Commentary on "Effect of temperature on a granular pile"

 

Antonio Coniglio¹, Massimo Pica Ciamarra²

Email: coniglio@na.infn.it

¹  Dip.to di Scienze Fisiche, Universit´a di Napoli “Federico II” and CNR-SPIN, Naples, Italy
²  CNR-SPIN, Universit´a di Napoli “Federico II”, Monte S. Angelo Via Cinthia, 80126 Napoli, Italy.

The author of [1] writes an interesting and critical review on the effect of temperature in a granular pile. First, it is shown the effect of temperature in several experiments. Then, it is shown how the compaction experiments, originally done by shaking granular material, can equally be performed by thermal cycling.

 

To a first approximation, one may expect that at the particle level the only effect of a temperature variation is a small particle volume change. If this is the case, in a thermal cycle the volume fraction of the system changes, and a thermal cycle could be seen as a compression cycle. This analogy suggests that the initial and final states of a cycle differ because of the disorder of the system. To understand this point, it is convenient to first consider the effect of a particle expansion cycle in a crystalline structure of spherical balls. In this structure, all inter-particle forces are equal in magnitude, and the net force acting on each particle is zero. On inflating the particles, all forces increase exactly by the same amount, and the net force acting on each particle never changes. Accordingly, particles keep their position and no motion occurs. Conversely, in the presence of disorder, the interparticle forces are different and vary by different amounts on inflating the particles. Particle swelling drives the system out of mechanical equilibrium, and induces motion. Using different words, one may say that ordered structures respond in an affine way to particle swelling, while disordered structures are characterized by a non affine response.

This picture, which is also valid for frictionless particles, becomes more involved in the presence of friction. In fact, one should consider that the microscopic origin of friction is in the asperities of the surfaces; the shearing of the frictional contacts induced by the thermal cycle, may cause them to break. In a series of thermal cycles, contacts repeatedly break, allowing the system to compact. One may also speculate that, apart from the the temperature variation which controls the relative volume change of the particles, an important control parameter in thermal cycles is the absolute value of the temperature. In fact, the height of the asperities is expected to decrease as particles become bigger.

The role of friction in granular materials has been extensively investigated in the literature (see, for instance, the paper by Song et al. [5] and a recent review [4]), and we have recently proposed a jamming phase diagram in a three dimensional space, where the axes are volume fraction, shear stress and friction coefficient [2]. In this line of research, the results of Ref. [1] are of particular interest; in fact, relating friction to temperature may allow to experimentally tune the friction coefficient and to validate different proposed theoretical scenarios.

We take the occasion to present a speculative picture regarding the role of friction in sheared granular systems, making an analogy between frictional sheared granular and thermal systems. In this pieture, we speculate an analogy between the ratio ¡ija and the ratio e/T. Here, ¡i and a are the íriction coefficient and the shear stress oí a granular system, while e measures the strength oí the attractive forcé between thermal partióles, and T is the temperature. We associate ¡i to e, as higher the fj,, stickier the contact, i.e. the greater the shear forcé the contact is able to sustain.

 

Figure 1: Pressure a_zz as a function of the volume fraction </>, for a = 2 10³, and ¡i = 0.1 in a small (main panel) and in a much larger (inset) pressure range. Circles correspond to measures taken when the system flows, and diamonds to measures taken in the jammed phase. Full symbols correspond to measures taken in the steady state, while the open circles for (pj₁ < </> < <f>j₂ correspond to measures taken in flowing metastable states which jam at long times.

We have investigated the limit of validity of this analogy performing molecular dynamics simulations of sheared granular systems in three dimensions. Partióles are confined between two parallel rough plates in the xy plañe, the bottom píate is fixed, while the other may move. We apply to them a shear forcé. Periodic boundary conditions are along x an y. Details of the numerical model and of the investigated system are in Ref. [2,3]. We vary the volume fraction (changing the number of partióles at constant volume), the shear stress a_xy and the friction coefficient ¡i.

Figure 1 illustrates a typical pressure versus volume fraction curve, for a fixed valué of the shear stress, where three transitions are enlighten. Here, a_zz is the normal forcé acting on the confining píate per unit surface. For </> < </>j_1; the system is in a steady flowing state. For <pj₁ < </> < <f>j₂, the system is found either in a metastable flowing state, or in a equilibrium disordered solid state able to sustain the applied stress. When in the metastable state, the system flows with a constant velocity for a long time, but it suddenly jams in an equilibrium solidlike state.¹ For (f>j₂ < </> < <f>j₃, the system quickly jams in response to the applied stress. For 4> > 4>J₃ the system responds as a solid to the applied stress. The equilibrium valué of the pressure a_ZZl marked by solid symbols in Fig. 1, increases in the flowing state, it discontinuously jumps to a different valué at <j)j₁, and grows for </> > <f>j₃. This scenario, and particularly the presence of a density range where the pressure is constant, suggests to interpret the í/^ -</>j₃ segment as a coexistent line. We have investigated the limit of validity of this scenario performing a number of simulations at different valúes of the friction coefficient ¡i. In the proposed analogy, low valúes of ¡i correspond to a high

Figure 2: Normal pressure of the confining píate a_zz as a function of the volume fraction </> at <T = 5 10². Different symbols correspond to <f>j₁ (squares) and <f>j₃ (circles), for different valúes of the friction coefficient, from ¡i = 0 (top) to ¡i = 0.8 (bottom). The shaded área can therefore be identified with the coexistence región.

The terms 'metastable' and 'equilibrium' are used to indicate states with a finite/infinite lifetime, respectively.

The ratio. For each valué of /x, we have estimated (pj^ji) and (pj₃(p), which are the two extrema of the coexistence line at that valué of ¡i. As show in Fig. 2, tríese two lines allow to identify the (analogous to the) región in the cr_zz<j) plañe. At ¡i = 0, (pJiip) = ^^(a⁴)) and the coexistence área ends in what should be the critical point, which here occurs at infinite temperature (as ¡i <x Í/T = 0). At finite friction, (pjtip) < 4>j₃(p), and coexistence lines are found, as shown in the figure for few valúes of ¡i. The coexistence área has a lower bound, which is found in the limit of high friction.

These results suggest that it is not unreasonable to associate the friction coefiicient of sheared granular systems to the inverse temperature of thermal systems. A deeper investigation is required to define the limits of validity of this analogy.

[1] T Divoux, Invited review: Effect of temperature on a granular pile, Pap. Phys. 2, 020006 (2010).

[2] M Pica Ciamarra, R Pastore, M Nicodemi, A Coniglio, Jamming phase diagram for frictional particles, arXiv:0912.3140v1 (2009).

[3] D S Grebenkov, M Pica Ciamarra, M Nicodemi, A Coniglio, Flow, ordering, and jamming of sheared granular suspensions, Phys. Rev. Lett. 100, 078001 (2008).

[4] M van Hecke, Jamming of soft particles: geometry, mechanics, scaling and isostaticity, J. Phys.: Condens. Matter 22, 033101 (2010).

[5] C Song, P Wang, H A Makse, A phase diagram for jammed matter, Nature 453, 629 (2008).