Papers in Physics, vol. 6, art. 060008 (2014)

Received: 10 October 2014, Accepted: 10 October 2014
Edited by: L. A. Pugnaloni
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.060008

www.papersinphysics.org

ISSN 1852-4249

Commentary on “Jamming transition in a two-dimensional open
granular pile with rolling resistance”

Roberto Arévalo1∗

I. Introduction

The paper “Jamming transition in a two-
dimensional open granular pile with rolling resis-
tance” studies the jamming transition using the
flow of granular particles from piles. In this com-
mentary I would like to give a perspective on the
results by Magalhães et al. [1] and previous related
studies obtained in silos.

We speak about jamming transition in a silo
when an arch forms at the outlet effectively arrest-
ing the flow. This is likely to happen when the
size of the outlet is a few times bigger than that
of the flowing particles. What factors control the
probability of arch formation? If we fix the na-
ture of the particles, it is found that the pressure
close to the outlet has a major influence. In Ref.
[2], it was shown that altering this pressure in a
two-dimensional silo can change the probability of
arch formation by two orders of magnitude. At the
same time, the flow rate or velocity field are modi-
fied only slightly.

In a flat-bottomed silo, pressure increases lin-
early from the free surface toward the bottom. At
a certain depth, however, pressure saturates and
remains constant until the base. This is the well
known Janssen effect. It is usually attributed to

∗E-mail: aroberto@ntu.edu.sg

1 Nanyang Technological University, School of Physi-
cal and Mathematical Sciences, Physics and Applied
Physics, 21 Nanyang Link, 637371, Singapore.

the friction of the walls that sustains part of the
weight of the column of particles. The depth at
which pressure saturates depends on properties of
the grains (like the Poisson ratio) and the geometry
of the silo.

This scenario for the silo contrasts greatly with
that of a pile, as the ones studied in Ref. [1]. Piles
have no lateral walls so no Janssen effect is present.
Further, the geometry of the pile itself assures that
pressure is not homogeneous. As it could be ex-
pected, pressure increases linearly from the exte-
rior as we move toward the center. Surprisingly,
the pressure is not maximum at the center of the
pile. Instead, it shows a drop that has been termed
“pressure dip” [3].

When one opens an orifice at the base of a silo,
the material inside outpours driven by gravity. As
stated, if the size of the outlet is a few times that of
the flowing particles, an arch may form that jams
the flow. As the size of the orifice grows, the prob-
ability of arch formation decreases. Eventually, for
relatively large orifices, no arches are observed dur-
ing the accessible time-scale and the flow seems
continuous.

Actually, the last statement is currently under
debate. Some authors have conjectured that there
exist two regimes in the silo discharge. One for ori-
fices with radius R < Rc, where Rc is some critical
value (in units of the particles’ radii. Since most
of the literature concerns disks in 2D or spheres
in 3D, I will refer here to the radii of the parti-
cles and that of the orifice). In this regime arching

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Papers in Physics, vol. 6, art. 060008 (2014) / R. Arévalo

probability is always bigger than zero and arches
are bound to appear. A second regime is claimed
to exist for R > Rc, in which the probability to ob-
serve a stable arch is zero and the flow is actually
continuous.

II. Silos, piles and jamming

Several works have been devoted in the last years
to try to settle the accuracy of this picture, and
to obtain the value of Rc for different geometries.
In Ref. [4], Zuriguel et al. experimentally study
the case of a cylindrical silo filled with spherical
particles as well as other shapes (small cylinders
with low aspect ratio and rice). They measure the
size s of the avalanches, defined as the number of
particles fallen between two clogs. It is found that
their data are well described by a power law of the
form

< s >= A/ (Rc −R)
γ
. (1)

This leads them to propose that the avalanche
size indeed diverges at Rc (whose value depends
slightly on the shape of the particles). For spheres,
they find Rc = 4.94 and γ = 6.9.

A finite size analysis was carried out in Ref.
[5]. To this end, the jamming probability function
J (N,R) was defined as the probability that the
flow clogs before N particles have fallen. In other
words, the probability to observe an avalanche of
size N for a given aperture size R. Fixing the value
of N, the authors compute the jamming probabil-
ity JN (R). For low values of N, JN (R) presents
a maximum value of 1 for low R and a minimum
value 0 for big aperture size. The transition be-
tween these values is fast but smooth, with a well
defined slope. Upon increasing N, the transition
becomes more abrupt and the slope steepens. A
limiting step function as N → ∞ would be the
hallmark of a real abrupt transition separating the
two regimes at Rc. The authors fit the data to
JN (R) = {1 − tanh [α (R−Rc)]}/2, which actu-
ally converges to the Heaviside function in the limit
α → ∞. However, there is no theoretical justifica-
tion for this expression, apart from the good fit.

An analogous research in 2D (using spherical
particles) was carried out in Ref. [6]. It was found
that the data on avalanche size could be fit with

the same expression than in the 3D case, with val-
ues Rc = 8.5 and γ = 12.7. Nevertheless, a careful
finite-size system analysis leads the authors to con-
clude that there are no two separated regimes in
this case. As in the 3D case, the jamming proba-
bility function J (N,R) is used. Also in this case,
the data from experiments show an apparent con-
vergence toward a step function. However, by a
subsequent theoretical analysis, the authors found
a limiting function which is smooth and shows a
doubly exponential dependence in R. The fact that
the experimental data could be fit by Eq. (1) is
explained taking into account that the maximum
measured avalanche is for R ' 5.5 while the pre-
sumed critical value is much higher, Rc = 8.5.

To et al. [7] have studied the jamming of disks
in 2D in a hopper geometry. Measuring the jam-
ming probability defined above, they find that their
data can be fitted with several different expres-
sions. First, by an independent theoretical analy-
sis of their experiment, an expression for JN (R) is
found which is formally analogous to the one pro-
posed by Janda et al., and does not contain any
divergence. However, their data can be well fit
by Eq. (1) and an exponential containing a diver-
gent term. Being the curves indistinguishable in
the range of data available, the conclusion is that
the question of the existence of a critical outlet size
cannot be settled.

Recently, Magalhães et al. have proposed a se-
ries of simulations [1, 8] addressing the existence of
a critical outlet for the flow of grains in a different
geometrical setting. They use a realistic soft par-
ticle molecular dynamics method with static fric-
tion, which is supplemented with rolling resistance
in their contribution to this volume. Briefly, the
protocol implemented is as follows: in 2D a pile of
grains is created by allowing particles to “rain” over
a base of a certain width L, which is used as a mea-
sure of the size of the initial piles. Once the pile is
stable, a hole of size R is opened at the center of the
base and particles flow through it. The pile ends
up in one of two possible final states: i) a stable
arch appears around the outlet blocking the flow;
ii) no arch is formed and the pile collapses com-
pletely, only a few particles remain at both sides of
the outlet. In both cases, the number of particles
fallen is recorded and, in case i), also the height h
of the final pile.

The authors propose that this scenario can be

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likened to a phase transition between the two
regimes. Taking h as an order parameter, the plot
h (R) shows three regimes. For small apertures, h
is essentially a constant which depends on the size
of the initial pile. For big R, the pile collapses and
so h = 0. The transition between the two values
is gradual for small piles and becomes more abrupt
for larger piles.

The fluctuations of h as a function of the outlet
size present a well defined peak at some value R∗

which depends on the size L of the initial pile. As
L grows, R∗ seems to converge to a limiting value,
and when extrapolating L → ∞, the authors find
a critical outlet size Rc = 5.0.

In this case, there is no attempt at a theoretical
analysis of the observations that could further sup-
port the extrapolation of h (R) to a step function.

In their contribution to this volume, the authors
revisit this problem with new simulations in which
they implement rolling friction, making them more
realistic. The main conclusion is that the picture
drawn in their first work holds, although a correc-
tion in the value of Rc is introduced.

III. A critical outlet?

Let us, at this point, critically address the results
summarized in the previous section. Regarding the
3D silo, the conclusion that there exists a criti-
cal outlet size is based on the good fit of the data
on avalanche size by Eq. (1). In principle, there
is no reason precluding the idea that avalanches
keep growing for large orifices, with the probability
of arch formation decreasing without ever reaching
zero. That this is not observed could be simply due
to the limitations of the experimental setup. The
observations were carried out keeping the height of
material inside the silo constant. This is required
to assure steady state conditions and, in particu-
lar, that there is always a Janssen effect present
(if the height of the material decreases too much,
pressure does not saturate). Under this conditions,
it was simply not possible to refill the silo in order
to measure avalanches bigger that a few million of
grains.

From a more theoretical perspective, Eq. (1)
might seem a bit unsatisfactory. In first place, coef-
ficient A lacks a fundamental interpretation, being
just the size of the avalanche for an orifice of size

R = Rc−1. In second place, the value of the expo-
nent γ is rather high compared to what one usually
finds in phase transitions; and this holds for both
the 2D and 3D silos. Third, the avalanche size
should approach zero as the outlet size approaches
one, in units of the diameter of the particles.

As a matter of fact, other expressions can be
found to fit the available data. For example <

s >= A(R−1)e(B·R
C) gives a good fit with A ' 3.4,

B ' 0.04 and C ' 3.8. This expression is not diver-
gent and goes to zero as the outlet size approaches
that of the particles. Although given without the-
oretical motivation, it shows that the present data
are not enough to settle the question of the exis-
tence of a critical outlet size. A better understand-
ing of the process of arch formation and how it is
affected by the flow is required to gain insight into
this problem.

In the case of 2D silos we find, in addition to a
wealth of experimental data and careful finite size
study, a theoretical insight into the process of arch-
ing. In both the independent analysis by To and
Janda et al. formally equivalent expressions for the
jamming probability are found. This reinforces the
conclusion that the avalanche size does not diverge
and there is no critical outlet. However, one must
consider i) that there are underlying asumptions
in the theoretical considerations, and ii) that the
available data can be well fit with diverging ex-
pressions. More (difficult to obtain) data for ori-
fices similar to the putative critical size would be
necessary to make a strong statement.

It is worth mentioning that the theoretical
derivation of the jamming probability in Ref. [6]
leads, when applied to a 3D silo, to an expression
that does not fit the corresponding experimental
data. This could be due to some assumption being
valid in 2D but not in 3D. Or, one can also con-
sider the possibility that there is no transition in
two dimensions, and a critical outlet size appears
in three dimensions. This is a perfectly possible
scenario, as we know from other phase transitions
whose existence depends on the dimensionality of
the system.

Magalhães et al. propose an analogous study in a
radically modified geometry. One can imagine that
removes the lateral walls of a 2D silo, then the col-
umn of particles collapses and ends up in a pile
of certain width. Under these conditions, there is

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Papers in Physics, vol. 6, art. 060008 (2014) / R. Arévalo

no Janssen effect, the pressure is not homogeneous
and the dynamics could be very different from that
of a silo. In any case, as mentioned in the intro-
duction, the probability of arch formation could be
drastically affected. Thus, it is not obvious that
the conclusions reached for silos should hold in the
case of piles.

Actually, the conclusion in the previous work,
Ref. [8], is that there exists a critical outlet for the
flow from piles. The obtained value Rc = 5 is far
below to Rc = 8.5, reported by Janda el al. for the
2D silo. Without taking into account the existence
or not of a critical outlet, this could be an indication
that the two setups are not comparable.

In their present contribution, Magalhães et al.
report a correction to Rc = 5.3 for a contact model
that includes rolling resistance (RR). This is rea-
sonable, since particles interacting with RR should
interlock more easily, leading to more stable arches.
Nevertheless, this new Rc value is still much smaller
than the one found for the 2D silo.

IV. Avalanche size and flow

It may seem difficult to imagine the formation of an
arch stable enough to arrest the flow in a silo with
a large aperture. Effectively, it appears that the
flow will sweep any incipient arch and will continue
undisturbed. However, the flow is a tunable feature
of the silo, known to vary with the square root of
gravity, as manifested in experiments [9–11] and
simulations [12].

Gravity can be modified in experiments up to a
certain point, and it is always possible in simula-
tions, so let us call Γ to the imposed body force on
the particles. One can imagine, then, that reduces
Γ, thus making the flow slower, even for large aper-
tures. Under these conditions, molecular dynamics
simulations [12] lead to the conclusion that the size
of the avalanches increases with the kinetic energy
of the system. This is so because, when reducing
the driving force, the particles in the outlet region
have more time to dissipate their kinetic energy
and form a stable structure that blocks the flow.
It is then conceivable that one can have a Γ small
enough to allow the particles form a stable arch
before being swept by the flow.

Should this picture be correct, there would not
be a critical outlet size. Arches would simply be

less likely to be stabilize as the flow was increased
by increasing R. The observation of arches would
be limited by the time window allowed by the ex-
periments or simulations. In order to shed some
light in the process of arching a low Γ, additional
simulations with a variable outlet size are currently
under way.

V. Conclusions

In this brief commentary, I have tried to give a per-
spective on recent results on the jamming of parti-
cle flows in silos and its relation with Magalhães et
al. contribution to this volume. It is commonplace
that silos get jammed. When the size of the aper-
ture is not much larger than that of the particles,
arches appear to block the flow. Upon increasing
the size of the outlet, arches become scarcer. And
for large orifices, they are not seen at all at ac-
cessible time scales. The observation of power-law
relations between the size of the avalanches and
the outlet size sparked the interest in considering
a critical-like transition between two regimes: a
jamming regime in which arches are bound to ap-
pear and block the flow, and a continuous regime
in which there are no arches. Both regimes would
be separated by a critical outlet size.

Magalhães et al. undertake an analogous re-
search changing the conditions of the reservoir from
which the particles flow. They use piles, which have
open boundaries, and introduce new complications
to be considered. If not transferable from one an-
other, the results in piles and silos will widen our
perspective on the clogging of granular particles.

Let me, very briefly, summarize my opinion on
the existence of a critical outlet:

• Experimental data in 3D silos are compatible
with a transition from a jammed to a contin-
uos flow regime at a certain value of the out-
let. However, descriptions not involving a di-
vergence are also possible. So far, we lack a
theoretical frame that justifies one view over
the other.

• Experimental data in 2D silos are also compat-
ible with both the existence and the absence
of a critical outlet. Some theoretical insights
may point toward a picture without a critical

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Papers in Physics, vol. 6, art. 060008 (2014) / R. Arévalo

outlet. However, one should take into account
that:

- These theoretical insights contain under-
lying assumptions.

- They are not always based on a complete
understanding of the arching process.

• Simulations in Refs. [1, 8] on the flow from
2D piles are compatible with the existence of
a critical outlet. There is not a theoretical pic-
ture in which to understand the results yet.

• Due to the disparities in the conditions of the
reservoir, it is not a priori clear that results in
2D silos and 2D piles should be comparable.

• Based on results of recent simulations in 2D,
one can speculate that an arch can always
block the outlet irrespective of the flow rate
and, hence of R. For large R, these arches are
just extremely unlikely.

Possible ideas to advance in the future could be:

• Obtain more experimental data for R close
to the transition in silos and piles. This
could eliminate alternative expresions for the
avalanche size.

• Carry out experiments in inclined silos to
mimic reduced gravity.

• Investigate the dynamics of particles in the
region close to the outlet. Here, simulations
should be extremely useful, given that one has
access to all the variables involved.

• Extend the study of piles to three dimensions
and diverse conditions, as reduced gravity.

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