Electrostatic enhancement of particle collision rates in atmospheric flows

Abstract: Collisional growth of tiny particles is a fundamental process governing the growth of cloud droplets and the aggregation of ash particles in volcanic plumes, with direct implications for precipitation formation, cloud lifetime, and ash plume dynamics. The particles in these scenarios often carry electric charges. In this study, we investigate the collision dynamics of a pair of like charged dielectric spheres subjected to a uniaxial compressional flow, an important linear flow that captures key features of atmospheric straining motions. Finite particle size leads to electrostatic interactions that deviate from the point charge approximation, resulting in far field repulsion and near-field attraction, which in turn generate nontrivial particle trajectories and critical collision thresholds. For certain combinations of charge and size, the interplay between hydrodynamic and electrostatic forces creates strong radially inward particle relative velocities that substantially alter particle pair dynamics and modify the conditions required for contact. For uncharged particles, collision efficiency increases monotonically with particle size ratio. However, in the presence of electrostatic forces with high charge ratio values, the collision efficiency exhibits a nonmonotonic dependence, attaining a maximum at small size ratios and decreasing as the ratio increases, with a crossover beyond which larger particles become less favorable for collision. These results demonstrate that the same polarity charges on finite sized atmospheric particles do not necessarily inhibit collisions. Instead, they can enhance collisional growth for specific charge and size ratio combinations, revealing counterintuitive pathways relevant to cloud microphysical processes and volcanic ash aggregation in electrified atmospheric environments.

• The paper shows that near-field polarization forces can overcome far-field repulsion, enhancing collisions between like-charged particles.
• The paper develops a pair-mobility formalism that integrates finite-size effects, non-continuum lubrication, van der Waals forces, and full electrostatic interactions.
• The paper highlights implications for cloud formation and volcanic ash aggregation by quantifying changes in collision efficiency across diverse charge and size regimes.

Electrostatic Enhancement of Particle Collision Rates in Atmospheric Flows

Introduction and Scientific Context

The collisional growth of small suspended particles underpins cloud formation, precipitation initiation, and volcanic ash aggregation. These processes govern atmospheric lifetimes, hydrometeor evolution, and dispersal dynamics. Electrostatic charging of suspensions—arising from cosmic ray ionization, triboelectric effects, and field-induced charging—significantly affects inter-particle dynamics in both clouds and volcanic plumes. This work systematically analyzes binary collisions of like-charged dielectric spheres in uniaxial compressional flows, providing a physically consistent treatment of finite-size effects, non-continuum lubrication, van der Waals interactions, and full electrostatics incorporating polarization and dielectric contrast.

The authors motivate their analyses through observations of electrified clouds and volcanic eruptions, where lightning rates, clustering, and enhanced aggregation suggest potent charge effects. Most notably, they highlight that same-sign charge interactions in finite-size particle pairs exhibit non-trivial behaviors—contrary to the classical expectation of purely repulsive dynamics—due to the interplay between far-field repulsion and near-field polarization-induced attraction.

Theoretical Formulation and Governing Parameters

The study is built on a complete pair-mobility formalism, valid for low Reynolds and Stokes numbers. The uniaxial compressional flow serves as a paradigm for local turbulent straining, with the far-field velocity given by $$\mathbf{U}^\infty(\mathbf{x}) = (\dot{\gamma}x_1,\,\dot{\gamma}x_2,\, -2\dot{\gamma} x_3)$$, and the compressional rate $\dot{\gamma}$ reflecting typical Kolmogorov-scale turbulent dissipation rates in atmospheric contexts.

Electrostatic interactions are modeled for dielectric spheres, capturing both monopole (classical Coulombic) and higher-order polarization-induced forces. Crucially, at small surface separations, like-charged dielectric particles can experience net attractive forces, a non-intuitive result due to finite-size and dielectric effects.

Non-continuum lubrication corrections are incorporated for particle separations below the fluid mean free path (Knudsen numbers $Kn = \lambda_0/a^*$ up to $10^{-2}$ in air), and van der Waals interactions are included via the retarded potential, parameterized by $$N_v = 2A_H/[3\pi \dot{\gamma}\mu_f \kappa(1+\kappa)a_1^3]$$.

The dominant control parameter for electrostatics is

$$N_e = \frac{q_1^2}{12 \pi^2 \mu_f \epsilon_0 a_1^4 \dot{\gamma} \kappa}$$

which measures the charge-mediated force relative to hydrodynamic strain. The authors provide robust semi-empirical estimates for $N_e$ in warm clouds ($10^{-5} \lesssim N_e \lesssim 10^{-1}$) and volcanic plumes ($O(1)$--$O(10^3)$).

Trajectory Topologies and Critical Electrostatic Regimes

The study deploys trajectory analysis, revealing that the inclusion of full electrostatics leads to a rich set of dynamical regimes. For tracers in compressional flow, uncharged particles follow streamlines into collision (Figure 1).

Figure 1: Topological modifications in colliding and deflected trajectories for charged particle pairs in compressional flow, illustrating rapid suppression of colliding domains at high $N_e$.

Introduction of monopole electrostatics generates both far-field repulsion and—when polarization effects are included—near-field attraction. The transition between collisional and non-collisional regimes is governed by a critical $N_e$ value, $(N_e)_c$, which depends on size ratio $\kappa$ and charge ratio $\beta = q_2/q_1$. For $N_e > (N_e)_c$, all trajectories are deflected and collisions are fully suppressed. The width of the repulsive band in $(\kappa,\beta)$ space is finite for dielectrics, in contrast to the conductor limit (Figure 2).

Figure 2: Variation of the critical electrostatic parameter $N_e$ with charge ratio for different size ratios; the dotted region marks the repulsive band and the corresponding abrupt changes in the suppression threshold.

In the attractive regime, increasing $N_e$ enhances the inward radial velocity and thus the collision rate, up to a limit where repulsion dominates and a sharp cutoff occurs. The radial velocity profiles and trajectory topologies across this transition are detailed in Figures 5–7.

Figure 3: Radial velocity ($v_r$) as a function of separation, showing the emergence of a positive (outward) zone at $N_e > (N_e)_c$, demarcating the non-collisional state.

Figure 4: Decomposition of flow and electrostatic contributions to $v_r$ in the repulsive regime; only their sum governs access to the collision sphere.

Collision Efficiency: Parameter Space Structure and Numerical Results

Collision efficiency $E_{12}$ is defined as the ratio of the actual collision rate (with all interactions) to the ideal geometrical rate. For uncharged spheres, efficiency increases monotonically with $\kappa$ due to enhanced lubrication and van der Waals effects. Electrostatics fundamentally alters these trends.

For small charge ratios, efficiency remains monotonic in $\kappa$; as $\beta$ increases, a crossover occurs—efficiency becomes non-monotonic, with a peak at intermediate size ratios (Figures 9, 11).

Figure 5: Collision efficiency as a function of $N_e$ for different $\kappa$ and $\beta$, showing the collapse to zero above critical $N_e$ and non-monotonicity at high charge ratios.

Figure 6: Collision efficiency versus $N_v$ at different $\beta$, demonstrating the decreasing role of van der Waals as electrostatics becomes dominant.

Contrary to classical expectation, identically charged, finite-size droplets or ash particles can exhibit enhanced collisions for specific charge–size combinations. In regimes where near-field polarization attraction prevails, collision rates can be boosted by up to an order of magnitude relative to the uncharged case for unequal-sized pairs.

Knudsen number dependence quantifies the weakening of lubrication resistance in rarefied regimes, relevant for submicron particles in the upper troposphere or distal volcanic plumes. The inclusion of van der Waals forces is found essential for uncharged or weakly charged systems but is eclipsed by electrostatics at high $\beta$.

Atmospheric and Volcanological Implications

The results have immediate consequences for cloud microphysics and tephra aggregation. In typical non-thunderstorm clouds, real droplet charges are much lower than the suppressive $(N_e)_c$ threshold (Figure 7), so charging can be beneficial to growth, bridging the $15$--$40\,\mu$m "size gap." In volcanic plumes, where particles can carry much higher charges owing to strong tribo- and fractoelectric effects, both enhancement and suppression regimes are accessible, depending on eruption energetics, ash size distribution, and local charge partitioning.

Figure 7: Maximum permissible cloud/ash particle charge as a function of size; atmospheric observations typically lie well below the suppression curve, enabling charge-enhanced aggregation.

Extensive lidar and radar measurements of rapid fine-ash fallout in volcanic eruptions are consistent with the strong increase in collision efficiency for moderate-$\beta$, high-asymmetry pairs, aligning with the predicted order-of-magnitude enhancements.

Limitations and Future Perspectives

The pair-mobility framework developed does not address particle anisotropy (ice hydrometeors, irregular ash), many-body interactions, or explicit turbulent intermittency. Nevertheless, the frozen-strain model provides key mechanistic insights and calibration for more complex stochastic or DNS-based approaches.

Extensions to non-spherical particles will need to incorporate orientation-dependent electrostatic torques and hydrodynamic coupling. Further, fluctuating velocity gradient fields in turbulence could induce intermittent excursions into electrostatics-dominated or suppressed-collision states.

Conclusion

This study delivers a comprehensive assessment of how charge, size ratio, and non-continuum effects combine to modify the collision dynamics of finite-sized dielectric spheres in atmospheric straining flows. The result that like-charged particles can both enhance and suppress collisions, depending on precise charge–size–flow regimes, overturns the simple narrative of monotonic collision inhibition. These findings have direct theoretical and practical consequences for modeling droplet coalescence in clouds, ash aggregation and fallout in volcanic plumes, and the design of atmospheric hazard prediction tools. Key future directions include generalization to anisotropic particles, inclusion of time-dependent turbulent straining, and development of high-fidelity collision kernels for electrified flows.