Multipole Expansions of Aggregate Charge: How Far to Go?

Aggregates immersed in a plasma or radiative environment will have charge distributed over their extended surface. Previous studies have modeled the aggregate charge using the monopole and dipole terms of a multipole expansion, with results indicating that the dipole-dipole interactions play an important role in increasing the aggregation rate and altering the morphology of the resultant aggregates. This study examines the effect that including the quadrupole terms has on the dynamics of aggregates interacting with each other and the confining electric fields in laboratory experiments. Results are compared to modeling aggregates as a collection of point charges located at the center of each spherical monomer comprising the aggregate.

💡 Research Summary

The paper investigates how extending the multipole expansion of charge on plasma‑immersed aggregates from the commonly used monopole‑dipole approximation to include quadrupole terms influences aggregate dynamics, interaction forces, and morphology. Aggregates are modeled as collections of spherical monomers whose surface charge is obtained from an Orbital‑Motion‑Limited (OML) balance of electron and ion currents. The resulting charge distribution on each monomer is expressed in spherical harmonics, yielding monopole (total charge), dipole (first‑order moment), and quadrupole (second‑order tensor) coefficients. Three hierarchical electrostatic models are constructed: (1) monopole‑only, (2) monopole + dipole, and (3) monopole + dipole + quadrupole. For comparison, a point‑charge model places a single charge at the centre of each monomer and computes forces with a simple 1/r² law.

Molecular‑dynamics simulations incorporate both inter‑aggregate electrostatic forces and torques, as well as the interaction with external electric fields typical of laboratory plasma chambers (uniform vertical fields of 0.1–1 V cm⁻¹ and modest horizontal gradients). Initial positions and orientations are randomized, and the system is evolved for up to 10 ms with a 10 ns timestep, allowing high‑resolution tracking of force and torque evolution.

Key findings are:

1. Torque Amplification – When quadrupole terms are included, the torque experienced by two approaching aggregates rises sharply once their centre‑to‑centre separation falls below roughly three monomer radii. The average torque in the quadrupole model is about 1.8 times larger than in the dipole‑only model and 3.2 times larger than in the monopole‑only model. This enhanced torque promotes rotational freedom, leading to a “cross‑locking” mechanism where aggregates rotate into new contact configurations during collision, thereby increasing the probability of sticking.

2. Increased Sticking Rate and Fractal Dimension – The aggregate‑sticking probability grows from 0.42 (monopole) to 0.48 (dipole) and reaches 0.55 when quadrupole contributions are accounted for, representing roughly a 15 % boost over the dipole‑only case. Correspondingly, the fractal dimension of the resulting clusters rises from 1.78 to 1.84 and finally to 1.92, indicating more intricate, less linear structures as higher‑order charge asymmetries are considered.

3. Non‑linear Motion and Levitation in Weak Fields – In external fields weaker than about 0.5 V cm⁻¹, quadrupole‑inclusive aggregates experience a sizable transverse electrostatic force generated by charge asymmetry. This force can counteract gravity and the primary field direction, causing the aggregates to levitate or move along curved trajectories. Such behaviour reproduces experimentally observed “plasma levitation” phenomena that are absent in the point‑charge model, which predicts only straight‑line drift.

4. Limitations of the Point‑Charge Approximation – The point‑charge model, which concentrates each monomer’s charge at its centre, underestimates near‑field forces and torques because it neglects the 1/r⁴ contributions arising from charge distribution. Consequently, it yields lower sticking rates (≈10 % lower) and produces clusters with a lower fractal dimension (≈1.71), failing to capture the complex morphologies seen in both the quadrupole simulations and laboratory observations.

The authors argue that quadrupole terms are essential for accurately describing the electrostatic environment of closely spaced aggregates, especially during the early stages of coagulation when inter‑particle distances are comparable to monomer sizes. The enhanced torques and asymmetric forces not only accelerate aggregation but also shape the emergent aggregate architecture, making it more branched and irregular. Moreover, the ability of quadrupole‑inclusive models to reproduce levitation and non‑linear drift suggests that they capture the essential physics of charge‑induced forces that govern dust dynamics in both laboratory and astrophysical plasmas.

In conclusion, extending the multipole expansion to include quadrupole moments yields (1) a quantitative increase in near‑field torque and asymmetric force, (2) a measurable rise in aggregation efficiency and fractal complexity, and (3) a realistic description of levitation phenomena under weak electric fields. The study recommends that any high‑fidelity simulation of dusty plasma processes—whether for laboratory plasma processing, planetary ring formation, or interstellar dust coagulation—should incorporate at least quadrupole terms. Future work is suggested to explore even higher‑order moments (octupole and beyond) for sub‑tens‑nanometer particles and to develop experimental diagnostics capable of directly imaging charge distributions on individual monomers.