How particle shape affects granular segregation in industrial and geophysical flows

Significance Granular materials like cereal, pharmaceuticals, sand, and concrete commonly organize such that grains segregate according to size rather than uniformly mixing. For example, in a jar of nuts, the largest ones are commonly found at the top. Here, we use computer simulations to explore how grain shape controls this phenomenon in industrial and natural settings. We find that even small differences in shape can substantially change the amount and style of segregation, with different effects depending on whether the system is wet or dry. This study demonstrates the importance of grain shape in different systems ranging from food and medicine production to geophysical hazards and processes such as landslides, river erosion, and debris flows on Earth and other celestial bodies.


Supporting Information Text
In this document we provide additional information for key aspects of the manuscript.In section I, we show the validation of our numerical model with an experimental setup with similar dimensions.In section II, we provide some information about how we determine the window size to compute the segregation level.In section III, we show instantaneous snapshots of the particles' final state for all the cases we used to produce our dry rotating drum results.In section IV, we show instantaneous snapshots of the final state of force networks for all the cases we used to reproduce the results described in section III.
I. Validation of the numerical model.We carried out an experiment to qualitatively validate our numerical model.The experimental setup (Fig. S1i) consisted of an acrylic drum with a diameter of D = 30cm, a stepper motor driver which rotates the drum and is controlled by an Arduino UNO through the GRBL project (https://github.com/grbl/grbl),a set of LED lamps, and a DSLR Nikon D7500 camera.The camera and the motor driver were automatically synchronized to record and save the movies.For this specific case, we placed small marbles (ds =4mm, yellow color in Fig. S1ii) at the bottom and then the bigger ones (d b =8mm, red color in Fig. S1ii) just above the smaller ones.We used this initial condition in order to simulate the exact initial condition for the experiment and the numerical model; however, note that it differs from the initial condition in our main model results, which begin with a fully mixed system.This initial condition would be very difficult to implement in a physical experiment.Finally, we ran the experiment for 140s at a rotation speed of 12RPM (similar conditions to the simulations carried out for the manuscript).From Figure S1ii and movie S1, we found that the behavior of the experiment and the numerical model is highly similar in terms of segregation levels and time to reach a final segregated state.
II. Determination of the window size for segregation level calculations.We studied the influence of the size of the subdomains detailed in Figure S2 in the calculation of the segregation level of the system.The segregation level is strongly dependent on the number and the size of subdomains we have in the system.Even though many studies focused in different techniques to study either level of segregation or mixing, there is no a defined recipe to follow when picking the correct window size.For our study we divided the domain into small squared sections in the x and y directions (∆ = dx = dy) throughout the width of the drum, then we varied the size of ∆ from 0.25(ds + d b ) (where the window is around half of the size of one species) to 10(ds + d b ) (where the window is around 20 times the size of one species).
Figure S3 shows the trends of the temporal evolution of the segregation level for different window sizes.In the case of a very small size (∆ =0.25(ds + d b )), the segregation level is very high because we are looking at a region where only one particle can fit inside, having thus a totally segregated system in that subdomain (S ≈ 1).For the case of a big size (∆ =10(ds + d b )), the region is big enough that we can encounter both species in same proportions, having thus a totally mixed system in that subdomain (S ≈ 0).However, for the cases where size is close to the sum of both species diameters (∆ ≈1(ds + d b )), the behavior of the evolution of the segregation level is similar.
To identify which size is better to compute the segregation level we have two references, 1) the initial condition for all the cases is that particles are randomly distributed which S should be around zero, and 2) the segregation level for spheres of equal size (ds = d b = 1) is zero as the system does not have either segregation by size or shape.Therefore, the best window size to compute the level of segregation is ∆ = 1(ds + d b ), because all the cases start in the same point at t=0s (Fig S3(d)) and the results are consistent with what we see qualitatively from the particle positions snapshots (Fig S5).Finally, to get the data that are in Figure 2 of the manuscript, we shifted the curves to the zero reference that is given by the case of V b /Vs=1.
Figure S4 shows the temporal evolution of the segregation level for all the tested cases and their exponential fits.
III. Final state of particles position for all the tested cases.Figure S5 shows instantaneous snapshots of particle positions for size segregation in both cases spheres and cubes.
Figure S6 shows instantaneous snapshots of particle positions for size and shape segregation in for cubes made by bonded particles and superquadrics.
IV. Determination of the depth of the flowing and solid-like layers in the rotating drum.To determine either the flowing layer or the solid-like layer in the rotating drum, first, we rotated the Cartesian coordinates of the particle positions in order to obtain the avalanche surface normal to the vertical axis.We calculated the time average velocity of the assembly of particles for all the cases studied in this work (color map in Figure S7(a)).From that, we horizontally averaged the color-map in order to obtain a vertical profile as shown in Figure S7(b) (magenta line).Finally, we computed the kink point in the vertical profile where it is possible to define the division of both layers.FL indicates the flowing layer; while SL indicates the solid-like layer.
With the vertical position obtained from the previous procedure, we calculated the depth of the avalanche layer (Fig. S7(b)), as well as mean values of vertical velocities(Fig.S8), packing fraction in both layers(Fig.S9).
V. Final state of force networks for all the tested cases.Figure S11 shows the network of force chains for mixtures of spheres and cubical particles at t=140s of simulation.
VI. Sizes of grains used in the numerical simulations.In the case of cubes made from bonded spheres, the size detailed in Tables S2 and S3 is the diameter of one single sphere.(Note that each cube contain 8 spheres).

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Fernando David Cúñez, Div Patel, and Rachel C. Glade

Table S1 . Size distribution of mixtures of spheres.
Fernando David Cúñez, Div Patel, and Rachel C. Glade

Table S2 . Size distribution of mixtures of cubes.
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Table S3 . Size distribution of mixtures of spheres and cubes (bonded).
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