Papers in Physics, vol. 2, art. 020007 (2010)

Received: 10 November 2010, Accepted: 12 November 2010
Edited by: I. Ippolito
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.020007

www.papersinphysics.org

ISSN 1852-4249

Commentary on “Effect of temperature on a granular pile”

Antonio Coniglio1∗, Massimo Pica Ciamarra2

The author of [1] writes an interesting and criti-
cal 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 shak-
ing 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 be-
cause of the disorder of the system. To understand
this point, it is convenient to first consider the ef-
fect 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 inter–particle forces
are different and vary by different amounts on in-
flating the particles. Particle swelling drives the

∗E-mail: coniglio@na.infn.it

1 Dip.to di Scienze Fisiche, Universitá di Napoli “Federico
II” and CNR–SPIN, Naples, Italy

2 CNR–SPIN, Universitá di Napoli “Federico II”, Monte
S. Angelo Via Cinthia, 80126 Napoli, Italy.

system out of mechanical equilibrium, and induces
motion. Using different words, one may say that
ordered structures respond in an affine way to par-
ticle swelling, while disordered structures are char-
acterized 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 re-
peatedly 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 con-
trol 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 jam-
ming 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 pic-
ture regarding the role of friction in sheared granu-
lar systems, making an analogy between frictional

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Papers in Physics, vol. 2, art. 020007 (2010) / A. Coniglio et al.

Figure 1: Pressure σzz as a function of the volume
fraction φ, for σ = 2 10−3, and µ = 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 φJ1 < φ < φJ2 correspond to measures
taken in flowing metastable states which jam at
long times.

sheared granular and thermal systems. In this pic-
ture, we speculate an analogy between the ratio
µ/σ and the ratio ε/T . Here, µ and σ are the
friction coefficient and the shear stress of a gran-
ular system, while ε measures the strength of the
attractive force between thermal particles, and T
is the temperature. We associate µ to ε, as higher
the µ, stickier the contact, i.e. the greater the shear
force the contact is able to sustain.

We have investigated the limit of validity of
this analogy performing molecular dynamics simu-
lations of sheared granular systems in three dimen-
sions. Particles are confined between two parallel
rough plates in the x–y plane, the bottom plate
is fixed, while the other may move. We apply to
them a shear force. 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
particles at constant volume), the shear stress σxy
and the friction coefficient µ.

Figure 1 illustrates a typical pressure versus vol-

Figure 2: Normal pressure of the confining plate
σzz as a function of the volume fraction φ at
σ = 5 10−2. Different symbols correspond to φJ1
(squares) and φJ3 (circles), for different values of
the friction coefficient, from µ = 0 (top) to µ = 0.8
(bottom). The shaded area can therefore be iden-
tified with the coexistence region.

ume fraction curve, for a fixed value of the shear
stress, where three transitions are enlighten. Here,
σzz is the normal force acting on the confining plate
per unit surface. For φ < φJ1 , the system is in a
steady flowing state. For φJ1 < φ < φJ2 , the sys-
tem 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
solid-like state.1 For φJ2 < φ < φJ3 , the system
quickly jams in response to the applied stress. For
φ > φJ3 , the system responds as a solid to the ap-
plied stress. The equilibrium value of the pressure
σzz, marked by solid symbols in Fig. 1, increases
in the flowing state, it discontinuously jumps to a
different value at φJ1 , and grows for φ > φJ3 . This
scenario, and particularly the presence of a density
range where the pressure is constant, suggests to
interpret the φJ1 –φJ3 segment as a coexistent line.

We have investigated the limit of validity of this
scenario performing a number of simulations at dif-
ferent values of the friction coefficient µ. In the pro-
posed analogy, low values of µ correspond to a high

1The terms ‘metastable’ and ‘equilibrium’ are used to
indicate states with a finite/infinite lifetime, respectively.

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Papers in Physics, vol. 2, art. 020007 (2010) / A. Coniglio et al.

T/ε ratio. For each value of µ, we have estimated
φJ1 (µ) and φJ3 (µ), which are the two extrema of
the coexistence line at that value of µ. As show in
Fig. 2, these two lines allow to identify the (analo-
gous to the) region in the σzz–φ plane. At µ = 0,
φJ1 (µ) = φJ3 (µ), and the coexistence area ends in
what should be the critical point, which here occurs
at infinite temperature (as µ ∝ 1/T = 0). At finite
friction, φJ1 (µ) < φJ3 (µ), and coexistence lines are
found, as shown in the figure for few values of µ.
The coexistence area 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 coefficient of sheared gran-
ular systems to the inverse temperature of thermal
systems. A deeper investigation is required to de-
fine the limits of validity of this analogy.

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

[2] M Pica Ciamarra, R Pastore, M Nicodemi,
A Coniglio, Jamming phase diagram for fric-
tional 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: ge-
ometry, 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).

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