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\begin{document}

\title{Experimental and Numerical Investigations of the Shear Behavior of
Binary Particle Blends}



\author[1]{Jiecheng Yang}%
\author[2]{Poom Bunchatheeravate}%
\author[2]{Subhash Thakur}%
\author[2]{Joseph Bullard}%
\author[3]{Jennifer Sinclair Curtis}%
\affil[1]{University of California Davis}%
\affil[2]{Vertex Pharmaceuticals Incorporated}%
\affil[3]{Universtity of California, Davis}%


\vspace{-1em}



  
  \date{May 26, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
In this study, a Schulze Ring Shear Tester and the Discrete Element
Method (DEM) are employed to investigate the effect of polydispersity on
the binary shear flows. Both experimental results and DEM simulations
show that the pre-shear stresses are greater for binary blends than for
monodispersed particles. The flowability of these mixtures is strongly
affected by the solid fraction, with minimal flow function values
correlating to maximum packing fraction. However, minimum flow function
values are not observed at the same packing fractions where the maximum
pre-shear stress occurs. The powder friction has a slightly higher
contribution to powder strength than powder adhesion, and the frictional
component follows the same trend as shear stress with mixture
composition. Using DEM, it is demonstrated that the addition of small
adhesive particles reduces the averaged angular velocity of the larger
particles, which makes a contribution to the larger shear stress for
binary blends.%
\end{abstract}%



\sloppy


\textbf{Experimental and Numerical Investigations of the}

\textbf{Shear Behavior of Binary Particle Blends}

Jiecheng Yang\textsuperscript{1*}, Poom
Bunchatheeravate\textsuperscript{2}, Subhash C.
Thakur\textsuperscript{2}, Joseph W. Bullard\textsuperscript{2}, and
Jennifer S. Curtis\textsuperscript{1}

\emph{\textsuperscript{1} Department of Chemical Engineering, University
of California Davis, Davis, CA 95616, USA}

\emph{\textsuperscript{2}Vertex Pharmaceuticals Incorporated, Boston, MA
02210, USA}

\emph{*corresponding author}

\textbf{Abstract:} In this study, a Schulze Ring Shear Tester and the
Discrete Element Method (DEM) are employed to investigate the effect of
polydispersity on the binary shear flows. Both experimental results and
DEM simulations show that the pre-shear stresses are greater for binary
blends than for monodispersed particles. The flowability of these
mixtures is strongly affected by the solid fraction, with minimal flow
function values correlating to maximum packing fraction. However,
minimum flow function values are not observed at the same packing
fractions where the maximum pre-shear stress occurs. The powder friction
has a slightly higher contribution to powder strength than powder
adhesion, and the frictional component follows the same trend as shear
stress with mixture composition. Using DEM, it is demonstrated that the
addition of small adhesive particles reduces the averaged angular
velocity of the larger particles, which makes a contribution to the
larger shear stress for binary blends.

\textbf{Keywords:} Binary shear flows; Ring shear test; Discrete Element
Method

\section*{1. Introduction}

{\label{introduction}}

Fundamental understanding of the flow behavior of bulk solids is
essential for granular material handling and the design of relevant
industrial processes and equipment, such as hoppers, silos and blenders.
The Schulze Ring Shear Tester (SRST) is one of the most popular testers
of bulk solids flow properties and it can measure the shear stresses
under various normal stresses. Therefore, the internal friction angle,
wall friction angle, unconfined yield strength, bulk density and flow
function of the tested granular material can also be
provided\textsuperscript{1,2}. In addition, numerical approaches such as
Discrete Element Method (DEM) are also increasingly applied to
investigate the flow behavior of granular materials. Detailed
information at the individual particle level, such as the particle
position, particle velocity and forces exerted on the particle, can be
obtained from DEM simulations, which is helpful for exploring the
mechanisms of particle behavior\textsuperscript{3,4}.

Both previous experimental and numerical research has shown that the
flow behavior of bulk solids could be influenced by many factors. The
effect of particle shape was investigated by Baran et
al.\textsuperscript{5} and results shown that the shear stress for
spherical particles was considerably lower than that of aspherical
particles. Shear stress for aspherical particles were found larger than
that for spherical ones, for both monodisperse\textsuperscript{6} and
binary mixtures\textsuperscript{7}. The internal friction angle was
increased by preventing particle rotation\textsuperscript{8}. The shear
stress was also found to increase with increasing coefficient of
friction between particles\textsuperscript{2,5,9}. On the other hand,
the shear cell size, shear rate, particle shear modulus, particle
Poisson's ratio, the coefficient of restitution and particle-particle
cohesion were argued to have insignificant influence on the shear
stress\textsuperscript{2,5,9}.

Among the factors affecting the flow properties of bulk solids, particle
size polydispersity is known to have an important effect on particle
flow behavior\textsuperscript{7,10--12}. Rule of Mixture (ROM)
approaches are often employed in predicting the flow behavior of binary
or higher-order powder systems. In ROM frameworks, the predicted
properties are calculated using the properties of constitutive
components, which are weighted by their composition of the final blend
using various averaging techniques. For example, the flow function
coefficient for binary mixtures of acetaminophen and starch were found
to decrease with increasing mass fraction of acetominophen, which
exhibits poorer flow behavior than starch as a pure
material\textsuperscript{13}. For binary systems, any property predicted
using ROM will always fall between those of the raw components. However,
this approach ignores many complex interactions inherent to mixtures of
particles, and even seemingly simple systems deviate from trends
predicted by ROM approaches.

Analogs to the systems evaluated here have previously been shown to
exhibit behavior that cannot be explained by simple ROM approaches.
Vallejo\textsuperscript{12} conducted laboratory tests on mixtures of
large (5000 \selectlanguage{greek}µ\selectlanguage{english}m) and small (400 \selectlanguage{greek}µ\selectlanguage{english}m) glass beads to evaluate the shear
strength of the resultant blends, and reported a strong dependence on
the relative concentration of the mass fraction of large and small
particles. In the range of 40-70\% mass fraction of large particles,
shear strength decreased with decreasing fraction of large particles.
However, shear strength was controlled by the frictional resistance of
the large particles if the mass fraction of large particles exceeded
70\%, whereas the effect of small particles dominated when below 40\%.
Vallejo argued that these limits resulted from changes in mixture
porosity and particle configuration.

In order to evaluate the effect of particle size polydispersity on the
flow behavior, experiments are set up to obtain shear stress data in a
simple flow configuration (SRST) using a model system of
well-characterized, binary mixtures of spherical glass beads. By varying
the ratio of small to large particles in the system, this approach
facilitates a detailed study of the effect of polydispersity on the flow
behavior of particle mixtures. Furthermore, in order to investigate the
fundamental underpinnings associated with the flow behavior (i.e. shear
stress trend), DEM is applied to model the shear flows of such mixtures
in the ring shear cell. First, the experimental data are used to
validate the model predictions. Then, via DEM, the effects of particle
rotation, particle contact type, particle contact number and force
network are investigated. By analyzing these effects, potential
underlying mechanisms giving rise to the variation in mixture
flowability are proposed and explored.

\section*{2. Experimental SRST}

{\label{experimental-srst}}

\subsection*{2.1 Material}

{\label{material}}

Glass beads in size ranges of 30-50\selectlanguage{greek}µ\selectlanguage{english}m (Polysciences Inc. 18901-100, lot
685012) and 150-210\selectlanguage{greek}µ\selectlanguage{english}m (Polysciences Inc. 05483-250, lot A760996) were
acquired from Polysciences Inc. A particle density of
2500kg/m\textsuperscript{3} is used in both the DEM simulation and
experimental analysis, consistent with averaged reported values from
vendor specifications.

Prior to testing, samples were screened using a \#100 US standard sieve
(150\selectlanguage{greek}µ\selectlanguage{english}m opening). Images of each bead type were taken at 200x
magnification using a Keyence VHX-2000 optical microscope. Binary blends
of small and large glass beads were prepared at ratios of 3:1, 1:1, and
1:3 on a mass basis, and prepared at a 30g scale. Weighed and screened
materials were transferred to a 135ml glass container and mixed using a
Turbula mixer at 32rpm for 5min.

\subsection*{2.2 Shear testing}

{\label{shear-testing}}

Ring shear testing of each blend was performed on the Schulze RST-XS
using the XS-MV4 shear cell. The shear cell is annular in shape with an
inner diameter of 32mm and an outer diameter of 64mm. The initial gap
between the base and the lid is 4mm; the size of this gap decreases as
pressure is applied to the shear cell and particle mixtures are
compressed. The shearing bar of the XS-MV4 cell is pyramidal in shape
and covers the surface of both the base and the lid.

Shear measurements were performed using a shear rate of
0.004s\textsuperscript{-1} and a 2000 Pa pre-shear load, with subsequent
consolidation loads of 400, 700, 1000, 1300, and 1600 Pa. For this work,
the shear data and the solid fraction data taken at the pre-shear
condition were used to compare to the DEM simulation. Two replicates
were performed for each blend condition, with new particle mixtures
prepared to generate the replicate data.

The segregation of the binary blend is very apparent. After blending,
there are regions in the glass jar where the small and large beads are
separated. Given the large size ratio (approximately 4.5), it is very
difficult to mix the particle into a random mixture. Segregation is an
inherent factor to the shear test and it was not possible to completely
eliminate from our experiment. The shear cell was filled by taking
multiple scoops of blend from the bulk mixture, with each scoop taken
from a random location within the blending vessel. Segregation is
observed even when filling the shear cell, the area of segregated
particle can be seen when preparing the shear cell. Figure 1 presents an
image taken on the surface of the shear cell before testing. All shear
tests were performed on the material as is. Due to the method of fill
for the shear cell (scooping from a mixture prone to segregation), the
content of the shear cell is emptied out after the shear test. The
content is again sieved through a \#100 US standard sieve, separating
them into smaller (\textless{}150\selectlanguage{greek}µ\selectlanguage{english}m) and larger ([?]150\selectlanguage{greek}µ\selectlanguage{english}m) segments. The
mass of each sample is then determined to verify the mixture
composition.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure 1. Segregation observed when preparing the XS-MV4 cell

To observe the behavior of individual particles in the shear cell, an
optical microscope with a 200x magnification lens (Keyence VH-Z20R) was
positioned above the shear cell, as shown in Figure 2. The resulting
images and videos detail the motion of particles visible through the gap
between the base of the shear cell and the lid. Although this approach
does not directly capture the motion of particles within the bulk system
inside the shear cell, it does provide an understanding of particle
behavior on the free surface of the blend.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Figure 2. Video capture of the shearing glass beads

\subsection*{2.3 Data analysis of shear test
results}

{\label{data-analysis-of-shear-test-results}}

The key outputs from ring shear measurements analyzed for each binary
blend include pre-shear stress (\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}), unconfined yield
strength (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{u}), flow function coefficient (FFc), packing
density, and linearized friction angle (\selectlanguage{greek}φ\selectlanguage{english}). The pre-shear stress,
\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}, is the first point of measurement on the material,
and represents the shear stress required to achieve steady state flow at
a specified pre-shear load, \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{pre}. In this work,
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{pre} was fixed at 2000 Pa for all systems evaluated. The
unconfined yield strength, \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{u,} is the stress required to
fail the material in an unconfined state after subjected to major
principal stress. This parameter gives an indication of the overall
strength of the powder. A physical representation of these parameters
are in Figure 3.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

Figure 3. Extracted parameters from the yield loci.

In order to better understand the contribution of pure friction and
adhesion to the unconfined strength, it can be decomposed into tensile
strength (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t}) and internal angle of friction (\selectlanguage{greek}\emph{φ} \selectlanguage{english})
using the equation\textsuperscript{14,15} as:

\(\sigma_{u}=\frac{1+sin\varphi}{1-sin\varphi}\sigma_{t}\) (1)

The equation assumes linearized yield loci (\selectlanguage{greek}τ \selectlanguage{english}and \selectlanguage{greek}σ \selectlanguage{english}plot).

The linearized friction angle represents the pure frictional component
of the material. On the other hand, the tensile strength
(\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t}) represents the adhesive component of the material,
which is the strength required to `pull' the material apart as shown in
Figure 4. The Flow function coefficient (FFc) is the ratio of the major
principal stress to the unconfined yield strength. Increasing values of
FFc relate to better flowability of the powder.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

Figure 4. Unconfined yield strength and tensile strength.

\section*{3. Numerical method and simulation
conditions}

{\label{numerical-method-and-simulation-conditions}}

\subsection*{3.1 Numerical method}

{\label{numerical-method}}

The translational and rotational motion of a particle is modeled by DEM
using Newton's second law as\textsuperscript{16}:

\(m_{i}\frac{\mathrm{d}u_{i}}{\mathrm{d}t}=\sum_{j}\left(F_{\text{ij}}^{n}+F_{\text{ij}}^{s}\right)+m_{i}g\)(2)

\(I_{i}\frac{\mathrm{d}\omega_{i}}{\mathrm{d}t}=\sum_{j}{R_{i}\times F_{\text{ij}}^{s}}\)(3)

where \(u_{i}\) and \(\omega_{i}\) are the translational
and angular velocity of particle and \(m_{i}\) and
\(I_{i}\) are the mass and moment of inertia of particle .
\(F_{\text{ij}}^{n}\) and \(F_{\text{ij}}^{s}\)are the normal and tangential
forces acting on particle. Particle adhesion/cohesion is considered by
implementing the adhesive/cohesive contribution into \(F_{\text{ij}}^{n}\)
and \(F_{\text{ij}}^{s}\)through JKR theory\textsuperscript{17} and the
Thornton and Yin model\textsuperscript{18,19}.

\textbf{3.2 Simulation conditions}

Large and small glass bead particles, described by the median sizes of
the actual particles, 180 and 40 \selectlanguage{greek}µ\selectlanguage{english}m respectively, are used in the
current study. Both particles are assumed to be spherical and possess
the same material properties. Initially, a number of large spheres and
small spheres are randomly generated in the simulation domain, periodic
in two dimensions, with a specified mixture composition. The size of the
domain is \(30d_{S}\times 30d_{S}\times 15d_{S}\), in which \(d_{S}\)is the diameter
of the small particle. Based on our previous work\textsuperscript{2}, it
is believed the simulated shear stress results are independent of the
domain size, although the domain size is smaller than the shear cell
size. Cases with mass fraction of large particles ranging from 0 to 1
are conducted. After all particles deposit on the bottom wall of the DEM
shear cell and achieve a negligible velocity, a normal pressure (2000
Pa) is specified on the top wall and a velocity (shear rate of 0.2
s\textsuperscript{-1}) is applied to the bottom wall to start the
shearing process as shown in Figure 5. The shear rate is higher than the
experimental one to reduce the computational expense but it also assures
a quasi-static state in which the shear stress is independent of shear
rate\textsuperscript{2,20}. Shearing bars (blades) on both top and
bottom walls are also created to mimic the configuration of the
realistic shear cell and prevent the sliding of particles. Simulations
run sufficient time steps until the shear process reaches a steady state
and the stresses are nearly constant with small fluctuations. The
particle properties in the simulations are given in Table 1.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

Figure 5. Sketch of the DEM shear cell.

Table 1. Particle Properties.\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
Properties & Values\tabularnewline
\midrule
\endhead
Large particle diameter (\selectlanguage{greek}µ\selectlanguage{english}m) & 180\tabularnewline
Small particle diameter (\selectlanguage{greek}µ\selectlanguage{english}m) & 40\tabularnewline
Particle density (kg/m\textsuperscript{3}) & 2500\tabularnewline
Young's modulus (GPa) & 16\tabularnewline
Poisson's ratio & 0.22\tabularnewline
Coefficient of restitution & 0.92\tabularnewline
Coefficient of sliding friction & 0.3\tabularnewline
Surface energy (J/m\textsuperscript{2}) & 0.005\tabularnewline
\bottomrule
\end{longtable}

\section*{4. Results and discussion}

{\label{results-and-discussion}}

\subsection*{4.1 Experimental data}

{\label{experimental-data}}

\subsubsection*{4.1.1 Relationship between flowability and packing
density}

{\label{relationship-between-flowability-and-packing-density}}

Figure 6 shows the relationship between FFc and packing density as a
function of blend composition. It can be observed that the FFc of any
binary mixture is lower than that of each of the primary components of
the system. A similar phenomenon was observed by
Vallejo\textsuperscript{12}, who attributed this behavior to the
increase in solid fraction of the blend and particle configuration. That
work investigated relatively larger sizes of glass bead mixtures, with
nominal sizes of small and large particles reported as 400 and 5000 \selectlanguage{greek}µ\selectlanguage{english}m,
respectively, and did so at consolidation stresses between 6 and 40
times larger than those employed here.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

Figure 6. Flow function coefficient and packing density of the binary
blends measured using the SRST.

While comparing FFc and packing density, indeed a correlation can be
found as suggested by Vallejo\textsuperscript{12}. This confirms that
for a mixture of similar materials such as glass beads, packing has a
significant influence on powder flowability. Increased packing is
associated with decreased flowability and lower FFc. The contribution of
adhesion and friction to FFc are further investigated.

\subsubsection*{4.1.2 Relationship between tensile strength and packing
density}

{\label{relationship-between-tensile-strength-and-packing-density}}

Figure 7 shows the key output from the shear test with its
interrelationships. Figure 7 (a) shows that the FFc relationship mirrors
that of the unconfined yield strength (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{u}) relationship.
In theory, the total strength of the material should be the summation of
the adhesion component and the frictional component of the system. The
adhesive or tensile component (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t}) is shown in Figure 7
(b). It is found that tensile strength (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t}), which
represents the stress required to fail the powder in tension, correlates
with the degree of packing of the powder. As the degree of packing of
the powder increases, greater forces are required to separate particles
from their neighbors, increasing the overall tensile strength of the
powder. The results are consistent with Rumpf's theory of tensile
strength \textsuperscript{21}.

\selectlanguage{greek}\subsubsection*{4.1.3 Relationship between internal angle of friction φ
and packing
density}\selectlanguage{english}

{\label{relationship-between-internal-angle-of-friction-ux3c6-and-packing-density}}

We further explore the relationship between the internal angle of
friction \selectlanguage{greek}φ \selectlanguage{english}and the packing density (Figure 7 (c)); \selectlanguage{greek}φ \selectlanguage{english}shows the most
deviation with the packing density. While \selectlanguage{greek}φ \selectlanguage{english}shows a linear relationship
with the packing density for 0\% to 50\% large glass beads, the results
deviate from the packing density as the large glass bead fraction is
increased from 50\% to 75\%. This seems counterintuitive, as \selectlanguage{greek}φ \selectlanguage{english}still
increases with decreasing packing density. On further increase of large
glass beads beyond 75\%, the \selectlanguage{greek}φ \selectlanguage{english}decreases with decreasing packing
density. This behavior is explored in sections 4.2-4.4 by simulating
steady state shearing using DEM.

The steady state shear stress (\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}) correlates well
with \selectlanguage{greek}φ \selectlanguage{english}(see Fig 7(c)). This can be described mathematically using the
Mohr Coulomb failure criterion:

\(\tau=\sigma tan\left(\varphi\right)+\tau_{c}\) (4)

Where \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{c} is the cohesion term which is directly related
to \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t} by a trigonometric relationship. For glass beads,
the (\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{pre}, \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}) point is colinear to
the yield loci. Based on equation 4, the \selectlanguage{greek}τ \selectlanguage{english}term is dominated
by\(\selectlanguage{greek}\text{σtan}\selectlanguage{english}\left(\varphi\right)\); therefore, the \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre} is \selectlanguage{greek}φ \selectlanguage{english}weighted
by 2000 Pa.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

a)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

c)

Figure 7. Plot of the key experimental output from the SRST. (a) FFc and
\selectlanguage{greek}σ\selectlanguage{english}\textsubscript{u}, (b) \selectlanguage{greek}σ\selectlanguage{english}\textsubscript{t} and packing density, and (c)
\selectlanguage{greek}φ \selectlanguage{english}and \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}.

\subsubsection*{4.1.4 Relative contribution of tensile strength and
friction on unconfined yield
strength}

{\label{relative-contribution-of-tensile-strength-and-friction-on-unconfined-yield-strength}}

Next, the relative contribution of the tensile and frictional components
on the unconfined yield strength are explored. The stress contribution
due to the frictional component \(\sigma_{f}\) is calculated by
subtracting tensile strength from the unconfined yield strength:

\(\sigma_{f}=\sigma_{u}-\sigma_{t}\) (5)

It is important to note that this frictional component is different from
the internal angle of friction and the shear stress. It is a fictitious
contribution of friction to unconfined yield strength in the normal
direction.

Figure 8 shows the relationship between the frictional and tensile
contributions to the total unconfined yield strength. Both the friction
and tensile strength contribute to the unconfined yield strength, and
the contribution of friction is slightly higher than that of tensile
strength. It is also noted that the frictional strength follows the
trend of internal angle of friction and \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre} as shown in
Figure 7c.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}

Figure 8. Contribution of friction and tensile strength on unconfined
yield strength.

\subsection*{4.2 Comparison of experimental and DEM
results}

{\label{comparison-of-experimental-and-dem-results}}

At the 2000 Pa pre-shear load, the DEM predictions for the shear stress
at steady state are compared to the experimentally measured
\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre}. This comparison is shown in Figure 9; higher
\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre} values represent poorer flowing systems. The
predicted stresses associated with the binary blends are larger than
those of the monodispersed particles, exhibiting the same trend as the
experimental results. However, the DEM predictions for
\selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre} are lower than the experimental values, possibly
due to neglecting the size distribution associated with the large and
small glass beads.

\textbf{Hosted file}

\verb`image12.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

Figure 9. Comparison of the \selectlanguage{greek}τ\selectlanguage{english}\textsubscript{pre} from experiments and
DEM simulations.

In order to compare the experimental packing density with the DEM
predictions, the experimental packing density is converted to solid
fraction. Solid fraction is calculated by dividing the packing density
with the true density of the glass beads, i.e. 2500
kg/m\textsuperscript{3}. The variation of solid fraction with
composition is shown in Figure 10. The solid fraction in both the
experiment and DEM simulations is calculated by recording the position
of the top boundary under the same 2000 Pa pre-shear load. For both
sizes of monodispersed spherical beads, the predicted and measured solid
fraction is similar with a value around 0.6. In addition, the solids
fraction for the binary mix is higher than that of the monodispersed
particles, similar to what was observed by Vallejo\textsuperscript{12}.
The DEM predictions of solid fraction are able to reproduce the
experimental measurements quite well.

\textbf{Hosted file}

\verb`image13.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

Figure 10. Comparison of the solid fraction from experiments and DEM
simulations.

\subsection*{4.3 Large particle rotation}

{\label{large-particle-rotation}}

Videos taken during the shear tests (available in the supplementary
information) reveal drastically different macroscopic flow behavior for
the monodispersed and binary mixtures. For the monodispersed smaller
sized beads, a bulk motion of the system, with coordinated movement of
particles in a wave-like fashion is observed. For the monodispersed
larger size beads, a start-stop and rotational motion of individual
particles is observed within the bulk. However, these individual
movements of large particles are not observed in the video with 0.50
large glass bead mass fraction, suggesting that at lower concentrations,
small particles hinder the ability for the larger particles to rotate.
The 0.75 system exhibited the highest degree of segregation of all the
mixtures tested. For the 0.75 system, in regions of low levels of
particle segregation, the macroscopic flow behavior of the binary
mixture resembles that of the 0.5 system. However, in regions of high
levels of particle segregation, with a higher concentration of large
glass beads near the surface, rotation of individual large glass beads
is observed. Time-lapse images of the video for the 0.75 system are
extracted and shown in Figure 11. In the images, a few particles are
outlined to indicate their movement with time. From these images, it is
evident that the highlighted large glass bead is rotating. As the larger
glass bead rotates, multiple small glass beads on the surface of the
larger particle can be identified. This behavior indicates that the
addition of small glass beads, even at low concentrations, can adhere to
the larger glass beads.

While spinning movement is observed for the large particles, the small
particles move as a bulk in a wave-like motion. Furthermore, for the
50:50 binary mixture, the larger particles show no evidence of spinning
movement, and the bulk flow is similar to that of the small particles.
Therefore, an explanation for the unique shear stress trend is proposed
based on this observed macroscopic flow behavior. The hypothesis is that
large particles can spin, rather than move transversally, minimizing the
bed dilatation during shearing. And, the addition of smaller, more
cohesive particles and the adhesion between large and small particles
minimizes or prevents the spinning of the large particles. This, in
turn, results in a larger shear stress of the binary blend required to
induce flow.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image14/image14}

\end{center}
\end{figure}

Figure 11. Time-lapse images of large particle rotation with two seconds
between images

In order to verify this hypothesis, the angular velocities of the large
particles in the various mixtures are examined via DEM simulations.
Figure 12 shows the averaged angular velocity of the large particles for
different blends. It is clearly found that the averaged angular velocity
for monodispersed large particles is significantly higher than those of
binary mixtures, demonstrating that the addition of smaller, more
cohesive particles reduces the average angular velocity of the large
particles. This reduction in angular velocity, making particle shearing
more difficult, consequently leads to an increase in the shear stress of
the binary particle mixture. While this reduced angular velocity will
give rise to an increased binary shear stress, it is believed that the
shear stress distribution of the binary mixture is a combined results of
many factors including, but not limited to, the rotation of the larger
particles. Therefore, the effect of particle interactions on the shear
stress is further explored.

\textbf{Hosted file}

\verb`image15.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

Figure 12. Averaged angular velocity of large particles for different
blends.

\subsection*{4.4 Particle interactions}

{\label{particle-interactions}}

In addition to the large particle rotation, different particle mixtures
may have different particle interactions that also contribute to the
shear stress difference. The contact numbers for three contact types,
i.e. the contact between large and large particles L-L, large and small
particles L-S, and small and small particles S-S, are shown in Figure
13. While only L-L contacts and S-S contacts exist for monodispersed
large and small particles, respectively, the number of S-S contact
dominate the numbers of the other two types of contacts. The number of
S-S contacts for the three binary mixtures decreases with increasing
mass fraction of large particles, as a lower large particle mass
fraction leads to a higher number of small particles and a higher number
of S-S contacts.

\textbf{Hosted file}

\verb`image16.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

Figure 13. Contact numbers for three contact types (L-L, L-S and S-S)
for different mixtures.

The contact numbers for the three contact types for the different
mixtures are normalized and shown in Figure 14. The L-L and L-S contact
numbers are normalized by the number of large particles in the given
mixture, and the S-S contact number is normalized by the number of small
particles. From the perspective of normalized contact number, the L-S
contact is the dominant type of contact for the binary mixtures,
indicating that for the current particle size ratio of 4.5, the large
particles in the binary mixtures are surrounded by small particles
sticking to the surface. The normalized L-S contact number for the
binary mixtures ranges from 45 to 12 and decreases with increasing mass
fraction of large particles. The significant difference in normalized
L-S contact number for various composition indicates that different
composition has different contact structure. This is especially true for
the contact structure around the large particles, which may also affect
the motion of particles and contribute to the shear stress difference.

\textbf{Hosted file}

\verb`image17.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

Figure 14. Normalized contact numbers for three contact types (L-L, L-S
and S-S) for different mixtures.

In order to further explore the effect of particle interactions on the
shear stress, detailed contact information is extracted from the DEM
simulations to illustrate the transmission of force in the system.
Contacts in a middle slice within the computational domain (one third of
the thickness of the domain), as shown in Figure 15, are displayed by
lines that connect the centers between two contacted spheres. The force
magnitude is represented by the thickness of the line.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image18/image18}

\end{center}
\end{figure}

Figure 15. Slice of the computational domain for displaying force
chains.

The force chains for L-L contacts are illustrated in Figure 16. For the
monodisperse large particles, L-L contacts uniformly exist throughout
the particle bed. However, for the binary system, the importance of L-L
contacts decreases with decreasing fraction of large particles, as an
increased number small particles are involved and play a more
significant role in force transmission.

\textbf{Hosted file}

\verb`image19.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

\textbf{Hosted file}

\verb`image20.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

(a) (b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image21/image21}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image22/image22}

\end{center}
\end{figure}

(c) (d)

Figure 16. Force chain plots for L-L contacts. (a) 25\% large particles;
(b) 50\% large particles; (c) 75\% large particles; (d) 100\% large
particles.

The force chains for L-L and L-S contacts are illustrated in Figure 17.
For the 100\% large system, the L-L contact is the only way for
transmitting the force. For binary systems (particularly for the 50\%
large and 25\% large mixtures), the L-S contacts are the primary vehicle
for force transmission between large particles, as large particles are
surrounded by many small particles. For the 75\% large particle systems,
small particles fill some gaps between the large particles and both L-L
and L-S contacts serve to transmit forces.

\textbf{Hosted file}

\verb`image23.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

\textbf{Hosted file}

\verb`image24.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

(a) (b)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image25/image25}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image26/image26}

\end{center}
\end{figure}

(c) (d)

Figure 17. Force chain plots for L-L and L-S contacts. (a) 25\% large
particles; (b) 50\% large particles; (c) 75\% large particles; (d) 100\%
large particles.

The force chains for L-L, L-S and S-S contacts are illustrated in Figure
18. For 100\% large and 75\% large particle systems, the effect of S-S
contacts is negligible due to the insufficient number of small
particles. However, for binary systems with a fraction of large
particles at 50\% and 25\%, the effect of S-S contacts on force
transmission is very significant. When the number of small particles is
sufficiently high, large particles are totally separated from each
other, immersed in a sea of small particles. The force chain plots of
Figure 18 (b) are also consistent with the experimental observation that
the bulk flow behavior of the 50:50 mixture is similar to that of small
particles only. For both of these particle systems, the S-S contacts are
the dominate mechanism for transmitting forces.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image27/image27}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image28/image28}

\end{center}
\end{figure}

(a) (b)

\textbf{Hosted file}

\verb`image28.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

\textbf{Hosted file}

\verb`image29.emf`  available at \url{https://authorea.com/users/319476/articles/449200-experimental-and-numerical-investigations-of-the-shear-behavior-of-binary-particle-blends}

(c) (d)

Figure 18. Force chain plots for L-L, L-S and S-S contacts. (a) 25\%
large particles; (b) 50\% large particles; (c) 75\% large particles; (d)
100\% large particles.

\section*{5. Conclusions}

{\label{conclusions}}

The flow behavior of binary mixtures of two sizes of glass beads was
investigated using both the Schulze Ring Shear Tester and the Discrete
Element Method. Both approaches demonstrate that the shear stresses
required to achieve steady state flow are greater for binary blends than
they are for monodisperse systems. Moreover, DEM simulations well
reproduce the experimental observation that the highest measured shear
stress is not associated with the highest measured solid fraction for
the range of binary mixtures studied. Experimental data further confirm
that the flow function coefficient (FFc) of these mixtures is strongly
affected by its solid fraction, with unconfined yield strength directly
correlating to this parameter. An analysis of the frictional and tensile
components of unconfined yield strength reveals that the powder friction
has a slightly higher contribution to powder strength than powder
adhesion, and that the frictional component follows the same trend as
steady state shear stress with mixture composition. Whereas maximum
tensile strength corresponded to maximum solid fraction, steady state
shear stresses and internal angle of friction values do not. DEM
simulation demonstrate that the addition of small adhesive particles
reduces the averaged angular velocity of the larger particles, which
makes a contribution to a larger shear stress and internal friction
angle for binary blends. Furthermore, particle contact type, contact
number and force network vary significantly for different blends. It is
believed that these factors also contribute to the shear stress
difference for blends of varying composition.

\section*{Acknowledgments}

{\label{acknowledgments}}

The financial support from Vertex Pharmaceuticals Incorporated is
gratefully appreciated.

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