Granular piston-probing in microgravity:

powder compression, from densification to jamming

1,2]Olfa D'AngeloCorresponding author<EMAIL_ADDRESS>3]Anabelle HorbCurrent address: Omnidea, Lda. – Polypark, Nùcleo Empresarial da Arruda dos Vinhos, Estrada da Quinta de Matos 4, 2630-179 Arruda dos Vinhos, Portugal.
3]Aidan Cowley
1,4]Matthias Sperl
4,1]W. Till Kranz

[1]Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft- und Raumfahrt (DLR), 51170 Köln, Germany.
[2]Institute for Multiscale Simulation, Universität Erlangen-Nürnberg, Cauerstraße 3, 91058 Erlangen, Germany.
[3]European Astronaut Centre (EAC), European Space Agency (ESA), 51170 Köln, Germany.
[4]Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany.

The macroscopic response of granular solids is determined by
the microscopic fabric of force chains, which, in turn, is intimately
linked to the history of the solid.
To query the influence of gravity on powder flow-behavior,
a granular material is subjected to compression by a piston in a closed container,
on-ground and in microgravity (on parabolic flights).
Results show that piston-probing densifies the packing,
eventually leading to jamming of the material compressed by the piston,
regardless of the gravitational environment.
The onset of jamming is found to appear at lower packing fraction
in microgravity
($\varphi^{\textrm{$\mu$-g}}_J = 0.567 \pm 0.014$)
than on-ground
($\varphi^{\text{gnd}}_J = 0.579 \pm 0.014 $).
We interpret these findings
as the manifestation of a granular fabric
altered by the gravitational force field:
in absence of a secondary load (due to gravitational acceleration) to stimulate reorganization
in a different direction to the major compression stress,
the particles' configuration becomes stable at lower density,
as the particles have no external drive
to promote reorganization into a denser packing.
This is coupled with a change in interparticular force balance
which takes place under low gravity, as cohesive interactions become predominant.
We propose
a combination of microscopic and continuum arguments
to rationalize our results.

§ INTRODUCTION

As humans make their way into space
and plan to travel
to sand-covered celestial bodies,
the influence of (reduced) gravity on granular phenomena becomes
a matter of concern <cit.>,
and new strategies are sought
to handle granular materials efficiently
in various gravitational environments <cit.>.
While the action of gravity
is intuitively expected to
weaken as objects become smaller,
the collective behavior of small particles composing granular systems is
significantly modified
by the
gravitational field,
challenging our expectation of powder response outside of Earth gravity.
Granular media are by definition composed of grains
big enough not to be subject to thermal motion <cit.> but small enough
for continuous material properties to emerge on the mesoscopic scale.
Such grains are, therefore, generally
sufficiently large for gravitational forces on each of them not to be negligible in comparison to the other forces in presence.
The flow behavior of granular matter can thus change dramatically with a change in gravitational acceleration,
such as that
affecting a body on a foreign celestial surface (e.g. Moon, Mars, or an asteroid), or subjected to microgravity conditions.

The aim of this article is to
the densification of a granular packing
in two scenarios:
1g, where Earth's gravity may accelerate particles relative to the fixed container, and
in mug,
on parabolic flights, where gravity cannot induce such relative acceleration. Note, that the term microgravity is widely used, although free-fall would be more precise <cit.>.

We propose a densification mechanism in the form of a piston
rising quasi-statically through a confined powder bed (see
Fig. <ref>), thereby compressing the material placed above.
The piston is of diameter inferior to that of the container.
Compression is continued until reaching a jammed state above the piston,
hence probing
the jamming point variations
the two environments.
The jammed state is characterized by
the appearance of a finite elastic modulus of the granular packing,
which drives the piston to a stop.
Such piston mechanism
could be used,
on-ground or in space,
to densify
shallow granular beds
and probe the percolation of the force network, possibly
before submitting the material to further processing.

In presence of gravity, granular flows can be caused by their
potential energy in gravity-driven flows, providing a granular
time-scale in the form of the time needed for a grain to fall by its
diameter.  In microgravity, particles do not fall towards the
minimum-height position available; instead, they cluster into floating
aggregates <cit.>, as the
interparticle cohesive forces become the predominant interaction.  If
no force field sediments the packing, particles or clusters are not
necessarily in contact with their closest
neighbors <cit.>.  Force
chains <cit.>, enabling force transmission
through lasting contacts in a dense granular medium, cannot appear
unless a contact network is established. Yet, such contact network
can also trigger flow arrest on the macroscopic scale, if it
percolates into a force network spanning the system, transmitting
force to its boundaries akin to a solid <cit.>.

The jamming transition <cit.>
describes the changeover between the flowing state, in which the
granular fabric deforms plastically,
and the jammed state, where
a mechanically stable contact network is formed,
which can withstand a certain amount of stress without reorganizing.
In other words,
the granular assembly as a whole
exhibits a finite elastic modulus.
If unpredicted,
the appearance of jammed regions can be
detrimental to
industrial processes
involving handling and transport of granular materials,
as it can draw entire production processes to a halt.
In sheared granular media, an anisotropic shear-jammed phase
can block powder flow
mono-directionally <cit.>.
It was suggested that for frictionless particles, this transition
already occurs at rlp <cit.>,
while a granulate in the state of rcp would
be isotropically jammed <cit.>.
However, the existence of a
fixed jamming packing fraction $\varphi_\text{J}$ (jamming point) has now been challenged <cit.>.

Besides evident dependence on particles'
shape <cit.>, (poly)dispersity <cit.> or mechanical properties <cit.>,
experimental and numerical studies have shown the
of $\varphi_\text{J}$,
with variations, even for a single granular material,
related to
the preparation protocol <cit.>
or compression rate of the granular bed leading to jamming
(namely, $\varphi_\text{J}$ decreases with increasing compression rate <cit.>).
Interparticular friction was shown to decrease the jamming
packing by decreasing particles' mobility in a dense
packing <cit.>.

It has been reported that when
gravitational forces are negligible,
the density of a granulate in rlp
decreases from
$\varphi^{\text{gnd}}_{\text{rlp}}=0.60$ to
$\varphi^{\textrm{$\mu$-g}}_{\text{rlp}}=0.55$ <cit.>
(for a granular material composed of hard monodispersed spherical particles with low or no friction in three dimensions).
However, to the best of our knowledge, the
impact of gravitational acceleration – and particularly of very low gravity $\sim \textrm{$\mu$-g}$ – on the packing fraction at the jamming transition
has hitherto not been studied experimentally.

Before reporting on the actual measurements, let us make a naive
We consider a piston placed at the bottom of
a container filled with granular material at a density $\varphi \approx 0.55$.
As in our setup, the piston has a diameter $d = \nicefrac{2}{3} D$, where $D$ is the container diameter.
When the piston rises
by a small height $\Delta h$,
it displaces the volume $\pi d^2\Delta h/4$ of powder above the piston.
This may either be achieved by compacting the powder to
a higher packing fraction above the piston,
or by pushing
powder beneath the piston.
1g, the material underneath the
piston will settle in an annular heap at its angle of repose
$\alpha$. At small piston displacements $h$, the volume of that heap
grows more slowly by a factor $2h / d\tan\alpha \ll 1$ compared to the
displaced material on the top. Consequently, we expect the piston to stop
shortly after the material is fully compacted. Under mug, however,
the displaced material does not have to settle in a
heap; it may instead
fill the entire volume freed below the piston. Consequently, one
may expect the piston to be able to rise indefinitely
without being stopped by a jammed phase.

§ RESULTS

§.§ Piston-probing

Piston-probing consists of applying a finite amount of normal stress
to a granular material in a confined space, by raising a piston inside a
container filled by this material.
A lateral space is left on the sides of the piston's platform,
allowing the granulate to yield and flow downwards, around the platform.
If applicable, the gravity vector points downward, against the rising motion of the piston.

The experimental setup is presented in
Fig. <ref>.
It consists of a transparent
pmma tube of diameter $D=\SI{54}{\mm}$ and height
$H=\SI{48}{\mm}$, inside which a piston of diameter $d=\SI{36}{\mm}$
can rise incrementally.
The piston is moved by a dc motor, driving a traveling-nut
linear translation stage that allows
the piston to rise vertically without rotating.
The translation stage can be placed in one of two
modes by the user: (i) a stationary mode, where the piston is actively
stabilized at its current position; and (ii) a traveling mode where
the piston is actively pulling upwards. The dc motor is
regulated through a velocity-controlled closed control loop that keeps
the velocity at its set value until the resistance of the packing
reaches a significant fraction of the motor's maximum torque,
at which point the
piston slows down and eventually stops.
The maximum torque was chosen such, that the experiment probes
the elastic modulus of the jammed packing, yet only minimally
deforms the particles which are much stiffer (see Methods).
An optical measurement of the piston height $h$ provides an indirect
assessment of its position, thus of its true rising speed.

Piston-probing setup.
(a) Piston-probing principle: a piston rises incrementally in a closed container, filled with powder at packing fraction $\varphi$, applying a normal stress $\Vec{\sigma}$ on the granular packing.
The pressure $p$ transmitted through the granular material is recorded by two pressure sensors placed on the lateral walls of the container (respectively recording $p_u$ and $p_l$, upper and lower pressures).
The direction of the gravitational acceleration $\Vec{g}$ is indicated by a downwards pointing arrow.
(b) Experimental setup, showing the container (here empty) in its support structure (photograph taken by the authors).
(c) Schematic including container's and piston's dimensions (in grey), and variables
recorded (in black).

Pressure sensors are placed inside the tube, on the lateral walls, and
record the pressure evolution as the piston compresses the granular
sample. One sensor is placed on the upper part of the container
($h_u \approx $ 36), and another one on the bottom part
($h_l \approx $ 12).  The sensitive area of the pressure
sensors measures 200□, which represents approx. 30 000
particles.  The pressure sensors are calibrated in-house (see
Methods for details), so the
pressure recorded corresponds to the pressure difference
due to the powder contained in the cell.
A camera below the
transparent container
allows to observe inside the setup
from below.
At the beginning of the experiment, the piston is generally at its lowest
position, i.e. the platform touches the bottom of the closed tube; it
is then programmed to rise step-by-step, each step being activated
by the user during approximately 20.
In some experiments the
platform is placed in the middle of the box, at its median position in
height (cf. results shown in Fig. <ref>(a) and (b)).
Hardware technical specifications are given
in Table <ref>,
and a list of all experiments reported is provided in Supplementary Material
(Supplementary Table 1).

References for hardware used on the piston-probing experimental setup.

Hardware     Model

dc motor     DSMP 320-24-0189-BF

(max. torque 0.69)

current sensor     Pololu ACS7174

pressure sensor      SingleTact capacitive pressure sensor

camera     YI 4K Plus Action Camera

(resolution $3840 \times 2160$ pixels)

The granular material used to fill the container is a monodispersed ps
powder with spherical particles of diameter 80
(see Methods
for further details).
Inside the closed container, the mass of
granular material inserted determines an overall packing fraction of
$\varphi_0 \approx 0.55$.
At the beginning of the experiment (that is, until reaching a jammed state),
the packing fraction is assumed to be sufficiently homogeneous
to be calculated from this initial packing fraction $\varphi_0$
and the piston's position in height, $h$.
Therefore, the
packing fractions given throughout this article, and in particular the packing fractions at the jamming transition $\varphi_{\text{J}}$, are global packing fractions;
the local packing fractions are not determined experimentally.
A calculation of formal uncertainty is provided in the Methods section.

Let us consider the most common case, where the piston is initially
at the bottom of the container ($h=0$).
The piston diameter being two thirds of
the container,
a lateral space is left around the piston,
through which the granular material can flow
downward to reach the bottom of the container (below the piston).
The dimensions of the
piston are chosen such, that the upward flow imposed by the piston
rise could be accommodated by a downward flow through the gap
around the piston. Maximum speed and force of the piston were chosen
to give the packing ample time to relax, yet allow for significant
piston motion within the limited time intervals of mug,
and to limit particle deformations in the jammed state.

During the initial instants of the experiment,
the powder distribution
under the piston can be observed through a transparent bottom
plate. At all stages of the experiment, the piston position $h$ and the
pressure on the outer walls at two heights, $p_u$ and $p_l$ (respectively upper and lower), are recorded.

This experiment flew on the 31st dlr
pfc in March 2018.
pfc provide periods of mug ($\sim 10^{-3} g$) of 22
as the plane (and its content) is in free-fall, alternating with
hypergravity ($\approx 1.8\,g$) phases (as the plane rises and swoops).
During the free-fall period, in the frame of the plane, the
directional force of gravity is replaced by approximately isotropic,
residual accelerations (termed $g$-jitter) on the level of $\sim 0.05\,g$.
Here $g=\SI{9.81}{\metre\per\second\squared}$ denotes the
standard acceleration of gravity.
For an overview of
parabolic flights as a gravity-related experimental platform, the reader
is referred to Ref. <cit.>.

Experiments conducted on parabolic flights are
reproduced on-ground,
using the same experimental setup
in the laboratory.
It should be noted that the flight time up to the first parabola, as
well as the first hypergravity phase might contribute to create an
initial state in mug that differs from their 1g counterpart.

A 2d toy model
reproducing a vertical section of our setup
with photoelastic particles <cit.> is used to help visualizing
the dependence of
the force chains
network on a secondary force field acting downwards
(see Supplementary Material, Supplementary Figure 1).

§.§ Time evolution of pressure and density

Pressure data from the first step up of the piston, measured on the upper half
of the container, 1g and in mug, is presented in
Fig. <ref>.
For all mug experiments, $t = \SI{0}{\s}$ corresponds to the start of the
mug period. For experiments conducted 1g,
it corresponds to
the start of the piston rise.

Time evolution of piston position and pressure in powder cell.
(a) Piston position in height as a function of time. Error bars represent the
experimental precision in optical measurement.
(b) Pressure recorded by the top sensor during the first step up of
the piston, 1g (blue disks, 1g) and in
mug (red triangles, mug). The
jamming times $t_{\text{J}}$ are marked by vertical dashed lines, as well as $\varphi_1$ (see body of the text) and five points of interest that
correspond to panel (c), images (i) to (v). (i) and (ii) are images from the
bottom of the experiment cell looking up in mug; (iii),
(iv) and (v) 1g. The piston is black and the
ps powder, white. The clearly visible piston at all
times in mug (i, ii, iii) indicates the absence of powder flow below the piston,
different from the behavior 1g (iv, v).

At first and until approximately 6, the piston is rising (Fig. <ref>(a)) but
the pressure does not undergo any change (Fig. <ref>(b)). In this initial phase, the
powder is compressed from the initial packing fraction $\varphi_0$ to a denser
configuration, $\varphi_1 > \varphi_0$. For both experiments,
only then does the pressure start increasing. A second characteristic
density, $\varphi_{\text{J}}$, is reached when the piston is brought to a halt
and the powder resists the maximum stress imposed on the sample. By
recording the piston's position at this instant, global estimates of
$\varphi_J$ have been obtained in four repetitions of the experiment,
each 1g and under mug conditions;
resulting $\varphi_{\text{J}}$ for each repetition are presented in
Fig. <ref>. Note that the jamming density is
systematically lower in mug compared to ground.
Statistical significance test
shows that albeit the small number of repetitions available,
this finding is statistically significant (see Methods section).

Packing fraction at
jamming $\varphi_{\text{J}}$, estimated from the height at which the
piston is driven to a halt by the resistance of the packing. Experiments are conducted in the same setup,
(a) 1g and (b) in mug. Error bars represent the formal accuracy (see Methods section for details on the calculation).

Above $\varphi_1$, the behavior of the packing depends on the
gravitational environment. 1g, the powder
undergoes a first large-scale reorganization at
$t^{\text{gnd}} =$ 10.5
(point (iii) in Fig. <ref>) which temporarily
releases pressure on the sensor, but with almost no powder flowing
under the platform, as seen in
Fig. <ref>(c) images (iii) and (iv).
The jamming packing
fraction $\varphi^{\text{gnd}}_{\text{J}}$ is reached at
$t^{\text{gnd}}_{\text{J}} =$ 12.5.  The position of the piston in height
correlates with the pressure measured on the wall of the container:
the piston is blocked and stops rising when a peak in pressure is reached.
At this point, the powder enters a creep regime of slow deformation
and eventually yields at $t^{\text{gnd}} =$ 18. As a result,
material flows under the platform, releasing the pressure on the
container wall, as seen
on Fig. <ref>(c), image (v). By placing the
piston higher in the tube, yielding can be enhanced to the point that
it repeatedly relaxes the pressure completely and allows the piston to
continue to rise (cf. Fig. <ref>(a)). Apart from small
dynamic excursions as material is flowing down, the pressure on the
wall below the piston is unaffected by the developments above the
piston. The creep response may extend over minutes (cf. Fig. <ref>) with smaller pressure
variations – indicative of continued internal
rearrangements – interspersed by yielding events that deposit more
material beneath the piston.

In mug, the granular behavior is quite different.
Approximately at the same time as for the ground experiment
($t^{\textrm{$\mu$-g}}=$ 6), the pressure rises on the sensor;
however it is not jerky as 1g, but a smooth and relatively
steep increase towards its saturation pressure, indicating an
effectively elastic response of the packing.  The maximum pressure is
reached earlier than on-ground: at $t_J^{\textrm{$\mu$-g}} =$ 9.3.
It should also be noted that at this instant
(point (ii) in
Fig. <ref>),
despite the platform
having risen by 1.7, almost no powder has flowed under the
platform: only the fringe of the piston's platform is slightly covered by less
than 1 of (white) powder.
This is attributed to gravity-jitter in the
horizontal plane, common on parabolic flights <cit.>.

In Fig. <ref>,
pressure evolution is presented for one step up of the piston,
1g (<ref>(a))
and in mug (<ref>(b) and <ref>(c)).
The pressure $p$
is recorded on the top and bottom part of the container (respectively $p_u$ and $p_l$).
In Fig. <ref>(a) and (b), at
$t = \SI{0}{\s}$ the piston is in the middle of the cell: the top
sensor records the pressure above the piston and the bottom sensor,
that under it.
In particular, for the experiment shown in
Fig. <ref>(b),
the piston, placed at half height of
the cell, is its initial position.
In Fig. <ref>(c),
the initial piston position
is at the bottom of the cell,
hence both sensors are placed above the piston.

Pressure evolution with piston rise,
(a) 1g and (b, c) in mug. The pressure $p_{u,\,l}$
(bottom graphs) is recorded by sensors in two
positions: on the top part of the cell ($p_u$: filled marks, 1g in
blue and mug in red) and on the bottom part of the cell
($p_l$: open marks, 1g in green and mug in orange).  On (a)
and (b), the piston starts in the middle of the cell (position
$h\approx \SI{20}{\mm}$), while on (c) it starts at the bottom of the
cell ($h = \SI{0}{\mm}$): on (a) and (b) the
pressure sensors are distributed one above the piston
and one below, while on (c) both sensors are above the piston.
On panel (a),
$t=\SI{0}{\s}$ represents the start of piston rise;
on panels (b) and (c),
$t=\SI{0}{\s}$ represents the start of the mug phase.
The pressure evolution exhibits repeated yielding 1g,
while the packing jammed in mug remains stable,
even at maximum load.
In the top graphs, the error bars represent the experimental precision in optical measurements.

While placing the piston higher in the tube does not facilitate
yielding as it does 1g,
occasionally, large scale rearrangements can be observed in the
initial buildup of pressure (see respectively Figs. <ref>(b) and <ref>(c)).
The latter Fig. <ref>(c) also shows that densification proceeds inhomogeneously
throughout the sample.
We observe
that until $t=\SI{10}{\s}$,
the upper sensor does not record any substantial signal change,
while the lower one already undergoes a pressure increase
of approx. $\SI{3}{\pascal}$ from
as the piston starts to rise in the container.
Our conjecture is that
at a point where the lower pressure sensor
already registers substantial stresses,
no force is yet transmitted to
the upper part of the packing,
as the contact network is not yet formed.

In Fig. <ref>,
the pressure is observed for a longer duration after the first step up of the piston, 1g (Fig. <ref>(a)) and in mug (Fig. <ref>(b)).
1g, four repetitions of the piston rise lead to
pressure build-up followed by collapse after $~\SI{5}{\s}$.
The inset is a close-up of the third step up of the piston.
Remarkably, there is no
noticeable creep in mug. The recorded pressure does not change
through repeated mug and hypergravity phases. In order to verify
that this is not due to malfunction of the pressure sensors, on a
different flight-day the pressure was recorded during mulyiple parabolas
without any motion of the piston
(remaining in its initial position at the bottom of the cell).
The result is presented in Fig. <ref>.
If the packing is left in its
initial unjammed state (no compression applied), the pressure
variations due to change in gravitational acceleration is clearly
visible, and corresponds to the temporal structure of the parabolic
flight: hypergravity–low gravity–hypergravity, surrounded by steady flight.
(The reader is referred to Fig. <ref>
for a longer pressure measurement without piston motion.)

Pressure evolution for a duration of $\approx \SI{200}{\s}$ after the initial
densification step leading to jamming.
Pressure is recorded by the top
sensor inside the powder container, (a) 1g and (b) mug. The inset of
panel (a) is a close-up of the pressure evolution during the
second step up of the piston,
where the time and pressure range expanded in the inset are highlighted on the main panel by a grey rectangle.
After an initial steep rise in pressure both 1g and in mug,
corresponding to jamming of the granular packing,
1g the packing yields under the stress applied by the piston rise.
On the contrary, the packing jammed in mug remains stable against hypergravity and
against pressure applied by the piston in the next mug period.

Pressure $p_l$ on the bottom sensor inside a cell filled
with powder, but where the piston does not move: the pressure
changes are solely due to changes in acceleration
during the parabolic flight maneuvers. Initial time $t=\SI{0}{\s}$ is not a significant instant.

Note that all the experiments reported in
Figures <ref> to <ref>
involve the same maximum piston load. However, the corresponding
pressure on the side walls depends on the distance between pressure
sensor and piston. In particular, the pressure increases with the
distance to the piston.

To summarize, the granular sample subjected to piston-probing undergoes the
following steps:

* Densification of the granular medium: the packing density
increases on top of the piston, leaving an empty volume under it.

* The granular medium reaches a critical packing fraction ($\varphi_1$) above the
piston, dense enough to transmit forces from the rising piston to the sensors on the container walls. Pressure on the side-walls increases and the piston
slows down.

enumi enumi$^{\,\textrm{$\mu$-g}}$. In
mug, the pressure increases continuously until the piston
stops at maximum load.
We label this point jamming point.
Note that this state is stable under
hypergravity as long as the piston is not lowered.

enumi$^{\,\text{gnd}}$. 1g, the pressure also increases until a maximum load is reached and
the piston stops. Again, it is this first stop of the piston that we label jamming point. Eventually, the packing yields and flows into the empty space below the piston.

§.§ Identification of regions in the packing

The piston-probing experiment conducted 1g and in mug
reveals a gravity-dependent response of the granular material on
two main aspects:

* Under compression, jamming occurs in mug at lower packing fraction than 1g.

* The packing eventually yields and flows 1g, due to the secondary (gravitational) force field; in its absence, i.e. in mug, the granular material arranges into a highly stable jammed state.

To understand the granular
physics behind the observed behavior and explain the gravity-dependence,
we need to have a closer look at the granular material
in the apparatus. To this end, it is helpful to conceptually
distinguish three different regions inside the tube,
represented in Fig. <ref>.
Region №1 is the
material above the piston, compacted as the piston rises; it is the region where the granular material eventually jams.
Region №3 is the
region below the piston, left empty as the piston rises.
In between, region №2 is the region around the piston,
which can flow into the empty region №3.

The task ahead can therefore be structured into: (1) understanding the
compaction phase of region №1 and the resulting granular fabric; (2)
understanding the stress distribution in region №1 that
determines the force on region №2; and, finally, (3),
understanding the yield criteria of region №2 in the
presence of an empty region №3.

Schematic representation of the regions appearing in the container as the piston rises,
(a) in mug and (b) 1g. 1g, the
powder forms a slope at its angle of repose $\alpha$ with the
horizontal. Stress magnitudes in arbitrary units are depicted by
red lines. The coordinate system used in
Sec. <ref> is shown above each schematic.
The height $h_J^{\textrm{$\mu$-g}, \, \text{gnd}}$ denotes the height at which the piston is stopped by the jammed material above it, respectively under mug and 1g conditions.
The difference in force chain configuration between mug and ground leads to drastically different vertical stresses in the outer region. The load $\Sigma$ on the material below the piston is, as a consequence, much lower in mug compared to ground.
See also Supplementary Material, Supplementary Figure 1.

§.§ Compaction of region №1

While characterizing the point of transition to jamming is still a
controversial topic <cit.>, the mesoscopic scale phenomenon at its origin is
generally accepted: compression-induced jamming occurs if strong force chains appear along
the major compression stress direction, in response to the load
itself <cit.>.  The presence or absence of a secondary
(gravitational) force field modifies the force distribution in the
different zones of the packing <cit.>.

At $\varphi_0$, prepared under ground conditions, the packing
assumes a configuration that is rigid against the unidirectional
gravitational body force,
as long as no other load is applied. This structure, however, is no longer rigid
against the unidirectional boundary force applied by the piston,
and the
packing can initially be compacted with negligible force.

Due to frictional interactions, vertical forces are
partially supported by the vertical outer walls of the tube. Force
chains, consequently, arch in the direction of the net force. When the
latter changes from the downward orientation due to gravity, to the
upward orientation imposed by the piston,
we can infer that
the force chains
change curvature. Where the forces balance, the force chains
buckle. For cohesionless particles, this would in itself allow the
packing to further collapse under the influence of gravity as the
unloaded contacts between particles are no longer constrained by
friction and/or cohesive interactions.

At this point it is instructive to estimate the maximal stress imposed
by gravity 1g.
From the analysis proposed by  <cit.>
(cf. Sec. <ref>),
we find a screening length
$\ell\approx\SI{26}{\milli\metre}$, which limits the gravitational
stress even at maximum depth $H$ to $\sigma_g \approx \SI{125}{\pascal}$
(cf. Eq. <ref>). As seen below in Sec. <ref>,
the cohesive energy density is at least on the order of
(equivalent to 10, as
stress and energy density have the same units).
Typical forces during buckling
of the force chains will therefore be unable to break most
contacts. The new
network of force chains will initially form by mostly
establishing additional contacts between particles, which necessarily
implies a densification of the packing.

As a result of the densification,
$\varphi_1$ is eventually reached.
We can infer the following mechanism on the microscopic scale:
the granular fabric supports load-bearing force chains again,
as the chains of contacts
above of the piston have percolated into a network transmitting the load imposed by the piston to the container's walls.
However, this network is not stable against the piston-imposed load:
the particles can still reorganize into denser configurations.

As the piston does not cover the whole cross-section of the tube,
we may conceptually distinguish
between an inner region №1 and an outer region
№1. The inner region directly transmits the forces of the
piston, while the outer region is indirectly affected through the
horizontal redistribution of stresses in the packing. Under gravity,
as soon as the stress due to the piston rise exceeds the gravitational stress,
forces are directed upwards while the outer region may still be forced
down by gravity. This create a shear force between inner and outer
that intermittently severs force chains. The resulting local force
balance may lead to large scale rearrangements that are at the origin
of the pressure drops observed 1g between $\varphi_1$ and
$\varphi_{\text{J}}$. Note that the stresses imposed by the piston considerably
exceed the gravitational stresses in the inner region for most of the
compression phase such that gravity is negligible there.

At $\varphi_{\text{J}}$, the packing has found a configuration that is stable
against the maximal piston force with respect to local rearrangements
of particles.
On longer time scales however,
for the experiments conducted 1g,
the evolving pressure
signal indicates an ongoing evolution of the force chain network under
constant load (see Fig. <ref>(a)).
Note that the pressure on the side walls is increasing
over time, which implies that the deflection of forces away from the
vertical also increases and the vertical normal stress decreases in
the upper parts of the piston. This aging or creep phenomenon can be
rationalized in terms of sgr <cit.>
and its adaption to jammed packings <cit.>. On a continuum
level, force chains entail a spatially heterogeneous stress
distribution. Global stress relaxations may therefore occur by chains
of local relaxations that proceed at a finite speed. In the
framework of Edward's ensemble of granular packing, the angoricity
is the intensive thermodynamic parameter conjugate to the stress; it
quantifies the sensitivity of the stress heterogeneities to the
imposed load <cit.>. Within
sgr, jamming occurs if

the angoricity $\beta$

falls below a critical value $\beta_J$. In the
jammed state, $\beta<\beta_J$, sgr predicts aging
<cit.>, compatible with our measurements.

For the mug experiments,
upon entering the mug phase of
each parabola, the cohesive forces among particles become
predominant <cit.> and create a packing where
once particles come into contact with their closest neigbours,
those contacts are likely to remain.
Upon compression and up to $\varphi_1$, the packing
collapses – like 1g, mostly by local buckling and by
preserving existing contacts – into a load bearing
configuration. However, no strong shear forces appear as the piston is
the single source of stress on the mesoscopic scale. Note that friction with the piston's
wall is generically smaller than internal friction of the granular
packing <cit.>. At the slow speed of the piston, the
wall friction therefore cannot generate shear forces between the inner
and outer region. Remarkably, under these conditions, the topology of
the force chain network seems to be independent of the magnitude of
the imposed stress. As a result, the measured pressure $p_{u, \, l}$ becomes
proportional to the strain imposed by the piston and the packing
behaves as an elastic solid with a well defined elastic modulus. As a
rough estimate from Fig. <ref>, let's
assume a strain of 1% during the buildup of pressure from
0 to 10. This yields an elastic
modulus on the order of 1, compatible with
theoretical predictions (see Methods Section).

On the time scale of the experiment, no creep can be observed in
mug. In terms of the angoricity, this implies a vanishing
sensitivity to the magnitude of the load, $\beta\to0$. This is in
line with the elastic regime observed before for $\varphi <
\varphi_J$. At this stage we can only deduce that a single boundary
force can lead to $\beta=0$ while the competition between said
boundary force and the gravitational body force leads to a finite
$\beta > 0$. We invite further studies that investigate this more

A 2d hands-on version of a section of this experiment using photo-elastic
discs allowed us to visualize the force chains distribution within the
granular medium; it qualitatively confirms the appearance of the three
regions sketched in Fig. <ref>. Results are
shown in Supplementary Material (Supplementary Figure 1).

§.§ Stress distribution in region №1

In order to understand the load the piston is indirectly enforcing on
region №2,
one has to understand the stress distribution in
region №1.
Of particular interest is
the normal
stress $\Sigma$ imposed by region №1 to region №2
through the gap around the piston.

The observation that the vertical free surface of the granular
material below the piston is unstable at heights of the order of
millimeters if prepared 1g,
yet remains stable even under
hypergravity conditions if prepared in mug,
is perplexing, at first. It is testament to the nontrivial stress
distribution in a granular sample. In the following,
we will propose a
continuum argument to rationalize this observation.

Following Janssen <cit.> (see
Sec. <ref>), one may, for the sake of the argument, assume
that the vertical, $\Sigma_{zz}$, and horizontal, $\Sigma_{rr}$,
normal stresses are the principal stresses. Unlike in a molecular
fluid, the two normal stresses are not identical, even in the absence
of gravity, and are not homogeneous throughout the material. Again
following Janssen, we restrict our discussion to a two dimensional,
vertical cross section of the experiment (which is in principle trivial to extend to 3d,
given the axisymmetry of our experimental setup).

At finite gravity but with an immobile piston, the experiment is
equivalent to the silo Janssen had in mind. We consider a silo with a circular cross-section of diameter $D$; in summary, the major
principal stress is $\Sigma_{zz}$ and the minor principal stress,
$\Sigma_{rr} = k\Sigma_{zz}$, is reduced by Janssen's constant $k < 1$.
Friction with the wall will screen the weight on a length scale
$D/4\mu k$.
Here, we focus on the radial rather than the vertical
structure of the problem, and just keep in mind that from one end to
the cylinder to the other, stresses will be renormalized by the
screening effect.

Let's begin by recalling that the normal stress due to gravity on the
gap around the piston,
$\Sigma\equiv\Sigma_{zz}(z=H) \leq \sigma_g \approx \SI{125}{Pa}$.
Now we
imagine a significant normal stress $\sigma\gg \sigma_g$ being applied
by the piston, such that we can neglect gravitational stresses in the
following. In addition, consider only the material directly above the
piston, for $r < d/2$. The Janssen effect will screen the stresses on a
length scale $\ell = d/4\mu k$, leading to a depth averaged vertical
normal stress $\overline\Sigma^<_{zz} = C_1\sigma$ and related to
this, an average horizontal normal stress
$\overline\Sigma^<_{rr} = k\overline\Sigma^<_{zz}$. Here
$C_1 = C_1(H/\ell) < 1$ is a dimensionless constant encoding the
geometry of the setup. From Fig. <ref>(c) we infer
that the pressure between at the two heights differs by about a factor
$2.5$ in the fully jammed state. Given the distance between the
sensors (see Fig. <ref>), we obtain an estimate
for the screening length, $\ell\approx\SI{26}{\milli\metre}$.

If the packing is prepared 1g, the major principal stress
will also be vertical in the outer region, $D/2 > r > d/2$, but the
geometry of the outer region will induce its own screening length
$L\approx D$. If we assume the applied stress to dominate over gravity,
$\sigma \gg \sigma_g$, then in the outer
region[$C_2(L) = C_2(H/\ell, H/L, \ell/L) < 1$ is another
dimensionless geometrical factor.]
$\overline\Sigma^>_{zz} = C_2(L)\overline\Sigma^<_{rr}/k$, while
$\overline\Sigma^>_{rr}(r) = \overline\Sigma^<_{rr}d/2r$. In
particular, the pressure recorded on the side wall
$p \approx \overline\Sigma^>_{rr}(D/2) = (C_1kd/D)\sigma$ and the stress
on the gap
$\Sigma_{\text{gnd}} \approx \overline\Sigma^>_{zz} = C_2(L)C_1\sigma$.  From
this we conclude $\Sigma_{\text{gnd}} \approx p/k$. Up to renormalizations, the
normal stress on the gap is essentially given by the stress applied
through the piston. The pressure recorded on the side wall, on the
other hand, is proportional to $k\sigma$.

However, if prior to applying a stress, the sample is prepared in
mug conditions, the compaction in the outer region is
effected by the horizontal normal stress, $\Sigma^<_{rr}$, exerted by
the inner region. Therefore the major principal stress is now
horizontal instead of vertical in the outer region and only a fraction
$k$ is transmitted to the vertical direction,
$\Sigma^>_{zz} = k\Sigma^>_{rr}$. This will shorten the screening
length in the outer region to $L' = k^2L$. For the average vertical
normal stress we now have $\overline\Sigma^>_{zz} =
kC_2(L')\overline\Sigma^<_{rr}$, i.e., $\Sigma_{\textrm{$\mu$-g}} \approx
\overline\Sigma^>_{zz} = k^2C_2(L')C_1\sigma$, while the pressure on
the side wall remains essentially unchanged. From this we conclude
$\Sigma_{\textrm{$\mu$-g}} \approx kp$ and most importantly $\Sigma_{\textrm{$\mu$-g}}/\Sigma_{\text{gnd}} \approx k^2$ independent of the applied load.

Notably, the load on the packing below the piston is significantly
reduced under mug conditions. The additional weight induced
stress in a subsequent hypergravity phase is heavily screened by the
short length scale $L' \approx k^2D \approx \SI{9}{\milli\metre}$, to
roughly $\gamma L' \lesssim \SI{50}{\pascal}$. This is well within the
stress fluctuations due to g-jitter. If the packing was stable to the
latter, it will not yield in hypergravity either.  As a rough
estimate, we infer from the maximum $p\approx\SI{11}{\kilo\pascal}$,

\begin{align}
\label{eq:7}
\Sigma_{\text{gnd}}&\approx \sigma \approx \frac{p}{k\left(1 -
e^{-H'/\ell}\right)} &\approx \SI{40}{\kilo\pascal},\\
\intertext{and}
\Sigma_{\textrm{$\mu$-g}}&\approx k^2\sigma \approx \frac{kp}{\left(1 -
e^{-H'/\ell}\right)}&\approx \SI{6.5}{\kilo\pascal},
\end{align}

where $H'=\SI{30}{\milli\metre}$ is the approximate distance between
the upper surface of the piston and the pressure sensor at maximum

Note that Mohr-Coulomb theory has nothing to say on the creep
behavior. From the slowly growing horizontal stress at constant
vertical load, we may infer that Janssen's constant is also slowly
growing in time and the packing evolves towards a slightly more
isotropic stress distribution.

§.§ Stability of region №2

First of all note that a vertical free surface of a granular packing
can only be stable due to cohesion, $c > 0$. Effective values for the
cohesive stress (or energy density) are hard to come by and do not
only depend on the particles but also on the preparation. However,
typical values for ordinary materials are on the order of several
kilopascal. Under gravity, the weight stress $\gamma h$ limits the
maximum stable height of a vertical surface to
$h_{\mathrm{max}} = 4c/\gamma(\sqrt{1 + \mu^2} - \mu)$
<cit.>. With
$\gamma_{\text{gnd}} \approx \SI{6}{\kilo\pascal\per\metre}$, one obtains stable
heights on the order of meters. It is obvious that we cannot expect
collapse of the free surfaces of a couple millimeters in height that
occur in the experiment even under hypergravity. In addition, we have
seen above that the load imposed by the piston on the gap, even under
the unfavorable mug conditions, by far exceeds the gravitational

Mohr-Coulomb theory also provides the maximum load that can be
sustained by a vertical wall,
$\Sigma_{\mathrm{max}} = 2c/(\sqrt{1 + \mu^2} - \mu)$
<cit.>. Given that the experiment yields 1g but not in mug, we can deduce a cohesion strength
on the order of a few kilopascal in line with expectations. Once
region №2 yields under ground conditions, the material will
settle in a heap with angle of repose $\alpha$ determined by the
material properties. This limits the amount of material that can be
deposited below the piston, as alluded to in the introduction.

§.§ Shift of the density at jamming

Concomitantly to the maximum height reached by the piston under the
different experimental environments, the critical packing fraction at
which jamming occurs under compression is found to be lower in
mug (on average $\varphi^{\textrm{$\mu$-g}}_{\text{J}} = 0.567 \pm 0.014$) than 1g
(on average $\varphi^{\text{gnd}}_{\text{J}} = 0.579 \pm 0.014$) (see Fig. <ref>).

Variability in the jamming packing fraction $\varphi_{\text{J}}$
has notably been studied
for increased frictional interactions between particles,
which was linked
to a reduction in the packing fraction at jamming <cit.>:
if the particles' contact number is
high enough, the jamming
transition happens at lower packing fraction for frictional particles
than for frictionless particles.
The proposed mechanism is the following:
increased frictional interactions
restrict the reorganization possibilities of the packing, resulting in
a higher stability of each configuration.
In other words, the
reduction of collective degrees of freedom available for the granular
fabric to reorganize into a denser packing under the applied stress
results in a jamming transition happening at lower packing fraction.
Similarly, it is our hypothesis that the lower packing fraction at
jamming observed in mug results from a decrease in collective
degrees of freedom and manifested in a reduced spatial heterogeneity
of the force network.

The absence of the force field due to Earth gravity
leads to another related phenomenon:
the cohesive interparticle forces,
insignificant when compared to the particles' weight,
become predominant in mug.
Stronger cohesive interactions
also reduce the freedom of motion of each individual particles,
hence of the granular packing to reorganize into a denser configuration as a whole.

This observation confirms results from one of the first rheology
experiments performed in mug, on board the Space Shuttle
A startup curve was measured for a
dense ($\varphi=0.65$), polydisperse powder, comparing 1g and
mug conditions.  The response in mug was jagged, and samples in mug
exhibited a much higher peak friction strength.

§ DISCUSSION

A granular piston-probing experiment was conducted 1g and in
mug (using parabolic flights as a low-gravity platform) to
probe the effect of the gravitational acceleration on the compression of a
granular medium. We find that the collective behavior of a granular
material reveals strong dependence on the gravitational environment.

As the piston rising densifies the granular material above it
(region №1),
an equivalent empty volume is created below the piston
(region №3).
Once the material in region №1 reaches a
packing sufficiently dense to transmit the load imposed by the piston,
a rise of pressure recorded on the container's walls indicates
the formation of a percolated network,
and the piston upward motion slows down.
Eventually, a jammed state is reached above the piston (region
№1): the load imposed by the piston is transmitted to the
container's walls and eventually the piston stops. 1g, the fabric of
the jammed packing allows for sufficient downward forces to overcome
cohesion in creep flow. The formation of an annular heap with a finite
angle of repose in region №3 eventually limits the powder
rearrangement and jams the piston. In mug, the fabric is
altered in such a way, that the downward forces are insufficient to
overcome cohesion. The piston jams even though region №3
still provides free space.

In our experiments conducted in mug, this jammed state is
reached while the piston is at lower height,
$h_{\text{J}}^{\textrm{$\mu$-g}} < h_{\text{J}}^{\text{gnd}}$, than for their repetitions
1g. We conclude that the packing fraction at jamming is
lower in mug than 1g.
We support this finding by two physical
explanations. Firstly, as the interparticular force balance changes in absence of
gravity, cohesive interactions become predominant.
Besides, 1g,
particles minimize their potential energy
by going to the lower position possible:
Earth gravity creates a
secondary force field to stimulate reorganization in a
different direction to the major compression stress.
Under mug, such reorganization stimulation does not take place.
As a result, in
mug, powder flow deteriorates;
the particles' configuration is more stable
or less likely to reorganize:
the packing fraction at jamming is lower in
mug than 1g.

In all experimental results presented,
the piston height $h$ is the parameter measured directly:
the global packing fraction $\varphi$ is
calculated assuming density homogeneity in the container.
Local packing fractions are not experimentally accessible in the current setup.
We also report
a lack of control over the initial state of the granular packing,
as the flight time preliminary to the first parabola and repeated hypergravity phases
might contribute to create an initial state specific to the mug experiment,
that is not reproduced in their 1g counterpart.
To minimize
the influence of repeated hypergravity phases on the granular packing,
only the first parabolas of each flight are used to calculate $\varphi_{\text{J}}$.
Besides, the granular samples
generally undergo
a slight positive acceleration along the $z$-axis at the initial instant of each parabola,
which is expected to counteract the compaction caused by hypergravity.

The authors hope that those results will spur interest and further
investigation on the influence of gravity on the packing fraction at
jamming in granular media.
If confirmed,
the deterioration of powder flow
and lower packing fraction at jamming
under reduced gravity
could have dramatic consequences for
powder handling on
sand-covered celestial bodies.

§ METHODS

§.§ Granular material

The granular material used for all experimental results presented is a
spherical, monodispersed ps powder of mass density
$\rho_b=\SI{1050}{\kilogram\per\cubic\metre}$ and diameter
$a=\SI{80}{\micro\meter}$ produced by the company Microbeads
under the name Dynoseeds <cit.>.
The powder was washed to ensure a clean, smooth surface, and sieved to
avoid outliers
(wet-sieving under ultrasound at frequency of
20 for a duration of
8 hours).
The surface state and sphericity of the granular material was verified using scanning
electron microscopy.
Micrographs before and after the cleaning process were taken, showing
the limited roughness and clean surface of the particles after cleaning.
Fig. <ref> shows the cleaned particles, as used in the experiments discussed.

Scanning electron microscopy of the granular material used. The micrographs presented are taken after completion of the cleaning process.

§.§ Bulk modulus of the packing: limits to stresses and speeds

For elastic particles in a given jammed packing, a further reduction
of volume, $\Delta V$, can be achieved by forcing the particles to
deform. Assuming an isotropic compression leads to small typical
deformations $\langle\xi\rangle/a$ relative to the particle radius $a$,
we can estimate the relative volume change as
$-\Delta V/V \approx 3\langle\xi\rangle/a$. Such a volume change is
related to the pressure $\Delta p = -K\Delta V/V$ by the bulk modulus
$K$.  <cit.> have shown that

\begin{equation}
\label{eq:1}
K = C\frac{\varphi z}{6\pi a}\times\frac{\partial F}{\partial
\xi}(\langle\xi\rangle),
\end{equation}

where $z$ is the average coordination number, $F(\xi)$ is the
inter-particle force, and $C$ is of order one for almost monodisperse
packings. Here we assume a nonlinear, Hertzian contact force

\begin{equation}
\label{eq:2}
F(\xi) = \frac{\sqrt2}{3}\times\frac{E_p}{1 - \nu^2}a^{1/2}\xi^{3/2}
\end{equation}

that is parametrized in terms of $E_p$ and $\nu$, Young's modulus and
Poisson's ratio of the particles' material, respectively. Using
Eq. (<ref>) in Eq. (<ref>), and noting that
$\varphi z/(1 - \nu^2) \approx 4$ around jamming where $\varphi \approx 0.6$,
$z \approx 6$, and for typical Poisson's ratios $\nu \approx 0.35$, we find

\begin{equation}
\label{eq:4}
K \approx \frac{E_p}{3\pi}\sqrt{2\langle\xi\rangle/a}
\approx \frac{K_p}{\varphi}\sqrt{\langle\xi\rangle/128a},
\end{equation}

where $K_p$ is the bulk modulus of a particle, and

\begin{equation}
\label{eq:3}
\frac{\Delta p}{E}
\approx \frac{\sqrt2}{\pi}\left(\langle\xi\rangle/a\right)^{3/2}.
\end{equation}

If we want to limit the deformations of the ps particles
($E(\mathrm{PS}) \approx\SI{3.3}{\giga\pascal}$ <cit.>),
to $\langle\xi\rangle/a\lesssim 10^{-3}$, we need to limit the pressure
to $\Delta p \lesssim \SI{45}{\kilo\pascal}$.

With the universal relation $G = 3K/5$ <cit.> for the shear modulus in a packing, the s-wave speed
$c_s = \sqrt{G/\rho}$, can be related to the speed of sound in a
particle, $c_p= \sqrt{3K_p(1 - \nu)/\rho_p(1 + \nu)}$ <cit.>
($c_p(\mathrm{PS})\approx\SI{2350}{\metre\per\second}$ <cit.>) by Eq. (<ref>) as:

\begin{equation}
\label{eq:8}
c_s \approx c_p\sqrt{\frac{1 + \nu}{5(1 - \nu)}}(\langle\xi\rangle/128a)^{1/4}
\approx \SI{80}{\metre\per\second},
\end{equation}

much larger than the piston speed. Note that these are order of magnitude
estimates only and many quantitative refinements would be possible.

§.§ Mohr-Coulomb material

Following  <cit.>, a two
dimensional Mohr-Coulomb material is characterized by three
parameters: the macroscopic friction $\mu$, the cohesion $c$, and the
weight density $\gamma = \rho^*g$ where $\rho^* = \rho_b\varphi$ is
the mass density of the material at a given packing fraction
$\varphi$ (where $\rho_b$ is the density of the particles' material). The Mohr-Coulomb failure criterion limits the shear
stress $\tau$ for a stable material depending on the normal stress
$\Sigma$, $|\tau|\leq \mu\Sigma + c$. One consequence of this yield
criterion is that normal stresses are not isotropic but that there is
a finite ratio $k<1$ between the minor and the major principal normal
stress, the Janssen constant.

Assuming that the major and minor principal stress axes are aligned
with the vertical and horizontal orientation of the container, one
obtains the vertical force balance

\begin{equation}
\label{eq:5}
\frac{d\Sigma_{zz}}{dz} \pm \frac{\Sigma_{zz}}{\lambda} = \gamma
\end{equation}

and the boundary condition $\Sigma_{zz}(h) = \sigma$ for an
external stress $\sigma$ applied at height $z=h$. The sign between the
first and the second term depend on the direction of the force. The
length scale $\lambda$, associated with the screening of forces
described by Eq. (<ref>), depends on the geometry of the
problem. For a major principal stress in the vertical orientation (the
classical Janssen analysis), $\lambda\propto1/k$, whereas for a
horizontal major stress, $\lambda\propto k$.
For a container of width $D$ and no external stress, Janssen
<cit.> obtained for the horizontal stress

\begin{equation}
\label{eq:6}
\Sigma_{rr}(z) = \frac{\gamma D}{4\mu}\left(1 - e^{-4\mu kz/D}\right)
\end{equation}

and for the vertical stress $\Sigma_{zz}(z) = \Sigma_{rr}(z)/k$.

§.§ Data processing

The pressure sensors placed inside the container were observed to
due to the constant pressure applied on the sensors by the granular
packing, once the material is placed inside the container.
This offset was found to be around 45 on all
experiments (both 1g and in mug).  However, this
offset pressure is not constant, but increases with the experiment's
duration, rendering exact quantitative pressure measurement
challenging.  This shift can be observed in
Fig. <ref>, where the raw pressure measured
$p_{\text{exp}}$ is shown: throughout the five parabolas described by the
Zero-g plane in the course of 1000, the pressure increases by approximately $+\SI{0.34}{\kilo\pascal}$.

Pressure $p_{\text{exp}}$ measured on
the bottom sensor inside an experimental cell filled with powder
but with no motion of the piston. Pressure changes are solely due
to changes in acceleration during the parabolic
flight maneuvers. The offset pressures found from the mug
phases are shown as dashed lines, labeled
$p_0^{a, \, b, \, c, \, d, \, e}$. The increase in offset pressure
throughout the experiments showed here is clearly visible,
amounting to $+\SI{0.34}{\kilo\pascal}$ in 1000.

In order to retrieve the pressure evolution $p$, from the pressure
measured $p_{\text{exp}} = p + p_{0}$, the offset pressure $p_{0}$ was
calculated per experiment (per parabola in the case of mug

Numerical values of the parameters describing our system: a ps powder in a pc container.
A plethora of ways are available to estimate the Janssen coefficient $k \in [0, \,1]$.
Here, we follow recommendations given in Ref. <cit.>. The ps properties are taken from the powder manufacturer data sheet <cit.>.
The powder density is defined as $\rho$* $= \rho_{b} \, \varphi$, ps density in bulk times the powder packing fraction.

Parameter          Unit                   Value      Absolute uncertainty
$\varphi$     packing fraction (initial)                  –       0.55	    –

$D$     container's diameter                 $5.40 \cdot 10^{-2}$	    $10^{-4}$

$d$     piston's diameter                 $3.60 \cdot 10^{-2}$	    $10^{-4}$

$m$     mass of powder                 $5.00 \cdot 10^{-2}$	    $10^{-4}$

$\rho_{b}$     density in bulk                         1050	    10

$\rho$*     powder density                         580	    10

$g$     gravitational constant                       9.81	   –

$\mu$     static friction coefficient                –     0.3	    –

$k$     Janssen coefficient             –     0.4	    –

For the mug experiments, $p_0$ is deducted from the steady-state
pressure (once a relatively stable value is reached after
transient decrease): under reduced gravitational acceleration and
before the start of piston rise (i.e. at $t=\SI{0}{\s}$), the expected
pressure is $p(t=\SI{0}{\s}) = \SI{0}{\pascal}$, as no pressure is
applied on the sensors. Therefore, we assume that the remaining
pressure measured equals the offset $p_{0}$, and use this value to
estimate $p= p_{\text{exp}} - p_{0}$.

For the measurements conducted 1g, the reference value used is the
pressure at the beginning of the experiment, before the piston starts
rising. At this point, the pressure recorded should be solely due to
the weight of the powder inside the container: using the mass of the
powder at the sensor's height, we calculate the expected horizontal
pressure $p_h\equiv\Sigma_{rr}(h)$ from Eq. (<ref>). Numerical
parameters used are in Tab. <ref>.  The
offset pressure $p_0=p_{\text{exp}} - p_h$ is calculated for each
experiment to obtain $p$.

§.§ Formal uncertainty

Formal uncertainty due to measurement limitations is calculated for
the packing fraction
from the mass $m$ of granulate,
the density $\rho_b$ of the particles' material (in bulk),
and the volume occupied by the granular material, which depends
on the diameter $D$ squared and height $h$ of the piston.
As each variable is measured independently,
we rely on propagation of uncertainty laws
and calculate
the relative error $\delta \varphi / \varphi$
as the sum of relative errors on each variable.

The absolute uncertainties for each measured variable are given in Tab. <ref>.
The height $h$ of the piston is
measured from image analysis using a
scale placed right behind the piston's axis,
which results in an absolute uncertainty of
$\delta h = \SI{0.5}{\mm}$.

The resulting absolute uncertainties $\delta \varphi$ are plotted as error bars in Fig. <ref>.
The maximum absolute uncertainty
$\delta \varphi^{\text{gnd}, \, \textrm{$\mu$-g}} = 0.027$,
which represents a relative error of 5%,
is taken as accuracy measurement for the average $\varphi^{\text{gnd}}_J $ and $\varphi^{\textrm{$\mu$-g}}_J $.

§.§ Statistical significance

To validate the claim that the gravitational environment influences the packing fraction at jamming
measured in our system,
a statistical significance test is essential. The test is done on the two series of four values
for $\varphi^{\text{gnd}}_J $ and $\varphi^{\textrm{$\mu$-g}}_J $ given in Fig. <ref>.
The null-hypothesis is that the cumulative distributions are identical,
$P(\varphi^{\text{gnd}}_J < \varphi_J) = P(\varphi^{\textrm{$\mu$-g}}_J < \varphi_J)$;
the alternative hypothesis is that
$P(\varphi^{\text{gnd}}_J < \varphi_J) < P(\varphi^{\textrm{$\mu$-g}}_J < \varphi_J)$.

All values given for packing fraction at jamming are calculated
solely from the experiment conducted
on the first parabola of each flight,
before any powder has flowed under the piston (region №1 in Fig. <ref>),
as shown in Fig. <ref>(a–e).
Therefore, only four repetitions of the experiment are available.
For small samples where normal distribution cannot be assumed,
a nonparametric statistical significance test is preferable.
We use the
sum of ranks test known as
Mann-Whitney U-test <cit.>.

We find a probability of the null-hypothesis of only 1.4% <cit.>.
We can therefore confidently conclude that for the limited amount of data available,
the probability of
the distributions of $\varphi^{\text{gnd}}_J $ and $\varphi^{\textrm{$\mu$-g}}_J$
being equal is low enough to be rejected,
which supports our conclusion that
the packing densities at jamming are different between ground and microgravity.

§ DATA AVAILABILITY STATEMENT

The datasets used in this article are publicly available in the Zenodo repository 7101542:

§ CODE AVAILABILITY STATEMENT

The codes used to generate the plots are available from the corresponding author upon reasonable request.

§ ACKNOWLEDGMENTS

ODA acknowledges financial support of the esa
npi contract 4000122340 on Physical Properties of
Powder-Based 3d-Printing in Space and On-Ground, and of
the dlr/daad Research Fellowship 91647576.  WTK
acknowledges funding by the dfg through grant number
KR4867/2. ODA and AH wish to express sincere gratitude to the
Spaceship eac program for financial support.
Miranda Fateri is thanked for support during the pfc.
Thorsten Pöschel, Philip Born, Matthias Schröter and Angel Santarossa are acknowledged for their critical reviewing of the manuscript.
Many thanks to the entire Novespace team for their help and patience,
and in particular to Yannick Bailhé.

§ AUTHOR CONTRIBUTIONS

ODA and AH designed and build the setup and performed the measurements. ODA and WTK analyzed and interpreted the data and wrote the manuscript. AC and MS supported the project. MS procured the flight opportunity. All authors revised the manuscript.

§ COMPETING INTERESTS

The authors declare no competing interests.

§ ADDITIONAL INFORMATION

Supplementary information accompanies this paper.