# Micro–macro transition and simplified contact models for wet granular materials

## Abstract

Wet granular materials in a quasistatic steady-state shear flow have been studied with discrete particle simulations. Macroscopic quantities, consistent with the conservation laws of continuum theory, are obtained by time averaging and spatial coarse graining. Initial studies involve understanding the effect of liquid content and liquid properties like the surface tension on the macroscopic quantities. Two parameters of the liquid bridge contact model have been identified as the constitutive parameters that influence the macroscopic rheology (i) the rupture distance of the liquid bridge model, which is proportional to the liquid content, and (ii) the maximum adhesive force, as controlled by the surface tension of the liquid. Subsequently, a correlation is developed between these microparameters and the steady-state cohesion in the limit of zero confining pressure. Furthermore, as second result, the macroscopic torque measured at the walls, which is an experimentally accessible parameter, is predicted from our simulation results with the same dependence on the microparameters. Finally, the steady- state cohesion of a realistic non-linear liquid bridge contact model scales well with the steady-state cohesion for a simpler linearized irreversible contact model with the same maximum adhesive force and equal energy dissipated per contact.

## Keywords

Rheology Wet granular materials DEM Micro-macro transition Cohesion## 1 Introduction

Granular media are collections of microscopic grains having athermal interactions through dissipative, frictional, or cohesive contact forces. External force leads to granular flow under the condition of applied shear stress exceeding the yield shear stress. After a finite shear strain, at constant rate, a steady state establishes with a typically lower shear stress, depending on both strain rate and pressure [1]. Most studies in granular physics focus on dry granular materials and their flow rheology. However, wet granular materials are ubiquitous in geology and many real-world applications, where interstitial liquid is present between the grains. Simplified models for capillary clusters [2, 3] and wet granular gases [4] were introduced before. The rheology of flow for dense suspension of non-Brownian particles have been studied in Refs. [5, 6, 7]. We study the local rheology of weakly wetted granular materials in the quasistatic regime with the discrete element method (DEM) using the open-source package MercuryDPM [8, 9] in a shear cell set-up, where the relative motion is confined to particles in a narrow region away from the walls, called shear band [10, 11]. We study partially saturated systems in the pendular regime, with a very low level of water content, where the formation of liquid bridges between particle pairs leads to development of microscopic tensile forces. Other forces such as the electrostatic double layer forces can occur between charged objects across liquids, typically dipolar as water. These forces are most active in systems with high surface area to volume ratio, such as colloids or porous materials. We neglect the effect of such forces in our system of rather large (\({\sim }\)mm) non-porous glass particles. The tensile forces generated at particle level result in cohesion at macroscopic scale. Earlier studies have been done for liquid bridge in the pendular regime to understand the effect of liquid bridge volume and contact angle on different macroscopic quantities like the steady-state cohesion, torque, and shear band properties [12, 13, 14, 15, 16]. Other studies for unsaturated granular media observe fluid depletion in shear bands [17, 18]. However, there is no theoretical framework or concrete model available yet that defines the exact correlation between the microparameters like the liquid bridge volume and the surface tension of the liquid with the steady-state cohesion.

The liquid bridge contact model is based on the experimental study of [19] where the capillary force was obtained by measuring the apparent weight of a moving upper sphere by a sensitive microbalance. The lower sphere was attached to a piezoelectric actuator which controlled the separation between the two surfaces. The distance between the two solid surfaces of the spheres was obtained from the position of the piezo-actuator. In order to develop a micro–macro correlation for the liquid bridge contact model, we initially study the structure of the microcontact model. How is the structure of the liquid bridge contact model affected by the microscopic parameters? How does this influence the steady-state cohesion? Here we study in detail on the effect of these parameters on the macro-results. For example, the effect of maximum interaction distance, or the distance at which the liquid bridge between two interacting particles ruptures, is studied by varying the liquid content. On the other hand, other parameters like surface tension of the liquid and contact angle affect the magnitude of force acting between the particles when in contact [14, 19]. Various surface tensions of liquids give a large scale variation of the capillary force and this allows us to study the effect of maximum force on the macroscopic properties. Furthermore, in the consecutive analysis, we re-obtain the macro-rheology results in the shear band center from the torque, torque being an experimentally measurable quantity.

The liquid bridge interactions between the particles are defined by the free-surface equilibrium shapes and stability of the bridge configuration between them [20, 21, 22]. Phenomenologically, even the simplified models of liquid bridges are quite complex in nature. In order to improve the computational efficiency for wet granular materials, we replace the non-linear interactions of liquid bridges with a simpler linear one. But in what way can a non-linear model like the liquid bridge contact model be replaced by a linear model? When can we say that the two different contact models are analogous? Therefore, we compare the realistic liquid bridge model with an equivalent simple linear irreversible contact model [23] that would give the same macroscopic effect.

The results in this paper are organized in three main parts. In Sect. 3.1 of this paper, we study the effect of varying liquid bridge volume and surface tension of the liquid on the macroscopic properties, the focus being to find a micro–macro correlation from this study. Most strikingly, we see a well-defined relationship between these microparameters and the macroproperties like the steady-state cohesion of the bulk material and macro-torque required under shear, neglecting the effect of fluid depletion in shear bands [17, 18] in quasistatic flow. In Sect. 3.2 of this paper, we show the derivation of macro-torque from the boundary shear stress. In this section we also compare this torque with the torque calculated from forces due to contacts on the wall particles. In Sect. 4 of this paper, we discuss about the analogy of two different contact models, with a goal to understand which parameters at microscopic scale would give the same macroscopic behavior of the system.

## 2 Model system

### 2.1 Geometry

*Split-bottom ring shear cell*The set-up used for simulations consists of a shear cell with annular geometry and a split in the bottom plate, as shown in Fig. 1. Some of the earlier studies in similar rotating set-up include [24, 25, 26]. The geometry of the system consists of an outer cylinder (radius \(R_\mathrm{{o}}\) = 110 mm) rotating around a fixed inner cylinder (radius \(R_\mathrm{{i}}\) = 14.7 mm) with a rotation frequency of \(f_\mathrm{{rot}}\) = 0.01 s\(^{-1}\). The granular material is confined by gravity between the two concentric cylinders, the bottom plate, and a free top surface. The bottom plate is split at radius \(R_\mathrm{{s}}\) = 85 mm into a moving outer part and a static inner part. Due to the split at the bottom, a shear band is formed at the bottom. It moves inwards and widens as it goes up, due to the geometry. This set-up thus features a wide shear band away from the wall, free from boundary effects, since an intermediate filling height (

*H*= 40 mm) is chosen, so that the shear band does not reach the inner wall at the free surface.

In earlier studies [1, 27, 28], similar simulations were done using a quarter of the system (\({0}^{\circ }\) \(\le \) \(\phi \) \(\le \) \({90}^{\circ }\)) with periodic boundary conditions. In order to save computation time, here we simulate only a smaller section of the system (\({0}^{\circ }\) \(\le \) \(\phi \) \(\le \) \({30}^{\circ }\)) with appropriate periodic boundary conditions in the angular coordinate, unless specified otherwise. We have observed no noticeable effect on the macroscopic behavior in comparisons between simulations done with a smaller (\({30}^{\circ }\)) and a larger (\({90}^{\circ }\)) opening angle. Note that for very strong attractive forces, the above statement is not true anymore.

### 2.2 Microscopic model parameters

Model parameters

Parameter | Symbol | Value |
---|---|---|

Sliding friction coefficient | \(\mu _\mathrm{{p}}\) | 0.01 |

Elastic stiffness | | 120 N m\(^{-1}\) |

Viscous damping coefficient | \({\gamma }_\mathrm{{o}}\) | 0.5 \(\times \) 10\(^{-3}\) kg s\(^{-1}\) |

Angular frequency | \(\omega \) | 0.01 s\(^{-1}\) |

Particle density | \(\rho \) | 2000 kg m\(^{-3}\) |

Mean particle diameter | \(d_\mathrm{{p}}\) | 2.2 mm |

Contact angle | \(\theta \) | \({20}^\circ \) |

#### 2.2.1 Bulk saturation and liquid bridge volume

The bulk material can be characterized by different states such as the dry bulk, adsorption layers, pendular state, funicular state, capillary state, or suspension depending on the level of saturation [30, 31]. In this paper we intend to study the phenomenology of liquid bridge between particles in the pendular state, where the well-separated liquid bridges exist between particle pairs without geometrical overlap. In this section, we discuss about the critical bulk saturation of granular materials and the corresponding liquid bridge volumes in the pendular state.

#### 2.2.2 Surface tension of liquid

### 2.3 Liquid bridge contact model

*k*is the elastic stiffness, \({\gamma }_\mathrm{{o}}\) is the viscous damping coefficient, and \(\delta \) is the overlap between the particles. The normal contact forces for the liquid bridge model are explained in Sect. 2.3.1

#### 2.3.1 Liquid bridge capillary force model

*r*, and the separation distance

*S*, \(S = -\delta \). With these parameters, we approximate the inter-particle force \(f_\mathrm{{c}}\) of the capillary bridge according to [19]. The experimental results are fitted by a polynomial to obtain the dependence of capillary forces on the scaled separation distance. During approach of the particles as indicated by the loading branch in Fig. 2, the normal contact force for this model is given by

*S*being the separation distance. The maximum capillary force between the particles when they are in contact (

*S*= 0) is given by

#### 2.3.2 Linear irreversible contact model

### 2.4 Dimensional analysis

Non-dimensionalization of parameters

Parameter | Symbol | Scaled term | Scaling term |
---|---|---|---|

Capillary force | \(f_\mathrm{{c}}\) | \({f_\mathrm{{c}}}^*\) | \(f_\mathrm{{g}}\) |

Particle overlap | \(\delta \) | \({\delta }^*\) | \(d_p\) |

Shear stress | \(\tau \) | \({\tau }^*\) | \(f_\mathrm{{g}}/{d_\mathrm{{p}}}^2\) |

Pressure | | \({P}^*\) | \(f_\mathrm{{g}}/{d_\mathrm{{p}}}^2\) |

Steady-state cohesion | | \({c}^*\) | \(f_\mathrm{{g}}/{d_\mathrm{{p}}}^2\) |

Liquid bridge volume | \({V_\mathrm{{b}}}\) | \({V_\mathrm{{b}}}^*\) | \({d_\mathrm{{p}}}^3\) |

Surface tension | \(\gamma \) | \({\gamma }^*\) | \(f_\mathrm{{g}}/{d_\mathrm{{p}}}\) |

Rupture distance | \(S_\mathrm{{c}}\) | \({S_\mathrm{{c}}}^*\) | \(d_\mathrm{{p}}\) |

Torque | \(T_\mathrm{{z}}\) | \({T_\mathrm{{z}}}^*\) | \({f_\mathrm{{g}}}{d_\mathrm{{p}}}\) |

Angular rotation | \({\theta _\mathrm{{rot}}}\) | \({\theta _\mathrm{{rot}}}^*\) | \(2{\pi }\) |

Adhesive energy | | \({E}^*\) | \({f_\mathrm{{g}}}{d_\mathrm{{p}}}\) |

## 3 Micro–macro transition

To extract the macroscopic properties, we use the spatial coarse-graining approach detailed in [40, 41, 42]. The averaging is performed over toroidal volume, over many snapshots of time assuming rotational invariance in the tangential \(\phi \)-direction. The averaging procedure for a three-dimensional system is explained in [41, 42]. This spatial coarse-graining method was used earlier in [1, 23, 27, 28, 42]. The simulation is run for 200 s and temporal averaging is done when the flow is in steady state, between 80 and 200 s, thereby disregarding the transient behavior at the onset of the shear.

### 3.1 Steady-state cohesion and its correlation with liquid bridge volume and surface tension

For dry cohesionless systems, the dependence of shear stress on pressure is linear without an offset, i.e., \(c^*\) = 0. In the presence of interstitial liquid between the particles in the pendular regime, cohesive forces increase with increasing liquid bridge volume. This results in a positive steady-state cohesion \(c^*\) as given by Eq. (18), see Fig. 5.

*a*= 0.9805 for \(\gamma = 0.020\) N m\(^{-1}\). In the next section, we study the dependence of this constant on the surface tension of liquid.

*p*= 2.1977 as obtained from the fitting shown in Fig. 9; the offset is very small and can be neglected.

This subsection shows that the macroscopic characteristics of the liquid bridge model are determined by the maximum interacting force between the particles and the rupture distance. The steady-state cohesion scales linearly with the surface tension of liquid, i.e., the maximum force between the particles. For a given maximum force, the cohesion scaled with the surface tension of liquid is also a linear function of the rupture distance of the liquid bridge.

### 3.2 Macroscopic torque analysis from the microscopic parameters

The strength, cohesion, and flow properties of granular materials are strongly influenced by the presence of capillary cohesion. Due to the cohesive properties of these wet materials, the shear stress increases and, as a result, partially saturated wet materials require higher torques for deformation (shear), e.g., in a shear cell. Loosely speaking, torque is a measure of the shear stress or force acting on the particles at the wall and thus can be used to find an estimate of shear stress in the shear band. To study solely the effect of capillary cohesion on the torque, the other parameters like the particle friction are kept very small in our simulations, with \(\mu _\mathrm{{g}}\) = 0.01. Earlier studies [13, 27, 43, 44] show that the average torque acting on the rotating part of the shear cell increases with increasing moisture content. In this section, we perform a detailed analysis of the macroscopic torque as a function of the microparameters in order to understand its connection with the steady-state cohesion of the material.

*z*-axis with frequency \(f_\mathrm{{rot}}\). All the particles forming the inner and outer wall are identified as \(\mathcal {C}_\mathrm {inner}\) and \(\mathcal {C}_\mathrm {outer}\), respectively. The macroscopic torque is calculated based on the contact forces on the fixed particles on the moving (outer) and stationary (inner) parts of the shear cell. Thus, the net inner and outer torque are calculated by summing up the torques for all the contacts with respect to the axis of rotation of the shear cell. The net torque is obtained from the difference between the outer wall torque and the inner wall torque. We multiply the total torque by a factor of \({2\pi }/({\pi /6})\) in order to get the torque for the whole system from the obtained torque of our simulations in a 30\(^\circ \) section. Thus, the torque is given by

*N*represents the number of particles, \(\mathbf {c}_{ij}\) is the position of the contact point, and \(\mathbf {f}_{ij}\) is the interaction force. Only the

*z*-component of the torque vector (\(T_z\)) is of interest as required for shearing the cell in angular direction.

Figure 11 shows \({{T}_z}^*\) as a function of \({{\gamma }^*}\) for different liquid bridge volumes. We observe that the resultant torque depends linearly in the surface tension of the liquid. The fit parameter *l* from the figure, the rate of increase of torque with surface tension, depends on the liquid bridge volume.

*t*is the fit parameter, see Fig. 11. Assuming \({{T}_z} = {{T}_z}^\mathrm{{macro}}\), an equivalent steady-state cohesion as obtained from the calculated torque can be given as follows:

*e*= 2.0062 is a fit parameter, see Fig. 12, and the offset is very small and can be neglected. Equation (28) shows equivalent steady- state cohesion as obtained from the torque is also linearly dependent on \({S_c}^*\). The fitting parameter

*e*of this equation shows a close similarity with the fitting parameter

*p*of Eq. (23). Alternatively, Fig. 13 shows a comparison of the two torques given by the scalar

*z*-component of Eqs. (24) and (26) for surface tension of liquid 0.020 N m\(^{-1}\). These results show that the steady-state cohesion and torque are related by Eq. (26).

*M*(a measure of the resultant arm-length times surface area) which depends only on the geometry of the system.

## 4 An analogous linear irreversible contact model for cohesive particles

As discussed in Sect. 3.1, the steady-state cohesion for the liquid bridge model is controlled by the rupture distance of the liquid bridge, which is proportional to the liquid bridge volume, and the magnitude of the maximum interaction force, which is governed by the surface tension of the liquid. Assuming that the non-linear liquid bridge capillary force can be replaced by a simple irreversible linear adhesive force between the particles with the same macrocharacteristics, we compare the steady-state cohesion of the two models in Sect. 4.1.

### 4.1 Equal maximum force and interaction distance

*g*= 2.1716 and

*h*\(\approx \) 0 for the liquid bridge contact model,

*g*= 2.0984 and

*h*= 0.2226 for the linear irreversible contact model.

So for a given liquid bridge volume and a given surface tension of liquid, the linear irreversible contact model with the same maximum force and same interaction distance has a higher cohesion.

### 4.2 Equal maximum force and adhesive energy

### 4.3 Different maximum force for the two contact models

## 5 Conclusion

We observed a correlation between the steady-state cohesion and the microscopic parameters of the liquid bridge model. The microparameters are the liquid bridge volume, the liquid surface tension, the contact angle (which was kept constant), and the size of particles (i.e., curvature, which was also not varied). A detailed study of the effect of liquid bridge volume and surface tension of the liquid was done in this paper. These microscopic parameters control the macroscopic cohesion in wet granular materials in different ways. The steady-state cohesion of the system is proportional to the maximum adhesive force, which varies linearly with the surface tension. On the other hand, the steady-state cohesion is also linearly dependent on the maximum interaction distance between the particles, which depends on the volume of the liquid bridge. From these results, we have obtained a good micro–macro correlation between the steady-state cohesion and the microscopic parameters studied.

We analyzed the effect of cohesion on the wall torque required to rotate the system at a given rate. The torque (experimentally accessible) and the steady-state cohesion of the system are proportional and show similar linear dependence on the microscopic parameters.

Finally, an analogy was established between the liquid bridge model and a simpler linear irreversible contact model; even though these two models have different micro–macro correlations, the steady-state cohesion for the two models is the same if the maximum force and the total adhesive energy dissipated per contact for the two models are matched, irrespective of the shape of the attractive force function acting between the particles. In this way one can always replace a non-linear liquid bridge force by a simpler, faster to compute, linear one, and obtain identical macroscopic properties in less computational time. Furthermore, results for the two types of contact models with equal energy and different magnitudes of the maximum force show that they have different steady-state cohesions. The adhesive energy is thus not the sole microscopic condition for the two contact models to have same steady-state cohesion. Instead, both adhesive energy and cohesion scale linearly with the maximum adhesive force. The scaled cohesion for the two contact models is same for equal scaled adhesive energy. In this way, we can determine the steady-state cohesion from the two microscopic parameters, the adhesive energy and the maximum force.

In this paper, our study was focused on the micro–macro correlations and comparing different contact models. It would be interesting to study the forces and their probability distributions for wet cohesive systems [1]. Future studies will aim at understanding the microscopic origin and dynamics of the contacts and liquid bridges throughout the force network(s) and also the directional statistics of the inter-particle forces inside a shear band. The effect of liquid migration on the macroproperties and a continuum description for wet, sheared granular materials will be studied in the near future.

## Notes

### Acknowledgments

We acknowledge our financial support through STW Project 12272 “Hydrodynamic theory of wet particle systems: Modeling, simulation and validation based on microscopic and macroscopic description.”

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