Ilya V Yakovlev

Vladimir I Zaikovskii

Mariya A Kazakova

Evgeniy S Papulovskiy

Olga B Lapina

Jean -Baptiste D'espinose De Lacaillerie

Crystal plane dependent dispersion of cobalt metal on metastable aluminas

Keywords: heterogeneous catalysis, 59 Co NMR, HRTEM, Internal Field NMR, DFT, metal support interaction, Fischer-Tropsch synthesis

Metallic Co nanoparticles, widely used and studied as supported heterogeneous catalysts for Fischer-Tropsch synthesis (FTS), display catalytic properties that can vary significantly depending on their size and crystal structure. In this work, we used 59 Co Internal Field NMR ( 59 Co IF NMR) complemented by high-resolution transmission electron microscopy (HRTEM) to demonstrate the influence of strong metal-support interaction on two noticeably different metastable alumina phases -γ-Al 2 O 3 and χ-Al 2 O 3 . According to 59 Co IF NMR and HRTEM, the metallic particles supported on χ-Al 2 O 3 were larger and displayed a significantly higher content of Co in its hcp phase, which are known to be more active and selective to C 5+ in FTS. The 1 H NMR chemical shifts of hydroxyl groups anchored to the (110) and (111) spinel crystal planes were calculated by DFT. It revealed that the hydroxyl coverage of γ-Al 2 O 3 facilitates the dispersion of Co precursor over the surface of the support, ultimately leading to the formation of smaller metal Co nanoparticles on γ-Al 2 O 3 than on χ-Al 2 O 3 .

Introduction

The catalytic properties of nanoparticles (NPs) strongly depend on their size, crystal structure and morphologyfactors that have a complex interconnection with each other but that are, to some extent, controllable by adjusting the synthesis conditions. An important case study of this dependence is metallic cobalt supported on oxides whose catalytic behavior depends strongly on the metal-support interaction in ways that are not yet fully understood. [START_REF] Van Deelen | Control of metal-support interactions in heterogeneous catalysts to enhance activity and selectivity[END_REF] In Fischer-Tropsch synthesis (FTS) for example, the choice of the support as well as possible decorations with various noble metal promoters are used to control the morphology and crystal structure of cobalt NPs in order to obtain the desired catalytic properties. The influence of these two properties of cobalt NPs on activity and selectivity towards C 5+ hydrocarbons has been extensively studied in the recent decades resulting in some generally agreed-upon requirements.

First, vast experimental data [START_REF] Ducreux | Microstructure of Supported Cobalt Fischer-Tropsch Catalysts[END_REF][START_REF] Enache | Situ XRD Study of the Influence of Thermal Treatment on the Characteristics and the Catalytic Properties of Cobalt-Based Fischer-Tropsch Catalysts[END_REF][START_REF] Karaca | Structure and catalytic performance of Pt-promoted alumina-supported cobalt catalysts under realistic conditions of Fischer-Tropsch synthesis[END_REF][START_REF] Gnanamani | Fischer-Tropsch synthesis: Activity of metallic phases of cobalt supported on silica[END_REF][START_REF] Lyu | Role of Active Phase in Fischer-Tropsch Synthesis: Experimental Evidence of CO Activation over Single-Phase Cobalt Catalysts[END_REF][START_REF] Liu | Crystallographic Dependence of CO Activation on Cobalt Catalysts: HCP versus FCC[END_REF] indicate that, between the two most common structures met in metallic cobalt, the hexagonal close-packed (hcp) crystal structure is more active and selective towards C 5+ than the face-centered cubic (fcc) one. Second, experimental data and theoretical calculations show that below a certain size (between 4 and 10 nm according to different sources) cobalt NPs become much less active in FTS and their selectivity shifts to CH 4 [START_REF] Tsakoumis | Evaluation of Reoxidation Thresholds for γ-Al 2 O 3 -Supported Cobalt Catalysts under Fischer-Tropsch Synthesis Conditions[END_REF][START_REF] Bezemer | Cobalt Particle Size Effects in the Fischer-Tropsch Reaction Studied with Carbon Nanofiber Supported Catalysts[END_REF][START_REF] Barbier | Characterization and Catalytic Behavior of Co/SiO 2 Catalysts: Influence of Dispersion in the Fischer-Tropsch Reaction[END_REF][START_REF] Tuxen | Size-Dependent Dissociation of Carbon Monoxide on Cobalt Nanoparticles[END_REF][START_REF] Rytter | On the selectivity to higher hydrocarbons in Co-based Fischer-Tropsch synthesis[END_REF][START_REF] Tsakoumis | Deactivation of cobalt based Fischer-Tropsch catalysts: A review[END_REF].

However, in many instances, how the desired cobalt size and structure are tailored by the support remains an open question.

During the preparation step, how the support directs the speciation of Co(II) ions and complexes from the precursor has been efficiently investigated by spectroscopic means (noticeably UV-visible spectroscopy [START_REF] Van De Water | Spatially resolved UV-vis microspectroscopy on the preparation of alumina-supported Co Fischer-Tropsch catalysts: Linking activity to Co distribution and speciation[END_REF] and EXAFS [START_REF] J.-B. D'espinose De Lacaillerie | Clause, Impregnation of gamma-Alumina with Ni (II) or Co (II) Ions at Neutral pH: Hydrotalcite-Type Coprecipitate Formation and Characterization[END_REF]), rationalized through coordination chemistry [START_REF] Lambert | The molecular approach to supported catalysts synthesis: state of the art and future challenges[END_REF] and modeled by DFT. [START_REF] Larmier | Tuning the Metal-Support Interaction by Structural Recognition of Cobalt-Based Catalyst Precursors[END_REF] Nevertheless, the interaction of the support with the larger Co nanoparticles obtained after reduction is paradoxically more difficult to characterize or model. [START_REF] Yang | Probing the electronic structures of Co n (n = 1-5) clusters on γ-Al 2 O 3 surfaces using first-principles calculations[END_REF] Co nanoparticles can be characterized by a variety of physical methods: X-Ray diffraction (XRD), electron microscopy techniques (transmission, scanning, or scanning transmission), magnetometry, magnetic resonance techniques (electronic ferromagnetic resonance -FMR, 59 Co Internal Field nuclear magnetic resonance -IF NMR). All of these techniques have their own experimental limitations and provide different kinds of structural information. In this work, we rely on a combination of 59 Co Internal Field NMR ( 59 Co IF NMR) and high-resolution transmission electron microscopy (HRTEM), a combination that proved effective in some of our previous works. [START_REF] Andreev | Magnetic and dielectric properties of carbon nanotubes with embedded cobalt nanoparticles[END_REF][START_REF] Kazakova | Co metal nanoparticles deposition inside or outside multi-walled carbon nanotubes via facile support pretreatment[END_REF][START_REF] Yakovlev | Superparamagnetic behaviour of metallic Co nanoparticles according to variable temperature magnetic resonance[END_REF][START_REF] Andreev | Espinose de Lacaillerie, Internal field 59 Co NMR study of cobalt-iron nanoparticles during the activation of CoFe 2 /CaO catalyst for carbon nanotube synthesis[END_REF] HRTEM is an indispensable technique providing unique information on the morphology of the particles close to the single-atom scale. However, the intrinsic limitation of this technique is its local character, making it necessary to extrapolate the data obtained from a limited set of micrographs. 59 Co IF NMR, on the contrary, is a bulk technique that was shown to provide information on the crystal and magnetic structures of Co nanoparticles [START_REF] Andreev | Magnetic and dielectric properties of carbon nanotubes with embedded cobalt nanoparticles[END_REF][START_REF] Kazakova | Co metal nanoparticles deposition inside or outside multi-walled carbon nanotubes via facile support pretreatment[END_REF][START_REF] Gossard | Ferromagnetic Nuclear Resonance of Single-Domain Cobalt Particles[END_REF][START_REF] Matveev | Cobalt nanoparticles with preferential hcp structure: A confirmation by X-ray diffraction and NMR[END_REF][START_REF] Andreev | A New Insight into Cobalt Metal Powder Internal Field 59 Co NMR Spectra[END_REF][START_REF] Andreev | Thermal stability and hcp-fcc allotropic transformation in supported Co metal catalysts probed near operando by ferromagnetic NMR[END_REF] as well as on their size distribution, [START_REF] Yakovlev | Superparamagnetic behaviour of metallic Co nanoparticles according to variable temperature magnetic resonance[END_REF][START_REF] Liu | Sampling the structure and chemical order in assemblies of ferromagnetic nanoparticles by nuclear magnetic resonance[END_REF] even though it is limited by the intrinsically low intensity of the NMR signal and the difficulty of spectral analysis.

We focus here on the role of two low-temperature metastable alumina phases, γ-Al surface sites [START_REF] Favaro | Experimental and ab initio infrared study of χ-, κ-and αaluminas formed from gibbsite[END_REF] and by having a higher surface density of weak Lewis acid sites. [START_REF] Yashnik | The Nature of Synergetic Effect of Manganese Oxide and Platinum in Pt-MnO X -Alumina Oxidation Catalysts[END_REF][START_REF] Yashnik | Effect of χ-alumina addition on H 2 S oxidation properties of pure and modified γ-alumina[END_REF] These differences have been suggested to explain different metal sintering behavior. [START_REF] Nazimov | Effect of alumina polymorph on the dehydrogenation activity of supported chromia/alumina catalysts[END_REF] Regarding specifically Co, the presence of χ-phase in γ-Al 2 O 3 resulted in higher dispersion of Co as well as higher CO hydrogenation activities of the Co/Al 2 O 3 catalysts compared to pure Co/γ-Al 2 O 3 samples. In samples of high Co loadings, the spherical-shape-like morphology of the χ-phase Al 2 O 3 provides a better stability of the Co particles. [START_REF] Pansanga | Effect of mixed γ-and χ-crystalline phases in nanocrystalline Al 2 O 3 on the dispersion of cobalt on Al 2 O 3[END_REF] This brief summary of the literature concerning Co nanoparticles on γ-Al 2 O 3 and χ-Al 2 O 3 reveals the importance of the nanoparticle-support interactions [START_REF] Tsakoumis | Evaluation of Reoxidation Thresholds for γ-Al 2 O 3 -Supported Cobalt Catalysts under Fischer-Tropsch Synthesis Conditions[END_REF][START_REF] Larmier | Tuning the Metal-Support Interaction by Structural Recognition of Cobalt-Based Catalyst Precursors[END_REF] that result from an interplay between the support surface structure and its morphology. Here, we show that HRTEM and 59 Co IF NMR reveal significant structural differences between the Co NPs supported on γ-Al 2 O 3 and χ-Al 2 O 3 . Finally, building on indirect DFT calculations, we link these differences to the differences in initial distribution of Co precursors on the catalyst surface.

Characterization techniques

The morphology of the Co nanoparticles was controlled using high-resolution transmission pulse sequence θ-τ-θ was used, with a θ pulse length of 1 μs and an interpulse delay τ of 8 μs.

Due to the intrinsically short T 1 nuclear spin relaxation time in ferromagnetic materials, a 33 ms delay between the echo sequences was used. In order to cover the entire frequency range, the spectra were recorded stepwise with 500 kHz carrier frequency steps. A custom automatic tuning and matching device was used to retain a constant quality factor over the entire recording range.

In order to determine the intensity of the spectrum at a chosen frequency, the echo signal recorded at this frequency was Fourier-transformed with adequate phase and baseline corrections such that a purely absorption spectrum was obtained. Then, the obtained signal was integrated over a roughly 400 kHz frequency range centered at the carrier frequency, after which the value of the integral was assigned as the intensity of the spectrum at said frequency. Radiofrequency irradiation power was varied at each frequency step to be able to account for the differences of enhancement factors between the bulk of the magnetic domains and the domain walls according to the method originally described by Panissod et al. [START_REF] Panissod | NMR analysis of buried metallic interfaces[END_REF] 1 H NMR spectra of the alumina supports were recorded at room temperature using a pulsed NMR spectrometer Bruker Avance 400 with a constant magnetic field of 9.4 T (400.13 MHz 1 H Larmor frequency). Prior to the 1 H NMR experiments, the samples were dehydrated for 4 hours at 450 °C and sealed inside glass ampoules to avoid rehydration from air. The samples were transferred into standard 4 mm ZrO 2 rotors under argon atmosphere right before the experiment.

The spectra were recorded under magic angle spinning condition (MAS, 10 kHz rotation frequency) using a single π/2 pulse (5 μs pulse length). 1024 scans were accumulated with a second delay. Tetramethylsilane was used as an external reference with a chemical shift of ppm.

Theoretical calculation details

To characterize the hydroxyl coverage of the alumina surfaces, we have conducted DFT calculations of 1 H NMR chemical shifts of hydroxyl groups on the two most prominent crystal planes dominating the surfaces of γ-Al 2 O 3 (the (110) plane) and χ-Al 2 O 3 (the (111) plane)

according to the cubic spinel model of the metastable alumina building blocks.

All calculations were performed using the CASTEP program. [START_REF] Yates | Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials[END_REF] The interactions between the core ions and the valence electrons were described using the projector augmented wave (PAW) pseudopotentials. [START_REF] Blöchl | Projector augmented-wave method[END_REF] The geometry calculations were performed at Г point with the exchange-correlation functional PW91. [START_REF] Perdew | Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[END_REF] A plane wave basis set cut-off energy of 450 eV with the self-consistent field threshold of 10⁻⁶ eV was used. The geometry calculations used the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm. [START_REF] Pfrommer | Relaxation of Crystals with the Quasi-Newton Method[END_REF] To mimic a flat alumina surface, a layer of alumina was created from the spinel structure with the lattice parameter a of 8.08534 Å. A model bulk crystal cell was cleaved along the (110)

and the (111) crystallographic planes. The (110) cell dimensions were 22.8688 × 16.1706 Å, while the (111) cell was 19.8048 × 11.4344 Å. The slab thickness was varied from cell to cell, from 3.5 to 8 Å, with vacuum spans of more than 10 Å to prevent periodic interaction. No additional restrictions were imposed during the structural relaxation. Extra hydroxyl groups were added to the surface Al sites in order to avoid dangling bonds and maintain electroneutrality.

The 1 H NMR parameters were computed by the GIPAW method [START_REF] Clark | First principles methods using CASTEP[END_REF] with cutoff energy of 600 eV and PBE (Perdew-Burke-Ernzerhof) functional. [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] The ultrasoft pseudopotentials are the ones supplied with CASTEP. [START_REF] Vanderbilt | Soft self-consistent pseudopotentials in a generalized eigenvalue formalism[END_REF]. A better visualization of distribution of Co nanoparticles on the surface of the support can be obtained from the energy-dispersive mapping images recorded using the HAADF-STEM technique (Fig. 2). In the corresponding EDS maps, areas where the Co K α1 line (6.930 keV) emerges from the background noise are figured in red, while those where the Al K α1/2 line (1.486 keV) emerges are in blue. Thus, at the spatial resolution of the HAADF-STEM image (about nm), Co appeared evenly distributed on the surface of the support and no Co particles could be evidenced by this method. Electron microscopy techniques revealed the morphology and structure of the samples, but due to their strictly local character, these methods are not optimal for studying the crystal structure of Co nanoparticles in the entire volume of the sample. On the other hand, 59 Co internal field NMR ( 59 Co IF NMR) is a technique able to discern between different types of crystal structure of metallic Co in the bulksomething that is more likely to relate to the overall efficiency of the catalyst. of Co with certainty because their appearance may be connected with the background noise in the EDX spectra. However, since noise is expected to be randomly distributed on the elemental maps, the areas that contain larger densities of points (shown with the white circles in Fig. 4b) can be associated with Co particles and do indeed coincide with the brighter areas of the HAADF-STEM image (Fig. 4a). at room temperature was shown to be equal to roughly 7.5 nm in diameter for isolated particles. [START_REF] Yakovlev | Superparamagnetic behaviour of metallic Co nanoparticles according to variable temperature magnetic resonance[END_REF] However, in non-isolated particles, magnetic ordering of the particles due to interparticle magnetic interactions (dipolar and exchange) affects the critical size for superparamagnetic/ferromagnetic transition in a way that is difficult to predict quantitatively. In practice, we can consider the particles to be isolated when the energy of dipolar interaction between the particles is much smaller compared to magnetic anisotropy energy [START_REF] Hansen | Models for the dynamics of interacting magnetic nanoparticles[END_REF], which is the case for Co particles about 5 nm in diameter and separated by at least the same distance.

Results

High-resolution transmission electron microscopy and EDS

Considering this, and according to the electron microscopy data, a significant fraction of Co nanoparticles in the 5% Co/γ-Al 2 O 3 and 5% Co/χ-Al 2 O 3 samples could be superparamagnetic and undetectable by 59 Co IF NMR at room temperature. Therefore, we conducted the 59 Co IF NMR experiments at 30 K where the critical particle size for superparamagnetic transition decreases to 3.5 nm. It can then be safely assumed that almost all Co particles in the studied samples are in a ferromagnetic state and contribute to the 59 Co IF NMR spectra.

As stated in the experimental section and according to a now well-established procedure, [START_REF] Yakovlev | Superparamagnetic behaviour of metallic Co nanoparticles according to variable temperature magnetic resonance[END_REF] the spectra were recorded at different radiofrequency powers in order to discern between the multi domain and single domain ferromagnetic particles. The distribution of optimal radiofrequency power (power at which the spectral intensity at the chosen frequency is maximum) for all spectra was practically uniform meaning that we have not detected any multi domain Co particles. This is in complete agreement with the observations made using HRTEM since multiple magnetic domains are not expected to occur in particles smaller than 70 nm. [START_REF] Leslie-Pelecky | Magnetic Properties of Nanostructured Materials[END_REF] Therefore, all of the obtained spectra were analyzed from the standpoint of single domain particles.

Experimental and predicted 1 H chemical shifts

The initial stage in the formation of Co nanoparticles on the surface of the support is the anchoring of Co(II), from the precursor ions, onto the available vacant sites. Using EXAFS and ab initio calculations of the adsorption energy of Co on different surfaces of α-Al 2 O 3 , Chizallet et al. [START_REF] Chizallet | Surface orientation dependent interaction of cobalt (II) precursors with alpha-alumina[END_REF] have recently demonstrated that the hydroxyl coverage of the surface and the location of the surface vacancies greatly influence the interaction energy of Co(II) with the alumina surface.

Determining ab initio the hydroxyls on the alumina surface susceptible of interacting with different amounts of Co(II) is obviously very informative but requires lengthy computations involving large sets of models with different types of alumina surfaces together with the necessary water to coordinate and stabilize the dangling groups. We tried instead a semi empirical approach based on correlating the experimentally observed and theoretically predicted 1 H NMR shifts.

In order to do so, the first step is to choose a reliable structural model of γ-Al 2 O 3 and χ-Al 2 O 3 surfaces. While modeling the surface of the stable trigonal α-alumina is relatively straightforward due to its completely refined crystal structure, the structures of metastable alumina remain disputed. The low temperature γ-Al 2 O 3 metastable phase, in particular, has attracted the highest attention due to its relevance as an heterogeneous catalyst and catalyst support. γ-Al 2 O 3 is generally considered to have a cubic spinel structure [START_REF] Prins | On the structure of γ-Al 2 O 3[END_REF][START_REF] Busca | The surface of transitional aluminas: A critical review[END_REF], even though the stoichiometry of Al 2 O 3 does not directly correspond to a spinel AB 2 O 4 . Such structural disparity has led to different models describing the distribution of Al ions and vacancies in the spinel cationic sublattice [START_REF] Prins | On the structure of γ-Al 2 O 3[END_REF]. Despite the differences of these models, the majority of researchers agree that the bulk structure of γ-Al 2 O 3 particles belongs to the cubic spinel type. However, the surface structures and bonding scheme of these particles remain debated; therefore a brief discussion of existing models is necessary before proceeding further.

The most widely used model of metastable alumina surfaces was proposed by Knözinger and Ratnasamy [START_REF] Knözinger | Catalytic Aluminas: Surface Models and Characterization of Surface Sites[END_REF]. It consists of bulk spinel structure cleavages along (100), ( 110) and (111) crystal planes. The reactivity of such surfaces in hydrated conditions mainly depends on the hydroxyl coverage obtained by addition of protons and hydroxyls to compensate for the charge and complete the coordination symmetry of surface Al and O sites [START_REF] Digne | Hydroxyl Groups on γ-Alumina Surfaces: A DFT Study[END_REF][START_REF] Digne | Use of DFT to achieve a rational understanding of acid-basic properties of γ-alumina surfaces[END_REF]. Further improvements of this model include reconsiderations of the bulk alumina structure to nonspinel models [START_REF] Krokidis | Theoretical Study of the Dehydration Process of Boehmite to γ-Alumina[END_REF], as well as accounting for the finite size effect [START_REF] Batista | Beyond γ-Al 2 O 3 crystallite surfaces: The hidden features of edges revealed by solid-state 1 H NMR and DFT calculations[END_REF][START_REF] Pigeon | Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces[END_REF] and for stacking faults of individual alumina crystallites [START_REF] Tsybulya | Nanocrystalline transition aluminas: Nanostructure and features of x-ray powder diffraction patterns of lowtemperature Al 2 O 3 polymorphs[END_REF].

The first consequence of the finite size of an alumina crystallite is the presence of edges between its surfaces, which leads to the occurrence of unique hydroxyl sites that are not hydrogen-bonded to other hydroxyl groups and remain highly reactive to chlorination, as was

shown by 1 H NMR and ab initio calculations by Batista et al. [START_REF] Batista | Beyond γ-Al 2 O 3 crystallite surfaces: The hidden features of edges revealed by solid-state 1 H NMR and DFT calculations[END_REF]. The second important effect of the finite particle size is linked to the synthetic routes used to obtain the metastable aluminas.

Most frequently, aluminum oxides are obtained through dehydration of corresponding hydrates, as was experimentally demonstrated by Stumpf et al [START_REF] Stumpf | Thermal Transformations of Aluminas and Alumina Hydrates -Reaction with 44% Technical Acid[END_REF]. In this case, the restructuring of individual particles is governed by topotactic transformations during which the structural features of the initial particles are conserved. Due to the layered nature of the alumina hydrates, the topotactic transitions result in differences between alumina surfaces that formally belong to the same crystal plane family (e. g. ( 110) cubic spinel plane) depending on basal/lateral position in the initial particle, as was demonstrated using DFT calculations in the recent paper by Pigeon et al. [START_REF] Pigeon | Revisiting γ-alumina surface models through the topotactic transformation of boehmite surfaces[END_REF].

Additionally, metastable aluminas are intrinsically nanostructured materials, thus the description of their structures requires to account for the defectiveness and stacking of the individual crystallites that also play an important role on the formation of the particle surfaces.

The approach developed by Tsybulya et al. accounts for the connectivity of individual spinel crystallites through planar defects lying in the (100), ( 110) and ( 111) crystal planes of cubic spinel [START_REF] Tsybulya | Nanocrystalline transition aluminas: Nanostructure and features of x-ray powder diffraction patterns of lowtemperature Al 2 O 3 polymorphs[END_REF]. This technique allowed modeling the structures of low-temperature aluminas as well as demonstrating the effect of abundant shearing planar defects that allow the existence of a cubic spinel material with a non-spinel stoichiometry. Later, this approach was also used by peak at 2θ = 43° in its XRD pattern that does not belong to the cubic spinel structure, even though the rest of the XRD pattern corresponds to the said structure. Additionally, the 27 Al NMR spectra reveal an Al [6] :Al [4] equal to roughly 3:1 that does not correspond to the spinel structure for which this ratio is expected to be 2:1 (as observed in 27 Al NMR spectra of γ-Al 2 O 3 ) [START_REF] Chandran | Alumina: discriminative analysis using 3D correlation of solid-state NMR parameters[END_REF].

However, recently, Yatsenko et al. [START_REF] Yatsenko | Low Temperature Transitional Aluminas: Structure Specifics and Related X-ray Diffraction Features[END_REF] have shown using the "crystallite stacking" approach that the non-spinel diffuse scattering peak of χ-Al 2 O 3 may appear due to planar defects of the anion sublattice of cubic spinel structure (the O 2-sublattice in ratioAB 2 O 4 spinel follows a regular fcc pattern, while such planar defects change its local ordering to hcp). This observation allows one to consider the particles of χ-Al 2 O 3 as having the cubic spinel structure when trying to model its surface in the same way as it was done for γ-Al 2 O 3 . Considering this, the surface of χ-Al 2 O 3 is mostly represented by the (111) spinel crystal plane, as was shown in [START_REF] Tsybulya | Nanocrystalline transition aluminas: Nanostructure and features of x-ray powder diffraction patterns of lowtemperature Al 2 O 3 polymorphs[END_REF]. Moreover, if we apply the same logic of topotactic transitions to the well-refined structure of gibbsite just as it was done for boehmite in the case of γ-Al 2 O 3 , the result for the most exposed crystal plane will be the same. Indeed, the hexagonally symmetric particles of gibbsite have a layered structure, with the layers of aluminum octahedral separated by hydrogen-bonded hydroxyl layers in the direction perpendicular to the [001] axis. Thus, the (001) crystal plane that has a hexagonal symmetry is the most prevalent for the gibbsite particles [START_REF] Ksenofontov | Structure refinement and thermal stability of gibbsite[END_REF][START_REF] Mercury | On the Decomposition of Synthetic Gibbsite Studied by Neutron Thermodiffractometry[END_REF]. According to Mitsui et al. [START_REF] Mitsui | Microstructural Transformation with Heat-Treatment of Aluminum Hydroxide with Gibbsite Structure[END_REF],

gibbsite particles retain their morphology during the dehydration into χ-Al 2 O 3 including the hexagonal symmetry of the (001) basal planes. Therefore, the retention of the particle morphology and the surface prevalence of the hexagonally symmetric planes (001) in the case of gibbsite and (111) in the case of χ-Al 2 O 3 allow us to assume that the (001) basal planes of gibbsite undergo a transition into the (111) planes of the dehydrated alumina.

In summary, the surface of metastable aluminas can be accurately described in terms of spinel crystal planes, mainly (100) and (110) with the minor presence of the (111) for γ-Al 2 O 3 nanoparticles, and (111) for χ-Al 2 O 3 .

Consequently, the γand χ-Al 2 O 3 surfaces (Fig. 6) were modeled by cleaving the crystal structure along the directions of the (110) of (111) crystal planes. We also note that such a representation of the particle surface as an infinite slab cannot possibly account for the edges formed between the crystallite surfaces or for the subtle features of the crystal planes formed during the topotactic transition. Nevertheless, such an approach can still be feasible and useful The hydroxyl groups were added to the spinel slabs to compensate for the charge and any dangling bonds that were formed during the cleavage (more details can be found in the Experimental section). After the relaxation of the structures without any additional constraints, we calculated the 1 H chemical shifts of all the hydroxyl groups now present on these model surfaces. The resulting sets of calculated chemical shifts ascribed to the (110) and (111) surfaces, as well as their comparison with the experimental 1 H NMR spectra, can be found in Fig. 7 (calculated spectra are shown with colored lines, while the experimental data are shown in black). Both the calculated sets of chemical shifts and the experimental 1 H 10 kHz MAS NMR spectra displayed a significant difference in the spectral intensity in the 5-10 ppm region.

Namely, the intensity in this region was zero for the (111) plane model, which agreed well with the low relative intensity observed in the same region of the experimental spectrum of χ-Al 2 O 3 .

Furthermore, edges between the spinel surfaces are present in both samples, resulting in additional terminal hydroxyl groups that give rise to a distinct line at 0 ppm in the experimental spectra (blue line in Fig. 7a,b), as was predicted by Batista et al. [START_REF] Batista | Beyond γ-Al 2 O 3 crystallite surfaces: The hidden features of edges revealed by solid-state 1 H NMR and DFT calculations[END_REF]. These hydroxyl groups were obviously missing from the perfect plane models and thus, had to be added to the decomposition of the experimental spectra manually. The experimental spectra were decomposed into the theoretical spectra for (111) and (110) planes as well as the line corresponding to surface edges until the best match was achieved. The resulting relative intensities of the theoretical spectra of crystal planes excluding the surface edges were: 50% is energetically favorable during the impregnation and drying stages of FTS catalyst synthesis. [START_REF] Larmier | Tuning the Metal-Support Interaction by Structural Recognition of Cobalt-Based Catalyst Precursors[END_REF] This interaction was linked to the formation of the mixed cobalt-aluminum oxide layer at the interface between the support surface and Co nanoparticle through structural recognition. Simply put, Co 2+ ions from the impregnation solution tend to occupy the vacancies created by removal of the octahedral Al 3+ sites, which leads to the creation of stable octahedral Co 2+ complexes and dispersion of Co over the surface of the support. This, in turn, may lead to the deactivation of small (less than 5.3 nm in diameter) Co nanoparticles during the onset of FTS or to a complete inactivity of the sample in the extreme case of very small (less than 3 wt.%) Co loading [START_REF] Tsakoumis | Evaluation of Reoxidation Thresholds for γ-Al 2 O 3 -Supported Cobalt Catalysts under Fischer-Tropsch Synthesis Conditions[END_REF].

The epitaxial relation between the reduced Co nanoparticles and the surface of χ-Al 2 O 3 that we observed by HRTEM in the reduced sample was necessarily a direct consequence of a strong particle-support interaction at some stage of the synthesis. However, the effects of this interaction were strikingly different in the cases of γand χ-Al 2 O 3 . The key differences between these supports lie firstly in the crystal planes dominating their surface and secondly in the morphology of the particles.

As apparent from our semi empirical modeling of alumina surfaces, γand χ-Al A prominent signal at 0 ppm in the 1 H NMR spectrum of χ-Al 2 O 3 is also worth noting because, according to work by Batista [START_REF] Batista | Beyond γ-Al 2 O 3 crystallite surfaces: The hidden features of edges revealed by solid-state 1 H NMR and DFT calculations[END_REF], this chemical shift range corresponds to the hydroxyl groups supported on the edge between the crystallite surfaces of the particle. Even though the relative content of these hydroxyl groups is high, they do not facilitate the anchoring of the Co 2+ ions due to their location. Indeed, the edges contain undercoordinated sites and cannot provide a suitable vacancy for an octahedral Co 2+ complex.

Thus, the modeling of the hydroxyl coverage helps rationalize the observations made using HRTEM and 59 Co IF NMR; namely, the smaller particle size and the prevalence of fcc metallic

Co in the sample supported on γ-Al 2 O 3 after calcination and reduction, when compared to the sample supported on χ-Al 2 O 3 . The same rationalization may also be made for the difference in moisture capacities that we observed during the sample preparation while the specific surface areas of the powders were very close. The role of surface hydroxyl coverage during surface wetting has been studied for many oxide materials including Al 2 O 3 [START_REF] Salmeron | Water growth on metals and oxides: binding, dissociation and role of hydroxyl groups[END_REF][START_REF] Deng | Adsorption of Water on Cu 2 O and Al 2 O 3 Thin Films[END_REF]. Dangling hydroxyl groups available for the formation of H-bonds with the water molecules were shown to facilitate the formation of a liquid water layer on the surface of all of the studied oxides. Thus, the higher moisture capacity of γ-Al 2 O 3 can also be ascribed to the presence of terminal hydroxyl groups on (110) plane, representing the majority of its surface. Nevertheless, it is reasonable to assume that the hexagonal symmetry of the surface of χ-Al 2 O 3 promotes the hcp phase formation which appeared to be favored, as deduced from the experimental NMR observations.

2 O 3

 23 and χ-Al 2 O 3 , in directing the dispersion and structure of Co nanoparticles. γ-Al 2 O 3 is a well-studied and widely employed support in Co FTS catalysts. It has a high specific surface area and demonstrates good stability under usual reaction conditions when stabilized by metal oxides. χ-Al 2 O 3 is the lowest temperature form of alumina in the gibbsite alumina series; the use of this metastable phase as a catalyst support has been investigated to a much lesser extent, even though it also demonstrates a relatively high specific surface area and thermal stability. Pansanga et al.have demonstrated that the use of a mixed χ-Al 2 O 3 + γ-Al 2 O 3 support leads to increased Co dispersion at high Co loadings, but this effect could be mostly due to the morphology of the particles and not necessarily to their surfaces structures[START_REF] Pansanga | Effect of mixed γ-and χ-crystalline phases in nanocrystalline Al 2 O 3 on the dispersion of cobalt on Al 2 O 3[END_REF]. The crystal structure as well as crystallite stacking of χ-Al 2 O 3 still remain an open question with several equally feasible viewpoints (see for instance a recent mini-review by Prins[START_REF] Prins | On the structure of γ-Al 2 O 3[END_REF]). Nevertheless, regardless of the preferred theory on the structure of bulk γ-Al 2 O 3 and χ-Al 2 O 3 , an important difference between these alumina phases lies in the different crystal faces that dominate the surface of the particles.[START_REF] Tsybulya | Nanocrystalline transition aluminas: Nanostructure and features of x-ray powder diffraction patterns of lowtemperature Al 2 O 3 polymorphs[END_REF][START_REF] Khabibulin | Surface Hydroxyl OH Defects of η-Al 2 O 3 and χ-Al 2 O 3 by Solid State NMR, XRD, and DFT Calculations[END_REF] As a result, χ-Al 2 O 3 differs from γ-Al 2 O 3 by the type and density of defect

Figure 1 (

 1 Figure 1 (a, b) displays typical HRTEM images of the sample 5% Co/γ-Al 2 O 3 at different magnifications. According to the micrographs, the γ-Al 2 O 3 support consists of large (1 μm) agglomerates of randomly oriented alumina nanocrystallites that are slightly elongated along the [100] axis. Cobalt nanoparticles (shown with white arrows in Fig. 1b) are poorly distinguished on the micrographs due to their relatively small size (5.5 nm mean diameter according to HRTEM image analysis). Several Bragg diffraction spots, corresponding to metallic Co crystallites, are clearly visible on the Fourier transform of Fig. 1b (Fig. 1c) along with the annular diffractions from the randomly oriented crystallites of γ-Al 2 O 3 . Interplanar distances in fcc and hcp Co are quite close, which makes the distinction between these crystal structures almost impossible here despite the presence of clearly defined diffraction spots.

Fig. 1 .

 1 Fig. 1. (a) and (b): HRTEM images of the 5% Co/γ-Al 2 O 3 sample at different magnifications. Metallic Co nanoparticles are visible as dense dark areas (additionally highlighted with white arrows on b). (c): spatial Fourier transform of b). Diffraction spots shown with arrows 1 and correspond to cobalt nanoparticles while the annular diffractions from γ-Al 2 O 3 are shown with arrow 3.

Fig. 2 .

 2 Fig. 2. HAADF-STEM image of the 5% Co/γ-Al 2 O 3 sample (a) and the elemental mapping of the area in the red box obtained with EDS (b). Al signals are shown in blue, Co signals are shown in red.

Fig. 3 .

 3 Fig. 3. (a) and (b): HRTEM images of the 5% Co/χ-Al 2 O 3 sample at different magnifications. Metallic Co nanoparticles are visible as dense dark areas. (c): Magnification of the area inside the red box of b). (d): Spatial Fourier transform of b). The diffraction spots corresponding to the Co metal and χ-Al 2 O 3 are denoted with white arrows. Such a close arrangement of diffraction spots in distance and angle suggests epitaxy between the Co nanoparticle and the surface of the support.

Fig. 4 .

 4 Fig. 4. HAADF-STEM image of the 5% Co/χ-Al 2 O 3 sample (a) and the elemental mapping of the area in the red box obtained with EDS (b), Al signals are shown in blue, Co signals are shown in red. White circles in (b) are used to locate the large Co NP assemblies identified as brighter areas in (a). Individual points on the EDS elemental maps do not indicate the presence

3. 2 59

 2 Co internal field nuclear magnetic resonance 59 Co IF NMR spectra reflect both the crystal and the magnetic structures of Co nanoparticles, which affect the hyperfine magnetic field at the location of the nucleus. It can be observed only in ferromagnetic particles. In superparamagnetic ones, the internal field fluctuates rapidly at the timescale of the NMR experiments and no signal is generated. Consequently, we have demonstrated earlier that the size of Co nanoparticles is an equally important factor affecting the 59 Co IF NMR spectra when particles below the nominal critical superparamagnetic size are present in the sample and invisible in the IF NMR spectra. The critical size below which particles isolated from each other are superparamagnetic (and thus are undetectable in IF NMR)

Fig. 5 . 59 Co

 559 Fig. 5. 59 Co IF NMR spectra of the 5% Co/γ-Al 2 O 3 (bottom) and 5% Co/χ-Al 2 O 3 (top) samples. T = 30 K. Spectra are normalized to the same maximal intensity. Echo measurement points are shown with symbols, while the lines show the result spectral decomposition into two Pseudo-Voigt profiles for fcc Co (green) and hcp Co (blue). The sum of these two profiles is shown in red.

  Pakharukova et al. to create 3D nanostructured models of γ-Al 2 O 3 particles and simulate the XRD patterns, which coincide with experimental XRD patterns and allow to explain the diffuse nature of several peaks[START_REF] Pakharukova | A study of γ-Al 2 O 3 from the viewpoint of 3D nanostructure[END_REF].Both the topotactic transition and crystallite stacking approaches do not contradict each other, but provide a complementary description of both the bulk and the surface of metastable alumina nanoparticles. Importantly, both of these approaches coincide in their prediction of the crystal planes that form the surface of γ-Al 2 O 3 nanoparticles, namely (100) and (110) crystal planes with the minor presence of the (111) plane. The much less investigated structure of the metastable χ-Al 2 O 3 obtained through dehydration of aluminum trihydroxide gibbsite remains an open question without any commonly accepted viewpoint. The source of this controversy lies in the presence of a diffuse scattering

  for the prediction of the general behavior of the alumina surface towards the hydrated Co ions in the precursor solution. By using the similar infinite slab model, Larmier et al.[START_REF] Larmier | Tuning the Metal-Support Interaction by Structural Recognition of Cobalt-Based Catalyst Precursors[END_REF] have demonstrated the so-called structural recognition of Co 2+ ions on the (100) and (110) surfaces of a cubic spinel model of γ-Al 2 O 3 . They have shown that the most energetically favorable mode of adsorption of Co 2+ ions is the one during which octahedral complexes including one or two bonds between Co 2+ and lattice oxygen are formed.

Fig. 6 .

 6 Fig. 6. Models of the (111) (top) and (110) (bottom) crystal planes of the spinel structure of crystallites forming metastable alumina phases. Al atoms are shown in silver, O atoms are shown in red, H atoms are shown in black. Green circles and lines denote the positions of the Al atoms that are missing from the next layer. Such sites can form vacancies available to Co 2+ ions.

( 111 )

 111 plane and 50% (110) plane for χ-Al 2 O 3 ; 30% (111) plane and 70% (110) plane for γ-Al 2 O 3 .Thus, the 1 H NMR data demonstrate that the surface of γ-Al 2 O 3 is mostly represented by the (110) crystal plane, while an equal mixture of (111) and (110) spinel crystal planes makes up the surface of χ-Al 2 O 3 . The difference between the compositions of these surfaces can play an important role during their interaction with the Co impregnation solution since hydroxyl coverages of the spinel crystal planes differ significantly: the hydration of the (111) spinel plane gives rise exclusively to bridging (μ 2 -and μ 3 -OH) groups, while the (110) spinel plane contains a mixture of terminal (μ 1 -OH) and bridging (μ 2 -OH) groups.

Fig. 7 . 4 Discussion

 74 Fig. 7. Decomposition of the experimental 10 kHz MAS 1 H NMR spectra of dehydrated -Al 2 O 3 (a, black line) and -Al 2 O 3 (b, black line) into the theoretically calculated spectra for (110) and(111) spinel planes (red and green lines, respectively) and the separate line corresponding to hydroxyl groups supported on the edges between the crystallite surfaces according to ref.[START_REF] Batista | Beyond γ-Al 2 O 3 crystallite surfaces: The hidden features of edges revealed by solid-state 1 H NMR and DFT calculations[END_REF] (blue line). The relative contributions of the (111) and (110) planes to the spectra, excluding the signal at 0 ppm, were: 50% (111) plane and 50% (110) plane in the case of χ-Al 2 O 3 ; 30% (111) plane and 70% (110) plane in the case of γ-Al 2 O 3 . Note, that the given values reflect the surface densities of the OH groups and correspond directly to the surface areas represented by these planes in the samples.

2 O 3

 23 differ by the nature of crystal planes at their surface thus leading to completely different types of hydroxyl coverage. The hydroxyl coverage of (111) plane (and correspondingly χ-Al 2 O 3 , Fig.6top) consists exclusively of bridging hydroxyl groups located under the potential vacancies (missing Al 3+ sites, shown with green circles in Fig.6, top), thus preventing them from stabilizing any adsorbed ion. For the (110) plane (dominating the surface of γ-Al 2 O 3 , Fig.6bottom) we observe a picture strongly resembling the C-μ 1 -μ 2 model presented by Chizallet et al.[START_REF] Chizallet | Surface orientation dependent interaction of cobalt (II) precursors with alpha-alumina[END_REF], with the main similarity being the mixed terminal-bridging hydroxyl coverage. This, in turn, is connected with the presence of a plane below the level of terminal hydroxyl where vacancies for octahedral complexes can form and provide direct anchoring to the lattice O 2-(Fig. 6, bottom). According to DFT calculations made by Chizallet et al., these vacancies provide a large gain in the absorption energy during the onset of Co(OH) 2 formation. Such increased strength of interaction between the metallic Co precursor and the surface of the support would inevitably lead to strong dispersion of Co 2+ ions over the surface of γ-All 2 O 3 that would, in turn, result in a large amount of hard-to-reduce Co species and a decrease of the metallic Co particle size.

Furthermore, the remarkable

  coincidence of the structural parameters of the metallic Co particles and the surface of χ-Al 2 O 3 cannot be ignored here. The hexagonally symmetric pattern of its surface could be directly explained by the topotactic transformations of gibbsite particles as mentioned above. This direct observation of the prominent (111) crystal plane in HRTEM images of χ-Al 2 O 3 serves as further proof for the spinel-based models of its crystal structure and may shed light on what can be called structural recognition. In the view of the (111) surface model presented in the preceding section (Fig.8), it can be inferred that the missing Al atoms (possible vacant sites for Co 2+ anchoring) also follow a hexagonal pattern. Moreover, the morphology of χ-Al 2 O 3 particles that includes relatively large and oriented crystallites would favor the formation of larger and well-ordered metal particles. However, the distance between any two closest "vacancies" (approximately 4.5 Å) is much larger than the Co-Co distance in the metallic cobalt (approximately 2.5 Å). Consequently, we cannot say that this simple vacancy model determines the way in which the Co nanoparticles are formed after reduction.

Fig. 8 . 5 Conclusions 6 Acknowledgements

 856 Fig. 8. View on the (111) plane from the above. Tetrahedral Al sites are depicted with red, octahedral Al sites are depicted with blue. Green dots represent the possible vacancies.

2 Experimental 2.1 Synthesis of Co/γ-Al 2 O 3 and Co/χ-Al 2 O 3 supported samples

  Low-temperature metastable alumina phases were used as the supports for the studied samples. γ-Al 2 O 3 was obtained by calcination of pseudoboehmite (Pural SB1, Condea Chemie, Hamburg, Germany) in air at 923 K for 6 h (1.5 K/min heating rate). The same procedure was used to obtain χ-Al 2 O 3 from gibbsite (Sigma-Aldrich, Merck, Darmstadt, Germany). XRD and27 Al solid-state NMR techniques (see Supplementary Materials) were used to confirm the purity of the obtained powders as in our previous works[START_REF] Yakovlev | Effect of carbon coating on the thermal stability of nanocrystalline χ-Al 2 O 3[END_REF][START_REF] Yakovlev | Stabilizing effect of the carbon shell on phase transformation of the nanocrystalline alumina particles[END_REF]. The resulting BET (Brunauer-Emmett-Teller) specific surface areas of γ-Al 2 O 3 and χ-Al 2 O 3 were roughly 175 m 2 /g and 150 m 2 /g respectively.

	the samples were reduced in a H 2 stream (50 ml/min) at 673 K for 4 hours (3 K/min ramping
	rate), purged with Ar for 20 min and sealed in airtight glass ampoules to avoid oxidation. The
	resulting samples were denoted as 5% Co/(γ-Al 2 O 3 or χ-Al 2 O 3 ).

In order to determine the moisture capacity of the obtained supports, they were dried at a temperature of 383 K overnight. Then, droplets of distilled water were added to 1 g of alumina under continuous stirring until the sample appeared uniformly wet. The resulting values of moisture capacity were 0.91 ml/g for γ-Al 2 O 3 and 0.50 ml/g for χ-Al 2 O 3 .

Co/Al 2 O 3 supported samples with different metallic Co content were obtained using incipient wetness impregnation with an aqueous solution of Co(NO 3 ) 2 •6H 2 O (98%, Sigma-Aldrich). The higher moisture capacity of γ-Al 2 O 3 allowed introducing up to 10 wt.% of Co into the sample with a single impregnation, while χ-Al 2 O 3 was able to take only up to 5 wt.% Co in a single impregnation step. Thus, to obtain comparable samples with similar impregnation protocols (meaning having the same Co content obtained by following identical impregnation protocols), 5 wt.% was finally chosen as the target Co content that could be obtained for both supports in a single impregnation step. The samples were impregnated with the Co nitrate solution, then dried at 383 K for 4 hours and calcined in an Ar flow at 623 K for 4 hours. Finally,


Christopher C Kyriazis
email: ckyriazis@g.ucla.edu

Jacqueline A Robinson
email: jacqueline.robinson@ucsf.edu

Sergio F Nigenda-Morales

Annabel C Beichman

Lorenzo Rojas-Bracho

Kelly M Robertson

Michael C Fontaine

Robert K Wayne

Barbara L Taylor

Kirk E Lohmueller
email: klohmueller@g.ucla.edu

Phillip A Morin
email: phillip.morin@noaa.gov

Models based on best-available information support a low inbreeding load and potential for recovery in the vaquita

Main text

In [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF], we employed genomic and simulation analyses to demonstrate that the critically endangered vaquita porpoise has a reduced potential for future inbreeding depression (inbreeding load), and is therefore not doomed to extinction by deleterious genetic factors. Garcia-Dorado and Hedrick (2022) (hereafter, GD&H) critique our analysis for not sufficiently demonstrating that the vaquita has a low inbreeding load and a good chance of recovery in the absence of continued incidental mortality in fishing gillnets.

Our conclusion that the vaquita likely has a very low inbreeding load is supported, first and foremost, by the finding that there are very few deleterious mutations that can contribute to inbreeding depression segregating in the vaquita due to its small historical population size.

There is widespread agreement that inbreeding depression is overwhelmingly due to the exposure of recessive deleterious mutations that are concealed as heterozygotes [START_REF] Charlesworth | The genetics of inbreeding depression[END_REF][START_REF] Hedrick | Understanding Inbreeding Depression, Purging, and Genetic Rescue[END_REF]. In the relative absence of such mutations, there is simply very little fuel for inbreeding depression. For example, our genomic analysis shows that vaquitas have ~17 heterozygous loss of function (LOF) mutations per individual, whereas the blue whale genome has ~248 of such mutations. Loss of function mutations can, in many cases, have severe effects on fitness because they disrupt the function of protein-coding genes (though see [START_REF] Macarthur | Loss-of-function variants in the genomes of healthy humans[END_REF]. Under the assumption that such LOF mutations are (partially) recessive, a low count of heterozygous LOF mutations as observed in the vaquita implies a low inbreeding load, whereas a high count as observed in the blue whale implies a high inbreeding load.

GD&H are correct that we do not precisely know the selection (s) or dominance (h) coefficients for putatively deleterious mutations, such that our identification of few segregating deleterious mutations in vaquitas offers only qualitative insight into inbreeding load. This limitation motivated our complementary simulation analysis, where we employ a distribution of s for new mutations that was inferred from our genomic dataset. Although we agree that our model, like any model, makes assumptions that should be critically evaluated, our model is informed by the best-available information on selection and demographic parameters in the vaquita. Below, we review some of the key components of the vaquita and our model that inform our conclusions.

What should be considered a "small" historical population size?

GD&H assert that inbreeding load in the vaquita cannot be low since the species has a large historical population size. Although the historical population size of vaquita may seem large relative to the very small recent effective population sizes (N e ) that are observed in endangered species, the historical size of the vaquita population is not large when considering the broader context of long-term N e in mammals or other taxa. Long term N e in mammals (defined here as N e =π/(4μ) where π is heterozygosity and μ is the mutation rate) is typically on the order of tens to hundreds of thousands, and rarely below 5000. For example, using published estimates of genome-wide heterozygosity in 42 species of mammals, we recently estimated a median long-term N e of 21,875 [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF]. Based on our estimate of the mutation rate in vaquitas of μ=5.8e-9 and estimate of π=9.04e-5 [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF], vaquitas are estimated to have a long-term N e of 3,896, the second lowest of the 42 species in this dataset [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF]). This finding is also supported by fitting more complex nonequilibrium demographic models to genomic datasets for the vaquita, which similarly estimate N e <5,000 going back tens or hundreds of thousands of years [START_REF] Morin | Reference genome and demographic history of the most endangered marine mammal, the vaquita[END_REF][START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF]). The species with the smallest long-term N e in this dataset, the Channel Island fox, has been previously shown to exhibit no phenotypic signs of inbreeding depression despite having experienced severe recent bottlenecks [START_REF] Robinson | Purging of Strongly Deleterious Mutations Explains Long-Term Persistence and Absence of Inbreeding Depression in Island Foxes[END_REF]. By contrast, North American gray wolves have a large long-term N e of ~92,000 (assuming μ = 4.5e-9 [START_REF] Koch | De Novo Mutation Rate Estimation in Wolves of Known Pedigree[END_REF]) and π = 1.65e-3 [START_REF] Robinson | Genomic signatures of extensive inbreeding in Isle Royale wolves, a population on the threshold of extinction[END_REF])), which may help explain numerous observed instances of severe inbreeding depression in the species [START_REF] Fredrickson | Genetic rescue and inbreeding depression in Mexican wolves[END_REF][START_REF] Räikkönen | Congenital bone deformities and the inbred wolves ( Canis lupus ) of Isle Royale[END_REF][START_REF] Robinson | Genomic signatures of extensive inbreeding in Isle Royale wolves, a population on the threshold of extinction[END_REF]. Thus, although the historical vaquita population size may seem large relative the current size of the population, it is still vastly smaller than the population sizes observed in most other species of mammals.

Although these long-term effective population sizes may not be representative of the current size of many threatened or endangered populations, they are essential for modelling the inbreeding load in a species. This is because long-term demographic processes have a major impact on patterns of segregating (recessive) deleterious variation, the key determinant of the inbreeding load. Though recent declines can influence patterns of segregating variation and inbreeding load, these effects often take tens or hundreds of generations to manifest. For example, as we show in our analysis, the recent and dramatic decline in the vaquita over the past ~35 years or ~3 generations does not appear to have impacted genetic diversity or inbreeding load in the species [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF]. This is likely the case for many other large mammals that have experienced declines over the past century, given the long generation times typical of these species. Thus, low inbreeding loads in species with long generation times are likely a product of small long-term historical population sizes, rather than recent humanmediated declines. However, the extent to which this is true for a given species will depend on the duration and severity of decline, something that can readily be assessed using simulations.

What should be considered a "typical" inbreeding load?

A useful approach for determining whether computational models of inbreeding depression are reasonable is to compare the predicted inbreeding load (B) from such models to those that have been estimated from natural populations (note that we report values in terms of the diploid inbreeding load [2B] in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF] but report the haploid inbreeding load [B] here to be consistent with GD&H). GD&H cite an average estimate from O'Grady et al. ( 2006) of B = 6, derived from an analysis of 16 existing inbreeding load estimates, as being a typical inbreeding load for wild populations. Based on this result, they then claim that our model-based prediction of B = 0.48 for the vaquita is unreasonably low. However, it has previously been shown that the inbreeding load estimate from O'Grady et al. ( 2006) is unreliable and upwardly biased, in part due to methodological issues associated with using generalized linear models with a logit link to estimate the inbreeding load [START_REF] Nietlisbach | Nonequivalent lethal equivalents: Models and inbreeding metrics for unbiased estimation of inbreeding load[END_REF]. Specifically, [START_REF] Nietlisbach | Nonequivalent lethal equivalents: Models and inbreeding metrics for unbiased estimation of inbreeding load[END_REF] found that the three highest inbreeding load values reported by O' Grady et al. (2006) are overestimates due to biased statistical models or issues with the original datasets (note that many of these same concerns apply to estimates reported in [START_REF] Hedrick | Understanding Inbreeding Depression, Purging, and Genetic Rescue[END_REF]). Moreover, the average estimate from O'Grady et al. ( 2006) is also likely to be upwardly biased due to their approach of summing together inbreeding load estimates from different life stages, an approach that ignores widespread pleiotropy for mutations underlying fitness [START_REF] Pickrell | Detection and interpretation of shared genetic influences on 42 human traits[END_REF][START_REF] Boyle | An Expanded View of Complex Traits: From Polygenic to Omnigenic[END_REF]. This approach can contribute to upward bias by potentially double or triple counting the effects of recessive deleterious mutations that contribute to inbreeding depression at different life stages (see [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF] for further discussion).

Based on 22 estimates that are deemed to be reliable and unbiased, Nietlisbach et al.

(2019) instead report a median inbreeding load for survival to sexual maturity in vertebrates of B = 2.25. Although our predicted inbreeding load of B = 0.48 is somewhat lower than this median, this is expected given the small historical population size and low levels of segregating (recessive) deleterious variation in the vaquita. Importantly, we note that this estimate of B = 2.25 is based on only 13 species, nearly all of which are birds, and should therefore be interpreted cautiously. Thus, obtaining additional high-quality estimates of the inbreeding load from wild populations represents a key area of future research that will enable better assessment of simulation models (see [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF], for further discussion). Nevertheless, based on available evidence, our model predicts an inbreeding load that is consistent with reliable estimates from natural populations.

How should we estimate selection and mutation parameters?

GD&H claim that our selection and dominance parameters are incorrect because they do not agree with estimates of selection and dominance parameters derived from experiments in Drosophila (e.g., [START_REF] Simmons | Mutations affecting fitness in Drosophila populations[END_REF][START_REF] Pérez-Pereira | Long-term exhaustion of the inbreeding load in Drosophila melanogaster[END_REF], 2022). However, such experimental estimates are well known to be biased towards strongly deleterious variation, as mutations with more mild effects cannot be observed in an experimental setting [START_REF] Davies | High frequency of cryptic deleterious mutations in Caenorhabditis elegans[END_REF][START_REF] Eyre-Walker | The distribution of fitness effects of new mutations[END_REF]. These experimental approaches are also limited due to issues of identifiability for selection and mutation parameters [START_REF] Lynch | Perspective: Spontaneous deleterious mutation[END_REF][START_REF] Eyre-Walker | The distribution of fitness effects of new mutations[END_REF][START_REF] Halligan | Spontaneous mutation accumulation studies in evolutionary genetics[END_REF]. In other words, the distributions of fitness in the experimental populations can be explained by a high mutation rate and a low strength of selection (s) or vice versa, making interpretation of these experimental results challenging.

Finally, experimental approaches are also only possible for laboratory organisms such as Drosophila or yeast, thus their relevance for understanding deleterious mutation parameters in natural populations of mammals such as the vaquita is unclear.

These limitations have motivated the increasing use of sequence-based estimates of the distribution of s, which leverage genetic variation datasets to estimate selection parameters while controlling for demography (Eyre-Walker and [START_REF] Eyre-Walker | The distribution of fitness effects of new mutations[END_REF]. Such approaches are widely employed in population genetics [START_REF] Eyre-Walker | The Distribution of Fitness Effects of New Deleterious Amino Acid Mutations in Humans[END_REF][START_REF] Boyko | Assessing the evolutionary impact of amino acid mutations in the human genome[END_REF][START_REF] Ma | Population Genomic Analysis Reveals a Rich Speciation and Demographic History of Orang-utans (Pongo pygmaeus and Pongo abelii)[END_REF][START_REF] Chen | Genetic diversity and the efficacy of purifying selection across plant and animal species[END_REF][START_REF] Huber | Determining the factors driving selective effects of new nonsynonymous mutations[END_REF][START_REF] Huber | Gene expression drives the evolution of dominance[END_REF][START_REF] Kim | Inference of the Distribution of Selection Coefficients for New Nonsynonymous Mutations Using Large Samples[END_REF][START_REF] Tataru | Inference of Distribution of Fitness Effects and Proportion of Adaptive Substitutions from Polymorphism Data[END_REF][START_REF] Huang | Inferring Genome-Wide Correlations of Mutation Fitness Effects between Populations[END_REF]) and give a much more complete picture of the full spectrum of deleterious mutations, from strongly to mildly deleterious. Nevertheless, these approaches do have limitations in that they are not well suited for estimating the proportion of lethal mutations [START_REF] Wade | Quantifying the fraction of new mutations that are recessive lethal[END_REF] and often assume additivity during inference (though see [START_REF] Huber | Gene expression drives the evolution of dominance[END_REF]). Addressing these limitations is an area of ongoing research.

Given these limitations of sequence-based approaches, we explored a variety of dominance models in our analysis, as well as models with an additional proportion of lethal mutations (see Fig. S21 and Table S6 in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF]. We also explored results when assuming a selection and dominance model proposed by [START_REF] Kardos | The crucial role of genome-wide genetic variation in conservation[END_REF] that is similar to that of Pérez-Pereira et al. (2022) (see [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF], for a detailed comparison of these models). In all cases, we found that recovery was still the likely outcome when assuming a 90% reduction in bycatch morality rates [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF]). In fact, we observed much lower predicted extinction rates under the model proposed by [START_REF] Kardos | The crucial role of genome-wide genetic variation in conservation[END_REF], perhaps due to more efficient purging in models with a high fraction of lethal mutations [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF].

In sum, models with a higher lethal mutation rate, like those favored by GD&H, also support our main conclusion that recovery is possible.

Finally, we note we did not consider the impact of non-coding deleterious mutations, as these mutations are generally inferred to be weakly deleterious (s on the order of 1e-3; [START_REF] Torgerson | Evolutionary processes acting on candidate cis-regulatory regions in humans inferred from patterns of polymorphism and divergence[END_REF][START_REF] Murphy | Broad-scale variation in human genetic diversity levels is predicted by purifying selection on coding and non-coding elements[END_REF][START_REF] Dukler | Extreme purifying selection against point mutations in the human genome[END_REF])) and therefore not highly relevant to modelling inbreeding depression. Indeed, the analysis of [START_REF] Pérez-Pereira | Long-term exhaustion of the inbreeding load in Drosophila melanogaster[END_REF] supports this assumption by suggesting that such deleterious mutations, though relevant over evolutionary timescales, are not particularly relevant in a conservation context. Future work should aim to refine estimates of the strength of selection against non-coding deleterious mutations and explore their potential impact on genetic load and extinction risk.

How can we validate simulation models?

Although the above verbal arguments serve as justification for the simulation analysis we present in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF], several approaches can be employed to more quantitatively validate and compare selection and dominance models informed by experimental versus sequence-based studies. These include comparing proposed models in terms of (1) how well they agree with patterns of genetic variation in sequencing datasets and

(2) how well they agree with reliable empirical estimates of the inbreeding load.

In [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF], we undertook this task using humans as a focal species. Humans are useful for this exercise because there are extensive genetic variation datasets [START_REF] Auton | A global reference for human genetic variation[END_REF], published demographic models (e.g., [START_REF] Gutenkunst | Inferring the joint demographic history of multiple populations from multidimensional SNP frequency data[END_REF][START_REF] Gravel | Demographic history and rare allele sharing among human populations[END_REF][START_REF] Li | Inference of human population history from individual whole-genome sequences[END_REF][START_REF] Tennessen | Evolution and Functional Impact of Rare Coding Variation from Deep Sequencing of Human Exomes[END_REF], estimates of mutation rates and coding sequence length [START_REF] Keightley | Rates and fitness consequences of new mutations in humans[END_REF], estimates of the distribution of s [START_REF] Eyre-Walker | The Distribution of Fitness Effects of New Deleterious Amino Acid Mutations in Humans[END_REF][START_REF] Boyko | Assessing the evolutionary impact of amino acid mutations in the human genome[END_REF][START_REF] Kim | Inference of the Distribution of Selection Coefficients for New Nonsynonymous Mutations Using Large Samples[END_REF], and estimates of the inbreeding load [START_REF] Bittles | The costs of human inbreeding and their implications for variations at the DNA level[END_REF] and number of segregating recessive lethals per individual [START_REF] Gao | An estimate of the average number of recessive lethal mutations carried by humans[END_REF]. Moreover, humans are much more closely related to the vaquita than Drosophila and have a long-term N e that is typical for mammals (N e =~17,000). In [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF], we leveraged these existing estimates of human demography, mutation rates, and coding sequence length to compare proposed selection and dominance models from Pérez-Pereira et al. ( 2022) and [START_REF] Kardos | The crucial role of genome-wide genetic variation in conservation[END_REF] to a model we previously presented in [START_REF] Kyriazis | Strongly deleterious mutations are a primary determinant of extinction risk due to inbreeding depression[END_REF], which is similar to that of Robinson et al. (2022). In [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF], we also propose a new model that better incorporates the impacts of recessive lethal mutations [START_REF] Wade | Quantifying the fraction of new mutations that are recessive lethal[END_REF].

When comparing these various selection and dominance models, we found that sequence-based models, such as the model used in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF], make predictions that are consistent with empirical estimates of the inbreeding load in humans, whereas models based on experimental approaches do not [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF]. For example, our model proposed in [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF] predicts an inbreeding load of B = 3.2 and ~0.9 recessive lethal mutations per diploid. These predictions are compatible with existing evidence in humans (note that [START_REF] Bittles | The costs of human inbreeding and their implications for variations at the DNA level[END_REF] estimate B = 0.7 for humans, though this is likely to be an underestimate as it is based only on juvenile mortality). By contrast, the model proposed by [START_REF] Pérez-Pereira | Reviewing the consequences of genetic purging on the success of rescue programs[END_REF] predicts an inbreeding load of B = 14, vastly exceeding available estimates in humans. The [START_REF] Pérez-Pereira | Reviewing the consequences of genetic purging on the success of rescue programs[END_REF] model also predicts ~12 recessive lethal mutations per individual, whereas available estimates are on the order of 0.6 mutations per diploid [START_REF] Gao | An estimate of the average number of recessive lethal mutations carried by humans[END_REF].

Comparing predicted patterns of genetic variation from these models to those observed in humans also provides support for sequence-based models. Specifically, the Pérez-Pereira et al. ( 2022) model predicts a large overabundance of rare mutations, with 72.8% of nonsynonymous mutations predicted to be singletons (variants with frequency 1/2n in a sample). However, only 56.8% of such mutations are observed to be singletons in the European sample from the 1000 Genomes dataset [START_REF] Auton | A global reference for human genetic variation[END_REF][START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF]. This large excess of rare mutations predicted by Pérez-Pereira et al. ( 2022) is a consequence of the extreme strength of negative selection in the model, which results in deleterious mutations being kept at low frequency. Importantly, the model we propose in [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF] predicts 57.3% of mutations to be singletons, in good agreement with observed patterns of genetic variation in humans [START_REF] Auton | A global reference for human genetic variation[END_REF].

Overall, these findings are consistent with previous findings that selection parameters derived from experimental studies in Drosophila and other taxa are biased towards strongly deleterious mutations [START_REF] Davies | High frequency of cryptic deleterious mutations in Caenorhabditis elegans[END_REF][START_REF] Eyre-Walker | The distribution of fitness effects of new mutations[END_REF][START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF]).

Thus, this analysis helps validate the use of sequence-based estimates of selection parameters, such as those employed in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF], given that they are consistent both with patterns of genetic variation and empirical estimates of the inbreeding load.

Other concerns

GD&H critique our simulations for ignoring stochastic environmental and demographic factors and not modelling a loss of adaptive potential in the vaquita. However, our analysis does incorporate demographic stochasticity, as we model survival and reproduction probabilistically. Although we do not model environmental stochasticity or loss of adaptive potential, the threat that these factors pose to the vaquita, if any, is entirely unknown. Moreover, we emphasize that the aim of our analysis was to test the assumption that the vaquita is doomed to extinction by inbreeding depression. We do not interpret our model projections beyond demonstrating the qualitative result that inbreeding depression alone does not impede recovery in the species, as we agree with the general view that population viability models should be interpreted cautiously in terms of their ability to accurately predict future population sizes [START_REF] Beissinger | On the Use of Demographic Models of Population Viability in Endangered Species Management[END_REF]. Though incorporating factors such as environmental stochasticity may influence our model predictions, they would not change our central conclusion that recovery remains possible.

Conclusions

In conclusion, we agree with GD&H that predictive models should be critically evaluated. Indeed, the critical evaluation of proposed mutation and selection models that we present in [START_REF] Kyriazis | Using computational simulations to quantify genetic load and predict extinction risk[END_REF] serves as justification for the type of sequence-based models we employed in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF]. Moreover, this analysis demonstrates that models based on experimental results, such as those proposed by [START_REF] Pérez-Pereira | Reviewing the consequences of genetic purging on the success of rescue programs[END_REF] and [START_REF] Kardos | The crucial role of genome-wide genetic variation in conservation[END_REF], are biased towards strongly deleterious variation and are not consistent with patterns of genetic variation or empirical estimates of the inbreeding load in humans. Nevertheless, our analysis in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF] found that recovery was still the likely outcome when assuming models with a much higher fraction of strongly deleterious variation. However, we emphasize that all predictive models should be interpreted cautiously, given that there is often a fair amount of uncertainty in parameter estimates that can be challenging to validate with orthogonal sources of information [START_REF] Beissinger | On the Use of Demographic Models of Population Viability in Endangered Species Management[END_REF]. For instance, having fieldbased estimates of the inbreeding load for the vaquita would represent a valuable source of additional information for our study, however, such data do not currently exist, and may never exist given the perilous circumstances of the species. Future work should aim to validate the sorts of models we employ in [START_REF] Robinson | The critically endangered vaquita is not doomed to extinction by inbreeding depression[END_REF] for wild species where field-based estimates of the inbreeding load exist. Such work could greatly strengthen conclusions drawn from such predictive models.

All modelling considerations aside, our conclusion that recovery is possible is also supported by all field surveys since late 1990s, including those in 2019 and 2021, where active healthy vaquitas with calves have been sighted [START_REF] Rojas-Bracho | More vaquita porpoises survive than expected[END_REF]. Despite an almost certain increase in gillnetting within the small range where vaquitas remain, vaquitas continue to survive at higher numbers than expected [START_REF] Rojas-Bracho | More vaquita porpoises survive than expected[END_REF]. The possibility of recovery is also supported by numerous examples of recovery for species that have dwindled below 20 individuals [START_REF] Wiedenfeld | Conservation resource allocation, small population resiliency, and the fallacy of conservation triage[END_REF], many of which were once thought to be doomed to extinction. Although inbreeding depression may in many contexts contribute to population decline, the naïve assumption that it will inevitably doom small populations is a dangerous view that is harmful to species conservation.

Acknowledgements

We are grateful to three anonymous reviewers for constructive comments on this manuscript. C.C.K. and K.E.L. were supported by National Institutes of Health (NIH) grant R35GM119856 (to K.E.L.). A.C.B. was supported by the Biological Mechanisms of Healthy Aging Training Program, NIH T32AG066574. S.F.N.-M. was supported by the Mexican National Council for Science and Technology (CONACYT) postdoctoral fellowship 724094 and the Mexican Secretariat of Agriculture and Rural Development postdoctoral fellowship. We dedicate this work to our mentor, colleague, and friend Bob Wayne, who passed away in December 2022.

Author contributions

C.C.K. wrote the manuscript with input from all coauthors.

Conflict of interest

The authors declare no conflict.