 repetitions of the pattern. The readout involves applying cycles of increasing strain amplitude ${\gamma }_{\mathrm{read}}$, starting from zero, and comparing the state of the system after each cycle to the state after training. Here mean-squared displacement (MSD) of particles is used to measure differences. The system returns to the same state when ${\gamma }_{\mathrm{read}}={\gamma }_2$, and shows a sharp change in behavior past ${\gamma }_{\mathrm{read}}={\gamma }_1$. Adapted from [2].

Figure 8

Shape memory. (a) When this bent titanium-nickel wire (top) is submerged in hot water, it spontaneously reconfigures into a paperclip (bottom). Adapted from [102]. (b) Simplified description of shape-memory alloys. Bottom row: When the sample is cooled, it undergoes a phase transition to the martensite phase without changing its macroscopic shape. The material may then be deformed to relatively large strains without changing the topology of the bond network (top). Heating up the sample recovers the original shape by restoring the cubic lattice (bottom left). (c) Optical microscopy image of the shape-memory alloy ${\mathrm{Au}}_{30}{\mathrm{Cu}}_{25}{\mathrm{Zn}}_{45}$. Colors (different shades) indicate different orientations of the martensitic phase. Adapted from [155]. (d) A medical stent made from a shape-memory polymer, shown expanding in a glass tube. Black rings were drawn for visualization. Adapted from [167].

Figure 9

Aging and rejuvenation. (a) Relaxation of the spin-glass ${\mathrm{CdCr}}_{1.7}{\mathrm{In}}_{0.3}{\mathrm{S}}_4$, measured by the imaginary part of the magnetic susceptibility. The reference curve is obtained when cooling from 25 K down to 5 K at a constant rate. Open symbols: If the system is held at an intermediate temperature, here 12 K, the susceptibility slowly drops (“aging”), but upon further cooling it returns to the curve it was previously following (“rejuvenation”). Remarkably, the same curve is traced out upon reheating at a constant rate (solid symbols). Adapted from [72]. (b) Left: Similar behavior is observed in a simple number-sorting algorithm, where nearest neighbors are swapped with a probability given by a Boltzmann factor. Right: Subtracting the reference curve (obtained for a constant cooling rate) makes the trend more clear. From [174].

Figure 10

Echo phenomena. (a) Spin echo produced with the pulse sequence described in the text. The timing of the two input pulses is indicated above a photograph of the oscilloscope trace; the subsequent peak is the echo. Note that there is no signal coinciding with the “$\pi$” pulse, as this simply flips the (dephased) spins at that instant. From [24]. (b) Quench echo in a simulated Lennard-Jones glass of 500 particles. The two quenches consist of setting the kinetic energy of the particles equal to zero instantaneously. Because they have potential energy, they will start to move again after each quench. Remarkably, this leads to an echo at a later time. From [119]. (c) Echo in a jammed amorphous solid due to anharmonic modes. The signal is an average of ${10}^4$ independent simulations, each composed of 1000 soft spheres in three dimensions. From [22]. (d) Echo in a system of capillary waves. A dish of water is disturbed by a face-shaped template, creating surface waves that radiate outwards. The entire dish is then accelerated downwards, causing the existing ripples to act as sources, sending a new set of “time-reversed” waves back to the site of the initial disturbance. The face reappears as the waves reconvene. From [8].

Figure 11

Kovacs effect. The top curve shows the general experimental protocol, whereby a perturbation is applied to a sample for a duration ${\tau }_{\text{train}}$, and then the size of the perturbation is reduced and held constant at a value closer to the initial one. The bottom curve shows the response, which peaks at a time ${\tau }_{\mathrm{peak}}$ that is proportional to ${\tau }_{\text{train}}$, indicating a memory of ${\tau }_{\text{train}}$. Experiments show such a memory of waiting time in a crumpled Mylar sheet ([92]), where the control parameter is the volume controlled by a piston, and the response is the force exerted by the sheet on the piston. See the image inset in Fig. 2.

Figure 12

Associative memory. (a) Partial or corrupted information may be used to retrieve a memory. (b) Schematic of Hopfield model, showing three sets of connections (solid, dashed, and dash-dotted edges) defined on the same set of neurons (vertices). For simplicity, only the bonds between the on neurons $(s_i=1)$ are drawn. Under the right conditions, the single set of weighted connections $J_{ij}=J_{ij}^{(1)}+J_{ij}^{(2)}+J_{ij}^{(3)}$ may be used to store the three individual memories.

Figure 13

Self-assembling memories. (a) Square building blocks may bind with up to four nearest neighbors on a lattice. Three different assembled structures ($A$, $B$, $C$) represent three memories. The memories are simultaneously stored by defining attractive interactions between blocks which are neighbors in any of the structures. (b) Simplified diagram of self-assembly regimes in simulations. Three behaviors (separated by heavy lines) are distinguished by the fate of an introduced seed for a desired structure. Building blocks are pixels in a given snapshot of the system (bordered by a thin square), each pixel being colored (shaded) according to the structure whose part it is forming in that snapshot. Below a critical number of stored structures $m_c$, and at low enough temperatures (bottom-left regime), a seed for the $B$ structure (irregular shape in initial snapshot on the left) leads to its self-assembly, i.e., the desired $B$ memory is successfully retrieved. Above storage capacity the same seed leads to an erroneous structure, bottom-right regime. When the thermal energy sufficiently exceeds the binding energy, all assembly is prevented. (b) Adapted from [117].

Figure 14

Forgetting and remembering initial conditions. (a) A drop of glycerol falls and pinches off from a faucet submerged in silicone oil. Near the top of the spherical drop, the dynamics become singular and have no dependence on the drop’s initial conditions. Nozzle diameter is 0.48 cm. From [28]. (b) Detail of a bubble pinching off from an underwater nozzle located just below the images. The black region is the rising air bubble and the surrounding white and gray is the water. The pinch-off is seen in the bottom frame where the black regions become disconnected. The dynamics become singular, but remember initial conditions strongly—in this case, the bubble’s release from an asymmetrically shaped nozzle. Scale bar is 0.5 mm and the images are $122\mu \mathrm{s}$ apart. From [79].