Soft Matter View Article Online
COMMUNICATION
View Journal | View Issue
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Size and density avalanche scaling near jamming Cite this: Soft Matter, 2014, 10, 2728
Roberto Are´valo and Massimo Pica Ciamarra*
Received 17th December 2013 Accepted 30th January 2014 DOI: 10.1039/c3sm53134a www.rsc.org/softmatter
The current microscopic picture of plasticity in amorphous materials assumes local failure events to produce displacement fields complying with linear elasticity. Indeed, the flow properties of nonaffine systems, such as foams, emulsions and granular materials close to jamming, that produce a fluctuating displacement field when failing, are still controversial. Here we show, via a thorough numerical investigation of jammed materials, that nonaffinity induces a critical scaling of the flow properties dictated by the distance to the jamming point. We rationalize this critical behavior by introducing a new universal jamming exponent and hyperscaling relationships, and we use these results to describe the volume fraction dependence of the friction coefficient.
While in simple crystals particles experience the same amount of deformation under a small, uniform applied stress, the disorder characterizing amorphous materials leads to a uctuating local deformation. When this nonaffine contribution to the displacement eld is of the same order of the affine displacement itself, it strongly inuences the elastic and rheological properties of the material and cannot be handled via perturbative approaches.1–7 In amorphous materials of technological interest such as foams, emulsions, polymeric suspensions and granular materials, this occurs close to the jamming volume fraction fJ marking the onset of mechanical rigidity upon compression.3 Indeed, nonaffinity can be characterized by a length scale that diverges as the jamming transition is approached,8 which is responsible, for instance, for the anomalous m/k f df1/2 scaling of the shear to bulk modulus ratio with the overcompression df ¼ f fJ. Nonaffinity also inuences the rheological properties at a nite shear rate, where its effects are known to depend on the energy dissipation mechanisms.9,10 Much less is known about the role of nonaffinity in the rheological properties in the athermal quasistatic shear (AQS) limit, which is of particular interest as it allows for the identication of the microscopic plastic events, and for the study of their
CNR-SPIN, Dipartimento di Scienze Fisiche, Universit` a di Napoli Federico II, Napoli, Italy. E-mail: [email protected]
2728 | Soft Matter, 2014, 10, 2728–2732
properties and correlations. In this limit, plastic events result from saddle node bifurcations11 that drive the irreversible rearrangement of an elementary unit of particles, generally known as a “shear transformation zone”, STZ;12 this elementary relaxation event might then trigger further rearrangements giving rise to avalanches, whose spatial features, such as their fractal dimension, are controversial.11,13,14 The triggering process is mediated by the elastic displacement eld produced by STZs, experiments and simulations (e.g. ref. 11, 15 and 16) suggested to be similar to that resulting from an Eshelby inclusion17 in a linear elastic solid. However, close to jamming nonaffine response is enhanced4,5 and linear elasticity is known to be no longer valid. Indeed, the displacement eld resulting from a point force4 or from a global deformation5 does not resemble that expected in a linear elastic solid. Analogously, one expects the displacement produced by an elementary plastic event not to resemble that produced by an Eshelby inclusion. Thus, qualitative and quantitative changes in the ow features are expected to occur as the jamming transition is approached. Recent AQS simulations18,19 of harmonic disks in two dimensions revealed that nonaffinity induces a critical behavior of some quantities characterizing the plastic ow, such as the average stress and energy drops, or the length of the elastic branches, which is dictated by the distance to the jamming point. However, it is not known whether this behavior is universal, as the role of the interaction potential and that of the dimensionality have not been explored, and the critical exponents have not been theoretically rationalized. Similarly, the effect of the increasing nonaffinity on the size scaling of the ow properties, which reveals the geometrical features of the avalanches, is unknown. Here we investigate the role of nonaffinity in the ow properties of amorphous systems via AQS shear simulations of harmonic and Hertzian particles, in both two and three dimensions, as a function of the distance to the jamming threshold, df, and of the system size, N. We show that the macroscopic ow properties are qualitatively unaffected by the degree of nonaffinity, as the dynamics exhibits the same size
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Communication
scaling regardless of the distance to the jamming point. Nonaffinity controls the critical scaling of the dynamics with df; we rationalize this scaling showing that the exponent characterizing the critical behavior of the length of the elastic branches is universal, and introducing hyperscaling relationships involving the critical exponents and the interaction potential. Finally, we use these results to infer the behavior of the friction coefficient close to the jamming. We consider 50 : 50 binary mixtures of N particles with diameters s1 ¼ 1 and s2 ¼ 1/1.4 and unit mass, in d ¼ 2 and in d ¼ 3 dimensions. Particles interact via a nite range repulsive s r a potential, V ðrÞ ¼ 3 for r < s, V(r) ¼ 0 for r > s, s1 corresponding to a harmonic (a ¼ 2) or to a Hertzian potential (a ¼ 5/2), with s being the average diameter of the interacting particles. 3 and s1 are our units of energy and length, respectively. We investigate different values of the volume fraction, considering systems with N ¼ 400, 800, 1600, and 3200 for harmonic and Hertzian particles in two and three dimensions. Additional simulations were run for harmonic particles in 2d, N ¼ 6400 and 12 800. These systems have been previously thoroughly investigated in the jamming context,3 and the pressure, the bulk and the shear modulus are known to scale with the overcompression as p f dfa1, k f dfa2, and as m f dfa3/2. Here we deform them via athermal quasistatic shear (AQS) simulations in which the strain is increased by small steps dg ¼ 105, and the energy of the system is minimized via the conjugate-gradient protocol aer each strain increment.20 Results are robust with respect to a factor of 10 change in dg. In the steady state regime of the AQS dynamics (g > 1), we have identied all plastic events, and measured the yield stress (stress right before the drop) sY, the stress Ds and the energy DE drops, the strain interval between successive events, Dg, and the instantaneous shear modulus, m. For each value of N and df, we have recorded from 103 to 104 plastic events. Our results support the presence of scaling relationships of the form hXi f NuXdfnX, where X is one of the investigated quantities, and uX and nX its size and density scaling exponents, respectively. Data supporting the validity of these relationships are illustrated in Fig. 1 and 2a. For the sake of clarity, we approximate the exponents with simple integer fractions to which the measured values agree within z0.02, and summarize their values in Table 1. The size scaling is of interest as a signature of the microscopic features of the plastic events and of their correlations,11,14,21 which build up through the elastic displacement eld induced by the STZs. For instance, if plastic avalanches have a fractal dimension D, then uDE ¼ D/d in d spatial dimensions.11,14 Previous studies have investigated the size scaling via AQS simulations of a variety of different systems, conrming the intensive character of the shear stress and of the shear modulus, usY ¼ um ¼ 0, and revealing a size dependence of other quantities associated with the plastic events. The energy-stress (DE f NDs) and the stress–strain (Ds f mDg) dependence lead to the relationships uDE uDs ¼ 1 and uDg ¼ uDs. These are always veried in the literature, and values compatible2,11,13,14,19,21 with uDE ¼ 1/2 or with22 uDE ¼ 1/3 have been reported.
This journal is © The Royal Society of Chemistry 2014
Soft Matter
Fig. 1 Scaling of the shear modulus (panels a and b), of the energy drops (panels c and d), of the yield stress and of the stress drops (panels e and f), for harmonic (left column) and Hertzian particles (right column). Symbols refer to N ¼ 1600 (squares), 3200 (circles), 6400 (up triangles) and 12 800 (down triangles). A superimposed cross distinguishes the 3d data. The data collapse occurs with no vertical shifts. Lines are power-law with exponents summarized in Table 1.
Fig. 2 (a) Universal scaling of the average length of the elastic branches. Symbols are as in Fig. 1. Data for Hertzian particles are shifted by a factor 2 for clarity. (b) Scaling of the Dg distribution. Data from 100 simulations obtained changing N, df and interaction potential collapse on the same master curve.
Our data of Fig. 1 and 2a show that both away form the jamming transition, where the system's response is affine, as well as close to the transition, where the response is highly nonaffine, the exponents assume values compatible with uDE ¼ 1/2, uDs ¼ 1/2 and uDg ¼ 1/2. These values depend neither on the interaction potential nor on the dimensionality. The df independence of the size scaling exponents suggests that the
Soft Matter, 2014, 10, 2728–2732 | 2729
View Article Online
Soft Matter
Communication
Values of the exponents describing the scaling as a function of the system size, u, and of the overcompression, n. The exponents do not depend on the dimensionality. Our numerical fits yield values consistent with the reported integer fractions, which satisfy the indicated relationships
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Table 1
hXifNuXdfnX
Harmonic
Hertz
Relationships
uDE uDs uDg nm nDs nsY nDE nDg
1/2 1/2 1/2 1/2 7/8 7/8 5/4 3/8
1/2 1/2 1/2 1 11/8 11/8 7/4 3/8
uDs ¼ uDE 1 uDg ¼ uDE 1 nm ¼ a 3/2 nDs ¼ nm + nDg nsY ¼ nm + nDg nDE ¼ nm + 2nDg
macroscopic plastic ow properties are only quantitatively affected by the nonaffine local response of the system. We now focus on the scaling with respect to the distance from the jamming point, df. For each investigated quantity and system size, we have performed a power law t to extract the critical exponent and the critical volume fraction, fj. The resulting values of fj are compatible within 103, consistent with the observation of a weak system size dependence under shear,23 and have average values fj ¼ 0.843 in 2d and fj ¼ 0.645 in 3d. These values are in good agreement with those reported for sheared systems,19,24 and we have used them to determine df ¼ f fj. Fig. 1 summarizes our results for the scaling of the shear modulus, of the average energy drops, of the average stress drops and of the yield stress. The data collapse and the numerical ts clarify that the critical exponents characterizing the df scaling of these quantities depend on the interaction potential, not on the dimensionality. In contrast, neither the interaction potential nor the dimensionality affect the scaling of the length of the elastic branches, we nd this scaling to be characterized by a universal scaling exponent, nDg z 3/8, as illustrated in Fig. 2a. The scaling relationships we have illustrated referring to the average values do actually work for the whole distributions, which scale as P(X) ¼ hXi gX (X/hXi). Here gX is a scaling function that depends on the considered quantity, but not on the dimensionality, the interaction potential or the volume fraction. Fig. 2b illustrates this scaling for the length of the elastic branches reporting the collapse of 100 datasets referring to different system sizes, densities and dimensionalities. Similarly, Fig. 3 shows the collapse of the distributions of the energy and of the stress drops. These two distributions have a roughly power law initial decay, which is followed by an exponential cutoff. In the case of the stress drop, the initial power law exponent is z1, as found19 for d ¼ 2 harmonic particles. For the energy drop we nd an initial power law exponent z0.7, in agreement with earlier results13,25 for d ¼ 2 harmonic particles. As an aside, we note that repulsive systems sheared via a spring mechanism are common experimental and numerical models for earthquakes,26–30 the particles representing the fault gouge in between the tectonic plates. These studies have found a different power law exponent for the energy drop distribution,
2730 | Soft Matter, 2014, 10, 2728–2732
Scaling of the distribution of the energy drops (a) and of the stress drops (b). As in Fig. 2b, data from 100 simulations referring to different N, df and a collapse on the same master curve.
Fig. 3
z1.7, in agreement with the well known Gutenberg–Richter law. This suggests that in purely repulsive systems inertia affects the scaling exponents and the scaling distributions, as recently observed in LJ systems.31 We now show the existence of hyperscaling relationships between the critical exponents. We begin by noticing that we nd no correlations between the measured quantities (not shown), so that hDsi ¼ hmDgi ¼ hmihDgi. Inserting the corresponding scaling relationships we recover the known relationship uDs ¼ uDg for the size scaling, and nd a relationship for the density scaling, nDs ¼ nm + nDg.
(1)
We then consider that, due to the quadratic relationship between energy and strain, the energy released in a plastic event 1 N is DE ¼ ½2sY Ds Ds2 . Given the scaling of the two terms 2r m in brackets, the condition DE > 0 can be expressed as N b dfnsY nDs . 1, and is always satised if nsY ¼ nDs.
(2)
Conversely, the condition could be violated at small or at large df. Given eqn (2), the above equation for the energy drop leads to the already known relationship for the size scaling, uDE uDs ¼ 1, and to a new relationship for the density scaling, nDE ¼ nm + 2nDs ¼ nm + 2nDg.
(3)
The validity of these relationships, which can be also derived from a simple dimensional analysis, is easily veried from the results summarized in Table 1. Regarding the dependence on the overcompression df, we have investigated 5 exponents, and derived three hyperscaling relationships. The only independent exponents are that of the shear modulus, which is known to be xed by the interaction potential,3 nm ¼ a 3/2, and that of the length of the elastic branches, we have found the scaling exponent to be universal, nDg ¼ 3/8. We nally consider the behavior of the pressure in our system. We rst notice that under shear at a constant volume
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Communication
the pressure is not constant because it has two contributions, one due to the overcompression, the other due to the strain. We therefore focus on the behavior of the pressure along the yield stress line, pY, which is of interest as related to the friction coefficient of the system. First, we notice that a power law t of our pressure data in the investigated volume fraction range gives a critical exponent that is slightly larger than that observed at zero applied shear stress, a 1, consistent with the previous results for 2d harmonic disks.19,24 However, the t gives a critical volume fraction that is smaller and not compatible with that resulting from the other power laws t. We interpret this as a signature of the fact that the yield pressure does not behave as a simple power law, being affected by both the compression and the shear stress. Indeed, when plotted as a function of df, as in Fig. 4a for 3d harmonic spheres, the yield pressure exhibits two regimes. At high compressions, it roughly scales as dfa1, as in the case of zero applied shear stress, while at small compressions it scales with a smaller exponent, q. Simulations with a smaller value of df, which are difficult to obtain due to the high computational cost of AQS simulations close to jamming, are needed to estimate q with condence. Nevertheless, dimensional analysis suggests q ¼ nsY ¼ a 9/8, which is a value compatible with our data. The behavior of the yield pressure leads to a crossover in the effective friction coefficient h f sY/pY. This is expected to approach a constant value close to jamming, as in previous numerical32 and experimental studies,33 as q ¼ nsY, and to scale as df1/8 away from the transition, regardless of the dimensionality and of the interaction potential. Fig. 4b shows that these predictions are veried by our numerical data. We have shown that the size scaling of the plastic ow features of amorphous materials is qualitatively unaffected by the distance to the jamming point. Close to jamming linear elasticity breaks, which suggests that concepts such as Eshelby like displacement elds might not play a fundamental role in the plastic ow, but more work is needed in this direction. Nonaffinity seems to quantitatively inuence the ow leading to a critical scaling of the dynamics with the distance to the jamming threshold, with exponents not depending on the dimensionality. We have rationalized this critical behavior introducing hyperscaling relationships between the exponents,
(a) Crossover in pressure dependence on the overcompression df, for 3d harmonic spheres. (b) Scaling of the friction coefficient for all considered system sizes and potentials. Symbols are as in Fig. 1.
Fig. 4
This journal is © The Royal Society of Chemistry 2014
Soft Matter
and a new universal jamming exponent. This universality suggests a critical behavior of the energy landscape itself. We thank R. Pastore for discussions and MIUR-FIRB RBFR081IUK for the nancial support.
References 1 F. Leonforte, A. Tanguy, J. P. Wittmer and J. L. Barrat, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2006, 97, 055501. 2 A. Tanguy, F. Leonforte and J. L. Barrat, Eur. Phys. J. E: So Matter Biol. Phys., 2006, 20, 355. 3 C. O'Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2003, 68, 011306. 4 W. Ellenbroek, E. Somfai, M. van Hecke and W. van Saarloos, Phys. Rev. Lett., 2006, 97, 258001. 5 W. Ellenbroek, M. van Hecke and W. van Saarloos, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 80, 061307. 6 A. Liu and S. Nagel, Annual Reviews of Condensed Matter Physics, 2010, 1, 347. 7 M. van Hecke, J. Phys.: Condens. Matter, 2010, 22, 033101. 8 M. Wyart, Ann. Phys., 2005, 30, 1. 9 P. Olsson and S. Teitel, Phys. Rev. Lett., 2007, 99, 178001. 10 B. Tighe, E. Woldhuis, J. Remmers, W. van Saarloos and M. van Hecke, Phys. Rev. Lett., 2010, 105, 088303. 11 C. Maloney and A. Lemaitre, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2006, 74, 016118. 12 M. L. Falk and J. S. Langer, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 57, 7192. 13 C. E. Maloney and A. Lemaˆıtre, Phys. Rev. Lett., 2004, 93, 016001. 14 N. Bailey, J. Schi¨ otz, A. Lemaˆıtre and K. Jacobsen, Phys. Rev. Lett., 2007, 98, 095501. 15 V. Bulatov and A. S. Argon, Modell. Simul. Mater. Sci. Eng., 1994, 2, 167. 16 V. Chikkadi, G. Wegdam, D. Bonn, B. Nienhuis and P. Schall, Phys. Rev. Lett., 2011, 107, 198303. 17 J. D. Eshelby, Proc. R. Soc. London, Ser. A, 1597, 241, 376. 18 C. Heussinger and J. Barrat, Phys. Rev. Lett., 2009, 102, 218303. 19 C. Heussinger, P. Chaudhuri and J. Barrat, So Matter, 2010, 6, 2050. 20 S. Plimpton, J. Comp. Physiol., A, 1995, 117, 1. 21 A. Lemaˆıtre and C. Caroli, Phys. Rev. Lett., 2009, 103, 065501. 22 E. Lerner and I. Procaccia, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 79, 066109. 23 D. V˚ agberg, P. Olsson and S. Teitel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 83, 031307. 24 D. V˚ agberg, D. Valdez-Balderas, M. Moore, P. Olsson and S. Teitel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 83, 030303R. 25 S. Tewari, D. Schiemann, D. Durian, C. Knobler, S. Langer and A. Liu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999, 60, 4385. 26 F. Dalton and D. Corcoran, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2001, 63, 061312. 27 M. Bretz, R. Zaretzki, S. Field, N. Mitarai and F. Nori, Europhys. Lett., 2006, 74, 1116. Soft Matter, 2014, 10, 2728–2732 | 2731
View Article Online
Soft Matter
31 K. Salerno, C. E. Maloney and M. Robbins, Phys. Rev. Lett., 2012, 109, 105703. 32 P. Peyneau and J. Roux, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2008, 78, 011307. 33 R. Lespiat, S. Cohen-Addad and R. H¨ ohler, Phys. Rev. Lett., 2011, 106, 148302.
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
28 K. Daniels and N. W. Hayman, J. Geophys. Res., 2008, 113, B11411. 29 M. P. Ciamarra, E. Lippiello, C. Godano and L. de Arcangelis, Phys. Rev. Lett., 2010, 104, 238001. 30 M. P. Ciamarra, E. Lippiello, L. de Arcangelis and C. Godano, EPL, 2011, 11, 54002.
Communication
2732 | Soft Matter, 2014, 10, 2728–2732
This journal is © The Royal Society of Chemistry 2014
COMMUNICATION
View Journal | View Issue
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Size and density avalanche scaling near jamming Cite this: Soft Matter, 2014, 10, 2728
Roberto Are´valo and Massimo Pica Ciamarra*
Received 17th December 2013 Accepted 30th January 2014 DOI: 10.1039/c3sm53134a www.rsc.org/softmatter
The current microscopic picture of plasticity in amorphous materials assumes local failure events to produce displacement fields complying with linear elasticity. Indeed, the flow properties of nonaffine systems, such as foams, emulsions and granular materials close to jamming, that produce a fluctuating displacement field when failing, are still controversial. Here we show, via a thorough numerical investigation of jammed materials, that nonaffinity induces a critical scaling of the flow properties dictated by the distance to the jamming point. We rationalize this critical behavior by introducing a new universal jamming exponent and hyperscaling relationships, and we use these results to describe the volume fraction dependence of the friction coefficient.
While in simple crystals particles experience the same amount of deformation under a small, uniform applied stress, the disorder characterizing amorphous materials leads to a uctuating local deformation. When this nonaffine contribution to the displacement eld is of the same order of the affine displacement itself, it strongly inuences the elastic and rheological properties of the material and cannot be handled via perturbative approaches.1–7 In amorphous materials of technological interest such as foams, emulsions, polymeric suspensions and granular materials, this occurs close to the jamming volume fraction fJ marking the onset of mechanical rigidity upon compression.3 Indeed, nonaffinity can be characterized by a length scale that diverges as the jamming transition is approached,8 which is responsible, for instance, for the anomalous m/k f df1/2 scaling of the shear to bulk modulus ratio with the overcompression df ¼ f fJ. Nonaffinity also inuences the rheological properties at a nite shear rate, where its effects are known to depend on the energy dissipation mechanisms.9,10 Much less is known about the role of nonaffinity in the rheological properties in the athermal quasistatic shear (AQS) limit, which is of particular interest as it allows for the identication of the microscopic plastic events, and for the study of their
CNR-SPIN, Dipartimento di Scienze Fisiche, Universit` a di Napoli Federico II, Napoli, Italy. E-mail: [email protected]
2728 | Soft Matter, 2014, 10, 2728–2732
properties and correlations. In this limit, plastic events result from saddle node bifurcations11 that drive the irreversible rearrangement of an elementary unit of particles, generally known as a “shear transformation zone”, STZ;12 this elementary relaxation event might then trigger further rearrangements giving rise to avalanches, whose spatial features, such as their fractal dimension, are controversial.11,13,14 The triggering process is mediated by the elastic displacement eld produced by STZs, experiments and simulations (e.g. ref. 11, 15 and 16) suggested to be similar to that resulting from an Eshelby inclusion17 in a linear elastic solid. However, close to jamming nonaffine response is enhanced4,5 and linear elasticity is known to be no longer valid. Indeed, the displacement eld resulting from a point force4 or from a global deformation5 does not resemble that expected in a linear elastic solid. Analogously, one expects the displacement produced by an elementary plastic event not to resemble that produced by an Eshelby inclusion. Thus, qualitative and quantitative changes in the ow features are expected to occur as the jamming transition is approached. Recent AQS simulations18,19 of harmonic disks in two dimensions revealed that nonaffinity induces a critical behavior of some quantities characterizing the plastic ow, such as the average stress and energy drops, or the length of the elastic branches, which is dictated by the distance to the jamming point. However, it is not known whether this behavior is universal, as the role of the interaction potential and that of the dimensionality have not been explored, and the critical exponents have not been theoretically rationalized. Similarly, the effect of the increasing nonaffinity on the size scaling of the ow properties, which reveals the geometrical features of the avalanches, is unknown. Here we investigate the role of nonaffinity in the ow properties of amorphous systems via AQS shear simulations of harmonic and Hertzian particles, in both two and three dimensions, as a function of the distance to the jamming threshold, df, and of the system size, N. We show that the macroscopic ow properties are qualitatively unaffected by the degree of nonaffinity, as the dynamics exhibits the same size
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Communication
scaling regardless of the distance to the jamming point. Nonaffinity controls the critical scaling of the dynamics with df; we rationalize this scaling showing that the exponent characterizing the critical behavior of the length of the elastic branches is universal, and introducing hyperscaling relationships involving the critical exponents and the interaction potential. Finally, we use these results to infer the behavior of the friction coefficient close to the jamming. We consider 50 : 50 binary mixtures of N particles with diameters s1 ¼ 1 and s2 ¼ 1/1.4 and unit mass, in d ¼ 2 and in d ¼ 3 dimensions. Particles interact via a nite range repulsive s r a potential, V ðrÞ ¼ 3 for r < s, V(r) ¼ 0 for r > s, s1 corresponding to a harmonic (a ¼ 2) or to a Hertzian potential (a ¼ 5/2), with s being the average diameter of the interacting particles. 3 and s1 are our units of energy and length, respectively. We investigate different values of the volume fraction, considering systems with N ¼ 400, 800, 1600, and 3200 for harmonic and Hertzian particles in two and three dimensions. Additional simulations were run for harmonic particles in 2d, N ¼ 6400 and 12 800. These systems have been previously thoroughly investigated in the jamming context,3 and the pressure, the bulk and the shear modulus are known to scale with the overcompression as p f dfa1, k f dfa2, and as m f dfa3/2. Here we deform them via athermal quasistatic shear (AQS) simulations in which the strain is increased by small steps dg ¼ 105, and the energy of the system is minimized via the conjugate-gradient protocol aer each strain increment.20 Results are robust with respect to a factor of 10 change in dg. In the steady state regime of the AQS dynamics (g > 1), we have identied all plastic events, and measured the yield stress (stress right before the drop) sY, the stress Ds and the energy DE drops, the strain interval between successive events, Dg, and the instantaneous shear modulus, m. For each value of N and df, we have recorded from 103 to 104 plastic events. Our results support the presence of scaling relationships of the form hXi f NuXdfnX, where X is one of the investigated quantities, and uX and nX its size and density scaling exponents, respectively. Data supporting the validity of these relationships are illustrated in Fig. 1 and 2a. For the sake of clarity, we approximate the exponents with simple integer fractions to which the measured values agree within z0.02, and summarize their values in Table 1. The size scaling is of interest as a signature of the microscopic features of the plastic events and of their correlations,11,14,21 which build up through the elastic displacement eld induced by the STZs. For instance, if plastic avalanches have a fractal dimension D, then uDE ¼ D/d in d spatial dimensions.11,14 Previous studies have investigated the size scaling via AQS simulations of a variety of different systems, conrming the intensive character of the shear stress and of the shear modulus, usY ¼ um ¼ 0, and revealing a size dependence of other quantities associated with the plastic events. The energy-stress (DE f NDs) and the stress–strain (Ds f mDg) dependence lead to the relationships uDE uDs ¼ 1 and uDg ¼ uDs. These are always veried in the literature, and values compatible2,11,13,14,19,21 with uDE ¼ 1/2 or with22 uDE ¼ 1/3 have been reported.
This journal is © The Royal Society of Chemistry 2014
Soft Matter
Fig. 1 Scaling of the shear modulus (panels a and b), of the energy drops (panels c and d), of the yield stress and of the stress drops (panels e and f), for harmonic (left column) and Hertzian particles (right column). Symbols refer to N ¼ 1600 (squares), 3200 (circles), 6400 (up triangles) and 12 800 (down triangles). A superimposed cross distinguishes the 3d data. The data collapse occurs with no vertical shifts. Lines are power-law with exponents summarized in Table 1.
Fig. 2 (a) Universal scaling of the average length of the elastic branches. Symbols are as in Fig. 1. Data for Hertzian particles are shifted by a factor 2 for clarity. (b) Scaling of the Dg distribution. Data from 100 simulations obtained changing N, df and interaction potential collapse on the same master curve.
Our data of Fig. 1 and 2a show that both away form the jamming transition, where the system's response is affine, as well as close to the transition, where the response is highly nonaffine, the exponents assume values compatible with uDE ¼ 1/2, uDs ¼ 1/2 and uDg ¼ 1/2. These values depend neither on the interaction potential nor on the dimensionality. The df independence of the size scaling exponents suggests that the
Soft Matter, 2014, 10, 2728–2732 | 2729
View Article Online
Soft Matter
Communication
Values of the exponents describing the scaling as a function of the system size, u, and of the overcompression, n. The exponents do not depend on the dimensionality. Our numerical fits yield values consistent with the reported integer fractions, which satisfy the indicated relationships
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Table 1
hXifNuXdfnX
Harmonic
Hertz
Relationships
uDE uDs uDg nm nDs nsY nDE nDg
1/2 1/2 1/2 1/2 7/8 7/8 5/4 3/8
1/2 1/2 1/2 1 11/8 11/8 7/4 3/8
uDs ¼ uDE 1 uDg ¼ uDE 1 nm ¼ a 3/2 nDs ¼ nm + nDg nsY ¼ nm + nDg nDE ¼ nm + 2nDg
macroscopic plastic ow properties are only quantitatively affected by the nonaffine local response of the system. We now focus on the scaling with respect to the distance from the jamming point, df. For each investigated quantity and system size, we have performed a power law t to extract the critical exponent and the critical volume fraction, fj. The resulting values of fj are compatible within 103, consistent with the observation of a weak system size dependence under shear,23 and have average values fj ¼ 0.843 in 2d and fj ¼ 0.645 in 3d. These values are in good agreement with those reported for sheared systems,19,24 and we have used them to determine df ¼ f fj. Fig. 1 summarizes our results for the scaling of the shear modulus, of the average energy drops, of the average stress drops and of the yield stress. The data collapse and the numerical ts clarify that the critical exponents characterizing the df scaling of these quantities depend on the interaction potential, not on the dimensionality. In contrast, neither the interaction potential nor the dimensionality affect the scaling of the length of the elastic branches, we nd this scaling to be characterized by a universal scaling exponent, nDg z 3/8, as illustrated in Fig. 2a. The scaling relationships we have illustrated referring to the average values do actually work for the whole distributions, which scale as P(X) ¼ hXi gX (X/hXi). Here gX is a scaling function that depends on the considered quantity, but not on the dimensionality, the interaction potential or the volume fraction. Fig. 2b illustrates this scaling for the length of the elastic branches reporting the collapse of 100 datasets referring to different system sizes, densities and dimensionalities. Similarly, Fig. 3 shows the collapse of the distributions of the energy and of the stress drops. These two distributions have a roughly power law initial decay, which is followed by an exponential cutoff. In the case of the stress drop, the initial power law exponent is z1, as found19 for d ¼ 2 harmonic particles. For the energy drop we nd an initial power law exponent z0.7, in agreement with earlier results13,25 for d ¼ 2 harmonic particles. As an aside, we note that repulsive systems sheared via a spring mechanism are common experimental and numerical models for earthquakes,26–30 the particles representing the fault gouge in between the tectonic plates. These studies have found a different power law exponent for the energy drop distribution,
2730 | Soft Matter, 2014, 10, 2728–2732
Scaling of the distribution of the energy drops (a) and of the stress drops (b). As in Fig. 2b, data from 100 simulations referring to different N, df and a collapse on the same master curve.
Fig. 3
z1.7, in agreement with the well known Gutenberg–Richter law. This suggests that in purely repulsive systems inertia affects the scaling exponents and the scaling distributions, as recently observed in LJ systems.31 We now show the existence of hyperscaling relationships between the critical exponents. We begin by noticing that we nd no correlations between the measured quantities (not shown), so that hDsi ¼ hmDgi ¼ hmihDgi. Inserting the corresponding scaling relationships we recover the known relationship uDs ¼ uDg for the size scaling, and nd a relationship for the density scaling, nDs ¼ nm + nDg.
(1)
We then consider that, due to the quadratic relationship between energy and strain, the energy released in a plastic event 1 N is DE ¼ ½2sY Ds Ds2 . Given the scaling of the two terms 2r m in brackets, the condition DE > 0 can be expressed as N b dfnsY nDs . 1, and is always satised if nsY ¼ nDs.
(2)
Conversely, the condition could be violated at small or at large df. Given eqn (2), the above equation for the energy drop leads to the already known relationship for the size scaling, uDE uDs ¼ 1, and to a new relationship for the density scaling, nDE ¼ nm + 2nDs ¼ nm + 2nDg.
(3)
The validity of these relationships, which can be also derived from a simple dimensional analysis, is easily veried from the results summarized in Table 1. Regarding the dependence on the overcompression df, we have investigated 5 exponents, and derived three hyperscaling relationships. The only independent exponents are that of the shear modulus, which is known to be xed by the interaction potential,3 nm ¼ a 3/2, and that of the length of the elastic branches, we have found the scaling exponent to be universal, nDg ¼ 3/8. We nally consider the behavior of the pressure in our system. We rst notice that under shear at a constant volume
This journal is © The Royal Society of Chemistry 2014
View Article Online
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
Communication
the pressure is not constant because it has two contributions, one due to the overcompression, the other due to the strain. We therefore focus on the behavior of the pressure along the yield stress line, pY, which is of interest as related to the friction coefficient of the system. First, we notice that a power law t of our pressure data in the investigated volume fraction range gives a critical exponent that is slightly larger than that observed at zero applied shear stress, a 1, consistent with the previous results for 2d harmonic disks.19,24 However, the t gives a critical volume fraction that is smaller and not compatible with that resulting from the other power laws t. We interpret this as a signature of the fact that the yield pressure does not behave as a simple power law, being affected by both the compression and the shear stress. Indeed, when plotted as a function of df, as in Fig. 4a for 3d harmonic spheres, the yield pressure exhibits two regimes. At high compressions, it roughly scales as dfa1, as in the case of zero applied shear stress, while at small compressions it scales with a smaller exponent, q. Simulations with a smaller value of df, which are difficult to obtain due to the high computational cost of AQS simulations close to jamming, are needed to estimate q with condence. Nevertheless, dimensional analysis suggests q ¼ nsY ¼ a 9/8, which is a value compatible with our data. The behavior of the yield pressure leads to a crossover in the effective friction coefficient h f sY/pY. This is expected to approach a constant value close to jamming, as in previous numerical32 and experimental studies,33 as q ¼ nsY, and to scale as df1/8 away from the transition, regardless of the dimensionality and of the interaction potential. Fig. 4b shows that these predictions are veried by our numerical data. We have shown that the size scaling of the plastic ow features of amorphous materials is qualitatively unaffected by the distance to the jamming point. Close to jamming linear elasticity breaks, which suggests that concepts such as Eshelby like displacement elds might not play a fundamental role in the plastic ow, but more work is needed in this direction. Nonaffinity seems to quantitatively inuence the ow leading to a critical scaling of the dynamics with the distance to the jamming threshold, with exponents not depending on the dimensionality. We have rationalized this critical behavior introducing hyperscaling relationships between the exponents,
(a) Crossover in pressure dependence on the overcompression df, for 3d harmonic spheres. (b) Scaling of the friction coefficient for all considered system sizes and potentials. Symbols are as in Fig. 1.
Fig. 4
This journal is © The Royal Society of Chemistry 2014
Soft Matter
and a new universal jamming exponent. This universality suggests a critical behavior of the energy landscape itself. We thank R. Pastore for discussions and MIUR-FIRB RBFR081IUK for the nancial support.
References 1 F. Leonforte, A. Tanguy, J. P. Wittmer and J. L. Barrat, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2006, 97, 055501. 2 A. Tanguy, F. Leonforte and J. L. Barrat, Eur. Phys. J. E: So Matter Biol. Phys., 2006, 20, 355. 3 C. O'Hern, L. E. Silbert, A. J. Liu and S. R. Nagel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2003, 68, 011306. 4 W. Ellenbroek, E. Somfai, M. van Hecke and W. van Saarloos, Phys. Rev. Lett., 2006, 97, 258001. 5 W. Ellenbroek, M. van Hecke and W. van Saarloos, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 80, 061307. 6 A. Liu and S. Nagel, Annual Reviews of Condensed Matter Physics, 2010, 1, 347. 7 M. van Hecke, J. Phys.: Condens. Matter, 2010, 22, 033101. 8 M. Wyart, Ann. Phys., 2005, 30, 1. 9 P. Olsson and S. Teitel, Phys. Rev. Lett., 2007, 99, 178001. 10 B. Tighe, E. Woldhuis, J. Remmers, W. van Saarloos and M. van Hecke, Phys. Rev. Lett., 2010, 105, 088303. 11 C. Maloney and A. Lemaitre, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2006, 74, 016118. 12 M. L. Falk and J. S. Langer, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 57, 7192. 13 C. E. Maloney and A. Lemaˆıtre, Phys. Rev. Lett., 2004, 93, 016001. 14 N. Bailey, J. Schi¨ otz, A. Lemaˆıtre and K. Jacobsen, Phys. Rev. Lett., 2007, 98, 095501. 15 V. Bulatov and A. S. Argon, Modell. Simul. Mater. Sci. Eng., 1994, 2, 167. 16 V. Chikkadi, G. Wegdam, D. Bonn, B. Nienhuis and P. Schall, Phys. Rev. Lett., 2011, 107, 198303. 17 J. D. Eshelby, Proc. R. Soc. London, Ser. A, 1597, 241, 376. 18 C. Heussinger and J. Barrat, Phys. Rev. Lett., 2009, 102, 218303. 19 C. Heussinger, P. Chaudhuri and J. Barrat, So Matter, 2010, 6, 2050. 20 S. Plimpton, J. Comp. Physiol., A, 1995, 117, 1. 21 A. Lemaˆıtre and C. Caroli, Phys. Rev. Lett., 2009, 103, 065501. 22 E. Lerner and I. Procaccia, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2009, 79, 066109. 23 D. V˚ agberg, P. Olsson and S. Teitel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 83, 031307. 24 D. V˚ agberg, D. Valdez-Balderas, M. Moore, P. Olsson and S. Teitel, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2011, 83, 030303R. 25 S. Tewari, D. Schiemann, D. Durian, C. Knobler, S. Langer and A. Liu, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1999, 60, 4385. 26 F. Dalton and D. Corcoran, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2001, 63, 061312. 27 M. Bretz, R. Zaretzki, S. Field, N. Mitarai and F. Nori, Europhys. Lett., 2006, 74, 1116. Soft Matter, 2014, 10, 2728–2732 | 2731
View Article Online
Soft Matter
31 K. Salerno, C. E. Maloney and M. Robbins, Phys. Rev. Lett., 2012, 109, 105703. 32 P. Peyneau and J. Roux, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2008, 78, 011307. 33 R. Lespiat, S. Cohen-Addad and R. H¨ ohler, Phys. Rev. Lett., 2011, 106, 148302.
Published on 31 January 2014. Downloaded by Brown University on 31/10/2014 13:22:09.
28 K. Daniels and N. W. Hayman, J. Geophys. Res., 2008, 113, B11411. 29 M. P. Ciamarra, E. Lippiello, C. Godano and L. de Arcangelis, Phys. Rev. Lett., 2010, 104, 238001. 30 M. P. Ciamarra, E. Lippiello, L. de Arcangelis and C. Godano, EPL, 2011, 11, 54002.
Communication
2732 | Soft Matter, 2014, 10, 2728–2732
This journal is © The Royal Society of Chemistry 2014
