Andrew J Ryan
email: ajryan@orex.lpl.arizona.edu

Daniel Pino Muñoz

Marc Bernacki

Marco Delbo

Full-Field Modeling of Heat Transfer in Asteroid Regolith: Radiative Thermal Conductivity of Polydisperse Particulates

Keywords: Regolith, Airless Bodies, Numerical Modeling, Thermal Conductivity, Thermal Radiation, Packed pebble bed Index terms: 6205-Asteroids, 5407-Surface materials and properties, 5134-Thermal properties, 0545-Modeling

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Introduction

Asteroids and airless bodies, such as the Moon, are commonly blanketed in loose and unconsolidated material known as regolith, which preserves a record of the geophysical history of the body [START_REF] Murdoch | Asteroid Surface Geophysics[END_REF]. The particle size, maturity, and depth of the regolith layer are thought to vary within the asteroid populations depending on the size, composition, shape, and age of these bodies [START_REF] Murdoch | Asteroid Surface Geophysics[END_REF]and references therein). Regolith physical properties also place important constraints on mission planning, particularly for safety during sample acquisition events. NASA's Origins, Spectral Interpretation, Resource Identification, and Security-Regolith Explorer (OSIRIS-REx, [START_REF] Lauretta | The unexpected surface of asteroid (101955) Bennu[END_REF]) and JAXA's Hayabusa2 [START_REF] Watanabe | Hayabusa2 arrives at the carbonaceous asteroid 162173 Ryugu-A spinning top-shaped rubble pile[END_REF] missions are currently studying the surface of carbonaceous asteroids (101955) Bennu and (162173) Ryugu, respectively. The prime objective for both missions is to collect regolith samples and return them to Earth. Infrared emission measurements of the asteroid surface by OSIRIS-REx's Thermal Emission Spectrometer (OTES, [START_REF] Christensen | The OSIRIS-REx Thermal Emission Spectrometer (OTES) Instrument[END_REF] and Visible and InfraRed Spectrometer (OVIRS, [START_REF] Reuter | The OSIRIS-REx Visible and InfraRed Spectrometer (OVIRS): Spectral Maps of the Asteroid Bennu[END_REF], and by Hayabusa2's Thermal Infrared Imager (TIR, [START_REF] Okada | Thermal Infrared Imaging Experiments of C-Type Asteroid 162173 Ryugu on Hayabusa2[END_REF], are currently being used to infer the physical properties of the regolith, such as particle size distribution, porosity, and composition. This is possible because the amplitude of diurnal temperature cycling of the regolith is directly related to the thermal inertia of the regolith, which in turn is controlled by the physical characteristics of the regolith.

OSIRIS-REx and Hayabusa2 recently revealed that asteroids Bennu and Ryugu are both covered in abundant large boulders with very little fine regolith surface cover, which increases the challenge of sample site selection [START_REF] Lauretta | The unexpected surface of asteroid (101955) Bennu[END_REF][START_REF] Dellagiustina | Properties of rubble-pile asteroid (101955) Bennu from OSIRIS-REx imaging and thermal analysis[END_REF][START_REF] Walsh | Craters, boulders and regolith of (101955) Bennu indicative of an old and dynamic surface[END_REF][START_REF] Sugita | The geomorphology, color, and thermal properties of Ryugu: Implications for parent-body processes[END_REF][START_REF] Watanabe | Hayabusa2 arrives at the carbonaceous asteroid 162173 Ryugu-A spinning top-shaped rubble pile[END_REF]. Preliminary infrared telescopic observations of these asteroids were interpreted to represent regolith with particle diameters ranging from a few millimeters to about a centimeter for Bennu [START_REF] Emery | Thermal infrared observations and thermophysical characterization of OSIRIS-REx target asteroid (101955) Bennu[END_REF][START_REF] Müller | Hayabusa-2 mission target asteroid 162173 Ryugu (1999 JU3): Searching for the object's spin-axis orientation[END_REF]. However, the images of the surfaces of Bennu and Ryugu from both space missions show that this is not case. The mismatch between regolith size predictions and surface imaging observations could be due to unexpectedly low thermal conductivity and density of the solid material composing the boulders of these asteroids [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF], limitations in the models used to interpret thermal infrared observations in term of regolith particle sizes, or a combination of both. In this work, we focus on the latter and seek to improve existing thermophysical models for asteroid regoliths.

Several theoretical models exist that predict the thermal conductivity of monodisperse particulate media as a function of particle size, packing structure, and physical parameters of the particles themselves, such as their thermal conductivity and emissivity (e.g., [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF][START_REF] Gundlach | Outgassing of icy bodies in the Solar System -II: Heat transport in dry, porous surface dust layers[END_REF], 2013;[START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF]. However, no such models have sought to describe polydisperse particulates, despite the prevalence of particle size mixtures in nature (except for where there are advanced natural sorting mechanisms in place). Furthermore, only some models (e.g., [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] have sought to incorporate particulate nonisothermality effects, which could be significant for regolith on Ryugu and Bennu, particularly if the geologic materials have unusually low densities and thermal conductivities.

Experimental studies have provided invaluable data necessary to calibrate theoretical models for particulate thermal conductivity, including some preliminary analyses of polydisperse samples [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF][START_REF] Sakatani | Thermal conductivity of lunar regolith simulant JSC-1A under vacuum[END_REF][START_REF] Ryan | Heat and Mass Transfer on Planetary Surfaces[END_REF]). However, such experiments are known to be difficult to perform and can produce results with uncertainties that are difficult to quantify, e.g. due to factors that are difficult to control or know, such as changes in particulate porosity in the vicinity of heat sources/sinks [START_REF] Presley | Thermal conductivity measurements of particulate materials: 1. A review[END_REF][START_REF] Ryan | Heat and Mass Transfer on Planetary Surfaces[END_REF]. This limits the number of experiments that can reasonably be performed and the utility of the results. Furthermore, there is often uncertainty in the physical aspects of the samples being studied, such as the nature of the packing structure, the magnitude of contact deformation between particles, and relevant physical parameters of the particles themselves, such as their thermal conductivity and emissivity, that can complicate the process of relating experimental results to theoretical models. This problem leads us to the methods presented herein, where advanced numerical methods are used to create a realistic, though idealized, simulation of a thermal conductivity experiment in which physical parameters related to the "experimental" samples are well controlled and thus known with certainty.

We present a thermophysical model for particulates in which a bed of spherical particles is fully rendered in a three-dimensional finite element (FE) mesh framework. This type of model enables rapid and well-controlled investigations of the fundamental parameters that affect particulate thermal conductivity. In this work, we present validation cases for solid and radiative thermal conduction through ordered and randomly packed monodisperse particulates. Then, we present the findings of a study of the radiative thermal conductivity properties in mixtures that follow power-law particle size frequency distributions (SFDs) commonly observed in boulders and coarse regolith on asteroid surfaces [START_REF] Bierhaus | The OSIRIS-REx Spacecraft and the Touch-and-Go Sample Acquisition Mechanism (TAGSAM)[END_REF]. Radiative thermal conductivity is selectively studied here because its magnitude tends to greatly exceed that of conduction through the solids for coarse particulates (> a few mm diameter, as explained in Section 2: Background and shown in Figure 1). Given the likely abundance of coarse materials on Bennu and Ryugu and the need to identify particle size thresholds below ~2 cm for sampling, the study of radiative thermal conductivity in coarse particulates mixtures is of high priority. We pay particular attention in these analyses to the effects of particle non-isothermality, given the initial findings by OSIRIS-REx and Hayabusa2 of geologic materials with potentially low thermal conductivity on Ryugu and Bennu (DellaGiustina and Emery et al., 2019;[START_REF] Sugita | The geomorphology, color, and thermal properties of Ryugu: Implications for parent-body processes[END_REF][START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF].

Background

Two modes of heat transfer exist in particulate media in a vacuum -radiation between particles and conduction through the solids at the contacts between particles [START_REF] Wesselink | Heat conductivity and nature of the lunar surface material[END_REF][START_REF] Watson | The thermal conductivity measurements of selected silicate powders in vacuum from 150 degrees-350 degrees K[END_REF][START_REF] Wechsler | Thermal Properties of Granulated Materials[END_REF][START_REF] Van Antwerpen | A review of correlations to model the packing structure and effective thermal conductivity of packed beds of mono-sized spherical particles[END_REF][START_REF] De Beer | A review of methods to predict the effective thermal conductivity of packed pebble beds, with emphasis on the near-wall region[END_REF]. These are typically represented in terms of their contributions to the bulk thermal conductivity of the regolith, where conductivity due to radiative heat transfer is denoted by kr and conductivity due to the contacts between the particles is denoted by ks. In the presence of an atmosphere, a third heat conduction term must be considered to account to heat conduction through pore-filling gas. The gas thermal conductivity, kg, tends to be the dominant heat transfer mechanism on Mars [START_REF] Presley | Thermal conductivity measurements of particulate materials: 1. A review[END_REF]. Although gas conduction will not be considered in this present work, a proper treatment of thermal conductivity due to radiation and particle contacts is still useful for improving models of regolith thermal conductivity on Mars, Earth, or other bodies with appreciable atmospheres.

For large, opaque particles, heat transfer by radiation is governed by the Stefan-Boltzmann law, which is influenced by the temperature, emissivity, and distance between radiating surfaces. Most theoretical formulations for radiative heat transfer in particulates begin by approximating the particles as a series of parallel, radiating plates separated by some representative distance for radiative transfer. The radiative thermal conductivity (kr) of such a configuration is proportional to the cube of the local mean temperature (e.g., [START_REF] Wesselink | Heat conductivity and nature of the lunar surface material[END_REF]. The representative distance for radiative transfer is often related to the size of the pore space between particles, which in turn is related to porosity and particle diameter, as is the case in the model for regolith thermal conductivity by [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF].

The regolith thermal conductivity model by [START_REF] Gundlach | Outgassing of icy bodies in the Solar System -II: Heat transport in dry, porous surface dust layers[END_REF]2013) uses the mean free path of the photon as the representative distance for radiative transfer. This formulation is based on theory for very small particles (<100 µm diameter), where particle transparency becomes important [START_REF] Merrill | Thermal conduction through an evacuated idealized powder over the temperature range of 100 to 500 K[END_REF]. As such, it should be used with caution for larger particulates, particularly given that Gundlach and Blum's model predicts a much stronger dependence between radiative thermal conductivity and porosity than most other theoretical models (e.g., [START_REF] Schotte | Thermal conductivity of packed beds[END_REF][START_REF] Watson | The thermal conductivity measurements of selected silicate powders in vacuum from 150 degrees-350 degrees K[END_REF][START_REF] Breitbach | The radiance heat transfer in the high temperature reactor core after failure of the afterheat removal systems[END_REF][START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF][START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF].

A more advanced formulation by [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] separately describes the radiative contribution of immediately adjacent spheres (short-range radiation) and nonadjacent spheres that still have some direct line of sight (long-range radiation). They incorporate approximations to describe the view factors-i.e., the proportion of radiation emitted from one surface that reaches another-between spheres for both of their radiative conductivity formulations. Furthermore, they demonstrate that the thermal conductivity of the solid material that composes the spheres can influence the radiative thermal conductivity, building upon work by [START_REF] Breitbach | The radiance heat transfer in the high temperature reactor core after failure of the afterheat removal systems[END_REF] and [START_REF] Singh | Effect of solid conductivity on radiative heat transfer in packed beds[END_REF]. In such cases, thermal gradients are present within individual particles, with the consequence being that the radiative thermal conductivity would not increase in proportion to the cube of the temperature but rather at a lower rate. [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] introduce an updated description of a non-isothermal correction factor, fk, that is related to a dimensionless parameter, Λs:

(Eq. 1)
where km is the thermal conductivity of the material that constitutes the particles (the thermal conductivity of a single particle), D is particle diameter, is the Stefan-Boltzmann constant, and T is the mean local temperature. If 1/Λs is greater than approximately 0.04, the particle nonisothermality can have a significant, measurable effect (>~1%) on radiative thermal conductivity. [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] provide correlation coefficients to relate Λs to the correction factor, fk in the direction of heat flow:

(Eq. 2)
When emissivity is unity, a1 = -0.3966, a2 = 0.7495, a3 = 0.5738, and a4 = 1.0484. When , fk = 1. These four correction factor coefficients were determined from numerical simulations of radiative heat transfer between two almost-touching hemispheres, implying that they are accurate for correcting short-range radiation non-isothermality. [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] use the non-isothermal correction factor for both short-and long-range radiative conductivity by simply multiplying each conductivity by fk but note that more work is needed to verify its validity for long-range radiation. Experimental and theoretical work describing particle thermal conductivity in the context of planetary regoliths has yet to incorporate nonisothermality in the radiative thermal conductivity, primarily owing to the normally small size of the particles under consideration (typically <1 mm) and the relatively high thermal conductivity of geologic materials (e.g., basalt: ~1-2.5 W m -1 K -1 [START_REF] Desai | Thermophysical Properties of Selected Rocks (CINDAS/TPRC Report 23)[END_REF]). Finally, it should be noted that even though the thermophysical properties of a planetary surface are determined in terms of thermal inertia, which is related to the square root of thermal conductivity, the effects of non-isothermality could still significantly affect the estimation of particle size on rubble pile asteroids like Bennu and Ryugu.

Having discussed the thermal conductivity of particulates due to radiation (kr), we now move our attention to the thermal conduction through the particle-to-particle contacts. The bulk thermal conductivity of particulates due to these contacts (ks) is directly related to but typically much smaller than the thermal conductivity of the individual particles themselves (denoted here by km for material conductivity). Although the thermal conductivity of most common geologic materials is relatively high (~2 to 4 W m -1 K -1 ), the flow of heat from one particle to the next is severely hindered by the very small size of the contacts between particles. The size of these contacts and the average number of contacts between particles (the coordination number) thus strongly influence the solid thermal conductivity of particulates. To estimate the size of the [START_REF] Gundlach | A new method to determine the grain size of planetary regolith[END_REF] and [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] use the model for interparticle forces due to adhesion and the resulting Hertzian deformation by [START_REF] Johnson | Surface energy and the contact of elastic solids[END_REF] ("JKR theory") to predict the particle contact radii that are used in their models for the thermal conductivity of particulates. The contact radii rc are typically very small compared with the radii of the particles Rp, with values for the ratio rc/Rp in the range of 0.0005-0.005 (depending on values for particle elastic moduli, surface energy, and gravity). As such, the magnitude of thermal conductivity by means of heat transfer between the particles via their contacts, ks, is typically several factors or even orders of magnitude smaller than the thermal conductivity of the particle material itself, km. [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] have proposed that ks and km are linearly related to each other by the ratio rc/Rp [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF]. Also, they and others (e.g. [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] have propose that ks is linearly related to the mean coordination number. Several model correlations between coordination number and porosity have been proposed (see van Antwerpen et al. ( 2010) for a review), but more work is needed on this topic to resolve discrepancies between models and to determine the effects of particle sphericity and roughness.

Λ s = k m 4Dσ T 3 σ f k = a 1 tan -1 a 2 ( 1 Λ s ) a 3 + a 4 1/Λ s < 0.01 contacts,
Figure 1 illustrates the relative contributions of radiative thermal conductivity and solid conductivity to the total thermal conductivity of particulates on Bennu, as predicted by the particulate thermal conductivity model by [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF]. The radiative thermal conductivity increases linearly with mean diameter, whereas solid thermal conductivity only changes modestly (or not at all, if adhesion is neglected). Therefore, the radiative thermal conductivity term comes to dominate for particles larger than a few millimeters in diameter.

All existing theoretical models for regolith thermal conductivity assume monodisperse particle SFDs (all particles have the same size). Some experimental work, however, has been conducted to determine whether these models could be used to approximate the thermal properties of particulates with polydisperse SFDs. [START_REF] Sakatani | Thermal conductivity of lunar regolith simulant JSC-1A under vacuum[END_REF] measured the thermal properties of the JSC-1 lunar regolith simulant (maximum diameter <=~1 mm) with a log-normal cumulative particle mass distribution and found a thermal conductivity approximately equivalent to that of the volumetric median particle size (~100 µm). However, it remains to be determined whether this relationship is applicable to other particle size distributions and whether it holds for both radiative and solid conductivity terms.

Figure 1.

Illustrative example of the predicted ratio of radiative to solid thermal conductivity (i.e. kr/ks) in monodisperse particulates on Bennu as a function of particle diameter (indicated next to the lines). Values were calculated with the model from [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF], using parameters assumed for asteroid Bennu, where particulate material properties are similar to CM chondrite Cold Bokkeveld [START_REF] Opeil | The thermal conductivity of meteorites: New measurements and analysis[END_REF], surface energy is 0.02 J m -2 , depth of burial is 1 cm, gravity is 75 µm s -2 , emissivity is 1.0, and porosity is 0.40. [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] and [START_REF] Gundlach | A new method to determine the grain size of planetary regolith[END_REF] models yield comparable predictions when the same or similar input parameters are used.

Methods and Model Validation

We study the thermal conductivity of particulates in a vacuum within a FE framework where constant heat flux is applied to one side of a three-dimensional, parallelepiped geometry of randomly packed spheres while a constant temperature boundary condition is applied to the opposite side (Figures 2 &3). Once the model temperatures reach steady state, the thermal conductivity of the particulates can be calculated from the temperature differential between the two opposing geometry boundaries where the flux and fixed temperature boundary conditions have been applied:

(Eq. 3)
Or in the case where the contribution of the two plates is removed:

(Eq. 4)

k = -q ∆ x ∆ T
where q is applied heat flux along the geometry boundary, is the total distance from one end of the sphere domain to the other, is the final steady-state temperature differential across the entire domain (including the end plates), is the total distance between the outer surfaces of the two plates, is the combined thickness of the two plates, and is the material thermal conductivity used in the simulation, which is the same for the plates as it is for the spheres. All adjacent boundaries are insulated such that the net heat flux occurs only in one direction.

Figure 2.

Cross-sectional illustration of packed sphere and end plate geometry that is used for steady-state thermal conductivity determination (Equation 4). Plate thicknesses are exaggerated and are not to scale. See Figure 3 for an example 3D geometry.

Multiple simulations for a single geometry are typically conductivity across a range of temperatures. In this work, the fixed-temperature plates were typically set to temperatures of 250, 300, 350, and 400 K. The prescribed flux values (q) were chosen based on the estimated bulk thermal conductivity of the geometry so that a thermal gradient (

) on the order of 15 K would be achieved at steady state. The resulting thermal conductivity that is determined at the end of the simulation is assumed to be representative of the mean temperature of the geometry at the final steady state. If the prescribed heat flux is too high, the thermal gradient at steady state would be very large, which makes it dubious to use the mean temperature to represent the bulk conductivity given the temperature dependence of radiative heat transfer. If the flux is too small, and the final temperature gradient value might be so small that precision is lost.

Heat conduction calculations for the tetrahedral domain mesh elements within the threedimensional model geometry are performed using an implicit diffusion equation solver that is part of the CIMLib scientific FE library [START_REF] Digonnet | Cimlib: A Fully Parallel Application For Numerical Simulations Based On Components Assembly[END_REF]. Radiative heat transfer between

k = ∆ x total -∆ x plates ∆ T q - ∆ x plates k m ∆ s sph ∆ T ∆ x total ∆ x plates k m ∆ T
triangular surface mesh elements is calculated following the Stefan-Boltzmann law in the following manner for element i of n elements: (Eq. 5)

where A is the area, Fij is the view factor from element i to j (i.e. the fraction of radiation leaving surface i that strikes surface j), and is the Stefan-Boltzmann constant. This value is added explicitly to the heat diffusion calculations, such that the surface temperatures from the previous time step are used for the calculation in Equation 5. A good discretization of the surface of the particles requires a large number of triangular surface elements. Thus, the number of possible pairs of surface elements that could "see" each other is high and requires an efficient computation of the view factors. The Intel Embree high-performance ray-tracing library meets these demands by rapidly tracing rays between all surface elements that share direct line of sight.

View factors for all non-obstructed elements are calculated using the single line integration method described by [START_REF] Mitalas | Fortran IV Programs to Calculate Radiant Energy Interchange Factors[END_REF]) (c.f. Walton, 1986). The surface mesh size is refined such that the precise calculation of partially obstructed view factors between elements is not necessary, given that these types of calculations tend to be slower and less accurate [START_REF] Walton | Calculation of Obstructed View Factors by Adaptive Integration[END_REF]. Emissivity is assumed to be unity, and as such, multiple reflections are not calculated. Our view factor calculations and explicit radiative heat transfer solver were validated with a geometric configuration with a known analytical solution -two concentric, hollow spheres are assigned fixed-temperature boundary conditions at the inner surface of the inner sphere and at the outer surface of the outer sphere (see Appendix A). The resulting steady-state temperatures of the outside of the inner sphere and the inside of the outer sphere were found to be within 1.5% of the analytical solution. The analytical expression for this test case is derived in Appendix A.

The geometries used in this study consist of three-dimensional beds of spheres that have been cropped to the shape of a parallelepiped box. Sphere coordinates are generated with the optimized dropping and rolling method described by [START_REF] Hitti | Optimized Dropping and Rolling (ODR) method for packing of poly-disperse spheres[END_REF], following userspecified SFDs. Plates are added to the top and bottom of the volume to create uniform surfaces for the heat flux and fixed temperature boundary conditions. The geometries are rendered and meshed using the Netgen automatic three-dimensional tetrahedral mesh generator [START_REF] Schöberl | NETGEN: An advancing front 2D/3D-mesh generator based on abstract rules[END_REF] (Figures 3 and4).

We performed a size convergence study with a simple cubic packing structure to determine the optimal representative volume element (i.e. parallelepiped width and height) where boundary effects are minimized. Model widths of only four particle diameters produced homogenized thermal conductivity results that were within ~5% of the convergence value for a much wider volume. Similarly, a model thickness of four to five sphere diameters produced thermal homogenized conductivity results within ~5% of the convergence value. As such, we conducted all further analyses with model geometries widths and heights of at least four sphere diameters. Commonly, we used larger geometries for random packings to capture a larger, more representative sampling of the packing. Most simulations thus contained between 120 and 175 spheres. Typical element counts were 500,000 to 3 million for volumetric (tetrahedral) elements and 130,000 to 500,000 surface (triangular) elements. View factor computation times ranged

Q i = n ∑ j=1 A i F ij σ ( T 4 i -T 4 j )
σ from a few minutes up to about 6 hours. Convergence in each heat flow simulation was typically achieved within 3-24 hours, depending on the number of elements and the number of processors used.

Figure 3. Meshed monodisperse sphere geometry colorized with final steady-state temperatures from a numerical simulation where a constant heat flux was applied to the bottom boundary plate and a fixed temperature (250 K) was imposed on the top plate. In this particular case, the spheres are not touching, so heat transfer between spheres is accomplished solely by radiation. In other conditions, as explained in the text and shown in Figure 4, we generated contact bridges between spheres to study the combined effect of solid and radiative heat transfer in a particulate medium. Contacts between spheres can be approximated by adding cylindrical bridges at each contact location with radii equal to the desired contact radius rc. It is necessary to use a fine mesh in the vicinity of the sphere contacts, given the narrowness of the geometry and the expectation of steep thermal gradients due to thermal constriction resistance (Figure 4). Following the numerical work by De Beer (2014), we use a minimum mesh element size that is approximately 0.2*rc. To validate the solid conductivity within our model, we performed a series of tests with ordered packing structures where the size of the contact neck, rc, was systematically varied (Figure 5). Our numerical results are in very good agreement with theoretical predictions by [START_REF] Siu | Effective conductivity computation of a packed bed using constriction resistance and contact angle effects[END_REF]. Simulations with sphere contacts are generally more computationally intensive than those where the spheres are not in contact owing to the increased number of mesh elements. Therefore, although we have demonstrated the utility of this model for studying solid conduction through particulates via sphere-to-sphere contacts, the remainder of this work focuses on heat transfer between non-touching spheres by radiation only. To study the thermal conductivity of particulates due to radiation only, we reduced the diameter of the spheres by 0.1-0.5% to avoid overlapping mesh elements. This was beneficial for computation times, as the removal of the contacts between spheres greatly reduced the mesh element count. We performed a sensitivity analysis to determine whether these slight reductions in sphere radius would significantly affect the radiative thermal conductivity. We found that 1% radius reduction lowered the thermal conductivity by <1%. The applied change to sphere radius is thus assumed to be negligible.

In Figure 6, we compare the radiative thermal conductivity of a dense random packing of two test cases with monodisperse spheres with 2-mm and 1-cm diameters to theoretical model predictions. The model by [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] predicts the radiative thermal conductivity results of the numerical simulations with an accuracy of approximately ±5%. Although in these cases their model overpredicted our results for smaller spheres and underpredicted our model for larger spheres, results from other test cases showed no systematic trend with particle diameter.

We conclude from these tests that the [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] model can be used to reliably predict the radiative thermal conductivity of a monodisperse sample, which we use to our advantage to compare polydisperse geometries to monodisperse geometries with equivalent radiative thermal conductivity values.

Figure 6.

Radiative thermal conductivity results from numerical simulations of two monodisperse particulate geometries with diameters of 2 mm (upper plot) and 1 cm (lower plot, also shown in Figure 3). These results are compared to theoretical model predicted values. Temperatures were taken as the mean temperature from the numerical simulations.

Result for Monodisperse and Polydisperse Particulates

To study the radiative thermal conductivity (kr) of particle size mixtures, we conducted numerical simulations of steady-state heat flow following the methodology described above where the spheres are not touching each other. We analyzed 16 randomly packed polydisperse particulate geometries with particle SFDs following a cumulative power law (Table 1). We used five different millimeter-to centimeter-scale particle size ranges, some of which were directly based on the size ranges described in [START_REF] Bierhaus | The OSIRIS-REx Spacecraft and the Touch-and-Go Sample Acquisition Mechanism (TAGSAM)[END_REF] for tests of the OSIRIS-REx Touchand-Go Sample-Acquisition Mechanism (TAGSAM) tests. These size ranges are relatively narrow due to the possible absence of fines on Itokawa [START_REF] Bierhaus | The OSIRIS-REx Spacecraft and the Touch-and-Go Sample Acquisition Mechanism (TAGSAM)[END_REF], which may also true for Bennu and Ryugu. Also, wider size ranges would require larger representative volumes, which could prohibitively increase the number of spheres and thus mesh elements. We generated packings for each size range by using various different cumulative power-law exponent values (e.g., Figure 7). We performed steady-state heat flow simulations at four different temperatures (approximately 250, 300, 350, and 400 K) to determine the radiative thermal conductivity as a function of temperature. We conducted all simulations with a solid material thermal conductivity value km of 1.0 W m -1 K -1 . Many of the simulations were recomputed with higher and lower km values of 10.0, 0.5, 0.1, and 0.05 W m -1 K -1 to study the non-isothermality effect (Equations 1 and 2).

Table 1. Particle diameter size ranges used for polydisperse particulate models and the powerlaw exponent values used for each size range. The second and third row size ranges correspond to the "7c" and "7d" size distributions used by [START_REF] Bierhaus | The OSIRIS-REx Spacecraft and the Touch-and-Go Sample Acquisition Mechanism (TAGSAM)[END_REF] for OSIRIS-REx TAGSAM sample collection testing. The geometries given in the last row (1.0-3.0 cm diameter) are shown in Figure 7.

An example set of results for one packing geometry as a function of temperature and km is presented in Figure 8. In the simulations where km is high (≥1.0 W m -1 K -1 ), the radiative thermal conductivity scales with the cube of the temperature, as predicted by theory. Lower values of km cause the radiative thermal conductivity to decrease, particularly for higher temperatures. In extreme cases (e.g., km = 0.05 W m -1 K -1 ), the radiative conductivity was found to increase nearly linearly with temperature, rather than with temperature to the third power. These observations are Particle diameter (mm) min max Power law exponent values 3.18 12.7 -2.5, -3.0, -3.5

3.18 15.88 -2.5, -3.0, -3.5

3.18 19.05 -2.5, -3.0, -3.5 6.35 19.05 -2.5, -3.0, -3.5 10.0 30.0 -3.0, -4.0, -6.0, -8.0 similar to the non-isothermality described by [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF]. Non-isothermality may be quantified as the ratio between the numerical radiative thermal conductivity results with non-isothermality (1/Λs<0.04) and results without non-isothermality (1/Λs>0.04). For example, we calculated non-isothermality correction factor (fk) values for the numerical radiative thermal conductivity results in Figure 8 by calculating the ratio between all points on the plot and the solid theoretical curve. The calculated values of fk from all polydisperse simulations and our monodisperse validation simulations are plotted in Figure 9 against 1/Λs (Equation 1). In the case of the polydisperse geometries, we used the effective particle diameters (described in the next paragraph) to calculate the values of Λs in Figure 9.

It is illustrative to express the conductivity results for these polydisperse geometries in terms of a monodisperse particle diameter that has the same thermal conductivity. We found that, when non-isothermality is negligible (i.e. 1/Λs<0.04) , the radiative thermal conductivity of each polydisperse geometry is equivalent to the radiative thermal conductivity of a monodisperse geometry with some effective particle diameter. To determine the effective particle diameter for each polydisperse geometry, we used the [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] model to calculate predicted monodisperse radiative thermal conductivity values. The model input diameter was allowed to vary as a free input parameter, whereas porosity and average coordination number were known from the polydisperse geometries. The determination of effective particle diameter was done only with the numerical thermal conductivity simulation solutions that we expected to be unaffected by non-isothermality-i.e., 1/Λs was less than ~0.04. We expect the uncertainty in the determined effective particle sizes to be approximately ±5%, on the basis of the goodness of fit between the Sakatani model predictions and our monodisperse numerical test cases. The resulting effective particle diameters are compared to volumetric mean particle size, volumetric median particle size, and the surface-volume mean particle diameter (Sauter mean) of each polydisperse geometry in Figure 10. The Sauter mean diameter, D32, was calculated as the average of the volume-to-surface ratio of n particles by: (Eq. 6)

where Di is the diameter of the ith particle. Figure 8. Numerical radiative thermal conductivity results for a mixture of particles having diameters 6.35-19.05 mm with an exponential slope of -2.5 as a function of temperature and solid particle material thermal conductivity km. The effective particle diameter for the case where km = 10.0 W m -1 K -1 is well fit by the theoretical model of [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF] with an effective particle diameter of 11 mm. It is clear that lowering the value of km can substantially reduce the radiative thermal conductivity of the particulates, particularly at higher temperatures. Mean temperature from the numerical simulations were used to plot the x-axis temperatures here. For most simulations, the range in temperatures ( ) between the two plates at steady state was less than 15 K. For simulations with very low values of km, was as high as 30 K. Λs values were calculated using the effective particle diameter for each geometry. Both lines are calculated using Equation 2, assuming that fk = 1 when the equation produces a value greater than unity (when ). Numerical test results are shown for all polydisperse geometries in Table 1 and a set of monodisperse validation cases (e.g. Figures 3 and6) at temperatures ranging from 250-400 K and with km values ranging from 0.05-10.0 W m -1 K -1 . . Comparison of the effective particle diameter determined from the radiative thermal conductivity numerical results to different expressions of the mean particle diameter for the 16 polydisperse geometries (Table 1). Effective particle diameter of a polydisperse particulate geometry is defined as the diameter of a monodisperse particulate geometry that has the same radiative thermal conductivity.

D 32 = ∑ n i=1 D 3 i ∑ n i=1 D 2 i

Discussion

The new values of fk versus1/Λs in Figure 9 provide compelling evidence that an isothermal correction factor correlation exists that is applicable to both monodisperse and polydisperse particulates. We find that the fk values are well fit by Equations 2 and 3 using the follow coefficient values: a1= -0.568 a2 = 0.912 a3 = 0.765 a4 = 1.035 The value of fk approaches unity as drops to 0.04. Any value of fk is assumed to be unity when , as shown in Figure 9. The original correction factor correlation proposed by [START_REF] Van Antwerpen | Multi-sphere Unit Cell model to calculate the effective thermal conductivity in packed pebble beds of mono-sized spheres[END_REF] underestimates the effects of non-isothermality that were observed in our simulations (Figure 9). As mentioned in Section 2, their correction factor values were determined from a numerical model of only short-range radiation between directly adjacent spheres and had not been validated for long-range radiation. Our correlation accurately accounts for both short-and long-range radiation together and appears to be valid under a wide range of particle SFDs. We are confident that our correlation is not simply recording a bias in our numerical results, given that it is composed of numerical solutions from geometries with multiple different length and width dimensions, temperature differentials, absolute temperatures, mesh sizes, and boundary flux values. However, all geometries tested here are dense random packings of spheres with unit emissivity, whereas the regolith on Bennu, Ryugu, and other airless bodies is surely composed of non-spherical particles, possibly with a much wider range of porosities and non-unit emissivity. We intend to consider non-sphericity, roughness, different packing densities, and lower emissivities in future studies.

Our non-isothermality results have important implications for the interpretation of thermal inertia on Bennu and Ryugu. Initial results from OSIRIS-REx and Hayabusa2 indicate that the rocks on both asteroids may have very low thermal conductivities. Recently, [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF] determined that the boulder on the surface of Ryugu has a thermal conductivity in the range of 0.06-0.16 W m -1 K -1 , based on measurements by the MARA radiometer aboard the Hayabusa2 MASCOT lander. If the particulates that compose the regolith in finer-particulate regions have similarly low thermal conductivity values, the non-isothermal correction factor described here would be necessary to correctly interpret the surface thermal inertia values. As an illustrative example, we performed a particle size inversion using a hypothetical regolith thermal inertia value of 200 J m -2 K -1 s -1/2 for Bennu (Figure 11). This value falls near the low end of thermal inertia values so far determined for Bennu [START_REF] Rozitis | Thermal Inertia Maps of (101955) Bennu from OSIRIS-REx Infrared Observations[END_REF], which could be indicative in some cases of particulate regoliths. To avoid model bias, we used three different models to produce particle size predictions for this thermal inertia value, where the thermal conductivity of the individual particles (km) is allowed to vary. We assume that particle density must also vary with km due to changes in microporosity within the individual regolith particles. Following the work by [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF], we use the average results from two models for meteorite thermal conductivity as a function of porosity to determine the regolith particle microporosity that would correspond to each x-axis value of km (Henke et al., 2019;Flynn et al., 2019). Density is then calculated by assuming an average CM meteorite grain density (2960 kg m -3 , [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF]. The predicted particle sizes are shown using the three models with and without the non-isothermality correction (Equation 2) applied to the radiative thermal conductivity term. For comparison, the thermal conductivity of the Cold Bokkeveld CM2 chondrite meteorite [START_REF] Opeil | The thermal conductivity of meteorites: New measurements and analysis[END_REF]) and the boulder on Ryugu [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF] are included as vertical lines/bars. Finally, the diurnal skin depth (i.e. the approximate depth at which the amplitude of the thermal wave is attenuated by a factor of 1/e) for a 200 J m -2 K -1 s -1/2 thermal inertia surface is shown as a thick gray line. The skin depth is calculated by: (Eq. 7)

1/Λ s 1/Λ s < ~0.04
The rotation period, P, for Bennu is 15466 seconds [START_REF] Lauretta | The unexpected surface of asteroid (101955) Bennu[END_REF], thermal inertia, TI, is assumed at 200 J m -2 K -1 s -1/2 , specific heat, , is 750 J kg -1 K -1 [START_REF] Biele | The specific heat of regolith material[END_REF], and bulk density, , is estimated as a function of km, using the average of two models by [START_REF] Henke | Thermal evolution and sintering of chondritic planetesimals III. Modeling the heat conductivity of porous chondrite material[END_REF][START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] (see Methods section in [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF], assuming a grain density of 2960 kg m -3 [START_REF] Flynn | Physical properties of the stone meteorites: Implications for the properties of their parent bodies[END_REF] and a regolith macroporosity of 0.40.

The diurnal skin depth line in Figure 11 can be seen as an approximate upper cutoff for the validity of the particle size determination. This is based on the well-known idea that regoliths composed of particles that that are much larger than the diurnal skin depth are expected to behave thermally like solid rock (e.g. [START_REF] Christensen | The Spatial Distribution of Rocks on Mars[END_REF][START_REF] Ryan | Asteroid Regolith Thermophysical Properties: Porosity and Skin-Depth Effects[END_REF]. Particle size predictions that exceed the diurnal skin depth should be considered indeterminate, such that the particles could be any size approximately larger than the diurnal skin depth. Interestingly, we see in Figure 11 that if the non-isothermality effect is ignored, one might conclude that a thermal inertia of 200 J m -2 K -1 s -1/2 is consistent with effective particle diameters in the range of ~10-25 mm, depending of course on assumptions of km, density, and macroporosity. However, once nonisothermality is considered, the predicted particle size increases markedly as km is reduced. If the particles have a km value that is similar to the Ryugu boulder from [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF] (grey shaded vertical bar), then then one can only conclude that the regolith is predominantly composed of particles larger than ~15 mm. It very well could just be boulders or a bedrock. Conversely, if we assume that the regolith particles have a km that is equivalent the Cold Bokkeveld meteorite, one could conclude that this thermal inertia observation is consistent with a regolith with an effective particle size near 10 mm. Example particle size prediction for a hypothetical surface on Bennu with a thermal inertia of 200 J m -2 K -1 s -1/2 at 265 K [START_REF] Rozitis | Thermal Inertia Maps of (101955) Bennu from OSIRIS-REx Infrared Observations[END_REF] as a function of the conductivity of the material that makes up the regolith particles (km). Three regolith thermal conductivity models are used both without and with the use of the non-isothermal correction factor to make particle size predictions. For comparison, the predicted diurnal skin depth is shown as a thick, gray line. For simplicity, only one macroporosity (0.40) is considered here. Increasing the assumed porosity would have the effect of further increasing the predicted particle diameters. Two reference km values are displayed with vertical lines/bars: The Cold Bokkeveld (CM2) meteorite [START_REF] Opeil | The thermal conductivity of meteorites: New measurements and analysis[END_REF] and the Ryugu boulder, estimated from MARA radiometer data [START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF]. Model input values: macroporosity=0.40; specific heat =750 J kg -1 K -1 (a); particle density is a function of thermal conductivity, using the average of two models (b,c) for meteorite thermal conductivity vs microporosity and assuming a CM grain density=2960 kg m -3 (c); surface energy=0.032 J m -2 (d); emissivity=1.0; Bennu mid-latitudes surface acceleration=40 µm s -2 (e); Poisson's ratio=0. (personal communication, 2018), measured values for Tagish Lake simulant.

In addition to our findings related to non-isothermality, we have also found that a polydisperse particulate generally has a radiative thermal conductivity that is equivalent to a monodisperse particulates with some effective particle diameter (Figure 8, solid line), at least in the case of particulates that follow cumulative power law SFDs. We also found that the effective particle diameter can be used in Equation 1 to calculate for the determination of the nonisothermality correction factor, fk, for a polydisperse regolith.

Figure 10 compares the effective particle diameters for 16 polydisperse geometries to various expressions for the mean and median particle diameter in a particle size mixture. The best correspondence, nearly 1:1, is found with the Sauter mean particle diameter. The Sauter mean diameter may be physically interpreted as the diameter that could be used to produce a monodisperse population of spheres that has the same surface area and the same volume (but not the same number of spheres) as the polydisperse sphere population [START_REF] Kowalczuk | Physical meaning of the Sauter mean diameter of spherical particulate matter[END_REF]. In the context of our work, this means that the radiative thermal conductivity of a polydisperse population of spheres is the same as the radiative thermal conductivity of a population of monodisperse spheres that has the same total mass (or volume) and same total surface area-suggesting that the total surface area present within a particulate volume is an important control on the radiative thermal conductivity. This is perhaps not surprising, given that heat diffusion through opaque particulates is largely limited by radiative heat transfer between the particle surfaces. If surface area is indeed a control, then replacing the spheres with any other particle shape would have the effect of increasing the total surface present within the system (if porosity is kept the same) and thus should decrease the radiative thermal conductivity. This idea is consistent with experimental datasets by [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF], in which they found that monodisperse, angular basaltic particles from the JSC-1A lunar simulant had a lower radiative thermal conductivity than spherical glass particles with equivalent diameters. Whether this is truly due to changes in surface area or instead to some other factor, such as changes in the length scales of the void spaces, is an interesting subject of investigation for a future study.

In Figure 10, there appears to also be a nearly 1:1 correspondence between effective particle diameter and volumetric median particle size. This correlation supports experimental results by [START_REF] Sakatani | Thermal conductivity of lunar regolith simulant JSC-1A under vacuum[END_REF], who found that the thermal conductivity of the JSC-1 lunar regolith simulant was equivalent to the volumetric median particle size for both radiative and solid thermal conductivity terms. However, the particle diameters deviate from this trendline in Figure 10 by up to ±15%. As such, thermal inertia measurements of a regolith with an unknown SFD could be used to estimate the volumetric median particle size of the regolith to within about ±15% of the true value. One could also use thermal data to estimate the Sauter mean particle size with a higher degree of accuracy, but the volumetric median particle size is perhaps a more useful metric because of its more common use in statistical reports.

Although these findings related to regolith particle SFDs might aid us in interpreting regolith properties from thermal inertia, we must again point out the importance of the relationship between regolith particle size and diurnal skin depth, particularly for analyses on fast rotators with apparently low thermal conductivity boulders like Bennu and Ryugu. If the effective particle size in some region is larger than the diurnal skin depth, the apparent thermal inertia is expected to be that of the solid rock, thus precluding particle size determination from thermal data. As seen in Figure 11, even the lower end of the thermal inertia values on Bennu (200 J m -2 K -1 s -1/2 ) corresponds to particle size predictions that approach or exceed this cutoff. Although this complicates or even prevents the interpretation of regolith thermal inertia in terms Λ s of particle size, it might present an opportunity to predict the thermal properties of the individual regolith particles: The OSIRIS-REx spacecraft will obtain very high-resolution photos and (ideally) ride-along OTES and OVIRS spectra of selected sample site. Should the site contain abundant fine-particulate regolith, the regolith particle SFD, particle shape, and packing configuration could be constrained from these data, providing a value ground truth for thermal models. An analysis like the one shown in Figure 11 could be repeated with constraints on the effective particle size in order to estimate the value of km. If km is low, the perhaps the regolith particles have a high degree of microporosity are derived from the low density/conductivity boulders. Conversely, if km is high-perhaps even higher than the surrounding boulders-then this could indicate that the regolith particles are stronger clasts weathered out of a weaker, lower density/conductivity matrix. Either scenario would provide interesting information regarding the nature of Bennu's boulders and regolith formation processes.

Conclusion

In this work, we build upon the fundamental knowledge of particulate regolith thermal properties in an effort to improve our ability to infer the physical properties (namely particle size distribution) of regolith from remote thermal measurements. This work was motivated by the ongoing OSIRIS-REx mission at asteroid Bennu. Radiative thermal conductivity was chosen as the focus here owing to the dominance of the radiative heat transfer (relative to heat conduction through the contacts between particles) for coarse, centimeter-scale regolith, which is the material of interest for sampling on Bennu.

We presented a new thermophysical model for asteroid regolith where the individual regolith particles are represented as randomly packed spheres in a FE mesh framework. Heat conduction and surface-to-surface radiation are modeled in a steady-state configuration to determine the bulk conductivity properties of the particulates. With this model, we determined the radiative thermal conductivity of a series of polydisperse particle geometries with cumulative power-law SFDs. We found that the thermal conductivity of the particulate material can have a very strong influence on the radiative thermal conductivity, particularly when material thermal conductivity (km) is very low, the temperature is high, and/or particles are large, owing to nonisothermality within individual particles (Figure 8). When this non-isothermality is negligible, the radiative thermal conductivity of polydisperse geometries scales normally with the cube of the temperature, as is also the case for monodisperse particulates. When non-isothermality is substantive, the radiative thermal conductivity has a weaker temperature dependence. The magnitude of non-isothermality for polydisperse and monodisperse particulates is larger than previously predicted, and new correlation coefficients (a1= -0.568, a2 = 0.912, a3 = 0.765, a4 = 1.035) were provided for a correction factor (Equation 2) for the radiative thermal conductivity. This correction factor can be applied to any model for regolith thermal conductivity (e.g. [START_REF] Sakatani | Thermal conductivity model for powdered materials under vacuum based on experimental studies[END_REF][START_REF] Gundlach | A new method to determine the grain size of planetary regolith[END_REF]) by multiplying the model-predicted radiative thermal conductivity term by fk. If the regolith particles on Bennu have very low thermal conductivity, particle predictions from thermal inertia could be significantly underestimated if non-isothermality is not taken into account in the regolith thermal conductivity model.

Finally, we found that the radiative thermal conductivity for each polydisperse simulation is identical to that of a monodisperse simulation with some effective particle diameter. A 1:1 correspondence exists between the effective particle diameter determined for the numerical results for each polydisperse geometry and the Sauter mean particle diameter (Figure 10). We conclude that amount of surface area per unit mass and volume within a particulate geometry is closely related to the total radiative conductivity, leading to the prediction that non-spherical particles should have lower radiative thermal conductivity values. The effective particle diameters of the mixtures also correspond within ±15% to the volumetric median particle diameter, enabling a convenient means of interpreting thermophysical data on the regolith of asteroid Bennu and other airless bodies, particularly when ancillary information about the regolith such as cumulative power-law slope and maximum particle size can be gained from image data. For the radiation validation test, we define a steady-state configuration with two concentric spherical shells with surface emissivity values of unity. We number the four surfaces as 1-4, starting from the inner surface of the inner spherical shell (Figure A1). Fixed temperatures are defined at surfaces 1 and 4, while surfaces 2 and 3 exchange energy via radiation. Once final steady state is achieved, the final temperatures of surfaces 2 and 3 are related to the net energy transfer between surfaces 2 and 3 by:

(1) where Q is energy transfer in watts, is the area of surface 2, is the view factor or configuration factor for surface 2 to surface 3, which in this case is equal to one, is the Stefan-Boltzmann constant, and T is the temperature of the surface denoted by the subscript.

To describe steady-state heat conduction through the spherical shells, we start with the one-dimensional form of the heat equation in spherical coordinates:

Q = A 2 σ f 2-3 (T 4 2 -T 4 3 ) A 2 f 2-3 σ (2)
The first integral gives:

(3) where is in integration constant. From Fourier's Law:

(4) we determine :

(5)

Taking the second integral of the heat equation (Equation 2) gives:

(6)

For the case of the inner spherical shell, we can evaluate this equation at r1 and r2 to determine the value of Cb:

(7)

We can substitute this into Equation 6 to find:

(8)

Evaluating at r = r2 provides an expression for T2:

(9) 1 r 2 d dt ( r 2 dT dr ) = 0 r 2 dT dr = C a C a Q = -k A dT dr C a C a = -Qr 2 k A T (r) = - C a r + C b T 1 = -C a r 1 + C b T 2 = -C a r 2 + C b C b = T 1 + C a r 1 = T 2 + C a r 2 T (r) = T 1 + C a ( 1 r 1 - 1 r ) T 2 = T 1 - Qr 2 2 k A 2 ( 1 r 1 - 1 r 2 )
Similarly, an expression for T3 is found to be:

(10)

Substituting Equation 9and 10 into Equation 1 gives us the final expression for net heat transfer in the system:

(11)

Equation 11 may be solved for Q numerically. In our work, we used the fsolve() function within the SciPy Python library. Once Q is found, the values of and are found from Equations 9 and 10 and can be compared to the results of the numerical test when steady state is achieved.

Figure 4 .

 4 Figure 4. Example mesh refinement in a case where the spheres are touching and the ratio of particle contact neck size rc to sphere radius Rp = 0.05.

Figure 5 .

 5 Figure 5. Solid conductivity ks (no radiation) normalized to the conductivity of the sphere material km in regular particulate packings as a function of the ratio of rc to Rp.

Figure 7 .

 7 Figure 7. Steady-state temperature solution examples for the 10-30 mm diameter particle size range with four different power-law exponent values: -8.0, -6.0, -4.0, and -3.0. The material solid thermal conductivity used in these simulations is km = 10.0 W m -1 K -1 , chosen to minimize non-isothermality within the individual spheres.

Figure 9 .

 9 Figure 9. Original non-isothermal correction factor proposed by van Antwerpen et al. (2012)(dashed line), shown with our values for the correction factor from numerical results (points) and a corresponding fit line (solid gray line). Λs values were calculated using the effective particle diameter for each geometry. Both lines are calculated using Equation 2, assuming that fk = 1 when the equation produces a value greater than unity (when ). Numerical test results are shown for all polydisperse geometries in Table1and a set of monodisperse validation cases (e.g. Figures3 and 6) at temperatures ranging from 250-400 K and with km values ranging from 0.05-10.0 W m -1 K -1 .

  Figure10. Comparison of the effective particle diameter determined from the radiative thermal conductivity numerical results to different expressions of the mean particle diameter for the 16 polydisperse geometries (Table1). Effective particle diameter of a polydisperse particulate geometry is defined as the diameter of a monodisperse particulate geometry that has the same radiative thermal conductivity.

  Figure11. Example particle size prediction for a hypothetical surface on Bennu with a thermal inertia of 200 J m -2 K -1 s -1/2 at 265 K[START_REF] Rozitis | Thermal Inertia Maps of (101955) Bennu from OSIRIS-REx Infrared Observations[END_REF] as a function of the conductivity of the material that makes up the regolith particles (km). Three regolith thermal conductivity models are used both without and with the use of the non-isothermal correction factor to make particle size predictions. For comparison, the predicted diurnal skin depth is shown as a thick, gray line. For simplicity, only one macroporosity (0.40) is considered here. Increasing the assumed porosity would have the effect of further increasing the predicted particle diameters. Two reference km values are displayed with vertical lines/bars: The Cold Bokkeveld (CM2) meteorite[START_REF] Opeil | The thermal conductivity of meteorites: New measurements and analysis[END_REF] and the Ryugu boulder, estimated from MARA radiometer data[START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF]. Model input values: macroporosity=0.40; specific heat =750 J kg -1 K -1 (a); particle density is a function of thermal conductivity, using the average of two models (b,c) for meteorite thermal conductivity vs microporosity and assuming a CM grain density=2960 kg m -3 (c); surface energy=0.032 J m -2 (d); emissivity=1.0; Bennu mid-latitudes surface acceleration=40 µm s -2 (e); Poisson's ratio=0.27 (f); Young's Modulus=5.63e9 Pa (f); mean burial depth=10 mm; Sakatani model tunable solid contact conductivity parameter ξ=0.63 (d); tunable radiative conductivity parameter ζ=0.85 (d). References for model inputs: a Biele et al. (2019), b Henke et al. (2016), c Flynn et al. (2018), d Sakatani et al., (2018), e Scheeres et al. (2019), f A.R. Hildebrand (personal communication, 2018), measured values for Tagish Lake simulant.

  Figure11. Example particle size prediction for a hypothetical surface on Bennu with a thermal inertia of 200 J m -2 K -1 s -1/2 at 265 K[START_REF] Rozitis | Thermal Inertia Maps of (101955) Bennu from OSIRIS-REx Infrared Observations[END_REF] as a function of the conductivity of the material that makes up the regolith particles (km). Three regolith thermal conductivity models are used both without and with the use of the non-isothermal correction factor to make particle size predictions. For comparison, the predicted diurnal skin depth is shown as a thick, gray line. For simplicity, only one macroporosity (0.40) is considered here. Increasing the assumed porosity would have the effect of further increasing the predicted particle diameters. Two reference km values are displayed with vertical lines/bars: The Cold Bokkeveld (CM2) meteorite[START_REF] Opeil | The thermal conductivity of meteorites: New measurements and analysis[END_REF] and the Ryugu boulder, estimated from MARA radiometer data[START_REF] Grott | Low thermal conductivity boulder with high porosity identified on C-type asteroid (162173) Ryugu[END_REF]. Model input values: macroporosity=0.40; specific heat =750 J kg -1 K -1 (a); particle density is a function of thermal conductivity, using the average of two models (b,c) for meteorite thermal conductivity vs microporosity and assuming a CM grain density=2960 kg m -3 (c); surface energy=0.032 J m -2 (d); emissivity=1.0; Bennu mid-latitudes surface acceleration=40 µm s -2 (e); Poisson's ratio=0.27 (f); Young's Modulus=5.63e9 Pa (f); mean burial depth=10 mm; Sakatani model tunable solid contact conductivity parameter ξ=0.63 (d); tunable radiative conductivity parameter ζ=0.85 (d). References for model inputs: a Biele et al. (2019), b Henke et al. (2016), c Flynn et al. (2018), d Sakatani et al., (2018), e Scheeres et al. (2019), f A.R. Hildebrand (personal communication, 2018), measured values for Tagish Lake simulant.

Figure A1 .

 A1 Figure A1. Hollow spheres radiation validation case. The four surfaces referenced in the text are labeled.

Acknowledgments

The authors acknowledge support from the Academies of Excellence on Complex Systems and Space, Environment, Risk and Resilience of the Initiative d'EXcellence (IDEX) Joint, Excellent, and Dynamic Initiative (JEDI) of the Université Côte d'Azur as well as from the Centre National d'Études Spatiales (CNES). This material is based in part upon work supported by NASA under Contract NNM10AA11C issued through the New Frontiers Program. We thank Christopher Edwards and an anonymous reviewer for providing useful comments that helped to improve the quality of this manuscript. We also thank Josh Emery, Jens Biele, and Cat Wolner for useful suggestions and edits. Solution tables, particulate geometry files, and source code for theoretical models utilized in this work are available on https://github.com/andrewryan2727/regolith-heattransfer


Axel Labattut

Sandra Abi Fayssal

Julien Buendia

Ibrahim Abdellah

Vincent Huc
email: vincent.huc@universite-paris-saclay.fr

Cyril Martini
email: cyril.martini@novecal.com

Emmanuelle Schulz
email: emmanuelle.schulz@universite-paris-saclay.fr.

Calixarene-supported Pd-NHC complexes as efficient catalysts for scalable Suzuki-Miyaura cross-couplings

Benzyloxycalix [8]arene is reported as a suitable macrocyclic support for Pd-NHC complexes, which catalyses C-C crosscoupling reactions with high efficiency. Resulting PEPPSI-derived complexes, issued from a scalable synthetic procedure avoiding any tedious purification, are indeed used as heterogeneous catalysts in ethanol to promote the Suzuki-Miyaura coupling between various aryl halides and arylboronic acids (or esters). Optimisation of the reaction conditions allowed the use of only a slight excess of base (K3PO4, 1.2 equivalents) and boronic acid (1.1 equivalents), and these conditions were also successfully applied to a large scale benchmark reaction (1 mole of substrate) using a low palladium loading (0.001 mol %). Furthermore, target products were easily isolated after simple filtration procedures with low residual Pd traces (down to 3 ppm) thanks to the insoluble nature of the macrocyclic catalyst in ethanol, thus proving the usefulness of this support to promote coupling reactions relevant to the fine chemical industry.

Introduction

The fine chemical industry is continuously looking for the most direct and economically viable methods to obtain highly functionalised valuable compounds. Catalytic processes are particularly desirable because they generally allow to lower the temperature and to reduce the reaction time, for an overall decrease in both production cost and impact on the environment. 1 The huge success of organometallic catalysis is well established because it meets these criteria, provided that the procedures in place prevent the contamination of the target products by metal traces.

2 Indeed, only very low metal content are tolerated inside ended pharmaceutical, agrochemical or cosmetic products, due to their toxicity. Furthermore, beyond these health and regulatory concerns, the whole chemical industry is impacted by metal contamination as the final composition of a reaction may dramatically be changed while using waste reagents.

3

Lowering the catalytic charge, which necessitates the use of increasingly reactive metal complexes, is therefore highly desirable. 4 Another solution relates to the development of innovative procedures, allowing easy removal of the catalyst from the reaction mixture. Obviously, nothing prevents the combination of these two solutions. This has led to the emergence of numerous studies describing the immobilisation of various types of homogeneous catalysts, leading to their elimination by simple filtration.

As a prerequisite for the use of a supported catalyst in the pharmaceutical industry, it must afford, in an immutable manner, the desired product with constant characteristics in terms of purity and yield. In this context, we addressed the immobilisation of transition metal complexes, our goal being the development of catalysts attractive for the fine chemical industry.

To this aim, preparation of N-heterocyclic carbene-based Pdcomplexes led to easy-to-handle and stable precatalysts for highly active species, particularly the PEPPSI type structures.

6 Some efforts were already devoted to anchor such complexes on various supports, some of them displaying an organic structure.

7

We have previously reported a practical and low-cost synthesis of a benzyloxycalix [8]arene intermediate and its transformation via a chromatography-free procedure into a support bearing eight well-defined PEPPSI-type Pd-NHC complexes. [START_REF] Huc | Catalyseurs supportés cycliques[END_REF] Advantageously, Suzuki-Miyaura cross-couplings 9 occurred in heterogeneous conditions, with the catalysts suspended in ethanol. They showed high TON and TOF for a large panel of functionalised aryl bromides, with low metal contamination after simple filtration.

10

We report here our efforts to optimise the amount of introduced material without increasing the reaction time, in order to develop reaction conditions more suitable for large scale transformations. A special emphasis is placed on the filtration processes to allow very low release amounts of palladium into the target products, while maintaining simple and cost-efficient procedures. The catalysts are also used for

2 | J. Name., 2012, 00, 1-3
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Please do not adjust margins Please do not adjust margins the preparation of some key precursors dedicated to the synthesis of valuable pharmaceutical compounds, to show their facile and broad applicability.

Results and discussion

We recently described a synthetic procedure allowing easy preparation of calix [8]arene-supported PEPPSI complexes.

10 Catalysts 1 and 2 (cat1 and cat2, Figure 1) were obtained in high global yield without the need for any tedious purification.

In this work, the synthesis of both catalysts was upscaled. They are indeed easily prepared from palladium chloride or palladium bromide, resulting in more than 7 g of palladium catalysts per batch (see supporting information). Their purification could be achieved by avoiding the use of silica gel chromatography, and by favouring precipitation and washing in various common organic solvents. Pure compounds were isolated on this large scale in 50 and 73 % yield, respectively. Throughout their whole synthesis, the calixarene-based catalysts remained soluble in the solvents used, allowing an easy control of the functionalisation steps, and leading to their precise characterisation. Perfectly reproducible analytical data was obtained for this procedure, delivering supported catalysts with a w/w metal content close to 15 %. Suitable crystals for X-ray analysis were obtained by vapour diffusion of diethyl ether into a dichloromethane/acetonitrile solution of cat1. 10 This catalyst resulting from the use of PdCl 2 was chosen to carry out most of the optimization tests reported here because of its advantageous preparation, allowing both cost and atom savings.

We have reported that both catalysts were particularly efficient to promote the Suzuki-Miyaura cross-coupling of a wide range of aryl bromides with arylboronic acids.

10

Interestingly, the best reaction rates and yields were obtained in ethanol as solvent, in which the catalysts were not soluble. It should also be noted that the use of pure water as solvent led to a very slow reaction, which nevertheless reached a suitable conversion after 24h stirring at room temperature.

Furthermore, ethyl acetate was also tested, as recommended solvent for use in sustainable chemistry, 11 in which cat1 was insoluble. A very low activity was recorded (14 % conversion in 2h) which could however be greatly accelerated by addition of a small amount of water (EtOAc/H 2 O 7/1; 60 % after 2h, 75 % after 24h), with this accelerating role of additional water being commonly observed in numerous Suzuki-Miyaura crosscouplings. [START_REF]Metal-Catalyzed Cross-Coupling Reactions[END_REF] Albeit not well understood, a significant effect of the calixarene macrocyclic structure on the reactivity of the anchored PEPPSI catalytic sites is particularly noticeable, depending on the solvent used. The screening of various bases demonstrated the efficiency of the low cost and easy-tohandle K 3 PO 4 as base. Thus, in a comparison experiment under these conditions, cat1 and its homogeneous analogue (see cat3 in Figure 1) were engaged in a benchmark Suzuki-Miyaura coupling reaction with a 0.01 mol % loading in palladium (scheme 1), for a better evaluation of their initial activity in ethanol.

Scheme 1. Suzuki-Miyaura benchmark reaction performed with cat1 or cat3

Tests were performed under these conditions for which > 95 % conversion was reached in both cases, after 2h at 80 °C in ethanol. GC sampling at the very beginning of the reaction proved however that a high conversion was indeed reached after only 2 minutes when the homogeneous PEPPSI catalyst cat3 was used. For cat1, 4 minutes were necessary for the same efficiency, highlighting nevertheless the very high activity of this insoluble catalyst, which remained comparable to that of its homogeneous counterpart.

Of major importance for industrial application is the evaluation of the palladium leaching after use of these supported systems. This was easily achieved due to the heterogeneous nature of the transformation in ethanol. The final product was indeed recovered after elimination of the insoluble species by simple paper filtration. ICP-MS analyses of the crude residue were performed to compare the leaching levels of cat1, cat2 and the homogeneous PEPPSI analogue of cat1 (see Figure 2 and supporting information). The palladium content inside 4-phenyltoluene reached only 10.6 ppm when cat1 was used. A five times higher amount of palladium was measured using cat2, while cat3 led to a much Please do not adjust margins Please do not adjust margins higher Pd content (516.5 ppm) detected in the final biaryl product.

Based on these promising results, further tests were conducted attempting to reach lower Pd contamination, while focusing on keeping simple purification procedures. The tests were performed on the reaction between 2-bromobenzonitrile and phenylboronic acid at a quarter-mole scale in the presence of cat1 (0.1 mol % Pd) in ethanol. After two hours at 80 °C, 95 % conversion was achieved. Purification of a one-gram sample by flash column chromatography allowed to isolate [1,1'biphenyl]-2-carbonitrile in 90 % yield. After a short prefiltration to remove most of the insoluble material, the filtrate was divided into equal parts, each of them undergoing an additional filtration before preparation of the ICP-MS samples (Figure 3).

Figure 3. Comparison of various types of filter media for very low palladium leaching content

With the short prefiltration, a moderate amount of Pd (54 ppm) was detected in the target functionalised cross-coupling product. Additional filtration on filters with different porosities (sintered glass filter, Whatman® paper, polypropylene) led to a decrease of this value (down to 22 ppm), but the limits authorised by the legislation could not be reached. Nevertheless, purification through Dicalite TM permitted the removal of 97.2 % of the introduced amount of palladium; this value reached 99.5 % via the use of a carbon filter, allowing recovery of the crude product containing only 3 ppm of Pd. Thus, immobilisation of PEPPSI-type Pd-catalysts on calixarenic structures led to the preparation of products almost rid of any palladium traces, after combining a short prefiltration step and a simple carbon filtration.

Both supported-catalysts cat1 and cat2 showed a similar reactivity which we already highlighted in our previous article, although cat2, with the bulkier diisopropyl-phenyl substituent on the NHC, proved to be more efficient for the transformation of demanding and sterically hindered substrates. 10,[START_REF] O'brien | [END_REF] Table 1 gives additional examples of Suzuki-Miyaura cross-couplings, varying first the aryl halide coupling partner. 14 To ensure high conversion while maintaining a short reaction time, tests were performed at a 0.5 M concentration, with 1.5 equivalents of phenylboronic acid and 2 equivalents of base. Typical trends in terms of reactivity were observed in the presence of aryl bromide derivatives bearing electron donating or withdrawing groups. For each example, the catalyst amount was adjusted, leading to the target products in excellent isolated yields within two hours (table 1, entries 1-2). The coupling of 1-bromo-2-chlorobenzene using cat2 occurred with complete conversion of the starting material (entry 3). Using only 1.1 equivalent of the boronic acid derivative led to the formation of 2-chlorobiphenyl in 91 % isolated yield, together with a small amount of the biscoupling product (See GC traces in the SI). Transformation of 1bromo-2-fluorobenzene led to 2-fluorobiphenyl in only 62 % yield, since the cross-coupling reaction was accompanied with non-negligible debromination (entry 4).

Table 1. Scope of the reaction with phenylboronic acid Ar-X Cat Pd (mol%) Conv (%) [a,b] Yield (%) [d] [a] Aryl bromide (1 equiv., 0.50 M), phenylboronic acid (1.5 equiv.), K3PO4 (2 equiv.), under argon atmosphere. [b] Conversion determined by GC and GC/MS analyses. [c] Use of 1.1 equiv. of phenylboronic acid. [d] Use of (2-methoxyphenyl)boronic acid (1.5 equiv.) as coupling partner.

1 1 0.
The reaction of a very hindered substrate (entry 5) required a higher catalyst loading to reach a moderate conversion. Cat2 was more efficient than cat1 and allowed the coupling of pentamethylbromobenzene, albeit complemented with some debromination of the reactant. Use of cat2 nevertheless afforded the desired coupling product in 58 % yield (see SI). As already reported, aryl chlorides were less reactive substrates, but some success (56 % yield) was met in the case of an electron-deficient derivative (entry 6). However, 1,2-dichlorobenzene reacted only hardly with 2-(methoxyphenyl)boronic acid, whose structure has been chosen to facilitate the purification procedure. In the presence of cat2, a moderate conversion was observed and 2-chloro-4'methoxy-1,1'-biphenyl was isolated in only 38 % yield due to This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins protodeborylation and formation of some bis-coupling product (entry 7, and SI).

The synthetic utility of cat1 was further demonstrated by coupling various functionalised boronic derivatives with different aryl halides, thus widening the scope of the procedure compatibility. All the tests were performed on a 1 mmol scale (0.5 M or 0.25 M) in the presence of 2 equivalents of base and 1.5 equivalents of the boronic partner (scheme 2). This additional screening showed that the crosscoupling reaction can also be conducted efficiently with an electron-poor boronic acid derivative bearing a CF 3 group (eq. 1). Bis-coupling occurred satisfyingly with a bis-pinacol phenyl boronate, provided that an excess of 4-bromotoluene is used (2.5 equiv., eq. 2).

Synthetic applications of cat1.

Furthermore, inspired by a recent report on the synthesis of a key intermediate for the preparation of Fluxapyroxad, a synthetic carboxamide with antifungal properties, 15 the crosscoupling between 5-bromo-1,2,3-trifluorobenzene and (2aminophenyl) boronic acid was evaluated. The reaction implied the use of cat1 with a 1 mol % loading in palladium, and allowed isolation of the target product (eq. 3) in 94 % yield. Reversing the coupling previously described in the literature by the direct use of the boronic acid holding an amine function has made it possible to reduce the number of steps for a simple access to this compound of interest. Harsher conditions were needed for the complete conversion of 2bromopyridine with (4-formylphenyl)boronic acid (eq. 4), since the reaction was performed in n-BuOH at 100 °C. The desired pyridinyl benzaldehyde coupling product, already used in the synthesis of Atazanavir, a HIV protease inhibitor, 16 was unfortunately isolated in only 51 % yield, because of the presence of unavoidable degradation compounds.

The standard conditions were further optimised with the constant aim of driving the reaction under optimal conditions in terms of atom economy. The amount of both boronic acid and base, together with the concentration of the substrate, were modified for the reaction at the mmole scale promoted by cat1 (see table 2). The volume of the solvent was therefore reduced (entries 1 to 4) affording high conversion in short reaction times. A 2 M concentration of aryl halide (entries 4 and 5) also proved to be successful, although the solubility limit of the reagents was reached under these conditions. Furthermore, the amount of phenylboronic acid could be adjusted to 1.1 equivalents, leading however to a slower reaction, which yielded full conversion in 6.5 hours (entries 6 and 7). Finally, the amount of base was also decreased down to 1.2 equivalents, and high conversion was still reached albeit reaction time had to be slightly increased (entries 8 and 9). By merging all optimised conditions to minimise the amount of reagents and reduce waste, we were pleased to obtain 80 % conversion for the room temperature transformation of 4bromotoluene at a 2 M concentration in EtOH with only 1.2 equiv. K 3 PO 4 and 1.1 equiv. phenylboronic acid in the presence of 0.5 mol % cat1 (see entry 10).

Worth mentioning is the sluggish aspect of the reaction mixture under these conditions, due to the reduced amount of solvent. Therefore, the reaction temperature was raised up to 80 °C to enhance mass transfer, together with a great reduction of the catalyst loading by a factor of 500, down to 0.001 mol % Pd. After two hours, the conversion reached 90 %. [a] 4-Bromotoluene (1 mmol, 1 equiv.), PhB(OH)2 (x equiv.), K3PO4 (x equiv.), under argon atmosphere. [b] Conversion determined by GC analysis. [c] Isolated yield.

Accordingly, a reaction was conducted on the mole scale in the presence of 0.001 mol % of palladium from cat1 at 80 °C, and fixing the concentration of the substrate at 2 M. It is noteworthy that a very low amount of catalytic species, which only features 6 ppm of palladium, is engaged in this transformation. This large-scale reaction behaved gratifyingly, and after two hours, 92 % conversion could be obtained, and the desired coupling product was further isolated in 77 % yield. Adjustment of the conditions thus allowed to obtain a very Please do not adjust margins Please do not adjust margins high TOF of 38500 for this Suzuki-Miyaura coupling, clearly demonstrating the valuable use of benzyloxycalix [8]arenesupported Pd-NHC complexes for the high yielded preparation of nearly metal-free coupling products.

Scheme 3. Suzuki-Miyaura reaction performed on the mole scale

The heterogeneous nature of the benzyloxycalix [8]arenesupported palladium catalysts was previously demonstrated by the absence of catalytic activity of the hot-filtered solution. 10 Additional tests were further performed to evaluate the recyclability of cat1. First, 4-bromotoluene was coupled with phenylboronic acid with full GC conversion (cat1, 0.5 mol % Pd, room temperature, 2 h in EtOH). Then, a new batch of substrates and base (same amount as used in the first run) was directly added in the reaction mixture without further treatment. Pleasingly, this second run afforded the desired product with the same GC conversion after 2 h reaction time. To further prove the recyclability of the calixarene-supported catalyst under these reaction conditions, another crude mixture from the same benchmark reaction (4-bromotoluene and phenylboronic, cat1, 0.5 mol % Pd, room temperature, 2 h in EtOH) was filtered after completion through a Dicalite TM pad, which was further washed with ethanol to ensure removal of the reaction product and the residual boronic salts. The catalyst was then recovered from the pad by washings with dichloromethane as solubilising solvent. The dichloromethane catalyst solution was evaporated, dried and then engaged in a new catalytic run, by adding ethanol and a new set of reagents. After 2 h reaction, a complete conversion was delightfully recorded, which proves that the recycled catalyst still exhibits high catalytic activity. Moreover, we should underline that this recycling procedure is easy and straightforward, which makes suitable for large scale applications.

Conclusions

To conclude, we report here a detailed study for the optimised use of benzyloxycalix [8]arene-supported PEPPSI-type NHCbased palladium complexes in Suzuki-Miyaura cross-coupling

We also here an upscaled synthesis of these calixarene-supported catalysts in high yields and without any fastidious purification procedure. These syntheses are perfectly reproducible, as is the Pd loading inside these catalysts. The coupling reactions were performed in ethanol, an industrially relevant solvent, in which the supported catalysts were insoluble. The extent of application of these catalysts is here demonstrated for the coupling of various substrates, for which the reaction conditions (in particular the temperature and the catalytic amount) have been optimised. The amount of base and boronic acid partner was also lowered to match at best the concept of atom economy, a key parameter for industrial applications. Some precursors for valuable pharmaceuticals were straightforwardly obtained via this procedure. The reaction was furthermore performed on the mole scale using a very low catalyst loading, without altering the efficiency of the process. Simple filtrations through a sintered glass filter followed by a carbon filter led to the product recovery with very low amount of residual palladium (3 ppm) as revealed by ICP-MS analyses. All in one, this work highlights a range of catalysts in line with a set of experimental conditions that combine all features needed for industrial applications in a green route. Further work is currently in progress to widen the structure of the easily accessible, welldefined and structurally-modular benzyloxycalix [8]arene macrocycle, for its use as support in various heterogeneous catalytic reactions involving other ligands and/or metallic salts, by taking advantage of its simple functionalisation. This journal is © The Royal Society of Chemistry 20xx

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Figure 1 .

 1 Figure 1. Calixarene-supported palladium PEPPSI catalysts and a PEPPSI analogue

10

 10

Figure 2 .

 2 Figure 2. Residual palladium content determined by ICP-MS

Table 2 .

 2 Optimisation of the reaction conditions

	Entry	K3PO4 (equiv.)	PhB(OH)2 (equiv.)	[4-Bromotoluene] (mol.L -1 )	Time (h)	Conv. (%) [a,b]
	1	2	1.5	0.17	2	93
	2 3	2 2	1.5 1.5	0.25 0.5	2 2	98 99 (93) [c]
	4	2	1.5	2	2	93
	5	2	1.5	2	24	98
	6	2	1.1	0.5	2	85
	7	2	1.1	0.5	6.5	99
	8	1.2	1.5	0.25	2	94
	9	1.2	1.5	0.25	6	96
	10	1.2	1.1	2	2	80

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Acknowledgements

The CNRS, the Ministère de l'Enseignement Supérieur et de la Recherche and the SATT Paris-Saclay are acknowledged for financial support (NOVECAL project). The authors also wish to thank the French National Research Agency (IdEx Paris-Saclay, ANR-IdEx-0039RA14, grant to C. M., the Charmmmat ANR-11-LABX-0039, grant to I. A., and the ANRT CIFRE, grant to S. A. F.) for their support.

Conflicts of interest

VH, CM and ES are co-founders of Novecal.