Soft Matter

PAPER

Role of disorder

aDepartment of Physics, California Polytech

93407, USA. E-mail: nkeim@calpoly.edubDepartment of Mechanical Engineering

Pennsylvania, Philadelphia, PA 19104, USA

† Electronic supplementary informa10.1039/c4sm02446j

Cite this: Soft Matter, 2015, 11, 1539

Received 4th November 2014Accepted 2nd January 2015

DOI: 10.1039/c4sm02446j

www.rsc.org/softmatter

This journal is © The Royal Society of C

in finite-amplitude shear of a 2Djammed material†

Nathan C. Keim*ab and Paulo E. Arratia*b

A material's response to small but finite deformations can reveal the roots of its response to much larger

deformations. Here, we identify commonalities in the responses of 2D soft jammed solids with different

amounts of disorder. We cyclically shear the materials while tracking their constituent particles, in

experiments that feature a stable population of repeated structural relaxations. Using bidisperse particle

sizes creates a more amorphous material, while monodisperse sizes yield a more polycrystalline one. We

find that the materials' responses are very similar, both at the macroscopic, mechanical level and in the

microscopic motions of individual particles. However, both locally and in bulk, crystalline arrangements

of particles are stiffer (greater elastic modulus) and less likely to rearrange. Our work supports the idea of

a common description for the responses of a wide array of materials.

I. Introduction

Connecting a material's response under stress with its micro-scopic structure—the arrangement of its constituent atoms orparticles—is a cardinal goal of materials science. In a crystallinesolid this is done by accounting for the essential symmetries ofthe material, plus a relative handful of lattice defects. Inamorphous or glassy materials, on the other hand, crystallineorder may be nearly absent, and properties such as acousticmodes can be dramatically different.1,2 Recently, Goodrich et al.3

proposed that these two cases delimit a continuum of disorder,but that over most of this continuum, a theory of amorphoussolids is more useful to describe innitesimal deformations andexcitations. For instance, with just O (1%) of particles deviatingfrom crystalline order, the behaviour of the ratio of elastic tobulk modulus G/B is much closer to that in a completelydisordered system than to that in a perfect crystal.

Does this picture extend to nite deformations, ones largeenough to rearrange particles? In amorphous solids, suchrearrangements occur in localised groups of O (10) particles,4–7

oen abstracted as shear transformation zones (STZs); theselocations can be thought of as pre-existing packing defects,though not in any way so straightforward as in a crystal.6 Inrecent experiments,7,8 we showed that under cyclic shear atnite strain amplitude g0 � 3%, an amorphous solid reached areversible plastic regime7—a steady state in which these regionswould rearrange, and then reverse, on each cycle. Because it

nic State University, San Luis Obispo, CA

and Applied Mechanics, University of

. E-mail: parratia@seas.upenn.edu

tion (ESI) available. See DOI:

hemistry 2015

isolates a stable, limited population of structural relaxations,this kind of experiment suggests a way to compare the micro-scopic signatures of plasticity among different materialstructures.

Here, we observe the reversible plastic regime in twodifferent packings that differ primarily in their amount ofdisorder, shown in Fig. 1(a) and (b). The packing made withbidisperse particle sizes more closely resembles an amorphousmaterial, while that made with monodisperse sizes more closelyresembles a polycrystal. Our experiments combine tracking ofmany (104) individual particles with shear rheometry. We ndthat the monodisperse packing, with fewer disordered regions,also has fewer rearranging regions. However, the responses ofeach material are otherwise closely similar. Our results indicatethat the conceptual picture proposed by Goodrich et al. may beextended to nite deformations, as also suggested by the recentsimulations of Rottler et al.9 Our ndings are also consistent intheir broad outlines with prior studies involving steadyshear.10–12

The model materials shown in Fig. 1(a) and (b) are packingsof polystyrene particles adsorbed at a water–decane interface.7,8

These particles have long-range electrostatic dipole repulsion,13

so that without touching they form a stable jammed material.The bidisperse packing is a mixture of 4.1 and 5.6 mm-diameterparticles, while the monodisperse packing includes the largespecies only. The materials are subjected to uniform sheardeformations in a custom-made interfacial stress rheometer(ISR).7,8,14,15 As shown in Fig. 1(c), a magnetised needle isembedded in the monolayer, parallel to the interface. Theneedle is centred between two vertical glass walls that form anopen channel. Electromagnets centre the needle and drive it,applying a uniform shear stress s(t) on the material between theneedle and the walls; thus, the rheometer is stress-controlled.

Soft Matter, 2015, 11, 1539–1546 | 1539

Fig. 1 Material and apparatus. Monolayers of (a) bidisperse and (b) monodisperse repulsive particles are adsorbed at a flat oil–water interface. (c)Interfacial stress rheometer (ISR) apparatus. A magnetic force on the needle shears the monolayer uniformly (velocity profile sketched).

Soft Matter Paper

Measuring the resulting strain g(t) allows us to computematerial rheology.

II. Results

In this section, we rst provide global descriptions of thematerials' particles and their rearrangements, followed by ananalysis of how rearrangements are localised and organised.

Fig. 2 Characterising particle packings. (a) Distributions of apparentparticle size (radius of gyration rg) in images of bidisperse (top) andmonodisperse (bottom) packings. Shaded rectangles above the curvesdefine “small” particles (in bidisperse only) and “large” particles (in bothpackings). (b) Pair distribution function g(r) of entire bidisperse packing,reflecting the separation of particles due to long-range repulsion.Inset: g(r) first peaks of (left to right) small particles only, small-largepairs, large particles only. Small and large peak positions are indicatedby vertical lines in main plot. (c) g(r) among large particles within eachpacking. Packings were prepared to match first peak positions closely,indicating very similar confining pressure on these repulsive particles.

A. Material composition and connement

Fig. 2(a) shows the sizes of the solid particles in each packing,measured as the radius of gyration of the particle image (seeMethods, below). While the large particles within each samplewere drawn from the same stock, the details of image analysiswere optimised separately for each packing, and so the rg oflarge particles differ slightly. Each packing also has a smallpopulation of abnormally large particles, which do not appearto play a disproportionate role in the dynamics.

Fig. 2(b) and (c) use the radial pair correlation function g(r) todescribe the arrangement of particles due to their long-rangerepulsion. The rst peak in g(r) represents the typical spacingbetween pairs of neighbouring particles, which we denote as a.Fig. 2(b) shows the overall g(r) of the bidisperse packings; theinset shows the contribution of pairs of each species. Thedistribution of repulsion strengths within a species was previ-ously studied by Park et al.16 Note that while the small and largespecies differ in solid size by�25%, their much greater effectivesizes differ by just �15%.

We control number density, and hence repulsion strengthand osmotic pressure, by changing the number of particlesdispersed into the 6 cm-diameter experimental cell. Our mate-rials may be considered athermal and jammed: we do notobserve thermal motion, and strong long-range repulsionsmean we are far beyond any jamming transition at whichparticles become under-constrained. Since the repulsive forcebetween particles falls off monotonically with separation,interparticle spacings are a proxy for the pressure that connesthe particles. Fig. 2(c) shows that our samples are at very similarpressures: the separation of neighbouring large particles(almost all particles in the monodisperse packing) is the samein the monodisperse and bidisperse samples to within 1%.Furthermore, the width of the rst peak is similar betweenpackings, despite the greater heterogeneity in the environments

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of particles in the bidisperse packing. This suggests that theconning osmotic pressure is a good control parameter forinterparticle interactions throughout the packing.

B. Rheology and rearrangements

Fig. 3(a) shows the oscillatory shear rheology of each material,as a function of strain amplitude g0 at 0.1 Hz. The materialshave similar responses, with the monodisperse packing being

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Fig. 3 Global measures of material response as a function of strainamplitude g0, for bidisperse (connected closed symbols) and mono-disperse packings (open symbols). (a) Oscillatory shear elastic (G0) andloss (G0 0) moduli are similar between the two packings. (b) The ratioG0 0/G0 highlights the yielding transition. Inset: loss modulus G0 0 dependsweakly on driving frequency, as shown here for bidisperse packing atg0 | 0.016. (c) Rearranging fractions of particles in the steady state,measured as defined in text: total (peak-to-peak), hysteretic, andirreversible (stroboscopic). Error bars on the irreversible points repre-sent standard deviation. g0

2 scaling is drawn for comparison. Total andhysteretic activity are much greater in the monodisperse packing. Theonset of yielding corresponds to a sharp increase in irreversible rear-rangement activity. Inset: total and hysteretic activity at g0 | 0.016(here averaged over just 2–3 cycles) show weak dependence onfrequency, with no clear trend.

Paper Soft Matter

stiffer (higher elastic modulus G0 at small g0). The inset ofFig. 3(b) shows that dissipation at low g0 depends weakly onfrequency.8 As g0 is increased, each material goes through arheological yielding transition, with G0 falling, and viscousdissipation rising (as measured by loss modulus G0 0). Theyielding behaviour, summarised by the ratio G0 0/G0 in Fig. 3(b),is nearly identical for the two packings.

Simultaneously with shear rheometry, our experiments alsotrack nearly all particles in a segment of the material. We use along-distance microscope, a high-speed camera at 40 frames pers, and in-house freely-available particle-tracking17 and analysis18

soware both to observe g(t) for rheometry, and to identifyrearrangements among �4 � 104 particles.7,8 To identify rear-rangements, we look for particle motions that are locally non-affine, as measured by the quantity D2

min.7,19 We compute D2min

between any times t1 and t2 by considering a particle and its twonearest “shells” of neighbours (within radius �2.5a), andnding the best affine transformation that relates their posi-tions at t1 and t2; D2

min(t1, t2) is the mean squared residualdisplacement aer subtracting this transformation, normalisedby a2. We dene a rearranging particle as one with D2

min(t1, t2)$0.015, a threshold comparable with one used for simulations ofdisordered solids.19 For an illustration of video microscopy,particle tracking, and D2

min, see ESI Movie 1.†

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Using these methods, we may measure the rate at which thematerial is altered by repeated cycles of driving. Before eachexperiment begins, we rejuvenate the material with 6 cycles oflarge-amplitude shearing (g0 � 0.5), then stop; once we resumeshearing with a smaller amplitude, we observe that the rate ofchange decays during a transient and reaches a relatively steadyvalue. The steady state appears to begin at roughly cycle 15 ineach 30-cycle movie; this timescale varies little with g0, incontrast to the divergences reported in some studies of disor-dered solids20–22 and sheared suspensions.23 The rate of irre-versible rearrangements over time is plotted in ESI Fig. 1.† Thatgure also shows the evolution of the monodisperse material atg0 ¼ 0.014, which is not shown in the present results; thatexperiment showed signicant irreversible change near the endof the recording, possibly due to an external disturbance.

Our analysis of rearrangements focuses on the steady state,and it attempts to capture all activity during a cycle: we detectboth total (peak-to-peak) rearrangements, due to the deforma-tion between a minimum in strain g(tmin) and the subsequentmaximum g(tmax); and irreversible (stroboscopic) rearrange-ments that are the net result of the full cycle, as delimited by(tmin + tmax � 2pu�1)/2. In general, a particle in the total set ofrearrangements, but not the irreversible set, is rearrangingreversibly. Finally, we identify the sub-population of reversiblerearrangements that are hysteretic, requiring a buildup of stressto activate.6,7,19,24 Within a cycle beginning at tmin, we obtain tonat the last video frame for which D2

min(tmin, t) < 0.015, and toff atthe last frame with D2

min(tmin, t) $ 0.015. These correspond toglobal strains gon at which the particle rearranges duringforward shear, and goff at which it is reverted during reverseshear. We consider a rearrangement hysteretic if gon � goff

exceeds the largest strain increment Dg between video framesin the movie, “on” and “off” are in the rst and second halves ofthe cycle respectively, and D2

min $ 0.015 for at least 50% of theintervening frames.7 The fraction of hysteretic particles isrobust to changes in the Dg threshold: increasing the thresholdby a factor of 10, or omitting it entirely, respectively decreases orincreases the fraction by only �10%. ESI Movie 1† features anexample of a hysteretic rearrangement.

Fig. 3(c) shows the fraction of particles in these three rear-ranging populations, as a function of g0, averaged over thesteady state. The measurements for the bidisperse packing areclose to previously-published results for bidisperse packings athigher osmotic pressure7 (not shown). The present data showmore rearrangements in the bidisperse packing than in themonodisperse packing, at all g0. However, the two packings'microscopic behaviours are otherwise strikingly similar.Reversible and hysteretic rearrangements occur in roughly thesame proportion to each other, with a steep onset around g0 ¼0.02, and thereaer scaling roughly as g0

2 even as the systemyields. Irreversible activity, on the other hand, is minimal [O (10)particles in the eld of view] and increases slowly below g0 �0.05, at which point the system appears to undergo a micro-scopic yielding transition to a strongly irreversible steady state,as has been observed for other so solids.7,8,20–22,25,25–29 We notethat levels for the monodisperse packing at g0 ¼ 0.06 are likelyreduced because of a nascent slip layer near the wall; however,

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Soft Matter Paper

measurements of that experiment are consistent with trends atlower g0, and so we have included them here.

C. Role of disorder

As noted above, the overall similarities in response between thetwo materials are despite their different distributions of effectiveparticle sizes, which result in different propensities toward localcrystalline ordering. To examine the packings' structures moreclosely, we compute the bond-order parameter j6, identifyingsixfold symmetry in the placement of a particle's neighbours:

j6 ¼1

Nr

XNr

n¼1

ei6qð~rn�~r0Þ (1)

where n runs over the Nr neighbours that fall within 1.5a of theparticle, and q(~rn �~r0) denotes the angle that the vector fromthe particle to its neighbour makes with a xed reference axis ı.The magnitude |j6| ranges from 0 to 1 andmeasures the degreeto which the particle's neighbourhood resembles a hexagonalcrystal, while the complex phase corresponds to the local latticedirector. Fig. 4(a) shows the distribution of |j6| within eachpacking, showing that the bidisperse packing has signicantlymore particles with local disorder.

The different prevalences of crystalline order are also clearwithin the portions of each packing shown in Fig. 4(b) and (c),drawn to represent the j6 of each particle. The monodispersepacking resembles a polycrystalline agglomeration of grains,each containing(100 particles, whereas the analogous regions

Fig. 4 Static crystalline structure and local deformation. (a) Magnitudemonodisperse (shaded region) packings. (b) j6 of bidisperse packing regio(large) or <0.9 (small); colour shows lattice director and is solely to guide tmonodisperse packing in Fig. 1(b), at g0 ¼ 0.038. There is a large differencC2

j(r), the probability that two particles r apart are in the same contiguouthe bidisperse (labeled “B,O”; “B,D”) and monodisperse (“M,O”; “M,D”) pacdisordered phase of the bidisperse packing (“B,D”) percolate their respeparticles with low (-) or high (C) local crystalline order |j6| that rearrangopen symbols: monodisperse. Particles in disordered environments are mpackings. (f) Local elastic deformation: elastic strain 3xy near particles that2g0,affine. Averages are plotted for particles with low and high |j6| as a fumore than the global strain), and high-|j6| particles appear “stiffer.” Adeformation (not plotted), and local structure becomes less relevant.

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in the bidisperse packing are much smaller and containnumerous defects. We can quantify this structure, and anyassociated length scales, by computing a 2-point cluster func-tion30,31 for local order, C2

j(r). We dene ordered and disor-dered phases using the threshold |j6|¼ 0.9. Among particles ofeach phase we then identify contiguous clusters of particles.Dened generally, C2(r) is the probability that any two particles,separated by distance r, lie in the same contiguous cluster (seeMethods section for details). It thus probes not only lengthscales, but also connectedness and percolation.30,31 In Fig. 4(d)we show C2

j(r) of the ordered and disordered phases withineach packing. There are dramatic differences: in the bidispersepacking, the disordered phase effectively percolates the entireobserved region (size �150a), while the ordered phase has acharacteristic length scale of �3 particles. In the monodispersepacking, it is the ordered phase that percolates. Here, extendednetworks of grain boundaries are suggested by the steep initialdrop-off of the disordered-phase C2

j(r) (the grain boundarywidth) and shallower secondary decay.

The red shaded markers in Fig. 4(b) and (c) indicate rear-ranging particles, ones with total D2

min$ 0.015. Rearrangementstend to be localised to particles with low |j6|.7 Similarly,Fig. 4(e) shows that particles with |j6| < 0.9 are more likely torearrange than those with |j6| $ 0.9. This trend is especiallypronounced in monodisperse packings.

Even when particles are not rearranging, we nd that theirmotion is correlated with |j6|. We compute the local horizontalshear strain 3xy by the same least-squares method as for D2

min,

distributions of bond order parameter j6 in bidisperse (black line) andn in Fig. 1(a), in the steady state at g0¼ 0.044. Dot size shows |j6|$ 0.9he eye. Particles that rearrange are highlighted in red (see text). (c) j6 ine in size and quality of crystalline grains. (d) Two-point cluster functions region, plotted for ordered (|j6|$ 0.9) and disordered “phases”withinkings. The ordered phase of the monodisperse packing (“M,O”) and thective materials; their counterparts have limited range. (e) Fraction ofe (total D2

min $ 0.015), as a function of g0. Closed symbols: bidisperse;ore likely to participate in rearrangements, especially in monodispersedo not rearrange (totalD2

min < 0.015) is normalised by global affine strainnction of g0. At low g0, low-|j6| particles appear “softer” (deformed byt high g0, much of the global shear is accomplished by non-affine

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Fig. 5 Material structure has little effect on spatial organisation ofrearrangements. (a) Map of normalised particle hysteresis, (gon � goff)/2g0, for one steady-state shear cycle of the bidisperse packing at g0 ¼0.032. A segment between the wall (bottom edge) and the needle (topedge) is shown. (b) Similar map for monodisperse packing, g0 ¼ 0.038.

Paper Soft Matter

and normalise it by the least-squares shear strain of the entirepacking, g0,affine x g0. Fig. 4(f) shows that at low g0 (belowyielding), 3xy tends to be�10% greater among particles with low|j6|. The local 3xy of a specic region is due both to its localelastic moduli and to the local stress on that region, and it isdifficult to separate their respective contributions. However, wemay expect the local elastic moduli to be most inuenced bylocal structure, while stress would be heavily inuenced by theproperties and deformations elsewhere in the material, topreserve force balance. The correlation of local 3xy with local |j6|therefore suggests that disordered regions themselves are atleast slightly “soer,” even when they do not rearrange.

As g0 is increased in Fig. 4(f), we see the effects of non-affinedeformation. Rearrangements locally accomplish shear defor-mation by a dissipative plastic process, rather than by accu-mulating elastic stress; they reduce bulk stress at a given strain.The result is that the non-rearranging portions of the system areunder less stress, and deform less, than they would if no rear-rangements happened. This is evidenced by the downwardtrend of 3xy/2g0,affine in Fig. 4(f). Additionally, the distinctionbetween low- and high-|j6| particles appears to vanish at largeg0, in contrast to the clear difference at small g0. The meaningof this change is unclear.

Although g0 is slightly higher than in (a), there are fewer rearrangingregions, consistent with Fig. 3(c). Note that gon � goff is nearly uniformwithin each cluster, suggesting that particles rearrange and reversecooperatively. (c) Two-point cluster function Cpl

2 (r) among clusters ofhysteretic particles (see text), for the data represented in (a) and (b).Radius r is normalised by typical particle spacing a. The last point ofeach curve indicates themaximal diameter of the largest cluster(s). Theshape and size of clusters are similar between the two packings.

D. Spatial organisation

As discussed above, plastic (i.e. hysteretic) rearrangements tendto involve discrete clusters of particles, which may be describedas shear transformation zones (STZs).6,7 In our discussion ofFig. 3(c), we noted that the number of hysteretically-rearrangingparticles is 2–3 times greater in the bidisperse packing than inthe monodisperse packing, at similar g0. The results in Fig. 5suggest that we should interpret this difference in terms of thenumber of rearranging clusters, not their shape or size. Wecolour each particle according to the extent to which its rear-rangement is hysteretic, gon � goff. Visual examination revealsthat clusters have comparable size in the two packings. Fig. 5(c)shows this more quantitatively using a 2-point cluster func-tion30,31 for these reversible plastic regions, Cpl

2 (r), which alsoreveals the maximum cluster diameter, given by the largest r forwhich Cpl

2 (r) > 0. We nd this broad similarity throughout thereversible plastic regime of 0.02 ( g0 ( 0.04, suggesting thatthe primary result of changing g0 or microscopic structure is tochange the number and placement of these regions.

III. Discussion

The comparisons we have presented are signicant in partbecause our experiments allow us to prepare differently-structuredpackings conned at similar osmotic pressure. We argue thatpressure is likely the best control parameter for this system: othercontrol parameters such as area fraction, particle overlap, andcoordination number are difficult to dene usefully in the pres-ence of long-ranged repulsions, and area fraction has a differentmechanical interpretation in ordered and disordered packings.1

At the smallest g0 we measure a G0 for the monodispersepacking that is �50% greater [Fig. 3(b)]. Perhaps more

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importantly, this discrepancy appears within the packings: evenas the entire packing deforms nearly elastically, regions withlocal crystalline order are deformed less than disorderedregions [Fig. 4(f)].

At larger g0, our experiments allow a more detailedcomparison of the mechanical properties of each packing,because of the observations afforded by the reversible plasticregime. The observed regions of hysteretic rearrangements lendthemselves to a theoretical description of material deformation,for the same reasons that favour shear transformation zones(STZs):6 they dissipate energy and change bulk rheology by theirhysteresis; they may be considered as two-state subsystems withlong-range interactions; and they arise out of the static micro-structure of the material. In such a model, the material'sresponse to large-amplitude or steady shear is built up out ofmany consecutive rearrangements, drawn from a continually-replenished population of STZs.6 Observations of the reversibleplastic regime thus connect static structure with yieldingbehaviour and steady shear, both in amorphous solids, andperhaps in polycrystals.10,11,32

Our results indicate that the reversible plastic regime, andthe type of rearranging regions it reveals, are shared by more-and less-disordered packings. This in turn hints at a commonunderstanding of the localisation of rearrangements in a rangeof materials. The idea of a common understanding is also

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supported by recent simulations by Rottler et al.,9 that suggestthat a method to predict localisation in bulk disordered pack-ings also works at lattice defects and grain boundaries. Notably,our observations show anecdotally that rearrangements tendnot to occur at lattice defects within crystallites, and are insteadfavoured at grain boundaries, as illustrated by Fig. 4(c).

Our work also complements comparative studies of solidsunder steady shear. Experiments with steadily-sheared foamshave shown that a small amount of disorder makes the owingmaterial's rheology qualitatively like that of an amorphoussolid.12 Furthermore, simulations of steady shear suggest that asin amorphous solids, particles in polycrystals rearrange withspatiotemporal heterogeneity11 and with similar sliding of parti-cles past one another.10 In a notable difference from our results,however, in simulations by Shiba et al.10 the scale of localisationin polycrystals may be much larger because of extended grainboundaries; for oscillatory shear we see no such trend (Fig. 5(c)).

IV. Conclusions

In this paper, we have investigated the response of materials tosmall but nite-amplitude deformations, considering how thisresponse depends on the degree of disorder in the materials'microscopic structure. We characterised the materials' steady-state response to cyclic shear, which can be controlled by astable population of rearrangements that occur and reverse oneach cycle. Through experiments on packings of particles withmonodisperse and bidisperse sizes at the same conningpressure, we have shown that despite clear differences inmaterial structure, the response is very similar, in bothmacroscopic rheology and microscopic particle motions. Areversible plastic steady state arises in each material for a rangeof strain amplitudes 0.02( g0( 0.05, rearrangements occur inclusters of O (10) particles, and these clusters are correlated withmore-disordered regions of the material. We nd just two majordifferences: at small amplitudes, a higher apparent elasticmodulus for particles with crystalline order, both locally(Fig. 4(f)) and at the bulk scale (Fig. 3(a)); and at larger ampli-tudes, a 2–3-fold greater population of rearrangements in themore-disordered material (Fig. 3(c)), occurring in clusters ofsimilar length-scale (Fig. 5). The consequences of this pop-ulation difference remain an open question, and could includethe details of rheology, the precise onset of irreversibility in thesteady state, and the materials' memory capacity.33,34

This work contributes to an understanding of materialplasticity and mechanical response as situated along acontinuum of disorder, from perfect crystals to maximallyrandom packings.35 Our experiments add to the evidence that acommon set of tools might be used to describe, and somedaypredict, the behaviour of a vast array of particulate solids withnearly any amount of disorder.

V. Experimental methodsA. Materials and sample preparation

Particles are sulphate latex (Invitrogen), with nominal diame-ters 4 and 6 mm. To prepare samples, we rinse and sonicate

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particles 4 times in deionised water, which yields more uniforminterparticle forces.16 They are then resuspended in a water–ethanol mixture (50% by volume), so that the suspension isviolently dispersed when added to the interface. To reduce polarcontamination that could screen repulsion, the decane super-phase (“99+%,” Acros Organics) is treated with activatedaluminium oxide powder (Alfa Aesar), which is then removed byltration (Qualitative no. 1, Whatman). The apparatus iscleaned before each experiment by repeated sonication andrinsing in deionised water and ethanol.

B. Particle sizes

The image radius of gyration rg of a particle is computed as

rgx

ðR0

r2Iðr; qÞr dr dqðR0

Iðr; qÞr dr dq

!1=2

(2)

where “x” represents a discrete (pixel-wise) approximation,and I(r, q) is the image intensity, at distance r and polar angle qrelative to the particle centroid. The integral is performed overthe portion of the image, with radius R, that encompasses theparticle.17,36

C. Quasistatic and 2D assumptions

For the material and our rheometry to be effectively 2-dimen-sional, the boundary conditions in the third dimension must beapproximately stress-free. This is quantied by the Boussinesqnumber Bq ¼ |h*|a/hl, where h* is the material's observedcomplex viscosity, a ¼ 230 mm is the needle diameter, and hl x10�3 Pa s is the oil and water viscosity.15 Here, Bq � 102, so thattypical stresses within the plane are much stronger than viscousdrag from the liquid bath. Our experiments may also beconsidered quasistatic, insofar as individual rearrangementsoccur on a timescale ((1 s) much shorter than a period ofdriving or the largest inverse strain rate (both�10 s). Consistentwith this assumption, we observe a weak dependence of rear-rangement activity on frequency (inset of Fig. 3(c)); other, longerrelaxation timescales may exist in the system, but we do notobserve their inuence.

D. Removal of spurious rearrangements

A temporary mis-identication of the particles can create theappearance of a rearrangement. Most such errors involve theabrupt, momentary displacement of a single, isolated particleby a large fraction of the interparticle spacing a. Genuine rear-rangements, by contrast, involve the motion of many nearbyparticles, especially when they generate D2

min far above thethreshold of 0.015. When counting rearranging particles forFig. 3(c), we therefore discard a particle if its D2

min exceeds themedian of its neighbours' by more than 0.2. This step nearlyeliminates spurious rearrangements without altering genuineones.

Because our apparatus features a long-distance microscope(Innity K2/SC) suspended over the sample and magnetic coils,

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Paper Soft Matter

the microscope is more sensitive to high-frequency transientvibrations than the sample itself. During the experiments withthe monodisperse packings, such vibrations intermittentlymademany particles blurred and indistinct. When the transientlasted for just a few video frames, the particle-tracking algo-rithm17 allowed us to safely discard those frames; when thetransient was more severe, we discarded the entire cycle, asevidenced in ESI Fig. 1.†

E. Computing C2c(r) and Cpl

2 (r)

Particles of interest are selected by their hysteretic rearrange-ments [for Cpl

2 (r)] or local sixfold symmetry |j6| [for C2j(r)].

Among these particles we nd clusters (in graph theory, con-nected components) that are connected topologically via near-est-neighbour relationships (i.e. separated by #1.5a). To avoidthe most tenuous clusters, we require that every cluster cansurvive the removal of any one particle.37 C2(r) is then dened asXi

NPiðrÞ=NSðrÞ, where NS(r) is the number of interparticle pair

distances of length r in the entire packing, and NPi(r) is the

number of pair distances of length r within the cluster i.30,31

Acknowledgements

We thank Denis Bartolo, Ludovic Berthier, Alison Koser, CarlGoodrich, Andrea Liu, Sal Torquato, and Martin van Hecke forhelpful discussions. This work was supported by the Penn NSFMRSEC (DMR-1120901).

References

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