Percolation of a cohesive fine particle in a static bed

Technical Summary: Percolation of a Cohesive Fine Particle in a Static Bed

Problem Statement
The percolation of fine particles (fines) through a static bed of larger particles is a fundamental process in industrial handling (e.g., pharmaceuticals, chemical processing) and geophysical flows (e.g., sediment transport, groundwater contamination). In non-cohesive systems, whether a fine particle passes through or becomes trapped is primarily governed by geometric constraints, specifically the ratio of large-to-fine particle diameters ($R = d_l/d_f$). A geometric trapping threshold exists at $R_t \approx 6.464$; when $R > R_t$, fines are theoretically expected to pass freely through the pore throats formed by the large particles. However, real-world systems often involve cohesive forces (van der Waals, electrostatic, liquid bridges) that alter contact dynamics. While previous studies have addressed non-cohesive percolation and the rheology of cohesive flows, there is a gap in understanding how cohesion influences percolation in the "free-sifting" regime where geometric constraints alone would permit passage. This study investigates the percolation of cohesive fine particles through a static bed of large particles where $R=7$ (above the geometric threshold), focusing on the single-particle limit where fines interact with bed particles but not with each other.

Methodology
The authors employ Discrete Element Method (DEM) simulations using the open-source software LAMMPS. The simulations utilize an improved Velocity-Verlet scheme suitable for large size ratios.

• System Configuration: A static bed of monodisperse large particles ($d_l = 4$ mm) is packed with a fraction $\phi = 0.6$. Fine particles ($d_f \approx 0.57$ mm, $R=7$) are released from rest above the bed and allowed to fall under gravity.
• Contact Models: Normal contact forces follow the Hertzian elastic model. Inelasticity is modeled via a constant restitution coefficient ($e_n$). Tangential forces use the Mindlin model with sliding friction ($\mu$) and rolling resistance.
• Cohesion: Cohesive interactions are modeled primarily using the Derjaguin-Muller-Toporov (DMT) model, which assumes a constant attractive force during contact and Hertzian repulsion. The strength of cohesion is quantified by the Bond number ($Bo = F_c / W$), defined as the ratio of the pull-off force to the weight of the fine particle. A limited number of simulations using the Johnson-Kendall-Roberts (JKR) model were also performed for comparison.
• Parameters: The study systematically varies the Bond number ($Bo$), sliding friction coefficient ($\mu$), restitution coefficient ($e_n$), and Young's modulus ($E$). The dynamics of $10^4$ independent fine particles are tracked to ensure statistical accuracy.

Key Results

1. Non-Geometric Trapping: Even when $R=7$ (geometrically permitting free passage), sufficiently large cohesion and friction lead to particle trapping. As $Bo$ increases, the characteristic percolation distance decreases, meaning particles become trapped at shallower depths.
2. Trapping Distribution: The vertical distribution of trapped particles follows an exponential decay. The characteristic trapping distance ($\lambda$) serves as a metric for percolation behavior. The inverse trapping distance ($d_l/\lambda$) increases monotonically with $Bo$.
3. Role of Friction and Cohesion:
   • At low $Bo$, sliding friction plays a significant role in trapping.
   • At high $Bo$, cohesion becomes the dominant trapping mechanism.
   • A "kink" is observed in the relationship between inverse trapping distance and friction. This kink shifts with $Bo$ and corresponds to a transition between regimes where friction dominates and where cohesion dominates.
   • Particle stiffness ($E$) influences trapping: softer particles (lower $E$) are more easily trapped, particularly at higher $Bo$.
4. Collisional Mechanism: The trapping process is decomposed into two sequential steps:
   • Adhesion Condition: Upon collision, a particle either rebounds or adheres. Adhesion occurs if the impact velocity is below a critical velocity ($v_{cr}$), which depends on $Bo$, $e_n$, and $E$.
   • Post-Contact Motion: If the particle adheres, it may slide and detach or remain trapped. A static equilibrium condition based on force balance determines if a non-rebounding particle remains stationary. This condition depends on a rescaled friction coefficient $\tilde{\mu} = \mu\sqrt{4Bo^2 - 1}$. If $\tilde{\mu} > 1$, the particle is likely to remain stationary; if $\tilde{\mu} < 1$, it is likely to slide and potentially detach.
5. Predictive Model: The authors develop a collisional model linking microscopic dynamics to macroscopic trapping distance. The inverse trapping distance is predicted by $1/\lambda = -k \ln(1 - P_t)$, where $k$ is the collision frequency and $P_t$ is the probability of trapping per collision (determined by the ratio of impact velocity to $v_{cr}$). This model accurately predicts trapping distances in the collision-dominated regime (high $Bo$).
6. Double Contact: In regimes with lower friction or cohesion, particles often slide and form double contacts (touching two large particles simultaneously). This configuration can lead to trapping even when single-contact conditions would suggest detachment, or conversely, allow detachment via a rebound from a second contact.

Significance and Claims
The paper establishes that the percolation of cohesive fine particles is not determined solely by geometric accessibility. Instead, particle-scale interaction dynamics—specifically the interplay of cohesion, restitution, friction, and impact velocity—can override geometric expectations, leading to trapping even when pore throats are geometrically wide enough for passage.

The authors claim to provide a foundational framework for understanding these mechanisms in static beds, which is a prerequisite for addressing the more complex problem of granular segregation in flowing mixtures of cohesive particles. By identifying the specific conditions (critical velocity, static equilibrium criteria) that govern adhesion and subsequent motion, the study offers a predictive capability for trapping distance based on fundamental material properties. The results highlight that in cohesive systems, the "fate" of a fine particle is governed by a sequence of local interactions (collision, adhesion, post-contact motion) rather than just the static geometry of the bed.