Lina M Romero Millán

Amel C Ghogia

Claire E White

Ange Nzihou
email: ange.nzihou@mines-albi.fr

Lina M Romero

Iron Nanoparticles to Catalyze Graphitization of Cellulose for Energy Storage Applications

The production of highly graphitic carbon from bioresources is an environmentally friendly approach to synthesize graphene for energy storage applications. Iron catalytic graphitization of cellulose, the most abundant biopolymer on earth, is an alternative approach as until now, cellulose has been classified as poorly graphitizable material. In this study, the impact of processing temperature and iron impregnation on the extent of graphitization of the cellulose-derived graphitic carbon nanostructure is uncovered by combining Raman spectroscopy, X -ray diffraction, t r ansmission e l ectron m i croscopy, a n d X -ray p a ir distribution function analysis. Raman spectroscopy is used in an innovative way to describe the evolution of the average graphitic phase size where the ash content misguides the X-ray diffraction analysis. A correlation was established between (i) the in-plane crystallite size L a and the I D″ /I G first-order ratio, (ii) the out-of-plane L c crystallite size and the I G /I tot′ second-order ratio, and (iii) the second-order Raman I G′ /I tot′ ratio and the average number of carbon layers per carbon crystallite. For iron-impregnated cellulose, phase quantification and analysis of the spatial distribution reveal highly crystalline rhombohedral graphite surrounded by a nanocrystalline carbon matrix. We explicitly show that traditionally non-graphitizable carbons can be used to form a graphite-like structure with multilayers of graphene sheets by careful addition of widely available nontoxic metal as catalysts. The study also shows that the impact of the catalyst is much more effective than the temperature in the nanostructure transformation. The proposed approach and the results obtained provide interesting insights that should stimulate further works aimed at extending the knowledge in the field.

■ INTRODUCTION

Widespread attention has been paid to the development of two-dimensional graphene due to its distinctive and extraordinary properties. [START_REF] Inagaki | Advanced Materials Science and Engineering of Carbon[END_REF][START_REF] Zhu | Graphene and Graphene Oxide: Synthesis, Properties and Applications[END_REF][START_REF] Balandin | Thermal Properties of Graphene and Nanostructured Carbon Materials[END_REF] Owing to these characteristics, graphene is considered a promising material for energy storage applications. [START_REF] Ren | Properties and Applications of Graphene in Functional Devices: A Concise Review[END_REF] In general, graphene synthesis can be performed using top-down and bottom-up methods. In the top-down approach, graphene is derived or exfoliated from graphite to form nanosized graphene sheets. [START_REF] Paton | Scalable Production of Large Quantities of Defect-Free Few-Layer Graphene by Shear Exfoliation in Liquids[END_REF][START_REF] Kumar | Top-down Synthesis of Graphene: A Comprehensive Review[END_REF] In contrast, bottom-up methods involve the growth of graphene from atomic-sized precursors on a support. [START_REF] Luong | Gram-Scale Bottom-up Flash Graphene Synthesis[END_REF][START_REF] Sun | Chemical Vapour Deposition[END_REF] This latter approach often requires sophisticated equipment and strict operating conditions, increasing the production cost in comparison to top-down methods. Considering these restrictions, top-down methods are presently the most common for the production of commercial graphene.

Top-down methods generally use synthetic graphite from fossil sources like coke, pitch, or coal as precursor for graphene synthesis, involving costly and energy-intensive preparation conditions with extremely high temperatures (>2000 °C). [START_REF] Wu | Controlled Synthesis of Graphitic Carbon-Encapsulated a-Fe 2 O 3 Nanocomposite via Low-Temperature Catalytic Graphitization of Biomass and Its Lithium Storage Property[END_REF] Greener methods have been explored that produce graphitic materials with lower environmental impact. For example, some biomass is a sustainable precursor that can be easily converted into a graphitic material using thermochemical decomposition processes. [START_REF] Thompson | Iron-Catalyzed Graphitization of Biomass[END_REF] However, cellulose, the main constituent in lignocellulosic biomass and the most abundant source of sustainable carbon, is a non-graphitizable material, and as a consequence, the resulting graphitic carbons may exhibit a randomly oriented carbon structure even after pyrolysis at high temperatures. [START_REF] Franklin | Crystallyte Growth in Graphitizing and Non-Graphitizing Carbons[END_REF] This characteristic may represent a barrier for the production of graphene, considering that highly ordered graphitic structures are required. Interestingly, different studies have demonstrated that transition metals may catalyze graphitization of other carbon materials. [START_REF] Thambiliyagodage | Catalytic Graphitization in Nanocast Carbon Monoliths by Iron, Cobalt and Nickel Nanoparticles[END_REF] Among them, iron is particularly interesting considering its abundance, relatively nontoxicity, and high catalytic activity toward carbon graphitization compared to other transition metals such as cobalt and nickel. [START_REF] Du; Deng | Iron Based Catalysts in Biomass Processing[END_REF][START_REF] Hoekstra | The Effect of Iron Catalyzed Graphitization on the Textural Properties of Carbonized Cellulose: Magnetically Separable Graphitic Carbon Bodies for Catalysis and Remediation[END_REF][START_REF] Li | Catalytic Graphitization of Coke Carbon by Iron: Understanding the Evolution of Carbon Structure, Morphology and Lattice Fringes[END_REF][START_REF] Gomez-Martin | Structural Evolution in Iron-Catalyzed Graphitization of Hard Carbons[END_REF][START_REF] Gai | Evolution of Biomass to Porous Graphite Carbon by Catalytic Graphitization[END_REF] Up to now, first-order characteristic bands, especially the D and G peaks of carbonaceous materials, are widely used in the literature to describe the structure evolution of different carbon materials. [START_REF] Thompson | Iron-Catalyzed Graphitization of Biomass[END_REF][START_REF] Thambiliyagodage | Catalytic Graphitization in Nanocast Carbon Monoliths by Iron, Cobalt and Nickel Nanoparticles[END_REF][START_REF] Gai | Evolution of Biomass to Porous Graphite Carbon by Catalytic Graphitization[END_REF] Generally, the D band is associated with the presence of defects in the graphitic structure (e.g., crystallite boundaries, defective carbon rings, impurities, etc.), [START_REF] Maslova | Determination of Crystallite Size in Polished Graphitized Carbon by Raman Spectroscopy[END_REF] while the G band is related to the in-plane vibration of sp [START_REF] Zhu | Graphene and Graphene Oxide: Synthesis, Properties and Applications[END_REF] -hybridized carbon atoms in graphitic materials. [START_REF] Muzyka | Characterization of Graphite Oxide and Reduced Graphene Oxide Obtained from Different Graphite Precursors and Oxidized by Different Methods Using Raman Spectroscopy[END_REF] The relationship between these peaks has proved to be suitable for studying graphitizable and graphitic materials [START_REF] Bernard | Raman and XRD Study of Anthracene-Based Cokes and Saccharose-Based Chars Submitted to High-Temperature Pyrolysis[END_REF][START_REF] Ferrari | Raman Spectroscopy of Graphene and Graphite: Disorder, Electron-Phonon Coupling, Doping and Nonadiabatic Effects[END_REF] but does not seem appropriate to evaluate the structure evolution of non-graphitizable feedstocks (e.g., cellulose) and the catalyst-enhanced graphitization process. [START_REF] Ferrari | Raman Spectroscopy of Amorphous, Nanostructured, Diamond-like Carbon, and Nanodiamond[END_REF][START_REF] Pawlyta | Raman Microspectroscopy Characterization of Carbon Blacks: Spectral Analysis and Structural Information[END_REF] In fact, it has been reported that the lattice region electronically affected by the presence of a defect may be around 2 nm. [START_REF] Mallet-Ladeira | Behavior of Raman D Band for Pyrocarbons with Crystallite Size in the 2-5 Nm Range[END_REF] In this respect, Ferrari and Robertson have shown that the I D /I G ratio may decrease during the structural evolution of carbon materials from nanocrystalline graphite to highly ordered graphite. However, they also found that this ratio may decrease with the increase in the structural disorder for carbon materials with a crystallite size below 2 nm. [START_REF] Ferrari | Interpretation of Raman Spectra of Disordered and Amorphous Carbon[END_REF] Moreover, other authors have shown that the D band can be even observed in perfectly oriented graphite, near the edges of the carbon crystallites, [START_REF] Pimenta | Studying Disorder in Graphite-Based Systems by Raman Spectroscopy[END_REF] and has been reported for carbon samples with in-plane crystallite size L a length as big as 30 nm. [START_REF] Ferrari | Raman Spectroscopy of Graphene and Graphite: Disorder, Electron-Phonon Coupling, Doping and Nonadiabatic Effects[END_REF] Consequently, the developing growth of graphitic zones in non-graphitizable materials by the effect of pyrolysis temperature or the presence of a catalyst may not be directly reflected in the D band, even though other Raman features confirm the graphitization of their carbon structure (i.e., second-order Raman band, I D″ ).

To make affordable highly graphitic carbons from bioresources using a scalable production process, it is necessary to effectively understand and quantify the structural development of these materials during pyrolysis in the presence of iron as catalyst. Several studies in the literature combine Raman spectroscopy and X-ray diffraction (XRD) to provide information about (i) defects in the carbon structure, (ii) stacking of layer, and (iii) average crystallite size for turbostratic carbons with short-range order or highly oriented and ordered materials. [START_REF] Ferrari | Raman Spectroscopy of Graphene and Graphite: Disorder, Electron-Phonon Coupling, Doping and Nonadiabatic Effects[END_REF][START_REF] Schuepfer | Assessing the Structural Properties of Graphitic and Non-Graphitic Carbons by Raman Spectroscopy[END_REF] A recent study showed that the structural development of carbon during iron-catalyzed graphitization can be rationalized by the formation/decomposition of Fe 3 C at lower temperature <1000 °C by combining in situ X-ray diffraction and X-ray scattering. However, there is a lack of comprehension about the nanostructure of catalystenhanced graphitic carbons and its evolution with temperature. [START_REF] Gomez-Martin | Structural Evolution in Iron-Catalyzed Graphitization of Hard Carbons[END_REF] Here, we propose a novel approach to rigorously characterize the carbon structure using second-order Raman bands, and we demonstrate the complementarity of Raman spectroscopy, X-ray scattering, and transmission electron microscopy (TEM) to effectively describe the nanostructure of highly graphitic carbons produced from the catalytic graphitization of cellulose. Cellulose before and after impregnation with iron nitrate solution has been pyrolyzed at different temperatures. From the analysis, we propose a set of empirical correlations to easily assess the average size of the graphitic crystalline domains based on a novel interpretation of the Raman spectra of highly graphitic carbons. We emphasize the validity of the secondorder Raman bands, scarcely addressed in the literature, to effectively assess the carbon layer stacking of highly graphitic carbons from a non-graphitizable precursor, extending the applicability of Raman spectroscopy as a versatile tool to quantify the structural development of catalyst-enhanced graphitic carbons. We also reveal the spatial heterogeneity of carbon phase(s) in the iron-containing samples, as assessed using TEM and X-ray pair distribution function (PDF) analysis, which reveal that in pyrolyzed cellulose iron catalyzes the formation of highly crystalline rhombohedral graphite that is dispersed in a nanocrystalline carbon matrix. This work should also open new perspectives in the production and utilization of sustainable graphene materials.

Impact of Pyrolysis Temperature and Iron Content on the Structure of Graphitic Carbon. The structure of the produced highly graphitic carbons was first analyzed using Raman spectroscopy. The normalized Raman spectra of select graphitic carbons at 1000, 1400, and 1800 °C from non-and iron-impregnated microcrystalline cellulose are presented in Figure 1. The spectra of the analyzed samples show the firstorder characteristic bands of carbonaceous materials between 1000 and 2000 cm -1 , with the peaks D near 1350 cm -1 and G near 1580 cm -1 , along with intensity D″ near 1500 cm -1 , being the most representative bands. In addition to the main D and G bands observed in the Raman spectra of the analyzed graphitic carbons, the impregnated samples also exhibited distinct peaks in the second-order region, between 2500 and 3000 cm -1 , while only a broad bump was observed in the nonimpregnated samples.

From the mathematical treatment of the first-order Raman spectra presented in the Methods section, the intensities of the observed bands were analyzed and compared as well as the ratios between them (i.e., I G /I tot , I D /I G , and I D″ /I G ) to determine the evolution of the carbon structure as a function of pyrolysis temperature and iron impregnation. In the literature, either the ratio of the band heights or the band areas have been used to describe the structure evolution of different carbon materials. [START_REF] Schuepfer | Assessing the Structural Properties of Graphitic and Non-Graphitic Carbons by Raman Spectroscopy[END_REF] However, considering that the band areas may be more sensitive to the discrepancies in the mathematical treatment of the Raman spectra (e.g., fitting functions, number of analyzed bands, bandwidths, etc.), only the band heights have been analyzed in this work. [START_REF] Zickler | A Reconsideration of the Relationship between the Crystallite Size La of Carbons Determined by X-Ray Diffraction and Raman Spectroscopy[END_REF] The experimental results showed that both the increase of pyrolysis temperature and the presence of iron had positive effects on the graphitization of the samples (Figure 2). For both non-and iron-impregnated samples, the temperature increase from 1000 to 1800 °C was associated with a decrease in the I D″ /I G ratio (Figure 2b). In the particular case of nonimpregnated samples, an average reduction of 31% in the I D″ /I G ratio was observed over the analyzed temperature range, suggesting a decrease of sp 2 and sp 3 amorphous structures with the thermal treatment. This effect was accentuated by iron, with an average reduction of 72% in the I D″ /I G ratio, observed for the iron-impregnated samples, suggesting a reduced amount of sp 2 and sp 3 amorphous structures and greater development of graphitic carbon (Figure 2a,b). It is worth noting that the organic composition and the mineral content of the cellulose did not show any significant effect on the graphitization of the samples. In contrast, no significant evolution was observed in the I G /I tot and I D /I G ratios with the temperature increase for nonimpregnated samples and a slight increase of 34% in the I G /I tot ratio for impregnated samples (see Table S3).

Accordingly, in the first-order region, the I D″ /I G ratio seems to be the most appropriate to describe the evolution of the carbon structure of the analyzed materials. Considering that the D″ band has been generally attributed to the presence of a random mixture of sp 2 and sp 3 amorphous carbons, the increased degree of structural order of the samples may be related to the decrease of this feature intensity. This impact is scarcely covered in the literature.

As was performed for the first-order bands, the second-order bands were deconvoluted according to the procedure presented in the Methods section, and the relationships between the intensities of the identified bands were analyzed. Particular attention was given to the ratio I G′ /I tot′ because the G′ peak around 2700 cm -1 is characteristic of graphitic materials and corresponds to the second-order Raman band of crystalline graphite. [START_REF] Pimenta | Studying Disorder in Graphite-Based Systems by Raman Spectroscopy[END_REF] The appearance of the G′ peak confirms the presence of highly oriented graphitic structures in the impregnated samples and, thus, the positive impact of iron impregnation on the carbon structure evolution. From the comparison of the first-and second-order Raman bands of all the analyzed samples, a linear relationship was found between the I D″ /I G and I G′ /I tot′ ratios (Figure 2c,d). Accordingly, the development of graphitic structures is related to a decrease in the I D″ /I G ratio and an increase in the I G′ /I tot′ ratio. These results confirm that under the analyzed experimental conditions of this investigation the reduction of sp 2 and sp 3 amorphous carbon is associated with the evolution of highly oriented graphitic carbon. Accordingly, the second-order ratio I G′ /I tot′ may also be a good indicator of the degree of crystallinity for highly graphitic materials from non-graphitizable feedstocks and those processed using the catalystenhanced graphitization approach.

Evolution of the Average Crystallite Size of Carbon. In addition to the analysis and comparison of the Raman bands and associated intensity ratios, the evolution of the carbon structure of the pyrolyzed carbon materials may also be described by the carbon crystallite size. [START_REF] Inagaki | Advanced Materials Science and Engineering of Carbon[END_REF] The crystallite lattice parameters include the height of the organized stacking of carbon layers L c , the lateral extension of the graphene sheets L a , and the average spacing between the layers d 002 . In this regard, the graphitization of the carbon matrix of the samples should be related to an increase in their carbon crystallite size. X-ray diffraction has been extensively used to characterize the structure of carbon materials and determine their apparent crystallite dimensions. As presented in the Methods section, the L c and L a lattice parameters can be deduced using the Debye-Scherrer equation, while the d 002 distance can be calculated from Bragg's law. The X-ray diffraction results obtained for the samples are presented in Figure 3a,b.

The X-ray diffraction analysis of the produced carbons confirmed the positive effect of the pyrolysis temperature, and particularly of the iron impregnation, on the graphitization of the carbon. It can be observed from Figure 3a that nonimpregnated samples are amorphous/nanocrystalline, showing broad (002) and (10) carbon peaks, with no significant evolution with the temperature increase. Only a slight structural improvement is suggested from the calculated L c and L a values. In particular, the temperature rises from 1000 to 1800 °C was related to an increase in the average crystallite size from 0.9 to 1.1 nm for L c and from 2.9 to 3.9 nm for L a .

In contrast, the shape of the X-ray diffractograms noticeably changes for iron-impregnated samples (Figure 3b), showing sharp carbon peaks along with those attributed to the ironcontaining phases. This confirms the significant impact of iron in enhancing the graphitization of carbon during pyrolysis. From 1000 to 1800 °C the average crystallite size of the samples increased from 4.7 to 6.4 nm for L c and from 5.4 to 7.2 nm for L a . From these results, it is clear that the effect of iron impregnation in carbon graphitization is much stronger than that of the temperature.

To better understand the evolution of carbon structure, the complementarity between X-ray diffraction and Raman spectroscopy to analyze the structure of nongraphitizable and catalyst-enhanced graphitic carbons was further explored in this study. More specifically, it was observed that the evolution of the carbon crystallite size, calculated from XRD analysis is inversely proportional to the first-order I D" /I G ratio and proportional to the second-order Raman I G′ /I tot′ ratio (Figures 3c and3d, respectively). The observed relationships suggest that Raman spectroscopy could also be used to give insight into the average crystallite size of catalyst-enhanced graphitic carbons. Several authors have already proposed the use of Raman to evaluate the crystallite size of highly ordered materials. The reported studies are mainly focused on the evaluation of the in-plane L a dimension from the first-order D and G Raman bands. Tunistra and Koening were the first to propose the use of the Raman intensity ratio I D /I G to calculate the average L a crystallite size. [START_REF] Tuinstra | Spectrum of Graphite[END_REF] The validity of this approach was subsequently confirmed by different authors, [START_REF] Knight | Characterization of Diamond Films by Raman Spectroscopy[END_REF] where crystallite L a sizes between 2 and 10 nm have been reported using several different Raman excitation energies. [START_REF] Cancado | Measuring the Degree of Stacking Order in Graphite by Raman Spectroscopy[END_REF][START_REF] Canado | General Equation for the Determination of the Crystallite Size La of Nanographite by Raman Spectroscopy[END_REF] Similarly, other reported studies have proposed the calculation of L a from the HWHM (half-width at half-maximum) of the G peak, fitted using a Breit-Wigner-Fano function (BWF). [START_REF] Mallet-Ladeira | A Raman Study to Obtain Crystallite Size of Carbon Materials: A Better Alternative to the Tuinstra-Koenig Law[END_REF] However, it is worth stressing that the existing correlations for L a have been established based on highly ordered graphitic materials and, as such, should be considered with caution. [START_REF] Schuepfer | Assessing the Structural Properties of Graphitic and Non-Graphitic Carbons by Raman Spectroscopy[END_REF] As already indicated above, we have demonstrated that the I D /I G ratio generally used to calculate the L a crystallite size is not suitable to describe the carbon structure evolution of non- graphitizable or catalyst-induced graphitic carbons and therefore should not be used to estimate the average carbon crystallite size. In contrast, the intensity ratios I G′ /I tot′ and I D″ / I G are more appropriate to describe these materials.

From the results in Figure 3c, eq 1 was derived to quantitatively describe the linear relationship between the Raman I D″ /I G first-order ratio and the lateral crystallite size L a calculated from analysis of the (10) diffraction peak using the Scherrer equation (considering the Warren correction). [START_REF] Alexander | Determination of Crystallite Size with the X-Ray Spectrometer[END_REF] This relationship suggests that the decrease of sp 2 and sp 3 amorphous structures in the samples is associated with an increase in the in-plane lattice size of the carbon crystallite. This is related to the development of highly graphitic structures induced mainly by the iron impregnation of the samples and, to some extent, the pyrolysis temperature. Thus, eq 1 provides an empirical correlation to describe the L a crystallite size evolution of non-graphitizable materials during the catalytic graphitization process such as cellulose.

= + L I I (nm) 7.05( / ) 7.69 a D G (1)
Concerning the out-of-plane L c crystallite size, it has been reported that the second-order Raman bands may provide information about the stacking of graphene layers in highly graphitic samples. Malard et al. showed that in the case of carbon AB graphitic stacking analysis of the second-order features allowed the identification of the number of graphene layers, from monolayer to four-layer graphene. [START_REF] Malard | Spectroscopy in Graphene[END_REF] In contrast, they pointed out that the identification of the number of layers in nanocrystalline and turbostratic carbons is not as straightforward, considering the random orientation of graphene sheets with respect to each other.

To the best of our knowledge, only one study has attempted to describe the L c crystallite size of partially disordered graphite from Raman spectroscopy data, [START_REF] Dresselhaus | Perspectives on Carbon Nanotubes and Graphene Raman Spectroscopy[END_REF] where the secondorder Raman bands (G′) were correlated to the L c parameter. However, the proposed mathematical treatment to fit these bands was not completely established and may not suitable to describe the evolution of the catalyst-enhanced graphitization process of carbons. Here, a linear relationship was observed between the I G′ /I tot′ ratio and the L c crystallite size determined by the analysis of the (002) diffraction peak using the Scherrer equation. From Figure 3d, it can be seen that higher I G′ /I tot′ ratios are associated with larger L c crystallite sizes and, consequently, a greater number of stacked carbon layers. In accordance, an empirical correlation can also be established to describe the average L c crystallite size of nanocrystalline and catalyst-induced graphitic carbons from their second-order Raman bands. With these data, an empirical correlation between the Raman I G′ /I tot′ ratio and the average number of carbon layers per carbon crystallite is also proposed in this work:

= L I I (nm) 10.03( / ) 1.30 c G tot (2)
= n I I 29.65( / ) 4.15 avg G tot (3)
These relationships are valid in the L a size range between 2 and 5 nm and the L c size range between 1 and 7 nm. Also, they correspond to Raman spectra measured with 532 nm excitation wavelength. From Figure 3c,d, it is worth noting that the crystallite size evolution of the analyzed samples is not the same in the in-plane and out-of-plane directions. In particular, the L c size increased from an average of 1 nm for nonimpregnated samples to 6 nm for iron-impregnated samples treated at 1300 °C. In contrast, the L a size evolved to a lesser degree from an average of 3 nm for nonimpregnated samples to around 5 nm for the iron-impregnated samples. These results may be related to the non-graphitizable nature of the cellulose and can be explained by the fact that for impregnated samples the iron nanoparticle controls the L a parameter because the curvature of the particle governs the curvature of the graphene layers (Figure 4a).

From the above analysis involving the different Raman bands and their ability to shed light on the degree of graphitization of cellulose, the empirical correlation proposed in this work for the calculation of the L a crystallite size is suitable particularly in the case of iron-catalyzed graphitic materials. In the comparison with the correlation of this work, those already existing in the literature established considering only the D and G Raman bands are only applicable for graphitic materials. [START_REF] Mallet-Ladeira | A Raman Study to Obtain Crystallite Size of Carbon Materials: A Better Alternative to the Tuinstra-Koenig Law[END_REF] The process of iron catalytic graphitization has been generally described as a dissolution-precipitation mechanism followed by particle growth, where the carbon precursor dissolves in the iron droplets during pyrolysis and subsequently precipitates as graphite during cooling. [START_REF] Thambiliyagodage | Catalytic Graphitization in Nanocast Carbon Monoliths by Iron, Cobalt and Nickel Nanoparticles[END_REF][START_REF] Neeli | Synthesis and Formation Mechanism of Iron Nanoparticles in Graphitized Carbon Matrix Using Biochar from Biomass Model Compounds as a Support[END_REF][START_REF] Sevilla | Graphitic Carbon Nanostructures from Cellulose[END_REF] This results in the formation of graphitic zones around the iron nanoparticles. As presented in Figure 3b, XRD analysis confirmed the presence of iron phases in the produced carbons in the form of α-Fe and γ-Fe, while the presence of iron carbides was not identified. Considering that the thermal expansion coefficient of iron is higher than that of graphite (∼12 to ∼6 × 10 -6 °C-1 , respectively), [START_REF] Carvill | Mechanical Engineer's Data Handbook[END_REF] pyrolysis may cause the nanoparticles to expand and break the graphitic shells, forming also carbon nanoribbons and graphitizing other areas of surrounding carbon. [START_REF] Hoekstra | Base Metal Catalyzed Graphitization of Cellulose: A Combined Raman Spectroscopy, Temperature-Dependent X-Ray Diffraction and High-Resolution Transmission Electron Microscopy Study[END_REF] As observed in Figure 4a, the growth of crystalline carbon takes place with a curved stacking of carbon layers around the metal nanoparticle. This curvature is related to the relatively short size of the average in-plane crystallite size (L a ), despite a large number of stacked carbon layers. This characteristic growth of graphitic structures is representative of iron catalystenhanced graphitization [START_REF] Neeli | Synthesis and Formation Mechanism of Iron Nanoparticles in Graphitized Carbon Matrix Using Biochar from Biomass Model Compounds as a Support[END_REF] and may explain the fact that the existing correlations in the literature are not suitable to describe the carbon structure evolution of the analyzed materials. Figure 4b,c shows the spatial distribution of carbon and iron phases in iron-impregnated samples. The simultaneous presence of carbon and iron phases in iron-impregnated samples was confirmed by EDX analysis (Figure 4d). TEM result in Figure 4e showed that the carbon graphitization around iron nanoparticles produces a nonhomogeneous structure at the nanometer scale, with highly crystalline zones adjacent to regions with reduced degrees of nano- crystallinity. The red Raman spectrum showing the G′ peak and red zone in the mapping correspond to the highly crystalline zones, while those in blue correspond to nanocrystalline carbon (Figure 4e).

These observations can explain the fact that in the analyzed range of iron concentrations higher concentrations in the raw samples resulted in a greater extent of structural ordering of the carbon materials, attributed to the presence of a higher number of iron particles enhancing the graphitization during pyrolysis.

Similarly, Figure 4e illustrates the heterogeneity of the carbon structure of the samples at the nanometer scale, where two distinct Raman spectra for adjacent regions in the same carbon particle exhibit different degrees of organization. The red zone spectrum corresponds to a well-developed crystalline structure, probably near an iron nanoparticle. In contrast, the blue zone spectrum is characteristic of carbon with a lower degree of nanocrystallinity and thus possesses only short-range order (limited to a couple of nanometers). An example of the presence of these zones in the produced graphitic carbon is presented in the TEM in the figure inset.

Impact of Iron on Graphitic Carbon Nanostructure. As shown by the X-ray diffractograms, the average crystallite size of cellulose-based carbons increased due to iron impregnation, but specific details on the impact of iron impregnation on the nanostructure of the carbon were not accessible using this technique. X-ray total scattering and subsequent PDF analysis is an ideal technique to elucidate the nanostructure of disordered materials. [START_REF] Egami | Underneath the Bragg Peaks: Structural Analysis of Complex Materials[END_REF] Figure S2 depicts a typical X-ray PDF data set for temperature-treated ironimpregnated cellulose, where the nanostructure of the carbon has been isolated from that of the iron (see the Methods section for the procedure used to remove non-carbon atomatom correlations from the X-ray PDF). Figure 5a-c shows the PDFs of the carbons contained in the iron-impregnated samples along with their multiphase least-squares refinement results (refinement details outlined in the Methods section). Refinements of the nonimpregnated carbon X-ray PDFs using the same least-squares approach revealed that the nanostructure of these carbons was amorphous (for 1000 °C) or nanocrystalline (for 1800 °C) with atom-atom correlations out to roughly 1.5-2.5 nm (see Figure 5d).

The anisotropic crystallite size of the nanocrystalline carbon phase was modeled in real space by considering a spherical graphite nanocrystallite where the atom-atom correlations along the c-axis (i.e., out-of-plane) were suppressed relative to those in the a-b plane (i.e., in-plane) via directionaldependent refinement of the atomic displacement parameters (ADPs). [START_REF] Poulain | Wide-Angle X-Ray Scattering Combined with Pair Distribution Function Analysis of Pyrolyzed Wood[END_REF] As such, the in-plane crystallite size (akin to the L a parameter) can be estimated from the size of the spherical graphite nanocrystallite, while the out-of-plane dimension is expected to be inversely correlated to the magnitude of the caxis ADP, where a larger c-axis ADP value implies a smaller out-of-plane nanocrystallite domain size (similar to the L c parameter). As seen from Figure 5f, the in-plane crystallite size increases from approximately 1.6 to 2.7 nm for nonimpregnated cellulose pyrolyzed at 1000 to 1800 °C, respectively. Comparison with X-ray diffraction results, where L a increased from 2.9 to 3.9 nm for the same treatment temperature range, reveals that the PDF-derived crystallite size is underestimated, likely attributed to the limitations of applying a spherically shaped crystallite during PDF analysis to represent anisotropic crystallites. Nevertheless, the magni-tude of increase from 1000 to 1800 °C (1.1 nm from PDF, 1.0 from XRD) is accurately predicted using PDF analysis.

One major finding of the PDF analysis was the identification and quantification of multiple distinct carbon phases in the iron-impregnated samples. Specifically, in addition to the nanocrystalline carbon phase there were found to be larger crystallites of rhombohedral graphite in all iron-impregnated samples along with crystalline hexagonal graphite for the sample treated at 1800 °C (Figure 5a-c). Interestingly, as seen in Figure 5e, the relative amounts of the phases in the ironimpregnated samples do not change significantly with treatment temperature. Instead, these samples roughly consist of 70 wt % nanocrystalline carbon and 30 wt % crystalline rhombohedral graphite. Hence, conventional X-ray diffraction failed to reveal that two distinct carbon phases were present in the iron-impregnated samples, and additional characterization is required to ascertain these details. In fact, the average crystallite size obtained from X-ray diffraction, and specifically the L c parameter, does not accurately reflect the complex structural details apparent in the TEM images (Figure 4a) where both highly crystalline zones of graphite are seen to be adjacent to regions with reduced degrees of nanocrystallinity. Nevertheless, X-ray diffraction and quantification of the L a and L c parameters can accurately capture the general trend of increased crystallinity with treatment temperature and iron impregnation and thus when combined with Raman spectroscopy is ideal for characterization of relative changes in degree of crystallinity for non-graphitizable biomass.

Iron impregnation not only instigates the formation of highly crystalline rhombohedral graphite, but it also catalyzes an increase of the in-plane crystallite size of the nanocrystalline carbon phase, particularly for the lower treatment temperatures (1000 and 1400 °C) as seen in Figure 5f via comparison of the non-and iron-impregnated samples. As alluded to above, PDF analysis of the nanocrystalline carbon phase underestimates the in-plane crystallite size, and thus the crystallite sizes reported in Figure 5f for the iron-impregnated samples are likely underestimated by about 1 nm. Nevertheless, from Figure 5f it is clear that the disordered carbon phase in all ironimpregnated samples is nanocrystalline as opposed to amorphous, and thus iron impregnation not only catalyzes the formation of highly crystalline graphite at 1000 °C but also increases the degree of ordering of the remaining carbon.

Lastly, PDF analysis may also shed light on the impact of iron impregnation and treatment temperature on the height of the organized stacking of carbon layers in the nanocrystalline carbon phase. Figure 5g reports the ADP value of the nanocrystalline carbon phase, which is expected to be inversely correlation with the number of stacked carbon layers, with a higher ADP corresponding to a smaller region of stacked layers. For the nonimpregnated samples this ADP value slightly increased with higher treatment temperature implying less layers in a coherent stack. However, this contradicts the L c parameter from X-ray diffraction which slightly increased from 0.9 to 1.1 nm; thus, the PDF-derived ADP values may be less sensitive to changes in dimensions of the stacked layers although Poulain et al. did find a correlation between decreasing L c parameter and increasing c-axis ADP for beech biomass as a function of pyrolysis temperature. [START_REF] Poulain | Wide-Angle X-Ray Scattering Combined with Pair Distribution Function Analysis of Pyrolyzed Wood[END_REF] Hence, the decrease of ADP values for the iron-impregnated samples is likely indicative of an increase in stacked layers for the nanocrystalline carbon phase in these samples, which shows that both the in-plane and out-of-plane dimensions of the nanocrystalline carbon phase found in the iron-impregnated samples increased as a function of pyrolysis temperature. It remains to be proven if the in-plane and out-of-plane dimensions of the highly crystalline rhombohedral graphite crystals also increase in size with treatment temperature, although the TEM images in Figure 4 indicate that this may indeed be the case.

This study exemplifies how complementary experimental techniques are required to fully understand the nanostructural details of complex carbon materials. For example, TEM showed the existence of two carbon phases in the ironimpregnated samples, while PDF analysis revealed that the crystal structure of the highly crystalline graphite phase was rhombohedral and provided a rough estimate of the relative phase fraction of the individual carbon phases. Raman spectroscopy, and specifically analysis of the first-and second-order bands, allowed for quantification of the relative degree of graphitization for carbons that are traditionally thought to be non-graphitizable, while its comparison with Xray diffraction enabled for estimation of the average in-plane and out-of-plane carbon crystallite size. The approaches outlined in this study and associated methods pave the way for future investigations of other complex carbon systems, with the aim to discover routes to instigate graphitization of renewable bioresources that until now have been considered non-graphitizable.

■ CONCLU SION AND OUTLOOK

Cellulose was used for graphitic carbon production from nonand iron-impregnated samples at different temperatures. A multiscale investigation addressing the bulk, molecular, and atomic scales was used to describe the carbon nanostructure. The combination of Raman spectroscopy and X-ray diffraction revealed that the effect of iron impregnation in carbon graphitization is much stronger than that of temperature. This effect of iron as a catalyst for carbon graphitization was confirmed by high-resolution transmission electron microscopy (HRTEM) and X-ray pair distribution function (PDF) analysis of iron-impregnated cellulose biochar, revealing the existence of highly crystalline graphitic domains (identified as rhombohedral graphite in PDF data) dispersed within a disordered arrangement of short-range stacked carbon layers. Such graphitic domains are absent in the nonimpregnated samples. This behavior can be rationalized by considering that the formation of multilayers graphene with a rhombohedral stacking sequence takes place directly on the iron nanoparticles. Using Raman spectroscopy in an innovative way, correlations allowing to calculate the parameters L a , L c , and n avg proposed in this work constitute a new approach to describe and quantify the structure of graphitic materials produced from cellulose because the presence of inorganic features in the Xray diffractograms could complicate or misguide the analysis of the carbon peaks. In addition to Raman, X-ray diffraction, and HRTEM, X-ray PDF analysis revealed that the amorphous carbon making up the nonimpregnated sample after pyrolysis at 1000 °C is nanocrystalline in the iron-impregnated sample, and thus iron impregnation not only catalyzes the formation of highly crystalline rhombohedral graphite but also an increase ordering of the short-range stacked carbon layers. The assessment of the nanostructure development of catalystenhanced graphitic carbons from bioresources is a crucial step to effectively advance the development of methods to produce graphene from sustainable and renewable resources for energy storage applications. Current work is devoted to the investigation of parameters such as the particle size and the shape of the initial cellulose to optimize the graphitization. The thermal, electrical, and mechanical properties of the graphitic matrixes obtained will be investigated and compared with carbon graphitized using other approaches developed in the literature.

■ METHODS Biomass Precursor. Microcrystalline cellulose (Sigma-Aldrich, CEL) was used in this study to analyze the graphitization during pyrolysis. Microcrystalline cellulose was impregnated with an iron nitrate solution prepared with Fe(NO 3 ) 3 •9H 2 O (Sigma-Aldrich, 99% purity). 20 g of cellulose was immersed in 200 mL of a 0.049 M solution under continuous stirring for 6 h. The samples were then filtered and dried at 105 °C for 24 h.

To determine the inorganic composition of the non-and ironimpregnated samples, 50 mg of ground sample was acid digested in closed vessels at 220 °C for 4 h. Acid reagents H 2 O 2 , HNO 3 , HF, and H 3 BO 3 were used according to EN 16967. Acid solutions were diluted with demineralized water to 50 mL and analyzed using a HORIBA Jobin Yvol Ultima 2 inductively coupled plasma optical emission spectrometer (ICP-OES). The CHNS composition of the samples was determined using a Themoquest NA 2000 elemental analyzer, and their ash content was calculated according to the standard EN ISO 18122. The inorganic and the CHNS compositions of non-and iron-impregnated carbon after pyrolysis were also determined by the same method described above. The compositions of all samples are summarized in Table S1.

Pyrolysis Experiments. Pyrolysis of the non-and ironimpregnated cellulose was performed in a horizontal laboratoryscale reactor, with an internal diameter of 20 mm, externally heated with an electric furnace. For each experiment, between 0.5 and 1 g of sample was placed inside the reactor using a high-temperature ceramic crucible. The samples were pyrolyzed at 1000, 1400, or 1800 °C with a heating rate of 8 °C/min under a nitrogen atmosphere with a gas flow of 1000 mL/min. The pyrolysis temperature was maintained for 1 h, and then the reactor was cooled to room temperature. Finally, the resulting produced carbon was collected, weighed, and stored for further characterization.

Characterization of Graphitic Carbons Using Raman Spectroscopy. Raman spectroscopy was used to analyze the carbon structure of the prepared samples. For this purpose, the carbon samples were mixed and ground with spectroscopic grade KBr in a weight proportion of 1:100 to prevent the sample from being heated during the acquisition of the Raman spectra. The mixtures were compressed using a 13 mm diameter die to form thin pellets for subsequent analysis. The spectra were acquired using a confocal Raman AFM WITEC Alpha 300AR microscope, using an excitation wavelength of 532 nm in the region between 200 and 4000 cm -1 . For each sample, at least three different spectra were collected (analysis surface near 25 μm 2 ), and the average data were calculated. Curve fitting of the Raman spectra was performed using MagicPlot software.

The first-order region between 800 and 2000 cm -1 was fitted using five Raman bands, as usually reported for carbon materials, and is summarized in Table S2. [START_REF] Sadezky | Raman Microspectroscopy of Soot and Related Carbonaceous Materials: Spectral Analysis and Structural Information[END_REF] Each band was represented using a Gaussian function, where the initial position was fixed for the iteration procedure according to the values in the table. No width limits were fixed for the Raman shift range of the D and G bands, while a maximum width was set for peaks I, D″, and D′ to avoid interference between these features and the D and G bands (width: 100, 70, and 30 cm -1 , respectively). In fact, a very large amplitude of peaks I, D″, and D′ may impact the height and amplitude of the main D and G features, hindering correct analysis of their evolution.

The second-order region between 2000 and 3500 cm -1 was fitted using four bands as reported in the literature. [START_REF] Vollebregt | Influence of the Growth Temperature on the First and Second-Order Raman Band Ratios and Widths of Carbon Nanotubes and Fibers[END_REF] For the analyzed samples, a good curve fitting was found using Gaussian functions, with initial positions at 2450 cm -1 for the I′ peak, 2700 cm -1 for the G′ peak, 2900 cm -1 for the D+G peak, and 3100 cm -1 for the 2D′ peak.

The general descriptions of the second-order features are also summarized in Table S2. In all cases, the maximum peak width was limited to 100 cm -1 to reduce overlapping and consider the contribution of each peak. No limits were fixed regarding the Raman shift range.

Characterization of Graphitic Carbons Using X-ray Diffraction. X-ray diffraction analysis was conducted using a Phillips Panalytical X'pert Pro MPD diffractometer and a Cu Kα radiation source (1.543 Å) with a voltage of 4 kV and a current of 40 mA. The diffraction patterns were collected for powder samples (particle size below 250 μm) between 2θ = 10°and 2θ = 70°with a step of 0.05°.

The structure of the carbon samples was determined from analysis of the X-ray diffraction patterns. Particular attention was given to the width of the carbon bands (002) and (10) which are related to the extent of atomic order of graphite-like carbon in the out-of-plane and in-plane direction, respectively. [START_REF] Weisbecker | Carbones Pyrolytiques Ou Pyrocarbones : Des Mateŕiaux Multiećhelles et Multiperformances[END_REF] To determine the carbon structural parameters, the (002) and ( 10) bands were fitted with HighScore software, using pseudo-Voigt functions. The instrument broadening in XRD analysis was removed in order to have accurate values of the L c and L a parameters. Thus, a standard NIST sample (silicon) was analyzed under the same conditions as those of the samples. The pseudo-Voigt functions were also used for standard peaks fitting. Because the Voigt function is a convolution of Gaussian and Lorentzian functions, and the Lorentzian function is the dominant cause of broadening, [START_REF] Balzar | X-Ray Diffraction Line Broadening: Modeling and Applications to High-Tc Superconductors[END_REF] the correction of the mid-width was performed using the Lorentzian shape (see eq 4).

The sample and standard peak widths at half-maximum were automatically calculated by the software, and these values were used to calculate the corrected peak width (eq 4). Then, this latter was used to calculate the average carbon crystallite domain size.

Here, β size is the corrected peak width at the half-maximum intensity in radians, β obs is the sample peak width at the half-maximum intensity in radians, and β inst is the peak width from instrument broadening at the half-maximum intensity in radians. Accordingly, the average crystallite size L of the samples and the average distance between carbon layers d 002 were expressed by the Debye-Scherrer equation (eq 5) and Bragg's law (eq 6), respectively:

= L K cos size (5) = d 2 sin 002 002 (6)
where θ is the peak position in radians, λ is the X-ray wavelength in nm, and K is a constant that generally depends on the shape of the crystallite. [START_REF]Theory and Practice[END_REF] The calculations were performed with K values of 0.89 for L c (graphitic planes stacking in the carbon crystallite) and 1.84 for L a (graphite-like atomic order in the single plane). [START_REF] Uvarov | Metrological Characterization of X-Ray Diffraction Methods at Different Acquisition Geometries for Determination of Crystallite Size in Nano-Scale Materials[END_REF]

Characterization of Graphitic Carbons Using High-Resolution Transmission Electron Microscopy (HRTEM).

The structure of the prepared carbons was observed using a JEOL JEM-ARM200F high-resolution transmission electron microscope. The elemental mapping of chemical species at the nanoscale was also performed using the microscope X-ray spectroscopy capability (EDX). Powder samples with particle sizes below 250 μm were dispersed in ethanol and mixed in an ultrasonic bath. The samples were then collected from the surface of the solution and deposited onto carbon support films before analysis.

Characterization of Graphitic Carbons Using X-ray Total Scattering and Pair Distribution Function Analysis. The nanostructure of the prepared graphitic carbons was determined using synchrotron-based X-ray total scattering and pair distribution function (PDF) analysis. Samples were loaded into 1 mm diameter polyimide capillaries and measured on the 11-ID-B beamline at the Advanced Photon Source, Argonne National Laboratory. Samples were measured in transmission geometry using a photon wavelength of 60 keV and a 2D image plate detector. High-resolution X-ray diffraction patterns were obtained using a sample-to-detector distance of ∼1000 mm, while X-ray total scattering was performed using a distance of ∼180 mm. Data reduction was performed using GSAS-II [START_REF] Toby | GSAS-II: The Genesis of a Modern Open-Source All Purpose Crystallography Software Package[END_REF] with CeO 2 as the calibrant to obtain 1D data sets. For the total scattering data, the structure factor and PDF were obtained using the PDFgetX2 software, [START_REF] Qiu | PDFgetX2: A GUI-Driven Program to Obtain the Pair Distribution Function from X-Ray Powder Diffraction Data[END_REF] where a Q max of 21.5 Å -1 was used for the sine Fourier transform. Instrument parameters were determined using nickel as the standard and the PDFgui software, [START_REF] Farrow | PDFfit2 and PDFgui: Computer Programs for Studying Nanostructure in Crystals[END_REF] where Q broad was found to be 0.0200 Å -1 and Q damp to be 0.0375 Å -1 .

Isolation of the graphitic carbon contributions in the ironimpregnated PDF data sets was performed using a direct subtraction method. [START_REF] White | Intrinsic Differences in Atomic Ordering of Calcium (Alumino)Silicate Hydrates in Conventional and Alkali-Activated Cements[END_REF][START_REF] Morandeau | In Situ X-Ray Pair Distribution Function Analysis of Accelerated Carbonation of a Synthetic Calcium-Silicate-Hydrate Gel[END_REF] Preliminary analysis of the high-r component of the PDF data sets (above 50 Å) using the real-space PDF modeling software PDFgui revealed that this region was dominated by contributions from α-Fe (ferrite), and no contributions from γ-Fe (austenite) were identified due to the relatively small amount of the γ-Fe phase (see Figure S1 for high-resolution X-ray diffraction patterns). The simulated X-ray PDF for α-Fe was generated using PDFgui and the iron-impregnated sample treated at 1800 °C, where the lattice parameters were refined over an r range of 50-60 Å together with the isotropic atomic displacement parameter (ADP) and scale factor. Subtraction of α-Fe from the iron-impregnated PDFs (to obtain the carbon contributions) was performed after the α-Fe simulated PDF data set was scaled to match that of the iron-impregnated PDF between 50 and 60 Å.

Identification and quantification of the carbon-based nanostructures found in the non-and iron-impregnated carbons were performed using the PDFgui software over an r range of 1-40 Å. Preliminary analysis revealed that the possible contributions to the graphitic carbon PDFs were from crystalline hexagonal and rhombohedral graphite and a disordered nanocrystalline phase based on hexagonal graphite. An amorphous carbon structure was only found to match the PDF of the nonimpregnated sample pyrolyzed at 1000 °C. [START_REF] Jurkiewicz | Modelling of Glass-like Carbon Structure and Its Experimental Verification by Neutron and X-Ray Diffraction[END_REF] The PDFs of the nonimpregnated samples pyrolyzed at 1400 and 1800 were better replicated using a nanocrystalline model. Note that crystallite sizes only below ∼8-10 nm could be assessed using the 11-ID-B beamline due to its instrument resolution. Therefore, phases identified as crystalline using PDF analysis in this investigation (i.e., α-Fe and crystalline hexagonal and rhombohedral graphite) will consist of crystallites that are larger than ∼8-10 nm, where other characterization techniques are required to determine their exact crystallite size. PDFgui software and a leastsquares approach were used for the quantification process. First, crystalline hexagonal and rhombohedral graphite phases were refined against the experimental PDFs of the carbons (iron-impregnated) over an r range of 30-40 Å. For both phases the scale factor, lattice parameters, and isotropic ADP were allowed to refine. Refinement of the crystallite sizes for these phases did not lead to an improved fit. A similar refinement for the nonimpregnated biocarbons was not performed due to the absence of discernible atom-atom correlations over this r range. Second, the nanocrystalline carbon phase (based on structure of hexagonal graphite) was refined over an r range of 1-40 Å. At first the parameters for crystalline hexagonal and rhombohedral graphite were kept fixed during the second stage of refinement. During this stage the following parameters for nanocrystalline carbon were allowed to refine: scale factor, lattice parameters, ADPs (c-axis separate from the a/b-axis), and spherical crystallite size. The separate refinement of c-axis ADP and a/b-axis ADP was necessary due to the inability of PDFgui to refine nonspherical crystallite shapes. By allowing the c-axis ADP to refine to a different value, the out-of-plane atom-atom correlations could be decoupled from the in-plane correlations, [START_REF] Poulain | Wide-Angle X-Ray Scattering Combined with Pair Distribution Function Analysis of Pyrolyzed Wood[END_REF] where a larger c-axis ADP value implies fewer graphene sheets making up the nanocrystallite domain. Subsequent refinement of all phases occurred together with their delta1 values (which account for in-phase vibration of atoms and the greater intensity of the first few atom-atom correlations). The level of agreement between the simulated and experimental PDF was provided by the R w value, where a smaller value implies better agreement.

Figure 1 .

 1 Figure 1. (a) Raman spectra of select graphitic carbons from nonimpregnated microcrystalline cellulose. Only a broad bump is observed for the second-order band of all the analyzed samples. (b) Raman spectra of carbons from iron-impregnated precursors. An evolution of the first-and second-order bands is observed, in contrast to the nonimpregnated samples. The second-order band is characteristic of graphitic materials.

Figure 2 .

 2 Figure 2. Evolution of structure for non-and iron-impregnated cellulose. (a) First-order Raman bands deconvolution of non-and iron-impregnated cellulose after pyrolysis. (b) Evolution of the first-order I D″ /I G ratio of pyrolyzed cellulose with processing temperature. Iron impregnation is mainly related to a decrease in the D″ feature and consequently in the I D″ /I G ratio. An increase in the G peak is also observed for the iron-impregnated sample. (c) First-and second-order Raman bands deconvolution of non-and iron-impregnated cellulose after pyrolysis. (d) Relationship between the first-and second-order Raman bands of carbon produced from non-and iron-impregnated cellulose and a TEM image which shows the stacking of the graphene layers. A linear correlation is observed between I D″ /I G and I G′ /I tot′ ratios.

Figure 3 .

 3 Figure 3. X-ray diffraction (XRD) patterns of graphitic carbon pyrolyzed at 1000, 1400, and 1800 °C: (a) graphitic carbon from nonimpregnated cellulose; (b) graphitic carbon from iron-impregnated cellulose. ( * ) α-Fe (ferrite), (•) γ-Fe (austenite), and indices (002), (10), and (004) are noted for the carbon peaks based on the crystal structure of graphite. (c) Relationship between the Raman I G′ / I tot′ ratio and the calculated L c crystallite size of the analyzed graphitic carbon. (d) Relationship between the Raman I D″ /I G ratio and the calculated L a crystallite size of the analyzed graphitic carbons. In both (c) and (d), the samples from nonimpregnated precursors are circled in green. The crystallite size values were calculated from XRD diffractograms.

Figure 4 .

 4 Figure 4. HRTEM image of the produced graphitic carbon nanostructure from iron-impregnated cellulose treated at 1800 °C. (a) The graphitic carbon is formed around the iron nanoparticles. (b) TEM image and (c) TEM-energy dispersive X-ray spectroscopy (EDX) map of the produced carbon showing the dispersion of iron nanoparticles (violet) in the carbon matrix (red). (d) EDX spectrum, where the presence of the copper on the energy-dispersive X-ray (EDX) spectrum comes from the TEM grids that is made of collodion carbon-covered copper film. (e) TEM shows the existence of two domains: in the red arrow, the growth direction of the carbon crystallite size L a and L c , and in the blue arrow, the nanocrystalline carbon with short-range order. Raman analysis shows two spectra which correspond to the two different zones (red and blue) on the spatial mapping.

Figure 5 .

 5 Figure 5. PDFs showing the carbon nanostructure of non-and iron-impregnated cellulose. The carbon nanostructure of iron-impregnated cellulose pyrolyzed at (a) 1000, (b) 1400, and (c) 1800 °C is quantitatively replicated using two (for 1000 and 1400 °C) or three (for 1800 °C) phases. All iron-impregnated samples are found to contain nanocrystalline carbon domains and highly crystalline rhombohedral graphite, while the sample treated at 1800 °C also contains a small amount of highly crystalline hexagonal graphite. (d) The nonimpregnated cellulose samples consist solely of amorphous (1000 °C) or nanocrystalline carbon (1800 °C). (e) Relative phase fraction (by mass) of the carbon phases in the iron-impregnated samples. (f) In-plane crystallite size (i.e., size of graphene sheets) of the non-and iron-impregnated samples obtained from the nanocrystalline carbon crystallite size. (g) c-axis atomic displacement parameter (ADP) of nanocrystalline carbon phase. A higher c-axis ADP value implies a reduced height of the organized stacking of carbon layers.

Funding

This work was supported by the French "Investissements d"Avenir" program under the Laboratory of Excellence, LABEX SOLSTICE, ANR-10-LABX -22-01 grant. This work was also financial supported by the Fulbright Commission in granting Professor Nzihou with a research scholarship at Princeton University.

Author Contributions

A.N. supervised, conceived the subject, and wrote the paper. L.M.R.M. and A.C.G. performed the graphitic biochar production, characterization, collected the data and analysis, and wrote the paper. C.E.W. wrote the paper.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

The authors thank colleagues at RAPSODEE Center-IMT Mines Albi (N. Lyczko), Centre de MicroCaractérisation Raimond Castaing (T. Hungria), and LPCNO (G. Ballon).


Vincent Guedj
email: vincent.guedj@math.univ-toulouse.fr

Chinh H Lu

DEGENERATE COMPLEX HESSIAN EQUATIONS ON COMPACT HERMITIAN MANIFOLDS

Keywords: Mathematics Subject Classification. 32W20, 32U05, 32Q15, 35A23 . Complex Hessian and Monge-Ampère equations, a priori estimates

In this note we provide uniform a priori estimates for solutions to degenerate complex Hessian equations on compact hermitian manifolds. Our approach relies on the corresponding a priori estimates for Monge-Ampère equations; it provides an extension as well as a short alternative proof to results of Dinew-Ko lodziej, Ko lodziej-Nguyen and Guo-Phong-Tong.

Introduction

Let X be a compact complex manifold equipped with a hermitian form ω X . We fix 1 ≤ k ≤ n = dim C X, ω another hermitian form, and µ X a volume form on X. The complex Hessian equation is

(ω + dd c ϕ) k ∧ ω n-k X = cµ X ,
where c > 0 is a positive constant and ϕ (the unknown) is a smooth (ω, ω X , k)subharmonic function, i.e. (ω + dd c ϕ) j ∧ ω n-j X ≥ 0 for all 1 ≤ j ≤ k. When k = 1 this reduces to the Laplace equation. When k = n this yields the complex Monge-Ampère equation which was famously solved by Yau [START_REF] Yau | On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation[END_REF] when ω is Kähler, and by Tosatti-Weinkove [START_REF] Tosatti | The complex Monge-Ampère equation on compact Hermitian manifolds[END_REF] in the hermitian setting.

From now on we assume that 1 < k < n. A unique solution has been provided by Dinew-Ko lodziej [START_REF] Dinew | Liouville and Calabi-Yau type theorems for complex Hessian equations[END_REF] when ω = ω X is Kähler; the hermitian setting has been resolved independently by Székelyhidi [START_REF] Székelyhidi | Fully non-linear elliptic equations on compact Hermitian manifolds[END_REF] and Zhang [START_REF] Zhang | Hessian equations on closed Hermitian manifolds[END_REF].

It is quite natural to consider such equations in degenerate settings, allowing the measure µ X = f ω n X to be merely absolutely continuous with respect to the volume form ω n X . Such degenerate complex Hessian equations have been intensively studied in the past fifteen years (see [Blo05, Lu13, DK14, DL15, LN15, Lu15, KN16, Char16, GN18, DDT19, BZ20, GP22] and the references therein).

In this note we go one step further by allowing the form ω to degenerate as well: we no longer assume it is positive, we merely ask that it is big, in analogy with the corresponding notion from algebraic geometry [START_REF] Boucksom | Monge-Ampère equations in big cohomology classes[END_REF], i.e. we assume there is an ω-psh function ρ with analytic singularities such that ω + dd c ρ dominates a hermitian form. We let Ω denote the Zariski open set where ρ is smooth.

One can not expect any longer that (ω, ω X , k)-subharmonic functions are smooth in X, but we show in Section 1.2 how to define the Hessian operator H k (f ) = (ω + dd c ϕ) k ∧ ω n-k X for such functions ϕ which are continuous in Ω and globally bounded on X.

In this context we obtain the following: Theorem A. Let ω be a semi-positive and big form and f ∈ L p (ω n X ) with p > n/k and X f ω n X > 0. There exists c > 0 and ϕ a bounded (ω, ω X , k)-subharmonic function, continuous in Ω, such that

(ω + dd c ϕ) k ∧ ω n-k X = cf ω n
X . This result extends the main result of [START_REF] Ko Lodziej | Weak solutions of complex Hessian equations on compact Hermitian manifolds[END_REF]. It applies in particular to the case when π : X → Y is the desingularization of a compact hermitian variety Y , and ω = π * ω Y is the pull-back of a hermitian form ω Y on Y .

As is often the case in similar contexts, the main difficulty lies in establishing uniform a priori estimates. This is the content of our second main result: Theorem B. Let ω 0 , ω be semi-positive and big forms such that ω 0 ≤ ω. Let ϕ be a smooth (ω, ω X , k)-subharmonic function solution to

(ω + dd c ϕ) k ∧ ω n-k X = f ω n X , where f ∈ L p (dV X ), p > n/k. Then Osc X (ϕ) ≤ C,
where C > 0 depends on ω 0 , n, k, p and an upper bound for ||f || L p .

The approach of Ko lodziej-Nguyen relies on a pluripotential theory for degenerate complex Hessian equations on hermitian manifolds (mimicking the corresponding results obtained by these authors and Dinew for the Monge-Ampère equation [START_REF] Dinew | Pluripotential estimates on compact Hermitian manifolds[END_REF], [START_REF] Ko Lodziej | Weak solutions to the complex Monge-Ampère equation on compact Hermitian manifolds[END_REF], [START_REF] Ko Lodziej | Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds[END_REF], [START_REF] Nguyen | The complex Monge-Ampère type equation on compact Hermitian manifolds and applications[END_REF]). An alternative PDE proof has been recently provided by Guo-Phong-Tong-Wang in [GPT21, GPTW21] when X is Kähler, and by Guo-Phong [START_REF] Guo | On L ∞ estimates for fully nonlinear partial differential equations on Hermitian manifolds[END_REF] when X is hermitian and ω is a hermitian form.

We provide in this note a proof which relies directly on the corresponding results for the complex Monge-Ampère operator. Given ϕ a solution of the k-Hessian equation with density f , we show that the ω-psh envelope of ϕ solves a Monge-Ampère equation whose density is bounded from above by f n/k . A lower bound on ϕ thus follows from the L ∞ -a priori estimates for the Monge-Ampère equation established in [START_REF] Guedj | Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds[END_REF].

Our method is new and also of interest when X is Kähler; it applies to more general densities (see Remark 3.2); in particular the results of Theorem A and Theorem B hold when f merely satisfies

X f n k | log f | n (log | log f |) n+δ < +∞,
for some δ > 0. Radial examples show that this condition is almost optimal.

It follows from Demailly's regularization theorem [START_REF] Demailly | Regularization of closed positive currents of type (1, 1) by the flow of a Chern connection[END_REF] that one can further assume that ρ has analytic singularities, in particular ω +dd c ρ is a hermitian form in some Zariski open set that we usually denote by Ω in what follows.

An important source of examples of big forms is provided by the following construction: if V is a compact complex space endowed with a hermitian form ω V , and π : X → V is a resolution of singularities, then ω = π * ω V is big.

1.1. (ω, ω X , k)-subharmonic functions. 1.1.1. Definition. Let U ⊂ X be an open subset and γ a hermitian form in U . A function h ∈ C 2 (U, R) is called γ-harmonic if dd c h ∧ γ n-1 = 0 pointwise in U.
We let H γ (U ) denote the set of all γ-harmonic functions in U . A function u : U → R ∪ {-∞} is γ-subharmonic if it is upper semicontinuous and for every D ⊂ U and every h ∈ H γ (D) the following implication holds

u ≤ h on ∂D =⇒ u ≤ h in D.
As shown in [HL13, Section 9] the γ-subharmonicity can be equivalently defined using viscosity theory: u is γ-subharmonic in U if and only if the inequality dd c u ∧ γ n-1 holds in the viscosity sense in U . The latter condition means that for every x 0 ∈ U and every C 2 function χ defined in a neighborhood V of x 0 , the following implication holds:

[u(x) ≤ χ(x), ∀x ∈ V and u(x 0 ) = χ(x 0 )] =⇒ dd c χ ∧ γ n-1 ≥ 0 at x 0 . When u ∈ C 2 (U ), it is γ-subharmonic if and only if dd c u ∧ γ n-1 ≥ 0 pointwise.
Definition 1.2. We let SH γ (U ) denote the set of all γ-subharmonic functions in U that are locally integrable. where ess sup B(x,r) u is the essential supremum of u in the ball B(x, r).

Definition 1.3. A function ϕ : U → R ∪ {-∞} is quasi-γ-subharmonic if locally ϕ = u + ρ where u is γ-subharmonic and ρ is smooth.
We say that a function ϕ is (ω, γ)-subharmonic if it is quasi-γ-subharmonic and (ω + dd c ϕ) ∧ γ n-1 ≥ 0 in the sense of distributions on X.

We let SH γ (U, ω) denote the set of all (ω, γ)-subharmonic functions in U .

If u ∈ C 2 (U, R) then u ∈ SH γ (U, ω) if and only if (ω + dd c u) ∧ γ n-1 ≥ 0 pointwise.
In general this inequality holds in the viscosity sense:

Lemma 1.4. If ϕ ∈ SH γ (U, ω) and χ is a C 2 function such that ϕ -χ attains a local maximum at x 0 ∈ U , then (ω + dd c χ) ∧ γ n-1 ≥ 0 at x 0 .
Proof. Fix a > 1. Let ρ be a quadratic function such that dd c ρ = aω(x 0 ). By continuity of ω, we have dd c ρ ≥ ω in a small ball B around x 0 . The function ρ + ϕ is then γ-subharmonic in B and the function ρ + ϕ -(ρ + χ) attains a local maximum at x 0 . It thus follows that, at x 0 ,

(aω + dd c χ) ∧ γ n-1 = dd c (ρ + χ) ∧ γ n-1 ≥ 0.
The conclusion follows by letting a ց 1.

Fix an integer 1 ≤ k ≤ n, and let Γ k (ω X ) denote the set of all (1, 1)-forms α such that α j ∧ ω n-j X > 0, j = 1, ..., k. By Gårding's inequality [START_REF] Gårding | An inequality for hyperbolic polynomials[END_REF]

, if α, α 1 , ..., α k-1 ∈ Γ k (ω X ) then α ∧ α 1 ∧ ... ∧ α k-1 ∧ ω n-k X > 0. In particular, α 1 ∧ ... ∧ α k-1 ∧ ω n-k X is a strictly positive (n -1, n -1)-form on X. By [Mic82], one can write α 1 ∧ ... ∧ α k-1 ∧ ω n-k X = γ n-1
, for some hermitian form γ. We let F k (ω X ) denote the set of all such hermitian forms.

Definition 1.5. A function ϕ : U → R ∪ {-∞} is (ω, ω X , k)-subharmonic if it is (ω, γ)-subharmonic in U for all γ ∈ F k (ω X ).
We let SH ω X (U, ω, k) denote the set of all (ω, ω X , k)-subharmonic functions on U which are locally integrable in U .

If u is of class C 2 in U then, by [Gar59], u is (ω, ω X , k)-subharmonic if and only if ω + dd c u ∈ Γ k (ω X ).
We refer the reader to [START_REF] Blocki | Weak solutions to the complex Hessian equation[END_REF] and [KN16, Section 2.3] for basic properties of (ω, ω X , k)-sh functions; they easily extend to our context.

Since ω X is fixed once and for all in this note, we let SH(X, ω, k) denote the set of (ω, ω X , k)-sh functions; they form a nested sequence

SH(X, ω) ⊃ • • • SH(X, ω, k -1) ⊃ SH(X, ω, k) ⊃ • • • ⊃ P SH(X, ω), where SH(X, ω) = SH(X, ω, 1) is the set of ω-sh functions, while P SH(X, ω) = SH(X, ω, n) is the set of ω-psh functions, for which ω + dd c ϕ ≥ 0 is a positive current of bidegree (1, 1) (see [GZ17, Chapter 8]). Observe that SH(X, ω, k) ⊂ SH(X, ω ′ , k) if ω ≤ ω ′ .
Since ω ≤ Aω X for some A > 0, any (ω, ω X , k)-sh function is also (Aω X , 1)-sh. We thus have the following compactness result (see [KN16, Lemma 3.3], [GN18, Section 9]).

Proposition 1.6. The set of normalized (ω, ω X , k)-subharmonic functions is compact in the L 1 -topology. In particular, there exists C > 0 such that for all ϕ ∈ SH(X, ω, k) normalized by sup X ϕ = 0,

X (-ϕ)ω n X ≤ C.
1.1.2. Smooth approximation.

Theorem 1.7. If ω is hermitian, any (ω, k)-subharmonic function is the pointwise limit of a decreasing sequence of smooth (ω, k)-subharmonic functions.

This result is a slight extension of [KN16, Lemma 3.20], which itself extends [LN15, Theorem 1.2] from the Kähler to the hermitian setting. We propose below an alternative and more direct proof.

Proof. Let u be a smooth function on X and let f be a smooth positive density such that

(ω + dd c u) k ∧ ω n-k X ≤ f ω n X .
For j ≥ 1, we solve the complex Hessian equation

(ω + dd c ϕ j ) k ∧ ω n-k X = e j(ϕ j -u) f ω n X .
As shown in [START_REF] Ko Lodziej | Weak solutions of complex Hessian equations on compact Hermitian manifolds[END_REF], the estimates established in [START_REF] Székelyhidi | Fully non-linear elliptic equations on compact Hermitian manifolds[END_REF] provide a smooth solution ϕ j which is (ω, k)-subharmonic on X. We want to prove that ϕ j converges uniformly to P ω,k (u), the largest (ω, k)-subharmonic function lying below u. The theorem follows from this fact, as shown in [START_REF] Lu | Degenerate complex Hessian equations on compact Kähler manifolds[END_REF][START_REF] Ko Lodziej | Weak solutions of complex Hessian equations on compact Hermitian manifolds[END_REF].

The classical maximum principle ensures that ϕ j ≤ u, hence ϕ j ≤ P ω,k (u). Fix now ϕ ∈ SH(X, ω, k) such that ϕ ≤ u. Let x 0 ∈ X be a point where the function ϕ -(1 + 1/j)ϕ j attains its maximum on X. By definition ϕ belongs to SH γ (X, ω), where γ is the hermitian form such that

γ n-1 = (ω + dd c ϕ j ) k-1 ∧ ω n-k X .
From now on the computations are made at x 0 . It follows from Lemma 1.4 that

(ω + (1 + 1/j)dd c ϕ j ) ∧ γ n-1 ≥ 0.
From this and Gårding's inequality [START_REF] Gårding | An inequality for hyperbolic polynomials[END_REF] we obtain j + 1

j (ω + dd c ϕ j ) k ∧ ω n-k X ≥ 1 j ω ∧ (ω + dd c ϕ j ) k-1 ∧ ω n-k X ≥ 1 j e j(k-1) k (ϕ j -u) f (k-1)/k g 1/k ω n X ,
where

gω n X = ω k ∧ ω n-k X . Let b ∈ (0, 1) be a positive constant such that g ≥ b k f on X. The above inequality yields (1 + j)e j(ϕ j -u)/k ≥ b, hence ϕ j -u ≥ k(log b -log(1 + j)) j .
Thus, for some uniform constant C ≥ 1 we have

1 + 1 j ϕ j -u ≥ 1 + 1 j (ϕ j -u) + u j ≥ -C log j j .
Since ϕ ≤ u, we also obtain

ϕ -1 + 1 j ϕ j ≤ C log j j .
This inequality holds on X, as ϕ -(1 + 1/j)ϕ j attains its global maximum at x 0 . Since this is true for all ϕ ∈ SH(X, ω, k) lying below u, we obtain

P ω,k (u) -C ′ log j j ≤ ϕ j ≤ P ω,k (u),
for a uniform constant C ′ , proving that ϕ j converges uniformly to P ω,k (u).

1.2. The Hessian operator.

1.2.1. Hermitian forms. We assume in this subsection that ω is a hermitian form. For (ω, ω X , k)-subharmonic functions ϕ of class C 2 , the complex Hessian operator is defined by

H k (ϕ) = (ω + dd c ϕ) k ∧ ω n-k X .
By Theorem 1.7, one can pointwise approximate from above any (ω, ω X , k)-sh function ϕ by a decreasing sequence of smooth (ω, ω X , k)-sh functions ϕ j .

If ϕ is moreover continuous on X, then the convergence is uniform, and it was shown in [START_REF] Ko Lodziej | Weak solutions of complex Hessian equations on compact Hermitian manifolds[END_REF] that the sequence of Hessian measures H k (ϕ j ) weakly converges to a unique positive Radon measure, independent of the approximants. One sets

H k (ϕ) := lim j→+∞ H k (ϕ j ) := (ω + dd c ϕ) k ∧ ω n-k X .
It is likely that one can define these operators for (ω, ω X , k)-sh functions that are merely bounded. This is indeed the case when ω X is closed, as one can proceed by induction and use integration by parts following Bedford-Taylor's construction [START_REF] Bedford | The Dirichlet problem for the complex Monge-Ampère operator[END_REF][START_REF] Bedford | A new capacity for plurisubharmonic functions[END_REF]. The situation is however delicate in the general hermitian setting, and more subtle than for the Monge-Ampère operator. Among the basic properties enjoyed by this operator are the following:

• ϕ → H k (ϕ) is continuous for the uniform topology;

• if u 1 , . . . , u k ∈ SH(X, ω, k) are continuous, the mixed Hessian operators

(ω + dd c u 1 )∧ • • •∧ (ω + dd c u k )∧ ω n-k X
are similarly defined and continuous;

• if u 1 , . . . , u k are smooth and (ω + dd c u j ) k ∧ ω n-k X = f j dV X then (1.1) (ω + dd c u 1 ) ∧ • • • ∧ (ω + dd c u k ) ∧ ω n-k X ≥ (f 1 • • • f k ) 1/k dV X .
These are a consequence of inequalities due to Gårding.

1.2.2. Big forms. We now no longer assume that ω is positive, but rather that it is semi-positive and big. Fix a quasi-plurisubharmonic function ρ : X → R ∪ {-∞} with analytic singularities such that ω + dd c ρ ≥ δω X , for some constant δ > 0.

We let Ω denote the Zariski open set where ρ is smooth. If ϕ ∈ SH(X, ω, k)∩L ∞ (X) is continuous in Ω, it follows from [KN16, Proposition 2.9] that for each smooth form γ, the mixed product (ω X +dd c ϕ) j ∧γ l-j ∧ω n-l X is a well-defined Radon measure in Ω with finite mass.

Definition 1.8. For ϕ ∈ SH(X, ω, k) ∩ L ∞ (X) ∩ C 0 (Ω) we set H k (ϕ) := k j=0 k j (ω X + dd c ϕ) j ∧ (ω -ω X ) k-j ∧ ω n-k X .
This operator is well-defined in Ω with finite total mass, so we consider its trivial extension to X. The definition is motivated by the identity

(ω + dd c ϕ) k ∧ ω n-k X = (ω X + dd c ϕ + ω -ω X ) k ∧ ω n-k X = k j=0 k j (ω X + dd c ϕ) j ∧ (ω -ω X ) k-j ∧ ω n-k X .
The operator H k is continuous along sequences that converge locally uniformly in Ω [KN16, Proposition 2.11]. The following result follows.

Proposition 1.9. If ϕ ∈ SH(X, ω, k) ∩ L ∞ (X) ∩ C 0 (Ω), then H k (ϕ) is a positive Radon measure.
Proof. Fix ε > 0 and approximate ϕ -using Theorem 1.7-by a decreasing sequence (ϕ j ) of (ω + εω X )-sh functions. Since (ω + εω

X + dd c ϕ j ) k ∧ ω n-k X ≥ 0, we infer from [KN16, Proposition 2.11] that (ω + εω X + dd c ϕ) k ∧ ω n-k X ≥ 0.
The conclusion follows by letting ε → 0, as we then obtain that (ω

+ dd c ϕ) k ∧ ω n-k X ≥ 0.
Lemma 1.10. Let ϕ, ψ be (ω, ω X , k)-sh functions which are continuous in Ω and bounded on X. Then

H k (max(ϕ, ψ)) ≥ 1 {ϕ>ψ} H k (ϕ) + 1 {ψ≥ϕ} H k (ψ).
In particular, if ϕ ≤ ψ then

1 {ψ=ϕ} H k (ϕ) ≤ 1 {ψ=ϕ} H k (ψ).
Proof. Fix t > 0. Since {ϕ < ψ + t} and {ψ + t < ϕ} are open in Ω, we have

H k (max(ϕ, ψ + t)) ≥ 1 {ϕ>ψ+t} (ω + dd c ϕ) k ∧ ω n-k X + 1 {ψ+t>ϕ} (ω + dd c ψ) k ∧ ω n-k X ≥ 1 {ϕ>ψ+t} (ω + dd c ϕ) k ∧ ω n-k X + 1 {ψ≥ϕ} (ω + dd c ψ) k ∧ ω n-k X .
Letting t ց 0 we obtain the first statement of the lemma. The second one follows straightforwardly from the first.

1.2.3. SH(X, ω, k)-envelopes. We consider two types of envelopes in this note.

Definition 1.11. If h : X → R is a continuous function, we set P ω (h) := sup {u ∈ P SH(X, ω), u ≤ h} , and P ω,k (h) := sup {u ∈ SH(X, ω, k), u ≤ h}.

Here is a list of basic properties of these envelopes.

• The measure (ω +dd c P ω (h)) n is concentrated on the set C = {P ω (h) = h}.

• The complex Hessian measure (ω +

dd c P ω,k (h)) k ∧ ω n-k X is concentrated on the contact set C k = {P ω,k (h) = h}. • If h is C 1,1 -smooth and ω is hermitian, then P ω (h) is C 1,1 -smooth and (ω + dd c P ω (h)) n = 1 C (ω + dd c h) n . • If h is C 1,1 -smooth and ω is hermitian, then P ω,k (h) is C 1,1 -smooth and (ω + dd c P ω,k (h)) k ∧ ω n-k X = 1 C k (ω + dd c h) k ∧ ω n-k X .
We refer the reader to [Ber19, Tos18, GLZ19, CZ19, CM21] for the proof of these results, as well as further information on these envelopes. 1.3. A priori estimates for the complex Monge-Ampère operator. We are going to use the following a priori estimates established in [START_REF] Guedj | Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds[END_REF].

Theorem 1.12. Let ω, ω 0 be semi-positive and big (1, 1)-forms such that ω ≥ ω 0 . Fix δ > 0 and g ∈ L 1+δ (dV X ) a non-negative density. If ϕ ∈ P SH(X, ω)∩L ∞ (X) satisfies (ω + dd c ϕ) n ≤ gdV X , then Osc X (ϕ) ≤ C for some C ∈ R + that depends on n, δ, ω 0 and an upper bound on ||g|| L 1+δ .

Since this result is not explicitly stated in [START_REF] Guedj | Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds[END_REF], we briefly sketch its proof.

Proof. We normalize ϕ so that sup X ϕ = 0. It follows from Skoda's uniform integrability theorem [GZ17, Theorem 8.11] that for 0 < α small enough, X e -α (2+δ)(1+δ) δ ϕ dV X ≤ C α is bounded from above independently of ϕ. Setting g = e -αϕ g, Hölder inequality then shows that g ∈ L 1+δ/2 with

||g|| L 1+δ/2 ≤ C 1/q α ||g|| 1/p L 1+δ
, where p = 1+δ 1+δ/2 and q = 2 + 2/δ. [GL21, Lemma 3.3] shows the existence of c(δ, ω) > 0 and u ∈ P SH(X, ω) such that -1 ≤ u ≤ 0 and

(ω + dd c u) n ≥ c(δ, ω) gdV X ||g|| L 1+δ/2 ≥ e αu c(δ, ω) gdV X ||g|| L 1+δ/2 . The function v := u -α -1 log[c(δ, ω)/||g|| L 1+δ/2 ] ∈ P SH(X, ω) thus satisfies e -αv (ω + dd c v) n = e -αv (ω + dd c u) n ≥ gdV X ≥ e -αϕ (ω + dd c ϕ) n .
The comparison principle [GL21, Corollary 1.14] finally ensures that v ≤ ϕ. This provides a uniform lower bound on ϕ, which only depends on δ, ω and an upper bound on ||g|| L 1+δ .

Domination principle

Fix ω a big form, ρ an ω-plurisubharmonic function with analytic singularities such that ω + dd c ρ ≥ δω X , δ > 0, and set Ω := {ρ > -∞}.

We fix a constant B 1 > 0 such that for all x ∈ Ω,

-B 1 ω 2 ρ ≤ dd c ω ρ ≤ B 1 ω 2 ρ and -B 1 ω 3 ρ ≤ dω ρ ∧ d c ω ρ ≤ B 1 ω 3 ρ . The existence of B 1 is clear since dω = dω ρ , d c ω = d c ω ρ and -Bω 2 X ≤ dd c ω ≤ Bω 2 X and -Bω 3 X ≤ dω ∧ d c ω ≤ Bω 3 X for some B > 0. The following is an extension of [GL21, Theorem 1.11]. Theorem 2.1. Assume ω is big. If u is a bounded (ω, ω X , k)-subharmonic func- tion which is continuous in Ω, then X (ω + dd c u) k ∧ ω n-k X > 0.
Proof. Fix ρ and B 1 as above. We set m = inf Ω (u -ρ). For s > 0 we set

φ = max(u, ρ + m + s) and U := {u < φ} = {u < ρ + m + s}.
Observe that U is relatively compact in Ω, and it is non empty for all s > 0, by definition of m. It follows from [KN16, Corollary 2.4] that, for any 0

≤ j ≤ k -1, dd c (ω j u ∧ ω k-j-1 φ ∧ ω n-k X ) ≤ C 1 k-1 m=k-2 m l=0 ω l u ∧ ω m-l φ ∧ ω n-m X ,
where C 1 is a uniform constant. On the open set U since ω φ ≥ δω X , we thus have

(2.1) dd c (ω j u ∧ ω k-j-1 φ ∧ ω n-k X ) ≤ C 2 k l=0 ω l u ∧ ω k-l φ ∧ ω n-k X .
Since u ≤ φ, Lemma 1.10 ensures that

1 {u=φ} ω j u ∧ ω k-j φ ∧ ω n-k X ≥ 1 {u=φ} ω k u ∧ ω n-k φ .
Noting that X \ U = {u = φ} we can write

U (ω j u ∧ ω k-j φ ∧ ω n-k X -ω k u ∧ ω n-k X ) ≤ X (ω j u ∧ ω k-j φ ∧ ω n-k X -ω k u ∧ ω n-k X ) = X dd c (φ -u) ∧ k-j l=0 ω l u ∧ ω k-j-l φ ∧ ω n-k X = X (φ -u)dd c k-j l=0 ω l u ∧ ω k-j-l φ ∧ ω n-k X .
In the last line above, we have used Stokes theorem. Since 0 ≤ φ -u ≤ s, using (2.1) we can continue the above inequalities as follows:

U (ω j u ∧ ω k-j φ ∧ ω n-k X -ω k u ∧ ω n-k X ) ≤ C 3 s k l=0 U ω l u ∧ ω k-l φ ∧ ω n-k X .
Summing up the above inequalities for j = 0, ..., k, one gets

(1 -C 4 s) k l=0 U ω l u ∧ ω k-l φ ∧ ω n-k X ≤ (k + 1) U ω k u ∧ ω n-k X .
If ω k u ∧ ω n-k X = 0 then, for s > 0 small enough, the above inequality yields

U ω k φ ∧ ω n-k X = 0, hence U has Lebesgue measure 0, which is impossible. Theorem 2.2. Assume ω is hermitian, u, v are continuous (ω, ω X , k)-subharmonic functions such that ω k u ∧ ω n-k X = 0 on {u < v}. Then u ≥ v.
Proof. By replacing v with max(u, v) and noting that {u < v} = {u < max(u, v)}, we can assume that u ≤ v. Our plan is thus to show that u = v. For b ≥ 1, we set

φ b := P ω,k (bu -(b -1)v). It is a negative (ω, k)-sh function. Since bu -(b -1)v is continuous on X, so is φ b . Observe also that b -1 φ b + (1 -b -1 )v ≤ u
with equality on the contact set C := {φ b = bu -(b -1)v}. Now by Lemma 1.10

1 C b -k ω k φ b ∧ ω n-k X + (1 -b -1 ) k ω k v ∧ ω n-k X ≤ 1 C ω k u ∧ ω n-k X , while C H k (φ b ) = X H k (φ b ) > 0 by Theorem 2.1. Since H k (u) = 0 on {u < v},
it follows that C ∩ {u = v} contains at least one point, say x. We then have

φ b (x) = bu(x) -(b -1)v(x) = v(x) ≥ inf X v; in particular sup X φ b ≥ inf X v.
Note that φ b is decreasing in b since u ≤ v. The limit φ := lim b→+∞ φ b is a (ω, k)-sh function which is not identically -∞. For any a > 0,

φ b ≤ bu -(b -1)v ≤ v -ba on the open set {u < v -a}, thus φ = -∞ on this set. Since φ is integrable on X, the set {u < v -a} must be empty, hence letting a → 0 + gives u ≥ v. Corollary 2.3. Assume ω is hermitian, u and v are continuous (ω, ω X , k)- subharmonic functions such that ω k u ∧ ω n-k ≤ cω k v ∧ ω n-k on {u < v} for some c ∈ [0, 1). Then u ≥ v.
Proof. Take b so large that (1 -b -1 ) k ≥ c, and consider φ := P ω,k (bu -(b -1)v). Then b -1 φ + (1 -b -1 )v ≤ u with equality on the contact set

C := {φ = bu -(b -1)v}.
By Lemma 1.10 we have

1 C b -k ω k φ ∧ ω n-k X + (1 -b -1 ) k ω k v ∧ ω n-k X ≤ 1 C ω k u ∧ ω n-k X .
Multiplying with 1 {u<v} and using the assumption

ω k u ∧ ω n-k ≤ cω k v ∧ ω n-k on {u < v} and (1 -b -1 ) k ≥ c, we then see that 1 C∩{u<v} ω k φ ∧ ω n-k = 0. Since ω k φ ∧ ω n-k is supported on C, we thus have {φ<v} ω k φ ∧ ω n-k = {bu-(b-1)v<v} ω k φ ∧ ω n-k = {u<v} ω k φ ∧ ω n-k = 0.
Invoking the domination principle (Theorem 2.2) we then get

φ ≥ v. It thus follows that bu -(b -1)v ≥ φ ≥ v, hence u ≥ v.
Corollary 2.4. Assume ω is hermitian and u and v are continuous (ω, ω X , k)subharmonic functions. Then

e -v (ω + dd c v) k ∧ ω n-k X ≥ e -u (ω + dd c u) k ∧ ω n-k X =⇒ u ≥ v.
Proof. Fix a > 0. On the set {u < v -a} we have H k (u) ≤ e -a H k (v). By Corollary 2.3, u ≥ v -a. Since it holds for all a > 0, we obtain u ≥ v.

Proof of the main theorems

3.1. A priori estimates. In this section we prove Theorem B, which is an extension of the main result of Ko lodziej-Nguyen [KN16]:

Theorem 3.1. Let ω 0 , ω be semi-positive and big forms such that ω 0 ≤ ω. Let ϕ be a smooth (ω, ω X , k)-subharmonic function satisfying

(ω + dd c ϕ) k ∧ ω n-k X ≤ f dV X ,
where f ∈ L p (dV X ), p > n/k. Then

Osc X (ϕ) ≤ C,
where C > 0 depends on ω 0 , n, k, p and an upper bound for ||f || L p .

Proof. We can assume that sup X ϕ = 0. Let u := P ω (ϕ) be the largest ω-psh function lying below ϕ. Let C denote the contact set C := {u = ϕ}. Then u is (ω, ω X , k)-sh with u ≤ ϕ hence Lemma 1.10 yields

1 C (ω + dd c u) k ∧ ω n-k X ≤ 1 C (ω + dd c ϕ) k ∧ ω n-k X ≤ f dV X .
Observe that the smooth function h := ω n X /dV X ≥ δ 0 is positive on X. Recall that the Monge-Ampère measure (ω + dd c u) n of u is concentrated on the contact set C. We set g := (ω + dd c u) n /dV X . It follows from Gårding's inequalities (1.1) that δ

1-k/n 0 g k/n ≤ h 1-k/n g k/n ≤ f, hence g ≤ C 0 f n/k . Since f ∈ L p with p > n/k, we obtain g ∈ L 1+δ with δ > 0.
Theorem 1.12 therefore provides a uniform bound Osc X (u) ≤ M.

This provides a uniform bound on u ≤ 0 if we can bound sup X u from below. Let q ′ be the conjugate exponent to p ′ = pk/n > 1. We obtain a uniform lower bound on sup X u by using [GL22, Proposition 3.4] and Hölder inequality, via

v - M (ω)(-sup X u) 1/q ′ ≤ X |u| 1/q ′ (ω + dd c u) n ≤ X |ϕ|ω n 1/q ′ X f p ′ ω n 1/p ′ ≤ C,
as X |ϕ|ω n is bounded from above independently of ϕ by Proposition 1.6. Together with the uniform bound for Osc X (u), this yields a uniform bound for |u|. This implies a uniform bound for ϕ since u ≤ ϕ and sup X ϕ = 0. Remark 3.2. We also have a uniform a priori bound when f satisfies the weaker integrability condition

X f n/k | log f | n |h • log • log f | n dt < +∞
where h is an increasing continuous function such that +∞ dt/h(t) < +∞. It follows indeed that g satisfies then Ko lodziej's optimal integrability condition X g| log g| n |h • log • log g| n dt < +∞, which ensures a uniform bound on Osc X (u) by [Kol98, Theorem 2.5.2]. We actually need here the extension of Ko lodziej's result to the case of semi-positive and big forms [START_REF] Eyssidieux | Singular Kähler-Einstein metrics[END_REF], as well as to the hermitian setting [START_REF] Dinew | Pluripotential estimates on compact Hermitian manifolds[END_REF][START_REF] Guedj | Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds[END_REF].

The same proof thus applies with a minor modification: to obtain a lower bound for sup X u, we replace Hölder inequality by the additive Hölder-Young inequality. If w denotes the convex weight w(t) ∼ t(log t) n (h • log • log t) n , w * is its conjugate convex weight, and (w * ) -1 denotes the inverse of w * , we obtain this way a uniform upper bound

v - M (ω)(w * ) -1 (-sup X u) ≤ X (w * ) -1 (u)(ω + dd c u) n ≤ X w • f dV X + X (-u)dV X ≤ C.
3.2. Stability estimates. We strengthen Theorem B by establishing the following stability estimate: Theorem 3.3. Assume ω is hermitian and ω ≥ ω 0 for some semi-positive and big form ω 0 . Assume 0 ≤ f, g ∈ L p (dV X ) with p > n/k, ϕ, ψ are continuous (ω, ω X , k)-subharmonic functions such that

(ω + dd c ϕ) k ∧ ω n-k X ≥ e ϕ f ω n X , and (ω + dd c ψ) k ∧ ω n-k X ≤ e ψ gω n X . Then (3.1) ϕ ≤ ψ + C f -g 1/k
p for some C > 0 depending on ω 0 , n, k and upper bounds for ||f || p , ϕ ∞ , | inf X ψ|.

In particular if ϕ, ψ are solutions of the corresponding equations, then

||ϕ -ψ|| ∞ ≤ C f -g 1/k p , so that ϕ -ψ is uniformly small if f is close to g in L p .
Proof. Our proof uses a perturbation argument from [GLZ18, LPT21], which goes back to [START_REF] Ko Lodziej | Some sufficient conditions for solvability of the Dirichlet problem for the complex Monge-Ampère operator[END_REF]. We solve the complex Monge-Ampère equation

(ω + dd c u) n = bhω n X , where h = |f -g| n/k f -g n/k p + 1,
sup X u = 0 and b > 0 is a constant. The existence of a continuous solution u ∈ P SH(X, ω) follows from [START_REF] Ko Lodziej | Weak solutions to the complex Monge-Ampère equation on compact Hermitian manifolds[END_REF][START_REF] Ko Lodziej | Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds[END_REF]. We can assume without loss of generality that ω X is a Gauduchon metric (i.e.

dd c (ω n-1 X ) = 0) which dominates ω. Thus u is ω X -psh with (ω X + dd c u) n ≥ (ω + dd c u) n ≥ bω n X since h ≥ 1. The mixed Monge-Ampère inequalities (see [Ng16]) yield (ω X + dd c u) ∧ ω n-1 X ≥ b 1/n ω n X , hence b ≤ 1 since X (ω X + dd c u) ∧ ω n-1 X = X ω n
X by the Gauduchon condition. Since h ∈ L q with q = pk/n > 1 and b ≤ 1, Theorem 3.1 ensures that u ∞ is uniformly bounded. It thus follows from [GL22, Proposition 3.4] that b ≥ δ 0 > 0 is uniformly bounded from below.

If f -g 1/k p is not small, the stability inequality (3.1) is trivially obtained by adjusting the value of C. In the remainder of the proof, we can thus assume without loss of generality that ε := b -1/n f -g 1/k p e sup X ϕ/k ≤ 1/2. (3.2) Since ω ≤ ω j and ||c j f j || L p is uniformly bounded from above, it follows from Theorem 3.1 that the ϕ j 's are uniformly bounded:

-C 1 ≤ ϕ j ≤ 0, for a uniform constant C 1 . Extracting and using compactness in L 1 (X) of normalized (ω, ω X , k)-sh functions, we can thus assume that c j → c > 0 and ϕ j converges in L 1 (X) and almost everywhere to a bounded function ϕ which is (ω, ω X , k)-sh, as j → +∞.

We next prove that the convergence ϕ j → ϕ is locally uniform in Ω. We set g j := c j f j e -ϕ j and g = cf e -ϕ . By Hölder inequality, g j -g q → 0 for some q > n/k. Fix j < l large enough. Since ω j ≥ ω l , we have (ω j + dd c ϕ l ) k ∧ ω n-k X ≥ e ϕ l g l ω n X . By Theorem 3.3, we infer (3.3) ϕ l ≤ ϕ j + C 2 g j -g l 1/k q , for a uniform constant C 2 . We next consider ψ j := (1 -ε)ϕ j + ερ -(2k + C 1 )ε, where ε = δ -1 2 -j < 1/2 when j is large enough. Since ω + dd c ρ ≥ δω X , we have

ω l + dd c ψ j ≥ ω + εdd c ρ + (1 -ε)dd c ϕ j ≥ εδω X + (1 -ε)(ω j + dd c ϕ j ) -2 -j ω X = (1 -ε)(ω j + dd c ϕ j ).
It follows that ψ j ∈ SH(X, ω l , k) and (ω l + dd c ψ j ) k ∧ ω n-k X ≥ e ϕ j +k log(1-ε) g j ω n

X ≥ e ϕ j -2kε g j ω n X ≥ e ψ j g j ω n X holds in Ω. The function ψ j,l := max(ψ j , ϕ l ) is (ω l , k)-subharmonic and continuous on X. Lemma 1.10 yields (ω l + dd c ψ j,l ) k ∧ ω n-k X ≥ e ψ j,l min(g j , g l )ω n X . Applying Theorem 3.3 we then get ψ j,l ≤ ϕ l + C 3 min(g j , g l ) -g l 1/k q ≤ ϕ l + C 3 g j -g l 1/k q , for a uniform constant C 3 . Together with (3.3), this implies ϕ j + δ -1 2 -j ρ -(2k + C 1 )δ -1 2 -j ≤ ϕ l + C 3 g j -g l 1/k q ≤ ϕ j + C 1 g j -g l 1/k q . Therefore, ϕ j converges locally uniformly in Ω to ϕ. Thus ϕ is continuous in Ω where it solves the complex Hessian equation

(ω + dd c ϕ) k ∧ ω n-k X = cf ω n X .
Remark 3.4. When the form ω is hermitian, the solution ϕ that we construct is continuous on the whole of X. Continuity of ϕ at points where ω vanishes is a delicate issue, even in the case of complex Monge-Ampère equations. We refer the interested reader to [START_REF] Guedj | Continuity of singular Kähler-Einstein potentials[END_REF] for a recent account.

The uniqueness of ϕ is also a subtle problem in the hermitian setting, see [START_REF] Ko Lodziej | Stability and regularity of solutions of the Monge-Ampère equation on Hermitian manifolds[END_REF] for a partial result in the case of the complex Monge-Ampère equation.

  If u ∈ SH γ (U ) then, by [HH72, Theorème 1, p.136], dd c u ∧ γ n-1 ≥ 0 in the sense of distributions. Conversely, by [Lit83, Theorem A], the latter condition implies that u coincides a.e. with a unique function û ∈ SH γ (U ) defined by û(x) := ess lim sup y→x u(y) := lim rց0 ess sup B(x,r) u,

Acknowledgements. This work is partially supported by the research project HERMETIC (ANR-11-LABX-0040), as well as by the ANR project PARAPLUI.

By the mixed Monge-Ampère inequalities (1.1) we have

We set C 1 = 2k -inf X ϕ and

The function ψ ε belongs to SH(X, ω, k) ∩ C 0 (X); its Hessian measure satisfies

p , for some uniform constant C 2 , as desired.

3.3. Proof of Theorem A. We approximate f in L p by smooth and positive densities f j . For j ≥ 1 we consider the hermitian forms ω j = ω + 2 -j ω X . It follows from [START_REF] Zhang | Hessian equations on closed Hermitian manifolds[END_REF][START_REF] Székelyhidi | Fully non-linear elliptic equations on compact Hermitian manifolds[END_REF] that there exist unique constants c j > 0 and smooth (ω j , k)-sh functions ϕ j such that (ω j + dd c ϕ j ) k ∧ ω n-k X = c j f j ω n X , and sup X ϕ j = 0.

We claim that the constants c j stay uniformly bounded, as j increases to +∞. To prove the claim we use the solutions to the Monge-Ampère equations by Tosatti-Weinkove [START_REF] Tosatti | The complex Monge-Ampère equation on compact Hermitian manifolds[END_REF]:

is uniformly bounded from above, by [GL21, Lemma 3.3], there exists a constant c > 0 independent of j and a bounded ω-psh function u such that ω n u ≥ cf n/k j

. By [GL21, Corollary 1.13], b j ≥ c. It then follows from Gårding's inequality that

X . Corollary 2.3 thus ensures that c j ≥ c k/n is uniformly bounded from below by a positive constant.

We next bound c j from above. By Gårding's inequality we have

The functions ϕ j belong to SH(X, ω 1 ) and are relatively compact in L 1 , hence integrating over X and using that X f 1/k j ω n X → X f 1/k ω n X > 0, we obtain a uniform upper bound for c j .