The charge of a particle varies with time according to
;
Equation
2
where k ranges over the different charging processes, and the current
of the kth charging process is denoted as Ik.[13]. The other variables on the right side:
= electron and ion currents to a moving
or stationary grain, 
= secondary electron
currents, and 
= photoelectron currents.
In a Maxwellian plasma with electron and ion energy distributions at the same temperature, the velocity and therefore, the charging current of electrons is higher -- grains are charged negatively. As a result, ions are attracted and electrons repulsed until the magnitude of both charging currents become equal and reach equilibrium negative charge. If the surface potential of the grain is positive, then the situation is reversed. In most cases, the electron flux dominates the ion flux and the dust grain is charged up negatively. Ion currents to a moving grain is a more general expression for the ion capture
The absorption of solar UV radiation releases photoelectrons and hence constitutes a positive charging current [14]. Its magnitude depends on the material properties of the grain, i.e. its photoemission efficiency, and on the grain's surface potential, which may, if positive, recapture a fraction of the photoelectrons. The spectrum of the photoelectrons released is often assumed to be a Maxwellian distribution with a temperatureT, which corresponds to an energy kT (~2 eV). The photoelectric current is given by equation 10 in [14].
;
Equation
3
Here a is the particle radius, qe is the charge on the electron,
kT is the average energy of the photoelectrons and 
, is the flux of photoelectron emission at 1 A.U. with 
for conducting magnetite (metals) grains and 
for dielectric olivine particles. The equilibrium potential in interplanetary space due
to the dominant photoelectron flux is Upe = +3 to +5 V.
The photoelectric current in reference [14] refers to Feuerbacher et al's detailed 1973 work [15]. Feubacher et al carried out calculations for grains of high (material: Al2O3) and low (material: C) photoelectric yields taking into account their full frequency dependence and the energy distribution of emitted photoelectrons.
The photoelectric threshold theoretically and observationally seems to increase with decreasing particle radii (see reference [13]). The smallest dust grains can be characterized by the physical properties of solid spherical particles.
The effect of electrons or ions bombarding grains with high energies can lead to an ionization of the grain material and ejection of electrons from the grain that is called "secondary electron emission." The three processes that can occur during secondary electron emission: 1) reflection, 2) backscattering and 3) true secondary emission are usually treated as distinct secondary electron yield processes by both electron impacts from solid particles as well as by ion impacts [16].
The flux of secondary electrons depends on the energy of the plasma electrons/ions E and on the surface potential. The number of secondary electrons (yield) depends also on the material properties of the charged grains, which is characterized by the secondary yield parameter d. If the secondary electron yield is greater than 1, then positive dust charging occurs. For some materials, the yield is greater than 1 for kTe = 50 eV, and for other materials, the yield is greater than 1 at kTe = 1000 eV and higher. The yield also increases with decreasing grain size. For example, the maximum emission yield from impact onto 0.01 micron-sized particles for both conductors and insulators is about 3.5 times higher than onto 1 micron-sized particles [17].
Secondary Electron Emission - Electron impact
Sternglass (1954) [18] published an expression for the secondary current by electron impact using the yield function:
Equation
4
[electrons];
where Em is a characteristic energy at which the release of a secondary electron peaks [19].
This extensively used formula for the secondary emission yield from semi-infinite slabs of material approximates the theoretical derivations of Bruining and Jonker in the 1950s. For small spherical grains (not a planar slab), secondary electrons are not limited to the point of incidence of the primary electron. Instead, it is possible for secondary electrons to exit from all points of the grain surface, thus increasing the yield over that determined by the Sternglass formula [17].
Meyer-Vernet (1982) [20] showed that the charge on a grain is not always unique:
the equilibrium potential may have multiple roots. We show some representative values for maximum yield 
and energy at maximum yield Em from Draine and Salpeter,
(1979) [21] in the next table.
| Material | density (g/cm3) | |
Em (eV) |
| Graphite | 2.26 | 1 | 250 |
| SiO2 | 2.65 | 2.9 | 420 |
| Mica | 2.8 | 2.4 | 340 |
| Fe | 7.86 | 1.3 | 400 |
| Al | 2.70 | 0.95 | 300 |
| MgO | 3.58 | 23 | 1200 |
| Lunar dust | 3.2 | ~1.5 | 500 |
Table 2: Some representative secondary electron emission parameters, from Draine and Salpeter, (1979) [21], Table 5.
Secondary-electron yield of electron impact onto other grain materials, such
as an ice grains, can influence that type of grain's charging also. We show some measured secondary electron yields,
the estimated maximum yield , and energy
at maximum yield Em, for four ices from Suszcynsky, D. et al (1993) [22] in the next table.
| Material | Energy Range (keV) | Measured Yield | |
Em (eV) |
| H20 | 2-30 | 1.2-0.1 | 3.2 | 406 |
| CO2 | 2-30 | 0.3-0.1 | 1.2 | 467 |
| NH3 | 2-30 | 0.7-0.1 | 2.2 | 362 |
| CH3OH | 2-30 | 0.35-0.05 | 1.6 | 235 |
Table 3: Measured secondary electron yields and estimated maximum yield
, and energy at maximum yield Em,
for four ices from Suszcynsky, D. et al (1993) [22], Table 1.
Smaller grains (conductors and insulators) have higher yields generally because
the excited secondary electrons have shorter distances to travel to reach the grain®s surface.
For the simulations of charged particles in Earth's magnetosphere by Horányi,
Juhász and and their co-workers [14,
23,
24,
25,
26], they assumed for conducting magnetite a = 1.5 and Em = 250 eV and for for dielectric olivine they
assumed a = 2.4 and Em = 400 eV.
Charging may become limited for cases of high negative and positive dust charges. Electron field emission begins when a compact solid particle has an electric surface field strength:
;
Equation
5
whereas ion field emission begins at about 10 times higher particle electric field values:
Equation
6
;
For noncompact particles, we need to consider the tensile strength of the material:
;
Equation
7
Typical tensile strengths, and corresponding field strengths, are:
Fluffy aggregates:
;
Equation
8
Icy materials:
;
Equation
9
Stony materials:
;
Equation
10
Glassy Materials:
;
Equation
11
Metals:
;
Equation
12
Figure 2: Limiting tensile strength, Ft, as a function of particle size, s, and surface potential Ud.
[The following is a contribution from J.A.M McDonnell.]
The equilibrium potential of isolated bodies in space and in charged particle environments is largely dominated by the surrounding plasma; the diffusion currents are high and thermal electrons are mobile and therefore able to compensate readily for extreme potentials. In sunlight the photoelectric current is a strong effect leading to an equilibrium between loss of photoelectrons and the attraction of thermal electrons (Graps and Grün, this document.). Such processes are concerned with the surface and surrounding of objects and results are similar for both insulators and for conductors. If we look within the outer shell of a spacecraft however, for example, where the surface potential is so established by equilibrium current balance, a new electrical regime pertains.
The lack of ambient plasma permits higher potentials and, although photoelectric effects are minor or absent, higher energy electron fluxes penetrate the outer skin; a fraction is absorbed. On dielectrics, electron migration to nearby conductors is limited by the (high) resistivity and high fields can be established within the material. In the situation pertaining to dielectric meteoroids or space debris, we have a situation perhaps midway between a conducting body and a satellite interior. The outer surface will, after equilibration, attain the potential similar to a conductor if the photoelectron and secondary electron yields are the same. But the interior may have a distributed charge.
We look at several aspects of this internal charge as applied to smaller bodies in orbit: firstly the magnitude of internal charging and the dimensional scaling of this effect and, secondly, possible effects of internal charging under non-equilibrium conditions.
We first find the incident current, and then calculate the fraction absorbed for a given size and the build-up and leakage of this after equilibrium. The ESA supported space environmental software tool SPENVIS permits study of this and it has been used, but a simple model is established which permits the effect to be considered in parametric form. The effect of internal charging is considered using the model in Figure 3.
If the particle is immersed in a charged particle environment the outer surfaces will be at potential Vs. The effect of internal charging on a cubic grain (dimension r metres) is considered by comparing the internal current I to other external currents which determine the overall potential in space. Internal charging currents will leak away to this outer potential, which is established elsewhere in this document. If part of the surface e.g. face I (Figure 3) is inaccessible by plasma or, if we look at a region in the interior of the cube, we find an induced potential caused by the internal current leaking away through the dielectric. If we looked at this potential established across the face of this cube to the opposite side maintained at the space potential by the competing photoelectron and plasma currents, we would find a differential potential of magnitude VD = iir /r Volts where r is the dielectric resistivity and ii is the absorbed or internal current.
Figure 3. A model for an "isolated" particle as a cube of dielectric
material with an electron flux incident from the left. An increased space potential VS is established,
but additionally, internal charging may lead to a differential voltage on a shielded (or interior region) on this
simple model (opposite the ground face).
Also shown in Figure 4 is the electron range, R and distance, a over which the current declines
linearly. The range R (g-cm-2) is the maximum penetration depth for electrons of energy E
(MeV), and R = 0.55 x E{1-(0.9841/1+3E)}. The linear distance a (g-cm-2) is a distance
over which the internal current decreases linearly, and a = 0.238 x E [40].
Figure 4. Graph of the electron range, R and distance, a over which the current declines linearly for energies of 10 keV to 7 MeV. To achieve effective internal charging, electrons must stop or be absorbed into the material. This occurs for lower energy electrons according to the graph, e.g., £ 0.3 MeV for a one millimetre thick Plexiglas particle.
Internal charge will leak away as equilibirum is approached and, before, the neutralistion by attracted ions either absorbed or redistributed within the plasma from the Debye shield so formed. Away from the surface, internal charging cannot be detected since a uniform distribution acts, as indeed a surface charge, as a single point source. Thus there is no effect on dynamics or detection by a space intrument e.g. GORID as a charge sensor. There would be, for all charges, the question of a non equilibirum charge change when e.g., if the Debye plasma shielding were to be turned off by an approach to an instrument. If a sensor front grid assembly were to prevent plasma entry and the charged particle continued, then the net charge would be detected; but this is not an effect of internal charge as such and applies to the surface charge also.
Figure 5. Amount of internal charging for various sizes of Plexiglas cubes
from incoming electrons of different energies.
Conclusion
- Internal charging leads to charge retention within the body of insulating particulates.
- The charge absorbed is a function of the size of the particles, dictated by the fraction absorbed.
- The charge density stored in the particle decreases with decreasing particle size.
- The time constant for discharge is the same at all dimensions.
- The time constant is long compared to photoelectric effects, especially for small particles.
- Ion neutralisation from the ambient plasma will readily compensate for the net charge by surface adsorption.
- The stored energy is miniscule compared to the impact plasma charge.
- Although potentials induced by internal charging can for larger, grains be significant, the effect will be mitigated in sunlight.
[The following is a contribution from J.A.M McDonnell.]
We have numerous treatments and results for the potential of a body in space (e.g. ref [33] and this report). A body in space is usually a conductor, and is likely approaching equilibrium under photoemission and plasma (electron) neutralisation. The surface potential is uniform, although the photoemission is only from the sunlit side. If the particle is an insulator, however, then the rear of the particle will not be at the same potential. We examine this case and find that an interesting effect can arise if the potential of the side of the insulator in shade is negative. Asking how this might arise, there are a number of situations:
- The whole particle may get a negative charge in shadow (e.g. an eclipse transit)
- The particle may approach the detector in the satellite shadow
Part of the particle will always be in shade. On the anti-sun side, we could still have a flux of electrons maintaining a moderate negative potential.
Equilibrium potentials (Equipotentials) are shown in Figure 6 for the case of varying potentials on the sunlit and shade side of the particle. Noting that the higher (absolute value) equipotentials will wrap around the opposite side, we find that even a modestly high negative potential on an insulator could inhibit the effect of photoemission on the other side by confining their trajectories and, unless lost by collision, they will form a space charge, but not contribute to the net emission. The energy of photoelectrons is the photon energy (dominated by the solar Lyman alpha Helium line) minus the work function. It leads to a net energy of some 5 volts relative to the emitting surface. The effect will lead to a retention of negative charge on the particles even as they move into sunlight, as distinct from the situation considered to date for conductors. This is termed "bootstrap charging".
We find that there is scope for effective closure permitting the bootstrap charging effect to operate. It operates for as long as there is a high negative area on some part of the particle. We ask next: How long can a negative potential be sustained on part, or all of the particle? The overall negative potential ž sustained by bootstrapping ž will attract plasma ions and surface neutralisation takes place unless the flux that caused the negative potentials is sustained. On exit from a shadow, where such negative potentials can be generated, the memory can be retained for as long as the electron flux is maintained on the dark side. But if that side rotates into the sun and, if the photoemission is at least nonzero, then that memory may be lost.
We see there is an opportunity for nonconductors to behave differently from conductors, but expectations concerning the average equilibrium potentials of nonconductors in respect to this effect may not be out of line with conductors after the photoelectric efficiencies and secondary emission coefficients are taken into account. The ambient plasma will neutralise negative potentials after the particle emerges from the region where negative potentials are generated. When considering how long the bootstrap effect can be sustained under sunlight, rotation is seen to be a key factor with insulators. If rotation is very fast, then the average equilibrium values may not be much different from a conductor. However, because rotation is a key to unlocking this bootstrap effect, the charge-up times will be dominated by the particle rotation time rather than electronic processes and could be relatively slow. The effect could persist well away from regions of negative potential.
We should note, therefore, that there is an effect which could:
- Lock non rotating particles into a negative potential state essentially indefinitely after a shadow/eclipse
- Lead to a positive charge-up time after shadow transit which is dominated by particle rotation rather than electronic/ion processes
There may be a population class, therefore, or a situation, where the potentials of detected particles will not be uniform and of one class. Anomalies may exist and some relatively "exotic" potential configurations maintained which are not typical of the immediate environment surrounding the detector.
Figure 6. Equipotentials around a particle, which is sunlit on one side resulting in a negative potential of 10V on the dark side and a positive potential of 5V on the sunlit side [-10V;+5V]. The negative potentials curl around to the opposite side and the zero potential surface closes at a large distance.
[The following is a contribution from J.A.M McDonnell.]
The GORID instrument has drawn attention to the possible role of charged microparticles in orbit; the opportunity to measure charges, although the inference of charge from the measured parameter set is not straightforward. The charge may affect the dynamics [33], and last section of this document], although we look here at the possible effect of high charges in geostationary orbit, which could fragment meteoroids [45]. Swarms and groups meteoroid fragments were first measured by the HEOS II instrument in a high eccentricity Earth orbit. These swarms and groups could well be the comminution of small bolides and/or ejecta from the Moon). We calculate some parameters and properties associated with fragmentation in geostationary orbit.
The prospect of high particulate charges has been studied by a number of authors in the context of the disruption of cosmic dust particles. These particles may be loose aggregates, because, indeed, field emission processes can evaporate very small particles. The disaggregation could be very relevant to geostationary orbit regions, where we could have swarms of (possibly even higher charged) particles.
Expansive forces on the particle are proportional to the square of the total charge (Q2)
but are spread over the particle area (4 pi R2) (R=particle radius). Since the potential V is (largely)
independent of R, we usually find that the critical strength limit has to match a given value of the surface charge
density 
,s which increases with
decreasing particle size due to the radius of curvature. This density is given by
s = E(R)/eo = V/Reo and hence, for given strength, all particles in
a size distribution able to reach the local equilibrium potential will be disrupted if they are below a critical
size RD (the critical disaggregation radius for a given strength). For field tensile yield, the strength
fracture of particle bonds at a lower critical radius RY may apply, and below this, a field emission
radius RF may apply for grain surfaces. These effects and limits are shown schematically in Figure 7
for potentials of 1 V and 1kV. Field strengths for common materials vary from 1 to 30 1010 V m-1.
For Tungsten we have a value of 2.1011 V m-1.
The consequences of disaggregation are dramatic; since components liberated are always smaller, the critical disaggregation potential is restored; and again and again until total disaggregation is complete. Particle strengths may then be exceeded if the cascade of fragments are beneath a yield radius RY and again field emission for fragments produced below RF. We examine aspects of this process from GORID®s geostationary orbit measurement scenario. Tabulated in the scenario for fragmentation into N component grains (assumed monomodal). A more typical size distribution would be n(r)dr a r-udr for a meteoroid distribution index e.g. where u ~ 4 and would be studied in the contract in the context of fireball disaggregation in geostationary orbit. This value of u gives a cumulative mass index of ž1 and a constant mass per magnitude. It is approximately equivalent to taking a 1 cm cube and distributing the same mass into 1012 particles of 1 µm in diameter.
We show in Table 4, a possible fragmentation sequence as a function of final grain size; the charge is calculated, which (after recharging) can then be accumulated on the whole swarm. If the disaggregation strength is low, the relative outward diffusion velocities will be only some metres/second; even for the tensile yield strength limit RY the velocity is only 40 m/sec. The disaggregated swarm dispersion time of 1 m sec thus leads to a spatial dispersion of only 20 cm for a particle disaggregation within 1 km of GORID. The scenario of high charges on GORID from a swarm of particles, comparable to the sensor dimensions, but disrupted in geostationary orbit or on approach to the EXPRESS II spacecraft is interesting; it needs study in view of the effective increase of charge considered by a swarm. This simple "isolated" particle treatment needs to be folded with the Debye shielding distance and the recharging time, however, in order to assess more realistic behaviour.
Figure 7 Schematic of critical field strengths for disaggregaton, yield
strength and field (ion) emission.
| |
Effective Radius |
Number of Products |
Charge per particle (no recharging) |
Charge / particle after recharging |
Total charge in Assembly |
|
|
R |
1 |
Q |
Q |
Q |
 |
R/2 1/3 |
2 |
Q/2 |
Q/21/3 |
22/3Q |
|
|
R/4 1/3 |
4 |
Q/4 |
Q/41/3 |
42/3Q |
|
|
R/N 1/3 |
N |
Q/N |
Q/N 1/3 |
N2/3Q |
Table 4: Disaggregation sequence due to electrostatic potential and the charge carried by products with comminution product recharging.
Electrostatic fragmentation on particles requires high potentials, varying from hundreds of volts at micron dimensions to kilovolts at centimetre scale. These potentials are not typical of average conditions but, are experienced in GEO fairly frequently (e.g., [43]). If experienced only once, in passage through the geomagnetic tail, a single bolide (agglomerate meteoroid) could provide enough fragmentation products to populate the GEO space above the natural background of sporadic micrometeoroids. Because the potential of a particle is independent of size, we find that once a ž perhaps infrequent ž high potential is achieved which exceeds the disagregaton strength, then the products will be subject to further disruptive forces of increasing strength as their size decreases. The disaggregation sequence is dependent on recharging which initially may limit the speed of the process for large particles; for smaller particles the recharging time constant is fast and the process is limited by the dispersion of the grains in the cloud to a distance beyond the Debye shielding distance.
Consideration of a conservative disaggregation sequence shows that the process should occur in a matter of only seconds for a centimetre scale object subjected to perhaps 10 kV potential and that during this time the cloud may have expanded to a scale of only some hundreds of metres.
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Charging Processes
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