# Lifting of Tribocharged Grains by Martian Winds

Maximilian Kruss  University of Duisburg-Essen, Faculty of Physics, Lotharstr.
1-21, 47057 Duisburg, Germany  Tim Salzmann  University of Duisburg-Essen,
Faculty of Physics, Lotharstr. 1-21, 47057 Duisburg, Germany  Eric Parteli
University of Duisburg-Essen, Faculty of Physics, Lotharstr. 1-21, 47057
Duisburg, Germany  Felix Jungmann  University of Duisburg-Essen, Faculty of
Physics, Lotharstr. 1-21, 47057 Duisburg, Germany  Jens Teiser  University of
Duisburg-Essen, Faculty of Physics, Lotharstr. 1-21, 47057 Duisburg, Germany
Laurent Schönau  University of Duisburg-Essen, Faculty of Physics, Lotharstr.
1-21, 47057 Duisburg, Germany  Gerhard Wurm  University of Duisburg-Essen,
Faculty of Physics, Lotharstr. 1-21, 47057 Duisburg, Germany

(Received 2021 April 20; Revised 2021 September 21; Accepted 2021 November 9)

###### Abstract

It is a long-standing open question whether electrification of wind-blown sand
due to tribocharging — the generation of electric charges on the surface of
sand grains by particle-particle collisions — could affect rates of sand
transport occurrence on Mars substantially. While previous wind tunnel
experiments and numerical simulations addressed how particle trajectories may
be affected by external electric fields, the effect of sand electrification
remains uncertain. Here we show, by means of wind tunnel simulations under air
pressure of 20 mbar, that the presence of electric charges on the particle
surface can reduce the minimal threshold wind shear velocity for the
initiation of sand transport, $u_{{\ast}{\mathrm{ft}}}$, significantly. In our
experiments, we considered different samples, a model system of glass beads as
well as a Martian soil analog, and different scenarios of
triboelectrification. Furthermore, we present a model to explain the values of
$u_{{\ast}{\mathrm{ft}}}$ obtained in the wind tunnel that is based on
inhomogeneously distributed surface charges. Our results imply that particle
transport that subsides, once the wind shear velocity has fallen below the
threshold for sustained transport, can more easily be restarted on Mars than
previously thought.

Mars, Aeolian processes

##  1 Introduction

Mars has an atmosphere that is almost two orders of magnitude lower in surface
pressure than Earth’s, and nevertheless Martian dunes are migrating daily and
producing Earth-like sand flux rates, while rovers and satellites have been
detecting dust devils and global dust storms on the red planet (Fisher et al.,
2005; Bridges et al., 2012; Bourke et al., 2019; Liuzzi et al., 2020; Heyer et
al., 2020). Indeed, while a breeze is sufficient to lift sand grains from the
ground at 1 bar on Earth, it requires a storm to do the same at a few mbar
$\rm CO_{2}$ on Mars (Bagnold, 1941; Greeley & Iversen, 1985; Merrison et al.,
2007; Rasmussen et al., 2015; Burr et al., 2020). This makes conditions for
particle lifting more critical on the Martian surface, as it might well depend
on minute details if a region is active or inactive with respect to particle
transport and if dust entrainment into the atmosphere is possible or not.

The physics of soil particle entrainment by wind forces is still today a
matter of research (Pähtz et al., 2020; Fu, L.-T., 2020). There have been a
number of ideas to explain how dust and sand can be moved more easily on Mars
(Neakrase et al., 2016). Previous experiments showed that dust aggregates are
more easily mobilized than sand (Merrison et al., 2007) and that sand can be
lifted from the soil in sporadic events of strong winds on the surface of Mars
(Swann et al., 2020). Furthermore, the low Martian gravity also seems to be
beneficial to particle lifting, since granular material settled under the
gravity of Mars produces granular packings associated with lower solid
fractions than their Earth counterparts (White et al., 1987; Musiolik et al.,
2018; Kruss et al., 2020).

However, one further ingredient in the physics of soil grain entrainment on
Mars is the rarified Martian atmosphere. Recently, Demirci et al. (2020)
showed that the transition from hydrodynamic flow to molecular flow around an
individual grain increases the necessary shear stress for smaller particles
further. For a less penetrable ground of small grains, lifting might be easier
owing to a pressure difference between subsoil and above-ground pressure as
vortices travel the surface (Greeley et al., 2003; Balme & Hagermann, 2006;
Neakrase et al., 2016; Koester & Wurm, 2017; Bila et al., 2020). Nevertheless,
a low pressure might not always be detrimental to particle lifting. As the
flow becomes molecular through the pores of the soil, thermal creep gas flow
sets in, following temperature gradients from cold to warm (Koester et al.,
2017). This can lead to a subsoil overpressure on the insolated Martian
surface (de Beule et al., 2014; Kuepper & Wurm, 2016; Schmidt et al., 2017).
Summarizing, these ideas on lifting turn the screws on the conventional forces
acting on grains: gas drag and pressure, gravity, and adhesion. However, there
is at least one more force that needs to be considered in this context, which
is electrostatics (Rasmussen et al., 2009; Esposito et al., 2016).

It is known that sand storms can produce large electric fields of the order of
100 kV/m close to the ground (Schmidt et al., 1998; Zheng, 2013; Zhang & Zhou,
2020). In addition, dust devils show electric fields (Franzese et al., 2018).
Volcano eruptions regularly come with lightning as a visible sign of charging
and charge separation (Aplin et al., 2014; Méndez Harper & Dufek, 2016). The
respective fields are strong enough to pick up charged grains. The
effectiveness of such electric fields in lifting sand grains has been tested,
for instance, through modeling and experiments (Kok & Renno, 2006; Pähtz et
al., 2010). Specifically, this previous work considered that the grains are
well conducting and subject to an external field, which induces charges on the
grains, thus producing an upward-pointing electrostatic force (Kok & Renno,
2006).

While this previous work relied on an externally applied electric field, sand-
driven electric fields are certainly self-generated by triboelectric effects
as the sand grains are hopping and colliding with each other within the
transport layer (Wurm et al., 2019; Méndez Harper et al., 2021). However, it
is by no means clear whether triboelectric charging lowers the minimal
threshold wind shear velocity for transport initiation,
$u_{{\ast}{\mathrm{ft}}}$, since charges may promote aggregation, thus making
particle lifting more difficult (Méndez Harper et al., 2017; Steinpilz et al.,
2020a, b; Teiser et al., 2021).

Here, we thus investigate the influence of tribocharging on
$u_{{\ast}{\mathrm{ft}}}$ by means of wind tunnel experiments operating at
Martian-like atmospheric pressure conditions (20 mbar). In contrast to
previous work, we do not apply any external electric field to the granular
bed. Instead, we use particle collisions to charge a granular bed, whereupon
we analyze the effect of this charge on $u_{{\ast}{\mathrm{ft}}}$. Moreover,
we assume that the ground is not able to conduct charges. As fields exist, a
conductive discharge does not occur on relevant timescales of seconds to
minutes at least. Our assumption is thus plausible, as regions of sand
activity and especially the surface of Mars are rather arid, with low humidity
making sand grains extremely bad conductors. Moreover, the charges of the
particles within the soil provide the source of the electric fields acting on
a particle that is at rest on the surface.

##  2 Experiment

In this section we describe the experimental setup developed to simulate the
entrainment of particles under Martian-like pressure conditions, the granular
material employed in our experiments, and the methods for charging and
discharging of this granular material.

###  2.1 Experimental setup and the wind tunnel

The main component of the experimental setup is a wind tunnel with an inner
tube diameter of 6.3 cm. The maximum Reynolds number inside the wind tunnel is
on the order of $\rm{Re}=500$, but the exact wind profile is not known. Fig. 1
shows the observation cell, which is located in a horizontal section of the
wind tunnel and hosts the granular bed. This cell is equipped with viewports
at its front and back sides, which enable visualization of the granular
dynamics with the camera through one side and illumination through the other
side. The entire wind tunnel acts as a vacuum chamber to simulate a thin
atmosphere such as the one of Mars.

Figure 1: Setup of the observation chamber. A charging unit is integrated into
a low-pressure wind tunnel, while in the experiments simulating uncharged
grains, a discharge unit is mounted on the upper flange.

The experiment is observed with a high-speed camera, which is combined with an
LED panel as bright field illumination (in backlight configuration) for a
visualization of the single grain trajectories. In our experiments, the camera
operates at 1000 fps with an exposure time of 1/5000 s, a focal depth of 1 mm,
and a spatial resolution of 25 µm.

The wind tunnel operates at 293 K and an air pressure of 20 mbar. This air
pressure is the lowest value suitable for our experiments. If we further
decrease the air pressure to simulate the mean atmospheric pressure on the
Martian surface (6 mbar), then the roots pump of our wind tunnel is not able
to produce strong enough surface winds for causing erosion of the granular bed
under Earth’s gravity. Therefore, all results in the present work have been
obtained using an atmospheric pressure of 20 mbar, or equivalently with an air
density of $0.023\,$kg/m3.

###  2.2 Granular material

The granular bed measures 3 cm x 3 cm with a thickness of about 1 cm in all
experiments. As a granular sample we choose two different materials.
Monodisperse soda-lime glass beads of a mean diameter of approximately 434 µm
with a standard deviation of $\pm\,$17 µm serve as model particles that allow
a theoretical model for charge-modified sand entrainment to be created. For
the exact size distribution we refer to Steinpilz et al. (2020a). After
vibrating the sample as described in Section 2.3, a packing fraction $\phi$ of
the bed of around 0.60 was measured.

In addition, Mars Global Simulant (MGS), a mineralogical analog to Martian
regolith, is used to evaluate the effect in a more realistic scenario. This
material was characterized in detail by Cannon et al. (2019). The size
distribution of the sand grains composing the sample is shown in Fig. 2.

Figure 2: Grain size distribution of the used sample of MGS.

###  2.3 Charging

To generate electric charges on the sample particles (tribocharging), the
granular material is placed into a 3D-printed container of PETG plastic, which
is subjected to controlled agitation produced by a voice coil. While the glass
samples are agitated in an uncoated plastic container, the sample container
for the sand measurements is covered with MGS grains to ensure the same
material contacts. To avoid spilling of the sample, the granular bed can be
lowered in the container for the duration of the shaking procedure and then
moved up again using the lift motor shown in Fig. 1. We perform different
types of agitation:

  * •

Charged beads #1: The glass beads are circulated for 2 hr in the container by
suitably setting the frequency of the voice coil to 19 Hz with a stroke of
around 1 mm (peak to peak). In this case, the particles move down at the
container walls and then up again in the bulk material.

  * •

Charged beads #2: The glass beads are shaken for 2 hr by applying controlled
vibrations to the container with an amplitude of around 1 mm and a frequency
of 15 Hz. At this frequency, the particles in the top layers bounce on the
surface without being mixed into the bulk material.

  * •

Charged sand: The MGS sample is shaken for 1 minute, which was found to be
sufficient to saturate the charges, with an amplitude of around 1 mm and a
frequency of 17 Hz.

Furthermore, to quantify the role of tribocharging on particle entrainment, we
need to compare the results obtained with the charged beds with the outcomes
from experiments performed without charging. In that case, the granular bed is
not shaken prior to the experiment. To reduce the charges as much as possible,
the glass beads and the sand sample are subjected to a plasma surface flow
(overall neutral) using an air ionizer that is directed onto the granular bed
from above. Thereafter, the system is closed and evacuated to 20 mbar.

###  2.4 Experimental procedure for estimating the minimal threshold wind
shear velocity $u_{{\ast}{\mathrm{ft}}}$

The goal is to identify the minimal flow speed in the wind tunnel that is just
sufficient to lift off single particles from the surface. We use Trackmate as
an automated particle tracker (Tinevez et al., 2017), which allows us to
measure the position and the velocity of moving glass beads in the system at
every 1 ms, based on the frame rate of 1000 fps in the experiments. Due to
their irregular shapes and wide size range, the sand grains are tracked
manually.

The wind velocity at different heights above the ground is estimated from the
trajectories of the hopping particles. Specifically, the following equation
gives the horizontal position $x(t,h)$ of a grain at time $t$ and height $h$
above the surface (Wurm et al., 2001),

$x(t,h)=\left[{u(h)-v_{0,x}}\right]\cdot\tau\cdot{\mbox{exp}}{\left({-\frac{t}{\tau}}\right)}+u(h)\cdot
t+x_{0}$ (1)

where $u(h)$ denotes the wind velocity at height $h$, $v_{0,x}$ and $x_{0}$
are the particle lift-off velocity and position, respectively, and $\tau$ is
the coupling time, i.e., the characteristic time needed by the particle to
adapt to the wind velocity.

It was shown by Merrison et al. (2008) that, under the air pressure conditions
associated with our experiments, the air flow near the granular bed is
laminar, so that the wind shear velocity can be estimated from the height
profile of the wind velocity with the equation

$u_{\ast}=\sqrt{\frac{\eta}{\rho_{\mathrm{f}}}\frac{{\mathrm{d}}u(h)}{{\mathrm{d}}h}},$
(2)

where $\eta=18\cdot 10^{-6}\,\rm{Pa}\cdot\rm{s}$ is the dynamic viscosity of
the air in the wind tunnel, $h$ is the height above the surface, and
$\rho_{\mathrm{f}}$ is the air density.

###  2.5 Estimating the particle charge in the experiments

In order to measure the charge of the particles, a Faraday cup is suitably
positioned at the downwind end of the observation cell to collect the
particles hopping along the bed. This Faraday cup is connected to an
electrometer, which allows us to measure the charge of the grains. To this
end, experiments are performed at the minimum threshold shear velocity
$u_{{\ast}{\mathrm{ft}}}$, so that only one particle at a time enters the cup,
thus enabling measurements of the charges of single particles. For details on
the charge measurement we refer to Genc et al. (2019) and Jungmann et al.
(2021). Each experiment is performed for about 10 minutes and no particles
with a charge magnitude smaller than $0.2\cdot 10^{5}\,$e can be detected. We
note that the charge measurement can only be carried out for the model system
of glass beads, as it requires well-defined, monodisperse particles. Since it
is not possible to determine the number of sand grains entering the cup at a
time, we cannot give charges of individual sand grains.

##  3 Experimental results

###  3.1 Charge measurements

Figure 3: Measured charges of glass beads for uncharged and both other cases.

The charge distribution of the glass beads in all three scenarios (uncharged,
charged #1, charged #2) is displayed in Fig. 3. The charge $q$ of the
particles is given in units of elementary charges of ${\mbox{e}}\approx
1.602\cdot 10^{-19}\,$C.

It has to be noted that the uncharged case still shows significant charges. We
attribute this to the motion of lifted beads that are not entrained by the
wind but fall back to the surface. Those particles hop over the bed surface
and charge before reaching the Faraday cup. In the charged cases, the shaking
procedure results in broader charge distributions with almost up to $10^{6}$ e
peak charge. Depending on the type of agitation, one charge distribution is
almost symmetrical, while the other contains predominantly positive charges,
which can be attributed to the dielectric difference between the glass
particles and the plastic container. The gaps in the distributions around
$q=0$ are due to the limited resolution of the electrometer.

Although we cannot measure the charge distributions of the sand particles,
these are assumed to be symmetrical, as charges are generated by the same
material tribocharging in the coated container. If the results of previous
experiments using larger grains are scaled to the sizes shown in Fig. 2,
charges of $10^{5}$ e to $10^{6}$ e can be expected (Wurm et al., 2019).

###  3.2 Minimal threshold wind shear velocity

Fig. 4 and 5 display the wind velocity measured in one experimental run at
different heights above the bed for the uncharged and the charged scenarios
(see Section 2.3). All experiments have been performed at the minimal
threshold flow speed conditions for transport. We see that the mean wind speed
increases approximately linearly with the height above the ground (as also
found by Merrison et al. (2008)), whereas this linear behavior is valid up to
a height of about $1.5\,$mm.

Table 1 displays the values of the wind velocity gradient,
${{\mathrm{d}}u}/{{\mathrm{d}}h}$, which is the slope of the dashed lines in
Fig. 4 and 5, and the associated threshold shear velocity
$u_{{\ast}{\mathrm{ft}}}$ obtained from these slopes using Eq. (2). As we can
see from this table, tribocharging has reduced the value of
$u_{{\ast}{\mathrm{ft}}}$ by about 16 % in the glass bead scenarios. The
realistic analog material shows a similar behavior of a lowered threshold for
charged grains. In this case, $u_{{\ast}{\mathrm{ft}}}$ decreases by 12 %. The
difference in the number of data points results from different measurement
times.

In order to ensure reproducibility, all experiments have been conducted twice.
Since the threshold value of charged sand is crucial for the application of
the results, this measurement has been conducted four times. The maximum
deviation from the values reported in Table 1 was around 0.02 m/s, which is on
the order of the uncertainty of $u_{{\ast}{\mathrm{ft}}}$ resulting from the
fits.

Figure 4: Velocity height profile for the three tested conditions in the model system of glass beads. Gray filled circles denote results from applying Eq. (1) to single grain trajectories, while averages and standard deviations are represented by the black circles and error bars, respectively. Dashed lines are fits to the data in the linear regimes. Figure 5: Velocity height profile for the uncharged and charged MGS grains. Again, gray filled circles denote results of single grain trajectories, while averages and standard deviations are represented by the black circles and error bars, respectively. Dashed lines are fits to the data in the linear regimes. Table 1: Wind velocity gradient ${{\mathrm{d}}u}/{{\mathrm{d}}h}$ and wind shear velocity $u_{{\ast}{\mathrm{ft}}}$ at the minimal threshold condition for transport for all experimental scenarios described in Section 2.3. | Data points |  $\frac{{\mathrm{d}}u}{{\mathrm{d}}h}$ [s-1] |  $u_{{\ast}{\mathrm{ft}}}$ [m/s]
---|---|---|---
Uncharged beads | 190 | $638\pm 16$ | $0.71\pm 0.01$
Charged beads #1 | 99 | $456\pm 55$ | $0.60\pm 0.04$
Charged beads #2 | 96 | $459\pm 16$ | $0.60\pm 0.01$
Uncharged sand | 140 | $1159\pm 60$ | $0.91\pm 0.02$
Charged sand | 115 | $882\pm 38$ | $0.80\pm 0.02$

##  4 Model for the minimal threshold wind shear velocity for aerodynamic
entrainment

The goal of this section is to shed further light on the experimentally
observed effect of electric charges on the minimum wind shear velocity
$u_{{\ast}{\mathrm{ft}}}$ required to initiate transport, by means of
modeling. Following Bagnold (1941), $u_{{\ast}{\mathrm{ft}}}$ can be estimated
from the balance of the lifting and resisting torques acting on a particle
protruding from the granular bed. Specifically, such a particle will be
entrained by the wind flow when it pivots around the point P in Fig. 6, which
is the point of contact with its supporting neighbor downwind. This balance is
computed next both for the uncharged bed and by considering tribocharged
particles. The models are based on the measurements with glass beads since
these represent well-defined spherical particles.

Figure 6: Schematical diagram of the main relevant forces acting on a particle
that is protruding from the granular bed considered in our experiments.
$F_{\mathrm{g}}$, $F_{\mathrm{drag}}$ and $F_{\mathrm{elec}}$ denote the
gravitational, drag, and electric forces, respectively, while the distances
$a$ and $b$ denote the moment arms of these forces relative to the pivoting
point P. Furthermore, $E_{0}$ denotes the near-surface electric field
generated by the granular bed, by considering that the particles have a net
positive charge $q_{\mathrm{eff}}>0$ (net negative charge,
$q_{\mathrm{eff}}<0$, would produce a downward-pointing $E_{0}$, but
$F_{\mathrm{elec}}$ would be still upward-pointing).

###  4.1 Modeling $u_{{\ast}{\mathrm{ft}}}$ for the uncharged granular bed

Bagnold (1941) obtained a simple expression for $u_{{\ast}{\mathrm{ft}}}$ by
considering the balance between the drag and gravitational torques,

$aF_{\mathrm{drag}}=bF_{\mathrm{g}}$ (3)

where the distances $a$ and $b$, defined in Fig. 6, are proportional to the
particle diameter $d$, while $F_{\mathrm{drag}}$ and $F_{\mathrm{g}}$ are the
drag and gravitational forces, respectively. Furthermore, by neglecting
buoyancy, $F_{\mathrm{g}}$ is given by the equation

$F_{\mathrm{g}}=\frac{\pi}{6}d^{3}{\rho}_{\mathrm{p}}g$ (4)

where ${\rho}_{\mathrm{p}}$ denotes the particle density and $g=9.81$ m/s2 is
gravity, while $F_{\mathrm{drag}}$ can be estimated using the equation

$F_{\mathrm{drag}}=K_{\mathrm{d}}{\rho}_{\mathrm{f}}d^{2}u_{\ast}^{2}$ (5)

with $u_{\ast}$ standing for the wind shear velocity and ${\rho}_{\mathrm{f}}$
denoting the air density. The values of the main parameters can be found in
Table 2. In Eq. (5), $K_{\mathrm{d}}$ is a dimensionless coefficient that
depends on the Reynolds friction number. By combining Eqs. (3), (4) and (5),
the following expression is obtained for the minimal threshold wind shear
velocity:

$u_{{\ast}{\mathrm{ft}}}=A_{\mathrm{d}}\sqrt{\frac{\rho_{\mathrm{p}}gd}{\rho_{\mathrm{f}}}}$
(6)

where the coefficient $A_{\mathrm{d}}$, called dimensionless threshold
friction velocity, encodes the dependence on
${\mbox{Re}}_{{\ast}{\mathrm{ft}}}$, the Reynolds number at the threshold
friction velocity.

We note that, in Eq. (3), we have not included attractive particle interaction
forces caused by interatomic or intermolecular (van der Waals) interactions,
since it has been shown that the effect of such forces on
$u_{{\ast}{\mathrm{ft}}}$ is small for particle sizes larger than 200 µm on
Earth (Demirci et al., 2020; Shao & Lu, 2000; Lu et al., 2005; Fu, L.-T.,
2020). However, various factors, such as the intermittent behavior of the wind
velocity and the thickness of the boundary layer, affect the dependence of
$A_{\mathrm{d}}$ on ${\mbox{Re}}_{{\ast}{\mathrm{ft}}}$. The derivation of an
expression for $A_{\mathrm{d}}$ that reliably accounts for these factors is
still a matter of research (Lu et al., 2005; Pähtz et al., 2020; Fu, L.-T.,
2020; Swann et al., 2020).

Table 2: Values of the main parameters of the model. Parameter | Symbol | Value
---|---|---
Particle diameter | $d$ | 434 µm
Particle density | $\rho_{\mathrm{p}}$ |  $2600\,$kg/m3
Air density | ${\rho}_{\mathrm{f}}$ |  $0.023\,$kg/m3
Height of the granular bed | $H_{\mathrm{bed}}$ |  $1\,$cm
Bed packing fraction | $\phi$ | $0.60$

Therefore, using Eq. (6), we estimate $A_{\mathrm{d}}$ from the value
$u_{{\ast}{\mathrm{ft}}}\approx 0.71\,$m/s obtained in the experiment with the
uncharged glass beads, which yields $A_{\mathrm{d}}\approx 0.032$. This value
of $A_{\mathrm{d}}$ is, thus, below the value of $A_{\mathrm{d}}$
characteristic of aeolian sand particles in the Earth’s atmosphere (about
$0.11$). Our finding aligns with recent observations (Swann et al., 2020; Fu,
L.-T., 2020) of lower-than-previously-thought $u_{{\ast}{\mathrm{ft}}}$ on
Mars — since early models of Martian aeolian processes assumed Earth’s value
of $A_{\mathrm{d}}\approx 0.11$ (for reviews, see Greeley & Iversen (1985);
Kok et al. (2012)).

In the next section, we extend our model to elucidate how an electric force
decreases the minimal threshold for aeolian sand transport initiation in the
experiments.

###  4.2 Modeling $u_{{\ast}{\mathrm{ft,elec}}}$ for the charged granular bed

The physics of electrostatic interactions in wind-blown particle systems is
still poorly known (Kok & Renno, 2006; Zhang et al., 2015). Furthermore, the
modeling of tribocharging in granular systems, as well as of the electric
forces between tribocharged particles, constitutes active matter of research
(Lacks & Shinbrot, 2019). Taking an additional electric force $F_{\rm{elec}}$
into account, the balance of the torques acting on the particle protruding
from the granular surface in Fig. 6 reads

$aF_{\mathrm{drag}}+bF_{\mathrm{elec}}=bF_{\mathrm{g}}$ (7)

where $F_{\mathrm{g}}$ and $F_{\mathrm{drag}}$ are given by Eqs. (4) and (5).
By substituting these expressions into Eq. (7), the following equation is
obtained for the minimal threshold wind shear velocity for entrainment of the
electrified grains,

$u_{{\ast}{\mathrm{ft,elec}}}=u_{{\ast}{\mathrm{ft}}}\sqrt{\mbox{max}{\left\\{0,\left[1-\frac{F_{\rm{elec}}}{F_{\rm{g}}}\right]\right\\}}},$
(8)

where $u_{{\ast}{\mathrm{ft}}}$, the corresponding threshold shear velocity
for the uncharged bed, is given by Eq. (6). In the following, we present three
different approaches to model the electric force on a soil particle and
compare the results with the experimental outcome.

####  4.2.1 Electric field of random net charges

The charge measurements revealed that the beads in the granular bed carry net
charges that create an electric field at the bed surface. Depending on the
sign of the charges and the resulting field, a particle protruding from the
surface may feel an additional attracting or repelling force. As the exact
charge configuration in the bed is unknown, we take a random charge
distribution as a first estimate to evaluate the force induced by net charges.

For simplicity, we model the granular bed based on a simple cubic geometry
with the respective dimensions of the experiment. The individual charges
$Q_{i}$ of each bead are assumed to be located in the center of the particle
and to be randomly distributed between $-10^{6}$ e and $+10^{6}$ e, which is
an overestimate, considering the measurement in Fig. 3. The electric field
$\vec{E}_{0}$ at a surface spot $\vec{r}_{\rm{s}}$ can then be calculated by
summing up the contributions of all charges $Q_{i}$ located at positions
$\vec{r}_{i}$ in the bed:

$\vec{E}_{0}=\frac{1}{4\pi\epsilon_{0}}\sum_{i}Q_{i}\frac{\vec{r}_{\rm{s}}-\vec{r}_{i}}{{|\vec{r}_{\rm{s}}-\vec{r}_{i}}|^{3}}.$
(9)

As only the vertical component of the field $E_{\rm{0,z}}$ is relevant to
determine the threshold wind velocity, the calculation can be limited to this
direction. The maximum upward force $F_{\rm{elec}}$ acting on a surface
particle with a charge $Q$ can be calculated according to

$F_{\rm{elec}}=Q\cdot\mbox{max}(E_{\rm{0,z}}),$ (10)

where $\mbox{max}(E_{\rm{0,z}})$ denotes the maximum vertical component of the
electric field generated in any surface point. Since the real charge
distribution within the bed, as well as on the bed surface, is not known, we
treat $Q$ as a model parameter that will be estimated from comparison with the
experimental data. Combining Eqs. (8) and (10) leads to the minimum threshold
wind velocity in the field of random charges. Fig. 7 depicts
$u_{{\ast}{\mathrm{ft,elec}}}$ in dependence of $Q$ for one simulation run.
Moreover, the horizontal dotted and dashed lines denote the experimental mean
values of the minimal threshold wind shear velocity obtained for the uncharged
bed ($\approx 0.71\,$m/s) and for the charged beds ($\approx 0.60\,$m/s),
respectively.

As we can see, the model predicts reduced values for the threshold wind
velocity for charged beds. To be consistent with the experimental results, a
charge of $Q=10^{8}$ e is required, which exceeds the observed charges by
several orders of magnitude. We are aware that this model is very simplified
regarding the exact geometry and charge distribution of the granular bed.
However, from the magnitude of the estimate, we can conclude that randomly
distributed net charges are not sufficient to explain the lower thresholds
observed in the experiments.

Figure 7: Modeled $u_{{\ast}{\mathrm{ft,elec}}}$ as a function of the charge
$Q$ of a protruding particle in the electric field of randomly distributed net
charges. The horizontal lines indicate the values of the threshold wind
velocities obtained in the experiments.

####  4.2.2 Electric field above a conducting plane

If each glass bead is assumed to carry the same effective charge, the
influence on the threshold wind velocity may be higher. In the extreme case,
we can regard the bed surface as a flat conducting plane. Such a plane
generates an electric field $E_{0}$ of magnitude (Griffiths, 1999; Jackson,
1999)

$E_{0}=\frac{\sigma}{2{\epsilon}_{0}},$ (11)

where $\sigma$ is the charge per unit area. Furthermore, we model $\sigma$
using the equation

$\sigma=\frac{n_{\mathrm{particles}}\cdot q_{\mathrm{eff}}}{A_{\mathrm{bed}}}$
(12)

where $n_{\mathrm{particles}}$ is the total number of particles in the
granular bed, $A_{\mathrm{bed}}$ is the bed surface area, and
$q_{\mathrm{eff}}$ is the effective (mean) charge of the particles within the
bed. Furthermore, $n_{\mathrm{particles}}$ can be estimated with the equation

$n_{\mathrm{particles}}=\phi\cdot\frac{A_{\mathrm{bed}}H_{\mathrm{bed}}}{(\pi/6)d^{3}},$
(13)

where $H_{\mathrm{bed}}$ is the thickness of the granular bed and $\phi$ its
packing fraction. Therefore, $H_{\mathrm{bed}}\,A_{\mathrm{bed}}$ gives the
volume of the bed, while the denominator of Eq. (13) denotes the volume of one
single particle.

Just as in the previous section, we assume that the electric force
$F_{\mathrm{elec}}$ on the protruding particle in Fig. 6 can be estimated
using the following equation:

$F_{\mathrm{elec}}=Q\cdot E_{0},$ (14)

where the model parameter $Q$ again denotes the characteristic charge value of
the protruding surface particle. Figure 8 shows the predicted values of
$u_{{\ast}{\mathrm{ft,elec}}}$ (solid lines), which are calculated using Eqs.
(8) and (14), as a function of $Q$ for different values of $q_{\mathrm{eff}}$.

Figure 8: Modeled $u_{{\ast}{\mathrm{ft,elec}}}$ as a function of the charge
$Q$ of a protruding particle in the electric field of the granular bed for
different effective charge magnitudes of the bed particles. The horizontal
lines indicate the values of the threshold wind velocities obtained in the
experiments.

As we can see from Fig. 8, the model would be consistent with the values
observed in the experiments, by applying an effective charge magnitude
$q_{\mathrm{eff}}$ and charge $Q$ of the surface particle of the order of
$15\cdot 10^{5}$ e. This result is closer to the measured charges in Fig. 3
than the first approach in Section 4.2.1, but the required charges are still
significantly higher than the observed values. Moreover, we note that the
effective charge $q_{\mathrm{eff}}$ of the bed particles is overestimated, as
the model does not account for the fact that both positive and negative
charges occur in the transport layer. For lower $q_{\mathrm{eff}}$, even
higher charges $Q$ would be required to explain the experimental observations.

####  4.2.3 Q-patch model

Electrostatically induced attraction or repulsion between two particles is not
only determined by net charges; multipoles also have to be considered
(Grosjean et al., 2020). Jungmann et al. (2018) found that the sticking
behavior of glass beads can be explained by a shift of the net charge from the
center to a near-surface spot. Steinpilz et al. (2020b) considered a complex
charge pattern on the surface with patches of negative and positive charge.
These Q-patches only measure a few microns but carry a significant amount of
charge comparable to the net charge of the particle.

Figure 9: Calculated $u_{{\ast}{\mathrm{ft,elec}}}$ as a function of the
charge $Q_{\rm{P}}$ of two patches at different distances $r_{\rm{P}}$
applying a Q-patch model. The horizontal lines indicate the values of the
threshold wind velocities obtained in the experiments.

Here we adopt this model and calculate the electric force acting on a
protruding particle based on a displaced center of charge. In this case,
Coulomb forces can be much stronger, as they decrease with the distance $r$ of
two charges according to $1/r^{2}$. If we assume that two beads in contact
interact over two equally charged patches in distance $r_{\rm{P}}$, the
Coulomb force reads

$F_{\rm{elec}}=\frac{1}{4\pi\epsilon_{0}}\frac{{Q_{\rm{P}}}^{2}}{{r_{\rm{P}}}^{2}},$
(15)

where $Q_{\rm{P}}$ quantifies the charge of the two patches. By substituting
this expression into Eq. (8), the threshold wind velocities in the Q-patch
model can be calculated. Since this model is based on charged patches near the
contact point of two beads, the exact geometry of the granular bed is of minor
importance to evaluate the lifting forces. Fig. 9 shows the results in
dependence of the charge $Q_{\rm{P}}$. As we can see, the charges required to
match the experiment are in the range of the measured charges in Fig. 3 for a
charge separation of 10 µm.

##  5 Discussion and application to Mars

Our results revealed values much lower than previously thought of the minimal
threshold wind shear velocity $u_{{\ast}{\mathrm{ft}}}$ in a Martian
environment using the model particles. Specifically, the experiments with an
uncharged bed of glass beads yielded $u_{{\ast}{\mathrm{ft}}}\approx
0.71\,$m/s, which is more than 3 times lower than the value of
$u_{{\ast}{\mathrm{ft}}}$ predicted by applying Bagnold’s equation to the air
pressure conditions and particle size and density of our experiments ($\approx
2.41\,$m/s). We have shown that the lower experimental threshold can be
explained by a smaller value of the Bagnold coefficient $A_{\mathrm{d}}$ (see
Eq. (6)). The threshold friction velocity obtained for the uncharged sample of
MGS is in the range of similar analog materials (Musiolik et al., 2018; Kruss
et al., 2020).

We note that the value of $u_{{\ast}{\mathrm{ft}}}$ assessed in our
experiments is the so-called threshold for sporadic, intermittent grain
movement, which corresponds to the transition from a static bed to the motion
of individual grains being lifted from the bed and performing a few hops over
it before coming to a halt. These bursts of grain motion are the precursors of
the state of continuous motion, in which saltating grains can be observed
continuously over the bed. The threshold for this state of continuous motion
has been called the general threshold and has been found to be 10 % larger
than the threshold for sporadic motion (Swann et al., 2020). Nevertheless,
because the bursts of grain hop provide the mechanism underlying the cascading
transport that leads to the continuous mode of transport, we follow previous
work (Swann et al., 2020) and adopt the onset of sporadic bursts as the
criterion to define the Martian fluid threshold, $u_{{\ast}{\mathrm{ft}}}$.

Furthermore, we have found that tribocharging can reduce the values of the
minimal wind shear velocity required for sediment transport substantially. Not
only is this effect observed with a model sample of monodisperse glass beads,
but also the MGS measurements show evidence of a lower threshold in a
realistic Martian environment. That is, sand grains can be lifted more easily
if electric charges are present. We did not investigate erosion rates or the
time needed for a potential cascading motion to set in. Since there are also
particle configurations where the additional electric force is attractive, the
erosion rate at the onset of motion is expected to be lower compared to an
uncharged bed at the corresponding threshold friction velocity. Nevertheless,
it is beneficial for erosion to already set in at lower wind velocities where
the uncharged bed is still at rest just as some wind gusts might initiate
saltation at lower wind speeds (Swann et al., 2020).

Based on the glass sample, we developed a model to predict
$u_{{\ast}{\mathrm{ft,elec}}}$ under consideration of an electric force (Eq.
(8)), for which we tested different approaches. It was shown that, as long as
charges are considered to be located in the center of the beads, neither the
Coulomb force of randomly distributed net charges nor the electric field of a
conducting plane can reproduce the observed values. Based on results of
Jungmann et al. (2018), Steinpilz et al. (2020b) and Grosjean et al. (2020)
among others, we also applied a model of charged surface patches. We note that
there is no detailed analysis of the surface charge pattern, i.e., the size of
the patches and the charges incorporated in them. However, the basic model is
the only one that can reproduce the order of magnitude of measured charges.
Therefore, we see the repulsion of charged surface patches as the most likely
mechanism to explain the observed lifting of glass beads. In fact, similar
ideas of charged patches on grains in close contact, though generated by
different mechanisms, are currently favored for ejecting dust grains from
atmosphereless bodies (Wang et al., 2016).

Our experiments may provide a basis to estimate electric charges on Martian
sands, e.g., in terrains where electrostatics may be a problem for landers on
Mars (Farrell et al., 2021). Furthermore, these insights will be useful to
improve models of sand transport on the Martian surface and to predict wind
conditions necessary for aeolian geomorphodynamics on Mars (Parteli &
Herrmann, 2007; Pähtz et al., 2014; Lämmel et al., 2018).

Given the large size of the glass beads considered in our study ($d=434$ µm),
van der Waals forces are negligibly small compared to gravitational forces.
However, in order to adapt our model to particles such as the used sand, van
der Waals forces should be included, since the cohesive interactions owing to
these forces become relevant for particles smaller than 70-100 µm and at
reduced gravity (Shao & Lu, 2000; Castellanos, 2005; Kok et al., 2012; Schmidt
et al., 2020; Demirci et al., 2020). Moreover, a theoretical approach to more
realistic conditions should also include broader size distributions.

##  6 Caveats

The experiments presented here come with some constraints. The side walls of
the plastic container in which the glass sample was vibrated were not covered
by glass beads. This might change the charging characteristics, as
triboelectrification is significantly influenced by the material of the
colliding particles (Lacks & Shinbrot, 2019). Two dissimilar materials tend to
carry charges of different sign and lead to broader charge distributions than
the interaction of same materials, which is also seen in the asymmetric
distributions in Fig. 3. However, even with a coated container and same
material contacts, in the case of sand particles, charge separation leads to a
significant reduction in the threshold shear stress. Without knowing the
charges in detail, their distribution is assumed to be symmetrical and is not
biased toward one sign. Nevertheless, the sand measurements show similar
evidence of a lower threshold to the glass beads.

Furthermore, the charging process is influenced by the size distribution of
the granular material. Same material tribocharging, e.g., typically leads to
positive charges on large grains, leaving the smaller ones with negative
charges (Waitukaitis et al., 2014). The monodisperse sample used here proved
suitable for first experiments and for developing a theoretical model. The MGS
sample, though, shows a broad size distribution and is more representative of
the Martian surface. Although it is not possible to measure the exact sizes of
moving particles owing to constraints of the optical system, it does not seem
that there is a favorable grain size for lifting. Particles around 100 µm in
size, as well as larger ones, are observed to be in motion. This finding also
provides some validation for the applied Q-patch model, as the effect can be
observed independently of the sign of the charges and their detailed
distribution. Regardless of the coating of the container, we observe a lower
threshold for charged samples. We note that only one size distribution was
employed in the experiments and that we did not quantify the charges. Whether
different size distributions of the particles affect the charge-induced
erosion process is subject to future work. We also have to note that in a full
flow, size-dependent charging and size sorting, e.g. lifting smaller grains
higher might result in large electric fields, which will have an impact on
particle lifting.

In addition to the sample properties, the pressure regime has to be
considered. Mars pressure could not be reached with the setup yet, as the wind
is not strong enough to lift particles under 20 mbar. Wurm et al. (2019)
showed that the charge gathering on colliding basalt particles is pressure
dependent and has a minimum around a few mbar owing to small-scale discharges.
According to their data, the width of the charge distribution drops by a
factor of around 2 going from 20 mbar to Martian pressure. The exact influence
on the charge $Q_{\rm{P}}$ expected in the charge patches is unknown, though
this might reduce the impact of electric forces on the lifting process on
Mars. Glass particles, in general, carry less charges than basalt and do not
reach charges high enough for pressure-dependent discharges to play a role
(Wurm et al., 2019; Steinpilz et al., 2020a). It is not clear to what extent
this behavior also applies to the sand sample, which is more basaltic.

##  7 Conclusion

In a wind tunnel experiment with tribocharged grains, we intended to evaluate
whether tribocharging during saltation might have an influence on particle
lift or not. It clearly does, as the threshold wind speed was reduced by more
than 10 % for both samples. It should be kept in mind that this was the first
setup with a number of constraints. Therefore, this number should not be taken
as the final face value. It is unlikely that our parameter combination did hit
the largest possible reduction factor in shear stress by chance, and in a
natural setting the shear stress might be reduced more strongly.

While strong winds are usually needed to initiate saltation on Mars, this
might be much easier after grains are charged triboelectrically. This does not
a priori guarantee that particles are always lifted at lower winds, as
charging is required first. The immediate conclusion would be that a wind that
subsided for a short time can continue saltation at a reduced threshold
afterward. How long the ground remains susceptible to this depends on the
discharge timescales. We can only speculate on this but at the dry, low-
pressure conditions on Mars, this might be hours, maybe more.

In any case, the experiments clearly show a potentially strong role for
saltation on Mars.

## acknowledgements

This project is supported by DLR Space Administration with funds provided by
the Federal Ministry for Economic Affairs and Energy (BMWi) under grant
numbers DLR 50 WM 1762 and DLR 50 WM 2142. This project has received funding
from the European Union’s Horizon 2020 research and innovation program under
grant agreement No 101004052. EJRP thanks the German Research Foundation for
financial support through the Heisenberg Grant, project 434377576.

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