Research ArticleCONDENSED MATTER PHYSICS

Crystalline shielding mitigates structural rearrangement and localizes memory in jammed systems under oscillatory shear

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Science Advances  12 May 2021:
Vol. 7, no. 20, eabe3392
DOI: 10.1126/sciadv.abe3392

Abstract

The nature of yield in amorphous materials under stress has yet to be fully elucidated. In particular, understanding how microscopic rearrangement gives rise to macroscopic structural and rheological signatures in disordered systems is vital for the prediction and characterization of yield and the study of how memory is stored in disordered materials. Here, we investigate the evolution of local structural homogeneity on an individual particle level in amorphous jammed two-dimensional (athermal) systems under oscillatory shear and relate this evolution to rearrangement, memory, and macroscale rheological measurements. We define the structural metric crystalline shielding, and show that it is predictive of rearrangement propensity and structural volatility of individual particles under shear. We use this metric to identify localized regions of the system in which the material’s memory of its preparation is preserved. Our results contribute to a growing understanding of how local structure relates to dynamic response and memory in disordered systems.

INTRODUCTION

Amorphous, jammed systems (1) are abundant in nature and used often to process and produce materials (2). The way in which these systems’ disordered multiscale structure (3) evolves under the application of stress (48), eventually resulting in catastrophic yielding (9), is an area of active investigation, with consequences for phenomena ranging from landslides and other forms of landscape evolution (10, 11) to cellular unjamming during tumor metastasis (12, 13). The identification of local structural characteristics that are coupled to dynamical response under stress in disordered systems is of particular interest (14), because foreknowledge regarding which local regions of a material might be susceptible to rearrangement under stress would enable the prediction, prevention, or tailored design of failure in disordered materials. However, the lack of obvious structural order in such systems and the often subtle nature of the relevant dynamics make it a challenge to draw conclusions about any causal relationship between structure and dynamics.

Memory encoding in amorphous materials is an especially intriguing stress response with a highly nontrivial relationship to heterogeneous structure. Inanimate objects are not typically associated with the very biological act of remembering their past, and yet the evolution of a disordered system can depend on its preparation history, resulting in embedded memories of previous events that can be read out by subsequent procedures (15). Material memory has been observed in a variety of forms, ranging from the simple to the complex, and exploring each of these mechanisms in full contributes to our understanding of the universality of memory in nature. For example, amorphous systems under oscillatory shear have recently been found to develop precisely cyclical particle trajectories (1621) that can encode single or multiple memories of the strain amplitude at which the material was prepared (2227). Materials may also simply remember the direction in which they were last deformed and express that memory via directional asymmetry in their response to subsequent stress, as in the case of the well-known Bauschinger effect observed in metals (28, 29) and amorphous materials (3032). Even in such a seemingly simple case, however, important open questions remain regarding the microstructural origin of response asymmetry (31). To understand the more complicated forms of memory in materials, it is crucial that the relationship between local structure and the simple memory encoded by shear response anisotropy in heterogeneous materials be fully elucidated.

Here, we provide a conceptual link between local structure and memory as encoded in the response of a two-dimensional (2D) dense particle system under oscillatory shear. Experiments are performed using a custom-made interfacial stress rheometer in which a dense (jammed and athermal) particle monolayer is sheared using a magnetic needle (19, 33). This method provides sufficient spatial and temporal resolution to probe structural rearrangement probability on the scale of individual particles while simultaneously measuring bulk rheological properties.

We show that these jammed, athermal, and disordered systems (Fig. 1, A and B) have memory of their preparation, exhibited via an asymmetry in local deformation with respect to shear direction below yielding. We use generic measures of structural homogeneity on an individual particle level to track the sample microstructure over time (Fig. 1, C and D) and find that global crystallinity is also asymmetric with respect to shear direction and thus encodes preparation memory. This memory is increasingly “erased” as strain amplitude increases beyond yielding. We next demonstrate that, on an individual particle level, correlations in crystallinity over time are reliable indicators of particle rearrangement. We stratify crystalline particles into subgroups according to their interiority within crystal grains (Fig. 1E) and find that the propensity for rearrangement occurs in a hierarchy according to this crystalline shielding metric, with particles most interior within grains rearranging least. Our results show that the likelihood of particle rearrangement depends on a continuum of interiority within crystal grains rather than a binary classification of grain boundary (or defect) versus interior as has been found previously (19, 3338). Last, we show that rearrangement asymmetry with respect to shear direction also occurs in a hierarchy according to crystalline shielding, with asymmetry being highest for particles most interior within grains. Thus, we conclude that sample preparation memory is spatially localized to the interior of crystal grains.

Fig. 1 The system and structural characterization methods used in this work.

(A) Snapshot of a 2D jammed amorphous bidisperse particle system. Scale bar, 100 μm. An arrow shows the direction of oscillating strain γ(t). (B) Oscillatory rheology also reported in Ref. (33). Connected symbols are measurements for the bidisperse systems considered in the Results, and unconnected symbols are measurements for the monodisperse systems mentioned in the Discussion and presented fully in the Supplementary Materials. (C) Schematic illustrating the environment matching method used to determine crystallinity. We identify neighbors, extract their environments, and then compare those environments to determine local structural homogeneity. (D) Visual rendering of a portion of a sample bidisperse experiment at γ0 = 0.068. Here, particles in crystalline grains of population greater than 1 are colored light blue. The inset is the radial distribution function collected over all particles during one cycle of the experiment, with the nearest-neighbor distance rcut marked in gray. (E) Visual rendering of the identical system snapshot with particles colored according to crystalline shielding Rnon−xtal. Disordered particles are colored purple. The inset is a histogram of Rnon−xtal collected over all particles during six cycles of the experiment.

RESULTS

Crystallinity oscillates with shear and reveals material preparation memory

We first investigate whether our systems retain memory of their preparation history by studying the symmetry of their deformation response with respect to shear direction. We measure local deformation according to how much the average nearest-neighbor shell stretches and find that deformation is asymmetric with respect to shear direction at low strain amplitude. Local deformation displays increasing symmetry as strain amplitude increases.

Figure 2A shows the eccentricity of the ellipse fit to the average nearest neighbor shell over multiple shear cycles for experiments at all strain amplitudes and Fig. 2C shows ellipse orientation, defined as the angle from the positive x axis to its major axis. Eccentricity and orientation signatures are calculated for each experiment from an ellipse fit to the average nearest-neighbor shell of all particles (even if they are not preserved over the entire experiment), defined as the boundary demarcated by the first peak of the 2D histogram of nearest neighbors accumulated over every particle. We show stroboscopic averages of each signal over the nontransient portion of each system trajectory, conservatively defined as the set of cycles well after the global crystallinity for each system, defined below, has reached steady-state oscillation. Signals for full trajectories are shown in fig. S3.

Fig. 2 Global signatures of crystallinity and local neighborhood deformation as a function of strain amplitude γ0 show asymmetry with respect to shear direction.

(A and C) Stroboscopic averages of local neighborhood deformation. The quantity ε is the eccentricity of the local neighborhood ellipse, and θ is its orientation. Error bars represent the standard error of the mean value at each time point. Increasingly dark colors correspond to increasing values of γ0. Values of γ0 for experiments at the lowest and highest strain amplitudes studied are shown. (B) Crystallinity in all systems, stroboscopically averaged. Error bars again represent the standard error of the mean value at each time point. Increasingly dark colors correspond to increasing values of γ0. Horizontal lines indicate the mean of each crystallinity signature. (D) Deviations in the crystallinity from its mean, ΔX(t), plotted against local neighborhood deformation ε(t) for example systems below and above yield. Shown are stroboscopically averaged quantities and corresponding error bars identical to those in (A) and (B). Light gray triangles mark all frames during the second shear half-cycle, for which θ(t) ≤ 90, and dark gray circles mark all frames during the first shear half-cycle, for which θ(t) > 90. (E) Two power spectral densities PX(ω) of nontransient crystallinity signatures as a function of γ0 for two distinct frequencies. The circles correspond to frequency ω*, which is the frequency of the needle oscillation; the squares correspond to frequency 2ω*, which is the second harmonic of the needle oscillation. Error bar estimation is described in the text.

At high strain amplitude, well above yield, eccentricity reaches approximately equal heights during the first and second halves of each cycle, and thus, local deformation is approximately symmetric with respect to shear direction. As strain amplitude decreases, however, eccentricity grows smaller (and thus local deformation is smaller) during the back half of each cycle, when θ ∼ 45. Asymmetry in deformation between the first and second shear half-cycles implies an anisotropy in the system that encodes the system’s history, and this memory is “erased” with increasing strain amplitude.

We next consider whether this asymmetrical response is exhibited by measures of structure in our systems. We calculate global crystallinity for each system over time and find that it also displays an asymmetry with respect to shear direction at low strain amplitude, thus seeming to also indicate material anisotropy and preparation memory. Global crystallinity at time t, X(t), is the fraction of particles in the system that are crystalline, as defined in the Materials and Methods. We note that we could instead have chosen to analyze a more continuous per-particle measure of structural homogeneity, Δi: the root mean square deviation between the environment of particle i and that of its neighbor, averaged over all of its neighbors. Another global measure of structural homogeneity is then Δ¯(t), where the average is taken over all particles in the system at each time. Calculation of Δ is explained in more depth in the Supplementary Materials, and results using this alternative definition, shown in fig. S5, are very similar to those presented here.

Figure 2B shows that global crystallinity X(t) oscillates in time with shear, in agreement with other studies that observed similar structural oscillations (39). The amplitude of the crystallinity oscillation increases with strain amplitude. We show stroboscopic averages of each signal over nontransient trajectories identical to those used to calculate Fig. 2 (A and C). Full signals, showing initial transient behavior, are shown in fig. S3. At low strain amplitude, there is an asymmetry in the crystallinity signal with respect to shear direction, and this asymmetry is erased as strain amplitude increases. This asymmetry erasure is evident in the power spectra of the signals via the periodogram estimate (Fig. 2E): We find that the power spectral density associated with twice the frequency of the needle oscillation, PX(2ω*), increases with strain amplitude, whereas PX(ω*) remains relatively stable. Each power spectral density is the mean of a set of PX(ω) values calculated over consecutive two-cycle windows of the relevant nontransient trajectory shown in fig. S4, and error bars represent the standard deviation of the mean. Full power spectral density distributions over all ω are also shown in fig. S4.

In general, global crystallinity decreases as deformation increases. This behavior can be seen in plots of deviation in the crystallinity from its mean, ΔX(t) ≡ X(t) − 〈Xt, against neighborhood ellipse eccentricity for systems below and above yield (Fig. 2D). Plots for all experiments are shown in fig. S6. As eccentricity increases, crystallinity dips, and this dip is more pronounced above yield. Below yield, crystallinity during the second half-cycle [for which θ(t) ≤ 90] remains distributed close to ΔX(t) = 0, due to asymmetry with respect to shear direction. In the experiment, we show that above yield, crystallinity during the second half-cycle dips even lower than crystallinity during the first half-cycle [for which θ(t) > 90].

Correlations in crystallinity over time indicate particle rearrangement under shear

A closer investigation of the structure of individual particles over time reveals that correlations in crystallinity are reliable indicators of individual particle rearrangement. To show this, we quantify crystallinity correlation via p(s, ts, t0), the conditional probability that a particle is in structure s at time t given that it was in the same structure s at time t0. In our analysis, either s = x, representing crystalline structure, or s = d, representing noncrystalline or disordered structure. We compare crystallinity correlation to a metric that measures particle rearrangement: Dmin2(t0,t) (30), the mean square deviation of the displacements of a particle and its neighbors from the best-fit affine deformation of those displacements (40). High values of Dmin2(t0,t) correspond to nonaffine deformations between times t0 and t, which manifest as particle rearrangements. We find that p(s, ts, t0) and Dmin2(t0,t) are inversely related: When p(s, ts, t0) is lower, Dmin2(t0,t) is higher and vice versa. This relationship implies that structural autocorrelation captures particle rearrangement dynamics.

To gain more insight into the influence of crystalline structure on rearrangement dynamics, we partition particles into groups according to their crystalline shielding level at time t0, as explained in the Materials and Methods, when calculating p(s, ts, t0) and Dmin2(t0,t). Figures S7 and S8 show fractions of particles at each crystalline shielding level as a function of time for all strain amplitudes; we observe that each signature oscillates distinctly in time, showcasing the evolution of crystal grain morphology during the shear cycle.

We find that rearrangements occur in a hierarchy according to shielding level, with more shielded particles prone to less rearrangement at all points of the shear cycle. This result indicates that degree of interiority within crystal grains has a substantial impact on rearrangement dynamics. As an example, we consider a sample experiment at γ0 = 0.068 (above yield) in Fig. 3A. The top panel of Fig. 3A shows the quantity p(s, ts,0) for each shielding layer as a function of t over one shear cycle, with t0 = 0 marking the beginning of the cycle. The bottom panel shows the quantity Dmin2(0,t) for the same experiment and identical values of t0 = 0 and t, where the average is taken over all particles in each shielding level at t0 = 0. Both panels show stroboscopic averages of each signal over nontransient trajectories. While particles in all shielding layers show rearrangement within the shear cycle according to both p(s, ts,0) and Dmin2(0,t), we find that more interior shielding layers generally show higher values of p(s, ts,0) and lower values of Dmin2(0,t), indicating less rearrangement.

Fig. 3 Correlations in crystallinity indicate individual particle rearrangement.

(A) Rearrangement measurements p(s, ts,0) (top) and Dmin2(0,t) (bottom) for a sample experiment at γ0 = 0.068. Signals are shown as a function of t over one shear cycle, and t0 = 0 marks the beginning of the cycle. Results are shown as stroboscopic averages, and error bars represent the standard error of the mean value at each time point. Each signal represents a shielding level according to the color scheme detailed in Fig. 1, and is calculated over particles in the appropriate shielding level at t0. Colors of the least shielded (disordered) and most shielded layers are shown for reference. (B) Stroboscopically averaged Dmin2(0,t) as a function of stroboscopically averaged p(s, ts,0), for all strain amplitudes and all shielding layers. Error bars are standard errors of the mean in both dimensions.

In addition, in all cases, p(s, ts,0) reaches a global minimum, and Dmin2(0,t) reaches a global maximum around t = 0.25 cycles (the time of the first strain extremum), while these signals reach local minima and local maxima, respectively, around t = 0.75 cycles (the time of the second strain extremum). Rearrangement is more evident during the first shear half-cycle; this asymmetry is further evidence of a difference in material response according to shear direction and hints that the system remembers its history.

We compile results for all strain amplitudes in Fig. 3C, which clearly evidences the inverse relationship between p(s, ts,0) and Dmin2(0,t) for all shielding levels. As strain amplitude increases, the minima in p(s, ts,0) decrease, and the maxima in Dmin2(0,t) increase, indicating increased rearrangement with strain amplitude. Curiously, shielding levels show a roughly log-linear relationship between log Dmin2(0,t) and [1 − p(s, ts,0)] across most strain amplitudes, with disordered particles and particles at grain boundaries showing a smaller slope between these quantities than the more interior crystalline particles. We may conclude in general that correlations in crystallinity over time, captured by p(s, ts, t0), indicate rearrangement of individual particles and that rearrangement occurs in a hierarchy according to interiority within crystal grains.

Last, we note that these results seem quite robust; alternate analysis of structural rearrangement using time correlations in the continuous structural homogeneity parameter Δi, introduced in the “Crystallinity oscillates with shear and reveals material preparation memory” section, leads to similar conclusions to those presented above (see fig. S9).

Asymmetry in crystallinity correlation with respect to shear direction is localized within crystal grains

Next, we demonstrate that particles more interior within crystal grains, or more shielded, have a rearrangement propensity that is more asymmetric with respect to shear direction, indicating that memory of material history is localized within crystal grains. To do so, we quantify the difference in rearrangement propensity with respect to shear direction via the correlation p(s, ts, t − 0.5), the conditional probability that a particle is in structure s (either crystalline or disordered) at time t given that it was in the same structure s half a cycle earlier. The quantity p(s, ts, t − 0.5) thus measures correlation between equivalent time points in each half-cycle, differing only in the direction of shear, since the applied shear is sinusoidal. As in the previous section, at each time t, we group particles according to their crystalline shielding level at time t0 = t − 0.5 and calculate p(s, ts, t − 0.5) over each particle subgroup. [Note that this correlation and the correlations presented in the previous section are 1D cuts through a full 2D probability distribution p(s, ts, t0). Full 2D distributions for nontransient portions of trajectories at all strain amplitudes are shown in figs. S10 and S12.]

Figure 4A shows p(s, ts, t − 0.5) for all shielding layers for experiments below (γ0 = 0.022) and above (γ0 = 0.068) yield. Signals are stroboscopic averages over nontransient system trajectories shown in fig. S12. The two minima in each signal show that particles in all shielding levels are least structurally autocorrelated when both t and t − 0.5 are times of strain extremum at 0.25 and 0.75 cycles. This bimodal nature stems from the fact that particles are most dynamically responsive to shear, rearranging most, at those times of strain extremum. However, for the more shielded layers in the experiment below yield, there is a notable asymmetry between the first and second halves of the signal. The correlation p(s, ts, t − 0.5) reaches a shallower minimum in the second half-cycle, around t = 0.75 cycles, than it does in the first half-cycle, around t = 0.25 cycles. This asymmetry implies that shielded particles are less responsive to shear during the second half-cycle than they are during the first half-cycle. The asymmetry is not as prominent for the least shielded layers and diminishes in the experiment above yield.

Fig. 4 More shielded crystalline particles are more asymmetric in structural response with respect to shear direction.

(A) Structural responses p(s, ts, t − 0.5) for particles at all shielding levels for experiments below (γ0 = 0.022) and above (γ0 = 0.068) yield, as solid and dotted lines, respectively. Stroboscopic averages are shown for clarity, and error bars denote the standard error of the mean value at each time point. (B) The ratio Pp(2ω*)/Pp(ω*) for all shielding layers as a function of γ0. Error bars are calculated via Taylor series propagation of the standard errors of the mean of each Pp(ω) quantity in the ratio. In both panels, signals are colored by shielding level according to the color scheme detailed in Fig. 1. Colors of the least shielded (disordered) and most shielded layers are shown for reference.

We quantify this asymmetry via Pp(2ω*)/Pp(ω*), the ratio of power spectra of each p(s, ts, t − 0.5) signal at 2ω* and ω*, where ω* is the frequency of the shear cycle. Figure 4B shows this ratio for each shielding layer at all strain amplitudes and provides further evidence that more shielded crystalline particles are more asymmetric with respect to direction in their response to shear. Power spectra are calculated via the periodogram estimate, and each value Pp(ω) is the mean of a set of such values calculated over consecutive two-cycle windows of the full nontransient p(s, ts, t − 0.5) signal shown in fig. S12. Power spectral densities for each frequency at all strain amplitudes are reported separately in fig. S11. The ratio Pp(2ω*)/Pp(ω*) is highest for the least shielded particles and lowest for the most shielded particles at all strain amplitudes. This finding implies that the structural rearrangement of the least shielded particles is most symmetric with respect to shear direction, and the structural rearrangement of the most shielded particles is least symmetric with respect to shear direction. Our results suggest a hierarchy of asymmetry in structural response according to crystalline shielding layer. Thus, response anisotropy, signifying the material’s memory of its history, is spatially localized within crystal grains.

DISCUSSION

We have demonstrated that jammed and athermal systems under oscillatory shear show an asymmetry with respect to shear direction in both local deformation and structural homogeneity. This asymmetric response is only erased at high strain amplitudes above yield. Per-particle autocorrelations in structural homogeneity are also asymmetric with respect to shear direction, and autocorrelations are especially asymmetric for particles that are most interior, i.e., shielded, within crystal grains. We believe that observed asymmetries are indicative of memory of the system’s history or preparation, and our findings imply that this simple form of memory is spatially localized within crystal grains.

Structural reversibility is destroyed only at strain amplitudes well above yield

We first address the system at the highest strain amplitude studied, γ0 = 0.157, which displays per-particle structural correlations that are qualitatively different than structural correlations of systems at lower strain amplitudes (even when those systems are still above yield). Figure 5 displays some of these signals; together, they show that particles at γ0 = 0.157 do not retain their distinct structural identities of crystalline or disordered in any notable capacity even over the course of one shear cycle. The system is in a state of structural irreversibility on the individual particle level, even while structural reversibility exists at lower strain amplitudes that are still above yield.

Fig. 5 Structural correlations for the system at γ0=0.157, the highest strain amplitude studied.

(A) Structural autocorrelations p(s, ts,0) as a function of t over 1.5 shear cycles, with t0 = 0 marking the beginning of the cycle. Results are shown as stroboscopic averages, and error bars represent the standard error of the mean value at each time point. Each signal represents a shielding level according to the color scheme detailed in Fig. 1 and is calculated over particles in the appropriate shielding level at t0. Colors of the least shielded (disordered) and most shielded layers are shown for reference. (B) Full 2D distributions of p(s, ts, t0) for three shielding layers, shown beneath colored bars that indicate shielding level at time t0 according to the color scheme detailed in Fig. 1. Dark and light gray lines indicate times of strain extrema during each first and second half-cycle, respectively. Dotted lines indicate a 1D slice through each distribution at p(s, ts, t − 0.5).

Figure 5A shows p(s, ts,0) for each shielding layer at γ0 = 0.157 as a function of t over 1.5 shear cycles, with t0 = 0 marking the beginning of the cycle. This structural autocorrelation is calculated identically to that shown in Fig. 3A for γ0 = 0.068, still above the yield strain. Notably, whereas the correlations in Fig. 3A appear periodic within one shear cycle and only show slow decay over longer time scales (figs. S10 and S12), correlations in Fig. 5A are never periodic and decay rapidly during the first cycle for all shielding layers, showing the destruction of maintained particle identities over the course of one cycle.

The decay in structural autocorrelation at γ0 = 0.157 can be seen more fully in 2D distributions of p(s, ts, t0) across t0 and t. Figure 5B shows distributions for particles in three crystalline shielding layers at time t0: disordered particles, grain boundary particles, and particles that are most interior within crystal grains. None of these 2D distributions display meaningful periodicity over any 1D cut through them, except the cut p(s, ts, t − 0.5), drawn as a black dotted line. This signal, explored thoroughly in the Results, measures correlations only within the half-cycle time window and indicates that the maximum period over which particles retain their structural identity is approximately half a cycle.

According to the oscillatory rheology shown in Fig. 1B, at the highest strain amplitude, γ0 = 0.157, the storage modulus G′ and loss modulus G′′ are closest to each other, approaching equality. We therefore posit that our particle-scale measurements of correlations in local structural homogeneity capture the macroscopic, rheological crossover point at which material behavior is equally dominated by elastic and viscous response.

Crystalline shielding in a monodisperse system with larger crystal grains

We also analyzed crystallinity and structural rearrangement in a monodisperse system (33), identical in preparation to the bidisperse one discussed in the Results. Because of the monodispersity, crystalline grains are larger, and thus, crystalline shielding is deeper. Our results, shown in figs. S13 to S15, are in agreement with those already presented. We find oscillating measures of global structural heterogeneity, both crystallinity X(t) and Δ¯(t) averaged over all particles, and an asymmetry in those measures with respect to shear direction as strain amplitude decreases, indicating that some material anisotropy and memory are preserved in the system (figs. S13, A and B, and S14). Deep crystalline shielding exists in the system (fig. S13, C to E), and particles of deeper shielding are generally less structurally volatile, with higher values of p(s, ts, t0) (figs. S13F and S15). Particles of deeper shielding are also generally more asymmetrical in their structural response with respect to shear direction: Crystalline particles that are more shielded have probability signals p(s, ts, t − 0.5) that are more dissimilar between each half of the shear cycle. As a result, more shielded layers generally have lower values of Pp(2ω*)/Pp(ω*) (fig. S13G). These results are not as clear-cut as those obtained from the bidisperse system, perhaps because in the monodisperse system, there are more crystalline layers, each with fewer particles, and statistics are consequently thinner. However, we can observe that the three most shielded layers have lower Pp(2ω*)/Pp(ω*) values than the three least shielded layers at all strain amplitudes except for the highest strain amplitude studied. Particles of deeper shielding thus, in general, form regions in which the system’s memory of its preparation is localized. At the highest strain amplitude, we observe qualitatively distinct structural correlation measures across all shielding layers (fig. S15). Correlations are not periodic even within one shear cycle and differ in behavior from correlations at all lower strain amplitudes, in a similar manner to that discussed in the “Structural reversibility is destroyed only at strain amplitudes well above yield” section. Thus, particle structural identities are not retained even over one shear cycle. The highest strain amplitude is again a point at which G′ and G′′ approach equality as shown in Fig. 1B, implying that our local correlation measures capture a macroscopic rheological transition.

Crystalline order and heterogeneous glassy dynamics

As mentioned in the Introduction, our finding that propensity for particle rearrangement occurs in a hierarchy according to particle interiority within crystal grains is an extension of the well-known result that, for amorphous solids subject to stress, particles on grain boundaries or within defect regions are more mobile than crystalline particles (19, 3338). In this section, we briefly discuss how this general result—that crystalline domains are less mobile under shear—is relevant in the context of supercooled glass-forming fluids and their dynamical heterogeneity (41). How structure informs heterogeneous dynamics in supercooled glass-forming systems is of long-standing interest, and observations have previously been made regarding the connection between structures liable to rearrangement under shear in athermal or thermal amorphous systems, and those liable to rearrangement due purely to thermal fluctuations [for a general review of structure and dynamical arrest, with a brief exploration of shear in this context, see Ref. (42)]. One relevant theory of glass formation developed by Tanaka and collaborators (43, 44) posits that, within supercooled systems, domains with medium-range crystalline order are longer lived and less mobile than other particles, contributing to dynamical heterogeneity. Lending credence to this idea are observations of less mobile crystalline regions in a wide variety of amorphous systems, ranging from monatomic liquids on hyperbolic surfaces (45), to 2D and 3D systems of repulsive and attractive bi- and polydisperse spheres (43, 46), to spin systems with antiferromagnetic crystalline ground states (47). Thus, a subportion of our findings—that particles with crystalline ordering (at the center of crystal grains) rearrange least under shear and are thus least “mobile”—fits well with this argument.

Steric favorability, rigidity, and memory

Recent work has postulated a yet more general mechanism for dynamical heterogeneity in amorphous systems either under shear or subject to thermal fluctuations. It has been proposed (48) that sterically favorable structures, or structures that are efficiently packed on a local length scale, may be the slow domains responsible for dynamical heterogeneity in supercooled liquids (and whose length scale couples to the characteristic length scale of cooperative motion in these systems). In some contexts, as in the case of weakly polydisperse or monodisperse disk systems in 2D, the authors of Ref. (48) note that sterically favored structures also happen to be those with crystalline (in this case, hexagonal) ordering, given by high absolute values of the hexatic order parameter ψ6. In support of this theory of steric favorability, a parameter quantifying steric unfavorability of local structures was found to correlate strongly with plastic rearrangement in athermal 2D systems under quasi-static shear (14), and it was also found that geometrically unfavored structures in metallic glasses are flexible motifs that encourage soft vibrational modes and local rearrangements under shear (49). In yet another context, it was found that the continuing adsorption of nanoparticles at a liquid interface, after the interface has already been significantly filled with particles, preferentially takes place at defect sites rather than within crystal domains, as defect sites are least sterically hindered and most forgiving of rearrangement at the interface (50).

Steric favorability offers a revealing lens through which our results may be interpreted. Particles of high crystalline shielding, at centers of crystal grains, are surrounded by multiple layers of steric hindrance and close-packing. Perhaps, our discovery of a hierarchy of rearrangement propensity according to interiority within crystal grains occurs because, as particles are embedded ever deeper inside grains, they are increasingly constrained by their neighborhood and thus are less likely to rearrange under oscillatory shear. Another recent work (51) lends further credence to this idea and offers a bridge to our findings related to memory storage within crystal grains. Authors of Ref. (51) found that locally favored icosahedral structures accumulate stress during the shear start-up of a dense soft solid en route to yielding, due to the overconstrained and stiff nature of the icosahedral domains. As a result, the icosahedral structures play a large role in shear-banding and eventual total fluidization of the material. Similarly in our system, the overconstrained nature of the crystalline domains (a consequence of their steric favorability) renders them measurably “stiffer” than the noncrystalline particles: At small strain amplitude, the local shear strain around nonrearranging high-∣ψ6∣ particles is smaller than that around nonrearranging low-∣ψ6∣ particles (33). Perhaps, our finding that preparation memory is stored within crystal grains is due to these particles’ stiffness and ability to accumulate stress during the material’s preparation. Our observed heterogeneity of memory would then be a consequence of the heterogeneity of local stiffness and could generalize to other systems regardless of whether or not they have crystalline ground states.

Conclusions

In this paper, we have presented new and accessible measures on the individual particle level that correlate with nontrivial phenomena on the macroscale. Our measures capture local structural homogeneity and its correlations over time and are quite distinct from other quantities such as Dmin2 (30) or T1 events (52) usually employed to investigate microstructural response under shear. We have found that, by simply measuring correlations in the degree to which an individual particle’s environment is similar to those of its neighbors, we can shed light on macroscale yield and memory effects in amorphous materials under oscillatory shear.

Our analysis indicates that the system-wide average of crystallinity on the particle level shows asymmetry with respect to shear direction and thus encodes the material’s memory of its preparation. Correlations over time in the structural homogeneity of individual particles are reliable indicators of particle rearrangement as measured by Dmin2, and these correlations also show asymmetry with respect to shear direction. The observed structural asymmetric response is not homogeneous throughout the system, however. Particles that are more interior within crystal grains, or more “shielded,” are generally less structurally volatile over time and least symmetric in their structural response with respect to shear direction. Thus, response asymmetry and consequent material memory are spatially localized in crystal grains.

In addition, we have found that structural correlations are qualitatively different at strain amplitudes for which macroscale rheological measures of elastic and viscous response approach cross over, implying that our local measurements indicate a behavioral transition usually only visible on a much larger length scale. Our work bridges the micro- and macroscales and thus will be useful for future experimentalists studying yield in amorphous systems who may have access only to information on one length scale, either microstructural or rheological. Our efforts add to the growing body of knowledge regarding the nature of microscopic rearrangement and macroscopic yield in disordered materials and help to illuminate how, and specifically where, certain types of memory are stored in these systems.

CITATION DIVERSITY STATEMENT

Recent work in several fields of science has identified a bias in citation practices such that papers from women and other minorities are undercited relative to other papers in the field (5358). Here, we sought to proactively consider choosing references that reflect the diversity of our field in thought, form of contribution, gender, and other factors. We obtained predicted gender of the first and last authors of each reference by using databases that store the probability of a name being carried by a woman or a man (53, 59); we supplemented these results with online research of individuals for whom automatic classification failed. By this measure (and excluding self-citations to the first and last authors of our current paper), our references within the main manuscript contain 69.0% man/man, 19.0% man/woman, 6.9% woman/man, and 5.2% woman/woman categorization, rounded to one 10th of 1%. The automated method is limited in that (i) names, pronouns, and social media profiles used to construct the databases may not, in every case, be indicative of gender identity and (ii) it cannot account for intersex, nonbinary, or transgender people. We look forward to future work that could help us better understand how to support equitable practices in science.

MATERIALS AND METHODS

Experiments

We first note that the analysis presented in this paper uses previously published experimental data (19, 33), and we only briefly describe the experiments here. Experiments are performed using a custom-made interfacial stress rheometer (ISR) to controllably impose shear deformation on 2D jammed particle suspensions. The ISR apparatus allows for the tracking of single particles (and hence microstructure characterization) while simultaneously measuring the suspension bulk rheological properties (e.g., viscous and elastic moduli). We will briefly describe the ISR used in our experiments, and further details can be found in Refs. (19, 33). The ISR is composed of a ferromagnetic needle trapped at a decane/water interface by capillary forces, between two vertical glass walls. These walls pin the interface to maintain a flat shearing channel that can be simultaneously imaged with a microscope. Particles are adsorbed at this interface, creating a 2D jammed particle suspension. Particle positions are identified and linked to form trajectories using the open-source particle-tracking software trackpy (60, 61). The positions of approximately 40,000 particles are tracked during shearing. Tracking data for the bidisperse and monodisperse systems presented here are from Refs. (19) and (33), respectively.

To obtain an interface’s rheological information, the needle is driven axially by a known, imposed, magnetic force generated by two Helmholtz coils. The displacement of the needle is measured using a microscope. A monolayer’s shear storage (G′) and loss (G′′) moduli are calculated from the imposed force and observed displacement (6264). Experiments access shear moduli over a range of strain amplitudes γ0, with 0.005 < γ0 < 0.16, at a fixed frequency of 0.1 Hz. Previous work (33) has shown that, at this frequency, the material stays in the linear response regime at low Reynolds number, and system response is accordingly independent of frequency. Before each experiment, the monolayer is prepared via six cycles of shearing at a large strain amplitude (γ0 ∼ 0.5). Shearing is then halted and resumed at smaller strain amplitudes for experimental data collection.

Particles are sulfate-coated latex (Invitrogen) and experience dipole-dipole repulsion due to charge groups on their surfaces (65). Rigorous cleaning methods are undertaken to ensure the reproducibility of particle interactions. More details regarding these preparations and the system itself are included in the Supplementary Materials.

All packings have high enough area fraction ϕ to be fully jammed without shear. Data from two main experimental systems are analyzed here: a bidisperse system (equal parts 4.1- and 5.6-μm diameters; ϕ ≈ 43%) is analyzed in the Results, and a monodisperse system (5.6-μm diameter; ϕ ≈ 32%) is analyzed in the Discussion and the Supplementary Materials. [We note that these experimental systems were examined in Refs. (19) and (33) with different analytical techniques to test distinct hypotheses.] A snapshot of a bidisperse 2D jammed amorphous particle system is shown in Fig. 1A. Oscillatory rheology measured for these systems is shown in Fig. 1B. In both systems, the rheological yield strain amplitude is approximately γ0 ∼ 0.03.

The particles are large enough that they do not experience detectable Brownian motion. Also, this system is jammed; without shear, the particles are fully arrested. Rheology (Fig. 1B) shows that in the low strain limit, the elastic modulus is over an order of magnitude higher than the viscous modulus. Therefore, it is an elastic jammed solid. The system is far above the jamming transition as determined by the strength of particle interactions (65) and the number density of the system. The number densities of the systems are approximately 16,500 particles/mm2 (bidisperse) and 14,300 particles/mm2 (monodisperse). Rheology experiments have shown that the bidisperse system unjams below a number density of approximately 13,800 particles/mm2 [see Ref. (19)].

Structural analysis

To investigate local structure in these amorphous particle systems, we use an environment matching method to characterize continuous structural homogeneity and discrete crystallinity. Software can be found in the open-source analysis toolkit freud (66). Our approach, schematically illustrated in Fig. 1C, characterizes whether particle environments are sufficiently similar to their neighbors’ environments, regardless of the structure of the environment itself. This method is but one of many available options (14) in the context of structural characterization for the purposes of dynamical prediction in amorphous materials; its advantages are that it is agnostic with respect to the nature of the structure itself, straightforward to calculate, and physically intuitive.

We define particle i’s environment (shown as green vectors in Fig. 1C) as the set of vectors {rim}, where rim points from the center of particle i to the center of particle m, and m is an index over i’s Mi nearest neighbors. We then inspect all environments of i’s neighbors. Let one such neighbor be labeled j. Particle j’s environment (shown as purple vectors in Fig. 1C) is defined as the set of vectors {rjm}, where rjm points from the center of particle j to the center of particle m′ and m′ loops over j’s Mj nearest neighbors. We then compare the environments of particle i and particle j by attempting to match these sets of vectors. Particle j’s environment “matches” particle i’s environment if we can find a one-to-one mapping such that ∣rimrjm∣< t for every mapping pair (m, m′) for some threshold t. Nearest neighbors of each particle are defined as those within a radial distance rcut = 11.04 μm for the bidisperse systems and rcut = 11.34 μm for the monodisperse systems, determined approximately by the minimum after the first peak of the radial distribution function g(r) calculated over all particles in each system. Figure 1D (inset) shows the radial distribution function g(r) of a sample bidisperse experiment at γ0 = 0.068, collected over one shear cycle. Environments of particles i and j are only compared if Mi = Mj, and thus, a one-to-one mapping is possible; otherwise, particles i and j are automatically deemed nonmatching. The threshold t = 0.2rcut was chosen for all systems because 0.2 or 0.3 times the approximate nearest-neighbor distance (rcut) has proven appropriate—neither too stringent nor too lenient—in other contexts (67, 68). Figure S2 explores the impact of threshold choice on crystallinity characterization as we explain next.

If the environments of two neighboring particles match, then they are designated members of the same crystal grain. Crystallinity is defined in this paper as the fraction of particles in crystal grains of size larger than 1, and particles are defined as crystalline if they are members of a crystal grain of size larger than 1. Figure 1D shows a snapshot of a sample bidisperse experiment at γ0 = 0.068, with all crystalline particles identified. Particles are drawn with radii equal to twice the measured image radii of gyration for ease of visualization. Crystalline structure in this system is hexagonal in nature, as reported in Ref. (33), and also as evidenced by histograms of the bond-orientational order parameter ∣ψ6∣, collected separately for crystalline particles and noncrystalline particles over one cycle of two example systems (fig. S1). The complex number ψ6(i)=1Nij=1Nie6iϕij, where Ni is the number of nearest neighbors of particle i and ϕij is the angle between rij and the vector (1,0), measures the sixfold orientational symmetry of particle i’s environment. The distribution of ∣ψ6∣ values for crystalline particles peaks near 1 in both experiments shown in fig. S1, implying strong hexagonal order, while ∣ψ6∣ for disordered particles is distributed evenly across all values between 0 and 1. The choice of matching threshold t influences the hexagonal quality of the identified crystal grains as shown in fig. S2A. We find that setting the threshold below t = 0.2rcut results in many particles being deemed disordered that nevertheless have high hexagonal ordering, whereas setting the threshold above t = 0.2rcut results in many particles being deemed crystalline that have low hexagonal ordering. Thus, t = 0.2rcut is a reasonable compromise that produces a well-defined bipartition of crystalline and disordered particles. Furthermore, we find that the choice of threshold, within reason, does not change our results pertaining to crystallinity reported in the “Crystallinity oscillates with shear and reveals material preparation memory” section (see fig. S2, B and C).

We quantify the shielding, or interiority of a crystalline particle within a grain, by Rnon−xtal, the distance of that particle to the nearest noncrystalline particle. Shielding is higher as Rnon−xtal increases. Example distributions of Rnon−xtal for all particles in six consecutive nontransient cycles of a sample bidisperse experiment at γ0 = 0.068 (Fig. 1E, inset) show clear peaks and valleys that inform the way in which we bin Rnon−xtal into four shielding levels {Ri}. The minima of Rnon−xtal are approximately the first three minima of g(r), shown inset in Fig. 1D, as one would expect. These minima are not influenced by strain amplitude, as shown in fig. S7, which displays distributions of Rnon−xtal for all particles over several cycles in experiments below and above yield. A rendering of the system with particles colored according to crystalline shielding illustrates the concept (Fig. 1E). Disordered particles, whose distance to the nearest noncrystalline particle is formally zero, are also shown and colored purple.

Because of occasional imaging or tracking errors, or to particles moving out of the imaging field of view, some particles are not preserved over the course of entire experiments. The fraction of preserved particles ranges from ∼0.97 (for the smallest strain amplitude) to ∼0.87 (for the largest strain amplitude) in the bidisperse system, and from ∼0.61 to ∼0.85 (not correlated with strain amplitude) in the monodisperse system. To eliminate any spurious effects due to the nonpreserved particles, we typically calculate structural signatures of all particles in every snapshot but only show those signatures of particles that are tracked and preserved over the entire experiment. Unless otherwise stated, reported results are for the preserved particles in each experiment.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/7/20/eabe3392/DC1

https://creativecommons.org/licenses/by-nc/4.0/

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: We thank M. Engel for the use of the software package inJaVis, used to visualize the particle systems. Funding: E.G.T., K.L.G., P.E.A., and D.S.B. are supported by the National Science Foundation Materials Research Science and Engineering Center at University of Pennsylvania (NSF grant DMR-1120901). K.L.G. and P.E.A. are additionally supported by the U.S. Army Research Office (ARO grant W911-NF-16-1-0290). E.G.T. and D.S.B. are additionally supported by the Paul G. Allen Family Foundation. Author contributions: E.G.T., K.L.G., P.E.A., and D.S.B. designed the research. E.G.T. and K.L.G. conducted the analyses. E.G.T., K.L.G., P.E.A., and D.S.B. wrote the manuscript. Competing interests: The authors declare that they have no competing interests. The views and conclusions contained in this document are solely those of the authors. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials.

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