Universality of Jamming Criticality in Overdamped Shear-Driven Frictionless Disks
UUniversality of Jamming Criticality in Overdamped Shear-Driven Frictionless Disks
Daniel V˚agberg, Peter Olsson, and S. Teitel Department of Physics, Ume˚a University, 901 87 Ume˚a, Sweden Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 (Dated: November 14, 2018)We investigate the criticality of the jamming transition for overdamped shear-driven frictionlessdisks in two dimensions for two different models of energy dissipation: (i) Durian’s bubble model withdissipation proportional to the velocity difference of particles in contact, and (ii) Durian’s “mean-field” approximation to (i), with dissipation due to the velocity difference between the particle andthe average uniform shear flow velocity. By considering the finite-size behavior of pressure, thepressure analog of viscosity, and the macroscopic friction σ/p , we argue that these two models sharethe same critical behavior.
PACS numbers: 45.70.-n 64.60.-i 64.70.Q-
Many different physical systems, such as granular ma-terials, suspensions, foams and emulsions, may be mod-eled in terms of particles with short ranged repulsivecontact interactions. As the packing fraction φ of suchparticles is increased, the system undergoes a jammingtransition from a liquid state to a rigid but disorderedsolid. It has been proposed that this jamming transitionis a manifestation of an underlying critical point, “pointJ”, with associated scaling properties such as is foundin equilibrium phase transitions [1, 2]. Scaling proper-ties are indeed found when such systems are isotropicallycompressed, with pressure, elastic moduli, and contactnumber increasing as power laws as φ increases abovethe jamming φ J [3]. When such systems are shearedwith a uniform strain rate ˙ γ , a unified critical scalingtheory has successfully described both the vanishing ofthe yield stress as φ → φ J from above, the divergenceof the shear viscosity as φ → φ J from below, and thenon-linear rheology exactly at φ = φ J [4].One of the hallmarks of equilibrium critical points isthe notion of universality ; the critical behavior, specifi-cally the exponents describing the divergence or vanish-ing of observables, depend only on the symmetry anddimensionality of the system, and not on details of thespecific interactions. For statically jammed states cre-ated by compression, where only the elastic contact inter-action comes into play, it is understood that the relevantcritical exponents are simply related to the form of theelastic interaction, and are all simple rational fractions[3]. In contrast, shear-driven steady states are formed bya balance of elastic and dissipative forces, and it is thusan important question whether or not the specific formtaken for the dissipation is crucial for determining thecritical behavior.In a recent work by Tighe et al. [5], it was claimedthat changing the form of the dissipation can indeed al-ter the nature of the criticality for sheared overdampedfrictionless disks. In contrast to earlier work [4], whereparticle dissipation was taken with respect to a uniformlysheared background reservoir, Tighe et al. used a col- lisional model for dissipation. They argued that thischange in dissipation resulted in dramatically differentbehavior from that found previously, specifically (i) thereis no length scale ξ that diverges upon approaching φ J ,and so behavior can be described analytically with amean-field type model; (ii) critical exponents are simplerational fractions; (iii) there is no single critical scaling,but rather several different flow regimes, each with a dif-ferent scaling. In this work we numerically re-investigatethe model of Tighe et al. and present results arguingagainst these conclusions. In particular we conclude thatthe two models have rheology that is characterized by the same critical exponents, and so are in the same criticaluniversality class.We simulate bidisperse frictionless disks in two dimen-sions (2D), with equal numbers of big and small diskswith diameter ratio 1.4, at zero temperature. The in-teraction of disks i and j in contact is V ij = k e δ ij / δ ij = r ij /d ij −
1, with d ij thesum of the disks’ radii. The elastic force on disk i is f el i = −∇ i (cid:80) j V ij , where the sum is over all particles j in contact with i . We use Lees-Edwards boundary con-ditions [6] to introduce a time-dependent uniform shearstrain γ ( t ) = ˙ γt in the ˆ x direction.We consider two different models for energy dissipa-tion. The first, which we call “contact dissipation” (CD),is the model introduced by Durian for bubble dynamics infoams [7], and is the model used by Tighe et al. [5]. Heredissipation occurs due to velocity differences of disks incontact, f disCD ,i = − k d (cid:88) j ( v i − v j ) , v i = ˙ r i . (1)In the second, which we call “reservoir dissipation” (RD),dissipation is with respect to the average shear flow of abackground reservoir, f disRD ,i = − k d ( v i − v R ( r i )) , v R ( r i ) ≡ ˙ γy i ˆ x. (2)RD was also introduced by Durian [7] as a “mean-field”[8] approximation to CD, and is the model used in many a r X i v : . [ c ond - m a t . s o f t ] S e p earlier works on criticality in shear driven jamming [4, 8–10].The equation of motion for both models is m i ˙ v i = f el i + f dis i . (3)Here we are interested in the overdamped limit, m i → m i = 0, inwhich case the equation of motion becomes simply v i = v R ( r i ) + f el i /k d ; we call this limit RD . In CD, becausethe dissipation couples velocities one to another, setting m i = 0 effectively requires inverting the matrix of con-tacts to rewrite the equation of motion in a form suit-able for numerical integration. Instead of that numer-ically difficult procedure, our approach here is to sim-ulate particles with a finite mass, and verify that themass is small enough for the system to be in the over-damped m i → . For oursimulations we use units in which k e = k d = 1, lengthis in units such that the small disk diameter d s = 1,time in units of τ ≡ k d d s /k e = 1, and particles of equalmass density, such that m i for a particle of diameter d i is m i = 2 mπd i /
4, with m = 1. In our SupplementalMaterial [11] we confirm that this choice is sufficient tobe in the m i → our simulations use N = 65536 particles, while for CD we use N = 262144,unless otherwise noted. For RD our slowest strain rateis ˙ γ = 10 − , while for CD we can reach only ˙ γ = 10 − .Before presenting our evidence that the two modelsRD and CD have the same critical rheology, we firstcomment on one quantity that is clearly very differentin the two models, the transverse velocity correlation, g y ( x ) ≡ (cid:104) v y (0) v y ( x ) (cid:105) . In RD g y ( x ) shows a clear min-imum at a distance x = ξ , and this length diverges asone approaches the critical point ( φ J , ˙ γ →
0) [4]. In CD however, it was found [5] that g y ( x ) decreases monotoni-cally without any obvious strong dependence on either φ or ˙ γ . This led Tighe et al. to conclude that there is nodiverging length ξ in model CD , that the only macro-scopic length scale is the system length L , and thus thereare no critical fluctuations. In our own work we have con-firmed this dramatic difference in the behavior of g y ( x ),but see our Supplemental Material for further comments[11].However the apparent absence of a diverging ξ in g y ( x )for CD does not necessarily imply that such a diverg-ing length does not exist. In the following we presentevidence for such a diverging ξ in CD by consideringthe finite-size dependence of the pressure p as a functionof strain rate ˙ γ at φ J . By a critical scaling analysis ofthe pressure analogue of viscosity, η p ≡ p/ ˙ γ , we furthershow that the rheology in CD is characterized by thesame critical exponents as is RD . Finally we considerthe macroscopic friction µ ≡ σ/p in the two models, with σ the shear stress, and show that they behave similarly.There is no sign of the roughly square root vanishing of µ at φ J that would be expected from the model of Tighe -4 -3 -2 p L RD : open symbols ! = 10 " . ! = 10 " . ! = 10 " . ! = 10 " . ! = 10 " . (a) = 0.8433CD : closed symbols -9 -8 -7 -6 -5 -4 -3 $ ! . (b) RD : open symbols $ ~ ! " z . z = 5.6 CD : closed symbols FIG. 1. (a) Finite size behavior of pressure p in model RD (open symbols) and model CD (closed symbols) at differentstrain rates ˙ γ . For ˙ γ = 10 − we only have results for modelRD . The crossover from power law behavior at small L toa constant at large L determines the correlation length ξ ,plotted vs ˙ γ in (b). We see that ξ is essentially identical inboth models, growing monotonically as ˙ γ decreases, reachingvalues as large as ξ ≈
20 for our smallest ˙ γ . As L varies, thenumber of particles varies from N = 24 to 4096. et al., but rather µ appears to be finite passing through φ J , as found in recent experiments on foams [12]. Finite-size dependence of pressure : We consider hereonly the elastic part of p which is computed from theelastic contact forces in the usual way [3]. If jammingbehaves like a critical point, we expect p to obey finite-size-scaling at large system lengths L [13], p ( φ, ˙ γ, L ) = L − y/ν P (( φ − φ J ) L /ν , ˙ γL z ) . (4)Exactly at φ = φ J the above becomes [14] p ( φ J , ˙ γ, L ) = L − y/ν P (0 , ˙ γL z ) . (5)For sufficiently small L , where ˙ γL z (cid:28)
1, we get p ∼ L − y/ν . For sufficiently large L , where ˙ γL z (cid:29) p be-comes independent of L and so p ∼ ˙ γ y/zν . The crossoveroccurs when L = ξ at ˙ γL z ≈ ⇒ ξ ∼ ˙ γ − /z , giving adiverging correlation length as ˙ γ → p vs. L for RD and CD at φ =0 . ≈ φ J for several different ˙ γ . Both models clearlybehave similarly. To determine the crossover ξ we fit ourdata to the simple empirical form p = C (1 + [ ξ/L ] x ) thatinterpolates between the two asymptotic limits. This fitgives the solid lines in Fig. 1(a). The resulting ξ is plot-ted in Fig. 1(b). We see that ξ is essentially identical inthe two models, growing monotonically as ˙ γ decreases,reaching values as large as ξ (cid:39)
20 for our smallest ˙ γ .Such a large length, many times the microscopic lengthset by the particle size, is clear evidence for cooperativebehavior [15]. Thus our results indicate a growing macro-scopic length scale in CD , just as was found for RD .The exponent z ≈ . L considered here [16], and the neglectof such corrections can skew the resulting effective expo-nents away from their true values at criticality. See ourSupplemental Material [11] for a more in depth discus-sion. Pressure analog of viscosity : We now seek to computethe critical exponents of the two models RD and CD ,to see if they are indeed in the same universality class.To do this we consider data at various packing fractions φ and strain rates ˙ γ close to the jamming transition. Weuse system sizes large enough ( N = 65536 for RD and N = 262144 for CD ), such that finite size effects arenegligible for the data presented here. As in our recentwork on RD [13], we consider here the pressure analogof viscosity η p ≡ p/ ˙ γ , since corrections to scaling aresmaller for p than for shear stress σ [13].To extract the jamming fraction φ J and the criticalexponents we use a mapping from our system of soft-core disks to an effective system of hard-core disks. Wehave previously shown [17] this approach to give excel-lent agreement with results from a more detailed twovariable critical scaling analysis [13] for RD . We use ithere because it requires no parameterization of an un-known crossover scaling function and so is better suitedparticularly to CD where the range of our data is morelimited (10 − ≤ ˙ γ ) as compared to RD (10 − ≤ ˙ γ ).This method assumes that the soft-core disks at φ and˙ γ can be described as effective hard-core disks at φ eff ( ˙ γ ),by modeling overlaps as an effective reduction in particleradius [17]. Measuring the overlap via the average energyper particle E , we take φ eff = φ − cE / y , (6)where y is the exponent with which the pressure rises as φ increases above φ J along the yield stress curve ˙ γ → c is a constant. We can then expressthe viscosity of this effective hard-core system as, η p ( φ, ˙ γ ) = η hd p ( φ eff ) = A ( φ J − φ eff ) − β . (7)Our analysis then consists of adjusting φ J , the exponents y and β , and the constants c and A in Eqs. (6) and (7),to get the best possible fit to our data.In Fig. 2 we show the results of such an analysis forCD . Panel (a) shows our raw data for η p vs. φ , for sev-eral different ˙ γ , in the narrow density interval around φ J that is used for the analysis. Panel (b) shows the resultfrom fitting η p for ˙ γ ≤ − to Eq. (7). Our fitted values φ J = 0 . β = 2 . ± . y = 1 . ± .
05 for CD areall very close to our earlier results for RD ( φ J = 0 . β = 2 . ± . y = 1 . ± .
01) [17] thus suggestingthat the critical behavior in CD is the same as in RD .In quoting the fitted values of φ J , β and y we note thatour results for RD include data to much lower strainrates 10 − ≤ ˙ γ as compared to CD where 10 − ≤ ˙ γ .For a more accurate comparison of the two models, weshould fit our data over the same range of strain rates ˙ γ .We therefore carry out a fitting to Eqs. (6) and (7) usingdata in the interval ˙ γ min ≤ ˙ γ ≤ ˙ γ max . In Fig. 3 we show × − × − × − × − × − × − × − ˙ γ (a) CD φ η p φ J = 0 . β = 2 . y = 1 . ∼ ( φ J − φ eff ) − β (b)CD φ eff = φ − cE / y η p FIG. 2. Pressure analog of viscosity η p ≡ p/ ˙ γ for model CD .Panel (a) shows the raw data for ˙ γ = 10 − through 10 − in anarrow interval of φ about φ J . Solid lines interpolate betweenthe data points. Panel (b) shows our scaling collapse of η p vs. φ eff , with φ J , β and the parameters in Eqs. (6) and (7)determined from the analysis of data with ˙ γ ≤ − . Thesolid line is the fitted power law scaling function. our results for φ J and β , where we plot the fitted valuesvs. ˙ γ min for two different fixed values of ˙ γ max = 1 × − and 2 × − . For RD we can extend this proceduredown to ˙ γ min = 10 − , while for CD we are limited to˙ γ min = 10 − . We see that for equivalent ranges of ˙ γ ,the fitted values of β agree nicely for the two models, forthe smaller value of ˙ γ max . We see that φ J for CD is justslightly higher than for RD . We cannot say whether thisis a systematically significant difference, or whether φ J would decrease slightly to match the value found for RD were we able to study CD down to comparably small ˙ γ .Whether or not the φ J of the two models are equal, orjust slightly different, the equality of the exponents β strongly argues that models RD and CD are in thesame critical universality class.To return to the results of Tighe et al. [5], we note thattheir value φ J = 0 . is clearly different fromour above value of 0 . − ≤ ˙ γ and so does not probe as close to the criticalpoint as we do here, and (ii) their analysis was based onthe scaling of shear viscosity η ≡ σ/ ˙ γ rather than thepressure viscosity η p . As we have noted previously [13]corrections to scaling for σ are significantly larger thanthey are for p , and without taking these corrections intoaccount, one generally finds a lower value for φ J , such aswas also found in the original scaling analysis of RD [4].Their lower value of φ J , and their higher window of strainrates ˙ γ , we believe are also responsible for the differentvalue they find for the exponent describing non-linearrheology exactly at φ J , σ ∼ p ∼ ˙ γ q ; they claim q = 1 / q = y/ ( β + y ) ≈ .
30 [17].
Macroscopic friction : Finally we consider the macro-scopic friction, µ ≡ σ/p . In Fig. 4 we plot µ vs φ forseveral different values of strain rate ˙ γ . We also showresults from quasistatic simulations [16, 18], representingthe ˙ γ → N = 1024 par-ticles to more explicitly compare with the results of Tighe − − − ˙ γ max CD RD × − × − ˙ γ min φ J − − − ˙ γ max CD RD × − × − ˙ γ min β FIG. 3. Comparison of critical parameters in models RD and CD for similar ranges of strain rate ˙ γ . Panel (a) showsthe jamming packing fraction φ J and panel (b) the viscosityexponent β , that result from fits of our data to Eqs. (6) and(7), for different ranges of ˙ γ min ≤ ˙ γ ≤ ˙ γ max and 0 . ≤ φ ≤ . φ of the data utilized inthe fit. We plot results vs. varying ˙ γ min for two different fixedvalues of ˙ γ max = 1 × − and 2 × − . The open symbolsare results for RD and the closed symbols are for CD . ForRD we have data down to ˙ γ min = 10 − , however for CD ourdata goes down to only ˙ γ min = 10 − . et al., who used a similar size system. While µ for themodels RD and CD differ slightly at the lower φ , we seethat near φ J they become essentially equal at the smaller˙ γ , and both RD and CD approach the quasistatic limitas ˙ γ →
0. We thus conclude that µ is finite as φ passesthrough φ J , consistent with recent experiments on foamsby Lespiat et al. [12].From our fit to η p in Fig. 2 we conclude that for bothmodels CD and RD the pressure along the yield stressline, i.e. ˙ γ → φ > φ J , vanishes upon approaching φ J as p ∼ ( φ − φ J ) y with y (cid:39) .
08. Our results inFig. 4 then argue that the shear stress along the yieldstress line, σ , vanishes similarly, so that µ stays finite.However the prediction of Tighe et al. is that the yieldshear stress vanishes as σ ∼ ( φ − φ J ) / . Were thisconclusion correct, we would expect µ ∼ ( φ − φ J ) . , vanishing as φ → φ J from above. Nothing in Fig. 4,where we see that µ = σ/p is a monotonically increasing function as φ decreases , suggests any such vanishing of µ ( φ J ). We thus conclude from Fig. 4 that the predictedscaling of σ by Tighe et al. is not correct, and moreoverthe two models CD and RD behave qualitatively thesame for both pressure p and shear stress σ .To conclude, we have examined the issue of the univer-sality of the jamming transition for overdamped shear-driven frictionless soft-core disks in 2D. We have con- ! ! ! quasistatic µ = " / p J $ . N = 1024 RD : open symbolsCD : closed symbols FIG. 4. Macroscopic friction σ/p vs φ for different ˙ γ for mod-els RD (open symbols) and CD (closed symbols), for a sys-tem with N = 1024 particles. Also shown are results fromquasistatic simulations representing the ˙ γ → sidered two different dissipative models that have beenwidely used in the literature: the collisional Durian bub-ble model CD and its mean field approximation RD .Contrary to previous claims [5] we find clear evidencethat CD does exhibit a growing macroscopically largelength ξ that appears to diverge as the jamming criticalpoint is approached. We further provide strong evidencethat CD and RD are in fact in the same universalityclass with the same critical exponents at jamming, andhave qualitatively the same rheological behavior moregenerally.We thank A. J. Liu, B. Tighe, M. van Hecke and M.Wyart for helpful discussions. This work was supportedby NSF Grant No. DMR-1205800 and the Swedish Re-search Council Grant No. 2010-3725. Simulations wereperformed on resources provided by the Swedish Na-tional Infrastructure for Computing (SNIC) at PDC andHPC2N. SUPPLEMENTAL MATERIALTransverse Velocity Correlation Function
The one quantity for which models RD and CD are clearly different is the transverse velocity correlationfunction, g y ( x ) ≡ (cid:104) v y (0) v y ( x ) (cid:105) . Defining the normalizedcorrelation, G y ( x ) ≡ g y ( x ) /g y (0), we plot in Fig. S1(a) G y ( x ) vs x , for several different values of strain rate ˙ γ , formodel RD at φ = 0 . ≈ φ J in a system of N = 4096particles. We see that G y ( x ) has a clear minimum at adistance x = (cid:96) , and that (cid:96) increases as ˙ γ → (cid:96) was interpretedas the diverging correlation length ξ . In CD however, itwas found [5] that G y ( x ) decreases monotonically with-out any obvious strong dependence on either φ or ˙ γ . InFig. S1(b) we plot G y ( x ) vs x , for several different ˙ γ ,at φ = 0 . ≈ φ J in a system of N = 4096 particles,confirming this result. ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − RD (a) x G y ( x ) ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − CD (b) x G y ( x ) FIG. S1. Normalized transverse velocity correlation function G y ( x ) = g y ( x ) /g y (0) at φ = 0 . ≈ φ J for a system of N = 4096 particles. Panel (a) is for model RD with shearrates ˙ γ = 10 − through 10 − . Panel (b) for model CD atshear rates ˙ γ = 10 − , through 10 − . As an alternative way to consider the difference inthis correlation between the two models, we now con-sider the Fourier transformed correlation g y ( k x ) = (cid:82) dx g y ( x )e ik x x , which we show in Figs. S2(a) and S2(b)for RD and CD respectively at packing fraction φ =0 . ≈ φ J . For RD we see a maximum in g y ( k x )at a k ∗ x that decreases for decreasing ˙ γ ; (cid:96) ∼ /k ∗ x givesthe corresponding minimum of the real-space correlation.For CD we show results only for the single strain rate˙ γ = 10 − since from Fig. S1(a) we expect no observabledifference as ˙ γ varies. We see an algebraic divergence g y ( k x ) ∼ k − . x as k x →
0. It is this algebraic divergencethat causes the real space G y ( x ) in CD to become solelya function of x/L as the system length L increases, as wasreported in Ref. [5].To try and give a qualitative understanding of thisdiffering behavior of g y ( k x ), we can consider how en-ergy is dissipated in each model. In RD the dissipa-tion is (1 /N ) (cid:80) i (cid:104)| δ v i | (cid:105) ≈ (cid:82) d k (cid:104) δ v ( k ) · δ v ( − k ) (cid:105) . ForCD , however, the dissipation is (1 /N ) (cid:80) i,j (cid:104)| v i − v j | (cid:105) ≈ (cid:82) d k (cid:104) δ v ( k ) · δ v ( − k ) (cid:105)| k | , where the sum is over onlyneighboring particles i, j in contact. Here δ v is the ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − RD (a) k x g y ( k x ) / " v y ˙ γ = 10 − slope: − . CD (b) k x g y ( k x ) / " v y FIG. S2. Fourier transform of the transverse velocity cor-relation function g y ( k x ) at φ = 0 . ≈ φ J . Panel (a) isfor model RD with shear rates ˙ γ = 10 − through 10 − .The peak in g y ( k x ), moving to smaller k x as ˙ γ decreases,is related to the minimum in the real space g y ( x ) moving tolarger x . The algebraic behavior in panel (b) for model CD at ˙ γ = 10 − , is consistent with the absence of any apparentlength scale, as reported in Ref. [5]. The number of particlesin these figures are N = 262144 except for the two smallestshear rates for RD for which N = 65536. non-affine part of the particle velocity. If we make anequipartition-like ansatz, and assume that as k → k , and both spatial directions x , y , con-tribute equally to the dissipation, we would then con-clude that for RD (cid:104) v y ( k ) v y ( − k ) (cid:105) ∝ constant, whilefor CD (cid:104) v y ( k ) v y ( − k ) (cid:105) ∝ /k . Noting that g y ( k x ) = (cid:82) dk y (cid:104) v y ( k ) v y ( − k ) (cid:105) , we then conclude that for RD wehave g ( k x ) ∝ constant as k x →
0, while for CD wehave the divergence g ( k x ) ∝ /k x . This saturation of g y ( k x ) for RD , as compared to the algebraic divergenceof g y ( k x ) for CD , is what is qualitatively seen in Fig. S2.The physical reason for this dramatic difference can beviewed as follows. For CD , since the dissipation dependsonly on velocity differences, uniform translation of a largecluster of particles with respect to the ensemble averageflow has little cost, thus enabling long wavelength fluc-tuations. For RD the dissipation is with respect to afixed background, so uniform translation of a large clus-ter causes dissipation that scales with the cluster size;long wavelength fluctuations are suppressed.That the observed divergence in CD is ∼ k − . x ratherthan the simple k − x predicted above, suggests that ourequipartition ansatz is not quite correct, and that thedifferent modes interact in a non-trivial way. That theexponent of this divergence is not an integer or simple ra-tional fraction suggests the signature of underlying criti-cal fluctuations, even though the correlation g y ( x ) itselfdoes not yield any obvious diverging length scale. Finite-Size-Scaling of Pressure
In Fig. 1 of the main article we showed data for thedependence of pressure p on system size L at differentstrain rates ˙ γ , at the jamming fraction φ J ≈ . ξ in both models RD and CR . Here we attempt a finite-size-scaling analysis of this data. We must note at theoutset, however, that our earlier work [13] demonstratedthat it is important to consider corrections-to-scaling toget accurate values for the exponents at criticality, andthat corrections-to-scaling are in fact large at the smallersizes L considered in Fig. 1 of the main article [16]. Sinceour data for p ( L ) is not extensive enough to try a scal-ing analysis including corrections-to-scaling, our resultsbased on a fit to Eq. (5) must be viewed as providing only effective exponents describing the data over the range ofparameters considered, rather than the true exponentsasymptotically close to criticality. We restate Eq. (5), p ( φ J , ˙ γ, L ) = L − y/ν P (0 , ˙ γL z ) . (S1)We can equivalently write the above in the form p ( φ J , ˙ γ, L ) = ˙ γ y/zν f ( L ˙ γ /z ) , (S2)using f ( x ) ≡ x − y/ν P (0 , x z ). We can now adjust theparameters q ≡ y/zν and z to try and collapse the datato a single common scaling curve. Plotting p/ ˙ γ q vs L ˙ γ /z we show the results for RD and CD in Figs. S3(a) and(b). For RD we find the effective exponents z = 6 . q = 0 . we find z = 6 . q =0 . z found in the present analysis arecomparable to the value z = 5 . f ( x ) → constant as x →∞ , which gives p ∼ ˙ γ q , q ≡ y/zν , in the limit of aninfinite sized system. ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − RD , φ = 0 . (a) L/ ˙ γ − /z , z = 6 . p / ˙ γ q , q = . ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − ˙ γ = 10 − CD , φ = 0 . (b) L/ ˙ γ − /z , z = 6 p / ˙ γ q , q = . FIG. S3. Scaling collapse of pressure according to Eq. (S2)for models RD and CD . The closeness of these fitted effective exponents for thetwo models is one more piece of evidence that RD andCD behave qualitatively the same, and do not have dra-matically different rheology as was claimed by Tighe etal. in Ref. [5].Finally we consider how the effective exponents foundhere compare to the true exponents asymptotically closeto criticality. From our most accurate analysis [13] ofthe critical behavior in RD , using a large system size N = 65536 and including the leading corrections-to-scaling, we have found the critical exponents q = y/zν =0 . ± .
02 and y = 1 . ± .
03, yielding zν = 3 . ± . q is in reasonable agreement with that found above from the finite-size-scaling analysis of p ( φ J , ˙ γ, L ).If we take the value of z ≈ ν ≈ .
65. Wenote that earlier scaling analyses [3, 4] that similarly ig-nored corrections-to-scaling found similar values for ν .However our recent [16] more detailed finite-size-scalinganalysis of the correlation length exponent, which in-cluded corrections-to-scaling, found that ν ≈
1, there-fore implying z ≈ . z foundhere from the finite-size-scaling of p is due to the strongcorrections-to-scaling that effect the correlation length atsmall L .As another way to see the effect of corrections-to-scaling on the correlation length, in Fig. S4 we plot ourresults for p vs L at φ = 0 . ≈ φ J , as obtained fromquasistatic simulations [16, 18] representing the ˙ γ → γ → p ∼ L − y/ν . If we fit the data at small L in Fig. S4 to apower law, we then find the exponent, y/ν ≈ .
79. Using y = 1 .
08 this then gives ν ≈ .
60, in rough agreementwith the value of ν obtained from the measured z of ourfinite-size-scaling of p with ˙ γ . If, however, we fit the dataat only the largest L to a power law, we then find theexponent y/ν ≈ .
11. Again using y = 1 .
08, we then get ν ≈ .
97, in better agreement with the expected ν ≈ L .
10 5010 − − y/ν = 1 . y/ν = 1 . p ∼ L − y/ν φ = 0 . RD , quasi-static L p FIG. S4. Pressure p vs system length L at φ J ≈ . L , giving an exponent y/ν ≈ .
79; solidline is a power law fit to the data at the largest L , giving anexponent y/ν ≈ . To conclude this section, although our finite-size-scaling of the pressure data in Fig. 1(a) of the main articleis effected by corrections-to-scaling, and so gives a largervalue for the dynamic exponent z than we believe is ac-tually the case at criticality, nevertheless the correlationlength ξ extracted from this data and shown in Fig. 1(b)demonstrates that RD and CD are behaving qualita-tively the same, and that both have a macroscopic lengthscale ξ that is growing (and we would argue diverging)as the jamming transition is approached. Effect of Finite Mass on Model CD − − − − − − − − − CD m = 10 CD m = 1 φ = 0 . . . N = 262144 (a) ˙ γ p − − − − − − − − − CD m = 1 CD m = 0 φ = 0 . . . N = 1024 (b) ˙ γ p FIG. S5. Pressure p vs. shear strain rate ˙ γ at packing fractions φ = 0 .
80, 0.8433, 0.85 for: (a) model CD with m = 1 and m = 10 for N = 262144 particles, and (b) model CD with m = 1 and model CD with m = 0 for N = 1024 particles. We wish to verify that the mass parameter m = 1,which we use in model CD, is indeed sufficiently smallso as to put our results in the overdamped m → , for the range of parametersstudied here. In Fig. S5(a) we show results for the elasticpart of the pressure p vs ˙ γ for model CD, with N =262144 particles, at three different packing fractions: φ =0 . φ = 0 . ≈ φ J , and φ = 0 .