Shear-Induced Reversibility of quasi-2D Colloids in Presence of Thermal Noise
aa r X i v : . [ c ond - m a t . s o f t ] O c t Shear-Induced Reversibility of 2D Colloidal Suspensions in the Pres-ence of Thermal Noise † Somayeh Farhadi ∗ , Paulo E. Arratia a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
The effects of thermal noise on particle rearrangements in cyclically sheared colloidal suspensions are experimentally investi-gated, using particle tracking methods. The experimental model system consists of polystyrene particles adsorbed at an oil-waterinterface, in which the particles exhibit small but non-negligible Brownian motion. We perform experiments on bidisperse (1.0and 1.2 µ m in diameter) colloidal samples with area fractions φ of 0.20 and 0.32. We characterize the reversibility of particlerearrangements, and show that unlike dense athermal systems, reversible clusters are not stable; once a particle enters into areversible trajectory, it has a non-zero probability of becoming irreversible in the following shearing cycle. This probability waspreviously found to be approximately zero for an analogous athermal system. We demonstrate that the stability of reversibilitydepends both on packing fraction, φ , and strain amplitude, γ . In addition, similar to previously studied athermal system, weidentify hysteresis in the dynamics of rearrangements for reversible particles, which indicates that such reversible rearrangementsare dissipative. However, at lower packing fractions, where the dynamics is moved closer to equilibrium by thermal noise, thishysteresis becomes less prominent. Particulate systems are ubiquitous in nature and in technol-ogy , and examples include pastes, paints, granular matter,foams, and metallic glasses . A common feature of thesematerials is their ability to flow under external load, whilemaintaining a solid (or jammed) state if unperturbed . Im-portantly, the macroscopic flow behavior of such particularsystems is a strong function of the material microstructure.Understanding the mechanism that governs the dynamics ofthe material microstructure is crucial in control and process ofthese materials.Colloidal suspensions have been widely used as a modelfor disordered molecular and atomic systems such as soft andmetallic glasses , where measuring individual particle posi-tions is a challenging task . In particular, colloidal suspen-sions can be very useful in the study of microscopic (particle-scale) fluctuations associated with mesoscale rearrangements,which subsequently affect the material properties such as bulkstiffness and plasticity . Particle fluctuations in colloidalsystems are governed by two distinct mechanisms: Brown-ian (thermal) motion and externally driven deformations (e.g.shear). A number of numerical simulations suggest that an ef-fective temperature exists for dense athermal systems, whichsets the energy scale of shear-induced fluctuations . Yet,with some notable exceptions , very few experiments havedirectly measured thermal fluctuations of colloids in presenceof shear. In this study, by shrinking the particle size to 1 µ m , where particles exhibit non-negligible Brownian motion, weadd a minimal thermal noise to particle dynamics, and studytheir trajectories under applied cyclic shear.In a recent study , the criteria for reversibility of rear-rangements under cyclic shear was comprehensively studiedfor a dense athermal colloid. In this manuscript, we character-ize the effects of thermal noise on the reversibility of particlerearrangements. We probe this effect for two systems with dif-ferent area fractions ( φ = .
32, and φ = . methods. In confocal microscopy, the time reso-lution of measurements is limited by scanning time, while forscattering the trajectories of individual particles are not avail-able, and instead only the correlated motion of a large groupof particles is measured. Here, by using a custom-made in-terfacial shearing apparatus , we shear and track nearly4 × particles adsorbed at an oil-water interface, and char-acterize reversibility, as well as the onset of rearrangement,for each particle. On the other hand, stroboscopic reversibil-ity gives us useful information on the fabric of configurational ||
32, and φ = . methods. In confocal microscopy, the time reso-lution of measurements is limited by scanning time, while forscattering the trajectories of individual particles are not avail-able, and instead only the correlated motion of a large groupof particles is measured. Here, by using a custom-made in-terfacial shearing apparatus , we shear and track nearly4 × particles adsorbed at an oil-water interface, and char-acterize reversibility, as well as the onset of rearrangement,for each particle. On the other hand, stroboscopic reversibil-ity gives us useful information on the fabric of configurational || nergy landscape. In particular, it manifests the existence ofenergy metabasins which restrict the dynamics of the system.Our data shows that the dynamics within these metabasins isstrongly affected in the presence of small thermal noise. The effects of thermal noise on particle rearrangements in col-loidal suspensions are investigated using a custom-made inter-facial shearing cell . In this apparatus, the particle suspen-sion is confined to a monolayer, which allows for the visual-ization of the evolution of the fluid microstructure by trackingindividual particles. As shown in Fig. 1(a,b), a thin magne-tized needle is embedded at the oil-water interface to be stud-ied, inside an open channel formed by 2 walls. An electromag-net forces the needle, creating a uniform shear stress on thematerial between the needle and the walls. The region fromwhich the data is acquired is near the center of the channeland sufficiently far from both tips of the needle, which avoidsthe non-uniform flow around the needle tips (more details be-low and elsewhere ).In order to create a well-controlled shearing flow, a smalldevice is built to hold and control the distance between twomicroscope cover-slips that serve as parallel walls (Fig. 1b).The device is placed inside a 10 cm diameter Petri dish andthen partially filled with DI water. A small amount of oil, de-cane 99+% from ACROS, is then carefully poured on top ofthe water, creating a thin oil layer and an (oil-water) interfacebetween the two coverslips. The magnetized needle is madefrom phosphate-coated carbon steel wire (from Mcmaster-Carr) and is 4 cm long, 0.15 mm in diameter, and 4 mg inmass. Prior to preparation, all of the parts are sonicated andrinsed with DI water, followed by a rinse with ethanol to avoidany aggregation inducing contaminations.Next, we inject particles to the oil-water interface. Due tothe high surface tension, particles are adsorbed and trappedat the interface, forming a stable particle monolayer .This monolayer is disordered because particle size distribu-tion is bidisperse, and also the charge distribution around eachparticle is non-uniform that results in asymmetrical particle-particle interaction. The magnetic needle is then carefullyplaced at the monolayer, and the needle’s weight is supportedby capillary forces. A static Helmholtz field keeps the nee-dle centered in the channel, while additional electromagnetsmove the needle back and forth, uniformly shearing the inter-face in the channel. The schematics of our custom made setupis shown in Fig. 1.The colloidal particles used in this experiment are sulfatelatex beads (8% w/v, Invitrogen). These microspheres arecharge-stabilized in DI water due to their surface sulphatetreatment. Once placed at the interface, they form dipoles witha long-range and repulsive interaction force. We use equal volumes of 1 µ m and 1.2 µ m particles to form bidisperse col-loidal mixtures. Here, bidispersity is used to model a disor-dered system, by preventing crystallization, and consequently,long-range ordering effects. We dilute 0 . . . By injecting different volumes of particle suspen-sion, we are able to control the area fraction at the interface. Inthis study, we provide data for two area fractions: φ = . φ = .
32. The area fractions are calculated as φ = A p A ,where A is a sampling region, and A p is the area spanning byparticles in that region. The particle monolayer is imaged witha long-distance inverted microscope (K2/SC Infinity Photo-Optical) and high-speed camera (Flare 4M180, IOIndustries);data is taken at 40, 60, or 80 fps. Within the recorded images(Fig. 1c), 1 µ m particles approximately span 6 pixels. We keepthe temperature constant (23 ◦ C) for all of the presented data,by confining the sample Petri dish in a glycerol-filled bath.For all experiments, we keep the needle frequency constantat f = G ∗ = | G ′ + iG ′′ | , where G ′ and G ′′ are respectively the elasticand loss moduli of the interface, and identify a region where G ∗ is independent of frequency . For our system, we findthat G ∗ begins to rapidly grow at approximately 0.2Hz, whichsets our upper limit for frequency. Also, as we move to lowerfrequencies below 0.05Hz, we begin to experience ambientvibrational noise. Due to these two issues, we chose 0.1Hzas our fixed experimental frequency. To consistently preparethe material for each experiment and to avoid memory effects,oscillatory forcing at large strain amplitude ( γ = .
0) is ini-tially performed for 3 cycles and then stopped. In the exper-iments presented here, the values of the strain amplitude ( γ )range from 0.01 up to 0.2. A sample video of the experimentis provided in the Supplementary Materials. The images arethen processed using trackpy , a python based particle track-ing package . For each data set, nearly 4 × particles areidentified and tracked. Particle trajectories are recorded up to60 cycles. We begin our analysis of thermal effects on particle rearrange-ments by characterizing particle diffusivity in the sample. Weuse particle mean squared displacement,
MSD y ( τ ) = < ( y ( t + τ ) − y ( t )) > , to characterize the particle diffusion in the qui-escent states (i.e., in the absence of shear). Here, y is the com-ponent of particle position perpendicular to the wall, and τ is | olding block SpacerCoverslipNeedle
OilWaterHelmholtz coils (a)(b) (c) y x
Fig. 1 a) Schematics of the shearing cell. b) Schematics of thespacer apparatus (top view). c) Mixture of 1 . . µ m particlesvisualized under microscope. The area fraction is φ = . the time-lapse. Fig. 2a shows the MSD y as a function of shearcycle τ for colloidal suspensions of volume fraction 0.20 and0.32; as expected, particles in the dilute suspension ( φ = . φ = . MSD y = D ∞ τ . Wefind that D ∞ = . d / c = ( . µ m / s ) for φ = . D ∞ = . d / c = ( . µ m / s ) for φ = .
20, where d is the particle diameter and c is the time scale of one shearingcycle.The relative importance of convection (or flow) to diffusionin a system is usually characterized by the P´eclet number. Wedefine a modified P´eclet number for sheared states as Pe ⋆ = | ˙ γ | d / D ∞ , which is the ratio of the flow time scale, | ˙ γ | , anddiffusion time scale, D ∞ d . Here, d is the particle diameter,and | ˙ γ | is the strain rate magnitude averaged over a completeshear cycle. For a sinusoidal strain imposed by the needle, γ = γ sin ( π T t ) , the average shear rate of the needle over one cycleis | ˙ γ | = γ T . This gives the range of modified P´eclet numbersfor the probed strain amplitudes presented in this manuscriptas: 0 . < Pe ⋆ < . φ = .
20, and 10 . < Pe ⋆ < . φ = .
32. While φ = .
32 could be considered in an athermalregime, for φ =
20, the thermal effects are non-negligible.Also note that
MSD / − y for φ = .
20 monotonically in-creases vs. time-lapse, which signifies a weak caging ef-fect ( α -relaxation). For the high packing fraction ( φ = . MSD y starts to plateau before reaching its diffusive regime.This plateau indicates that caging starts to dominate the dy-namics, and at φ = .
32, the system is close to its glass tran-sition point .Next, we quantify particle diffusivity in the lateral directionin the presence of shear by computing the root-mean-squaredisplacement rmsd y for a time scale equivalent to one shear-ing cycle (i.e. f = s ). Fig. 2b shows the quantity rmsd y as a τ (cycles) -2 M S D y ( d ) -2 φ =0.20 φ =0.32y=0.082xy=0.007x γ r m s d y ( d ) φ =0.20 φ =0.32 Fig. 2 a) Mean squared displacement of particles in quiescent state,measured for transverse to shear direction (from the y component ofdisplacements). b) Root mean squared displacement in y direction,transverse to shear. Here, rmsd y s are calculated for a time equivalentof one shearing cycle. The length unit, d, is 1 µ m , or the diameter ofsmall particles, and the time unit is one shearing cycle (10 s ). function of strain amplitude γ for the φ = .
20 and φ = . rmsd y increases monoton-ically with strain amplitude for both samples. However, theincrease is much weaker for the system with lower packingfraction ( φ = . φ = .
32 sam-ple show larger displacements under shear than a more dilutesample; in the limit of very low shear (or strain amplitude),the opposite is observed.A key advantage of this interfacial shear cell is that one canobtain detailed imaging of colloidal suspension microstructurewhile undergoing cyclic shear. We will now focus on stro-boscopic rearrangements, which we define as particles thatchange neighbors after completing a shearing cycle. Thechange in neighbors is measured stroboscopically by samplingat times t n = t + n πω − , n = { , , , ... } , so that we comparethe beginning and end of a full period of driving that straddles t n and t n + (Fig. 5c). Particle rearrangements are characterizedby D min = Σ ( r ′ i − E r i ) , which quantifies the mean-squared de-viation of particle displacements from the best-fit affine defor-mation of particles during a time interval ∆ t . Sufficientlyhigh values of D min denote rearrangements . The associa-tion of high values of D min with rearrangements enables us toidentify the events in which particles change neighbors, and inother words, break their cages. For a given particle p , r ′ i and r i ||
32 sam-ple show larger displacements under shear than a more dilutesample; in the limit of very low shear (or strain amplitude),the opposite is observed.A key advantage of this interfacial shear cell is that one canobtain detailed imaging of colloidal suspension microstructurewhile undergoing cyclic shear. We will now focus on stro-boscopic rearrangements, which we define as particles thatchange neighbors after completing a shearing cycle. Thechange in neighbors is measured stroboscopically by samplingat times t n = t + n πω − , n = { , , , ... } , so that we comparethe beginning and end of a full period of driving that straddles t n and t n + (Fig. 5c). Particle rearrangements are characterizedby D min = Σ ( r ′ i − E r i ) , which quantifies the mean-squared de-viation of particle displacements from the best-fit affine defor-mation of particles during a time interval ∆ t . Sufficientlyhigh values of D min denote rearrangements . The associa-tion of high values of D min with rearrangements enables us toidentify the events in which particles change neighbors, and inother words, break their cages. For a given particle p , r ′ i and r i || re the positions of its neighbouring particles before and afterapplying a deformation. Usually, the neighbouring group ofparticles is chosen as all of the particles centered within 2.5 σ aa relative to the center of particle p , where σ aa is the dis-tance corresponding to the first peak of pair correlation func-tion, g ( r ) . Note that σ aa , which represents the average inter-particle spacing, becomes larger as the packing fraction, φ , isreduced. E is the least squares fit, which transforms r i to r ′ i affinely. With this definition, D min is a measure of nonaffinityof particle displacements, and its value quantifies the extent ofrearrangement for a given particle.Fig. 3 shows snapshots of the spatial distribution of strobo-scopically rearranging particles for a colloidal suspension ofvolume fraction φ = .
20 for γ = .
017 (Fig. 3a) and γ = .
158 (Fig. 3b). Particles colored in red undergo irreversiblerearrangements while particle colored in blue undergo re-versible motion (i.e., no stroboscopic rearrangements). Here,we take D min > .
015 as the threshold for rearrangement (thisthreshold is set by the noise level in our D min calculation).For low strain amplitude ( γ = . γ = . D min ( C rD min ) for particlesundergoing irreversible rearrangements (colored red). Thequantity C rD min for different values of strain amplitude γ , for φ = .
20 and φ = .
32, are shown in Fig. 4a and Fig. 4b, re-spectively. For φ = .
20, the correlation length remains nearlyidentical as γ is increased, except for the largest amplitudecase. However, for φ = .
32, the correlation length is highlydependent on the strain amplitude. This data shows that aswe move to a denser state, stroboscopic rearrangements aremostly governed by external driving (i.e. shear), and are lessaffected by thermal noise. Similar behavior was previouslyobserved in simulations as the system was moved from sheardominated to thermal dominated regime .We have shown that even at large strain amplitudes, thereare still quite a few stroboscopically reversible particles orreversible rearrangements (Fig. 3b). Do these particles un-dergo rearrangements within a given cycle? Here, we inves-tigate the existence and statistics of such rearrangements bymeasuring the quantity D min for rearrangements between thetwo peaks of the sinusoidal strain cycle, or the “peak-to-peak D min ”. For this, we study particle positions at times t n = t + n πω − , n = { , , , ... } (Fig. 5c). We find that (for allexperiments presented here) a fraction of particles which ap- xy Fig. 3
Rearranging (Red), and nonrearranging (blue) particles for γ = .
017 (left), and γ = .
158 (right). The packing fraction is φ = .
20, and both distributions are plotted for cycle 15. A fractionof particles are still reversible, even for a relatively high strain valueof γ = . × pear stroboscopically reversible, do rearrange ( D min > . φ = .
20 and φ = .
32 as a function of strain amplitude. Wefind that for most cases, the fraction of particles undergoingpeak-to-peak rearrangement is larger than stroboscopic rear-rangement; that is, a significant number of particle rearrange-ments are reversible. As the strain amplitude is increased, thefraction of particles undergoing both types of rearrangementsalso increase. Note that the fractions of reversible particles(fraction of stroboscopic rearrangements subtracted from frac-tion of peak-to-peak rearrangements) are relatively smallerfor φ = .
20 compare to φ = .
32 (Fig. 5a,b). Also, thepeak-to-peak rearrangements behave very similarly for bothpacking fractions, while the stroboscopic rearrangements arehighly dependent on the packing fraction, particularly for thelow strain amplitudes. This suggests that, while cyclic driv-ing at small strain amplitudes self-organizes the system to re-versible states, thermal noise counter-affects the organization,and makes the dynamics irreversible.Previous studies on a jammed athermal system, identifiednon-rearranging (reversible) particles which were dissipatingenergy, and were referred to as ‘plastic reversible’ . Aplastic reversible particle undergoes a rearrangement whichcompletely reverses within a strain cycle, yet the rearrange-ment path has hysteresis. We now investigate whether the re-versible rearrangements found here are also plastic. In orderto identify hysteresis, we define the strain at which a givenparticle starts to rearrange as γ on , and the strain at which theparticle stops rearranging as γ o f f . Consider a strain cycle γ ( t ) (Fig. 5c). We then take the γ ( t = ) as the referenceframe, and calculate D min values for all of the frames in thecycle with respect to this reference frame. Using the threshold D min = .
015 as the threshold for rearrangement, we obtain γ on > γ ( ) as the last strain where D min < . γ o f f asthe first strain where γ o f f > γ on , and D min < . γ on and γ o f f to be in the first and second halves of the γ | ( σ aa ) γ (cid:5)=(cid:6).(cid:7)(cid:8)7 γ (cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15) γ (cid:16)(cid:17)(cid:18)(cid:19)(cid:20)5(cid:21) γ (cid:22)(cid:23)(cid:24)(cid:25)(cid:26)(cid:27)(cid:28) γ (cid:29)(cid:30)(cid:31) !8" γ φ ,3:;< (a) r ( σ aa ) C r D m i n -1 γ =0.018 γ =0.027 γ =0.048 γ =0.063 γ =0.097 γ =0.151 φ =0.32 (b) Fig. 4
Spatial correlation of D min ( C rD min )for φ = .
20 and φ = . γ is increased. σ aa is the length scale associated with the firstpeak of g ( r ) , which quantifies the average interparticle spacing foreach system. function. Note that for a hysteretic particle, γ on − γ o f f = R rev , hyst = N rev , hyst N rev , where, N rev is the number ofall reversible particles and N rev , hyst is the number of all re-versible particles which exhibit hysteresis in their rearrange-ment path. We have excluded the first 3 transient cycles fromthe statistics. The data clearly indicate the existence of plasticreversible paths even in the presence of thermal noise. How-ever, the fraction of plastic reversible particles is considerablysmaller for φ = .
20 compared to φ = .
32. And, althoughirreversible particles are more frequent for φ = .
20, the hys-teresis of rearrangements for its reversible particles is signifi-cantly lower than for φ = .
32. A possible explanation of thisbehavior is that, since the thermal fluctuations are stronger atlower packing fractions, the dynamics is closer to equilibrium,and hence, non-dissipative. Similar to a glass forming liquid,as the packing fraction is lowered below glass point, the cagesare broken, and the particles escape the arrested dynamics anddiffuse. This diffusive process allows the system to explore alarger region in configurational energy landscape, and conse-quently, moves the system closer to equilibrium. By increas-ing the packing fraction, thermal equilibration is hindered and -2 -1 r >a rr ? n g i n @f r Act i o n -2 -1 pBEk - FG - HIJKsL r M b NOPQR i ST =0.20 γ -2 -1 R r e v , h ys t -4 -2 U =0.32 V =0.20 γ WXY Z[\] γ -2 -1 -2 -1 ^ =0.32 γ r _‘ rr b n d i n hj r lqu i v n wxyz{|}~(cid:127)(cid:128)(cid:129)(cid:130)(cid:131)(cid:132)(cid:133)(cid:134)(cid:135)(cid:136)(cid:137)(cid:138)(cid:139)(cid:140)(cid:141)(cid:142) (b) (cid:143)(cid:144)(cid:145)(cid:146)(cid:147)(cid:148) Fig. 5
Fraction of stroboscopic and peak-to-peak rearrangingparticles for (a) φ = .
20, and b) φ = .
32. c) Fraction of hystereticreversible particles ( R rev , hyst ).d) schematics of peak-to-peak andstroboscopic sampling. the system regains out-of-equilibrium and athermal properties.For dense athermal systems, once a particle enters into a re-versible cycle it remains reversible in the upcoming cycles .In contrast, thermal noise allows the particles to escape re-versible cycles with a certain probability (see SupplementaryMaterials). Fig. 6a shows the transition probabilities (TP)between the reversible/irreversible states for φ = .
20, and φ = .
32 as a function of strain amplitude. The TPs are mea-sured by counting the number of particles transitioning froma particular state at cycle t to another state at cycle t +
1, andaveraged over all cycles. We exclude 3 initial cycles to avoidtransient behavior (TPs do not change significantly over non-transient cycles). The notation is as follows: p [ I t + | R t ] is thetransition probability (TP) of a particle undergoing reversiblerearrangement at time t to switch to an irreversible rearrange-ment at time t +
1; similarly, p [ R t + | I t ] is the transition proba-bility (TP) of a particle undergoing irreversible rearrangementat time t to switch to an reversible rearrangement at time t + S = p [ R t + | I t ] − p [ I t + | R t ] (Fig. 6b). Wefind that reversible cycles at lower values of γ and irreversiblecycles at higher γ s are more stable. However, we also observethat at lower packing fraction, where thermal noise is signifi-cant, the stabilities of both reversible and irreversible particles(respectively, at lower and higher γ ) are hindered in compari-son with the denser system. This is consistent with our earlierobservation (Fig. 4) that the dynamics of reversibility is less ||
1; similarly, p [ R t + | I t ] is the transition proba-bility (TP) of a particle undergoing irreversible rearrangementat time t to switch to an reversible rearrangement at time t + S = p [ R t + | I t ] − p [ I t + | R t ] (Fig. 6b). Wefind that reversible cycles at lower values of γ and irreversiblecycles at higher γ s are more stable. However, we also observethat at lower packing fraction, where thermal noise is signifi-cant, the stabilities of both reversible and irreversible particles(respectively, at lower and higher γ ) are hindered in compari-son with the denser system. This is consistent with our earlierobservation (Fig. 4) that the dynamics of reversibility is less || -2 -1 (cid:150) r (cid:151) b (cid:152) b i (cid:153) i (cid:154)(cid:155) (cid:156)(cid:157)(cid:158) (cid:159)(cid:160) ¡¢£⁄¥ƒ§ ¤' “«‹› φ =0.32 γ -2 -1 fi r fl b (cid:176) b i – i †‡ ·(cid:181)¶ •‚ „”»…‰(cid:190)¿ (cid:192)` ´ˆ˜¯ φ =0.20 -2 -1 ˘ -1-0.500.51 φ =0.32 φ =0.20 γ ˙¨(cid:201) (b) Fig. 6 a) Transition probabilities between reversibility states as afunction of strain amplitude, γ . Note that the probability ofirreversible rearrangement increases as γ is increased. b) S r = p [ R t + | I t ] − p [ I t + | R t ] quantifies the stability of reversiblestates. affected by external driving as we increase the thermal noisein the system.One can understand the reversible-irreversible TPs withinconfigurational energy landscape framework. Initially, a lo-cal energy minimum is occupied (Fig. 7a). Application ofshear distorts and flattens the local minimum such that thesystem jumps to a nearby minimum state . For athermalparticles, by reversing shear, the energy landscape resumes itsoriginal shape, and the system jumps back to the initial localminimum (reversible regime). This implies the existence ofa metabasin in the energy landscape which confines the dy-namics to a small region in configurational space . By in-creasing the shear amplitude, the distortion of the energy land-scape is large enough, such that the system escapes from themetabasin and the trajectories become irreversible. If, in addi-tion to shear, the particles are also thermally activated, the sys-tem has a finite probability to escape the metabasin (Fig. 7b).As the thermal energy increases, the probability of escapingthe metabasin, and hence irreversibility, also increases. In this manuscript, we presented an experimental investiga-tion on particle rearrangements of a 2D colloidal system un-der cyclic shear in the presence of small but finite thermalnoise. To our knowledge, this is the first experimental studythat directly measures thermal fluctuations under shear for alarge number of particles (4 × ) and for relatively longtimes. Experiments are performed using a custom-made in-terfacial shearing cell, and colloidal samples with area frac-tions φ = .
20 and φ = .
32. The thermal noise in the col-loidal suspensions was quantified by the diffusion constant, D ∞ , extracted from the long time mean squared displacements Shear forwardShear reversed Shear forwardShear reversed
Themal activation (a) (b)
Fig. 7
Schematics of the state dynamics as a result of shear inducedactivation for a) athermal, and b) thermal systems. (Fig. 2).The relative importance of thermal motion to applied flowin our system is quantified using the modified P´eclet number,which ranges from 0.8 to approximately 80 and straddles thethermal and athermal regimes. We find that minimal levelsof thermal noise, which is usually negligible in steady shear,has non-negligible effects on the reversibility of particle re-arrangements under cyclic driving even for relatively large Pe ⋆ ( ∼ showed thatthe instant vanishing of flow at turning points, which movesthe system towards low P´eclet regime, could affect the rheol-ogy of particle suspensions in strongly nonlinear flows. Thisvanishing P´e at turning points (i.e. oscillation peaks), andhence the dominance of diffusion over shear, could also bea possible explanation for the dynamics we observe, specif-ically the fraction and the instability of reversible rearrange-ments, compared to an athermal system. We also show thateven though the thermal noise is slightly lower in the quies-cent state of the denser system ( φ = . φ = . D min , which quantifies the mean-squared deviation of particleposition at time t from the best-fit affine deformation of itsneighborhood during a time interval ∆ t . Rearrangementsare characterized by having high D min values. We find that thecorrelation length of rearrangements remains almost constantfor the low packing fraction case φ = .
20, while for φ = . | s, plastic reversibility is diminished as we move the system(slightly) towards thermal equilibration, here by lowering φ .We also observed that, for thermally activated systems, un-like jammed athermal systems, the particles escape from re-versible states with a certain probability which depends bothon packing fraction and strain amplitude. We find that for suf-ficiently small strain amplitudes, the reversible cycles are rel-atively more stable for higher packing fractions. This suggeststhat as the liquid is moved close to its glass transition point,the combination of thermal energy and shear is not sufficientto overcome the energy barrier and rearrange the particles to-wards equilibrium.We would like to thank N. Keim, A. Yodh, and A. Liu forhelpful discussions and suggestions, as well as Madhura Gur-jar for the experimental setup design. This work was partiallysupported by NSF-Penn-MRSEC-DMR-1120901. Acknowl-edgment is made to the Donors of the American ChemicalSociety Petroleum Research Fund for partial support of thisresearch. References , 301-322 (2010).3 M. Chen, Annu. Rev. Mater. Res. , 445-469 (2008).4 M. Van Hecke, J. Phys. Condens. Matter , 1-75 (2015).5 C. Royall, S. Williams, Phys. Rep. , 1-75 (2015).6 L. Berthier, G. Biroli, Rev. Mod. Phys. , 587 (2011).7 H.W. Sheng, W.K. Luo, F.M. Alamgir, J.M. Bai, E. Ma,Nature , 419-425 (2006).8 D. Chen, D. Semwogerere, J. Sato, V. Breedveld, E.R.Weeks, Phys. Rev. E , 011403 (2010).9 T.K. Haxton, A.J. Liu, Phys. Rev. Lett. , 195701 (2007).10 I.K. Ono, C.S. OHern, D.J. Durian, S.A. Langer, A.J. Liu,S.R. Nagel, Phys. Rev. Lett. , 095703 (2002).11 V. Chikkadi, P. Schall, Phys. Rev. E , 031402 (2012).12 E. Leutheusser, Phys. Rev. A , 2765 (1984).13 L. Berthier, J.L. Barrat, Phy. Rev. Lett. , 095702 (2002).14 R. Yamamoto, A. Onuki, Phys. Rev. E , 3515 (1998).15 N.C. Keim and P.E. Arratia, Soft Matter , 6222 (2013).16 N.C. Keim, P.E. Arratia, Phys. Rev. Lett. , 028302(2014).17 N.C. Keim and P.E. Arratia, Soft Matter , 1539 (2015).18 E.R. Weeks, D.A. Weitz, Phys. Rev. Lett. , 095704(2002).19 M.T. Dang, D. Denisov, B. Struth, A. Zaccone, P. Schall,Eur. Phys. J. E , 44 (2016).20 L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D. El Masri, D. L’Hˆote, F. Ladieu, M. Pierno, Science , 1797(2005).21 E. Weeks, D. Weitz, Chem. Phys. , 361 (2002).22 V. Prasad, D. Semwogerere, E.R. Weeks, J. Phys. Con-dens. Matter , 113102 (2007).23 E.R. Weeks, J.C. Crocker, AC. Levitt, A. Schofield, D.A.Weitz, Science , 627 (2000).24 G. Petekidis, A. Moussad, P. N. Pusey, Phys. Rev. E ,051402 (2002).25 trackpy: http://dx.doi.org/10.5281/zenodo.34028.26 M.L. Falk, J.S. Langer, Phys. Rev. E , 7192 (1998).27 V. Chikkadi, S. Mandal, B. Nienhuis, D. Raabe, F. Varnik,P. Schall, Europhys. Lett. , 56001 (2012).28 D.J. Lacks, Phys. Rev. Lett. , 225502 (2001).29 I. Regev, J. Weber, C. Reichhardt, KA. Dahmen, T. Look-man, Nat. Commun. , 9805 (2015).30 M.D. Ediger, P. Harrowell, J. Chem. Phys. , 080901(2012).31 V.J. Anderson, HNW. Lekkerkerker, Nature , 811(2002).32 R. M¨obius, C. Heussinger, Soft Matter , 4806 (2014).33 S. Reynaert, P. Moldenaers, J. Vermant, Phys. Chem.Chem. Phys. , 6313 (2007).34 C.F. Brooks, G.G. Fuller, C.W. Frank, C.R. Robertson,Langmuir , 2450 (1999).35 S.S. Schoenholz, A.J. Liu, R.A. Riggleman, J. Rottler,Langmuir , 031014 (2014).36 R. Candelier, A. Widmer-Cooper, J.K. Kummerfeld, O.Dauchot, G. Biroli, P. Harrowell, D.R. Reichman, Phys.Rev. Lett. , 135702 (2010).37 A.S. Khair, J.Fluid. Mech. , R5 (2016). ||