Methodology for quantifying particle charge statistics in electric fields of gas insulations
Hans-Christoph Töpper, Simon Scherrer, Lucio Isa, Christian M. Franck
TL;DR
This paper introduces a new method to measure and understand how charged particles behave in high-voltage gas insulation systems.
Contribution
A novel experimental methodology is introduced to quantify particle charge statistics and their motion in high-voltage gas insulation systems.
Findings
Particle diameter is the most influential factor in determining charge magnitude.
Non-Gaussian charge distributions are observed, attributed to adhesive forces between particles and electrodes.
Mirror-charge-induced motion patterns are identified as a new term in particle motion equations.
Abstract
Electrically charged particles in high-voltage gas insulation systems can lead to field distortion, partial discharges, and ultimately, insulation breakdowns. Despite their significance, particle charges have rarely been studied in this context. This work therefore characterizes the electric charges of metallic and dielectric particles with diameters ranging from 1 to 170 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}m. The charge magnitudes measured at different field strengths using tracking velocimetry range from approximately 1 fC to 10 pC. The particle diameter is observed to be the most influential factor, while intrinsic particle material…
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Figure 9- —Swiss Federal Institute of Technology Zurich
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Taxonomy
TopicsHigh voltage insulation and dielectric phenomena · Power Transformer Diagnostics and Insulation · Aerosol Filtration and Electrostatic Precipitation
Introduction
Micrometer-sized particles of various materials with different shapes and diameters are ubiquitously present^1,2^. As a result, they occur in the electric fields of high-voltage gas insulation systems, where they can acquire electric charges. These charges can cause insulation performance-diminishing effects that add to the safety margins of power grid high-voltage gas insulations, limiting their efficiency^3^. Regarding laboratory-scale insulation systems used in research, particles can unintentionally influence results by randomly reducing their performance^4^. To understand this, a dedicated investigation is required, as the charge accumulation capability is not an intrinsic material property and is not directly derivable from theoretical models. Quantifying charge magnitudes, polarities, and charging rates of micrometer-sized particles are essential to assessing their potential impact on the performance and reliability of gas-insulated electric power grid components.
Performance-reducing effects of particles in gas insulations
In addition to the design parameters of high-voltage gas insulations, such as the gas type^5^, gas pressure^6,7^, or electrode geometry^8,9^, the presence of contaminating particles has long been recognized as critical. Due to their charge, they can cause gas insulation performance-diminishing effects such as electric field distortion^10^, corona discharge^11,12^, and even breakdown initiation^13^. Regarding the latter, breakdown strength reductions of multiple tens of percent can occur under various insulation operating conditions^14,15^. Adding to the criticality is that these effects can become more pronounced at higher insulation gas pressures^10^. These are investigated for next-generation systems as they allow for more compact designs^16,17^, which potentially makes particles more critical in the future.
Origins and characteristics of contaminating particles
Particles enter gas insulation systems during manufacturing and assembly, when they are opened for maintenance, and during operation. Even precautions such as clean room manufacturing^18^, dry ice blasting^19^, and laser cleaning^20^ do not result in their complete removal. This is due to the airborne-driven omnipresence, small diameters that complicate detection within the large and complex interiors of insulation systems, and adhesive forces causing them to stick to the surfaces. Generally, contaminating particles can be divided into metallic and dielectric particles, as these material classes differ in their origin, size ranges, and intrinsic material properties.
Metallic particles in the upper micrometer range up to the low millimeter regime can originate from machining in production^21^, mechanical vibrations during shipment and operation^22^, thermal joint expansion^23^, or switching arcs ablating material from the electrodes^15^. Smaller metallic particles with diameters in the low micrometer regime, originate from natural atmospheric dust^24^ or sources like mining^24^ and vehicle brake wear^25^. Overall, these particles are made from materials such as aluminum^26,27^, brass^27^, copper^10,26^, iron^4,25^, and stainless steel^12^.
Micrometer-sized dielectric particles are omnipresent as they originate from natural^28^, industrial^29,30^, and urban sources^31^ in addition to being transported along long-range paths around the world^1,2^. Natural-origin particles, such as desert sand, were reported to have diameters between approximately 100 nm and 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m in the Sahara Desert^32^, and approximately 30 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 300 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m in the Mojave Desert^33^. Natural-origin aerosols fall in slightly smaller size ranges with diameters between approximately 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 20 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m^34^. Natural-origin articles are made up of a variety of structures such as silicates, carbonates, and minerals^35^. The dielectric particles emitted by industrial processes with diameters between approximately 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 100 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m^36^ fall in a similar size range. For these, multiple materials such as iron oxides, including hematite and magnetite, spinels such as zinc iron oxide, and minerals such as zinc oxide, potassium chloride, or calcium hydroxide occur^37^. Caused in urban areas and including similar materials^38^, their diameters fall in the range of approximately 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m to 100 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m in residential neighborhoods^39^ and on roads^40^.
Collectively, this shows that particles potentially contaminating gas insulation systems originate from various sources, have a wide range of materials, and possess diameters predominantly in the low micrometer range. Despite thorough cleaning precautions, their removal is challenging if not practically impossible. Moreover, all particles generated during the operation inside the insulation are unavoidable. This requires a detailed understanding and quantification of their charges acquired in the electric fields, as it causes the performance-diminishing effects that can render power grid components inoperable.
Electric charges of particles in electric fields of gas insulations
In the context of gas insulation breakdown analyses and partial discharge investigations, electric charges of millimeter-sized metallic particles were experimentally approximated in bulk in one instance by measuring the electric current caused by them between the electrodes^10^. Other than that, charges were theoretically calculated. The latter approach^11,23,41^ used pre-assuming empirical expressions to obtain a single charge polarity and magnitude for spherical or wire-shaped particles. For both geometries, the formulas state that the charge magnitude depends on the particle geometry, that it is proportional to the electric field strength, and that only one polarity occurs given a constant field direction. In addition, empirical correction factors linearly increasing or decreasing the charge as a function of the electric field strength were included^42,43^. Although these formulas have been applied frequently, only a few analyses explicitly mention the calculated charge magnitudes. For a 1 mm aluminum sphere in a 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} AC field, a charge of approximately 1 nC was stated^44^. A metallic wire-shaped particle with a length of 10 mm and a diameter of 0.15 mm in a 132 kV gas-insulated bus duct was reported to have a charge of 59 pC^43^. For the same shape and length but with a 0.2 mm diameter in a 100 kV bus duct, 35pC were reported^43^. In the same high-voltage system rated between 100 kV and 175 kV, the charges of the wires made of aluminum and copper were described to fall in the range between 20 pC and 60 PC^43^. In addition, the charges of various spheres and rods were approximated to fall in the range of 10 pC to 1 nC^45^. Charge data is hardly present for metallic particle geometries below the millimeter range. Experimentally determined magnitude and polarity dynamics have only been reported for particles with diameters of 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m across four different metallic materials^46^. The charging currents and charging mechanism were not quantified or described in any of the investigations. Across all analyses, characterizations beyond the material type, geometry, and occasionally the charge, such as adhesion measurement based on atomic force microscopy (AFM) quantifying the particle-electrode interaction, or particle morphology investigations using scanning electron microscopy (SEM), were not included.
Similarly, fundamental particle characterization techniques were not applied for dielectric particles in electric fields of gas insulation systems, and their charges have only been described once. The same work providing data on micrometer-sized metallic particles showed for four different particle materials and diameters of 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m, that the magnitudes and polarities possessed a broad scatter^46^. None of the charge formulas used for metallic particles were applied and are possibly unsuitable, given their empirical derivation, and since the dielectric particles have diameters approximately two orders of magnitude lower^2,28,30^ and differing material properties. Despite missing charge data in this context, studies in other fields have shown the charge accumulation potential of dielectric particles. This includes their charges in gas-solid fluidized beds resulting from triboelectric^47^ or ion-adsorbing mechanisms^48^. Other investigations show the charging of atmospheric aerosol particles^49^, which also happens in the processing of electrostatic powder coatings^50^.
Research gap on particle charges in gas insulation systems
A critical knowledge gap exists about particle charges in the electric fields of gas insulation systems. This is due to the absence of analyses dedicated solely to charge characterization and the lack of an experimental methodology. First, available quantitative data on electric particle charge polarities and magnitudes are limited. Charges have been reported for a heterogeneous set of millimeter-sized metallic particles. The charges of smaller particles were only reported for several metallic and dielectric materials with a diameter of 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m^46^. As a result, a systematic link between particle properties and accumulated charge is lacking, making charge-based evaluations of their impact on insulation reliability impossible. Second, the reported charges of the millimeter-sized metallic particles are based on an experimental bulk analysis or assumed to be static and fixed inputs. Theoretical and empirical models provide only deterministic magnitudes and a positive or a negative polarity. The wide variability in particle charge has only been reported for particles with diameters of 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m, making it unclear whether the particle charge is stochastic in different diameters. Third, there is no description of how particle properties affect their interactions with the electrode and influence charge accumulation. Adhesive forces, particularly pronounced at the micrometer scale, have not been measured in this context or postulated to influence charge accumulation. Therefore, the role of adhesion and its dependency on particle properties remain unquantified. Fourth, it is unclear if the particle charges are constant during motion or whether the charge interacts with the electrodes by inducing image charges. The latter can alter particle motion beyond commonly assumed forces. Lastly, the charging currents and charging mechanisms of particles in electric fields of gas insulation systems have not been described, leaving both the mechanisms of charge accumulation and the timescale of its potential impact on insulation unresolved.
New combined approach for particle charge characterization
In this work, we present a new fundamental experimental methodology to investigate the electric charges of micrometer-sized dielectric and metallic particles (see Fig. 1). It provides data on the statistics of charge magnitudes, polarities, and charging currents across different particle diameters and materials.
Metallic and dielectric particles are placed in the electric field of a laboratory-scale gas insulation setup, which creates a uniform direct current (DC) field in ambient air. The measurement of the charges accumulated in this field builds on previous work that introduced a non-intrusive in-situ technique^46^. The method combines high-speed imaging, particle tracking velocimetry, and force analysis to determine the charges and charging currents of individual particles in motion. This also includes detecting charge loss during motion and mirror charges induced in the electrodes. It also allows for measuring charging rates at electrodes at different potentials. By measuring multiple particles simultaneously, the method yields charge distributions that can be analyzed statistically. Previous work explained the minimum charge magnitudes for various particle materials^46^. However, the broad statistical distributions with high charge magnitudes and their possible origins remain unquantified. We hypothesize that these features arise from interactions between the particles and the electrode surface. At this particle size, the surface forces can dominate the body forces due to large surface-to-volume ratios. This creates short-range adhesive forces that can far exceed the gravitational force and typical inertial forces. For a particle to detach and move in an electric field, the electric force must overcome both gravity and adhesion, directly linking adhesion to charge accumulation. To quantify this relationship, we measure adhesion forces for metallic and dielectric particles using AFM. In addition, we analyze their surface topographies using scanning electron microscopy. Based on this, the relationship between adhesion, surface topography, and measured charge is investigated. Together, these results provide a fundamental characterization of particle charges and offer a new, physically plausible explanation for charge dynamics with particularly high charge magnitudes.Fig. 1A schematic overview of the experimental approach shows the particle preparation steps and measurement techniques. (a) The particle batches are sieved to obtain defined diameter ranges with distinct mean values. (b) Particles are placed on the lower electrode using a grounded spatula, ensuring exposure to a uniform electric field and preventing particle-particle interference. (c) Applying a high DC voltage to the upper electrode and grounding the lower electrode creates a uniform electric field. The particles acquire electric charges and move up and down perpetually. (d) The particle motion within the uniform field is captured with a high-speed camera with a narrow depth of field. Based on the extracted accelerations and forces for each particle, the particle charges and charging current are determined. (e) The adhesion between the particles and the electrode surface is measured using an AFM.
Results
The results provide quantitative data on the electric charges of micrometer-sized particles in a uniform DC field at two field strengths (field intensities). By examining various particle materials and diameters, we can describe their influence on charge magnitude and polarity and compare them to theoretical force-based expectations. Measuring the charging currents as a function of the particle diameter shows that the main charging mechanism can be attributed to contact electrification. We approximate the non-Gaussian charge distributions by kernel density estimations (KDE) to quantify the probability of observed charge magnitudes and compare characteristics such as the width, skewness, and peak positions. As an explanation for the charge distributions, we measure the particle-electrode adhesion. Its contribution to the charge variability is shown by comparing theoretical detachment thresholds with measured charge magnitudes, highlighting how the intrinsic material properties and particle geometry influence the particle-electrode adhesion and how it relates to the charge. Particularly high charges likely resulting from the adhesion are observed to cause a complex particle behavior in the electric field. This includes charge loss during motion and the influence of image charges on the trajectories.
Influence of the particle diameter on the electric charge and charging current
ZnO is a widespread particle material originating from environmental, urban, and industrial sources. Measuring its charge for diameters from 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m to 170 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m covers most of the described contamination particle size spectrum. Through sieving, this range was divided into seven size classes with average diameters from 12.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m to 145 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m.
At 5KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} and for the upward-moving particles (cf. Fig. 2a), only negative polarities are observed with a broad charge scatter. The whiskers of each box plot show the maximum and minimum measured charge across 500 individual data points. The lower and upper end of the box show the lower quartile (Q1) and upper quartile (Q3), and the line inside the box shows the median value. Overall, this shows that the surface area across all diameters is sufficient to hold charge for motion. Thereby, the minimum and maximum magnitudes are separated by approximately three orders of magnitude. Except for the smallest particles, the lowest charge magnitude for each size class corresponds to gravitational charge qmin, which sets the motion threshold based on the particle weight (cf. Eq. (13)). Charges below this threshold are not possible. The maximum charge magnitude strongly depends on the diameter, approximately log-linearly increasing by more than two orders of magnitude from the lowest to the highest particle diameter class. Notably, the maximum charge of each diameter class is significantly higher compared to qmin and reaches more than ten times that threshold in the smallest diameter class. When the field strength is doubled to 10KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (cf. Fig. 2b), the motion threshold and charges are slightly reduced across all particle diameter classes. Significant differences, however, are not observed, as the minimum charge magnitudes are close to the motion threshold, and the maximum charge magnitudes are also much higher compared to qmin. With regard to the maximum charge qmax describing the charge at which the electric field strength at the particle surfaces equals the breakdown strength of air, it is observed that almost all measurements fall below this value. For the higher of the two field strengths and the smallest diameter class, few particles are observed to have slightly higher charge magnitudes.
For the downward-moving particles at 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (cf. Fig. 2c), both positive and negative polarities are observed across all particle diameter classes with a broad overall scatter. Comparing them with the upward-moving particles does not show notable charge magnitude differences. Increasing the particle diameter also results in significantly higher charge magnitudes, however, with positive and negative polarity. A similar observation can be made for the downward-moving particles at a field strength of 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (cf. Fig. 2d).
Comparing the charge magnitudes of particles with the average diameters of 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 100 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m at the two field strengths shows that doubling the diameter increases the charge magnitudes by approximately an order of magnitude, while doubling the field strength has a much lower and inverse effect. This implies that the field strength influences the charge magnitude, but is secondary to the particle diameter. This, however, only holds given that enough charge for motion can be accumulated on the particle surface and that the charge surpasses the motion threshold.Fig. 2. The charge of ZnO particles with mean diameters between 12.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 145 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m at two field strengths is shown for the upward and downward motion. n gives the number of data points per diameter. (a) At a field strength of 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} , the upward-moving particles possess a broad spectrum of negative charges. (b) At 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} , the particles show slightly lower charge magnitudes while the spread slightly increases. (c) The downward-moving particles at 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} possess charges with negative and positive polarities with different magnitudes. (d) The same can be observed for the downward-moving particles at 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} .
It must be noted that both particle size and particle–electrode adhesion contribute to the observed charge distributions. However, the size variation of each diameter class produces only a modest spread. As an example for the 25 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m to 35 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m size class, the minimum charge not accounting for adhesion (cf. Eq. (13)) for the largest and smallest particles varies by a factor of approximately 2.7. This is roughly a factor of three smaller than the observed charge scatter. However, without the exact intra-bin size distribution, a strict numerical differentiation of variance between size and adhesion is not possible. Further supporting the dominating adhesive force influence is that the number of individual charge measurements is approximately an order of magnitude higher compared to the number of particles in the electric field. This means that the same particle must obtain varying electric charges.
The corresponding charging currents of the ZnO particles were determined on the basis of the net charge transfer during contact with the grounded and positive high-voltage DC (HVDC) electrodes and the respective contact times (cf. Eq. (1), (2)). The results of fifty measurements at each electrode for each diameter show a broad variability across the particle size classes. However, they are all charged in the order of milliseconds. This means that their influence on the insulation system can take effect immediately from a practical application perspective. Thereby, the contact with the lower grounded electrode results in a negative charge, and the contact with the upper positive HVDC electrode results in a positive charge. Figure 3a shows the charging currents at a field strength of 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} . The whiskers of each box plot show the maximum and minimum measured charging rate, the lower and upper ends of the box show the lower quartile (Q1) and upper quartile (Q3), and the line inside the box shows the median. It can be observed that the charging currents at both electrodes generally increase with the particle diameter. The median values of the smallest and largest particle diameters are separated by approximately one order of magnitude, while the spread results in an overlap of the extreme values. Significant differences between the two electrodes are not observable. This shows that the particles acquire net negative and net positive charges at approximately the same rate. Doubling the field strength to 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} , as shown in Fig. 3b, results in a wider charging current spread across all particle diameter classes. However, the imminent influence on the insulation from a practical perspective is not changed. While a clear field strength influence is not observable, lower minimum and higher maximum charging currents occur overall, covering almost four orders of magnitude. Simultaneously, a slight difference in the charging current magnitudes between the two electrodes can be observed. The particles are charged positively at the HVDC electrode faster than they are negatively charged at the grounded electrode. Thereby, a changed dependency on the particle size is not observed compared to the lower field strength, suggesting that, while the electric field strength influences the charging current, the dominant factor is the particle diameter.
At 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (cf. Fig. 3c), the charging current normalized by the theoretical contact area based on the Derjaguin, Tuller, and Toporov (DMT) model is approximately the same for the two electrodes (cf. Eq. (5)). This model, indicated by the Tabor parameter, extends the standard Hertz contact theory for small-scale elastic contacts by adding adhesive attraction as an external force. The power-law fit of the class medians (cf. Eq. (6)) is essentially constant with overlapping 95% confidence intervals (CI), indicating no systematic diameter dependence. At 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} (cf. Fig. 3d), the normalized current is larger at the HVDC electrode than at the grounded electrode. The fitted trend is slightly increasing with diameter, but the CI includes a zero slope and therefore does not provide statistically significant evidence for a diameter dependence. For reference, the theoretical contact-area curve is also shown. If the charging current at either electrode were scaled with the contact area, the measurements and fits would follow that curve, which they do not. Instead, the normalized currents yield an approximately constant per-area transfer rate across diameters. These observations likely indicate contact electrification being the dominant charging mechanism, which also aligns with the observation from the high-speed imaging. The latter shows that particles move up or down with one charge magnitude and polarity and only acquire the opposite polarity when in contact with an electrode. The transferred current scales with the contact area, and after normalization, the charge transfer per unit contact area is approximately invariant. Small asymmetries, such that a larger normalized current occurs at the HVDC electrode, likely indicate the influence of factors such as differences in impact dynamics or locally varying electrode surface roughness. However, the wide scatter should caution against over-interpretation.Fig. 3. The charging currents and charging currents normalized by the DMT contact area of ZnO particles are shown at two field strengths for both electrodes. (a) At 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} , the charging current magnitude generally increases with increasing particle diameter at both electrodes. (b) 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} results in a wider charging current variability, while it does not change the influence of the particle diameter. (c) The charging current normalized by the theoretical contact area is shown at 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} . The power law fit shows a reasonable confidence interval for charging at both electrodes and it does not follow the shape of the theoretical contact area curve. (d) A slightly increasing normalized charging current is observed with increasing particle diameter at 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} .
Approximation of the statistical charge distributions
To quantify the statistics of the particle charges, their distributions are approximated using a kernel density estimation (KDE) including a 90% CI (cf. Eq. (3), (6)). The number of data points per diameter class and field strength is 500. Given the observed non-Gaussian shape, a mean response-based approach with a standard deviation would not show the details of the charge characteristic. This is potentially critical because varying charges directly translate to varying influences on the insulation, such as field distortion and partial discharges driven. An approximation of the charging current distributions at the two electrodes is not made. This is due to their magnitudes being high relative to the charge magnitudes. From a practical perspective of gas insulation performance, particles are charged immediately, and no criticality difference emerges from the variability.
The KDEs in Fig. 4 show the statistical character of the ZnO charges as a function of the particle diameter class, electric field strength, and motion direction. Differentiating between the negatively charged upward-moving and mostly positively charged downward-moving particles is necessary, as the charge magnitudes between them may not be physically possible^46^. An extrapolation of the KDEs past the measured minimum and maximum charges is not shown, as extensions past these thresholds might not be physically plausible. A minimum negative charge magnitude is required for upward motion, and the maximum charge can be limited by the breakdown strength of the surrounding air or capacity of the particle.
The charge distributions of the smallest particles moving upward show a negative skewness of approximately -1, indicating a longer left tail and a concentration of charge values on the right, as shown in Fig. 4a. The relatively high kurtosis of approximately 3.5 suggests a leptokurtic distribution with a sharper and heavier tail compared to a normal distribution. This results in a higher probability for low charge magnitudes and a much lower probability for high charge magnitudes. The minimum and maximum values of the distributions at field strengths of 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} and 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} are separated by a factor of approximately ten. The downward-moving particles of the same diameter class, as shown in Fig. 4b, have a narrower distribution with lower maximum charge magnitudes. While the distribution for 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} mirrors the distribution of the upward-moving particles, the distribution at 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} is more similar to a standard normal distribution. The probabilities of the maximum charge magnitudes, however, are similar. For the largest particle diameter class, the upward charge distributions at the two field strengths maintain a similarly skewed shape. This can be seen in Fig. 4c. The high kurtosis shows that the probability for lower charge magnitudes is higher than for higher charge magnitudes. The latter, however, have much higher magnitudes compared to the smaller particles. For the downward motion shown in Fig. 4d, the distributions for both field strengths mirror the upward motion behavior with some particles crossing the 0 C line.
This means that independently of the particle size class and field strength, the highest charge magnitudes for the positive and negative polarity occur with the lowest probability, while lower magnitudes are observed more frequently. Thereby, the negative extreme values are more critical than the positive ones as they are higher. This is due to gravity, which has to be compensated by the Coulomb force during the upward motion and reduces the charge during the downward motion (cf. Eq. (13), (15)). This statement can be made with relatively high certainty, as the 90% CI clearly shows this trend.Fig. 4. The charge distributions of ZnO particles are approximated using KDEs with a CI at two field strengths. (a) The charge distribution of upward-moving particles is shown for the diameter class of 12.5 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m. (b) The charge distribution of the downward-moving particles is shown for the same particle diameter class. (c) Particles from the 145 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m diameter class show a similarly shaped charge distribution with higher charge magnitudes. (d) For the downward motion of the same diameters, similar but mirrored distribution shapes obtained.
To assess if the charge distributions of ZnO particles are representative of other particles that potentially contaminate gas insulations, further micrometer-sized particles made from different metallic and dielectric materials were analyzed at a field strength of 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} . The number of data points for each material is 500. Their properties such as the density, electric conductivity, work function, and band gap vary significantly (cf. Table 1) to evaluate the influence of the intrinsic material properties. Figure 5 shows the measured charges and the KDEs of the charge magnitudes for upward-moving particles. To show the variability, the 5^th^, 10^th^ percentile and the mean value are also shown. As this motion direction was shown to have higher charge magnitudes, it is potentially more critical to the insulation system. This shows, that the particles possess large enough surface areas to hold charge for motion across material classes.
Although the charge magnitudes differ among the materials, the skewness and kurtosis of the distributions are comparable across all materials. This means that lower charge magnitudes occur more frequently, while higher magnitudes have much lower probabilities. The behavior is similar to that of the ZnO particles and indicates that the particle material is not an explicit predictor of the charge accumulation potential and its distribution. Regardless of whether the particle is metallic or dielectric, it is observed to acquire an electric charge varying by approximately an order of magnitude. Another commonality is that the highest particle charge magnitudes occur with the lowest probability.
The \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 2\sigma$$\end{document} interval around the mean charge value shows, that it includes the bulk of the charges but that such a usually applied description is insufficient as charges occur outside this range. This makes the charge behavior a phenomenon not describable robustly by common mean-based approaches. Additionally, the interval is cut short on the right edge. It is not symmetrical around the mean value and extends further to the left as the right is limited by the minimum charge the particles have to acquire for detachment.
As already stated for ZnO particles, a quantitative differentiation between the size and adhesion variation influencing the charge scatter cannot be done. Nevertheless, size variation alone cannot account for the width of the charge distribution. Particularly, the extreme charges observed in the distribution tails are consistent with enhanced adhesive forces. We therefore conclude that adhesion is the dominant mechanism responsible whenever the rare, but exceptionally high charges occur.Fig. 5. The KDEs with 90 % CI are shown for metallic and dielectric particles at 10 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} . The charges are negative as upward-moving particles are considered. For each distribution, two percentiles, and the mean value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} are marked. Around the latter, the KDE falling in the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm 2\sigma$$\end{document} standard deviation range is shown. (a–c) The charge distributions of metallic particles are shown. (d–f) A similar charge behavior is observed for the charges of dielectric particles.
Adhesive particle-electrode interactions translating to charge variability
The measured charge variability can be linked to the variation in particle-electrode adhesion. In particular, the adhesive force must be overcome in addition to the gravitational force in order for a particle to move in the electric field. The adhesion therefore sets the detachment threshold. For upward motion, the Coulomb force must exceed the sum of the adhesive and gravitational force (cf. Eq. (12)). For downward motion, the Coulomb force must exceed the adhesive force minus the gravitational force (cf. Eq. (14)). Therefore, a variation of the adhesive forces can directly translate into charge variability of moving particles.
The adhesive force results from microscale interactions such as van der Waals forces and capillary forces, which easily exceed gravitational forces for micrometric particles. The variability in adhesion can derive from differences in the particle size, shape, surface roughness, and local topography of the electrode, as all these factors affect the area of contact between the particle and the electrode. To quantify the influence of some of these factors, we measured the adhesion of two types of particles on an electrode surface using AFM. The results for silica particles with a diameter of approximately 25 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and vanadium particles with diameters of approximately 80 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m show that adhesion does not depend on whether the particle is metallic or dielectric but that it is rather determined by particle surface characteristics and geometric aspects. For each particle type, 2,800 individual adhesion measurements were obtained in total.
Figure 6 shows the distributions of adhesive forces of irregularly shaped vanadium particles on a metallic electrode surface. A broad variation is observed both within individual particles and across multiple particles. As shown by Fig. 6a–g for a single vanadium particle measured at different electrode positions, the adhesive forces range from 110nN to 290nN. The range of minimum and maximum adhesion broadens further when characterizing multiple particles at seven positions each, as shown in Fig. 6h–k. The minimum values differ by up to 100nN, and the maximum values are separated by 290nN. The overall distribution compiled from all particles and positions displayed in Fig. 6l shows that the adhesion ranges between 65nN and 570nN, which significantly exceeds the gravitational force. Most measured adhesive forces fall between approximately ten and thirty times the gravitational force. Much higher adhesive forces above 400nN occur with a probability of only approximately 2 %.Fig. 6. Distributions of adhesive forces of vanadium particles on an electrode surface. (a–g) Adhesion forces for a single particle measured multiple times at seven locations on the electrode. (h–k) Adhesion forces for four different particles. Each particle was measured in seven different positions. (l) The overall adhesion spectrum obtained by combining the data from all particles and positions. The vertical lines compare the adhesion to the gravitational force of the particles. (m) SEM scan of the vanadium particles showing their irregular shape. The inset shows the details of the surface topography.
An SEM image of the particles reveals an irregular but similar morphology among the vanadium particles, as seen in Fig. 6m. The zoom-in of the particle surface shows how the irregular surface leads to variations in the contact area between the particle and electrode. Since adhesive forces scale with the contact area, adhesion can vary depending on the orientation of the particle on the electrode^51^. Consequently, even similarly sized particles of the same material can have different adhesive forces towards the electrode surface. As these must be overcome for the detachment from the lower electrode in addition to the much lower and, for one particle size and material constant, gravitational force (cf. Eq. (13)), it creates a varying lift-off charge threshold for the particles. At the upper electrode, the same holds except that the gravitational force supports detachment as it is oriented in the downward direction (cf. Eq. (15)). Given its much lower magnitude compared to the varying adhesion, it is the particle electrode interaction that governs the detachment and resulting charge magnitude.
Furthermore, the SEM analysis demonstrates that the irregular shape of the particles complicates the calculation of surface charges based on the measured charge magnitudes. The total surface area is difficult to determine and likely much larger than the projected area due to the surface roughness of the particle. Even precise surface area characterization techniques such as the Brunauer-Emmett-Teller method would likely not allow to describe single particles precisely enough^52^. In addition, local variations in the surface curvature give rise to non-uniform electric fields, which in turn result in a heterogeneous surface charge distribution. As a result, the calculated surface charge of such a particle would have a high uncertainty. This shows that the measured charge of individual particles serves as a more robust descriptor.
We similarly measured adhesive forces for particles of a dielectric particle material, choosing SiO_2_ microparticles as an example. As shown in Fig. 7 a to g, the adhesion of a single particle varies significantly with the contact location. Across seven positions, we measured values between 6 nN and approximately 440 nN. When extended to multiple particles, an even broader distribution is observed, as seen in Fig. 7h to k. Although the minimum values differ by approximately 10 nN, the maximum values range from 370 nN to 780 nN. The aggregated measurements reveal that high adhesion values between 300 nN and 800 nN occur only with a probability of approximately 2 %. This is, on average, much less compared to what we measured for the vanadium particles. However, the maximum values are higher. This shows the variability among different particles and the practically impossible transfer from one adhesion measurement to another. Similarly, however, is the significant exceedance of the gravitational force and the following implication for the charge threshold required for detachment.
The SEM scan of the particles reveals their nearly perfect spherical shape and smooth surface., see Fig. 7m. Nonetheless, the adhesion measurements reveal a heterogeneity that is likely ascribed to nanometric variations of surface topography and chemical composition, potentially from surface contaminations and adsorbed water.Fig. 7. Distribution of adhesive forces between SiO_2_ particles and the electrode surface. (a–g) Adhesion of a single particle measured at seven different positions on the electrode. (h–k) Adhesion of four different particles, each set at seven different positions. (l) Distribution combining the measurements of all particles and positions. Different magnitudes of the gravitational force are shown as comparison. (m) SEM micrograph showing particle shape and surface.
To investigate the relationship between adhesion and electric charge, the adhesive force spectra (cf. Fig. 6 l, 7 l) were translated to the detachment charge (cf. Eq. (13)), the particle charge required to compensate for the gravitational and adhesive forces and to enable particle motion. This translation is shown in Fig. 8a for vanadium particles and in Fig. 8b for silica particles. As adhesive forces of higher than 0 nN were measured for vanadium, the lowest detachment charge is higher than expected from only the gravitational charge qmin. Above that, the adhesion range results in a detachment charge range. For the silica particles, adhesive forces of close to 0 nN were measured, which translates to a minimum detachment threshold of approximately qmin. Above that, a detachment charge range is obtained. Comparing this with the measured upward-moving charge magnitudes shown in 8c for vanadium and in 8d for silica shows that there is a partial to full overlap. The vertical lines described as multiples of qmin indicate how much higher the charge is compared to the minimum detachment charge based only on gravity (cf. Eq. (13)).Fig. 8. The theoretical translation of the measured adhesive forces into detachment charges and a comparison with measured charges are shown. (a) For vanadium particles, the adhesive force range due to particle electrode interaction translates to a charge detachment threshold range. The measured absolute charges during the upward motion overlap partly with the theoretical detachment threshold. (b) Similarly, the comparison is made for SiO_2_ particles. The adhesion spectrum translates into a charge detachment threshold spectrum. The measured absolute charges overlap with the adhesion-caused detachment range at the lower end of the spectrum.
The measured charge spectra do not overlap perfectly with the calculated gravitation- and adhesion-based detachment threshold. While vanadium particle charges were observed that are lower than those expected based on the measured minimum adhesion, no charges were found to exceed the measured adhesion for both particle materials. The former observation can be attributed to high charging rates repelling particles from the electrode before they come into full contact, which would be similar to a resting position that results in the highest adhesion. The latter observation highlights the decisive role of adhesion resulting from particle-electrode interaction. As the particle capacity might not be large enough to compensate for the full detachment threshold spectrum, they cannot acquire charges high enough for motion. An opposite explanation might be that particle charges at the upper detachment threshold spectrum might cause charge loss, such as partial discharges, which discharge the particles before they can move.
However, the comparison indicates that the adhesive forces could lead to charge magnitudes even higher than those measured. Since the charge behavior is stochastic with low probabilities for higher charges, they might occur if the particles are exposed to the electric field longer. This suggests that, as adhesion governs the detachment, it is also the driving factor in rare but extreme particle charges.
Charge phenomena due to high adhesive forces
The detailed analysis of particle trajectories reveals that particle charges cannot reach arbitrarily high levels. This is shown in Fig. 9a for upward-moving ZnO particles from the 145 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m size class at a field strength of 5 KV cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{-1}$$\end{document} . The acceleration and therefore charge remains constant for most moving particles during their motion for charges lower than qmax. This is associated with the lower spectrum of the adhesive forces. However, a small fraction of particles exhibit high initial charges caused by high adhesive forces. During the particle motion, their acceleration and charge is observed to decrease. The initial charge of each particle is higher than qmax. This can be attributed to a gradual charge loss, which reduces the upward motion-driving Coulomb force. Such a motion pattern is only observed for upward-moving negatively charged particles where such electron detachment is possible. The charge detachment rate increases with an increasing initial charge magnitude. Thereby, the timescale is much longer compared to other charge loss mechanisms, such as more rapid partial discharges.
Moreover, for these particularly high charges, an increase in acceleration is observed right after the detachment. A zoom-in of this is shown in Fig. 9b for a particle. This phenomenon is likely attributable to the presence of mirror charges induced within the electrode, which generate an additional force directed to the electrode surface (cf. Eq. (11)). This force diminishes to a negligible magnitude after the initial few hundred micrometers, at which point the field detachment results in an acceleration decrease.
This shows that while most particles have trajectories based on constant charges, high adhesive forces can induce a complex and position-dependent motion behavior. Given the low probability of the highest measured charges, this occurs only on a rare basis. As a result, the overall particle charge characterization is only slightly affected by this.Fig. 9. The trajectorial data shows that high adhesive forces can cause charge loss during the particle motion and induce mirror charges in the electrodes. (a) Low accelerations during the gap crossing are approximately constant corresponding to constant charge magnitudes and low adhesive forces. Few particularly high charges are observed attributed to high adhesive forces. They decrease as electron loss appears to takes place. The charge for the maximum acceleration is noted for these particle. (b) A detailed view of the measured accelerations after detachment shows the influence of induced mirror charges.
Discussion
Particle charge dynamics in the context of the gas insulation literature
This study contributes to the quantitative characterization of the electric charges of micrometer-sized particles in the context of gas insulation systems. Unlike previous work relying on empirical calculations and assumed static particle charges^10,11,27^, the used method allows for a direct quantification of the dynamic charge characteristics. This includes the magnitudes, polarities, and charging currents of individual particles. Given that the particle motion can be captured by a high-speed camera and that all forces are quantitatively described, the approach is transferable to other particle types, electric field configurations, and electric stresses. As the observations and description of the particle behavior are consistent across different particle materials, diameters, and field strengths, it is not expected that fundamental differences will arise for varying conditions.
The results show non-Gaussian charge distributions spanning up to more than an order of magnitude for the individual particle materials, diameters, and field strengths. Thereby, extreme charge magnitudes occur with the lowest probability. At the same time, these potentially have the strongest effect on the insulation system as they result in the highest field distortion and are most likely to cause partial discharges. This contradicts conventional deterministic assumptions for similar experimental setting. There, fixed particle charge magnitudes and polarities were used for motion and breakdown studies^10,11,14,17^, which could mean that actual charges were underestimated. A direct transfer of findings from other fields such as the broader aerosol and triboelectric literature on particle charges^47,48^ that sometimes explicitly mentions the charge stochasticity^53^ is not possible because of differences in regard to influencing conditions such as the pressure and temperature of the surrounding gas, electric field characteristics, and electrode properties such as roughness or conductivity.
Within the scatter of the charge, one observable pattern, however, is qualitatively in agreement with the gas insulation literature on particle charges, which states that the charge magnitude increases with the particle diameter^16,17^. Since the empirical formulas used in the literature were only applied to much larger metallic particles, their transferability to the investigated much smaller particles of dielectric and metallic materials is likely not possible.
Charging mechanism of particles in electric fields of gas insulations
Contact electrification is likely to be the dominant charging mechanism at both electrodes in the chosen experimental conditions. The charging current scales with the contact area, while the per-area transfer rate is approximately independent of particle diameter. This indicates that the particle charge is transferred during brief electrode contacts rather than gradual accumulation from electrons or ions present in the gap. The measured charging times are on the order of milliseconds, and the observed polarity change occurs only when particles physically contact an electrode, which is consistent with contact area-controlled charge transfer. From a practical insulation operation perspective, these results demonstrate that particles can acquire substantial charge essentially instantaneously upon electrode contact.
The physical processes enabling contact electrification depend strongly on aspects such as the electronic transport properties, the surface chemistry, and intrinsic material properties (cf. Table 1). For metallic particles, charge transfer occurs through rapid electron exchange and Fermi level equilibration during contact. The metal electrode and the metal particle form an electrical micro-junction, enabling fast equilibration and resulting in nearly uniform surface and bulk charging. In contrast, semiconductors and insulators exhibit low bulk conductivity on the timescale of a contact, and charge exchange can therefore be assumed to be inherently localized. ZnO behaves effectively like an electrical insulator during the short contact events, with charge primarily transferred to or from surface-localized states and traps. Adsorbed layers of water present in the experimental conditions and ubiquitous contaminating molecules provide mobile ions at the interface, which can strongly modulate the microscopic transfer pathway. This increases the charge transfer variability and enables ion-mediated transfer in addition to electron trapping. Together, these material-dependent processes explain why contact electrification yields different phenomenologies for metallic and for dielectric materials.
Assessing the potential contribution of free electrons or ions in the electrode gap indicates that such species are unlikely to be the principal contributors of the observed particle charges under the tested conditions. The measured particle charges range from about 1 fC to about 10 pC. The latter corresponds to roughly 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^3$$\end{document} to 6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\times$$\end{document} 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^7$$\end{document} elementary charges per particle. The applied uniform field strengths remain well below the air breakdown threshold for uniform gaps and below the field strengths required for significant field ionization or sustained gas conduction without sharp emitters. In addition, the high-speed imaging shows that the charge polarity and magnitude are established during contact events rather than through a continuous charging current through the gas. Taken together, the charge magnitudes and the gap field strength make it unlikely that a dense population of free electrons or gas ions in the bulk can deliver the measured net charges to individual particles. Local ion or electron contributions, however, can be present at microscopic contact interfaces or in adsorbed water films and may assist or bias the contact transfer process.
A distinction must be made between metallic and dielectric particles regarding the charge distribution after their detachment from the electrodes. For conductive particles, the charge can redistribute quickly over the particle surface and bulk due to the high electric conductivity. For the practically non-conducting particles, the deposited charge can be assumed to remain locally confined. The resulting local surface field can therefore be much higher for uneven surface topographies. As a result, the charging behavior becomes highly sensitive to contact geometry and surface trap state density. With the local surface field approaching the air breakdown limit at asperities, even when the macroscopic applied field is lower than the breakdown field of the surrounding gas, electron field detachment or localized discharges can be triggered. This reduces the particle charge during the motion. The observations of rare high charge events that decay during motion are consistent with such localized field-driven relaxation mechanisms.
Collectively, these aspects show that the charging mechanisms of the particles are complex and influenced by various parameters. This leads to a variability in charging rate, transfer efficiency, charge persistence, and maximum charge accumulation capacity. However, variability in the charging rate alone is unlikely to be the primary origin of the measured charge distributions. Instead, adhesion is postulated to be the dominant cause because the threshold charge required for motion is set by a force balance rather than by the specific charge transfer mechanism.
For upward motion, the Coulomb force, directly proportional to the charge, must overcome both the particle’s weight and the adhesive force at the contact. During downward motion, adhesion remains the opposing force while gravity assists it. A polarity reversal at the upper electrode is therefore implausible because a particle would detach and fall once its Coulomb force drops below the gravitational force, without requiring a sign change of the particle. The appearance of broad charge and adhesion distributions for metallic and dielectric particles, despite their different microscopic charging physics, supports adhesion as the common and dominant factor. The net charge magnitude ultimately determines whether motion occurs, while the charging mechanism governs how quickly that magnitude is reached.
Adhesive force influencing particle behavior and translating to the charge
The adhesive interactions between particles and electrodes are shown to be strongly pronounced in a laboratory-scale gas insulation setup. They significantly exceed the gravitational force by a factor of thirty to fourty. The adhesion was quantified using an AFM, with values reaching up to several hundred nanonewtons, which is in agreement with AFM measurements in other contexts^54,55^. Quantifying this force shows that existing particle charge models that only consider gravitational detachment thresholds significantly underestimate the real charge magnitudes^41^. Given that the previously studied particles are larger, however, this underestimation might be not as critical as for the investigated particles. In general, the adhesive forces should still be incorporated in the particle motion equation. This was not done in the currently available models^16,56^, and neglecting them would require an experimental validation of them being significantly lower than the other forces.
Interestingly, irregularly shaped metallic vanadium particles and nearly perfectly spherical dielectric silica particles show a similar adhesive character in both magnitude and variability. Thereby, the adhesive forces are observed to vary for a single particle across the same and different positions on the electrode surface, and it also varies among different particles. As these forces increase the detachment threshold for particle motion, they can translate directly to the charge magnitude and cause variability. Since they are significantly higher than the gravitational force, they offer a new explanation for the occurrence of particularly high charge magnitudes, which cause effects such as electron detachment and mirror charges. These findings imply that intrinsic particle material properties such as electric conductivity, permittivity, work function, and band gap are less decisive for the charge accumulation process than previously thought^26^. Similarly, the particle geometry does not seem to play the most important role. This implies that instead, nanoscale surface characteristics, contamination, humidity, and electrode roughness may be the primary factors influencing adhesion and, therefore the charge accumulation.
At a microscopic level, the measured adhesion force is defined by the extent of the contact, a quantity often termed ’the true contact area’, and the strength of the adhesion per unit area. The large variability of adhesion forces from the AFM measurements thus demonstrates that, even though different interfaces may have different adhesion strength, the experimental behavior is largely dominated by variability originating from nanoscale variations of contact area between the particle and the substrate.
Consequently, introducing the adhesion between particles and electrodes to charge studies and quantifying the particle-electrode interaction provides a new perspective, as it might be the governing mechanism behind setting the particle charge magnitude. It also suggests that the electrode surfaces inside gas insulation systems may alter the particle charges due to variations in contamination or roughness. This would result in a range of outcomes, including alterations in field distortion, partial discharges, and breakdown initiation. As the adhesion variability, however, is not directly predictable and magnitudes can only be estimated, only measurements can quantify the adhesive forces. This, however, is specific to one particle material and diameter, one electrode material and surface finish, and is dependent on ambient conditions such as the humidity. For further investigations of particles in high-voltage insulations, particularly in specific insulation gases at higher pressures, AFM may become an important characterization technique.
Complex particle behavior through electron loss and mirror charges
A novel aspect is the experimental observation of electron field detachment and mirror charge forces. These phenomena occur only occasionally, as they require high charge magnitudes which appear with the lowest probability. Additionally, electron detachment can only be seen for negatively charged upward-moving particles. Observing this shows that a particle charge can only be determined when the entire motion path between the electrodes is quantified. Both phenomena have not been described in the gas insulation literature that focuses on particles, but are well-documented in related fields. For instance, surface charge relaxation through field emission has been discussed in high-vacuum systems^57^. To what degree this potentially interacts with the insulation gas is not directly accessible, but should be considered when investigating aspects such as the presence of free electrons, field distortion, or calculating particle motion. Regarding the latter, the presence of mirror charges induced in the electrodes adds a force term to the particle motion equation. This additional term has been described in contexts outside gas insulations, such as aerosol physics^58^. The former shows that even constant charge magnitudes can lead to nonlinear motion over time and position. While the additional force term and the nonlinearity of particle motion have not been explicitly described in the context of gas insulations, some analyses accounted for the image charge force^16,41^. There, the motion-driving Coulomb force based on an approximated constant charge was multiplied by a factor of 0.832 to 0.836. This constant force reduction for the entire motion pattern might hold for one specific particle position, but should rather be expressed as a function of the charge magnitude and location when calculating the trajectories (cf. Eq. (11)). Particularly, since it acts opposite to the motion direction for the first half of the electrode gap crossing, which reverses for the second half. From a practical perspective of gas insulations, the influence of the mirror charge can be neglected. This is because their effect on the particle motion is only relevant within a few particle diameters from the electrode surface and therefore does not alter the overall charge dynamics across the insulation gap.
Limitations regarding power grid gas insulation operation conditions
The experiments presented in this work were conducted in ambient air at atmospheric pressure and in a uniform electric DC field. Large-scale gas-insulated systems employed in the electrical power grid, however, typically operate with specialized insulating gases at elevated pressures, various electrode materials, and under various electrical stress types. These factors are known to influence particle charging mechanisms, charge relaxation, and adhesion forces. Consequently, the absolute charge magnitudes, adhesion values, and charge loss thresholds reported here cannot be directly transferred to application-relevant insulation systems. The present study, therefore, primarily establishes a fundamental experimental methodology and reveals key statistical trends in particle charging and adhesion under well-controlled laboratory conditions. These results provide a physically grounded basis for future investigations, but quantitative extrapolation to high-pressure gas insulation systems will require dedicated measurements in representative gas environments and operating regimes.
Conclusion
This work provides a new fundamental experimental methodology and quantitative analysis of the electric charges of micrometer-sized particles in a uniform DC field. Using a specifically developed non-intrusive in-situ tracking velocimetry approach, the charge magnitudes, polarities, and charging currents of individual particles were measured. Thereby, metallic and dielectric particle materials were covered with diameters representative of real-world contamination in gas insulation systems. As the method is broadly applicable, it enables future data acquisition across a wider range of particle characteristics, electrode configurations, gas types, and field strengths.
The charges of particles with diameters between 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 170 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m range from approximately 1 fC to 10 pC in both polarities. Charging currents range from picoamperes to nanoamperes and scale with the particle diameter, pointing to contact electrification as the dominant charging mechanism. Thereby, no difference in the charge accumulation potential was observed between dielectric and metallic particles. Across all materials, larger diameters resulted in higher maximum charge magnitudes, whereas the influence of the electric field strength on the charge is significantly less pronounced. For a given particle material, diameter range, and field strength, a substantial variation in charge was observed. It covered up to more than an order of magnitude in both polarities. The skewness and kurtosis of the underlying distribution indicate that low charges dominate statistically, but rare high-charge events may pose the greatest risk to insulation performance. This shows that electric particle charges should be described probabilistically. The particularly high charges in the distribution are observed to cause electron detachment and induce mirror charges in the electrodes. This challenges the assumption of constant particle charges during motion and requires adding a new force term to the motion equation.
The charge variability is closely tied to statistical adhesive particle-electrode interactions, where measured adhesive forces, reaching up to multiple hundred nanonewtons, far exceed gravitational forces. The adhesion influences the minimum detachment charge threshold for particle motion in the electric field. Adhesion spectra measured for vanadium and silica particles provide a mechanistic and qualitative support for the transferability of the adhesion variability to other particle materials with diameters on the micrometer scale. This demonstrates that AFM-based characterization of particle-electrode interactions is crucial for understanding particle charges. Especially, since particle characteristics such as the material and shape are not sufficient descriptors for the adhesive behavior.
Outlook
The new experimental methodology on the electric charges of micrometer-sized particles and their quantification raises so far unaddressed questions about their influence on the performance of gas insulation systems. First, dedicated studies should establish how the particle charge influences performance-diminishing effects such as partial discharges or gas breakdowns. This requires experiments in which systems are thoroughly cleaned to a baseline and then tested after a single particle diameter is introduced. These test should investigate how the charge, motion, and performance reduction are connected. This should be performed under real-world insulation operation conditions with gases, pressures, electric field types, and electrical stresses used for power grid components. Second, the suitability of existing metallic particle traps to capture micrometer-sized dielectric particles should be investigated. The trapping effectiveness depends on charge-driven particle motion, which is similar for all particle materials and diameters. The dielectric particles may either be collected or remain within the electric fields or sticking to the surface of the insulation systems. From these, they may unpredictably lift-off with particularly high charges. Third, measuring the criticality and effectiveness of capturing them could form the basis for defining cleanliness and filtration specifications so that all particles above the critical diameter are removed or prevented from reaching critical electric field regions. Fourth, the economic trade-offs resulting from this must be evaluated. Introducing increased cleanliness standards to the gas insulation manufacturing and assembly will increase their costs. These increased costs should be compared with the reduced design safety margins, lower material costs, smaller insulation volumes, and lower equipment footprints. A cost-benefit analysis would determine whether the higher production costs related to the particle removal are justified.
Methods
This study combines two experimental techniques and a theoretical physics-based model to characterize particle charge dynamics in a uniform HVDC electric field in ambient air. It focused on particle materials and diameters that have commonly been reported across all dust types and potentially contaminate gas insulations of electric power grid components. First, the particle charges and their trajectories were quantified using a specifically developed particle tracking velocimetry approach. The charging currents of individual particles could be determined based on this. To explain the observed particle charges and complex motion patterns, a theoretical force-based description is provided. This includes the introduction of previously neglected mirror charge and adhesive forces. Thereby, the latter is essential in explaining the observed charge variability and is measured using an AFM.
Particle materials, sizes, origins, and intrinsic properties
The particle materials under investigation were selected dielectric and metallic materials commonly reported in the natural, urban, and industrial dust^1,2,24,25^. Chromium (Sigma-Aldrich, USA) occurs in urban and industrial dust. Copper (Sigma-Aldrich, USA) is part of the urban, but particularly the industrial dust. Vanadium (Sigma-Aldrich, USA) can mostly be found in environmental, sometimes in industrial dust. Zinc oxide (Carl Roth, Switzerland) occurs in environmental and industrial dust. Alumina (Thermo Scientific, Germany) frequently occurs in environmental, urban, and industrial dust. Tin oxide (Carl Roth, Switzerland) is part of urban and mostly industrial dust. Silica (Kromasil, Sweden) can mostly be found in environmental and industrial dust. While this covered only a small fraction of dust particle materials, they possessed a wide range of intrinsic physical material properties. In addition, their diameters fell within the range of most frequently described sizes in dust studies considering different particle origins. This is intended to ensure that the results are directly applicable to real-world scenarios.
It must be noted, however, that practical limitations constrained the selection of materials and size classes, specifically by the commercial availability of continuous diameter ranges for the materials. In most cases, producing or sourcing certain sizes and size classes was prohibitively expensive. As a result, a complete factorial experimental design regarding the material and diameter was not attainable. This resembles the real-world variability in dust particle materials and sizes, where uniform property sets are precluded under any circumstances. Despite this, should the charge characteristics of different materials or diameters be of interest, the approach can be used for their characterization due to its universal applicability.
Table 1 shows the investigated dust particle materials and their propertiese^59^. For ZnO, where multiple size classes were obtainable, size-dependent charging behavior was explored. For the other materials, single or more limited size fractions still contribute valuable comparative data, especially for the middle of the reported dust particle diameters. The size ranges were obtained by sieving (Micro Sieve, Bel-Art Scienceware) the particle batches from the manufacturers. This took approximately 15 min on a vibrating plate (Gipsrüttler Hertz 80, Sirio Dental) for each material.Table 1. Overview of the investigated particle materials with their diameter ranges and physical properties. The latter were taken from the PubChem materials data base.TypeMaterialSize / \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} mDensity / g/cm \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{3}$$\end{document} Conductivity / S/mWork function / eVBand gap / eVDielectricZnO1–255.6 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 10^{-13}$$\end{document} 5.33.325–3535–4545–6060–8080–120120–170SnO \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} 1–256.45 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 10^{-5}$$\end{document} 4.72.5Al \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} O \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_3$$\end{document} 25–353.97 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 10^{-14}$$\end{document} 4.38.8SiO \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$_2$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 25$$\end{document} 2.2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\times 10^{-17}$$\end{document} 5.09.0V \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 80$$\end{document} 6.11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.0\times 10^{6}$$\end{document} 4.3–MetallicCr25–357.19 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$7.9\times 10^{6}$$\end{document} 4.5–Cu25–358.96 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5.96\times 10^{7}$$\end{document} 4.7–
Electric charge measurement using tracking velocimetry
The measurement of the charges was performed using an in-situ, non-intrusive technique. It is based on high-speed imaging, which is evaluated using automated particle tracking velocimetry, as detailed in prior work^46^.
Two horizontally aligned Rogowski-profiled electrodes with a diameter of 80 mm each were used to create a uniform electric field with vertically oriented field lines. The measured average surface roughness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_a$$\end{document} was \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(0.9 \pm 0.1)\ \mu \text {m}$$\end{document} (VK-X260K, Keyence) and the gap between the electrodes was set to (10.0000 ± 0.0005) mm using a gauge block. The lower electrode was grounded, while a high positive DC voltage of (5 ± 0.05) kV and (10 ± 0.05) kV was applied to the upper. Therefore, a self-built two-stage Greinacher cascade was used, which was fed by a transformer connected to a manually operated DC source. The setup was enclosed in a Plexiglas safety box to eliminate air drafts potentially disturbing the particle motion. The air inside the box was at ambient pressure at a temperature of (21 ± 2) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^\circ$$\end{document} C, and a relative humidity of (55 ± 3) %.
Three focused halogen lamps illuminated the electrode gap for particle visibility. A high-speed camera (Fastcam SA-Z, Photron) with a 100 mm macro lens (AT-X M100 PRO D Macro, Tokina) captured particle motion at 20,000 frames per second. The lens was adjusted for a narrow depth of field of approximately 200 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m, focusing on moving particles in the central vertical plane of the electric field.
A grounded metal spatula was used to place tens of particles spatially separated on the lower grounded electrode within the depth of field of the high-speed camera. This neutralized any initial particle charge. The particles remained in contact with the grounded surface for several minutes before the electric field was applied. The Greinacher cascade reached the target voltage within milliseconds, after which the camera was triggered to capture the steady-state particle motion, which includes upward and downward motion. Only individually moving particles were tracked in the high-speed video to ensure statistical independence. Thereby, the previously specifically developed tracking velocimetry method^46^ was used, relying on sub-pixel accuracy to ensure a reliable detection of particles in the single shots. From the positions on the consecutive frames, their trajectories were reconstructed. This allowed to quantify their position, velocity, and acceleration as a function of time and position between the electrodes. Based on this data and the motion equation, the electric charges were calculated, which is shown in Fig. 10. A total of 500 individual particle charges was obtained for each experimental setting. This number is approximately an order of magnitude higher than the number of particles. Due to the perpetual up and down motion of the particles, the charge of each is measured for each up and each down movement. After each experiment, the electrodes and insight of the safety box were cleaned using wipes (Science Precision Wipes, Kimtech) and a solvent (Acetone, Rotisolv, GC Ultra Grade, Carl Roth).Fig. 10. The key steps of the particle charge measurement^46^ are shown. (a) The particle just detached from the lower electrode after the electric field was applied. (b) The particle moves upward due to its accumulated charge, which creates an upward-oriented electric force. (c) The particles are partially blurred on the high-speed videos capturing their motion. (d) The sub-pixel particle detection allows for the detection of their centers despite the relatively low image resolution. (e) The center is tracked along the motion from the lower to the upper electrode. Based on the extracted acceleration, the charge is calculated.
Charging current and mechanism characterization at ground and high-voltage electrode
The charging currents of the particles at the grounded and high-voltage electrodes were determined based on the result of the tracking velocimetry algorithm. For the ZnO particles across all diameter ranges, the measured upward charge, downward charge, and contact times with the grounded and HVDC electrode were used to calculate the charging current. The charge magnitudes and polarities of the upward and downward motion were automatically quantified by the particle tracking algorithm, while the contact times with the electrodes were manually extracted from the high-speed videos.
For a particle i, the transferred charge was calculated as the difference between charge before and after contact with an electrode. This value was then divided by the contact time at the respective electrode. For the charging current at the lower ground electrode \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{q}_{\text {grounded}}^{(i)}$$\end{document} , this was done as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{q}_{\text {grounded}}^{(i)} = \frac{q_{\text {down}}^{(i)} - q_{\text {up}}^{(i)}}{t_{\text {grounded}}^{(i)}} \end{aligned}$$\end{document}and for the charging current at the upper HVDC electrode \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{q}_{\text {HVDC}}^{(i)}$$\end{document} as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{q}_{\text {HVDC}}^{(i)} = \frac{q_{\text {up}}^{(i)} - q_{\text {down}}^{(i)}}{t_{\text {HVDC}}^{(i)}}. \end{aligned}$$\end{document}Here, the charge during the upward motion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{\text {up}}^{(i)}$$\end{document} , the charge during the downward motion \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{\text {down}}^{(i)}$$\end{document} , and the respective contact times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\text {grounded}}^{(i)}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{\text {HVDC}}^{(i)}$$\end{document} were sampled from the measurements. This method captured the stochastic nature of the charging process without assuming any specific distribution type or shape.
To investigate if contact electrification is the dominant charging mechanism, the measured charging currents are normalized by the theoretical contact area predicted for a sphere of radius r that is indented by a fixed depth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} by the DMT adhesion theory. This model describes small-scale elastic contacts by extending the Hertz contact model through incorporating adhesive forces that act outside the actual contact area.
The contact radius for a particle with indentation depth \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} is calculated as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} a = \sqrt{r\delta }, \end{aligned}$$\end{document}and the corresponding contact area is
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_{\textrm{contact}} = \pi a^{2} = \pi R \delta , \end{aligned}$$\end{document}where r is the particle radius and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta$$\end{document} is the indentation depth. For the calculation of the contact area across the diameters, the mean diameter of size class is chosen. Based on this, the normalized charging rate can be calculated as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \tilde{I} = \frac{I}{A_{\textrm{contact}}}, \end{aligned}$$\end{document}where I is the absolute measured charging rate. This is applied to the charging rate at the lower ground and upper high-voltage DC electrode. The median of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{I}$$\end{document} within each size class is calculated to fit a power law in log–log space using ordinary least squares. The fit model is
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \log _{10}\!\big (\textrm{median}(\tilde{I})\big ) = \beta + \alpha \log _{10}(d), \end{aligned}$$\end{document}which corresponds to
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{median}(\tilde{I}) \propto d^{\alpha }, \end{aligned}$$\end{document}where d is the particle diameter, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} is the fitted power-law exponent, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta$$\end{document} is the intercept. The least squares estimator for the slope \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document} is calculated as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\alpha } = \frac{\displaystyle \sum _{i=1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\displaystyle \sum _{i=1}^{n}(x_{i}-\bar{x})^{2}}, \end{aligned}$$\end{document}with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_{i}=\log _{10}(d_{i})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y_{i}=\log _{10}\!\big (\textrm{median}(\tilde{I}_{i})\big )$$\end{document} , and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{x},\bar{y}$$\end{document} their sample means. A 95% confidence interval for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\alpha }$$\end{document} is computed from the standard error as followed.
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{SE}(\hat{\alpha }) = \sqrt{\frac{\hat{\sigma }^{2}}{\displaystyle \sum _{i}(x_{i}-\bar{x})^{2}}}, \end{aligned}$$\end{document} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{\sigma }^{2} = \frac{1}{n-2}\displaystyle \sum _{i}\big (y_{i}-\hat{\beta }-\hat{\alpha }x_{i}\big )^{2}, \end{aligned}$$\end{document}and the CI is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\alpha } \pm t_{n-2,0.975}\,\textrm{SE}(\hat{\alpha })$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_{n-2,0.975}$$\end{document} is the Student–t quartile for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-2$$\end{document} degrees of freedom. This fit allow visualize and compare the normalized charging rate as a function of the particle diameter against the theoretical contact area. If the fit shows a constant normalized charging current as a function of the particle diameter with a reasonably narrow confidence interval, contact electrification can be assumed to be the driving charging mechanism.
Approximating charge distributions using kernel density estimation
To estimate the distribution of the measured charges for a specific particle material, diameter class, and electric field strength, a kernel density estimation is used. This provides a smooth, continuous approximation of the probability density function without assuming a predefined distribution model.
The set of n measured charges of upward-moving or downward-moving particles can be written as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{x_1, x_2, \dots , x_n\}$$\end{document} . For these, the kernel density estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f}(x)$$\end{document} at a point x is defined as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \hat{f}(x) = \frac{1}{n h} \sum _{i=1}^{n} K\left( \frac{x - x_i}{h} \right) , \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h > 0$$\end{document} is the bandwidth, a smoothing parameter that controls the width of the kernel, and K(u) the kernel function, a symmetric function that integrates to one. An Epanechnikov kernel was used as it efficiently minimizes the mean squared error and accounts for local variations. It is defined as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} K(u) = {\left\{ \begin{array}{ll} \frac{3}{4}(1 - u^2), & \text {if } |u| \le 1, \\ 0, & \text {otherwise}. \end{array}\right. } \end{aligned}$$\end{document}The kernel density estimation was calculated for the charge data of one particle material, size, and field strength. Thereby, it was differentiated between particles moving upward and downward. The interval for all estimations was set to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ \min (x_i), \max (x_i)\right]$$\end{document} as an extension beyond the observed data range may physically not be feasible. This, however, does not mean, that higher and lower charges values cannot occur.
To assess the uncertainty of the estimation, a confidence interval was computed using bootstrap resampling. From the original set of n charge values, 1,000 bootstrap samples were generated by random sampling with replacement. For each, a separate kernel density estimate \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{f}^{(b)}(x)$$\end{document} was calculated using the same parameters as for the original estimation.
This resulted in a collection of density estimates
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \hat{f}^{(1)}(x), \hat{f}^{(2)}(x), \dots , \hat{f}^{(B)}(x) \right\} \end{aligned}$$\end{document}evaluated over the same interval. At each evaluation point x, the empirical 5^th^ and 95^th^ percentiles across all bootstrap estimates were taken to define a 90% confidence interval as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {CI}_{90\%}(x) = \left[ \hat{f}_{5\%}(x), \hat{f}_{95\%}(x) \right] . \end{aligned}$$\end{document}Forces, motion equation, and charge limits of the particles
Multiple forces act on particles in the set uniform HVDC field. These include the commonly mentioned gravitational force, the electric force due to the electric field, and the viscous drag from the surrounding air. Newly introduced in the context of particles in gas insulation systems are the electrostatic mirror charge force, which is particularly pronounced near the electrode surface, and the adhesive force when a particle is in contact with an electrode. Together, the forces describe the particle motion and can be used to obtain the charge threshold for the motion to occur.
The gravitational force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F}_{g}$$\end{document} acts vertically downward and is given by
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_g = m_p g, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_p$$\end{document} is the particle mass and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g$$\end{document} is gravitational acceleration.
The electric force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F}_{e}$$\end{document} driving the particle motion is given by
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_e = q E, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q$$\end{document} is the particle charge and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${E}$$\end{document} the electric field strength. In the chosen setup, the electric field lines are oriented vertically meaning that this force points upward for negatively charged particles and downward for positively charged particles.
The viscous drag force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F}_{d}$$\end{document} opposing the motion is described by Stokes’ law as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_d = 6 \pi \mu r v, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} is the dynamic viscosity of the air, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document} is the particle radius, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v$$\end{document} is the vertical velocity. This linear dependence on the velocity applies under low Reynolds number conditions, which are typical for micrometer-sized particles in air.
The electrostatic mirror charge force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F}_{m}$$\end{document} becomes relevant when a charged particle is close to a conductive electrode. It results from the interaction between the charge of the particle and its induced image charge in the electrode and is given by
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F_m = \frac{1}{16\pi \varepsilon _0} (\frac{q}{x})^2, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _0$$\end{document} is the vacuum permittivity and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x$$\end{document} is the distance of the particle to the electrode. Although often negligible at lower to moderate charge levels and distances larger than a few times the particle diameter, this force can significantly alter near-surface trajectories and result in non-linear motion despite a constant charge.
The net force acting on the particle during the motion between the electrodes is the sum of all above forces. Applying Newton’s second law gives the motion equation as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} m_p a = q E + m_p g + 6 \pi \mu R v + \frac{1}{16\pi \varepsilon _0} (\frac{q}{x})^2, \end{aligned}$$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a$$\end{document} is the particle acceleration. The sign of the forces then depends on the motion direction of the particles. Potentially occurring charge changes during the motion must thereby be expressed as a function of influencing parameter such as the charge itself or the electric field strength.
Before the particle is moving in the electric field and the motion equation applies, it is in contact with one of the two electrodes. In this case, a force newly introduced in this context must be considered. Due to particle-electrode interactions which can be strongly pronounced at the investigated diameters, an adhesive force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F_a$$\end{document} occurs. It is oriented toward the electrode surface and makes the particle stick. Thereby, it increases the electric force required for detachment and motion to occur. Consequently, it directly translates to an increase of the particle charge. For describing this detachment threshold, however, a differentiation must be made between the lower and upper electrode.
At the lower electrode, the particle must be lifted upwards against both gravity and adhesion. The electric force must overcome both, which leads to
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| qE \right| > m_p g + F_a. \end{aligned}$$\end{document}The absolute value is written for the electric forces due to the setup specifics in which a negative charge is required for upward motion. For the forces, only their magnitudes are of relevance as the point in the same direction in this case. Solving this for the required particle detachment charge magnitude \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left| q_{d,\text {lower}} \right|$$\end{document} results in
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left| q_{d,\text {lower}} \right| > \frac{F_a + m_p g}{E}. \end{aligned}$$\end{document}The minimum detachment charge is referred to as qmin and is obtained for an adhesive force of 0 N. It is also called the gravitational charge as motion occurs just by overcoming the weight of the particles and sets the particle motion threshold.
At the upper electrode, gravity assists the particle detachment from the electrode surface. The electric force and gravitational force both act in the downward direction whereas the adhesive force is oriented upward. Consequently, the gravitational force and electric force must exceed the adhesive force. This is expressed as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \ qE > F_a - m_p g. \end{aligned}$$\end{document}The electric force is not denoted as an absolute value as the charge for the downward motion is positive. The direction of the other forces is accounted for by subtracting the gravitational force from the adhesion. Solving this for the particle detachment charge magnitude \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{d,\text {upper}}$$\end{document} gives
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_{d,\text {upper}} > \frac{F_a - m_p g}{E}. \end{aligned}$$\end{document}These equations define the threshold for detachment from each electrode and particle motion to occur. Once the particle accumulates sufficient charge, it begins to move in the electric field, meaning that the detachment charge is identical to the motion charge, should no effects like electron field detachment or additional electron accumulation occur. The previously stated Eq. (11) then describes the motion. It must be noted, however, that these equations do not describe the charge acquisition process. Particularly, the motion equation only applies if enough charge accumulation occurs.
To obtain a theoretical limit for the maximum particle charge qmax, the breakdown field strength of air is used. The surface field strength Esurf is calculated as the sum of the charge-induced field and the enhancement of the background field due to relative permittivity of the particle. Imposing that the surface field strength does not exceed the the air breakdown strength, gives
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} q_{\max }=4\pi \varepsilon _0\,r^{2}\bigl (E_{\textrm{bd}}-f_{\textrm{d}}\,E_{\textrm{}}\bigr ), \end{aligned}$$\end{document}where the dielectric pole factor fd is calculated as
\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f_{\textrm{d}}=\frac{3(\varepsilon _{r}-1)}{\varepsilon _{r}+2} \end{aligned}$$\end{document}with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{0}$$\end{document} is the vacuum permittivity of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$8.8542\times 10^{-12}\ \mathrm {F\,m^{-1}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r$$\end{document} the particle radius, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\textrm{bd}}$$\end{document} the breakdown field strength of air at approximately \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2.5\times 10^{6}\ \mathrm {V\,m^{-1}}$$\end{document} , \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} the applied uniform field, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon _{r}$$\end{document} the relative permittivity of the dielectric particle material. For a metallic particle, fd would be set to its maximum value of 3.
Adhesive force measurement using atomic force microscopy
The adhesion between the particles and electrode surfaces is measured for the first time in the context of gas insulation systems. Colloidal probes were fabricated to measure the adhesion of individual particles, as shown in Fig. 11. Tipless AFM cantilevers (HQ: CSC36 / tipless / no Al and HQ: CSC35 / tipless / no Al, MiKroMasch, Bulgaria) were cleaned with UV ozone (Ossila Ltd., UK) for 10 minutes. The spring constant of each cantilever was determined using the Sader method^60^. The resonance frequency and quality factor of each cantilever were determined using an atomic force microscope (AFM) before attaching a particle. The cantilever dimensions were provided by the supplier. Vanadium and silica particles were dispersed on a clean microscope glass slide (Thermo Scientific, Switzerland) along with a small drop of UV-curable adhesive (Norland Optical Adhesive 63, USA). The glass slide and the cantilever were then mounted on the stage of an optical microscope (BX41, Olympus, Japan). Using a micropositioning system (Lang GmbH, Germany) and a sharp tungsten needle (Imina Technologies, Switzerland), a small amount of glue was transferred to the tip of the cantilever. Another clean needle was then used to position an individual particle at the cantilever tip. The adhesive was cured using a UV light (EM10 UV, 365 nm, Nitecore, Germany) for 1 minute. Examples of vanadium and silica particles attached to the cantilever can be seen in Fig. 11a and b. Adhesion measurements were performed using an AFM (Nanowizard III, JPK, Germany for the vanadium particles and Asylum Research MFP-3D, Oxford Instruments, USA for the silica particles). Prior to measurement, the polished metal substrate was thoroughly rinsed with acetone and dried with nitrogen before being mounted on the AFM stage. The cantilevers were not cleaned after attachment of the particles to maintain their native surface. Four separate colloidal probes were prepared for each particle type (vanadium and silica). For each cantilever, adhesion was measured at seven different locations on the substrate, several millimeters apart. At each location, a grid of 100 force curves was acquired within a 50 x 50 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} m square area. This way, 2,800 adhesion measurements were obtained for each particle material. The force setpoint was 200 nN and the cantilever was retracted 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} m from the surface at a speed of 1 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu$$\end{document} m/s. The adhesion force was determined from the force-distance curves as the pull-off force recorded during cantilever retraction, as seen in Fig. 11c.Fig. 11SEM scans of the particles glued to an AFM cantilever and an example of a force-distance curve. (a) A vanadium particle, attached to an AFM cantilever. (b) A SiO_2_ particle, attached to an AFM cantilever. (c) A typical force-distance curve is shown based on a SiO_2_ particle. During the approach, the cantilever experiences a positive force as the particle is pushed onto the electrode surface. Upon retraction, the particle stays in contact due to adhesion until the stiffness of the cantilever causes the particle to detach. The adhesion is obtained from the minimum value of the retraction part.
Based on the adhesion measurements, their statistical descriptors were calculated for a quantitative comparison. The mean absolute adhesion forces were obtained for each particle by pooling measurements across all probed positions, providing a first-moment estimate of the average interaction strength. The most probable adhesion force was determined from the maximum of a kernel density estimate of the absolute adhesion distribution using Eq. (3), thereby capturing the dominant contact state while reducing sensitivity to the distribution tails. The overall width of the adhesion distributions was quantified using the standard deviation of the absolute adhesion forces, reflecting the variability of interaction strengths within each particle. The distribution asymmetry was characterized through the skewness of the absolute adhesion distributions, enabling assessment of the relative contribution of rare, high-adhesion events. Differences between particle-resolved adhesion distributions were quantified using the one-dimensional Wasserstein distance, which compares the full distribution shapes and retains the physical units of adhesion force.
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