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# Insights into the rheology of cohesive granular media

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 2, 2020 (received for review December 11, 2019)

## Significance

An uninterrupted flow of powders is the key to smooth production operations of many industries. However, powders have more difficulty flowing than coarse, granular media like sand because of the interparticle cohesive interactions. What precisely controls the “flowability” of powders remains unclear. Here, we address this issue by performing numerical simulations of the flow of cohesive grains. We show that the cohesiveness during flow is not only controlled by the interparticle adhesion, but also by the stiffness and inelasticity of the grains. For the same adhesion, stiffer and less dissipative grains yield a less cohesive flow, i.e., higher “flowability.” This combined effect can be embedded in a single dimensionless number—a result that enriches our understanding of powder rheology.

## Abstract

Characterization and prediction of the “flowability” of powders are of paramount importance in many industries. However, our understanding of the flow of powders like cement or flour is sparse compared to the flow of coarse, granular media like sand. The main difficulty arises because of the presence of adhesive forces between the grains, preventing smooth and continuous flows. Several tests are used in industrial contexts to probe and quantify the “flowability” of powders. However, they remain empirical and would benefit from a detailed study of the physics controlling flow dynamics. Here, we attempt to fill the gap by performing intensive discrete numerical simulations of cohesive grains flowing down an inclined plane. We show that, contrary to what is commonly perceived, the cohesive nature of the flow is not entirely controlled by the interparticle adhesion, but that stiffness and inelasticity of the grains also play a significant role. For the same adhesion, stiffer and less dissipative grains yield a less cohesive flow. This observation is rationalized by introducing the concept of a dynamic, “effective” adhesive force, a single parameter, which combines the effects of adhesion, elasticity, and dissipation. Based on this concept, a rheological description of the flow is proposed for the cohesive grains. Our results elucidate the physics controlling the flow of cohesive granular materials, which may help in designing new approaches to characterize the “flowability” of powders.

Many industrial (wet granulation, food processing, construction, etc.) and geophysical (landslides, mudflow, etc.) processes involve the flows of an assembly of cohesive grains. The cohesion between the grains has different origins. Van der Waals or electrostatic forces are responsible for cohesion in fine grains (1, 2). Liquid capillary (3⇓–5) or solid bridges (6) between the grains give rise to cohesion in large grains. In all cases, the cohesion introduces an additional complexity to granular materials—the flows of cohesive grains are intermittent and less homogeneous (7, 8) in comparison with coarse, cohesionless grains, leading to frequent jamming of industrial units. It is, therefore, necessary to a priori characterize and quantify the capability of flow, so-called “flowability,” of a powder to yield better handling. Different methods are used in industrial contexts for this purpose (1, 9⇓–11). A first method measures the tapped bulk density and the freely settled bulk density of a powder to define the Hausner ratio (or Carr index), which is the ratio of the two. A powder with high Hausner ratio is shown to have poor “flowability.” A second one employs a series of measurements, using the Hosokawa powder tester, comprising angle of repose, aerated bulk density, tapped bulk density, etc., to define a weighted “flowability” index, which ranges from 0 to 100. Very cohesive powders yield “flowability” indices close to zero and the free-flowing ones close to 100. Other methods estimate the macroscopic cohesion from the yield loci of a powder using shear testers (Jenike shear tester or ring shear tester) for various preconsolidation normal stresses, which are useful in understanding the arch formation in silos. All these methods, carried out in the quasistatic limit, are useful for comparing the macroscopic properties of different powders and for characterizing their plastic behavior. However, they do not provide any information about the flow dynamics. Understanding the concept of “flowability” from a physical point of view is still a challenge.

The flow dynamics of rigid, cohesionless grains, interacting solely by contact and friction, is less complex in comparison with cohesive grains, as shown by numerous experimental and numerical studies (12). Flow rules have been evidenced and constitutive laws have been proposed for different flow regimes (13, 14). In the dense flow regime, the rheology of the grains of diameter d and density

In this article, we examine the flow dynamics of cohesive grains down a rough inclined plane using intensive discrete numerical simulations to gain physical insights about the “flowability” of powders. The chosen configuration has inhomogeneous stress distributions and, hence, turns out to be very rich to explore the rheology of cohesive granular materials. We perform a detailed parametric study of the flow to reveal that the flow is significantly affected by the stiffness and the inelasticity of the grains, unlike in the case of cohesionless granular media. We introduce the concept of a dynamic “effective” adhesive force to take into account the effect of the material properties, along with the interparticle adhesion, which is shown to control entirely the flow dynamics. We then define an “effective” cohesion number based on this force, which replaces the cohesion number defined above to form a pair of constitutive relations. The rheology is shown to be well described in this framework.

## Results

### Simulation of the Flow of Cohesive Grains Down a Rough Inclined Plane.

We investigate the flow of frictional, inelastic, cohesive grains down a rough inclined plane (Fig. 1A) using three-dimensional (3D) discrete element method simulations; an in-house code is used, which is validated by comparing the mean velocity and volume fraction profiles of (cohesionless) monodisperse spheres with those in the study of ref. 17. The grains are spherical and have an average size d, with a polydispersity of 20% and an average mass m. The rough base (shown in red in Fig. 1*A*) comprises a packed bed of the same grains of height 1.8d. The simulation box has length

The interparticle contact forces were computed by using the Hookean spring-dashpot model with a frictional slider (Fig. 1*B*) and a Johnson–Kendall–Roberts (JKR)-like (31), yet nonhysteretic, adhesive force model (24). The normal contact force (*1*) elastic force *2*) viscous force *3*) adhesive force *C*. Note that the adhesive interaction model is short-range, meaning that the force vanishes when two grains are not in contact, unlike in wet capillary bridges. The tangential contact force (*SI Appendix*, section SI 1), where the (nonhysteretic) adhesive force is assumed to be independent of the area of contact and is constant. The other one comprises the Hookean spring-dashpot model and a DMT-like, yet hysteretic (*SI Appendix*, section SI 2), force model, where the grains experience a constant attractive force only during the detachment of a contact, quite similar to a capillary bridge model, but without introducing a finite distance for the detachment.

Using the Hookean-JKR (first) model, the dynamics of two identical (**1** are zero, and the balance between the attractive adhesive force and the repulsive elastic force (Eq. **1**) then yields an equilibrium overlap

All of the equations are made dimensionless by using d as the length scale, *SI Appendix*, section SI 3.

### Not Only the Interparticle Adhesion, but Also the Stiffness and Inelasticity of the Grains Affect the Flow.

Fig. 2*A* shows a typical velocity profile for the cohesive grains for an intermediate value of adhesion *C*). The shear rate then gradually increases toward the base (Fig. 2*C*). The inertial number profile (*C*) and is not uniform over the depth of the pile, unlike in the cohesionless case (17, 33). The volume fraction profile (*B*) (34) with a highdensity region in the plug, unlike in the cohesionless case (17, 33). The cohesive grains possess a finite yield stress, which is reached at a finite depth in the flowing layer. This, therefore, explains the plug formation near the free surface for the flow of cohesive grains.

We now examine the effect of the interparticle adhesion *D*) at a given inclination angle *E* and *F*. The free-surface velocity increases, and the thickness of the plug decreases with increasing the stiffness, keeping other parameters fixed (Fig. 2*E*). The plug completely disappears for a sufficiently high value of stiffness. A similar observation is made while increasing the quality factor, i.e., decreasing the dissipative nature of contact (Fig. 2*F*). These observations clearly indicate that the bulk cohesion is not solely controlled by the interparticle adhesion

### A Scaling for the Dynamic “Effective” Adhesive Force.

We infer from Fig. 2 *D*–*F* that decreasing the stiffness or decreasing the quality factor are equivalent to increasing the adhesion. Hence, we seek for an expression of the dynamic “effective” adhesive force as**2** and look for the best collapse of the data in each case by trying out different combinations of a and b. For each chosen combination of a and b, the two master curves (*SI Appendix*, Fig. S3). Finally, two well-defined master curves, *SI Appendix*, Fig. S4). We notice two distinct regions in the figure, separated by a vertical dashed line at *1*) a plug-less region on the left, where *2*) a plug-full region where *SI Appendix*, Fig. S5) to be controlled by the dynamic “effective” adhesive force given by a similar equation (*SI Appendix*, Eq. **4**). The origin of the scaling will be discussed in the last section. We also perform some simulations using the hysteretic contact model. We again find the flow dynamics (at a given angle) to be dependent on the stiffness (*SI Appendix*, Fig. S6), signifying that in this case as well, the contact parameters, along with the interparticle adhesion, determine the bulk cohesion. We examine below if this dynamic “effective” adhesive force is relevant in the description of the rheology.

### Flow Cessation Is Controlled by the “Effective” Adhesive Force, but Flow Initiation Is Controlled by the “Actual” Adhesive Force.

A first step toward exploring the rheology is to study the yield criteria of our model cohesive material, i.e., to study the stress conditions under which the flow stops (dynamic yielding) or starts (static yielding). The yield criteria are usually described by using a cohesive Mohr–Coulomb model stipulating that, on the plane of incipient failure, the shear stress *A*. The location of the “yield point” shifts with changing the inclination angle. Thus, a yield locus for a given set of particle properties (*A*, *Inset* shows different yield loci (shown by symbols; only a few are shown for clarity) for different sets of particle properties. All of the yield loci are well approximated by straight lines, which are the best fits of the Mohr–Coulomb model. The slope of each straight line gives the dynamic friction coefficient *y* axis, the dynamic cohesive stress *A*), showing again that *B*, *Inset* shows different yield loci (shown by symbols) for different sets of particle properties. The striking observation is that two yield loci for two different *B*. The variation of **5** yields

### Bulk Rheology Is Described by an “Effective” Cohesion Number.

The last step toward exploring the rheology is to go beyond the yield criteria and analyze how the shear stress varies with the shear rate. The inclined plane configuration serves as a rheometer to enable us to measure the local shear rate *A*) shifts upward with increasing **3** into the Mohr–Coulomb model and using the definition of μ and *B*) shifts downward with increasing *SI Appendix*, section SI 8). The same function provides a reasonable fit of our data, as shown in *SI Appendix*, Fig. S7.

## Discussion and Conclusion

In this work, we have examined the flow of cohesive grains down an inclined plane using discrete element method simulations. We use a simplified adhesive interaction law, characterized by a minimum pull-off force

We have not succeeded in understanding the scaling. However, the existence of an interplay between the interparticle adhesion and the mechanical properties of the grains can be evidenced, considering the dynamics of a binary collision. When two cohesive grains collide, one can show that they remain glued together if the relative kinetic energy before impact is less than a critical value (*SI Appendix*, section SI 9) given by

We have shown that the initiation of the flow is controlled by a static cohesive stress proportional to *SI Appendix*, Fig. S8, we present some preliminary rheological data obtained in a normal stress-imposed shear cell (see *SI Appendix*, section SI 10 for simulation details). The data of μ and ϕ for two different sets (*SI Appendix*, Fig. S8), showing the generality of the proposed description.

The relevance of the “effective” cohesion number in defining the flow dynamics opens perspectives to analyze the behavior of cohesive granular media in other configurations, and we give one example below. We carry out additional simulations to measure the packing fraction in a pile of cohesive grains for various (*Inset*). More configurations need to be studied to be able to understand to which extent this concept of “effective” cohesion number is valid for the flow of cohesive grains, which might help in developing new approaches for the characterization of powders.

Although the concept of “effective” adhesion is based on a simplified model of adhesion, we find that it is generic for other kinds of adhesive interactions; for example, (hysteretic) capillary and electrostatic adhesion. In these cases, the particles experience an attractive force, even without a physical contact over a small separation distance, and this distance plays an equivalent role of the stiffness and influence the “effective” adhesion. The bulk cohesion, in studies (26, 46) using capillary bridge models, was noted to decrease when decreasing this separation distance (named as the “rupture distance,” beyond which a capillary bridge breaks). This outcome can be understood considering the argument of energy proposed above—the work needed to separate two bonded grains, in this case, is proportional to the rupture distance.

One last remark concerns the limit of rigid particles. The scaling reported above suggests that the “effective” adhesion will go to zero in the rigid limit, implying that a direct comparison of our results with those using contact dynamics simulations (27), in which the grains are treated as perfectly rigid, is difficult. We speculate that, in these simulations, the size of the time step could be crucial and could play a role similar to the stiffness in our soft-particle approach.

The main idea conveyed in this article that the “effective” adhesion is not controlled solely by the interparticle adhesion, but also dependent on material properties is a first step toward a better understanding of the flow of cohesive granular media, which may benefit engineering and geophysical communities to understand the long-standing issue of “flowability” of cohesive powders.

All data presented in this article are openly available in the Zenodo repository, https://zenodo.org/record/3699632#.XmJ048tKgaw.

## Acknowledgments

This work was supported by ANR Grant ANR-17-CE08-0017 under the Cohesive Powders Rheology: Innovative Tools Project; “Laboratoire d’Excellence Mécanique et Complexité” Grant ANR-11-LABX-0092; and Excellence Initiative of Aix-Marseille University-A*MIDEX Grant ANR-11-IDEX-0001-02, funded by the French Government “Investissements d’Avenir Program.” We thank Y. Forterre for useful comments on the manuscript. Center de Calcul Intensif d’Aix-Marseille is acknowledged for granting access to its high-performance computing resources for running some of the simulations.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: olivier.pouliquen{at}univ-amu.fr.

Author contributions: S.M., M.N., and O.P. designed research; S.M. performed research; S.M., M.N., and O.P. analyzed data; and S.M. and O.P. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Data deposition: All data presented in this article have been deposited on the Zenodo open data site, https://zenodo.org/record/3699632#.XmJ048tKgaw.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1921778117/-/DCSupplemental.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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