Lags in Desorption of Lunar Volatiles

Because the exosphere of the Moon is a macroscopic manifestation of interactions between gas atoms and the surface (e.g., Stern 1999), exospheric measurements can be used to extract information about the gas-surface bond. With some exceptions (H₂, He, Ne), gas particles accumulate, or adsorb, on the cold surface at lunar night and desorb from the warmer dayside soils, with the local time of the density peak constraining the temperature dependence of the desorption rate and the strength of the bond. Exosphere models are tasked with retrieving the strength of gas-surface interaction from such measurements by repeatedly simulating adsorption and desorption events until loss of these atoms and molecules by photoionization or dissociation. They tend to overestimate the desorption rate because they do not account for the porosity of the powdery surface, and must compensate by using a higher effective activation energy for desorption than measured in laboratories to fit exospheric measurements (e.g., Grava et al. 2015; Hodges 2016). This has been interpreted as indicating that experiments conducted in laboratories do not represent the pristine conditions found on the Moon. While this conclusion may be fair, we demonstrate that an improved treatment of the porous surface in exosphere models reduces these discrepancies. Furthermore, we show that accurate knowledge of the distribution of adsorption energies from experiments is a necessary but not sufficient condition for high-fidelity exosphere models. Additional uncertainty is introduced by the role of surface diffusion, whose effect on desorption has been ignored by previous exosphere models.

We used two exemplary species, argon and water, to illustrate the range of outgassing delays caused by diffusion. Argon bonds very weakly with the surface (e.g., Grava et al. 2015; Kegerreis et al. 2017, and references therein), while the interaction between water and the lunar surface is stronger (Poston et al. 2015; Jones et al. 2020a). Argon gas is a product of radioactive decay of ⁴⁰K in the lunar soil and has repeatedly been measured in the lunar exosphere, both during the Apollo days and more recently by the Lunar Atmosphere and Dust Environment Explorer (LADEE; Benna et al. 2015; Hodges & Mahaffy 2016). To interpret the existing data Kegerreis et al. (2017) implemented a simple model to account for diffusion of argon adsorbates. They assumed that some argon atoms penetrate randomly into the ground at night, only to reemerge during the daytime with a uniform distribution in delay times when the regolith warms. The argon density peak at sunrise, and the day to night ratio near sunrise, could be simulated if the trapping fraction of a few parts per thousand atoms was assumed on every bounce. They concluded that diffusion provides an alternative to assuming very high activation energies for desorption (Hodges 2016; Hodges & Mahaffy 2016). A sophisticated model of a granular surface is used here to quantify how Ar moves between the subsurface and vacuum.

Desorption of adsorbed water has been presumed to be the cause of slope changes in reflectance spectra measured over a lunar day by the Lunar Reconnaissance Orbiter (LRO; Hendrix et al. 2019). Water gas may be generated by recombinative desorption of solar wind-implanted hydroxyl (Jones et al. 2018), and by meteoroid impacts, as shown by the detection of water-group products in the lunar exosphere during meteor showers (Benna et al. 2019). The gas form, water or hydroxyl, was not determined from these LADEE measurements. If water, the LADEE measurements imply that the water does not accumulate on the nightside and it lacks signatures of repeated adsorption–desorption cycles, perhaps because of dissociative adsorption to the regolith (Gun'ko et al. 1998). Whether or not water is continuously present in the lunar exosphere, in this study we use water as an example of an adsorbate that interacts with the lunar surface with a wide distribution of binding energies (Jones et al. 2020a). On such heterogeneous surfaces the residence time of different molecules on the surface will differ considerably. By analogy to volume diffusion controlling the timescale of hydrogen release from grain rims when they contain a distribution of crystal defects (e.g., Farrell et al. 2017), surface diffusion has been hypothesized to slow thermal desorption of adsorbates from heterogeneous planetary surfaces (Killen et al. 2018). This effect has not been considered in exosphere models and is quantified for the first time in this study.

A detailed description of the processes taking place on an isolated astrophysical grain is provided by He & Vidali (2014). Additionally, when a grain is enveloped in a grain pile, some desorption events lead to readsorption and some lead to diffusion deeper into the powder. And, finally, surface diffusion, i.e., the thermal migration of adsorbates between adjacent sites, or local minima of the adsorption potential on a grain, enables particles to find grain contact points and slowly travel onto other grains (e.g., Sarantos & Tsavachidis 2020).

Three random sphere-packings were computer-generated (Kloss et al. 2012) to simulate the disordered structure of lunar regolith. Each packing had width of 1 × 1 mm², depth of ∼5–7 mm, and consisted of 30,000 spherical grains. The grain size distribution for each packing was based on three Apollo and Luna samples that were representative of size distributions for mature (72141), submature (24109), and immature (67481) lunar soils. ³ The samples were truncated at one standard deviation, with a most likely particle diameter of ∼29 μm, a mean particle size of 44 μm, a maximum grain size of 374 μm, and a void fraction of 0.5 for the sample of intermediate maturity. Although piles of spherical grains provide better representation of the granularity of lunar regolith for gas calculations than the current state-of-the-art models, most lunar regolith particles are not spheres, and future work should include more appropriate particle geometries.

An obstructed random walk of 20,000 randomly distributed test particles was initiated in these packings to simulate gas transport. The spaces between grains in these simulations provided realistic diffusion paths. Similar approaches have previously been used to study gas transport in porous media under atmospheric pressures (e.g., Zalc et al. 2003; Hlushkou et al. 2013), but under lunar conditions the test particles collide only with grains or escape to vacuum. Particles were either deposited all at one time for isothermal simulations (Figure 1), or with a time profile for temperature-programmed desorption (TPD) simulations (Figures 2–3). Particles in the continuous dosing scenario (TPD runs) were uniformly distributed in time from sunset to sunrise to simulate adsorption during the long lunar night. For time-dependent runs the temperature was changed for every time step in order to simulate the heating rate experienced by patches of the lunar surface. Continuous linear heating or cooling from a lunar thermal model (Hurley et al. 2015) was used, but with a more gradual temperature ramp-up near the terminator due to realistic shadowing (Williams et al. 2017). No lateral or vertical temperature gradients were assumed, and their additional effects on the lifetime of adsorbates (e.g., Davidsson & Hosseini 2021) should be assessed in future work. Particles were removed when they escaped the medium, and particles remaining in the grain pile were permitted to undergo diffusion. Desorption events were recorded. We thus calculated a realistic initial condition for the distribution of adsorbates with depth as the surface element arrives at the dawn terminator.

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On every time step we separate the adsorption, desorption, and surface diffusion events. At the beginning of the step we insert new particles (for the time-dependent dosing simulations) and we test for each particle whether desorption is to take place. If desorbed, we mark with ray tracing whether the particle hits another grain, including perhaps many successive collisions with other grains until it comes to a stop (Knudsen diffusion), and if it escapes to vacuum it is removed from the simulation. If no desorption occurs, we check the probability for surface diffusion, and for those particles that will undergo hopping events we update the binding energy and location. When the sites have different adsorption energies, we randomly select the binding energy after each sticking event or surface diffusion hop by importance sampling of the binding energy distribution. This means that there is no correlation between activation energies of successive jumps. The distance traveled along the spherical grain from an adsorbate's current position within a time step of 1–4 s is simulated probabilistically as described in Sarantos & Tsavachidis (2020) (see also Hołyst et al. 1999).

Sticking is one of the parameters enhancing the retentiveness of soil. Higher sticking probability per collision of a desorbed atom with intervening grains leads to readsorption onto adjacent grains, prolonging the time to escape from the granular bed. The adsorption probability during single collisions with grains is described by a sticking coefficient, S, while the probability of reflection is 1-S. Values of S for gases on lunar regolith have not been reported, so we used values for simple metal oxides. The sticking probability of Ar on MgO decreases exponentially with gas incident energy, E, for gas speeds >250 m s⁻¹ (Dohnalek et al. 2002). Using their empirical relation S(E) = exp^−a(E-E0), where a = 0.202 mol kJ⁻¹ and E₀ = 2.08 kJ mol⁻¹, and integrating over a Maxwell–Boltzmann distribution for thermally accommodated gas, we find population-averaged sticking probabilities of S ≈ 1 at T = 50 K, S ≈ 0.76 at T = 200 K, and S ≈ 0.55 at T = 400 K. The high sticking probabilities used in our simulation are consistent with the efficient energy accommodation of lunar argon as indicated by LADEE measurements of the scale height (Hodges & Mahaffy 2016). The sticking probability of water on CaO single crystals is constant at temperatures 120–300 K and near 0.85–0.9 (Seifert et al. 2021). Similarly, a temperature-independent sticking coefficient near unity has been measured for low-coverage water on MgO single crystals at temperatures 100–250 K (Stirniman et al. 1996). In the results presented here we assumed S = 0.85 for single collisions of water with the lunar surface at all temperatures.

The adsorption time of a test particle between desorption events is calculated as −ln(1−ξ)/R_des, where R_des is the desorption rate and ξ a random number that is uniformly distributed between zero and one. This numerical scheme generates exponentially distributed waiting times for desorption with mean residence time τ = 1/R_des. The desorption rate increases with grain temperature, T, R_des = v₀ × exp(−E_des/KT). Desorption parameters for Ar were adopted from the work of Grava et al. (2015), who simulated the decay of argon density from sunset to sunrise using Apollo data. The surface interaction was empirically described by an unusually low pre-exponential factor v₀ = 2 × 10⁹ s⁻¹ and E_des = 0.281 eV, values adopted here, although other values of v₀ and E_des will give similar results (Kegerreis et al. 2017). The kinetic parameters describing the interaction of water with lunar grains were obtained by Jones et al. (2020a). A pre-exponential factor of v₀ = 10¹³ s⁻¹ and a wide distribution of binding energies between 0.6 and 1.9 eV, with the most likely value 0.63 eV, were derived from TPD experiments on Apollo Highlands sample 14163, and were adopted in these simulations.

The surface diffusion of adsorbates is simulated as a random walk between neighboring sites of potential energy with a jump rate, R_hop = v₀ × exp(−E_diff/KT). The barrier to surface diffusion, E_diff, is uncertain and is usually expressed as E_diff = αE_des, where α < 1 is a surface corrugation ratio. For complex surfaces and weakly adsorbed species α ∼ 0.7–0.8 (e.g., Perets et al. 2007). Values around 0.5–0.6 are reported for gases adsorbing on glass (Gilliland et al. 1974). Theoretical estimates of Ar tracer diffusion on oxides, carried out by Riccardo & Steele (1996), suggested α ≈ 0.45, the value assumed for the argon simulations here. As to water, its mobility on complex oxide surfaces is uncertain. On the one hand, high mobility of adsorbed molecules along the surface is expected from the finding that their entropies near desorption are 2/3 of their gas phase entropies, implying that only motion perpendicular to the grain surface is restricted (Campbell & Sellers 2012; Weaver 2013). Theoretical calculations on some model oxide surfaces confirm this expectation for water. For instance, the motion of an isolated water molecule adsorbed on MgO was estimated by McCarthy et al. (1996), indicating that adsorbed water molecules are very mobile on this surface (α ≈ 0.25). On the other hand, water molecules on many oxides are partially dissociated, and the dissociation products are shown to be immobile (e.g., Matthiesen et al. 2009). Yet for dissociation to occur, some mobility of water molecules is necessary to even reach the active sites (Schaub et al. 2001). For these reasons, a wide range of values (α = 0.25–1) for water surface diffusion have been assumed in our work. When there is a distribution of sites this ratio, α, was assumed to be constant at all sites regardless of the depth of the adsorption well.

The model of gas–solid interaction adopted here is suitable to extended regions of the Moon. Adsorbate–adsorbate repulsive or attractive interactions can be neglected if the average separation between adsorbates is large. Hendrix et al. (2019) inferred that less than 0.01 of a water monolayer (1 ML ∼ 10¹⁵ cm⁻²) exists on the lunar surface. The adsorbate abundance, C_ads, in flux balance with the argon gas density, n_g, can be estimated as C_ads = n_g × v_g × t_res. To support a gas density n_g = 2 × 10⁴ cm⁻³ (Benna et al. 2015) for a thermally accommodated gas of mean speed v_g with the Grava et al. (2015) parameters for the surface residence time, ${t}_{\mathrm{res}}$, the surface coverage of adsorbed argon must be C_ads ∼ 10¹²–10¹³ cm⁻², i.e., 1 in every 100–1000 sites near sunrise are occupied. In many gas-surface systems abundances higher than a tenth of a monolayer must be inferred before coverage-dependent kinetic parameters must be considered (e.g., Zhdanov 1991). Furthermore, gas atoms and molecules are deposited not only on the top 1 or 2 grains of the porous regolith, but up to 10 grains into the subsurface (Figure 1(a); see also Kulchitsky et al. 2018). Thus, the grains are essentially bare over large portions of the lunar surface, and the noninteracting test particle approach is valid.

3.1. Adsorbed Argon

3.2. Adsorbed Water

We find two sources of gas lag corresponding to two forms of diffusion. When all sites are equivalent, the desorption lag is due to Knudsen diffusion followed by readsorption. The time dependence of the adsorbate reservoir beyond the half-life, and hence the exosphere longevity, is not correctly described by an exponential decay because thermal desorption and Knudsen diffusion form a negative feedback loop inside a powder (Figures 1–2). To describe the removal of gas and derive meaningful activation energies for desorption without explicitly treating the multiple types of diffusion covered here, expressing the gas release as a second-order reaction with respect to surface concentration is a reasonable approximation in global exosphere models.

We suggest that the thermal desorption flux be modeled as F_out(t) ≈ 0.25 × R_des × σ × C_ads(t)², where F_out is the instantaneous outgassing flux corresponding to an adsorbate coverage C_ads, R_des is the single-grain desorption rate, and σ = 10⁻¹⁵ cm² is the cross section of an adsorption site. The factor 0.25 denotes the reduction of the yield from flat surfaces due to readsorption (Figures 1–2). This parametric change will describe how volatile adsorbates dissipate after sudden changes (e.g., crossing the terminator, following a natural or manmade impact, or following the descent of a lunar lander).

Surface diffusion, another diffusion mechanism ignored in exosphere models, has measurable macroscopic effects. For strongly bound gases, such as alkalis at the Moon, surface diffusion assists adsorbates to escape microscopic shadows, enabling more adsorbates to be degassed by photons but limiting the photodesorption rates (Sarantos & Tsavachidis 2020). For volatiles the effect of surface diffusion is more pronounced when a wide distribution of binding energies is assumed. Thermal hopping of adsorbates between physisorption (shallow) and chemisorption (active) sites, i.e., diffusion in the energy dimension, affects the desorption rate in a complicated way, suppressing desorption at lower temperatures (He & Vidali 2014), but enhancing desorption over a lunar day depending on the maximum dayside temperature of the soil. With this redistribution mechanism, desorption at sunrise is slower and occurs at later local times for species like water (Figure 3). More generally, when surface diffusion has a lower barrier than desorption, the occupancy of desorption energies is not evolved correctly with time in exosphere models, and the modeling error (e.g., local time of the peak density) grows as a wider distribution of binding strengths is assumed. Kinetic parameters for adsorbate mobility and desorption are equally needed from experiments.

The demonstrated effects of Knudsen and surface diffusion apply to obtaining upper limits for all lunar volatile gases. Volatiles such as CO, methane, and CO₂, which adsorb weakly with the nightside soil, will experience slower outgassing and peak at later local times. Given binding energies and surface mobility, the modeled output is less sensitive to remaining parameters (sticking coefficient, grain size distribution). Only the timeline and magnitude of the gas delays change with the strength of the gas-surface bond, not the general trends described here (higher-than-first-order desorption from granular medium, nonlinear effect of mobility on outgassing from heterogeneous surfaces). For instance, the second-or-higher-order desorption effect under diffusive conditions also appears in the simulation of alkali gas removal from loosely packed beds at Mercury's surface temperatures (Figure 3 of Sarantos & Tsavachidis 2020). Although the strength of that bond, 1.85 eV, is much stronger than the argon-surface bond, 0.28 eV, the granular medium produces the same slow desorption. For these reasons, the effects presented here do not merely describe specific gas-surface systems, but can be generalized to all exosphere-surface interactions involving thermal desorption.

This research was supported by the NASA Solar System Workings program and the Solar System Exploration and Research Virtual Institute (SSERVI) through LEADER. The granular beds for these simulations were produced with LIGGGHTS-Public v3.6.0. The Jones et al. (2020a) data were retrieved from Jones et al. (2020b). Output of simulations depicted in Figures 1–3 is available at Zenodo (doi: 10.5281/zenodo.5231999).