# Shape matters: Competing mechanisms of particle shape segregation

D. Hernández-Delfin  Departamento de Física y Matemática Aplicada, Universidad
de Navarra, P.O. Box. 177, E-31080 Navarra, Spain  BCAM - Basque Center for
Applied Mathematics, Mazarredo, 14 E48009 Bilbao, Basque Country – Spain.
D.R. Tunuguntla  Multiscale Mechanics, Department of Thermal and Fluid
Engineering, Faculty of Engineering Technology, MESA+, University of Twente,
P.O. Box. 217, 7500 AE Enschede, The Netherlands  T. Weinhart  Multiscale
Mechanics, Department of Thermal and Fluid Engineering, Faculty of Engineering
Technology, MESA+, University of Twente, P.O. Box. 217, 7500 AE Enschede, The
Netherlands  R.C. Hidalgo<EMAIL_ADDRESS>Departamento de Física y
Matemática Aplicada, Universidad de Navarra, P.O. Box. 177, E-31080 Navarra,
Spain  A.R. Thornton<EMAIL_ADDRESS>Multiscale Mechanics, Department
of Thermal and Fluid Engineering, Faculty of Engineering Technology, MESA+,
University of Twente, P.O. Box. 217, 7500 AE Enschede, The Netherlands

###### Abstract

It is well-known that granular mixtures that differ in size or shape segregate
when sheared. In the past, two mechanisms have been proposed to describe this
effect, and it is unclear if both exist. To settle this question, we consider
a bidisperse mixture of spheroids of equal volume in a rotating drum, where
the two mechanisms are predicted to act in opposite directions. We present the
first evidence that there are two _distinct_ segregation mechanisms driven by
relative _over-stress_. Additionally, we showed that for non-spherical
particles, these two mechanisms can act in different directions leading to a
competition between the effects of the two. As a result, the segregation
intensity varies non-monotonically as a function of $AR$, and at specific
points, the segregation direction changes for both prolate and oblate
spheroids, explaining the surprising segregation reversal previously reported.
Consistent with previous results, we found that the kinetic mechanism is
dominant for (almost) spherical particles. Furthermore, for moderate aspect
ratios, the kinetic mechanism is responsible for the spherical particles
segregation to the periphery of the drum, and the gravity mechanism plays only
a minor role. Whereas, at the extreme values of $AR$, the gravity mechanism
notably increases and overtakes its kinetic counterpart.

##  I Introduction

Anyone who has shaken a box of cereal has observed segregation/de-mixing. This
is the so-called Brazil-nut problem, which still has many open questions
despite being heavily studied Rosato et al. (1987); Gajjar et al. (2021). In
general, segregation occurs due to differences in physical properties, ranging
from shape and size to the coefficient of restitution and sliding friction.
However, in sheared dense granular flows, it is the difference in size that
primarily drives segregation Gray (2018).

Size-based segregation in sheared dense granular flows has recently attracted
a lot of scientific attention, utilizing a variety of different approaches and
geometries. For example, van der Vaart et al. van der Vaart et al. (2015)
experimentally studied particle size segregation in a shear box, looking at
individual small and large particle dynamics in a system under oscillatory
shear. In contrast, Jing et al. Jing et al. (2017) used particle simulations
in a periodic chute to show that individual large particles carry higher
contact forces. This over-stress (or _over-pressure_) acts as a mechanism that
drives these large particles upwards.

The idea of large particle over-stresses measured by Jing et al. Jing et al.
(2017) (and many others) had previously been theoretically postulated by Gray
& Thornton Gray and Thornton (2005a); Thornton et al. (2006). However, they do
not express the over-stress in terms of particle-size, which left open two
questions: How does the over-stress scale with particle size-ratio; and, how
do you define size? In the original model, the over-stress is assumed to be
proportional to the hydrostatic load (pressure), which for most geometries
scales with gravity. However, in dynamic cases, Fan & Hill Fan and Hill
(2011a, b) showed a second source of over-stress called kinetic stress, which
is due to difference in velocity fluctuations and does not scale with gravity.

For polydisperse mixtures of spheres, Tunuguntla et al. Tunuguntla et al.
(2016) developed a novel micro-macro analysis technique, based on coarse-
graining (CG) Goldhirsch (2010); Weinhart et al. (2012); Weinhart et al.
(2013a); Artoni and Richard (2015, 2019). This is required as it consistently
splits the stress of a contact between different sizes of particles, which is
not guaranteed with simpler methods, e.g., binning. CG is used to obtain
macroscopic properties of granular flows from microscopic details. This
technique is widely used as a micro-macro mapping procedure presenting
advantages compared to other, simpler methods, namely binning and the method
of planes Tunuguntla et al. (2017). Using the technique, Tunuguntla et al.
Tunuguntla et al. (2017) showed that, in bidisperse mixtures flowing down
inclined planes, the over-stress in the kinetic stress is far greater than the
contact stress. However, both act in the same direction. Hence it is very hard
to distinguish the effects and raises the questions: Is there one or two
segregation mechanisms, and what happens for non-spherical particles?

Lu and Müller Lu and Müller (2020) investigated mixtures of spherical and non-
spherical particles in a rotating drum. They explored mixtures with different
blockiness; always, non-spherical particles segregated towards the center.
Various other authors have also studied segregation in rotating drums with
shapes such as cuboids and spheres Pereira and Cleary (2017) and rods Zhao et
al. (2018); Jones et al. (2020). In these studies, the particles differed in
volume and mass. Recently, He et al. He et al. (2019) numerically studied a
bidisperse mixture of ellipsoids in a rotating drum, keeping the particle
volume constant. Under these conditions, segregation is exclusively a result
of the differences in shape. Astonishingly, they showed that the segregation
direction changes depending on the shape differences. They attributed the
results to two competing segregation mechanisms. The first one is related to
the fact that spheres have better flowability than non-spherical particles,
making spheres flow along the strongly sheared drum periphery while ellipsoids
deposited at the center as they dissipate more energy in the flowing layer.
They described a second mechanism by observing that particles tend to orient
more against the flow direction as they become more non-spherical, allowing
spheres to percolate towards the core as ellipsoids offer less resistance.

Figure 1: Sketch of the numerical system; the dashed line represents the
boundary between the flowing layer and the solid body. In addition, the inset
shows a representation of the particle shapes
($AR\in\\{0.1,0.5,1.0,2.5,4.0\\}$).

In this work, inspired by the segregation reversal reported by He et al. He et
al. (2019) for equal-volume particles and equipped with the state-of-art
particle analysis tool MercuryCG Tunuguntla et al. (2016), we thoroughly
analyse the segregation behaviour of spheroids for different aspect ratios
within the existing theoretical frameworks. Remarkably, we demonstrate 1) the
segregation reversal is predicted by the idea of an over-stress, and 2) there
are two distinct stress segregation mechanisms, which even compete with each
other.

##  II Numerical model (DEM)

The model consists in a non-dimensional system involving a bidisperse mixture
of spherical and ellipsoidal particles in a rotating drum, subjected to a
gravitational field of magnitude $g=1$. Figure 1 illustrates a sketch of the
numerical setup, a drum is of radius $R=15d$ and depth $L=6d$, with periodic
boundary conditions in the transverse direction ($y$-axis). Gravity acts in
$z$-direction. The main goal of the simulations is to study segregation as a
result of differences in shape only, the material densities of both
constituents are equal, and spheroid semi-diameters $a$, $b$, and $c$ are
chosen in order to keep equal volumes, and therefore equal masses. Thus, each
sphere ($a=b=c$) has diameter $d=1$ and mass $m=1$, respectively. Similar to
He et al. He et al. (2019), we define an aspect ratio $AR=a/c$ and set
$a=AR^{2/3}/2$ and $b=c=AR^{-1/3}/2$, such that the spheroids have the same
volume as the spheres. If $AR<1$, the mixture is made of spheres and prolate
spheroids; if $AR=1$, both species are spheres; otherwise, the bed is made of
spheres and oblate spheroids. The studied range of $AR$ was from $0.1$ to
$4.0$ at $0.1$ intervals, with a total number of $N=2000$ particles, half of
which belong to each one of the constituents.

The drum walls are constructed by placing spherical particles of diameter
$d/2$ onto a cylindrical surface in a $L\times 2\pi R$ grid and forcing them
to rotate at a given angular velocity around the $y$-axis. Constructing the
drum from particles creates a rough surface with sufficient friction to avoid
slipping between the drum and the bulk particles. The rotation velocity
$\omega$ is set such that the non-dimensional Froude number $F=\omega^{2}R/g$,
which characterizes the ratio of centrifugal to gravitational forces, is
$F=0.01$. This value is close to the upper limit of the rolling regime, which
is one of the six categories of rotating drum flows and one of the most
relevant for industrial applications Govender (2016). Under this regime, the
granular flow involves two distinct zones, a thin flowing layer at the surface
and a passive solid body rotation below.

Simulations were carried out in MercuryDPM Weinhart et al. (2020), an open-
source code for discrete element modeling (DEM) particle simulations. To model
forces among particles, the linear spring-dashpot contact model is used, with
a restitution coefficient $e_{r}=0.1$, collision time $t_{c}=0.05$, and
Coulomb friction coefficient $\mu=0.5$. In order to find the contact point and
compute the overlap between two particles, we consider spheroids and spheres
as a subset of superquadric particles. The implementation is similar to
Podlozhnyuk et al. (2017), describing the surface of superquadrics with
$n_{1}=n_{2}=2$ for ellipsoidal shapes; inset in Figure 1 illustrates particle
shapes for $AR\in\\{0.1,0.5,1.0,2.5,4.0\\}$. As a starting point, the $N$
particles are randomly placed inside the drum, and we wait until the system is
relaxed before starting to rotate the drum. Setting the time step as $\Delta
t=t_{c}/50$, we analyze the results after ten rotations, at which time He et
al. He et al. (2019) observed appreciable differences regarding the components
segregation.

##  III Obtaining continuum fields

The previously mentioned continuum models are all expressed in terms of
Eulerian continuum properties like stress and volume fraction. However, the
DEM algorithm resolves the discrete Lagrangian trajectories of all the
simulated particles, providing their location and contact network with a given
time resolution. We employ a coarse-graining methodology to post-process this
discrete data to the required continuous fields which are required for the
theoretical models Goldhirsch (2010); Weinhart et al. (2013a); Artoni and
Richard (2015, 2019). Analogous to Tunuguntla et al. (2017), the simulated
system includes three different types of constituents (bulk spheres) type-$s$,
(bulk spheroids) type-$e$, and (boundary) type-$b$. The interstitial pore
space is considered as a zero-density passive fluid. Following Tununguntla et
al. Tunuguntla et al. (2017) we define the subsets of spherical particles,
$\mathcal{F}^{s}$, elliptical particles, $\mathcal{F}^{e}$, boundary
particles, $\mathcal{F}^{b}$, and their union
$\mathcal{F}=\mathcal{F}^{s}\cup\mathcal{F}^{e}\cup\mathcal{F}^{b}$, where
each particle $i\in\mathcal{F}^{\nu}$ has a mass $m_{i}$, a center of mass
$\mathbf{r}_{i}$ and velocity $\mathbf{v}_{i}$.

According to Tunuguntla et al. (2017); Goldhirsch (2010); Weinhart et al.
(2013a); Artoni and Richard (2015, 2019), the mass density corresponding to
type-$\nu$ constituent, $\rho^{\nu}(\mathbf{r},t)$, at $\mathbf{r}$ and time
$t$ is defined by

$\rho^{\nu}\left(\mathbf{r},t\right)=\sum_{i\in\mathcal{F}^{\nu}}m_{i}\psi\left(\mathbf{r}-\mathbf{r}_{i}(t)\right)$
(1)

and $\psi\left(\mathbf{r}-\mathbf{r}_{i}(t)\right)$, henceforth $\psi_{i}$ for
simplicity, is an integrable coarse-graining function. In the present work, we
choose the Lucy-polynomials with a width of 1, i.e., the diameter of the
spherical particles, as recommended by Tunuguntla et al. (2016).

To satisfy the mass and momentum conservation equations, expressions for the
other type-$\nu$ partial quantities are

$\displaystyle\begin{aligned}
\mathbf{u}^{\nu}&=\frac{1}{\rho^{\nu}}\sum_{i\in\mathcal{F}^{\nu}}m_{i}\mathbf{v}_{i}\psi_{i},\\\
\boldsymbol{\sigma}^{k,\nu}&=\sum_{i\in\mathcal{F}^{\nu}}m_{i}\mathbf{v}_{i}^{\prime}\mathbf{v}^{\prime}_{i}\psi_{i},\\\
\quad\boldsymbol{\sigma}^{c,\nu}&=\sum_{i\in\mathcal{F}^{\nu}}\sum_{j\in\mathcal{F}}\mathbf{f}_{ij}\mathbf{b}_{ij}\psi_{ij},\end{aligned}$
(2)

where $\mathbf{v}^{\prime}_{i}$ is the fluctuation of the velocity of particle
$i$ with respect to the mean field,
$\mathbf{v}_{i}^{\prime}=\mathbf{u}-\mathbf{v}_{i}$, $\mathbf{f}_{ij}$ and
$\mathbf{b}_{ij}$ denote the force between particle $i$ and $j$ and the branch
vector between particle $i$ and its contact point with particle $j$,
respectively. Finally, $\psi_{ij}$ is the lineal integral along
$\mathbf{b}_{ij}$,
$\psi_{ij}=\int_{0}^{1}\psi(\mathbf{r}-\mathbf{r}_{i}+s\mathbf{b}_{ij})ds$,
which distributes the contact stress proportionally, subjected to the
corresponding branch fraction of each constituent.

Following the previous procedure, all relevant macroscopic fields were
extracted from the DEM data. The fields were averaged over the depth of the
drum and in time for a period of 1.5 rotations, starting from the tenth
rotation, with a temporal resolution $\Delta t_{CG}=0.01\sqrt{d/g}$.

##  IV Results and discussion

Figure 2: (a) Snapshots of the mixtures $AR=\\{$0.1,0.5,2.5,4.0$\\}$ after ten
rotations ($t\approx$ 2433). The spherical particles are colored gray, while
elongated particles are red. (b) The concentration of spheres $\phi_{s}(x,z)$
obtained from density CG-fields for the same $AR$ values of (a).

We performed DEM simulations of a rotating drum by exploring the impact of the
particle aspect ratio $AR$ on the segregation process. To this end, a simple
visual inspection of the system can tell whether the particles tend to group
by shapes after a while. Figure 2 a) depicts the mixture states after ten
rotations for $AR=\\{$0.1,0.5,2.5,4.0$\\}$. As can be clearly seen,
segregation is evidenced in all these cases at different degrees. The most
extreme mixture ($AR=$ 0.1) shows that most of the spherical particles are in
the core of the drum, while non-spherical ones are in the periphery. In case
$AR=$ 0.5, the segregation inverts despite being less intense. The inversion
of the segregation also manifests in mixtures that contain prolate ellipsoids.
Similar segregation reverse was previously found, e.g., the called Reverse
Brazilian Effect (RBE) Hong et al. (2001); Breu et al. (2003) or the recent
result in reference Duan et al. (2021), where the inversion is obtained by
varying size and density simultaneously. Here, the most striking about this
result is that the segregation reverse occurs only with varying the particle
shape (see also Ref.He et al. (2019)).

Here we will define two types of volume fraction the normal volume fraction
per unit $\varphi^{\nu}$ mixture volume, which is simply the local ratio of
volume of constituent $i$, over total volume $\varphi^{\nu}=V^{i}/V$. From
this we can define volume fraction per unit granular volume that is
$\phi^{\nu}=\varphi^{\nu}/(\varphi^{s}+\varphi^{e})$, which has the property
$\phi^{e}+\phi^{s}=1$. Figure 2 b) shows as color-map the volume fraction per
solid volume of spheres $\phi^{s}(x,z)$. It resembles the spatial distribution
of spheres and ellipsoids, depicted in Figure 2 a). One can notice that
$\phi^{s}$ is non-homogeneous, with higher values in the core of the drum for
mixtures with extremely elongated spheroids ($AR=$ 0.1 and $AR=$ 4.0). In
contrast, the opposite behavior is observed for $AR$ values close to 1, where
the mixture is homogeneous.

Steeping forward, using the quantities $\phi^{\nu}$ and the bulk density
$\rho(\mathbf{r})$, we adapt the method of Arntz et al. Arntz et al. (2008,
2014) to quantify segregation accurately. From statistical mechanics, the
mixing entropy is defined as

$\tilde{M}=\sum_{\nu\in\\{s,e\\}}\int_{\mathbb{R}^{3}}\rho(\mathbf{r})\phi^{\nu}(\mathbf{r})\ln{\phi^{\nu}(\mathbf{r})}\differential\mathbf{r}.$
(3)

Next, we define a segregation index $S$ that measures not only the segregation
intensity but also the direction in which segregation occurs in a rotating
drum (inwards or outwards),

$S=\left\\{\begin{array}[]{ll}M-1&\text{if spheres segregate inwards},\\\
1-M&\text{if spheres segregate outwards}.\end{array}\right.$ (4)

Thus, a positive value of $S$ indicates that spheres tend to segregate away
from the core.

Figure 3: The segregation index $S$ as a function of the aspect ratio $AR$,
which has been computed for each of the three half rotations and averaged. The
red circles illustrate the aspect ratios where spheres segregate to the center
of the drum, whereas the blue triangles indicate that spheres go to the
periphery. For $AR=$ 1.0, a particular marker is assigned (black diamond). In
all cases, error bars account for the standard deviation of the mean value.
Figure 4: Drawing representing the velocity field $\mathbf{u}$ (black
streamlines) and its perpendicular field
$\hat{\mathbf{y}}\crossproduct\mathbf{u}$ (red arrows). Complementary, gravity
$\mathbf{g}$ and its projection along a vector
$\mathbf{n}=\hat{\mathbf{y}}\times\mathbf{u}/{|\mathbf{u}|}$ are represented,
$g_{\rm n}=\mathbf{g}\cdot\mathbf{n}$.

Going further, we show the spatially-averaged segregation index Eq. 4 as a
function of $AR$ in figure 3. For $AR$ values close to 1.0, the values of $S$
are positive (labelled with blue triangles), indicating that the spheres tend
to segregate to the drum periphery, whereas $S$ is negative for extreme $AR$
values (labelled with red circles); thus, the spheres segregate towards the
core.

For the case of prolate spheroids, $S$ reaches a maximum local value at
$AR\approx$ 2.5 and flips direction at $AR\approx$ 3.1, suggesting that a new
effect starts to play a role in the behavior of the system. Similarly, for
oblate spheroids, $S$ reaches a maximum local value at $AR\approx$ 0.5 and
flips direction at $AR\approx$ 0.4. At $AR=$ 1 (labeled with a black diamond),
$S\approx$ 0, a result expected beforehand because segregation must not occur
for similar constituents. These results are similar to He et al. (2019) and
indicate that the segregation changes direction at specific $AR$ values.

###  IV.1 Continuum analysis of particle segregation

Figure 5: Every row illustrates the profiles of $(f^{{\rm k},\nu}-\phi^{\nu})$
(left column), $(f^{{\rm c},\nu}-\phi^{\nu})$ (center column), and $(f^{{\rm
c},\nu}-f^{{\rm k},\nu})$ (right column) as a function of $\phi^{\nu}$ for
oblate $AR\in\\{0.1,\,0.2,\,0.4,\,0.5,\,0.7\\}$ and prolate
$AR\in\\{1.5,\,2.3,\,2.5,\,3.7,\,4.0\\}$ ellipsoids, respectively.

Next, we quantify the segregation mechanisms and clarify the origin of the
change in the segregation direction, employing the theoretical framework
introduced by Fan and Hill Fan and Hill (2011b). This formulation is based on
the partial momentum and mass conservation equations for individual
constituents of a mixture Morland (1992),

$\displaystyle\partial_{t}\rho^{\nu}+\nabla\cdot\left(\rho^{\nu}\mathbf{u}^{\nu}\right)$
$\displaystyle=0,$ (5a)
$\displaystyle\rho^{\nu}\left(\partial_{t}\mathbf{u}^{\nu}+\mathbf{u}^{\nu}\cdot\nabla\mathbf{u}^{\nu}\right)$
$\displaystyle=-\nabla\cdot\boldsymbol{\sigma}^{\nu}+\rho^{\nu}\mathbf{g}+\boldsymbol{\beta}^{\nu},$
(5b)

where $\rho^{\nu}$, $\mathbf{u}^{\nu}$, and $\boldsymbol{\sigma}^{\nu}$ are
the density, velocity, and stress of the constituent $\nu$, respectively.
$\mathbf{g}$ is the gravity vector, and $\boldsymbol{\beta}^{\nu}$ is the
inter-constituent drag force that obeys
$\sum_{\nu}\boldsymbol{\beta}^{\nu}=\boldsymbol{0}$. We use the coarse-
graining formulae from section III in order to directly obtain these mixture
variables from the discrete particle data.

We consider systems in which the velocities and partial densities approach
steady state long before the segregation profile equilibrates, such that the
temporal derivatives in (5b) become negligible. Moreover, we assume that
segregation occurs in the direction of the shear plane orthogonal to the
barycentric velocity, $\mathbf{u}=\sum_{\nu}\rho^{\nu}\mathbf{u}^{\nu}/\rho$.

Thus, if the drum is rotating around the $y$-axis, then the direction of
segregation is $\mathbf{n}=\mathbf{\hat{y}}\times\mathbf{u}/{|\mathbf{u}|}$
(see Fig. 4). In the flowing layer (where gradients perpendicular to
$\mathbf{n}$ can be neglected), multiplying (5b) by $\mathbf{n}$ yields

$\nabla_{\rm n}\sigma_{\rm n}^{{\rm c},\nu}+\nabla_{\rm n}\sigma_{\rm n}^{{\rm
k},\nu}=\beta_{\rm n}^{\nu}+\rho^{\nu}g_{\rm n},$ (6)

where $\sigma_{\rm n}^{{\rm c},\nu}$ and $\sigma_{\rm n}^{{\rm k},\nu}$ denote
the contact and kinetic parts of the stress component pointing in the
$\mathbf{n}$, respectively. The gradient $\nabla_{\rm
n}=\nabla\cdot\mathbf{n}$ represents the derivative in $\mathbf{n}$ direction,
and $g_{\rm n}=\mathbf{g}\cdot\mathbf{n}$ is the projection of $\mathbf{g}$
along $\mathbf{n}$.

In addition to the previously defined partial quantity $\phi^{\nu}$, we define
a _kinetic_ and _contact stress fraction_ , i.e., the stress per unit granular
stress, as $f^{{\rm k},\nu}=\sigma_{\rm n}^{{\rm k},\nu}/\sigma_{\rm n}^{{\rm
k}}$ and $f^{{\rm c},\nu}=\sigma_{\rm n}^{{\rm c},\nu}/\sigma_{\rm n}^{{\rm
c}}$, as was done in previous contributions Gray and Thornton (2005a); Gray
and Chugunov (2006); Tunuguntla et al. (2017). If the stress fraction of a
constituent exceeds the value $\phi^{\nu}$, we all this condition _over-
stress_ , else _under-stress_.

The last ingredient of the model is the form taken by $\beta_{\rm n}^{\nu}$;
similar to Hill and Tan Hill and Tan (2014), we neglect diffusive remixing and
propose the following drag terms

$\beta_{\rm n}^{\nu}=\sigma_{\rm n}^{{\rm c},\nu}\nabla_{\rm n}\left(f^{{\rm
c},\nu}\right)+\sigma_{\rm n}^{{\rm k},\nu}\nabla_{\rm n}\left(f^{{\rm
k},\nu}\right)-\rho^{\nu}c\left(u_{\rm n}^{\nu}-u_{\rm n}\right)$ (7)

The first two terms ensure that, as in Darcy’s law, the percolation process is
driven by intrinsic rather than partial stress gradients. The third term is a
linear drag law similar to that provided by Morland Morland (1992) for the
percolation of fluids, where $c$ is an inter-constituent drag coefficient,
which we assume to be constant. This form of the drag may appear complicated,
but it is both theoretically justified, e.g., Gray and Thornton (2005b); Hill
and Tan (2014) and confirmed in particle simulations Weinhart et al. (2013b);
Bancroft and Johnson (2021). Of particular note is the work of Bancroft and
Johnson (2021) as they extended the drag model to give a closed form
expression for $c$; however, here we simply give it as to determined
parameter, $c$.

Substituting (7) in (6), the relative percolation velocity is

$c\phi^{\nu}(u_{\rm n}^{\nu}-u_{\rm n})=\underbrace{\dfrac{(f^{{\rm
c},\nu}-f^{{\rm k},\nu})}{\rho}\nabla_{\rm n}\sigma^{k}_{\rm
n}}_{\Phi_{k}}+\underbrace{\dfrac{(f^{{\rm c},\nu}-\phi^{\nu})}{\rho}g_{\rm
n}}_{\Phi_{g}}$ (8)

where $\Phi_{k}$ and $\Phi_{g}$ quantify the effects of kinetic and gravity
mechanisms. Note that $\Phi_{k}$ is not present in the original Gray &
Thornton model Gray and Thornton (2005a). Before we compute $\Phi_{k}$ and
$\Phi_{g}$, it is convenient to obtain the profiles of $(f^{{\rm
k},\nu}-\phi^{\nu})$, $(f^{{\rm c},\nu}-\phi^{\nu})$, and $(f^{{\rm
c},\nu}-f^{{\rm k},\nu})$ as a function of $\phi^{\nu}$.

Figure 6: From top to bottom: averaged gravity, $\langle\Phi_{g}\rangle$
stress; kinetic segregation, $\langle\Phi_{k}\rangle$; and total
stress,$[\langle\Phi_{g}\rangle+\langle\Phi_{k}\rangle]$; for formulations see
(8). The blue triangles are where the spherical particles segregate inwards
and the red spheres outwards. Showing only the combination of the two
mechanisms predictions the correct tread.

To perform the analysis related to the quantification of the segregation
mechanisms, we averaged the CG-fields on the domain where the density of the
mixture is different from zero. Fig. 5 illustrates the aforementioned profiles
for oblate $AR\in\\{0.1,\,0.2,\,0.4,\,0.5,\,0.7\\}$ and prolate
$AR\in\\{1.5,\,2.3,\,2.5,\,3.7,\,4.0\\}$ ellipsoids, respectively. One can see
that the values of $(f^{{\rm k},\rm s}-\phi^{\rm s})$ for spheres are positive
for all $AR$, indicating that spherical particles support more kinetic stress
than their relative concentration. This result agrees with previous
observations that spheres possess a better flowability and implies that
kinetic sieving alone is unable to explain the change in the segregation
direction. Moreover, $(f^{{\rm c},\rm s}-\phi^{\rm s})$ is near zero for
$AR=0.5$ and $2.5$ and strictly negative for $AR=0.1$ and $AR=4.0$, indicating
that spherical particles support equal or less contact stress than their
relative concentration. To our knowledge, this is the first time that research
evidences this finding.

In figure 5, we plot the kinetic $(f^{\mathrm{k},\nu}-\phi^{\nu})$ (left),
contact overstresses $(f^{\mathrm{c},\nu}-\phi^{\nu})$ (middle), and their
difference $(f^{\mathrm{c},\nu}-f^{\mathrm{k},\nu})$ (right). It should be
noted that the difference (right) is pre-factor for the kinetic mechanics,
$\Phi_{k}$ in Eq. (8), and the contact over-stress (right), the gravity
mechanism $\Phi_{g}$ in Eq. (8). Both mechanisms are influenced by the contact
over-stress and that the kinetic and contact overstress are always
complementary: mixtures that are contact-overstressed are simultaneously
kinetic-understressed, and vice versa. Thus, a high contact over-stress also
leads to a high prefactor of $\Phi_{\mathrm{k}}$, that is, the magnitude (not
direction) of both mechanisms is highly influenced by the contact stress. This
is why previously it has been hard to determine if there are one or two
different segregation mechanisms.

We denote the averaging over the flowing layer by $\langle\Phi_{g}\rangle$ and
$\langle\Phi_{k}\rangle$, respectively. Fig. 6 displays both terms and the
averaged total contribution, multiplied by $c$ as a function of $AR$.
Comparing $\langle\Phi_{g}\rangle$ and $\langle\Phi_{k}\rangle$ shows that the
kinetic mechanism dominates where spheres segregate to the periphery of the
drum (blue triangles). However, where the spheres segregate to the core of the
mixture (red circles), the gravity mechanism increases as the constituents of
the mixture become more different, and the absolute values of
$\langle\Phi_{g}\rangle$ are larger than the $\langle\Phi_{k}\rangle$ ones.
This implies that for positive values of $S$, segregation occurs mainly
because of the kinetic stress gradient, and the gravity mechanism is
negligible. In contrast, for negative values of $S$, the gravity mechanism
drives the segregation, even though $\langle\Phi_{k}\rangle$ has considerable
values. Finally, the averaged total contribution
$\langle\Phi_{g}\rangle+\langle\Phi_{k}\rangle$ correlates with the behavior
of $S$, and the sign changes of the total contribution occur at the same $AR$
values as the sign change in $S$. So, in order to correctly predict the
direction of segregation, both mechanisms are required. To the best of the
authors’ knowledge, this represents the first system where these two effects
have been shown to be in competition.

##  V Conclusions

Our numerical and theoretical analysis accurately explains particle-shape
segregation patterns in rotating drums. We present the first evidence that
there are two _distinct_ segregation mechanisms driven by _over-stress_ in the
contact and kinetic stresses, as was suggested by Fan & Hill Fan and Hill
(2011b). In fact, we show that for non-spherical particles, these two
mechanisms can act in different directions leading to a competition between
the effects of the two. This explains the surprising segregation reversal
reported by He et al. He et al. (2019). Particularly, in rotating drums, the
segregation intensity varies non-monotonically as a function of $AR$, and at
specific points, the segregation direction changes for both prolate and oblate
spheroids. Remarkably, our analysis predicts the location of these transition
points quantitatively, quantifying the relative momentum interchange between
the species. Consistent with previous results Fan and Hill (2011b); Tunuguntla
et al. (2017) we found that the kinetic mechanism is dominant for (almost)
spherical particles. For moderate $AR$ values, the kinetic mechanism is
responsible for spherical particles segregating to the periphery of the drum,
and the gravity mechanism plays only a minor role. Whereas, at the extreme
values of $AR$, the gravity mechanism increases rapidly and dominates its
kinetic counterpart.

D.H. acknowledges Asociación de Amigos de la Universidad de Navarra,
MercuryLab, and Fundación Caja Navarra-Caixa. This work was partially funded
by Ministerio de Economía y Competitividad (Spanish Government) through the
Project PID2020-114839GB-I00 MINECO/AEI/FEDER, UE. R.C.H. acknowledgments the
European Union’s Horizon 2020 Research and Innovation programme under the
Marie Sklodowska-Curie grant agreement CALIPER No 812638. A.T. acknowledges
the support of the Dutch Research Council, STW-Vidi project 13472. T.W.
acknowledges the support of the Dutch Research Council, NWO-TTW grants 15050
and 16604. Finally, we acknowledge the contributions of I.F.C. Dennissen, who
started the superquadrics implementation and the early simulations on this
topic. Denissen (2019).

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