High-Temperature Ionization in Protoplanetary Disks

High-Temperature Ionization in Protoplanetary Disks


We calculate the abundances of electrons and ions in the hot ( K), dusty parts of protoplanetary disks, treating for the first time the effects of thermionic and ion emission from the dust grains. High-temperature ionization modeling has involved simply assuming that alkali elements such as potassium occur as gas-phase atoms and are collisionally ionized following the Saha equation. We show that the Saha equation often does not hold, because free charges are produced by thermionic and ion emission and destroyed when they stick to grain surfaces. This means the ionization state depends not on the first ionization potential of the alkali atoms, but rather on the grains’ work functions. The charged species’ abundances typically rise abruptly above about 800 K, with little qualitative dependence on the work function, gas density, or dust-to-gas mass ratio. Applying our results, we find that protoplanetary disks’ dead zone, where high diffusivities stifle magnetorotational turbulence, has its inner edge located where the temperature exceeds a threshold value  K. The threshold is set by ambipolar diffusion except at the highest densities, where it is set by Ohmic resistivity. We find that the disk gas can be diffusively loaded onto the stellar magnetosphere at temperatures below a similar threshold. We investigate whether the “short-circuit” instability of current sheets can operate in disks and find that it cannot, or works only in a narrow range of conditions; it appears not to be the chondrule formation mechanism. We also suggest that thermionic emission is important for determining the rate of Ohmic heating in hot Jupiters.

accretion, accretion disks, planetary systems: protoplanetary disks, solar system: formation

1 Introduction

Protoplanetary disks’ evolution is governed by the extraction of orbital angular momentum needed for accretion to proceed. Magnetic forces are among the prime candidates to drive the extraction. Modeling the disks therefore requires knowing how readily the magnetic fields diffuse through the material. Hence the ionization state of the gas is key.

In particular, the magnetorotational instability (MRI) is suppressed unless the electron fraction is (Jin 1996; Gammie 1996), where is the number density of molecules. Launching disk winds requires (Bai & Stone 2013). The gas ionization likewise bears on the existence of lightning (Desch & Cuzzi 2000) and current sheets (McNally et al. 2013), two proposed mechanisms for transiently heating and melting chondrules (millimeter-sized inclusions in chondritic meteorites) in the solar nebula. Assessing the extent of the MRI and the heating of chondrules requires knowing the ionization rates, especially in the disks’ denser midplane regions.

Ionization in the cooler parts of disks, where temperatures  K, has been studied in detail. Several non-thermal processes are important at these low temperatures, with ionization rates per volume often expressed as . Far-ultraviolet (FUV) photons from the central star can ionize gas at a rate at 1 AU, but are attenuated by column densities (Perez-Becker & Chiang 2011). Similarly, X-rays from the star can ionize at rates at 1 AU, but are attenuated by column densities (Glassgold et al. 1997; Igea & Glassgold 1999; Ercolano & Glassgold 2013). Galactic cosmic rays, if not shielded by stellar winds (Cleeves et al. 2013a), can ionize at rates , but are attenuated by column densities (Umebayashi & Nakano 1981). In the innermost few AU of a protoplanetary disk, the expected column densities are , so that the dominant non-thermal ionization sources are particles from radioactive decays of species like , leading to (Umebayashi & Nakano 1981); or (if present), leading to (Cleeves et al. 2013b). In the disks’ cold midplane regions, this is the expected ionization rate, and it is quite low.

In chemical equilibrium in the dense midplane gas, the electron density is found by balancing the ionization rate against adsorption, and subsequent recombination, of electrons on grain surfaces. Assuming monodisperse grains with radius , internal density , and dust-to-gas mass ratio , the equation describing this balance is


(Sano et al. 2000), where is the number density of grains, the sticking coefficient of electrons colliding with grains, and other variables have their usual meanings. From this we find (assuming , and ). For a typical midplane gas density inside a few AU, (Weidenschilling 1977), , a value too low to be of significance to the MRI. Recognition of this fact led Gammie (1996) to suggest that protoplanetary disks’ interiors, especially the parts within a few AU of their stars, are “dead zones” where the MRI cannot act.

Jin (1996) and Gammie (1996) were the first to recognize that very close to the central star, protoplanetary disks may be hot enough to sustain thermal ionizations, recoupling the magnetic field to the gas. Specifically, they pointed out that collisional ionization of gas-phase potassium atoms at temperatures would generate free electrons and ions. The threshold temperature of 1000 K is based on calculating the abundances of ions and electrons using the Saha equation:


Here and are the statistical weights of the (ground states) of and , the Boltzmann constant, Planck’s constant, the electron mass, and the first ionization potential energy of potassium. Assuming and that the abundance of K relative to in a solar-composition gas is (Lodders 2003), half of the K atoms are ionized when , for a density likely to occur near the midplane. More significantly for the MRI, when , meaning substantial ionization would be expected if potassium were in the gas phase at this temperature. According to Lodders (2003), the 50% condensation temperature of potassium is 1006 K (at , or ; the pressure dependence is weak). So it is fairer to say that above 1000 K the Saha equation predicts significant thermal ionization of potassium, and for temperatures between about 850 and 1000 K, thermal ionization may or may not lead to significant ionization.

Other species contribute less strongly to thermal ionization than potassium. Because of the exponential dependence on the first ionization potential (), one might expect other alkali atoms to contribute to thermal ionization. The first ionization potentials of Li, Na, K, Rb, and Cs are, respectively, 5.39, 5.14, 4.34, 4.18, and 3.89 eV. These elements’ abundances relative to are, respectively, , , , , and (Lodders 2003). The first ionization potentials (FIP) and cosmochemical abundances of many elements are depicted in Figure 1. Repeating the analysis above, the temperature at which each alkali element contributes enough electrons to yield (at ) is 1149, 975, 865, 932, and 852 K, respectively. Considering that cesium is in the gas phase above 800 K whereas potassium is not (Lodders 2003), one might expect Cs to be the dominant contributor of ions and electrons at temperatures between 850 and 1000 K; however, the Cs is times less abundant, so that even if all Cs were ionized and only 0.1% of all K atoms were in the gas phase, potassium would still dominate over cesium. Thermal ionization of potassium is likely to be more important than that of other alkalis, and potassium is in any case an excellent proxy for other alkalis. Throughout this paper we will consider the case of potassium to represent the contributions of all the alkali elements to thermal ionization.

Figure 1: Cosmochemical abundances of various elements and their first ionization potentials (FIPs). Abundances (on a scale where the number of H atoms is ) are derived by Asplund et al. (2005) from analyses of carbonaceous chondrites (“meteoritic”) or from spectrometry of the Sun (“photospheric”). Producing free electrons by thermal ionization requires the presence of elements with high abundance and low FIP, i.e., those elements toward the plot’s lower right corner. The exponential sensitivity of the ionization fraction to FIP means potassium (K) is most favored to provide free electrons by thermal ionization, with Na, Rb, and Cs potentially also important—if these elements occur in the gas phase rather than locked up in minerals.

A criticism of this thermal ionization picture is that the Saha equation is valid only in local thermodynamic equilibrium, or LTE (Desch 1998). It relies on the collisional ionization of the gas-phase K atoms, e.g., , being in detailed balance with its inverse reaction, three-body gas-phase recombination, . But under conditions typical for protoplanetary disks, electrons are more likely to recombine with ions on the surfaces of grains. Desch (1998) argued that the electron density must be set by balancing collisional ionizations against collisions of electrons with dust grains. Once adsorbed on the grain surfaces, electrons and ions quantum tunnel from one adsorption site to the next, wandering across the surface until they find each other and recombine. The resulting neutral K atoms can then escape the grain. Dust grain surface-catalyzed recombinations are much faster than gas-phase recombinations, as the frequency of colliding with dust grains is (at K and , and assuming grains with radius and a dust-to-gas mass ratio 0.01), whereas the frequency of recombining with an ion is , where is the recombination coefficient (see below), and we have assumed the ion density matches the electron density, and . Not until do gas-phase recombinations dominate over grain-catalyzed recombinations. The dominance of grain effects invalidates the use of the Saha equation for potassium as a means of calculating the electron density.

The dominance of grain processes also points to an additional ionization source, not previously considered in the context of protoplanetary disks. The process that is the inverse of electrons and ions colliding with grains, resulting in neutral atoms leaving the grain, is the emission of electrons or ions following collisions of neutral atoms with grains. These processes are more commonly called thermionic emission and ion emission. When solids are heated to temperatures , they emit electrons and/or ions. The emission of electrons depends on the work function of the solids comprising the grains, which also affects whether evaporating atoms such as potassium are emitted as neutral atoms or ions. Including the thermionic and ion emission completely changes the role of dust grains in the ionization of the gas: instead of mopping up free charges, they generate them.

In this paper we examine the effects of thermionic and ion emission on the ionization state of high-temperature, dusty gas. In §2 we discuss the microphysical mechanisms of thermionic and ion emission, as well as the evaporation and condensation of K atoms, and provide quantitative rates and likely input parameters. In §3 we create a simple model treating evaporation and condensation of K atoms and ions, including ion emission and thermionic emission. We show that, because of the high work function of silicate grains, ion emission creates signficant ionization at temperatures around 800 K, possibly as low as 600 K. In §4 we discuss the impact on several aspects of protoplanetary disks. In particular, we address where the dead zone’s inner edge should lie, what temperatures are needed for disk gas to diffuse into the star’s magnetosphere, and whether including this new ionization source can enable the “short-circuit instability” of McNally et al. (2013). Thermionic and ion emission profoundly affect the ionization state of high-temperature gas in disks.

2 High-Temperature Chemical Network

2.1 High-Temperature Ionization Effects

In the dusty gas of protoplanetary disks, we cannot always use the Saha equation, and instead must check the rates of various ionization and recombination processes. What’s more, protoplanetary disks are marked by transient temperature changes, and we are interested in how the ionization responds to the changing conditions. Thus motivated, in this section we lay out several processes’ kinetic rates and construct a chemical network to find the steady-state abundances of electrons and ions as well as the timescales for the ionization chemistry to reach equilibrium.

Our chemical network includes five species: electrons, molecular or atomic ions, K atoms bound to dust grains, gas-phase neutral K atoms, and gas-phase K ions. Energetic particles (cosmic rays, radioactive decay products, etc.), and at high temperatures also gas-phase reactions, ionize the gas molecules; ions and electrons recombine in the gas phase. We also include dust grains, calculating their steady-state charge. All the other species can collide with and condense on or chemically react with the grains. The grains can lose K atoms at high temperatures, and can eject electrons and ions by the thermionic effect and by ion emission, respectively. Before discussing the chemical network, we review these chemical processes.

2.2 Gas-Phase Collisional Ionization and Recombination

As discussed above, the molecules’ ionization can come from energetic particles such as Galactic cosmic rays, X-ray photons, or particles from radioactive decays. The number of ionizations per volume per time is defined as . We are most interested in protoplanetary disks’ interiors, so we assume that the cosmic rays and X-rays are shielded, leaving only ionization by the decay of radioactive elements. If short-lived radionuclides are not present, then decay of , as well as , and , dominate, yielding (Umebayashi & Nakano 1981). If the short-lived () radionuclide is present at levels inferred from meteorites (MacPherson et al. 1995), then (Umebayashi & Nakano 2009). Our focus here is on high-temperature ionization, so we arbitrarily take in what follows. (If ionizations by are significant, the ion and electron densities would rise sharply above 800 K instead of 700 K.) The ions quickly exchange charge with other molecules to produce ions such as , which then also exchange charge with free atoms, creating ions such as . We assume these steps are faster than the other reactions, so each ionization of quickly yields an atomic ion. We do not consider as one of these ions, because other species like Mg are more abundant. Instead we assume ions are produced mainly by thermal ionization.

Atomic ions and free electrons can recombine in the gas phase. We assume a rate per volume , where , measured in K (Oppenheimer & Dalgarno 1974). Ions and free electrons can also be adsorbed onto grain surfaces, at rates




where and reflect modifications to the collision cross section due to the grain charge (Draine & Sutin 1987).

Thermal ionization of alkali atoms occurs if they are in the gas phase and suffer collisions with ambient molecules, mostly . This process can be represented by the reaction

In the gas phase, electrons and ions can radiatively recombine, as follows,

but in the regions of protoplanetary disks where temperatures are (the midplane regions well inside 1 AU), the gas densities are likely high enough that three-body recombinations dominate:

We justify this statement below.

Literature on the collisional ionization and recombination rates is sparse. Pneumann & Mitchell (1965) present a kinetic rate, but cite no measurements or sources. Their rate also appears to conflict with that provided by Ashton & Hayhurst (1973). Since the latter is experimentally determined by analyzing the chemical reactions in flames, it is our preferred value. Following Ashton & Hayhurst (1973), collisional ionization takes place at a rate per unit volume




and is measured in K. Here for K, and . The 3-body gas-phase recombination rate is




and again is measured in Kelvin (Ashton & Hayhurst 1973).

The two coefficients and must be related to each other. In the absence of dust, these would be the only rates affecting the ionization fraction, and should be in detailed balance with each other in local thermodynamic equilibrium. The ionization fractions found using these coefficients thus should conform to the Saha equation. Hence, , and , and therefore we expect


The values from Ashton & Hayhurst (1973) yield


consistent with the Saha equation.

Recombinations by 3-body reactions proceed at a rate , which is to be compared to the rate of radiative 2-body recombinations, . For typical radiative recombinations, (Oppenheimer & Dalgarno 1974). At , , and if , then . For densities , radiative recombinations of and electrons are less significant than 3-body recombinations, but we include both rates in our calculations.

2.3 Thermionic Emission

Thermionic emission, the ejection of electrons from heated solids, is a long recognized effect and the basis for the cathode ray tube. The rate at which electrons are emitted, per surface area of solid, is described by Richardson’s law,


where is a dimensionless number of order unity depending on the solid material, and is typically (Crowell et al. 1965).

Thermionic emission is sensitive to , the work function of the material – that is, the energy needed to make an electron escape the solid. As the grains lose electrons and develop a positive charge , removing more electrons requires extra work to overcome the Coulomb force. The effective work function is


For conductors, the escape takes place from the top of the conduction band, and the work function is easily measured and commonly tabulated. For example, the work function of graphite is 4.62 eV (Jain & Krishnan 1952), Fe has a work function of 4.31 eV, and that of Ni is 4.50 eV (Fomenko 1966).

Unfortunately, the grains likely to occur in the solar nebula and other protoplanetary disks contain many insulating materials, making determining the work function difficult. The best analogs to the grains in the solar nebula, and presumably in other protoplanetary disks as well, are the unaltered (pre-accretionary) matrix grains in chondrites. These grains are composed of crystalline, Mg-rich olivines [] and pyroxenes []; relatively Fe-rich amorphous silicates [] and []; and a smaller fraction (a few percent) of grains of metallic Fe,Ni and their sulfides (Scott & Krot 2005). Metallic FeNi grains are conducting, with a work function presumably  eV, but the silicates are insulators. Because of this, work functions of silicates are difficult to measure, and few data exist in the literature. Quartz () has a work function 5.0 eV, and mica (a phyllosilicate) has a work function 4.8 eV (Fomenko 1966), but the authors were unable to find any other direct measurements for specific silicate minerals. Far better than quartz and mica as analogs for solar nebula dust, however, is the JSC-1 lunar regolith simulant (McKay et al. 1993), composed of a mixture of 40 wt% ferromagnesian olivine [], 40 wt% plagioclase [ and ], the remainder comprised of Fe- and Ti-rich minerals like ilmenite (), anatase [], magnetite [], hematite [] and pseudobrookite []. Trigwell et al. (2009) examined how JSC-1 simulant is charged when brought into contact with different substances, and found it was negatively charged by Al (work function 4.28 eV), Cu (4.65 eV) and stainless steel (5.04 eV), but positively charged by the organic polymer PTFE (work function 5.75 eV). The acquired charge correlates with the work function of the other material, and the correlation strongly suggests JSC-1 itself has an effective work function somewhere between 5.0 and 5.4 eV. Feuerbacher et al. (1972) examined the photoelectric effect using JSC-1 simulant and found the threshold for photoelectron emission implies an effective work function close to 5.0 eV. Based on these measurements, we adopt a work function of 5.0 eV for solar nebula dust grains.

With the work function defined, the rate of electron production per volume due to thermionic emission from dust is


Again, for silicates we adopt . Since data are lacking for the coefficient , we adopt .

The inverse of thermionic emission is electrons sticking on dust grains, which occurs at a rate . In the absence of gas-phase processes affecting electrons, thermionic emission and its inverse could be expected to be in detailed balance, with and


Note that since the electrons’ production and destruction rates both scale with the grain surface area, the grain density and radii drop out of the balance equation. Solving for yields


This is identical in form to a Saha equation, involving a charged and neutral species and with statistical weights and , provided . Potassium ion and electron densities essentially follow a Saha equation, but with an energy equal to , which in practice is typically in the range of 3 to 4 eV (see below), not the ionization potential . In this way the dust grains fundamentally alter the gas ionization state, even in the high-temperature limit, where their effects are described by a Saha-like equation.

2.4 Evaporation of Neutral and Ionized Potassium Atoms

In addition to electrons, atoms can also emit ions. For the reasons outlined above, most relevant are ions. At high temperatures, potassium exists mostly in the gas phase, with K atoms that meet dust grains quickly vaporizing because of thermal vibrations. A single K atom bound to a grain surface vibrates on the lattice with a frequency . We adopt a typical value , appropriate for K bound to Pt (Hagstrom et al. 2000). In any vibration cycle, there is a probability of vaporization , where is the activation energy, equal to the binding energy, typically several eV. Defining as the number of K atoms in the grain minerals per volume of nebula, the vaporization rate per volume is


Balancing these vaporizations is the condensation of K atoms onto the grain surfaces. The total rate at which K atoms collide with dust grains is , where


being the focusing factor appropriate for ions (Draine & Sutin 1987) and the mass of a potassium atom, and


The ratio of bound K atoms to gas-phase K atoms can be found be balancing these two rates:


The fraction of potassium atoms in the gas phase (versus bound in grains) is strongly temperature dependent.

At low temperatures, potassium is unlikely to be found in the gas phase because it will chemically react with silicates and be bound into minerals. In particular, potassium resides in feldspar () below its 50% condensation temperature (Lodders 2003). To mimic this behavior, we assume the binding energy of K atoms in silicates is , so that at . Here we take a gas density , a dust-to-gas ratio , a grain density , a grain radius , a sticking coefficient , and neglecting the charge state of the K atoms.

At temperatures above 1006 K, potassium atoms will evaporate from the surface, but whether they leave as neutral atoms or as ions depends on their ionization potential and the work function of the grains, according to the Saha-Langmuir equation. Among departing potassium atoms, the ratio of ions to neutrals obeys


The fraction of leaving K atoms that are positive ions is then


The rate at which ions evaporate from grains (per volume of nebula) is therefore , and the rate at which neutral K atoms do so is .

2.5 Chemical Network

We now construct a chemical network to calculate the densities of the five species, (free electrons), (gas-phase atomic ions), (grain-adsorbed K atoms), (gas-phase K ions), and (gas-phase neutral K atoms), as well as the charge on each dust grain. The time derivatives of these quantities are as follows.


It is straightforward to confirm that the total number of potassium atoms (neutral, ionized, and bound) is conserved in such a model, i.e., , where we assume (Lodders 2003). We seek a steady-state solution by setting all time derivatives to zero. Note that the five densities are, through the cross section modifications , all functions of the grain charge , the sixth variable. The sixth equation needed to close the system corresponds to charge neutrality. If grains’ mean charge is ,


In practice, we vary till charge neutrality is achieved and the other densities are in steady state.

More specifically, we find the steady-state solution for gas of density and temperature as follows. At a particular value of , we calculate the cross section modifications for ions, electrons, and neutral atoms, using the approximation formulas of Draine & Sutin (1987). We assume the gas and dust temperatures are identical; in protoplanetary disks this is an excellent assumption except for the uppermost surface layers (Glassgold et al. 2004). With the coefficients known, we then solve for the species’ steady-state abundances. Given guesses for the densities of and , we find the electron density by first solving for :


This value is then substituted into the equation for the electron density,


to derive a quadratic equation for :


Once is found, is found using the above equation, as are the other densities, as follows:


yields the density, in terms of which




In practice, at each guess for we iterate between solving for and solving for , and , then updating . Performing 10 such iterations allows us to compute the net charge for the given . At very large the net charge is positive, while at very large it is negative, so it goes to zero at some finite . We use a bisection method with 80 steps to find the value of that yields both charge neutrality and ionization equilibrium.

3 Results

3.1 Canonical Case

We have solved for the abundances of electrons, ions ( and ), as well as neutral and bound K atoms, using the chemical network described above. We start with our canonical case, characterized by , grains with a dust-to-gas mass ratio 0.01, uniform radius , and work function . Figure 2 shows the fractional abundances of electrons, , as well as , , , and , as functions of temperature. In cold gas, , the ion and electron densities come from the balance between ionizations by energetic particles and collisions with dust grains. Because of their slower thermal velocities, ions take longer to reach the grains and are more abundant, with , compared to for electrons. The very low abundances are a consequence of the low ionization rates associated with the radioactivities. Essentially all potassium is bound into solids, with few ions or neutral atoms in the gas phase. As the temperature increases, however, and especially enter the gas phase. Some come directly from evaporation of ions from grain surfaces; because , most K atoms leave as ions. In addition, once the temperatures exceed about 750 K, electrons are produced effectively by thermionic emission. Above 1000 K, the potassium atoms evaporate from solids; most takes the form of neutral atoms in the gas-phase. Only in gas hotter than about 1500 K do the ions compare in abundance to the neutral atoms.

Figure 2: Abundances of electrons (solid red curve), atomic (Mg) ions (dotted blue curve), ions (dashed green curve), gas-phase atoms (dash-dot curve), and bound K atoms (dash-dot-dot-dot curve), as functions of temperature , for the canonical case (gas density , dust-to-gas mass ratio 0.01, grain work function . At low temperatures, non-thermal ionizations (from radioisotopes) dominate, but emission of ions from dust grains is significant for . Evaporation of atoms from grains, and their subsequent thermal ionization, contribute significantly to and electron densities for . For , thermal ionizations and recombinations dominate and are nearly in thermodynamic equilibrium. For comparison, we also plot the electron abundance derived using the Saha equation (dashed red curve). The Saha equation overpredicts the ionization at temperatures K, and underpredicts the ionization at higher temperatures.

The bolded curve in Figure 3 shows the charge on dust grains as a function of temperature. Below about 700 K, when the ionization rate is low, grains typically hold only a few electron charges. The value of is such that charging by electron and ion collisions balance each other, including the effects of charge on the collision rates (i.e., , ). As the temperature increases, however, ion emission comes to dominate the charging, and the grains quickly become negatively charged; at 900 K the charge is about . Because adsorption of gas-phase ions is unimportant, the grains charge until is reduced enough that ion emission is suppressed and thermionic emission enhanced. In other words, ions are emitted until approaches unity, which requires . At 900 K, . As the temperature increases, collisions of ions with grain surfaces begin to matter, and the grain charge is reduced, approaching the thermodynamic limit where , or (at 1500 K).

Figure 3: Charge on dust grains (in units of elementary charge ) as a function of temperature, for the canonical case but with various grain work functions. From top to bottom, , , … . The bolded curve corresponds to . For low work functions (, thermionic emission dominates the charging, and dust grains are positively charged; as temperature increases, thermionic emission becomes more effective, and grains become more positively charged. For large work functions (), ion emission dominates, and grains become negatively charged; as the temperature increases, collisions of ions onto the grains becomes more effective, and grains become less negatively charged.

The relative abundances of neutral and ionized K atoms also approach the thermodynamic limit only at high temperatures. The bolded curve in Figure 4 shows what effective ionization potential energy must exist so that the Saha equation would yield the correct ratio for , i.e.,


At high temperatures it tends to approach the actual value of for potassium, , but at lower temperatures is lower, reflecting the fact that many K atoms evaporate as ions from grain surfaces, according to the Saha-Langmuir equation. At all temperatures, but especially below 800 K, emissions from dust grains generate more ions than the Saha equation would indicate.

Figure 4: Effective ionization potential, , needed for the Saha equation to return the correct value of , as a function of temperature, for various values of the work function, . The value of (in eV) is used to label several of the curves. The case is bolded. The horizontal dashed line refers to , the value tht would obtain in the absence of species other than potassium. Because of emission of charges from dust grains (thermionic emission of electrons at low , ion emission of at large ), the gas charge state is always greater than expected based on a pure Saha equation for potassium; that is, . The effect is most pronounced and sensitive to at low temperatures . For , for between 2 and 6 eV. For work functions close to the ionization potential of potassium (), .

Figure 5 shows the timescales for the steady-state electron density to be affected by various ionization / recombination processes, as functions of temperature. For example, since is the rate per volume of nebula at which electrons are produced, . Likewise, the timescale for recombination of electrons with atomic ions is defined as . At low temperatures, , the dominant loss mechanism is adsorption of electrons onto dust grains, while the dominant production mechanism is ionizations of gas-phase molecules by energetic particles. These balance each other and the electron density turns over on timescales of roughly 10 seconds. At temperatures above 800 K, thermionic emission of electrons from grains dominates, and the electron density increases. Ionizations by energetic particles become less important compared to thermionic emission. But because the electron density is greater, the timescale for overturning the electron population rises, to above 1 hour at about 780 K, to about 1 day at 870 K, and peaking at  s at 900 K. At higher temperatures still, the timescales for both adsorption and thermionic emission decrease, eventually falling below about 1 hour again for .

Figure 5: Characterisic timescales for various processes affecting the electron abundances, as functions of temperature, for our canonical case. The dominant removal processes is always electrons colliding with grains (‘adsorption’, solid curve). At low temperatures (), ionization of by energetic particles (’ionize, ’, dashed curve) is the dominant charging process. At higher temperatures, emission of electrons from grains (’thermionic’, long dashed curve) dominates. At high temperatures the timescales for thermal ionization of K (dashed-dotted curve) and recombinations of ions and electrons (dotted curve) become shorter but never dominate. Recombination of atomic ions and electrons is slower than all other processes and is not plotted. Notably, for temperatures , the chemical timescales are longer than hours.

3.2 Variation with Work Function

We now investigate the effect of varying the work function of the solids making up the grains, keeping other parameters the same as our canonical case (, dust-to-gas ratio ). Figure 3 shows the charge on each dust grain (in units of ) as a function of temperature for various . At low work function, , thermionic emission of electrons is very effective, whereas any potassium atoms leaving are likely to become neutral atoms. The grains are positively charged. At higher temperatures, the rate of thermionic emission is greater, and the dust grains are increasingly positively charged. For high work function, , thermionic emission is ineffective, but emission of is effective. At higher temperatures, grains emit ions more effectively and so are more negatively charged. At temperatures to 1000 K, however, ion emission is as effective as it can be. At still higher temperatures, gas-phase ions’ collisions with dust grains are increasingly important, and the grains become less negatively charged.

Figure 4 shows the effective ionization potential, , needed for the Saha equation to yield our calculated value of , as a function of temperature, for various work functions . At low temperatures this ratio depends sensitively on , as the ion and electron abundances are controlled by grain processes. Especially for high work functions, , ion emission dominates and increases above the levels predicted using the Saha equation. At low work functions, thermionic emission increases . In gas hotter than about 800 K, thermal ionizations dominate and is driven toward the ionization potential ; however, this is only a trend, and the effective ionization potential falls in the range . The departure of from is most pronounced for that are far from ; for , . Notably, though, never appreciably exceeds , meaning the dust makes the gas better-ionized than would be predicted using the Saha equation alone.

Figure 6 shows the abundances of various species if the work function is increased to . Compared to the case with , the abundances as a function of temperature are quite similar, with the notable differences that is much reduced, and rises with much more steeply. This underscores that what is driving up for