Papers in Physics, vol. 2, art. 020006 (2010)

Received: 10 August 2010, Accepted: 28 October 2010
Edited by: I. Ippolito
Reviewed by: A. Coniglio, Universitá di Napoli “Federico II”, Napoli, Italy.
Licence: Creative Commons Attribution 3.0
DOI: 10.4279/PIP.020006

www.papersinphysics.org

ISSN 1852-4249

Invited review: Effect of temperature on a granular pile

Thibaut Divoux,1∗

As a fragile construction, a granular pile is very sensitive to minute external perturbations.
In particular, it is now well established that a granular assembly is sensitive to variations
of temperature. Such variations can produce localized rearrangements as well as global
static avalanches inside a pile. In this review, we sum up the various observations that have
been made concerning the effect of temperature on a granular assembly. In particular, we
dwell on the way controlled variations of temperature have been employed to generate the
compaction of a granular pile. After laying emphasis on the key features of this compaction
process, we compare it to the classic vibration-induced compaction. Finally, we also review
other granular systems in a large sense, from microscopic (jammed multilamellar vesicles)
to macroscopic scales (stone heave phenomenon linked to freezing and thawing of soils) for
which periodic variations of temperature could play a key role in the dynamics at stake.

I. Introduction: a granular pile as a
fragile construction

A granular pile can be described as a bunch of hard
and frictional grains for which the thermal ambi-
ent agitation is negligible [1]. Indeed, the poten-
tial energy of a grain of density ρ assessed over a
displacement equivalent to its diameter d satisfies
ρgd4/kBT ' 1011 � 1. Thus a granular pile is
an athermal system, and one needs to inject en-
ergy inside the pile to trigger any reorganization of
the packing. The accessible packings can then be
seen as jammed states, i.e. minima of the potential
energy, in some energy landscape (Fig. 1) and the
energy one injects makes it possible to overcome
the energy barrier that separates two configura-
tions from one another. Various methods have been
used, over the passed 20 years, to provide energy

∗E-mail: Thibaut.Divoux@ens-lyon.fr

1 Université de Lyon, Laboratoire de Physique, École Nor-
male Supérieure de Lyon, CNRS UMR 5672, 46 Allée
d’Italie, 69364 Lyon cedex 07, France.

to a granular pile, among which mechanical vibra-
tions [2–4] and shear [5,6] are the most widespread.
Nevertheless, this description of a granular pile in
terms of an athermal system hinders one of its ma-
jor characteristics: namely its fragility [1, 7]. The
sensitivity of a granular pile to minute external per-
turbations has first been pointed out in the case of
frictionless hard spheres [8–11]. A heap of such
spheres presenting a slight polydispersity has been
shown to be isostatic and thus very sensitive to
external perturbations. In the case of frictional
spheres, jamming and isostaticity no longer go hand
in hand and the way the packing has been build
plays a crucial role on its stability [12]. Neverthe-
less, the pile pictured as a contact network, can be
decomposed in two subnetworks: a network gather-
ing strong contacts (weak contacts resp.) involving
grains which carry a force larger (lower resp.) than
the average force in the packing [13]. The key con-
tribution of both this “strong contact” network and
the surface roughness of the grains to the fragility
of the pile is very well illustrated by the Scalar
Arching Model (SAM) [14]. This model, inspired
from the q-model developed by Liu, Coppersmith

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Figure 1: Sketch of the energy landscape associ-
ated to the dynamics of a glass, below the glass
transition. In the case of a granular pile, the ther-
mal agitation is negligible and one has to inject
energy to overcome the energy barrier separating
the jammed states. Reproduced with permission
from [85]. Copyright Wiley-VCH Verlag GmbH &
Co. KGaA.

et al. [15, 16] only takes into account the weight of
the grains and the solid friction between the grains
following Coulomb’s law. The control parameter
of the simulation is the friction coefficient between
two grains, denoted Rc. In this model, looking at
a static pile, Claudin and Bouchaud demonstrated
that a relative variation of Rc as small as 10−7

triggers large scale reorganizations inside the pile,
named static avalanches (Fig. 2), emphasizing the
fragility of a pile to minute perturbations, despite
its athermal nature [14]. Of course, in a laboratory
experiment, controlling and varying the friction co-
efficient between the grains during an experiment is
impossible. Nonetheless, as suggested in [14], vari-
ations of temperature might perturb the pile at the
scale of the surface roughness of the grains, hav-
ing an effect equivalent to a change of the friction
coefficient. Indeed, one can assess that a granular
heap of size L (typically a few centimeters) sub-
mitted to variations of temperature of amplitude
∆T experiences a dilation δL = κgL∆T , where
κg stands for the thermal expansion coefficient of
the grains. Dilations corresponding to the surface
roughness scale lead to ∆T ' 0.1◦C for standard
glass beads (κg = 10−6 K−1). Such an amplitude is
easily accessible and for instance daily variations of

temperature in the lab are already of roughly a few
degrees. This is the topic of this brief review, where
we focus on the effect of temperature on a granular
assembly. The content of this review goes as fol-
lows: In section II we sum up the first experimen-
tal observations of uncontrolled temperature varia-
tions that have pushed for further experiments un-
der controlled variations (section III). In section IV,
we dwell on the use of cycles of temperature to in-
duce the compaction of a granular pile in a delicate
way, in particular we discuss the role of both the
amplitude and the frequency of the imposed cycles.
Section V deals with enlarged “granular” systems
and extend the scope of the results presented in
previous parts. Finally, section VI proposes some
outlooks to this method of thermal cycling as a
general method to probe granular systems.

Figure 2: Force chains in a two-dimensional gran-
ular pile (200×200) following the Scalar Arching
Model. The white dots label the grains involved in
a force chain in the initial force network, whereas
the black dots label the grains involved in the force
chains network after a relative change of 10−7 of the
friction coefficient Rc. One observes that the reor-
ganization takes place in the whole pile, despite the
perturbation taking place at the scale of the grain
surface roughness. Reprinted figure with permis-
sion from [14]. Copyright (1997) by the American
Physical Society.

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II. From undesired variations of
temperature ...

The effect of temperature variations over a gran-
ular pile has first been reported as a hindrance
to perform reproducible measurements. Those ex-
periments were not dedicated to probe the conse-
quences of temperature variations on a granular as-
sembly, and the role of temperature is simply as-
sessed. Nonetheless, several issues were raised in
this seminal work, and are highlighted here.

i. Sound in Sand

The influence of temperature is first mentioned in
the early 90’s in a work dealing with sound propa-
gation in sand [17–19]. An acoustic wave is gener-
ated inside a granular pile and recorded a few cen-
timeters away. C. Liu and S. Nagel observed that
“a temperature change of only 0.04 K inside the
pile, produced by the change of the ambient tem-
perature, or by a local heater, could cause a fac-
tor of 3 reversible change in the measured vibration
transmission” [17]. One can indeed assess the grain
dilation δd associated to such variations of temper-
ature ∆T to be δd = κg∆T d ' 2 nm. In agree-
ments with the reversibility of the effect, such an
amplitude is on the one hand negligible compared
to the typical surface roughness of the beads used in
their experiments (probably 100 nm for glass beads
[32]) and, on the other hand less than the typical
deformation of the beads inside the heap (roughly
10 nm assuming a Hertz-like contact). However, at
the time of these experiments, it was still unclear
whether the temperature or the gradient of tem-
perature were leading to such observations. Exper-
iments performed by Clément and co-workers later
confirmed the key role of the gradient, and brought
to the fore the role of the container dilation on the
reorganization process [20].

ii. Apparent mass fluctuations

In 1997, E. Clément and co-workers reproduced
Jansen experiments [20, 21], which consists in mea-
suring the apparent mass of a granular pile confined
in a tube. A piston at the tube bottom, imple-
mented on an electronic scale, makes it possible to
measure the apparent mass of the pile. Although
the experiment is in good agreement with Jansen’s

Figure 3: Apparent mass and temperature varia-
tions. The system (see text) consists in a vertical
cylinder (radius 2.0 cm) filled with glass beads (typ-
ical diameter 3 mm). A drift in the temperature
of 0.4◦C leads to several reorganizations inside the
pile and thus to several variations of the apparent
mass of the system. Reprinted from [20].

prediction [22], they noticed that their data pre-
sented fluctuations as high as 20 % in the saturated
limit, where the measured mass becomes indepen-
dent of the amount of beads poured inside the tube.
Part of this fluctuations were attributed to temper-
ature fluctuations and the authors emphasized that
“the origin is mixed since it can be due to the dila-
tion of the boundaries and the resulting action on
the piston or it can be due to the dilation of the
grains, themselves inducing spontaneous rearrange-
ments of the force network ” [20]. In particular, the
contribution of the boundary dilation will be ad-
dressed in section IV. Figure 3 illustrates that even
a drift in the temperature of 0.4◦C is sufficient to
trigger rearrangements which lead to measurable
mass fluctuations. However, no systematic mea-
surements were performed on this Jansen config-
uration and it would still be of great interest to
assess the effect of temperature variations on ap-
parent mass of a pile [23].

III. ... to controlled variations of
temperature

Since then, temperature variations have been used
to generate minute perturbations inside a granular
medium. Local heaters placed at different positions

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Figure 4: Sketch of the experimental setup used to
measure the effective thermal conductivity λg. The
granular material consists in glass beads (typical
diameter 300 µm). The copper tube dimensions are
the followings: inner diameter 1 cm; outer diameter
5 cm; length 15 cm. Reprinted from [26].

inside a static pile have been shown to produce
very different disturbances on sound propagation
underlining the key role of force chains in the prop-
agation process [18]. Moreover, small thermal per-
turbations have been used to generate large non-
Gaussian conductance fluctuations in a 2D pack-
ing of metallic beads. Perturbations are induced
by a 75 W light bulb standing a few centimeters
away from the packing and experiments were per-
formed with light turned on for packing prepared
with light off, and vice versa. Gathered in bursts,
such conductance fluctuations were interpreted as
the signature of individual bead creep rather than
collective vault reorganizations [24, 25]. In partic-
ular, the probability density function P (∆t) where
∆t denotes the waiting time between two successive
bursts of conductivity, follows a power law:

P (∆t) ∼ ∆t−(1+αt)

The exponent for stainless steal beads is found to
be αt ' 0.6 independent of both the strength of the
perturbation and of the external stress. The value
of αt is only governed by the surface roughness
of the beads which has been confirmed by AFM
measurements of the Hurst exponent of the bead

surface [24]. In agreement with the SAM [14] dis-
cussed in section I, the origin of the fluctuations of
conductivity originate in local micro-contact rear-
rangements.

More recently [26], temperature variations have
been applied by means of a nickel wire (diameter
rw =100 µm) connected to a power supply, and
which crosses a granular medium partially filling
a copper tube (Fig. 4). The tube is maintained
at a constant temperature Te (precision 0.1◦C).
The imposed current I and the resulting voltage
U are simultaneously measured in order to esti-
mate the heating power P = U I and the wire tem-
perature Tw which is deduced from the resistance
Rw = U/I (Fig. 4). This setup makes it possi-
ble to measure the effective thermal conductivity
λg of the granular material as Tw − Te ∝ P/λg.
One obtains λg = 0.162 W/m/K which is com-
patible with the value expected for a pile of glass
beads (λglass = 1.4 W/m/K) surrounded by air
(λair = 0.025 W/m/K) [27]. More interestingly,
the authors also observe that for the same mate-
rial λg depends on the preparation : One measures
λg ' 0.162 W/m/K if the system is tapped prior to
the measurement and λg ' 0.156 W/m/K if not.
It is then particularly interesting to consider the

Figure 5: Thermal conductivity λg vs. number of
cycles n. The upper curve corresponds to a pile
which has been gently tapped prior to the experi-
ment. The lower curve corresponds to a loose pile.
Note that the conductivity is larger for a larger
density. Moreover, the conductivity of the loose
material increases significantly when one imposes
the thermal cycles (typical grain diameter 300 µm,
∆T = 40◦C). Reprinted from [26].
.

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Figure 6: Change in packing fraction after multi-
ple thermal cycles (cycle temperatures: red, ∆T =
107 ± 2◦C; blue, ∆T = 41 ± 1◦C) for glass spheres
in a plastic cylinder. Lines represent fits of the
data by the sum of two exponentials introducing
a short (resp. long) timescale associated to in-
dividual (resp. collective) rearrangements of the
grains [34,35]. Reprinted with permission from [28].
Copyright (2006) by the Nature Publishing Group.

behavior of the sample when subjected to several
temperature cycles. When the measurements are
repeated several times, one observes that the ther-
mal conductivity of the loose sample along with
its packing fraction significantly increase with the
number n of imposed cycles. By contrast, the con-
ductivity of the tapped sample only slightly fluctu-
ates around a constant value (Fig. 5). Such results
clearly emphasize that one can induce compaction
by thermal cycling. This issue is addressed in the
following section.

IV. Compaction induced by thermal
cycling

The compaction of a granular pile has been studied
in two different configurations: (i) both the grains
and the container are submitted to cycles of tem-
perature [28–30], and (ii) the grains alone experi-
ence the cycles [31, 32]. In both cases, the packing
fraction increases under thermal cycling. We first
review the role of the amplitude and frequency of
the cycles for both cases. Second, we gather some
insights on the dynamics at the grain scale.

i. The role of the cycling amplitude and fre-
quency

Chen and co-workers [28, 30] examined the change
in packing fraction for glass, polystyrene or high
density polyethylene spheres (typical diameter
1 mm) contained in polymethylpentene plastic or
borosilicate glass cylinders (diameter ranging from
14 to 102 mm) in response to thermal cycling. One
thermal cycle is conducted by placing both the
grains and the container into an oven until the ther-
mal equilibrium is reached, before letting them re-
lax at ambient temperature. The typical cycling
period is about 10 hrs and has not been varied.
They observed that the packing fraction increases,
even after one cycle [28]. This effect is indepen-
dent of the sample depth and width (14-102 mm),
more efficient for higher cycling amplitudes (Fig. 6)
and, contrary to their first guess, also independent
of the relative coefficients of thermal expansion of
the grains and the container [30], as confirmed by
numerical simulations [33]. This indicates that the
compaction mechanism is local as suggested by the
observations discussed in section III and thus linked
to the solid friction between the beads, and between
the beads and the container.

Two other setups have been used to impose cy-
cles of temperature: the first one consists of a ver-
tical glass tube filled with spherical glass beads
[26, 29]. The temperature cycles are imposed by
means of a heating cable (40 W/m) directly taped
on the outer surface of the tube wall. Here again
both the container and the grains are thus submit-
ted to the cycles. The free surface of the material
is imaged from the side with a video camera which
makes it possible to measure accurately the column
height H from the images and to perform a time re-
solved study of its dynamic. In such a setup one can
control both the amplitude ∆T of the cycles, and
the penetration length lp ≡

√
2λ/(ρ Cω) through

the frequency ω/2π of the cycles (λ and C respec-
tively denote the thermal conductivity and heat ca-
pacity of a typical glass-grains pile). Prior to each
experiment, the granular column is prepared in a
low-density state thanks to a dry-nitrogen upward
flow. The top of the column is then higher than
the field imaged by the camera and one sets the
amplitude of the cycles, ∆T , to the largest accessi-
ble value, ∆T = 27.1◦ C. The column flows under
thermal cycling, and the preparation of the sample

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Figure 7: Height variation hn vs. number of cycles
n. One observes first an exponential behavior at
short time followed by a logarithmic creep at long
time. The black curve corresponds to the test func-
tion htn ≡ h0 + he exp (−n/nc) + hl ln(n). Inset -
Oscillations of the column height associated with
the temperature cycles : An and δn are respec-
tively defined to be the amplitude of the increase
and the drift of hn at the cycle n (H = 140 cm,
2π/ω = 600 s and ∆T = 10.8◦C.). Reprinted fig-
ure with permission from [29]. Copyright (2008) by
the American Physical Society.

ends when the top of the column enters the obser-
vation field. At this point, the granular column is
“quenched”: the amplitude of the cycles is set to
the chosen value ∆T lying between 0 and 27.1◦C,
which defines the origin of time t = 0. The gran-
ular column is subsequently submitted to at least
1000 cycles.

Role of cycling amplitude- Here, the cycling pe-
riod (10 minutes) is chosen to ensure that the as-
sociated thermal penetration length lp ' 6 mm is
about the tube radius so that all the grains expe-
rience the dilation process. Figure 7 reports the
variation of the column height defined as hn ≡
H(2πn/ω) − H(0), where n denotes the number of
imposed cycles. For a large ∆T (typically more
than 3◦C), the column systematically compacts
(Fig. 7) at each cycle [Fig. 7, (inset)]. Follow-
ing Barker and Mehta [34, 35] like Chen and co-
workers [28, 30], one can try to fit the height de-
cay by the sum of two exponentials: the result
is not satisfying and the compaction dynamics is
better accounted for by the following test function:

Figure 8: Amplitude An (defined in Fig. 7) versus
the number of cycles n. The data are successfully
accounted for by An = ∆T [a0 + b0 ln(n)] with a0 =
1.0 µm·K−1 and a0 = 0.13 µm·K−1 (initial column
height: 140 cm, cycling period: 2π/ω = 600 s and,
from top to bottom, ∆T =27.1, 16.2 and 10.8◦C).
Such an increase in the dilation amplitude with the
cycles of temperature is a signature of the aging
the granular pile experiences under thermal cycling
[26]. Reprinted figure with permission from [29].
Copyright (2008) by the American Physical Society.

htn ≡ h0 + he exp (−n/nc) + hl ln(n) (Fig. 7) [32].
First, this response to the thermal quenching is
strikingly similar to the one the system exhibits to
vanishing step strain perturbations [36]. Second,
the long time logarithmic behavior of the column
height leads to an inverse logarithmic evolution for
the packing fraction φn which is strongly reminis-
cent of the way the packing fraction of the granular
pile submitted to tapping evolves in the absence of
any convection [2]. Another remarkable feature of
this time resolved study is that one can also observe
that An, defined as the amplitude of the increase
of hn at the n-th cycle (Fig. 7), is proportional
to the cycling amplitude ∆T and increases loga-
rithmically with n (Fig. 8). The more the column
compacts, the higher is the dilation amplitude of
the total height of the column at each cycle. This
is a signature of the aging of the pile under thermal
cycling, like the increase of the thermal conductiv-
ity (section III).
In the limit of small cycling amplitude ∆T (here
below ∆Tc = 3◦C), one observes that the column
is not flowing regularly anymore, but evolves by
successive collapses (typical amplitude of a tenth

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of grain diameter) separated by rest periods (ran-
domly distributed) [Fig. 10 (top)]. Such a dif-
ferent compaction process from the one observed
for high cycling amplitude has been linked to the
surface roughness of the glass beads. Indeed, for
∆T < ∆Tc the dilation of a grain is smaller than
the surface roughness, and thus, the beads behave
as smooth particles whose dilation induces only lo-
calized rearrangements. Thus one has to wait sev-
eral cycles of temperature before observing a large
scale collapse of the column level, as a cumulative
effect of several local reorganizations. On the con-
trary, for ∆T > ∆Tc the dilation of a grain is larger
than the surface roughness, and thus the beads be-
have as rough particles. One cycle of temperature
is more likely to generate a large scale reorganiza-
tion [29].

Role of cycling frequency- The influence of the
frequency is discussed in [32], but still remains to
be fully explored. On the one hand, for frequencies
shorter than the one discussed above, the penetra-
tion length is larger than the tube radius. The
compaction process is more efficient, and the dila-
tion amplitude An is constant, independent of the
age of the system. On the other hand, for larger
frequencies, the penetration length is shorter than
the tube radius. The column is observed to flow
continuously while small amplitude settlings may
occur (Fig. 9). The interpretation proposed in [32]
is the following: the penetration length is shorter
than the tube radius, the grains in the center of the
column are not submitted to the cycles of temper-
ature and thus behave as a solid body that expe-
riences some stick-slip motion due to the periodic
dilations of the grains close from the walls of the
container. Indeed, the column dynamics over a few
cycles [Fig. 9, (inset)] displays a very similar behav-
ior to a tilted monolayer of grains on an inclined
plane (see Fig. 3 in [37]). In this case, the flow
results from a competition between a gravitational
flow and the formation of arches [37]. However, a
full study on the role of the frequency, and in par-
ticular on the influence of harmonics in the shape
of the cycles (squares, tooth-like signals, etc.) on
the compaction dynamics still remains to be done.

The second setup that has been used to impose
temperature cycles is very similar to the one pre-
viously discussed in figure 4. A thin wire crosses a
granular column in its center, while this time the
container is a thin glass tube [31]. For large enough

Figure 9: Evolution of the column height hn ver-
sus the number n of cycles of temperature. Inset:
zoom on the evolution of hn over 24 cycles of tem-
perature: the column settles by jumps separated
by periods of continuous flowing (H = 140 cm,
T = 2π/ω = 150 s et ∆T = 9.5◦C). Reprinted
from [32].

frequencies, the penetration length is shorter than
the tube radius which makes it possible to cycle
the grains close to the wire and let the container
at rest. It is here relevant to compare the way the
dilation of the container modifies the compaction
process in the limit of low amplitude cycles. In the
case where both the grains and the container are
submitted to variations of temperature, the com-
paction process is linear in the number n of applied
cycles [Fig. 10 (top)], whereas, in the case where
the grains alone are submitted to cycles of tempera-
ture and the container is fixed, the compaction pro-
cess goes logarithmically with n [Fig. 10 (bottom)].
Both experiments were conducted under the same
cycling frequency, and for comparable cycling am-
plitudes (resp. ∆T = 2.8 and 1◦C). Those results
clearly indicate that the dilation of the container
plays a key role in the global compaction dynamics
of the granular column. However, up to now, the
full contribution of the container dilation has not
been characterized. The answer should be easily
obtained with the set-up presented in figure 4.

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Figure 10: Height variation hn of the column versus
the number of cycles of temperature. (Top) Curve
obtained in the case where temperature cycles were
applied to both the container and the grains. On
average, hn decreases linearly with the number of
cycles and the column settles by jumps separated
by rest periods (H = 140 cm, 2π/ω = 600 s,
and ∆T =2.8◦C). Reprinted figure with permis-
sion from [29]. Copyright (2008) by the American
Physical Society. (Bottom) Curve obtained in the
case where cycles of temperature are applied to the
grains alone by means of a hot wire crossing the
pile (see Fig. 4). The column also settles by jumps,
but the height variations goes as the logarithm of
the number of cycles. Reprinted from [31].

ii. Dynamics at the grain scale under ther-
mal cycling

One of the key features of the motion induced by
thermal cycling in a granular pile is that a grain
stays in contact with its neighbors and that the
global motion is really slow (hours and days typ-
ically). It makes it possible to use experimen-
tal techniques developed to study creep motion,
like dynamic light scattering (DLS) [38], successive

snapshots [29] or 3D scanning [39] to follow the in-
dividual grain trajectories. Recently, L. Djaoui and
J. Crassous used a DLS setup to follow the evolu-
tion of a granular pile under thermal cycling [40].
This method was successful at extracting both lin-
ear and non-affine displacement of the grains [41],
as recently done under mechanical shear [42], thus
confirming that controlled temperature variations
are a delicate way to generate small displacements.
Slotterback and co-workers have been following the
dynamics induced by thermal cycling at the grain
scale [39]. In their experiment, the column of grains
is immersed in an index-matching oil containing a
laser dye which makes it possible to produce 3-
D images of the system at the end of each cycle
by means of a laser sheet scanning method. Par-
ticle displacements in this jammed fluid correlate
strongly with rearrangements of the Voronoi cells
defining the local environment about the particles.
This might be a proof for stringlike cooperative mo-
tion as already pointed out in quasi-2D air driven
granular flows [43], and more generally in super-
cooled liquids [44, 45]. However, it is rather diffi-
cult to compare these results to those obtain for
dry grains [28, 29]. First, because in the case of the
immersed experiment, a weight is placed on top of
the granular column to apply a controlled vertical
force, whereas in the dry case the top column is free
of any weight. Second, the presence of the intersti-
tial fluid lubricates the contacts between the grains,
and certainly leads to a more homogeneous prop-
agation of temperature front through the pile [46].
Indeed, the thermal conductivity of oil is roughly
10 times higher than the thermal conductivity of
air which reduces the role played by force chains in
the heat conduction. Those three effects speed up
the compaction process compared to the dry case:
the steady state packing fraction is obtained after
only 10 cycles in the immersed system (Fig. 11).

iii. A comparison with the tapping experi-
ments

Let us here recall that tapping experiments consist
in imposing vertical vibrations to a container full
of grains. In practical terms, a low frequency sig-
nal (usually 10 < f = ω/2π < 100 Hz) [47] is used
to generate a “tap” of amplitude a, two successive
taps being separated by a duration ∆t long enough
to be considered as independent (roughly ∆t ' 1 s).

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Figure 11: Packing fraction vs the number of cy-
cles of temperature for various temperature differ-
entials. The system consisting of glass beads im-
mersed in an index matching oil inside a cylinder,
is subjected to thermal cycling via a water bath. A
weight is placed on top of the bead packing to ap-
ply a controlled vertical force. Both this weight and
the interstitial fluid might explain the rapid com-
paction compared to what is observed in Fig. 7.
Reprinted figure with permission from [39]. Copy-
right (2008) by the American Physical Society.

The key control parameter is the reduced accelera-
tion Γ defined as Γ ≡ aω2/g [48] and one can distin-
guish between three different regimes. For low val-
ues of the reduced acceleration (0 ≤ Γ ≤ Γ∗ ' 1.2),
where Γ∗ denotes the critical acceleration at which
the grains lose contact with the bottom of the con-
tainer, the compaction process is extremely slow
[49]. This regime is very similar to that observed
for thermal cycling: the geometry of the pile is
essentially frozen and the force network can still
evolve by slowly depleting the most fragile contacts
[50, 51]. Besides, under such small vibrations, the
packing is aging: the behavior of the pile is first
dominated by grain motion and then by the con-
tact force variations at larger timescales [51]. How-
ever, in this range of acceleration, no phenomeno-
logical law was proposed to describe the evolution
of the packing fraction and/or the column height
that could be compared to the results obtained un-
der thermal cycling. For intermediate values of the
acceleration (Γ∗ ' 1.2 ≤ Γ ≤ Γc ' 2), the com-
paction process is more efficient and takes place

over timescales accessible in the lab [2,49]. At each
tap, the packing loses contact with the bottom of
the container, and crushes back which generates a
shockwave responsible for the compaction of the
pile. A convection roll might take place inside the
container - roughly, if the ratio of the container size
to the typical bead diameter is large enough, a sig-
nature of which can be found on the free surface
of the pile that presents a slope [52]. Such convec-
tion rolls are not observed in a pile submitted to
cycles of temperature as confirmed by the results
observed under tapping for higher range of acceler-
ation. Indeed, for larger values of the acceleration
(Γ > Γc ' 2), the compaction takes place homoge-
neously in the whole shaken sample [2, 53], but the
dynamics strongly depends on the presence or the
absence of convection rolls inside the pile [4,53,54].
Without any convection roll, the packing fraction
φn evolves in a similar way to what was found for
thermal cycling in [29]:

φ∞ − φn ∝
1

ln(n)
(1)

where φ∞ is the steady state packing fraction ob-
tained at long time. This phenomenological expres-
sion, first proposed in [2], has been justified the-
oretically using various approaches: analogy to a
parking process [55,56], “tetris-like” models [57,58],
excluded volume approach [59] and void diffusion
models [60, 61]. A common feature of these models
is the geometrical frustration: the denser the pack-
ing, the harder it becomes to insert grains from the
top. Such a frustration process seems to be also
at stake in the compaction induced by thermal cy-
cling, which still remains to be properly put into
equations. Still, under tapping, if convection rolls
develop inside the pile, the compaction dynamics
follows a Kohlraush-Williams-Watts (KWW) law:

φ∞ − φn ∝ exp
[
−(n/nf )β

]
(2)

where nf and β are two parameters which depend
on the acceleration Γ [49, 53, 62]. Such a law pops
up naturally if one considers the global dynamics at
stake as a superposition of several processes, each
with a proper and well defined characteristic time.
Indeed, one can see here the compaction process as
the sum of the reorganization of several groups of
beads, each with different sizes and different relax-
ation times. One can also emphasize that the global

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Papers in Physics, vol. 2, art. 020006 (2010) / T. Divoux

dynamics is not controlled by the biggest (and thus
slowest) groups of beads, but by the fastest individ-
ual beads. These are the grains that can quickly
relax individually and jump over large distances
(roughly their radius), and so during a single tap,
that control the compaction dynamics [63]. This
result indicates once again that there are certainly
no convection rolls inside a granular pile submit-
ted to cycles of temperature as neither the KWW
law nor rare jumps of individual beads are observed
experimentally in this case [64].

Figure 12: (Top) Scheme of the capillary contain-
ing the onion gel, the location of the field of view
and the orientation of the parallel and perpendic-
ular axis. (Bottom) Portion of a typical image
of the sample taken by light microscopy between
cross polarizers; The arrow points to a large onion.
Reprinted figure with permission from [66]. Copy-
right (2009) by the American Physical Society.

V. Enlarged “granular” systems

In this section, we tackle two other systems differ-
ent from a simple dry granular assembly and for
which temperature cycles might play a crucial role
in the observed dynamics.

i. Thermally driven aging in a polydisperse
vesicle assembly

The closest case from what we have been presenting
in this article consists in a dense packing of poly-
disperse multilamellar vesicles, or “onions”, sub-
mitted to (unavoidable) temperature fluctuations

[65, 66]. Such a system, a water-based mixture of
surfactants and block-copolymer, is fluid below 8◦C
but forms a phase of jammed vesicles at ambient
temperature (here 23.4◦C) which is known to ex-
perience aging [67, 68]. The system is loaded in a
glass capillary which is flamed sealed and placed
under a microscope. Images are taken every 15 sec
during 24 h (Fig. 12). The temperature is con-
trolled within 0.09◦C. Mazoyer and co-workers ob-
served that the unavoidable temperature fluctua-
tions induce local mechanical shears in the whole
sample due to local thermal expansion and con-
traction, which is a scenario very similar to the
one proposed in [29] for dry grains. Here, each
image [Fig. 12 (Bottom)] is divided into small re-
gions of interest (ROI) and the authors look at
the translation motion ∆R(tw, τ ) of each ROI for
pairs of images taken at two different times (tw
and tw + τ ). Both the global parallel displacement
〈∆R//〉(tw) [Fig. 13 (b)] and the relative parallel
displacement [〈∆r2

//
〉]1/2(tw) [Fig. 13 (a)] present

strong correlations with the temperature fluctua-
tions ∆T [Fig. 13 (c)] [65]. Furthermore, looking
at individual trajectories of ROIs, the authors iden-
tify two kinds of dynamical events: reversible and
irreversible rearrangements. The first class is due
to the contraction-elongation of the sample because
of temperature fluctuations, and correspond to a
shear deformation along the long axis of the capil-
lary. The second class of events that are irreversible
occurs as the result of repeated shear cycles. The
motion resulting from these ultraslow rearrange-
ments is ballistic (the initial growth of ∆r// is pro-
portional to τ ), with a velocity that decreases expo-
nentially as the sample ages. Such results could be
easily checked in dry and immersed granular sys-
tems [29, 39] to see how far the analogy between a
jammed vesicle assembly and a granular pile goes.
Also, it raises the generality of such dynamics gov-
erned by temperature fluctuations. In the case of
dilute colloidal systems [69], this scenario does not
hold, as no correlation could be observed between
temperature fluctuations and rearrangements. In-
deed, one expects temperature fluctuations to play
a key role in divided and athermal systems being
rather concentrated or solid-like. It might therefore
be relevant to find out other systems presenting
temperature-induced strain fluctuations, in order
to find how general the properties of these ultra-

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Papers in Physics, vol. 2, art. 020006 (2010) / T. Divoux

slow rearrangements are.

Figure 13: Age dependence of (a) the square root
relative parallel displacement, (b) the global paral-
lel displacement and (c) the temperature variation
∆T over a lag time τ . Reprinted figure with permis-
sion from [65]. Copyright (2006) by the American
Physical Society.

ii. Stone heave phenomena

Cyclic freezing and thawing of soils can cause stones
and particles embedded in those soils to move and
relocate mainly depending on the initial void ra-
tio (defined as the fraction of sample filled by
empty space). The stones move vertically upward
(Fig. 14) in dense soils (low void ratio) and ver-
tically downward in loose soils (high void ratio).
A full interpretation of this phenomenon is still
lacking, but in the case of stone heave, the mech-
anism invoked lays on the propagation of the frost
front in the soil: the freezing from the top leads to
the growth of the ice beneath the stone, due to its
higher thermal conductivity than that of the sur-
rounding soil and in turn, causes the stone heave.
A review of this topic can be found in [71]. A re-
markable effect is that the void ratio also evolves
toward a critical value (in the range of 0.29-0.34)
under the successive freezing and thawing cycles
[71]. Thus, a dense (loose resp.) soil will become
looser (denser resp.) under cycles of freezing and
thawing. This is exactly what experiences a gran-
ular assembly submitted to a mechanical shear:
its packing fraction will tend to a critical value
[72]. Here again, as for onions and dry granular
assemblies, the temperature-induced shear controls
the dynamics. Such an analogy suggests that one

could shed some new lights on the stone heave phe-
nomenon by looking at the way a large intruder
(simulating the stone) in a granular pile (simulat-
ing the soil) behaves under thermal cycling (which
produces a shear [29]). Such a work has been tack-
led by Chen and co-workers in [30]. They observed
that, under thermal cycles, the intruder (aluminum
or brass) either does not move, or sinks inside the
pile (polystyrene beads in a borosilicate glass con-
tainer) if the intruder density overcomes a certain
threshold. The pressure exerted by the intruder on
the pile seems to be the relevant parameter, despite
the ratio of the container size to the grains diam-
eter which might play a crucial role on the force
network inside the pile, has not been varied. Also,
no experiments have been performed varying the
initial packing fraction or the initial position of the
intruder in the granular packing to mimic the ob-
servations performed on stone heave [71, 73]. Still,
such a simple setup is certainly relevant to extract
the key ingredients of the stone heave phenomena,
and deserves further use. A comparison of the re-
sults with those obtained for vibrated grains (Brazil
nut effect [3, 74]) would also be interesting.

Figure 14: Raised stone in a paved parking lot
in Lule̊a, Sweden. Reprinted from [70], Copyright
(2000), with permission from Elsevier.

VI. Summary, open questions and
outlooks

i. Summary

Temperature variations, even of low amplitude, in-
duce the reorganization of a granular assembly and

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Papers in Physics, vol. 2, art. 020006 (2010) / T. Divoux

its slow compaction, and lead to the aging of the
system, a signature of which can be found in the
slowing down of the dynamics and in the evolution
of the key properties of the system (increase of the
effective thermal expansion and thermal conduc-
tion coefficients, etc). In this sense, temperature
variations induce aging in a pile the same way mois-
ture [75,76], constant applied stress [77] and chem-
ical reaction [78] do. However, the mechanism at
stake lays on a pinning-depinning transition at the
grain scale generated by the shear-induced succes-
sive contractions and dilations cycles. Temperature
variations are thus a local and very delicate way to
perturb and induce displacement in a granular as-
sembly.

ii. Open questions and outlooks

First, the role of several parameters still remains
to be assessed: the mean temperature T around
which the variations of amplitude ∆T are imposed
might play a role on the reorganization process in-
side the pile. Because of the presence of the con-
tainer and the fact that the granular pile is a de-
formable medium, the mean temperature impacts
on the initial stress distribution and on the space
available for the grains. Its role fully remains to
be investigated. Also, the parameters relevant in
the case of vibrated grains are to be studied here.
Namely, the shape of the grains [79–82], their sur-
face roughness [83, 84], their polydispersity [3], etc.
But also, the size ratio of the grains and the con-
tainer [53]. Such a study might help to compare
more accurately tapping and thermal cycling ex-
periments. It could then be relevant to mix the
two types of solicitation on a granular assembly to
test the statistical framework proposed by Edwards
[4, 85]. Indeed, recent experimental results lead to
think that the steady state packing fraction reached
by a shaken granular column (and function of Γ) is
a genuine thermodynamic state, within this theo-
retical framework [86, 87]. It might thus be rele-
vant to study the fluctuations of packing fraction
[88] around its equilibrium value, induced by cycles
of temperature. Also, moisture is another relevant
parameter that would deserve an exhaustive exper-
imental study. Indeed, the existence of a liquid
bridge between two adjacent grains is expected to
increase the effective thermal conductivity [46] and
thus to reinforce the role of the force chains inside

the pile.
Second, the aging phenomenon observed under

thermal cycling needs to be better characterized.
In particular, how can we compare the aging effects
observed in vibrated granular piles and described
in [89] to the one described here ? Another way to
put it would be to ask if we can loosen a sample by
means of cycles of temperature.

Third, more insights on the individual grain dy-
namics is needed in the dry case. Do the re-
sults of Slotterback and co-workers discussed in sec-
tion IV for immersed grains still hold in the dry
case ? The techniques already used for vibrated
dry granular assembly like X-rays tomography [90]
or γ-rays absorption [49] would certainly provide
some answers to these questions. Also, numeri-
cal simulations either simply based on successive
dilation/contraction of the grains to mimic the ef-
fect of temperature, or taking into account the heat
conduction through the granular media [33, 91, 94]
would spare time and unravel the key parameters
in the dynamics at the grain scale.

Acknowledgements - I am indebted to my PhD
advisor, J.-C. Géminard, for introducing me to this
flourishing topic and for countless enlightening dis-
cussions. I also thank S. Manneville and M.-A.
Fardin for carefully reading this manuscript as well
as E. Bertin and S. Ciliberto for fruitful discus-
sions.

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