Unifying same- and different-material particle charging through stochastic scaling

TL;DR

The paper tackles the long-standing challenge of predicting triboelectric charging in particle-laden flows, where experiments show variable impact charges, charge reversal, and bipolar charging that conventional models fail to capture. It introduces the stochastic scaling model (SSM), a unified, physics-based framework that uses a single well-characterized reference impact to scale the stochastic charge-transfer statistics to diverse particle-wall and particle-particle contacts via ratios such as $N_w/N_{w0}$, $N_i/N_0$, and $A/A_0$, while treating wall transfer deterministically and particle transfer as a skewed-normal random variable. The approach, grounded in a small set of measurable inputs ${\mu_0,\sigma_0,\gamma_0,\Delta Q_{0,\mathrm{min}}}$, reproduces variable impact charge, charge reversal, and size-dependent bipolar charging in large-scale CFD simulations with exceptional efficiency (less than 0.01% of CPU time). The model connects to, and extends, existing theories (condenser, surface-state, mosaic) by expressing them as special cases of SSM, offering a practical path to simulating electrostatics across a wide range of particle-laden flows. Overall, the SSM provides a scalable, physically grounded framework to predict and analyze triboelectric charging in both natural phenomena and industrial processes.

Abstract

Triboelectric charging of insulating particles through contact is critical in diverse physical and engineering processes, from dust storms and volcanic eruptions to industrial powder handling. However, many experiments over the years have consistently revealed counterintuitive charging patterns, including variable impact charge under identical conditions, charge sign reversal with repeated impacts, and bipolar charging of differently sized particles. Existing computational models cannot predict these patterns; they either rely on oversimplified heuristics or require inaccessible detailed surface properties. We present a stochastic scaling model (SSM) for particle charging that unifies same-material (particle-particle) and different-material (particle-wall) charging in a single theoretical framework. The model grounds in a physics-based stochastic closure by the mean, variance, skewness, and minimum impact charge measured in a highly-controlled reference experiment. To test the SSM, we implemented it in an open-source Lagrangian-Eulerian CFD solver. When simulating 300 000 insulating particles transported by turbulent wall-bounded flows, the SSM takes less than 0.01% of the CPU time. By scaling the statistical parameters of the reference impact to each collision, the new model reproduces the complex charging patterns observed in experiments without requiring surface-level first-principles inputs. The SSM offers a physically grounded route to large-scale simulations of electrostatic effects across many fields of particle-laden flows.

Figures (15)

• Figure 1: (a) The classical condenser model predicts impact charge between different materials ($\Delta Q_{iw}$) according to their contact potential difference, resulting in monotonic charging of a particle until it reaches its saturation charge ($Q_\mathrm{sat}$). The model fails to reproduce several experimentally observed charging patterns: (b) stochastic variability in impact charge under identical conditions, (c) charge reversal, where the direction of net charge transfer between materials can invert after repeated contacts, and (d) size-dependent bipolar charging in contacts between particles of the same material but different sizes (adapted and reprinted with permission).
• Figure 2: (a) Particle and wall before (left), during (middle), and after contact (right). The white circles indicate the contact areas. Small dots are charge carriers on the surfaces' charging sites; their surface density is $c$. A ratio of $\alpha$ charging sites are active. Brown charging sites are inactive, and their charge carriers remain during contact on the surfaces. Black sites transfer charge carriers from the wall to the particle and are thus no longer available for subsequent transfer. As the wall is grounded and conductive, these sites instantly regenerate. On the particle, active charging sites (green) transfer charge carriers to the wall with a probability of $p$, reducing the particle’s reservoir of transferable charge carriers. (b) The same mechanism applied to particle-particle contact. During impact, green charge carriers transfer between the two insulating particles, depleting the number of transferable carriers on both surfaces.
• Figure 3: Three possible impact charge distributions (Types A, B, and C) for contacts between fresh, uncharged, insulating particles and a conductive wall (sketched for negative skewness). Each distribution type allows two distinct interpretations (labeled 1 and 2) regarding the polarity of the transferred charge carriers $\epsilon_w$ from the wall and $\epsilon_p$ from the particle. Types A1 and B1 appear to be most representative of the materials in this study; nevertheless, the model is constructed in a general form.
• Figure 4: (a) The impact charge distribution from reference experiments (black, Type B1) provides the statistical parameters $\mu_0$, $\sigma_0$, $\gamma_0$, and $\Delta Q_{0,\mathrm{min}}$. We decompose this distribution into the transfers (blue) from the wall and the particle. The SSM scales these parameters (green arrows) to each contact: (b) For particle-wall contacts, to $\mu_w$, $\mu_i$, $\sigma_i$, and $\gamma_i$; (c) For particle-particle contacts, to $\mu_i$, $\sigma_i$, $\gamma_i$, $\mu_j$, $\sigma_j$, and $\gamma_j$. (For clarity, skewness values $\gamma_0$, $\gamma_i$, and $\gamma_j$ are not shown in the plots.) Superimposing the contributions gives the impact charge distribution (black) per contact.
• Figure 5: Statistical parameters of repeated particle-wall impacts predicted by the SSM ($A/A_0 = \text{const}$, $\gamma_0 = 0$, Type B1; see \ref{['fig:concept2']}). (a) Probability distributions of the impact charge for uncharged particles ($\alpha_w/\alpha_0 = 1$) with depleting charging sites on the surface. (b) Distributions for particles maintaining their initial charging site density ($c/c_0 = 1$) while their charge increases. (c) Mean and (d) standard deviation of the impact charge as functions of $\alpha_w/\alpha_0$ and $c/c_0$.