Tom F. O’Hara*,
David P. Reid,
Gregory L. Marsden and
Karen L. Aplin
Faculty of Science and Engineering, University Walk, Bristol, BS8 1TR, UK. E-mail: tom.ohara@bristol.ac.uk
First published on 19th December 2024
The triboelectric charging of granular material is a long-standing and poorly understood phenomenon, with numerous scientific and industrial applications ranging from volcanic lightning to pharmaceutical production. The most widely utilised apparatus for the study of such charging is the Faraday cup, however, existing analysis of the resulting measurements is often simplistic and fails to distinguish charging due to particle–particle interactions from charging occurring through other mechanisms. Here, we outline a modular approach for interpreting these measurements, enabling triboelectric phenomena to be explored in greater detail. Our approach fits approximated charge distribution shapes to experimental Faraday cup traces. The fitting process uses measured size distributions in combination with simplified models of charge distribution and particle dynamics to predict the relative charging contributions. This modular approach allows scope for adaptation of each aspect to fine-tune the process to specific application cases, making the technique broadly generalisable to any insulating granular material. An example case of volcanic ash showed that samples from the Grímsvötn volcano charged with a greater proportion of particle–particle interactions than ash from Atitlán. Experimental validation is provided using sieved fractions of volcanic ash, where the broader size fractions were found to exhibit greater particle–particle charging. Non-particle–particle charging was also shown to scale with particle size as ∝ d−0.85±0.03p, roughly scaling with the particles' effective surface area.
The description of charge transfer is well understood for metals, compared to the insulators responsible for typically stronger triboelectric charging. On contact, metals transfer electrons according to their well-defined work functions. This results in a net negative charge remaining on the metal with a higher work function upon separation. However, the mechanisms behind triboelectric charging in insulators remain poorly understood. Proposed mechanisms include electron or ion transfer, mechanically induced bond breakages, and bulk material movement, each supported and refuted by varying evidence.8,9 Some empirical understanding can be gained by placing materials in a “Triboelectric Series”, in which the polarity of charge obtained from the rubbing of two materials can be predicted.10 Unfortunately, this approach is highly system-dependent, with changes in environmental factors such as temperature and humidity changing the magnitude or even polarity of charging.11–16 It also fails to explain the charging between the same material, such as identical grains that have been shown to charge with a varying size dependence.17
More reliable and well-understood measurements are required to better understand the system dependence of charge transfer in granular materials. Faraday cups are the most widespread and commonly used apparatus for charge measurements of powders.1,18–20 A Faraday cup is a typically cylindrical and shielded conductive vessel that exhibits “electrification by induction” as observed by Faraday in 1849.21 When a charged species impacts the cup, charge is transferred by conduction to the inner electrode, which can be measured using a connected electrometer.22 These charge measurements have a vast array of applications from ion beam characterisation and plasma diagnostics to the charged granular material relevant to this work.23,24 A Faraday cup can be employed to measure the net charge of a sample landing in the cup, as in this work. Measurements can also be obtained purely from electrostatic induction in cases where particles do not make contact, which may be referred to as a “Faraday pail”, although the terms of “cup” and “pail” are used inconsistently in the literature.18,25 The charge carrier may also pass freely through the system, allowing for the measurement of a charged particle's velocity or be used as part of an array of cups to analyse subsets of the sample.22 Electrometers are used to obtain the transferred or induced charge, by measuring the voltage or current and calculating the charge.15
Faraday cups can be used to study the triboelectric charging of single-particles or small clumps of particles charging against different surfaces.19,26 Other techniques involve impacting individual particles on plates, which can be made of the same material to emulate same-material particle charging.27 Charge measurements can also be achieved by measuring the charge on the impacted plate or with particle tracking velocimetry (PTV) where particles in an applied external electric field have their charge deduced from displacement, tracked by a high-speed camera.28,29 PTV can also be employed to probe individual particle–particle collisions.30
When looking at bulk powders, Faraday cups are usually able to detect the polarity and magnitude of powders' net charge, or subsets of sizes of the powder, but are not yet able to characterise the overall charge distribution with respect to size.31 Other techniques can therefore be employed to assist in resolving charge measurements spatially or by size. Electrostatic charge separation, which sorts particles by applying an external electric field, is widely used in industries such as recycling and coal processing and can complement Faraday cup measurements.32–34 PTV approaches can also be applied to groups of particles and have recently detected charges on individual grains up to 76 times greater than their samples' average, which traditional Faraday cup methods would produce.35,36 This work aims to formulate a model in which contributions to Faraday cup charge measurements of granular material can be separated utilising the temporal resolution provided by powders falling, using relatively low-cost apparatus at room temperature and pressure.
Existing techniques for analysis of powder traces from Faraday cups predominantly consist of taking the net difference before and after the charged granular material enters the cup, or by using the range of values acquired.18,37 Here “trace” refers to the variation of a derived quantity, primarily specific charge in this work, over time. (The quantity measured is voltage, which is linearly related to charge via the instrument capacitance, discussed in Section 2). In some cases, the maximum may be assumed to be an asymptotic value when the smaller particles remain dispersed in the air as aerosol and may take far longer to deposit, if at all.1 These measurements of net charge are quick and easy and can be carried out with individual particles or larger assemblies, however, the shape of the trace contains more information on the granular charging than has typically been extracted previously.1,19
Simply using the net charge yields a single value that, in many cases, is taken to indicate the relative extent of triboelectric charging, its polarity, or both. However, in cases where particle–particle charging is being investigated, this charge could be due to either contact with the container or from a pre-existing charge distribution that could have been generated by handling (for example, loading the sample into the measurement apparatus). Some works aim to study particle–particle charging in isolation by minimising particle–wall charging by reducing contact with container walls, which may be grounded, employing fluidized beds, or by using a “fountain-like” flow in a low-pressure environment.1,38,39 Powders can be exposed to positive and negative ions to neutralise some pre-existing charge.40,41 However, it is very difficult to entirely negate the effects of pre-existing charges and particle–wall contacts under most conditions. For instance, a granular sample with an initially positive trace from particle–particle charging and a negative charge from loading could yield a net result similar to or less than a case with less particle–particle charging but the same polarity as the non-particle–particle charge. A few example traces from samples of volcanic ash that will be used later in this work can be seen in Fig. 3 to demonstrate some of the different shaped traces that can be obtained. The goal of this paper is to develop a more sophisticated approach that feeds the entire Faraday cup trace into a model that predicts the extent of particle–particle charging separated from the other sources of charge, exploiting information stored in the overall shape of the trace. The aim is not to simulate the triboelectric charging of a specific case in detail, but rather to create a general methodology, suitable for a range of applications, using simple yet justified assumptions which can give information on the extent of the measured charge arising from particle–particle interactions.
In this paper, a new modular approach for interpreting the Faraday cup measurements of triboelectrically charged granular material is outlined, applied, and verified. The experimental apparatus is detailed in Section 2 and then illustrated with example traces before specifying the numerical and simulation techniques employed at each step in the interpretation methodology. In Section 3 the approach is initially applied to samples of volcanic ash, and then sieved fractions of ash are used to provide experimental validation for the new interpretation methodology, before investigating the scale-up of the non-particle–particle charging for simplified material. Finally, the work is concluded with a discussion of the limitations and potential application areas in Section 4.
In the examples used here, two charge distributions are defined, one for the pre-charging and one for the self-charging, each giving rise to an expected contribution to the final Faraday cup trace. However, the principle of this model is easily generalisable to a greater number of charge generation mechanisms. For example, if the expected shape of the size dependencies for pre-charging arriving from different sources were known separately then the same model would, in principle, be able to resolve the relative charging arising from each source.
Fig. 2 A schematic of the ash charge apparatus for this work, adapted from Houghton et al. with the Faraday cup highlighted in red.1 |
The geometry of the tube the particles fall in is not particularly important for this approach, but the key quantity that will be used in Section 2.4 for the particle dynamics model is the drop height, which in this case was 37.25 cm. Any inductive effects before contact with the cup is assumed to be negligible (though they could be included subsequently). The cylindrical delivery tubes had heights of 13 mm and diameters of 7 mm. This means the available volume for a sample was 500 mm3, hence the dropped samples had masses of grams. For each trace, the variation in mass can be adjusted for by dividing by the mass of the sample dropped, yielding the specific mass in each case if required. The Faraday cup was connected through BNC connectors to a Keithley 6514 system electrometer. Short rigid BNC connectors were utilised to minimise noise in the low-level measurements.42 The output of the electrometer was plugged into a USB-6210 Data Acquisition (DAQ) device from National Instruments that was processed using LabVIEW® 2024-Q1 before post-processing with the outlined model. The DAQ was also connected to a DHT11 humidity and temperature sensor via an Arduino Mega 2560 to record the environmental conditions of each drop. The temperature and humidity were not controlled but were recorded for reference at the time of each drop with common values around 24 °C and 40%, respectively. Over a 2 h measuring period typical variations of around 1 °C and 2% relative humidity were observed, which did not significantly affect the measurements.
When converted to surface area distributions the smaller modes become more significant and are further amplified in their respective number distributions. This conversion assumes that the particles are spherical and uses the measured optical diameter, even though the aerodynamic diameter is the more relevant factor. Conversions can be made between the two quantities either utilising the particles' shape factor or empirically through measurements. In this case, no conversion was made as the relatively small adjustment (usually around 20%44) would make an insignificant difference to the broad size distributions which span multiple orders of magnitude. The Mastersizer and Camsizer were chosen due to the operating size ranges of 10 nm to 3.5 mm and 0.8 μm to 8 mm, respectively. Multi-modal distributions with the form
(1) |
For a spherical particle the drag force (FD) exerted on the particle by the surrounding fluid, which is air in this case, can be expressed as
(2) |
(3) |
(4) |
(5) |
The only other force that is assumed be acting on the particles is the force due to gravity (Fg), which naturally acts downwards with Fg = mpg, where g is the acceleration due to gravity and the mass of the particles mp is related to their diameter and density (ρp) by . In this work, ρp was assumed to be constant throughout each sample and was measured using the Anton Paar Ultrapyc 3000® gas pycnometer.
These forces can be resolved and used with Newton's second law to find each particle's acceleration. Each time step the position and velocity of the particles can be updated, recalculating Re, CD, and FD. For numerical stability, the time step must be updated to hold the Courant–Friedrichs–Lewy (CFL) condition at one. The CFL condition is the ratio of the simulation time step to the system's characteristic length scale. In this case, the particle can move roughly its diameter within each time step. At the very beginning, the particles start stationary, meaning the CFL condition number would be zero so a minimum time step of 0.1 ms is also implemented. Overall, this allows for the total time for each particle size to reach the cup to be determined, with the results shown in Fig. 5b. The aerodynamic diameter here is equated to the geometric diameter, as a spherical approximation is made, although empirical measurements could be made to account for this discrepancy.
Only particles that land within the final trace time need to be calculated, as the rest will not land and hence be shown on the trace. In the case of Fig. 5b, only particles larger than 20 μm contribute to the predicted trace; however, for longer traces or higher densities, smaller particles are also included in the calculations. Additionally, as particles approach and pass below 10 μm in size they would begin to behave diffusively and deposit on the container walls or remain dispersed in the air rather than impacting the Faraday cup.49
A slightly more involved approach is required for the self-charging (particle–particle interactions). In this work, an event-driven molecular dynamics (EDMD) hard-sphere approach was adopted, which reduces computational expense. Due to the unknown nature of tribocharging,32 this work does not assume the nature of the charge carrier. Therefore, a complex charge transfer mechanism, such as electron tunnelling, as employed by Kok et al. cannot be used.50 Instead, the simple case of transferring a single charge from a high-energy to a low-energy state is assumed. This can be thought of as negative (e.g. an electron) initially, although again when the fitting takes place only the shape is important and the polarity may be flipped, such that the charge carrier could be of either polarity. As the charge carriers are not known, this approach seems more robust than arbitrarily assuming one sign.
In this EDMD model, distributions of 300 particles were used, as a trade-off between accuracy and computational cost, matching the size distribution described in Section 2.3. These particles were then randomly placed in a box such that their density was held constant, at a packing fraction of 0.1, by calculating the required box length for each volume of particles. A packing fraction of 0.1 is large enough to prevent particles from being trapped locally, yet small enough to avoid excessive interactions with the walls, which would reduce computational efficiency. This value is also consistent with the range used in previous studies.50–52 If particles overlapped then one of the overlapping particles was replaced elsewhere in the box until there were no overlaps present. The time until a particle–particle collision event (tcol) between each pair of particles, such as i and j, can be calculated from the initial distance rij = rj(t) − ri(t) and velocity vij = vj(t) − vi(t) vectors by
(6) |
After an elastic collision, the change in the particles' velocity was calculated using the particles' respective masses (mi and mj), their velocities and their unit vector of separation ij. By first calculating the effective mass meff and impact speed vimp,
(7) |
J = (1 + ε)meffvimp, | (8) |
(9) |
A hard wall treatment was used in this case for simplicity, meaning if a particle collided with a wall its velocity vector in the direction of the wall would be inverted, although a periodic boundary condition could also be used.53
Initially, these particles were assigned random velocities before being allowed to equilibrate for a few thousand steps, such that they could reach a thermodynamic equilibrium (defined as their kinetic energies fitting a Maxwell–Boltzmann distribution) before the charge transfer mechanism was turned on. The final arbitrary charges were then fit to a polynomial of the form f(dp) = adbp + cddp + e, where dp is the particle diameter and a, b, c, d, and e are fitting parameters, such that a > 0 and c < 0. For the positive term it was found that b was usually around 2 as at large diameters the surface area (∝ dp2) dominates the fitting. For the negative term d was usually around −1, which is close to what may be expected from the increased collisions of smaller particles due to their greater velocity at the same kinetic energy but slightly offset by the smaller collision diameter reducing the effective collision cross-section. The fit with constraints b = 2, d = − 1, and e = 0 shows very good agreement with the data, giving an R2 value of 0.996 in Fig. 6a. Overall, this fit is in line with expectations where the small particles are negatively charged and large particles are positively charged for a negative charge carrier, such as an electron.32,54
As the size distribution is represented as frequency density and the charge distribution shape is known as a function of size, the two can be multiplied to give a charge frequency density distribution, demonstrated in Fig. 6b. The calculated charge frequency density represents the total relative amount of charge expected over all particles of a given diameter. In the case of Fig. 6b, at around 20 μm there would be a net-zero charge on the particles arising from particle–particle interactions. Particles any smaller than this would be predicted negative (if the charge carrier is negative) and any bigger would be predicted positive. This distribution is useful in this work as integration between size limits will give the net charge predicted for that specific size range. For self-charging, there should naturally be no net charge across the whole distribution due to the conservation of charge, meaning the total integral of the charge frequency density distribution should be zero. In reality, the distributions integrate to near zero but not exactly due to the discretisation of the particle sizes used in the EDMD simulation. If an infinite number of particles were used the integral of the arising charge frequency density distributions of self-charging should limit to zero. In the case of Mount St. Helen's, the total integration deviates from zero by 2.2% as a fraction of the functions' total variation. This process was then repeated for expected charge distribution arising from pre-charging sources (Q ∝ dp2) to obtain the respective pre-charging frequency density distribution.
The methodology outlined here assumes that the pre- and self-charged populations do not interact, simply because the nature of an interaction is not well understood. However, due to its modular nature, our model is flexible enough to allow for such interactions in future.
For each ash sample, the smallest particle fraction (dp < 63 μm) was removed. This is because the smallest particles are harder to model, due to some falling in the diffuse regime so will stick to walls and form agglomerates. The charge on some samples, such as Grímsvötn ash, appeared to decay after being dropped due to the conductivity of air. This can be accounted for by calculating the expected decay at each time step from the equation ΔQ = Q0(1 − e−t/τ), where the change in charge (ΔQ) after a time step (t) is found using the charge before the decay (Q0) and the time constant of air (τ).55 ΔQ can be subtracted from the measured trace to get the trace without decay. The conductivity of air in the lab has previously been measured as 2.85 × 10−14 S m−1, yielding a time constant of 310.5 s.2 This charge decay in air is only observed in certain materials, possibly due to the particles' electrical conductivity or surface chemistry playing a role in the decay of the static charge. For the fits seen in Fig. 7 the R2 value is minimised, prioritising the immediate drop where there is more deviation from the fit. However, a minimisation of the total residuals, which prioritises fitting the later tail of the trace can also help indicate the extent of pre- and self-charging.
From the charge trace fits, we can now determine the extent of self- and pre-charging contributions. One approach to assigning an overall value to the ratio of these components' relative contributions is to take the ratio of the integrals of the respective absolute size frequency density distributions. For Grímsvötn ash the ratio of pre-:self-charging is 2.7:1, indicating the two types of charging have initially the same polarity with 27% of the charging originating from particle–particle interactions. For Atitlán, this pre-charging only makes up 7% of the total charging. Overall, the greater self-charging indicates that the ash from Grímsvötn is expected to charge more from particle–particle interactions in a less-bounded system such as a volcanic vent. However, the total charging (including pre-charging) is still significant in some cases as it may give information on the saturation charge of the sample. The larger degree of self-charging for Grímsvötn ash is not inconsistent with observations of many registered lightning events per day upon eruption of the Grímsvötn volcano.56 However, it is hard to make a direct comparison between laboratory charging investigations and observed lightning events due to the many other variables and charging mechanisms involved, such as fractoelectrification and “dirty thunderstorms”.5,57
By applying the modular analysis approach for the broad sieving fraction (63 < dp < 125 μm) of Grímsvötn ash, as seen in Fig. 8c, the pre:self-charging ratio was found to be 2.4:1. This indicates that nearly a third of the charging originated from self-charging. For the slightly narrower fraction (80 < dp < 125 μm) the ratio was 2.9:1 indicating around a quarter of the charging was from self-charging. This trend continued to the much narrower fraction of (80 < dp < 125 μm) with a ratio of 1:4.8 indicating only around a sixth of the charging is arising from self-charging. The analysis demonstrates that the broader size fractions show both a greater proportion and magnitude of their electrification arising from particle–particle interactions. This is expected based on the increased asymmetry in particle size, providing evidence for the validity of the new analysis.
The measured labradorite self-charging was found to fit a power-law function as shown in Fig. 9b with an R2 value of 0.98, with a power of −0.85 ± 0.03. This is likely due to the increased contact area between the particles and the container walls or any other handling apparatus. To approximate this effect we could assume the container walls are effectively flat and the particles are mono-disperse spheres of radius r. Then the approximate amount of each particle's surface area that is within a distance (L) of the surface is 2πrL. If the spheres are close packing hexagonally in 2D against an area of the flat surface, then the packing fraction is constant at giving particles per area of the flat surface (A). Therefore, the total effective area (Aeff) can be approximated by:
(10) |
This trend explains why the smallest size fraction in Section 3.2 displayed the greatest overall magnitude of charging despite having a smaller self-charging component than the broader size fraction. More investigation is needed to include other variables such as accounting for differently shaped particles to probe the relation to contact area and the dependence of the self-charging on average particle size.
The outlined approach is broadly generalisable to any insulating granular material, although the experimental setup outlined in this work operates best for materials with average particle diameters (dp) in the range 20 μm < dp < 200 μm (for particles with a density of around 1.5 to 3.0 g cm−3). This operating range could be altered by utilising a different drop height, whereby a taller experimental setup would allow for particles of greater diameter or lower density to be distinguished. The modular nature of the approach also allows for fine-tuning to each application case. Overall, this novel interpretation methodology will allow for greater insights into the granular triboelectric charging behaviour for a host of natural and industrial applications.
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