# Comparative Study to Evaluate Mixing Efficiency of Very Fine Particles
## 2 Numerical Scheme
### Dem  
The DEM is a numerical technique for tracking the motion of a large number of particles. In DEM simulations, particles experience both contact and non-contact forces. Non-contact forces, such as gravity and ambient fluid drag, act continuously, regardless of collisions, and primarily influence inertial motion. Contact forces, by contrast, arise from collisions and are typically decomposed into normal and tangential components. These forces are calculated iteratively at each timestep and play a critical role in energy dissipation and particle trajectory evolution.  
In this study, normal contact forces were modeled using the HLS approach, which accounts for elastoplastic behavior during particle collisions. The HLS model separately defines the loading stiffness during compression and the unloading stiffness during rebound. Plastic deformation is represented by the residual overlap after unloading, while the area between the loading and unloading curves corresponds to the energy loss due to internal friction, heat, or sound. Figure 2 depicts the normal force-overlap relationship, which is mathematically formulated in Equations (1)-(6).  
To model tangential contact forces, we employed the LSCL model, which improves upon the basic Coulomb friction model by incorporating sliding stiffness and tangential overlap. The conventional Coulomb model defines tangential forces as simply the product of the friction coefficient (\(u\)) and the normal force (\(F_{n}\)), disregarding contact area deformation and elastic recovery. In contrast, the LSCL model considers tangential stiffness and captures dynamic frictional behaviors more accurately. Figure 3 presents a conceptual definition of overlap and stiffness, and the corresponding force formulations are provided in Equations (7)-(9).  
The HLS model computes the normal contact force \(F_{n}{}^{t}\) based on the direction in which the overlap changes. If the normal overlap increases (\(\Delta S_{n}\geq 0\)), the force is determined  
Figure 3: Definition of overlap and stiffness.  
Figure 2: Overlap–normal force relationship in HLS model.  
using the loading stiffness \(F_{nl}\), whereas if it decreases (\(\Delta S_{n}<0\)), the unloading stiffness \(F_{nu}\) is used with a limiting lower bound defined by a damping parameter \(\gamma\):  
\[F_{n}^{t}=min(K_{nl}S_{n}^{t},\ F_{n}^{t-\Delta t}+K_{nu}\Delta S_{n})\quad if \ \Delta S_{n}\geq 0, \tag{1}\]  
\[F_{n}^{t}=max(\ F_{n}^{t-\Delta t}+K_{nu}\Delta S_{n},\ \gamma K_{nl}S_{n}^{t})\quad if \ \Delta S_{n}<0, \tag{2}\]  
\[\Delta S_{n}=S_{n}^{t}-S_{n}^{\Delta t}. \tag{3}\]  
Here, \(F_{nl}\) and \(F_{nu}\) are defined as  
\[\frac{1}{K_{nl}}=\frac{1}{K_{nl,p1}}+\frac{1}{K_{nl,p2}}, \tag{4}\]  
\[K_{nu}=\frac{K_{nl}}{\epsilon^{2}}, \tag{5}\]  
\[K_{nl,p}=E_{p}L, \tag{6}\]  
where \(K_{nl,p1}\) and \(K_{nl,p2}\) represent the stiffness of the particle, \(L\) is the particle size, \(E_{p}\) is the bulk Young's modulus, and \(\epsilon\) is the restitution coefficient, which represents the energy conservation during a collision.  
The LSCL model computes the tangential force \(F_{\tau}^{t}\) as follows: the preliminary elastic tangential force is first calculated using the sliding stiffness \(K_{\tau}\) and the change in tangential overlap \(\Delta S_{\tau}\); then, this value is limited by the Coulomb threshold \(\mu F_{\tau}^{t}\):  
\[F_{\tau}^{t}=min\big{(}\big{|}F_{\tau,\rho}^{t}\big{|},\ \mu F_{n}^{t}\big{)}, \tag{7}\]  
\[F_{\tau}^{t}=F_{\tau}^{t-\Delta t}-K_{\tau}\Delta S_{\tau}. \tag{8}\]  
The sliding stiffness \(K_{\tau}\) is defined in terms of the tangential stiffness ratio \(r_{K}\) and the normal loading stiffness:  
\[K_{T}=r_{K}K_{nl}. \tag{9}\]  
These formulations allow the LSCL model to capture the energy dissipation and dynamic friction in tangential contact scenarios more accurately.