Electrostatic enhancement of particle collision rates in atmospheric flows

TL;DR

This work analyzes how electrostatic forces modify collisions between like-charged dielectric particles in a uniaxial compressional flow, a model capturing atmospheric straining. By combining finite-size electrostatics with hydrodynamic interactions and short-range forces, it reveals that the collision efficiency $E_{12}$ can be nonmonotonic in the electrostatic-to-hydrodynamic ratio $N_e$, with a critical threshold $(N_e)_c$ that depends on size ratio $\kappa$ and charge ratio $\beta$. The study shows that near-field attraction can enhance collisions for certain parameter ranges, while strong charging can suppress them, and it identifies distinct trajectory topologies arising from attractive or repulsive near-field regimes in dielectric spheres. These findings have implications for cloud droplet growth and volcanic ash aggregation in electrified flows, highlighting regimes where modest charging and size asymmetry promote rapid collisional growth and where strong charging can inhibit it. Extensions to anisotropic particles and turbulent flows are proposed to further develop physically grounded collision kernels for atmospheric and plume microphysics.

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.

Figures (14)

• Figure 1: Relative pair-trajectory topology for the 'charged tracer' problem in a uniaxial compressional flow without hydrodynamic interactions. The blue circle denotes the collision sphere. (a) In the absence of charge, trajectories follow the background flow and intersect the collision sphere, leading to collisions. (b) with Coulombic repulsion at $N_{e}=10$, only a subset of trajectories reach the collision sphere, while others are deflected. (c,d) For $N_{e}>16$, electrostatic repulsion dominates over the compressional flow, and all trajectories are diverted away from the collision sphere, preventing collisions.
• Figure 2: Schematic of two like-charged particles of size $a_{1}$ and $a_{2}$ in an uniaxial compressional flow. $r$ is the distance between the center of the spheres, $\theta$ is the angle between the compressional axis and the position vector $\textbf{r}$. Sphere $1$ is the satellite sphere, sphere $2$ is the test sphere and sphere $3$ is the collision sphere of size $a_{1}+a_{2}$. Representative trajectories followed by a satellite sphere relative to the test sphere are shown in blue. We denote the unit vectors in the $r$ and $\theta$ directions as $\hat{\boldsymbol{e}}_{r}$ and $\hat{\boldsymbol{e}}_{\theta}$.
• Figure 3: (a)Variation of the scaled ideal collision rate, $K_{12}^{(0)}/[n_1 n_2 \dot{\gamma}(a_1+a_2)^3]$, with the scaled electrostatic parameter $\tilde{N}_e$ for a like-charged tracer pair under monopole–monopole repulsion ($\mathscr{K}=0$). The collision rate decreases monotonically and vanishes at $\tilde{N}_e=16$, defining the critical value $(N_e)_c$ for the chosen $\kappa,\beta$. (b) Electrostatic force map in the $(\kappa,\beta)$ plane, compared with the roots of Eq. (\ref{['Eq:roots_of_inequality']}), charge–size scalings colgate1970chargepruppacher1998microphysics, and the repulsive boundary for conducting spheres lekner2016regions. For dielectric spheres, repulsion occurs within a finite band in parameter space, shown by the dark region when the full electrostatic interaction is retained; truncation to the monopole–dipole term yields the blue curves corresponding to Eq. (\ref{['Eq:roots_of_inequality']}).
• Figure 4: Variation of critical $N_{e}$ as a function of charge ratio $\beta$ for different $\kappa$ when $Kn=10^{-2}$ and $N_{v}=0$, for particle-pair with dielectric constant $k=k_1=k_2=80$. The dotted vertical lines indicate the width of the repulsive band for which the near-field is repulsive at separation $\xi=10^{-3}$. The sudden variation in the slope lies within the repulsive region. Note that the width of the repulsive band increases as $\kappa$ is increased.
• Figure 5: The radial component of the relative velocity $v_{r}$ as a function of the separation distance for values of $N_{e}$ less than, equal to and greater than the critical value of $(N_{e})_{c}$ for $\kappa=0.5$, $\beta=10$.