Yongzhen Qi

Ying Huang

Ziliang Gao

Celine Chen

Andrea Perego

Hakan Yildirim

Madeleine Odgaard

Tristan Asset

Plamen Atanassov

Iryna V Zenyuk
email: iryna.zenyuk@uci.edu

Insight into carbon corrosion of different carbon supports for Pt-based electrocatalysts using accelerated stress tests in polymer electrolyte fuel cells

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Polymer electrolyte fuel cells (PEFCs) using hydrogen as a fuel show higher fuel efficiency and decrease in pollutant emissions, which provides a promising way to combat climate change [START_REF] Manoharan | Hydrogen fuel cell vehicles; Current status and future prospect[END_REF]. The major PEFCs applications are developed in the transportation sectors to replace the internal combustion engines (ICEs). Other applications of the PEFCs focus on forklift power sources and electric vehicles range extenders [START_REF] Kast | Clean commercial transportation: Medium and heavy duty fuel cell electric trucks[END_REF][START_REF] Liu | Comparison of well-to-wheels energy use and emissions of a hydrogen fuel cell electric vehicle relative to a conventional gasoline-powered internal combustion engine vehicle[END_REF]. However, due to the high voltage and transient operations, fuel cell lifetime is shortened due to the corrosion mechanisms [START_REF] Chalk | Key challenges and recent progress in batteries, fuel cells, and hydrogen storage for clean energy systems[END_REF][START_REF] Wilson | Surface Area Loss of Supported Platinum in Polymer Electrolyte Fuel Cells[END_REF][START_REF] Taniguchi | Analysis of electrocatalyst degradation in PEMFC caused by cell reversal during fuel starvation[END_REF],

reducing PEFCs vehicles lifetime compared to ICEs. U.S. Department of Energy (DOE) set the lifetime target of 8000 hours, 150000 equivalent driving miles with less than 10% performance loss for light-duty vehicles [START_REF] Cells | Multi-Year Research, Development, and Demonstration Plan[END_REF][START_REF] Borup | FC135 : FC-PAD : Fuel Cell Performance and Durability Consortium[END_REF]. To meet the lifetime targets, electrochemical characterization and morphology changes resulting from catalysts corrosion and carbon support corrosion must be investigated.

Carbon corrosion occurs within the PEFC catalyst layers that consist of carbon-based support and Pt or Pt-alloy catalysts. It is especially critical at the cathode catalyst layer due to: (i) preferable oxidizing operation condition, (ii) more water participation and Pt catalyzing the carbon corrosion in the operating potential range. The corrosion mechanism of carbon corrosion in the catalyst layer is related to carbon oxidation by water [START_REF] Borup | Scientific aspects of polymer electrolyte fuel cell durability and degradation[END_REF][START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF]:

C + 2H 2 O → CO 2 + 4H + + 4e -, E 0 = 0.207 V RHE (1)
Transient startup and shutdown (SUSD) operation usually shows 1.4-1.6 V voltage difference between cathode and anode, which favors the carbon corrosion to occur [START_REF] Shen | Study on the processes of start-up and shutdown in proton exchange membrane fuel cells[END_REF][START_REF] Brightman | In situ mapping of potential transients during start-up and shutdown of a polymer electrolyte membrane fuel cell[END_REF]. In addition, H2 starvation at the anode often leads to cell voltage reversal and, thus, carbon corrosion occurs at the anode [START_REF] Taniguchi | Analysis of electrocatalyst degradation in PEMFC caused by cell reversal during fuel starvation[END_REF][START_REF] Mandal | Understanding the voltage reversal behavior of automotive fuel cells[END_REF]. Based on the knowledge of these mechanisms, two carbon corrosion accelerated stress test (AST) protocols were developed and recommended by U.S. DOE and the U.S. DRIVE Fuel Cell Tech Team (FCTT). The first AST testing protocol published by U.S DOE in 2007 consists in having a cathode constant voltage hold of 1.2 V under H 2 /N 2 (anode/cathode) atmosphere [START_REF] Garland | DOE Fuel Cell Program: Durability Technical Targets and Testing Protocols[END_REF], and the second AST testing protocol published by U.S.

DRIVE FCTT in 2013 consists in cell voltage cycling between 1-1.5 V with a sweep rate of 500 mV s -1 under H 2 /N 2 (anode/cathode) atmosphere, [START_REF] Author | Fuel Cell Technical Team Roadmap[END_REF]. However, due to formation of passive oxides during AST, the corrosion rate of the former is 100-150 times slower than that of the latter [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF]. Thus, in this work, the 1-1.5 V cycling carbon corrosion AST was used.

Selection of catalyst layer materials and operation conditions are critical to extend the durability of PEFCs. For the carbon support type, despite presenting excellent Pt nanoparticles dispersion, high surface area carbon (HSAC) shows the highest carbon oxidation rate due to the preferential oxidation of the more disordered domains of the carbon support [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF][START_REF] Castanheira | Carbon corrosion in proton-exchange membrane fuel cells: Effect of the carbon structure, the degradation protocol, and the gas atmosphere[END_REF]. On the other hand, graphitic carbon (GrC) supports show more resistance to electrochemical carbon corrosion with the lowest corrosion rate since it has the greatest graphitic carbon content, which is more robust than disordered amorphous carbon [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF][START_REF] Castanheira | Carbon corrosion in proton-exchange membrane fuel cells: Effect of the carbon structure, the degradation protocol, and the gas atmosphere[END_REF][START_REF] Miyazaki | Novel graphitised carbonaceous materials for use as a highly corrosion-tolerant catalyst support in polymer electrolyte fuel cells[END_REF][START_REF] Oh | Corrosion resistance and sintering effect of carbon supports in polymer electrolyte membrane fuel cells[END_REF][START_REF] Vinod Selvaganesh | Graphitic carbon as durable cathode-catalyst support for PEFCs[END_REF]. Another factor, which impacts carbon corrosion rate is Pt content. It has been shown that Pt catalyzes carbon corrosion in the operation voltage range of 1-1.5 V, most likely via the formation of oxygen-containing surface groups (CO ad ) absorbed on the metal surface followed by their oxidation into CO 2 [START_REF] Maass | Carbon support oxidation in PEM fuel cell cathodes[END_REF][START_REF] Linse | The effect of platinum on carbon corrosion behavior in polymer electrolyte fuel cells[END_REF]:

C + H 2 O ⇌ C -O ad + 2H + + 2e - (2)
Pt + H 2 O ⇌ Pt -OH ad + H + + e -

C -O ad + Pt -OH ad ⇌ Pt + CO 2 + H + + e -

Pt/HSAC with most disordered amorphous carbon usually shows better Pt dispersion because Pt nanoparticles preferentially deposit within the amorphous domains. However, this also deteriorates carbon support corrosion due to Pt catalyzing carbon corrosion. Apart from material selection, operating conditions also play an important role in carbon corrosion in PEFC. Mittermeier et al. [START_REF] Mittermeier | PEM Fuel Cell Start-Up/Shut-Down Losses vs Relative Humidity: The Impact of Water in the Electrode Layer on Carbon Corrosion[END_REF] showed the carbon corrosion rate increased with relative humidity (RH). Overall, carbon corrosion in PEFC is still one of the key challenges, thus requiring both an optimization of the catalyst layer design and of the operation conditions to be mitigated.

Carbon corrosion causes irreversible damages to the membrane electrode assembly (MEA).

Electrochemical surface area (ECSA) is reduced due to coalescence and sintering of Pt particles [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF], as well as, due to collapse of the catalyst layer porous structure [START_REF] Star | FIB-SEM Tomography Connects Microstructure to Corrosion-Induced Performance Loss in PEMFC Cathodes[END_REF][START_REF] Hegge | Impact of Carbon Support Corrosion on Performance Losses in Polymer Electrolyte Membrane Fuel Cells[END_REF]. Despite many studies focusing on carbon support degradation, only few compared different carbon support types using cycling AST protocol adopted by DOE and it is still unclear how electrochemical performance changes during AST. Macauley et al. [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF] focused on carbon support degradation using 1.2 V constant voltage hold, and they estimated significantly different degradation rates for different carbon support types at given cell potential. Hegge et al. [START_REF] Hegge | Impact of Carbon Support Corrosion on Performance Losses in Polymer Electrolyte Membrane Fuel Cells[END_REF] studied carbon support degradation after voltage cycling from 1.0-1.5 V at 500 mV/s, and they focused on the changes of catalyst layer porosity and gas diffusivity during carbon corrosion over 1,000 AST cycles.

Saha et al. [START_REF] Saha | Correlating morphological changes to electrochemical performance during carbon corrosion in polymer electrolyte fuel cells[END_REF] conducted a detailed study on carbon corrosion of HSAC support in PEFC. It showed electrochemical changes during carbon corrosion including ECSA, Pt loading and catalyst layer porosity, etc. However, in their work only one type of carbon support was studied.

Few studies give a comprehensive electrochemical characterization comparison between different carbon support types. To better understand the effect of carbon corrosion on different carbon supports, in this work, lifetime of Pt supported on HSAC and GrC supports was examined, and a detailed electrochemical characterization comparison was performed. Analytical and electrochemical tools were used to quantify the amount of Pt dissolved, resulting loading, catalyst layer thickness, surface groups content and degree of carbon corrosion. Electrochemical measurements were conducted at the beginning of life (BOL), after 100, 500, 1,000, 2500, 5000 and 10,000 AST cycles to assess polarization behavior, ECSA, etc. along with an extensive array of physicochemical and microscopic characterization methods, aiming to provide a complete picture of the changes undergone by the different catalytic layers during the AST.

Materials and methods

Material preparation.

Catalyst inks and MEAs were custom-made by IRD Fuel Cells (Albuquerque, NM, United

States) for the purpose of this study, and the ink formulation and preparation are in accordance with best industrial practices. For all membrane electrode assemblies (MEAs), PFSA membranes with thickness of 18 µm were used. The anodes for all MEAs were loaded with 0.15 mg cm -2 of Pt while the cathodes had Pt loading of around 0.53 mg cm -2 and 0.648 mg cm -2 for Pt/HSAC and Pt/GrC, respectively. Pt loading discrepancy between these two samples is due to difficulty of achieving the same Pt loadings during the manufacturing process given the supports are different.

In this work, Pt/HSAC and GrC supports were used on cathode sides as different carbon supports and a detailed electrochemical characterizations comparison was conducted. Two MEAs with each carbon support were investigated to ensure the reproducibility of the study.

Cell assembly.

A 5 cm x 5 cm Freudenberg H23C6 with micro porous layer (MPL) was used as the gas diffusion layer (GDL). A 50 cm 2 14 channel serpentine flow-field with channel and land width of 0.5 mm x 0.5 mm was used as flow-field. Cell active area was controlled as 25 cm 2 and a 5.5 cm x 5.5 cm Teflon fiberglass sheet layer of 180 μm thickness on both sides between the MEA and the bipolar plate as hard-stop gasket was used to achieve a 20% compression of the GDL.

In Situ Electrochemical Diagnostics.

All the experiments were done using a Biologic VSP potentiostat and a Scribner 850e fuel cell hardware. Cells first were activated by conducting voltage cycling break-in followed by voltage recovery process to hydrate the catalyst layer and to remove the contaminants.

Electrochemical characterization was performed after the activation process, as described below.

Cell voltage break-in and recovery.

A cell break-in process was performed by using 0.8/1.5 standard liter per minute (SLPM) ultrahigh purity (UHP) H2 and compressed air at anode and cathode, respectively at 80 °C and 100%

RH with an absolute backpressure of 150 kPa. A series of constant voltage holds with 200 repeated cycles were performed by applying constant voltages to the cells at 0.8 V, 0.5 V and 0.2 V for 30 seconds, respectively.

During voltage recovery the cell was held at 0.1 V for 7200 seconds using 0.45/0.25 SLPM UHP H2 and air at the anode and cathode, respectively at 40 °C and 150% RH with an absolute backpressure of 150 kPa. Previous study showed recovery process was an effective way to remove the surface contaminants and increased the electrochemical surface area (ECSA) [START_REF] Kabir | Elucidating the Dynamic Nature of Fuel Cell Electrodes as a Function of Conditioning: An ex Situ Material Characterization and in Situ Electrochemical Diagnostic Study[END_REF].

Carbon corrosion accelerated stress test.

Carbon corrosion ASTs for PEMFC components were performed using the protocol published by U.S. DRIVE FCTT in 2013 [START_REF] Author | Fuel Cell Technical Team Roadmap[END_REF]. Briefly, the potential was cycled between 1 and 1.5 V at 500 mV s -1 with a triangular wave pattern. Cell temperature was 80 °C under fully humidified H2/N2 flow.

𝐻 2 /𝐴𝑖𝑟 polarization curves.

The polarization curve measurements were performed in galvanostatic mode, on the anodic direction from 2 to 0 A cm -2 (2, 1.8, 1.6, 1.4, 1.2, 1.0, 0.8, 0.6, 0.4, 0.2, 0.1, 0.08, 0.04, 0.02, 0 A cm -2 ). at 80 °C, 100% RH with an absolute backpressure of 150 kPa. Each step of the polarization curve was held for 180 s. The polarization curve measurements were conducted in anodic direction with the reactant gas stoichiometry factors of 2/2 at the anode/cathode, respectively. During the polarization curve measurements, high frequency resistance (HFR) was measured under the same operating condition using a built-in function of the potentiostat, which performs a single point impedance calculation at 1 kHz.

Cyclic voltammetry.

The cell was humidified by purging 0.5/0.5 SLPM pure N2 without back pressure at 60 °C and 100% RH for 1 hour. A cathode surface cleaning step was performed by conducting a repeated cyclic voltammetry (CV) under 0.2/0.3 SLPM H2/N2 with a voltage range of 0.05 V to 1.05 V at 200 mV s -1 scan rate for 30 cycles. Subsequently, CV measurements with 100 mV s -1 and the same voltage range and same gas flow rates were conducted.

CO displacement and CO stripping methods.

Prior to CO displacement, a surface cleaning CV step of 20 cycles using 1/1 SLPM of 5% H2 on anode and pure N2 on cathode was performed. After that, the cell was held at 0.4 V for 7 min without any changes of gas composition and gas flow rates. This was first followed by a 2 min N2 purge at the cathode, after what the gas was instantly switched to 2% CO. The cell was then held under the same constant voltage with 2% CO gas purging continuing for 5 min to establish another constant current baseline. Because of the residual air gas within the system, the first CO displacement measurement was contaminated by the ORR. Hence the result of the first measurement was discarded. After that, CO displacement experiments were conducted from 0.1 V to 0.4 V in increments of 0.1 V. After CO displacement, 1 SLPM pure N2 was purged at cathode side for 20 min to remove the residual CO gas in the cell. Subsequently, a CV under 1/1 SLPM 5% H2 and N2 with a potential sweep range from 0.05 V to 1.05 V at 40 mV s -1 scan rate was performed and repeated 5 times.

Electrochemical impedance spectroscopy (EIS) method.

The cell was held at 0.2 V for 5 min using 0.2/0.3 SLPM of pure H2 on anode and pure N2 on cathode without back pressure at 60 °C and 100% RH. Subsequently, a 10 mV AC potential was applied to the cell over a frequency range from 500 kHz to 100 mHz with 8 points per decade recording rate. EIS measurements were repeated twice to provide reproducibility.

For Pt/HSAC, the aforementioned electrochemical characterization was performed for beginning of life (BOL), after 100 AST cycles, after 500 AST cycles and after 1,000 AST cycles, which was considered as the end of life (EOL) for the cell. For Pt/GrC, due to lower carbon corrosion rate [START_REF] Maass | Carbon support oxidation in PEM fuel cell cathodes[END_REF], more AST cycles were performed. Electrochemical characterization was performed for beginning of life (BOL), after 100, 500, 1,000, 2500, 5000 and 10,000 AST cycles, which was considered as the EOL of the cell.

2.3. Ex Situ Electrochemical Diagnostics.

Scanning electron microscope (SEM) sample preparation and imaging.

Cross-sectional MEA samples using HSAC and GrC supports were embedded in epoxy resin, polished and imaged with a FEI Magellan 400 scanning electron microscope detector operating at 3.00 kV accelerating voltage. SEM measurements were repeated for BOL and EOL for each carbon support type, and catalyst layer thicknesses were measured from the SEM micrographs.

Scanning transmission electron microscopy (STEM).

At BOL, a JEOL Grand ARM300F microscope, equipped with two spherical aberration correctors, was used to perform the aberration-corrected scanning transmission electron microscopy (AC-STEM) characterization for both Pt/HSAC and Pt/GrC MEAs. The TEM imaging was performed at 300 kV.

Transmission electron microscopy (TEM).

After AST, the catalyst layers were scraped from the MEA to make powders. The powders were then dispersed in ethanol by sonication. A few drops of the suspension were deposited onto holey carbon-coated copper grids. The images were obtained using a JEOL JEM-2800 microscope at 200 kV.

Particle size analysis.

The TEM images were processed with Fiji ImageJ [START_REF] Schindelin | Fiji: An open-source platform for biological-image analysis[END_REF] using thresholding and the ROI manager. The overlapped particles were removed. The particles were converted into solid ellipses after the processing. The measure function of Fiji Image J provides the length of the major and minor axis of the ellipses. The particle size was calculated by taking the square root of the product of the length of major and minor axis. TEM is a good way to show the morphological change of the particle. However, TEM only provides the particle size distribution in a small field of view (usually less than 100 nm × 100 nm). Also, the particle size might be overestimated if the overlapping particles were not removed and has a large uncertainty depending on how the thresholding was done. To decrease the uncertainty, ROI manager was used to remove the overlapping particles and thresholding value was carefully chosen based on the grayscale histogram of each image.

Identical location μ-X-ray Fluorescence (XRF).

To ensure identical locations were mapped before and after the AST, a 25-μm PTFE template was prepared, and, using the MEA as a reference, a hole was cut out in the template near the center of the active area. The diameter of the hole was the same as the diagonal of a 2048 μm by 2048 μm area. Ex situ μ-XRF measurements were performed using the XGT-9000 Horiba XRF microscope. A focus distance of 1 mm and a 10 µm capillary with 1024 pixels in both x and y directions were used to map an area of 2048 μm by 2048 μm, resulting in a resolution of 2 µm × 2 µm. The x-ray energy was 50 keV, and the reflected x-ray signal to the detector included Pt contribution from both the anode and the cathode as the x-ray penetrates through the entire MEA. Therefore, the total average Pt loading is the sum of anode and cathode loadings, and it was quantified using a calibration curve generated from 10 µm capillary maps of samples with known Pt loading. 2D colormaps were generated in MATLAB using the quantified total average Pt loading and raw fluorescence images from the microscope.

X-ray photoelectron spectroscopy (XPS).

The GDL of the MEA was removed. The sample was mounted on the sample loader and the cathode side was facing up. The XPS measurement was conducted on the cathode side of the MEA. XPS was performed on a Kratos AXIS Supra spectrometer with a monochromatic Al Kα source. CasaXPS software was used to analyze the data. The fitting was based on our previous calculations and publications [START_REF] Artyushkova | Structural correlations: Design levers for performance and durability of catalyst layers[END_REF][START_REF] Patel | Investigating the effects of proton exchange membrane fuel cell conditions on carbon supported platinum electrocatalyst composition and performance[END_REF]. For the C 1s, a linear substation was used. The graphitic C spectra were fitted with an asymmetric modified 20% Gaussian/80% Lorentzian line shape, and other spectra were fitted with a 70% Gaussian/30% Lorentzian line shape. For the Pt 4f, a Shirley substation was used, and all spectra were fitted with an asymmetric modified 70% Gaussian/30% Lorentzian line shape.

Raman spectroscopy.

All Raman spectra were obtained using a Horiba LabRAM-HR Evolution Confocal Raman Microscope at 633 nm laser with a 100x objective and a 600 g/mm grating. Twenty individual 30s spectra were summed.

X-ray diffraction (XRD).

XRD patterns were collected on a Rigaku Ultima III X-Ray Diffractometer with a Cu Kα Xray generator. PDXL software was used to analyze the data. Measurement was conducted on the MEA with cathode facing up. The crystalline sizes of the samples were calculated by applying whole powder pattern fitting method after the phase identification.

2.4. Calculation methods.

Double layer capacity.

Double layer capacity (C dl ) is considered as the capacity summation of the four different interfaces within the catalyst layer: (i) Pt/ionomer, (ii) Pt/water, (iii) carbon/ionomer and (iv) carbon/water [START_REF] Iden | An in situ technique for analyzing ionomer coverage in catalyst layers[END_REF]. Investigating C dl change during the carbon corrosion AST provides more insights on interfacial properties change between solid/ionic phase materials. In this work, C dl was calculated using the following equation:

I = C dl × dV dt (5)
where I denotes the upper/lower current in the double layer region (0.4-0.5 V) from CV measurements.

dV dt represents the voltage sweep rate of CV measurement, here 100 mV s -1 .

Effective ionic conductivity.

Effective ionic conductivity was derived by using a H2/N2 EIS fitting model based on Qi et al.'s work [START_REF] Qi | Interpreting Ionic Conductivity for Polymer Electrolyte Fuel Cell Catalyst Layers with Electrochemical Impedance Spectroscopy and Transmission Line Modeling[END_REF]. The model is based on Obermaier et al.'s model [START_REF] Obermaier | A Comprehensive Physical Impedance Model of Polymer Electrolyte Fuel Cell Cathodes in Oxygen-free Atmosphere[END_REF], which modeled the equivalent circuit of catalyst layer based on transmission line model. In this work, catalyst layer ionic conductivity was derived by inputting the frequency, cell active area, catalyst layer thickness, imaginary impedance and real impedance etc. into the fitting model.

CO displacement charge.

During CO displacement technique, CO bonds onto Pt, leading to the displacement of adsorbed ion species at the Pt surface, for example H + and SO 3 -groups, which results in charge transfer and current flow. According to Equation (6), negative current signal was recorded when constant cell potential, which was greater than the potential of zero charge (PZC) of Pt was applied. CO displacement charge was the integration of CO displacement peak in the current-time plot.

Pt -An + CO + e -→ Pt -CO + An -

where Pt-An is anion adsorbed on Pt surface. Here at 0.4 V SO 3 -group of ionomer is absorbed anion species, which follows conventions of Qi et al. [START_REF] Qi | Understanding Platinum Ionomer Interface Properties of Polymer Electrolyte Fuel Cells[END_REF] and Garrick et al. [START_REF] Garrick | Characterizing Electrolyte and Platinum Interface in PEM Fuel Cells Using CO Displacement[END_REF].

CO stripping charge.

CO stripping charge was calculated by integrating the CO stripping peak from CO stripping measurements then divided by the scan rate of 40 mV s -1 . The CO stripping mechanism is a multiple steps process that includes, for example, the formations of OH ads and COOH ads , which followed Langmuir-Hinshelwood mechanism [START_REF] Maillard | Size effects on reactivity of Pt nanoparticles in CO monolayer oxidation: The role of surface mobility[END_REF]:

H 2 O + * ↔ OH ads + H + + e - (7)
CO ads + OH ads → COOH ads [START_REF] Borup | FC135 : FC-PAD : Fuel Cell Performance and Durability Consortium[END_REF] COOH ads → CO 2 + H + + e -+ * [START_REF] Borup | Scientific aspects of polymer electrolyte fuel cell durability and degradation[END_REF] where * denotes the free surface site.

Electrochemical surface area.

Electrochemical surface area (ECSA) was calculated from the H2 under potential deposition (H upd ) region of CV:

ECSA H upd (m 2 g -1 ) = Q H upd (C) 210 μC cm 2 × Area (cm 2 ) × L pt ( mg cm 2 )
× 10 5 [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF] where Q H upd (C) is charge integration over the H upd region and area (cm 2 ) denotes the cell active area. L pt denotes the Pt loading (mg cm -2 ). 210 μC cm 2 is the charges per H upd monolayer [START_REF] Binninger | Determination of the Electrochemically Active Surface Area of Metal-Oxide Supported Platinum Catalyst[END_REF].

2.4.6. 𝑆𝑂 3 --group coverage.

SO 3 -group coverage was calculated using the following equation [START_REF] Garrick | Characterizing Electrolyte and Platinum Interface in PEM Fuel Cells Using CO Displacement[END_REF]:

θ SO 3 -= 2 × q dis q strip × 100% ( 11
)
where q dis is the CO displacement charge, q strip denotes the CO stripping charge. There is a multiplier 2 in the numerator since CO requires two electrons to oxidize CO to CO2. The coverage is an averaged and the SO 3 -group coverage corresponds to an average ratio of Pt atoms that are covered by SO 3 -groups over all electrochemically active Pt atoms.

Results and discussion

Polarization curves, voltage performance and HFRs.

Fig. 1a shows the polarization curves of MEA with Pt/HSAC support over the duration of 1,000 carbon corrosion AST cycles. After 100 AST cycles, Pt/HSAC showed a slight loss in performance compared to the BOL, with a voltage loss of 54 mV at 1 A cm -2 from 0.644 V to 0.59 V (Fig. 1b).

From HFR results shown in Fig. 1a, after 100 AST cycles, the HFR of the cell decreased from 56.4 mohms cm 2 to 53.9 mohms cm 2 due to the better contacts between GDL/catalyst layer and catalyst layer hydration. After 500 AST cycles, a significant performance loss was shown in Fig. 1a. However, a similar HFR of the cell of 55.9 mohms cm 2 was measured after 500 AST cycles (Fig. 1a), which indicates that the performance drop might be due to loss of the ECSA or due to the collapse of the catalyst layer porous structure leading to a mass transport problem, but not due to intrinsic losses in the ionic/electronic conductivity. At the EOL, the performance loss further deteriorated to 0.247 V at 0.2 A cm -2 (Fig. 1a). In addition, the HFR increased to 73.3 mohms cm 2 (30% increase compared to BOL), this being likely induced by contact loss between catalyst layer and the GDL, which indicates that at the EOL, the cell had both ohmic and mass transport issues leading to significant loss of performance.

Cell performance drop for Pt/GrC was much more moderate since graphitized carbon support is more robust. From the results shown in Fig. 1c, cell performance deteriorated 14 mV after 1,000

AST cycles and voltage at 1 A cm -2 was 0.656 V (Fig. 1d). HFR shown in Fig. 1c after 1,000 AST cycles decreased from 52.8 mohms cm 2 to 50.8 mohms cm 2 compared to BOL due to the better contacts at GDL/catalyst layer interfaces and further hydration of the catalyst layer, which is similar to what was observed for Pt/HSAC. Cell performance maintained after 2500 AST cycles and a 1 mV voltage decrease was found. After 5000 AST cycles, the cell degraded and a voltage loss of 63 mV at 1 A cm -2 was observed compared to BOL (Fig. 1c and Fig. 1d). Apart from that, Pt/GrC MEA experienced voltage loss of 69 mV at 0.8 A cm -2 and 129 mV at 1.5 A cm -2 , which did not meet the DOE target of 40 mV loss at 0.8 A cm -2 and 40 mV loss at 1.5 A cm -2 [START_REF] Cells | Multi-Year Research, Development, and Demonstration Plan[END_REF]. The performance decreased significantly at the EOL shown in Fig. 1c, which was after 10,000 AST cycles. Cell voltage decreased to 0.561 V at 0.2 A cm -2 at the EOL due to the loss of ECSA and porous structure collapse of catalyst layer. However, in Fig. 1c, HFR remained constant during the AST, which indicates the contacts between catalyst layers and GDLs remains mostly intact. For Pt/GrC, (c) MEA performance at 80 ℃, 100% RH, 150 kPa in H2/Air with stoichiometric anode and cathode flows of 2 and 2 and HFR at BOL, after 100, 500, 1,000, 2500, 5000, and 10,000 AST cycles (EOL), (d) voltage at 1

A cm -2 .

Cathode catalyst layer thicknesses.

Averaged cathode catalyst layer thicknesses at BOL and EOL for Pt/HSAC and Pt/GrC are shown in Fig. 2c and Fig. 2f, respectively according to the cross-section SEM measurement results for Pt/HSAC and Pt/GrC at BOL and EOL (Fig. 2a, Fig. 2b, Fig. 2d and Fig. 2e). Cathode catalyst layer thickness of Pt/HSAC decreased from 11.4 μm at BOL (Fig. 2a) to 3.7 μm after 1,000 AST cycles at EOL (Fig. 2b). For Pt/GrC, thickness of cathode catalyst layer decreased from 18.7 μm at BOL (Fig. 2d) to 5.91 μm after 10,000 AST cycles at EOL (Fig. 2e). Thickness reduction for Pt/HSAC due to the carbon support oxidation also explained the increase of HFR (Fig. 1a). At EOL, HFR for Pt/HSAC increased 30% due to the reduction of catalyst layer thickness, which resulted in insufficient contacts between GDL and catalyst layer. However, for Pt/GrC, even though catalyst layer thickness decreased to 5.91 μm at EOL, it still provided sufficient contacts at the catalyst layer/GDL interface, which resulted in a maintained HFR (Fig. 1c).

Cathode double layer capacities, CV measurements and ECSAs.

Catalyst layer and interfacial properties were measured for MEAs with both HSAC and GrC supports during AST. For Pt/HSAC, C dl was measured using the Equation ( 5) and shown in Fig. 3a. According to Iden and Ohma's work [START_REF] Iden | An in situ technique for analyzing ionomer coverage in catalyst layers[END_REF], C dl was determined as the summation of four interfaces within the catalyst layer, which are Pt/ionomer, Pt/water, carbon/ionomer, and carbon/water. At BOL, Pt/HSAC shows a much higher C dl of 119.6 mF cm -2 (Fig. 3a) compared to the 37.8 mF cm -2 of Pt/GrC (Fig. 3c) because of the greater surface area of carbon black support due to the internal porous structure (amorphous domain) [START_REF] Garrick | Characterizing Electrolyte and Platinum Interface in PEM Fuel Cells Using CO Displacement[END_REF]. C dl for Pt/HSAC increased to 152.2 mF cm -2 after 100 AST cycles and further increased to 158.3 mF cm -2 after 500 AST cycles because the carbon corrosion led to a roughening of the carbon surface, and thus a higher surface area was exposed to ionomer and water. At EOL, C dl decreased to 128.6 mF cm -2 , indicating, a collapse of the porous network and thus, a decrease accessibility of the carbon support to ionomer/water. A total capacity increase of 9 mF cm -2 (7.5%) at EOL was shown in Fig. 3a. For Pt/GrC, C dl shown in Fig. 3c remained at around 37.4 mF cm -2 after 100 AST cycles, followed by a gradual increase with the AST cycling number. Eventually a significant increase of 26.8 mF cm -2 (69.8%) at EOL was observed, reaching 64.2 mF cm -2 . The increase of C dl for Pt/GrC was much greater than that of Pt/HSAC, as the oxidation of the ordered graphitic carbon logically resulted in the creation of a more amorphous, porous and disorganized carbon surface, leading to a significant C dl increase compared to Pt/HSAC. Fig. S1 show the CV measurements for Pt/HSAC and Pt/GrC during AST, respectively. H upd peaks decreased as a function of cycle number, as expected for both HSAC and GrC supports. The decrease of H upd peak for Pt/GrC was much smaller compared to Pt/HSAC because Pt/HSAC had higher carbon oxidation rate at amorphous domains where Pt particles were well dispersed and catalyzed carbon corrosion reaction. ECSA results calculated from H upd peaks for both carbon support types are shown in Fig. 3b and Fig. 3d. Reductions of ECSAs show consistency with the decrease of H upd peaks. Pt/HSAC, at the BOL showed higher ECSA value of 43.5 m 2 g -1 (Fig. 3b) compared to 26.2 m 2 g -1 for Pt/GrC (Fig. 3d) because of better Pt dispersion within the internal porous carbon structure. ECSA for Pt/HSAC (Fig. 3b) decreased most rapidly at first 100 AST cycles from 43.5 m 2 g -1 to 36.2 m 2 g -1 , then slightly decreased to 24.7 m 2 g -1 (41.8%) after 500 AST cycles. Eventually, ECSA decreased to 20.7 m 2 g -1 and a total ECSA loss of 22.8 m 2 g -1 (52.4%) was observed for Pt/HSAC at the EOL. For Pt/GrC (Fig. 3d), reduction of the ECSA was much more moderate due to protective carbon support. ECSA remained constant after the first 100 AST cycles and was reduced by 4.23 m 2 g -1 (16.1%) after the first 1,000 AST cycles, then followed by a decrease of 8.03 m 2 g -1 (30.6%) after 5,000 AST cycles, which met DOE target of a 40% loss of initial surface area after 5,000 AST cycles [START_REF] Cells | Multi-Year Research, Development, and Demonstration Plan[END_REF]. Eventually, ECSA for Pt/GrC decreased by 44% and reached 14.7 m 2 g -1 after 10,000 AST cycles (EOL). ECSA was also calculated and summarized using CO stripping method for both Pt/HSAC and Pt/GrC after certain AST cycles in Fig. S2, and it showed good consistency compared to using H upd charge integration.

At BOL shown in Fig. S2a, it shows 45.6 m 2 g -1 ECSA for Pt/HSAC and 28.4 m 2 g -1 for Pt/GrC respectively. For Pt/HSAC at EOL, Fig. S2a shows the calculated ECSA of 17.7 m 2 g -1 using CO stripping method, which is close to 20.7 m 2 g -1 using H upd charge integration. For Pt/GrC, after 1,000 AST cycles, ECSA was 25.8 m 2 g -1 using CO stripping method, compared to 22 m 2 g -1 using H upd charge integration (Fig. S2b). After 10,000 AST cycles, it shows 11.1 m 2 g -1 ECSA using CO stripping method, compared to 14.7 m 2 g -1 using H upd charge integration. Possible reasons for ECSA reduction might be: i) Pt nanoparticles detachment due to carbon corrosion, ii)

Pt dissolution during the AST as Pt 2+ , iii) Pt nanoparticles growth due to 2D/3D Ostwald ripening and coalescence, and iv) burying of the Pt nanoparticles during the catalyst layer structural collapse and the resulting loss in H + /O2 accessibility. Reduction of the ECSA also partially explains the performance decrease shown in Fig. 1a and Fig. 1c for Pt/HSAC and Pt/GrC, respectively.

Cumulative average Pt loading maps.

Local averaged Pt loading distributions at the BOL and EOL for Pt/HSAC and Pt/GrC are shown in Fig. 4c and Fig. 4f, respectively according to identical location micro XRF maps for BOL (Fig. 4a and Fig. 4d) and EOL (Fig. 4b and Fig. 4e). Again, in this work, the X-rays penetrate both the anode and cathode catalyst layer and therefore the cumulative average Pt loading of both anode and cathode were shown. Fig. 4c shows that average Pt loading of Pt/HSAC reached 0.675 mg cm -2 at the BOL (Fig. 4a) and decreased to 0.577 mg cm -2 at the EOL (Fig. 4b). For Pt/GrC, averaged Pt loading shown in Fig. 4f decreased from 0.798 mg cm -2 at BOL (Fig. 4d) to 0.732 mg cm -2 at EOL (Fig. 4e). Pt loss for Pt/GrC after 10,000 AST cycles was similar compared to that loss for Pt/HSAC after 1,000 AST cycles, which again indicating graphitized carbon support is much more robust and protective. For both Pt/HSAC and Pt/GrC, cumulative averaged Pt loadings decreased at their EOLs, which indicates that Pt loss might be one of the reasons causing the reduction of ECSA.

Cathode Pt particle sizes.

Cathode side Pt particle sizes for Pt/HSAC and Pt/GrC at BOL and after certain AST cycles were determined by TEM/STEM and XRD (Fig. S3-S4) and summarized in Fig. 5. The Pt particle sizes increased after AST in both samples, which is due to the Pt particle coalescence and sintering. Based on TEM/STEM results shown in Fig. 5a, the average Pt sizes of Pt/HSAC increased from 2.25 nm at BOL to 3.96 nm after 1,000 AST at EOL. The average Pt sizes of Pt/GrC raised from 2.25 nm at BOL to 8.27 nm after 10,000 AST cycles at EOL. The TEM/STEM data also provide particle size distributions and reveal the morphological changes of the Pt particles (Fig. S3). After certain AST cycles, the Pt particles in the Pt/GrC at EOL demonstrated a dumbbell-like shape. As comparison, the particle shapes in Pt/HSAC at EOL are closer to spheres, which are more similar to the Pt in pristine samples. XRD measurement results (Fig. S4, Table . S1 and Table. S2) were summarized in Fig. 5b, and it shows Pt particle size increased from 2.5 nm to 4.2 nm (68%) after 1,000 AST cycles at EOL for Pt/HSAC, while for Pt/GrC Pt particle size grew from 3.4 nm to 5.4 nm (59%) after 10,000 AST cycles at its EOL.

Overall, the average particle sizes gained by whole powder pattern fitting based on XRD results are in good alignment with the ones gained from TEM/STEM (Fig. 5d). However, the difference between the sizes gained different measurements were larger when the Pt sizes are bigger. This is because of the difference between these measurements. EM is detecting the size of Pt particles individually based on the morphology images of a small field of view. The average size is calculated with an assumption that the particles were in ellipse shape. XRD is measuring the diffraction of the whole samples and provides the crystalline sizes. In our case, the diffraction pattern of the Pt might be affected by the amorphous carbon inside the sample [START_REF] Adams | Particle size determination of supported catalytic metals: Platinum on silica gel[END_REF].

Also, in the Pt/GrC at EOL, there is a high probability that most of the crystallites aggregated and thus the XRD underestimated the results.

For Pt/GrC, after 1,000 AST cycles, ECSA reduced 16.1% with an increase of Pt particle size of 44% (Fig. 5c). Comparing to Pt/HSAC, whose ECSA reduced 52.4% with a 68% increase of Pt particle size (Fig. 5b). More than 2 times higher ECSA loss (Fig. 2) but with only 50% greater particle size growth and similar Pt loss from XRF measurements (Fig. 7) for Pt/HSAC strongly suggested that a greater Pt detachment occurred on HSAC support during the AST cycles.

During carbon corrosion, the growth of the detached nanoparticles being thus stopped. In addition, the overall better dispersion of the Pt/HSAC catalyst at BOL limited the coalescence of the Pt/HSAC vs. the Pt/GrC, where the Pt nanoparticles were in closer proximity at BOL. Furthermore, it is believed that the mainly graphitic nature of the GrC support resulted in a

Cathode surface chemistry.

In this work, high-resolution XPS spectra was performed to study the cathode surface chemistry for both Pt/HSAC and Pt/GrC at BOL and EOL. Fig. S7 shows the relative percentages of different elements, surface carbonaceous moieties and surface Pt species before and after the AST. The curved fittings of high-resolution C 1s spectra and the summarized quantitative results are shown in the Fig. S5 and Table . 1. The curved fittings for Pt 4f spectra and the summarized quantitative results are shown in Fig. S6 and Table . 2.

The percentages of the sp 2 graphitic carbon peak for Pt/GrC and Pt/HSAC were different. For Pt/HSAC (Fig. S7b) after 1,000 AST cycles, no obvious change in the graphitic carbon concentration was observed (from 46.6% at BOL to 46.3% at EOL). However, for Pt/GrC (Fig. S7a) at EOL, concentration of graphitic carbon dropped from 56.3% to 46.1% on the surface after 10,000 AST cycles. The sp 3 carbon concentration corresponding to C-C increased for Pt/GrC after AST was accompanying with the decrease of the sp 2 graphitic carbon concentration might because of the degradation of graphitic carbon (probably happen at the edge). The changes in the total amount of carbon-oxygen groups (including C-O, C=O and COOH) can express the extent of the carbon corrosion. For Pt/GrC, the carbon-oxygen groups amount did not change much for the entire AST process. For Pt/HSAC, the total amount of the carbon-oxygen group increased from 18.4% to 24.1% at EOL (1,000 AST cycle). This indicates that the carbon support of the Pt/HSAC corroded after 1,000 AST cycles.

The ratio of CF2 and CF can reveal the ionomer's morphology. For Pt/GrC, the ratio of CF2/CF changed from 0.26 to 0.81 after 1,000 AST cycles, and then to 0.69 at EOL after 10,000 AST cycles. For Pt/HSAC, ratio of CF2/CF changed from 0.15 to 0.83 after 1,000 AST cycles.

This indicates that the ionomer rearranged in both cases. The C-F concentration reduction usually correlates to the ionomer degradation. Here for Pt/HSAC, relative amount of C-F dropped from 20.3% to 11.0% after 1,000 AST cycles. However for the Pt/GrC, relative amount of C-F decreased from the 16.3% to 11.3% after 1,000 AST cycles, and then increased to 18.8% at EOL after 10,000 AST cycles due to the exposure of the hydrophilic C-F side chains of the ionomers. The Pt environment of the samples had been changed after the AST, as shown in Fig. S7c-d and Table . 2. For both samples, the relative concentration of metallic Pt increased while Pt-C decreased when compared the samples at BOL and EOL. This phenomenon is consistent with the Pt size grew observed in the TEM and XRD. Pt species relative concentrations were similar when comparing Pt/GrC (Fig. S7c) to Pt/HSAC (Fig. S7d) at their EOLs (1,000 AST cycles for Pt/HSAC and 10,000 AST cycles for Pt/GrC). However, when comparing both samples after the 1,000 AST cycles, the relative concentrations of Pt species of Pt/GrC did not change as severely as the one of Pt/HSAC. This result also shows the consistency to the slight ECSA reduction for Pt/GrC after 1,000 AST cycles. Raman measurement was performed on cathode sides of both Pt/HSAC and Pt/GrC MEAs.

The results for the range of 1,000 to 1800 cm -1 are shown in Fig. S8. The D band (~1330 cm -1 ) is related to the disordered carbon of the sample, while the G band (~1580 cm -1 ) is related to the graphitic carbon. An additional D' band (~1614 cm -1 ) appears when the disordered graphitic lattice exists [START_REF] Sadezky | Raman microspectroscopy of soot and related carbonaceous materials: Spectral analysis and structural information[END_REF][START_REF] Ferrari | Raman spectroscopy of graphene and graphite: Disorder, electron-phonon coupling, doping and nonadiabatic effects[END_REF].The ratio of the intensities of the D band to the G band (D/G ratio) is shown in Table . 3. For the pristine samples, the D/G ratio of the Pt/HSAC (Fig. S8a) sample was higher, meaning that the carbon in Pt/HSAC sample was more disordered than the Pt/GrC sample. The G band of the Pt/HSAC sample was shifted slightly since its graphitic carbon was highly disordered.

Inversely, the D/G ratio of the Pt/GrC sample (Fig. S8b) was less than 1, meaning that the sample contained more graphitic carbon than the disordered one. Additionally, a clear D' band is shown in the spectra of the Pt/GrC sample since its G band was sharp. This sharp G band indicated that the Pt/GrC sample contained the graphitic carbon with higher degree of crystallinity [START_REF] Sahoo | Platinum-decorated chemically modified reduced graphene oxide-multiwalled carbon nanotube sandwich composite as cathode catalyst for a proton exchange membrane fuel cell[END_REF]. After 1,000 AST cycles, the D/G ratio of Pt/HSAC (Fig. S8a) increased to 1.256, meaning that more disordered carbon appeared after the AST. Thus, the amorphous degree of the Pt/HSAC was getting higher. The D/G ratio of Pt/GrC after 1,000 AST cycles (Fig. S8b) reached to 1.156 with the sharp D' peak existing. When the Pt/GrC hits the EOL (10,000 AST cycles), the D/G ratio increased to 1.372. The G peak shifted to lower Raman shift and the dispersion increased. This is the sign of nanocrystalline graphite transformed to sp 2 amorphous carbon [START_REF] Ferrari | Raman spectroscopy of graphene and graphite: Disorder, electron-phonon coupling, doping and nonadiabatic effects[END_REF]. This phenomenon supports the conclusion gained from the XPS results, where the concentration of the graphitic carbon of the Pt/GrC decreased at its EOL. Considered the sp 3 carbon detected on the surface is way less than 30%, the carbon should be mainly at sp 2 amorphous status. Cathode catalyst layer ionic conductivities were measured and shown in Fig. 6a by using a H2/N2 EIS fitting model according to Qi et al. [START_REF] Qi | Interpreting Ionic Conductivity for Polymer Electrolyte Fuel Cell Catalyst Layers with Electrochemical Impedance Spectroscopy and Transmission Line Modeling[END_REF] for both Pt/HSAC and Pt/GrC at their BOLs and EOLs. In that work, cell active area, frequency, real part of impedance, imaginary part of impedance and catalyst layer thickness were input into the developed H2/N2 EIS impedance fitting model and effective ionic conductivity was derived. For Pt/HSAC, the impedance Nyquist plots at BOL and EOL are shown in Fig. S9a and Fig. S9b, respectively, and an example impedance fitting at BOL and EOL using the 1D H2/N2 EIS fitting model are shown in Fig. S10a and Fig. S10b, respectively. According to the fitting results, cathode catalyst layer ionic conductivity decreased from 4.2x10 -2 S cm -1 at BOL to 1.97x10 -2 S cm -1 (53.1%) after 1,000 AST cycles at EOL. For Pt/GrC, the impedance Nyquist plots at BOL and EOL are shown in Fig. S9c and Fig. S9d, respectively, and an example impedance fitting at BOL and EOL using the 1D H2/N2 EIS fitting model are shown in Fig. S10c and Fig. S10d, respectively. Ionic conductivity of Pt/GrC cathode catalyst layer was 5.2x10 -2 S cm -1 at BOL, which was higher compared to Pt/HSAC, since the graphitic carbon supports have better ionomer continuity and uniformity than disordered HSAC supports [START_REF] Ramaswamy | Carbon Support Microstructure Impact on High Current Density Transport Resistances in PEMFC Cathode[END_REF]. At EOL, catalyst ionic conductivity increased to 1.88x10 -2 S cm -1 . The reduction of ionic conductivity for both Pt/HSAC and Pt/GrC was due to the carbon corrosion leading to the further increase of discontinuity of ionomer and water distributions within the catalyst layer, causing higher tortuosity of H + transport. Apart from that, Pt loss during the carbon corrosion resulted in worse continuity of ionic pathways with catalyst layer, leading to more tortuous H + transport pathways.

Cathode SO 3 -group coverages at 0.4 V for both Pt/HSAC and Pt/GrC were calculated and shown in Fig. 6b using CO displacement (Fig. S11a for Pt/HSAC and Fig. S11b for Pt/GrC) and CO stripping (Fig. S11c and Fig. S11d for Pt/HSAC and Fig. S11e and Fig. S11f AST cycles, which again indicates that Pt/GrC was much more robust compared to Pt/HSAC. Fig. 7c shows that cumulative average Pt loading of Pt/HSAC reached 0.675 mg cm -2 at the BOL (Fig. 7a) and decreased to 0.577 mg cm -2 after 1,000 AST cycles (Fig. 7b). For Pt/GrC, cumulative averaged Pt loading shown in Fig. 7f decreased from 0.750 mg cm -2 at BOL (Fig. 7d) to 0.628 mg cm -2 after 1,000 AST cycles (Fig. 7e). Again, more than 2 times higher ECSA loss (Fig. 8b) but with only 50% greater particle size growth and similar Pt loss from XRF measurements (Fig. 7) for Pt/HSAC strongly suggested that a greater Pt detachment occurred on HSAC support during the AST cycles. Pt/HSAC, catalyst layer ionic conductivity decreased by 53.1% after 1,000 AST cycles at EOL.

However, for Pt/GrC, the decrease of catalyst layer ionic conductivity was around 2.5 times smaller compared to Pt/HSAC, which was 21.9% after 1,000 AST cycles. According to the comparisons of CO displacement and CO stripping measurements shown in Fig. S11(a,c,d) and Fig. S13e-f, cathode SO 3 -group coverage comparison at 0.4 V for both Pt/HSAC and Pt/GrC at BOL and after 1,000 AST cycles were calculated and shown in Fig. 8e). SO 3 -group coverage decreased by 50.9% for Pt/HSAC, compared to 11.8% for Pt/HSAC, which was around 4 times smaller.

This comprehensive work again proves that MEAs containing low surface area carbon support are more robust. Macauley et al. [START_REF] Macauley | Carbon Corrosion in PEM Fuel Cells and the Development of Accelerated Stress Tests[END_REF] also showed that graphitized carbon support had 9

times lower carbon corrosion rate compared to HSAC support.

Conclusion.

PEFC MEAs custom-made by IRD Fuel Cells with HSAC and GrC supports were subjected to the carbon corrosion AST published by the U.S. DRIVE FCTT. In this work, spectroscopic techniques were performed and the performance changes after ASTs were measured. 1,000 AST cycles were conducted on Pt/HSAC and 10,000 AST cycles were conducted on Pt/GrC due to different carbon corrosion rates. Polarization curve results showed that Pt/GrC did not meet DOE target of 40 mV loss at 0.8 A cm -2 and 40 mV loss at 1.5 A cm -2 after 5000 AST cycles. Carbon corroded during the AST and formed rougher carbon surfaces and created more contacts with ionomer and water, which resulted in increases of C dl for Pt/HSAC (7.5%) and Pt/GrC (69.8%) at their EOLs. The Pt/GrC support was more ordered and organized, however after AST, a nonnegligible fraction of disordered carbon was generated, leading to a more significant increase of C dl . ECSA was lost during AST for both Pt/HSAC and Pt/GrC because of Pt loss, Pt detachment from the carbon support and Pt particle size growth. Cumulative Pt loading decreased 14.5% and 8.3% for Pt/HSAC and Pt/GrC after AST at their EOLs, respectively. For Pt/HSAC, Pt detachment might be one of the major reasons leading to the 52.4% reduction of ECSA. However for Pt/GrC, after 5000 AST cycles, a ECSA loss of 30.6% was observed, which met the DOE target of less than 40% ECSA loss compared to the BOL. After 10,000 AST cycles, the increase of particle size and Pt loss caused 44% loss of ECSA at the EOL. After 1,000 AST cycles at EOL, for Pt/HSAC, catalyst layer thickness reduced by 67.5% leading to a 30% increase of HFR due to insufficient contacts between GDL and catalyst layer. For Pt/GrC after 10,000 AST cycles, a catalyst layer thickness reduction of 68.4% was observed whereas the HFR almost remained unchanged due to sufficient contacts between GDL and catalyst layer after AST. After AST, both Pt/HSAC and Pt/GrC showed higher D/G ratio from Raman measurements, meaning that the carbon status became more disordered. Cathode catalyst layer ionic conductivity for Pt/HSAC and Pt/GrC decreased 53.1% and 63.8% after AST at their EOLs, and it was found that the SO 3 -group coverage decreased 50.7% and 73.5% for Pt/HSC and Pt/GrC, respectively. An in-situ and ex-situ electrochemical characterizations comparison between Pt/HSAC and Pt/GrC after 1,000 AST cycles was displayed. Graphitized carbon support was much more robust compared to HSAC support, and it showed 2 times less ECSA reduction, 2.5 times less catalyst layer ionic conductivity reduction and 4 times less SO 3 -group coverage drop.

Fig. 1 .

 1 Fig. 1. Performance and HFRs of MEAs using HSAC (a and b) and GrC (c and d) support during carbon corrosion AST. For Pt/HSAC, (a) MEA performance at 80 ℃, 100% RH, 150 kPa in H2/Air with stoichiometric anode and cathode flows of 2 and 2 and HFR at BOL, after 100, 500, and 1,000 AST cycles (EOL), (b) voltage at 1 A cm -2 .

Fig. 2 .

 2 Fig. 2. Cross-sectional scanning electron microscope (SEM) and averaged cathode catalyst layer thickness reductions for Pt/HSAC (a-c) and Pt/GrC (d-f) during AST. Cathode catalyst layer thicknesses for Pt/HSAC using SEM at (a) BOL and (b) EOL. (c) Thickness reduction at EOL for Pt/HSAC. Catalyst layer thicknesses for Pt/GrC using SEM at (d) BOL and (e) EOL. (f) Thickness reduction at EOL for Pt/GrC.

Fig. 3 .

 3 Fig. 3. Double layer capacitance, CV measurements and ECSAs of MEAs using HSAC (a-b) and GrC (c-d) supports during carbon corrosion AST. For Pt/HSAC, (a) cathode double layer capacitance, (b) cathode ECSA at BOL, after 100, 500, and 1000 AST cycles (EOL). For Pt/GrC, (c) cathode double layer capacitance, (d) cathode ECSA at BOL, after 100, 500, 1000, 2500, 5000, and 10000 AST cycles (EOL).

Fig. 4 .

 4 Fig. 4. Identical location micro X-ray fluorescence (XRF) maps for Pt/HSAC at (a) BOL and (b) EOL. (c) Loadingfrequency distribution for Pt/HSAC at BOL and EOL. Identical location micro X-ray fluorescence (micro XRF) maps for Pt/GrC at (d) BOL and (e) EOL. (f) Loading-frequency distribution for Pt/GrC at BOL and EOL.

Fig. 5 .

 5 Fig. 5. Cathode Pt particle size measurement result using scanning transmission electron microscopy (STEM), transmission electron microscopy (TEM) and X-ray diffraction (XRD) for Pt/HSAC and Pt/GrC. (a) Scanning transmission electron micrographs of mean particles size at BOL and transmission electron micrographs of mean particles size at EOL, (b) X-ray diffraction of mean particles size at BOL and after 1,000 AST cycles (c) X-ray diffraction of mean particles size at BOL and EOL for Pt/HSAC and Pt/GrC and (d) cathode Pt particle size determined from XRD as a function of Pt particle size determined from electron micrographs (EM). The dashed line represents a one-to-one correlation.

Fig. 6 .

 6 Fig. 6. (a) Cathode catalyst layer ionic conductivity reduction for and (b) cathode catalyst layer SO 3 -group coverage for Pt/HSAC and Pt/GrC at BOL and EOL.

Fig. 7 .

 7 Fig. 7. Identical location micro X-ray fluorescence (XRF) maps for Pt/HSAC at (a) BOL and (b) after 1,000 AST cycles. (c) Loading-frequency distribution for Pt/HSAC at BOL and after 1,000 AST cycles. Identical location micro X-ray fluorescence (micro XRF) maps for Pt/GrC at (d) BOL and (e) after 1,000 AST cycles. (f) Loading-frequency distribution for Pt/GrC at BOL and after 1,000 AST cycles.

Fig. 8d compares

  Fig. 8d compares the cathode catalyst layer ionic conductivity between Pt/HSAC and Pt/GrC

Fig. 8 .

 8 Fig. 8. Electrochemical characterizations and ex-situ measurements comparison between Pt/HSC and Pt/GrC at BOL and after 1,000 AST cycles. (a) MEA performance and HFRs at 80 ℃, 100% RH, 150 kPa in H2/Air with stoichiometric anode and cathode flows of 2 and 2, (b) cathode ECSA, (c) cathode catalyst layer thickness, (d) cathode catalyst layer ionic conductivity and (e) cathode catalyst layer SO 3 -group coverage.

Table . 1

 . . Relative concentrations for surface C species for cathodes of Pt/HSAC and Pt/GrC before and after AST

		Binding	Pt/GrC		Pt/GrC		Pt/GrC		Pt/HSAC		Pt/HSAC
	(%)	energy (eV)	BOL	Std.	cycles 1,000	Std.	EOL	Std.	BOL	Std.	EOL	Std.
	sp 2	~284 56.3	1.87	56.2	0.68	46.1	1.47	46.6	1.52	46.3	1.25
	C-C	~285 4.3	1.30	3.4	0.72	5.4	2.11	4.6	1.89	1.6	0.81
	C*	~286 6.4	1.21	6.3	0.54	4.8	1.01	6.0	1.39	6.3	0.58
	C-O	~287 3.7	1.05	3.9	0.35	2.9	0.95	5.2	0.98	4.7	0.70
	C=O ~288 3.4	0.92	3.3	0.31	4.1	0.50	5.9	0.61	5.2	0.76
	COOH ~289 3.7	1.11	3.3	0.26	3.2	1.11	7.3	1.59	14.2	1.42
	C-F	~291 16.3	0.85	11.3	0.6	18.8	3.23	20.3	1.20	11.0	2.77
	C-F2 ~292 4.3	1.23	9.1	0.72	12.9	3.25	3.2	0.96	9.1	2.32
	C-F3 ~293 1.7	0.98	3.3	0.51	1.8	0.83	1.0	0.75	1.7	1.17

Table . 2

 . . Relative concentrations for surface Pt species for cathodes of Pt/HSAC and Pt/GrC before and after AST

		Binding	Pt/GrC		Pt/GrC		Pt/GrC		Pt/HSAC		Pt/HSAC
	(%)	energy (eV)	BOL	Std.	cyc 1,000	Std.	EOL	Std.	BOL	Std.	EOL	Std.
	Pt 0	~70.9 53.2	3.09	57.1	1.54	65.2	3.86	49.3	3.79	63.3	3.12
	Pt-C	~71.7 30.1	3.62	27.1	3.07	17.6	4.78	32.4	4.01	18.0	2.89
	Pt-O	~73.3 16.7	2.01	15.8	2.61	17.2	3.78	18.3	2.66	18.7	3.57
	3.7. Cathode carbon characterization.

Table .

 .

	3.8. Cathode catalyst layer ionic conductivities and sulfonic acid group coverages
	3. Cathode Raman D/G ratio of Pt/GrC and Pt/HSAC before and after AST
	Sample	D/G ratio
	Pt/GrC BOL	0.980
	Pt/GrC after 1,000 AST cycles	1.156
	Pt/GrC after 10,000 AST cycles (EOL)	1.372
	Pt/HSAC BOL	1.081
	Pt/HSAC after 1,000 AST cycles (EOL)	1.256

  According to the combined results of the ECSA loss, Pt loss and detachment leading to the ionomer detachment might be the principal reason for the 50.7% reduction of SO 3 group coverage at the EOL. Pt/GrC shows a higher SO 3 -group coverage of 24.1% at BOL compared to Pt/HSAC because Pt particles within HSAC support mesopores were not in contact with ionomer. Recent studies showed that Pt nanoparticles within mesoporous carbon support are buried not too far into the pores but at the same time were not contacted by SO 3 -groups[START_REF] Ramaswamy | Carbon Support Microstructure Impact on High Current Density Transport Resistances in PEMFC Cathode[END_REF][START_REF] Clark | Side chain flexibility in perfluorosulfonic acid ionomers: An Ab initio study[END_REF][START_REF] Ott | Ionomer distribution control in porous carbon-supported catalyst layers for high-power and low Pt-loaded proton exchange membrane fuel cells[END_REF].

	for Pt/GrC)
	methods. For Pt/HSAC, SO 3 -group coverage decreased from 13.8% at BOL to 6.8% after 1,000
	AST cycles at EOL.

Apart from that, Qi et al.

[START_REF] Qi | Understanding Platinum Ionomer Interface Properties of Polymer Electrolyte Fuel Cells[END_REF]

also showed that SO 3 -group coverage was less dependent to Pt loading, but more dependent to carbon support type. After 10,000 AST cycles, SO 3 -group coverage for Pt/GrC reached 6.39%, which again might be because ionomer detached Pt surface caused by Pt loss and detachment.

4. Discussion: in-situ and ex-situ characterizations comparison between Pt/HSAC and Pt/GrC after 1,000 AST cycles

  An additional MEA using GrC support was subjected to study both in-situ and ex-situ electrochemical characterizations after 1,000 AST cycles. Then a further detailed comparison was published and shown in Fig.8between Pt/HSAC and Pt/GrC to investigate the different levels of degradations of different carbon supports after same AST cycles. Fig.8acompares the MEA performance and HFR results between Pt/HSAC and Pt/GrC at BOL and after 1,000 AST cycles. Pt/HSAC experienced a rapid cell performance deterioration after 1,000 AST cycles while a much more moderate voltage drop of 14 mV at 1 A cm -2 was found for Pt/GrC. Apart from that, after 1,000 AST cycles, HFR for Pt/GrC decreased from 52.8 mohms cm 2 to 50.8 mohms cm 2 due to better interfacial contacts between components and further cell hydration. However, HFR for Pt/HSAC increased from 56.4 mohms cm 2 to 73.3 mohms cm 2 because of thickness reduction of catalyst layer, which is shown in Fig.2. Fig.8bshows the cathode ECSA comparison. After 1,000 AST cycles, ECSA of Pt/HSAC decreased from 43.5 m 2 g -1 to 20.7 m 2 g -1 (52.4%). For Pt/GrC, ECSA decreased from 26.2 m 2 g -1 to 22 m 2 g -1 (16.1%) after 1,000 AST cycles, which is much less compared to Pt/HSAC. Catalyst layer thickness comparison is shown in Fig.8c. Cathode catalyst layer thickness for Pt/HSAC reduced by 7.7 μm (67.5%) after 1,000 AST cycles. However according to SEM measurement results shown in Fig.S12, cathode catalyst layer thickness reduced from 18.7 μm to 14.6 μm (21.9%) after 1,000

Acknowledgement

The authors acknowledge funding from the National Science Foundation CAREER award 1652445. The authors acknowledge the use of facilities and instrumentation at the UC Irvine Materials Research Institute (IMRI). XPS work was performed using instrumentation funded in part by the National Science Foundation Major Research Instrumentation Program under grant no. CHE-1338173. The XRF and Raman work was done at the HIMaC 2 's Analytical Laboratory, a user facility operated by the Horiba Institute for Mobility and Connectivity, University California, Irvine.


Wilfried Blanc

Bernard Dussardier

Mukul C Paul

Er-doped oxide nanoparticles in silica-based optical fibers

Erbium-doped materials are of great interest in optical telecommunications due to the Er 3+ intra-4f emission at 1.54 µm. Erbium-Doped Fiber Amplifiers (EDFA) were developed in silica glass because of the low losses at this wavelength and the reliability of this glass. Developments of new rare-earth doped fiber amplifiers aim to control their spectroscopic properties: shape and width of the gain curve, optical quantum efficiency, .... Standard silica glass modifiers, such as aluminum, give very good properties to available EDFA. However, for more drastic spectroscopic changes, more important modifications of the rare-earth ions local environment are required. To this aim, we present a fiber fabrication route creating rare-earth doped calco-silicate or calco-phospho-silicate nanoparticles embedded in silica glass. By adding alkaline earth elements such as calcium, in low concentration, one can obtain a glass with an immiscibility gap. Then, phase separation occurs under an appropriate heat treatment. We investigated the role of two elements: calcium and phosphorus (a standard silica modifier). Under Scanning Electron Microscopy analyses, nanoparticles are observed only when calcium is incorporated. The size of the particles is determined to be around 50 nm in preform samples. The nature of these particles depends on phosphorus content: without P, electron diffraction pattern reveals an amorphous particle while it is partially crystallized when phosphorus is added. Besides, through Energy Dispersive X-Ray techniques, we succeeded to show that erbium ions are located into the nanoparticles.

Introduction

Among the rare-earths, erbium is especially interesting because the 13 I 3/2 → 15 I 3/2 transition at ≈1.54 µm coincides with the lowest attenuation window of silica glass fiber; hence, erbium doped glasses can be used to amplify attenuated signals in optical fiber telecommunication systems. While Erbium-Doped Fiber Amplifier (EDFA) was developed 20 years ago, extensive studies are still carried out to improve its properties. In particular, linear dimensions should be reduced although the severe limitations imposed by the poor rare earth solubility into silica matrix.. Moreover, spectral bandwidth should be increased to improve the Wavelength Division Multiplexing (WDM) applications. To solve these problems, various approaches based on nanoparticles incorporation have been proposed.

Over the past several decades considerable work have been carried out for incorporation of rare-earth oxide nano-crystallites into different glass hosts. Different processes have been developed such as cosputtering technique 1 , pyrolysis 2 , ion-implantation 3 , laser ablation process 4 and sol-gel process 5 . Another process which was recently developed by Liekki, Finnish company, is the direct nano-particles deposition process 6 . All these processes were related to the outer vapour deposition except the sol-gel process which involves longer time for fabrication of preform. A lot of work was also studied for development of suitable silica and non-silica based glass host for doping of rare-earth ions to improve their lasing and amplification properties . In most of the cases core glass is made of non-silica based or very low silica-based materials made by double crucible furnace melting procedure but they are not very much reliable for the system comparable to the high silica based rare-earth doped fibers. Otherwise, there are reports on thin films containing semiconductor and metal nanoparticles acting as sensitizers, absorbing the incident light with a high cross-section and exciting erbium co-doped ions in the nanoparticle vicinity through energy transfer.

Here we propose for the first time to incorporate Er into oxide nanoparticles in optical fibers to improve the spectroscopic properties of erbium ions. To our knowledge only few studies on nanostrutured silica fibers were dedicated to metal ions properties 7 . In our samples, oxide nanostructures are prepared by adjusting the composition of usual modifiers in the core of the fiber. A glass with an immiscibility gap is obtained and two phases are then formed with high and low silica content. The role of calcium on the preparation of the nanostructures is discussed in this contribution.

Experimental details

Preforms were fabricated by the conventional Modified Chemical Vapor Deposition (MVCD) technique. In this process, gaseous chlorides (SiCl 4 , GeCl 4 , POCl 3 ) are passed through a rotating silica tube, heated by an external burner in translation along the tube. Due to the high temperature, chlorides oxidize, forming particles which deposit on the inner wall. This porous layer turns into a glassy layer when the burner passes over it (the temperature is around 1500 °C). In the final stage, the tube is collapsed into a rod at a temperature higher than 1800°C. This rod can be drawn into fibers. Opto-geometric properties of the perform (or fiber) are determined by adjusting the composition and number of layers. In our samples, phosphorous and germanium concentrations are ~1 mol% and 2 mol%, respectively.

Erbium and calcium ions were incorporated through the solution doping technique. An alcoholic solution (of desired strength of ErCl 3 .6H 2 O or CaCl 2 .6H 2 O) is soaked for 2 hours in the unsintered core layer. After removing the solution, the layer is dried and sintered. Erbium concentration is estimated through absorption spectra to be around 1000 ppm. Three calcium concentrations in the solution were prepared: 0, 0.01 and 0.1 mol/l. The core diameter was measured with a preform analyzer (York Technology P 101 ) to be 1 mm.

The field emission scanning electron microscope (FESEM) image of the core glass of highly polished preform section of thickness around 2mm was taken. The morphology of such type of core glass was also studied using TEM image of some of the preform samples. The preform TEM specimens were mechanically polished and dimpled to thickness of about 10 µm. The final thinning of the sample to electron transparency was carried out using an Ar ion mill. The TEM examinations were performed with a 200 kV JEM-2000FX transmission electron microscope equipped with Evex ultra-thin window X-ray detector.

Results and discussion

When the calcium concentration is increased, the color of the central core of the preform turns from transparent to milky. This variation is explained by the structural changes of the core. For preforms with calcium concentration higher than 0.01 mol/l, nanoparticles are observed. A TEM picture of such a preform is presented on figure 1. We can clearly observed polydisperse spherical nanoparticles with a mean diameter of 50 nm. Smaller particles of 10 nm are visible. The size of the biggest particles was around 200 nm (not shown in figure 1). When the calcium concentration decreases, the size distribution of the particles is nearly identical but the density is lowered. The composition of the sample was investigated by EDX analyses. When nanoparticles are analyzed, a same amount of Ca, P and Si is found while only Si is detected outside of the particles. Germanium seems to be homogeneously distributed over the entire glass. The structure of the particles was determined with electron diffraction pattern along the TEM analyses. When calcium and phosphorus are incorporated, the oxide particles are amorphous while they are partially crystalline when there is no phosphorus. As the resistance of the structures against the electron diffraction beam irradiation damage is low, the exact nature of the crystal phase could not be determined. When erbium is added to the composition, it is found to be inserted into the particles as it is presented on figure 2. The formation of these nanoparticles can be discussed on the basic principle of the phase separation phenomena and crystal growth mechanisms which normally occur in bulk silicabased glass. On the basis of thermodynamical data such as activity coefficient, entropy of mixing, enthalpy of mixing and Gibbs-free energy change, the phase diagram of the SiO 2 -CaO binary compound was derived using Factstage software (figure 3). A miscibility gap is found when the CaO concentration is between 2 and 30 mol%. It coincides with the experimentally determined miscibility gap region of binary SiO 2 -CaO glass 8 . The prerequisite for phase separation in multi-component glasses is the existence of an immiscibility region in their phase diagram. Glasses with compositions within the immiscibility region however form a clear and transparent glass when their melts are quenched rapidly below the glass transition temperature, T g . Below this temperature the viscosity is too high to allow for the necessary rearrangements of structural groups required for phase separation. However when such glasses are reheated to temperatures in the range between T g and the upper critical temperature of the immiscibility dome, sub-liquidus phase separation occurs. Two principal mechanisms of phase separation have been proposed so far, phase separation by spinodal decomposition and by nucleation and growth. Due to the spherical shape of the particles, one may think that the last mechanism prevails [START_REF] Zarzycki | Glasses and the vitreous state[END_REF] . The opaqueness observed when calcium concentration increases is a consequence of light scattering. This induces high loss and has to be avoided for telecommunication applications. Tick discussed parameters to be controlled and mentioned that the particle size must be less than 15 nm and the particle-size distribution must be narrow [START_REF] Tick | [END_REF] .

Conclusion

In this contribution we have investigated a new route to prepare erbium-doped oxyde nanoparticles into preforms. By adding calcium into silica-based glass, we successfully prepare nanostructures through phase separation. The mean diameter is around 50 nm with a high calcium and phosphorus content. Erbium ions are found to be located into nanoparticles. This would lead to improve spectroscopic properties of the rare-earth and benefit to the telecommunication systems.

Figure 1 :

 1 Figure 1: TEM image from sample doped with Ca and P

Figure 2 :

 2 Figure 2: Energy Dispersive X-ray Analyses spectra of the preform sample doped with Ca and Er. The area analyzed corresponds to the nanoparticle (a) and outside (b)

Figure 3 :

 3 Figure 3: Miscibility-gap in the derived phase-diagram of binary SiO 2 -CaO glass

Acknowledgements

This work was realized in the frame of a P2R project (Programme de Recherche en Réseau, CNRS (France) and DST (India)). We thank Ph. Vennéguès at CRHEA (CNRS, Valbonne, France) and S.C. Cheng (Corning, New-York, USA) for the TEM images and EDX analyses.


Bishal Deb
email: bishal@gonitsora.com.

Alan D Sokal
email: sokal@nyu.edu

Coefficientwise Hankel-total positivity of the Schett polynomials

Keywords: Schett polynomials, Jacobian elliptic functions, total positivity, production matrix, exponential Riordan array, checkerboard exponential Riordan array

We prove the coefficientwise Hankel-total positivity of the even and odd subsequences of Schett polynomials X n (x, y, z).

Introduction

A half-century ago, Schett [START_REF] Schett | Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions[END_REF][START_REF] Schett | Recurrence formula of the Taylor series expansion coefficients of the Jacobian elliptic functions[END_REF] introduced implicitly, and then Dumont [START_REF] Dumont | A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions[END_REF][START_REF] Dumont | Une approche combinatoire des fonctions elliptiques de Jacobi[END_REF] introduced explicitly, the sequence of polynomials

X n (x, y, z) = yz ∂ ∂x + xz ∂ ∂y + xy ∂ ∂z n x , (1.1)
which we shall call the Schett polynomials initialized in x; clearly they are homogeneous polynomials of degree n + 1 with nonnegative integer coefficients, and are symmetric in y ↔ z. Schett [START_REF] Schett | Properties of the Taylor series expansion coefficients of the Jacobian elliptic functions[END_REF][START_REF] Schett | Recurrence formula of the Taylor series expansion coefficients of the Jacobian elliptic functions[END_REF] showed that the polynomials X n (x, y, z) unify and generalize the Taylor coefficients of the Jacobian elliptic functions sn, cn, dn.

It is easy to see that for even (resp. odd) n, the polynomial X n (x, y, z) is divisible by x (resp. by yz) and that the quotient is a polynomial in x 2 , y 2 , z 2 . We therefore define the reduced Schett polynomials

X n (x 2 , y 2 , z 2 ) =        1 x
X n (x, y, z) for n even 1 yz X n (x, y, z) for n odd (1.2)

Then X 2k and X 2k+1 are homogeneous polynomials of degree k in x 2 , y 2 , z 2 . It is also not difficult to show that these two polynomials coincide when x = 0: that is, X 2k (0, y 2 , z 2 ) = X 2k+1 (0, y 2 , z 2 ).

Total positivity and production matrices

The main purpose of the present paper is to prove the coefficientwise Hankel-total positivity of the even and odd subsequences of Schett polynomials. Recall first that a finite or infinite matrix of real numbers is called totally positive (TP) if all its minors are nonnegative. Background information on totally positive matrices can be found in [START_REF] Karlin | Total Positivity[END_REF][START_REF] Gantmakher | Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems[END_REF][START_REF] Pinkus | Totally Positive Matrices[END_REF][START_REF] Fallat | Totally Nonnegative Matrices[END_REF]; they have applications to many areas of pure and applied mathematics. 1 In particular, it is known [9, Théorème 9] [15, section 4.6] that an infinite Hankel matrix (a i+j ) i,j≥0 of real numbers is totally positive if and only if the underlying sequence (a n ) n≥0 is a Stieltjes moment sequence, i.e. the moments of a positive measure on [0, ∞). However, this is only the beginning of the story, because our main interest [START_REF] Sokal | Coefficientwise Hankel-total positivity[END_REF] is not with sequences and matrices of real numbers, but rather with sequences and matrices of polynomials (with integer or real coefficients) in one or more indeterminates x: in applications they will typically be generating polynomials that enumerate some combinatorial objects with respect to one or more statistics. We equip the polynomial ring R[x] with the coefficientwise partial order: that is, we say that P is nonnegative (and write P ⪰ 0) in case P is a polynomial with nonnegative coefficients. We then say that a matrix with entries in R[x] is coefficientwise totally positive if all its minors are polynomials with nonnegative coefficients; and we say that a sequence a = (a n ) n≥0 with entries in R[x] is coefficientwise Hankel-totally positive if its associated infinite Hankel matrix H ∞ (a) = (a i+j ) i,j≥0 is coefficientwise totally positive. Most generally, we can consider sequences and matrices with entries in an arbitrary partially ordered commutative ring; total positivity and Hankel-total positivity are then defined in the obvious way. Coefficientwise Hankel-total positivity of a sequence of polynomials (P n (x)) n≥0 implies the pointwise Hankel-total positivity (i.e. the Stieltjes moment property) for all x ≥ 0, but it is vastly stronger.

Our first main theorem is:

Theorem 1.1 (Coefficientwise Hankel-total positivity of the Schett polynomials).

(a) The sequence X 2n (x2 , y 2 , z 2 ) n≥0 of even reduced Schett polynomials is coefficientwise Hankel-totally positive in x, y, z.

(b) The sequence X 2n+1 (x 2 , y 2 , z 2 ) n≥0 of odd reduced Schett polynomials is coefficientwise Hankel-totally positive in x, y, z.

We remark that the z = 0 specialization of Theorem 1.1(a) follows from a continued fraction due to Stieltjes [21, p. H17] and Rogers [16, p. 77] together with some general theory [START_REF] Sokal | Coefficientwise Hankel-total positivity[END_REF]. 2 But the general case of Theorem 1.1 is considerably more subtle.

Our proof of Theorem 1.1 will be based on the method of production matrices [START_REF] Deutsch | Production matrices[END_REF][START_REF] Deutsch | Production matrices and Riordan arrays[END_REF]. Let P = (p ij ) i,j≥0 be an infinite matrix with entries in a commutative ring R; we assume that P is either row-finite (i.e. has only finitely many nonzero entries in each row) or column-finite. Now define an infinite matrix A = (a nk ) n,k≥0 by a nk = (P n ) 0k . We call P the production matrix and A the output matrix, and we write A = O(P). The two key facts here are the following [14, section 9.2]: if R is a partially ordered commutative ring and P is totally positive, then O(P) is totally positive [START_REF] Pétréolle | Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity[END_REF]Theorem 9.4], and the zeroth column of O(P) is Hankel-totally positive [START_REF] Pétréolle | Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity[END_REF]Theorem 9.7].

We will prove Theorem 1.1 by exhibiting production matrices that generate the even or odd reduced Schett polynomials as the zeroth column of their output matrix, and then proving the total positivity of those production matrices. The production matrices will be quadridiagonal unit-lower-Hessenberg matrices: that is, they will have a superdiagonal with all entries equal to 1, and then a diagonal, a subdiagonal, and a second subdiagonal; all other entries are equal to 0. In the language of [START_REF] Pétréolle | Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity[END_REF], these are the production matrices for a 2-branched J-fraction. Similar but simpler quadridiagonal production matrices arose in [START_REF] Deb | Lattice paths and branched continued fractions, III: Generalizations of the Laguerre, rook and Lah polynomials[END_REF]. In particular, we will prove the following two results: Proposition 1.2 (Production matrices for even and odd Schett polynomials).

(a) The quadridiagonal production matrix P(x, y, z) = (p ij ) i,j≥0 defined by p n,n+1 = 1 (1.3a)

p n,n = (2n) 2 x 2 + (2n + 1) 2 (y 2 + z 2 ) (1.3b) p n,n-1 = (2n) 2 (2n -1) (2n -1)x 2 (y 2 + z 2 ) + (2n + 1)y 2 z 2 (1.3c) p n,n-2 = (2n) 2 (2n -2) 2 (2n -1)(2n -3)x 2 y 2 z 2 (1.3d) p n,k = 0 if k < n -2 or k > n + 1 (1.3e)
generates the even reduced Schett polynomials X 2n (x 2 , y 2 , z 2 ) n≥0 as the zeroth column of its output matrix O(P).

(b) The quadridiagonal production matrix Q(x, y, z) = (q ij ) i,j≥0 defined by

q n,n+1 = 1 (1.4a) q n,n = (2n + 2) 2 x 2 + (2n + 1) 2 (y 2 + z 2 ) (1.4b) q n,n-1 = (2n) 2 (2n + 1) (2n + 1)x 2 (y 2 + z 2 ) + (2n -1)y 2 z 2 (1.4c) q n,n-2 = (2n) 2 (2n -2) 2 (2n + 1)(2n -1)x 2 y 2 z 2 (1.4d) q n,k = 0 if k < n -2 or k > n + 1 (1.4e)
generates the odd reduced Schett polynomials X 2n+1 (x 2 , y 2 , z 2 ) n≥0 as the zeroth column of its output matrix O(Q).

Proposition 1.3 (Total positivity of the production matrices).

(a) The quadridiagonal production matrix P(x, y, z) defined in (1.3) is coefficientwise totally positive in x, y, z.

(b) The quadridiagonal production matrix Q(x, y, z) defined in (1.4) is coefficientwise totally positive in x, y, z.

We note that the production matrices P(x, y, z) and Q(x, y, z) coincide when x = 0; this is consistent with the fact that X 2n (0,

y 2 , z 2 ) = X 2n+1 (0, y 2 , z 2 ).
Combining Propositions 1.2 and 1.3 with the general theory of totally positive production matrices [14, Theorem 9.7] proves Theorem 1.1.

Exponential generating functions of the output matrices

We can also give exponential generating functions for the output matrices O(P) and O(Q), using the theory of exponential Riordan arrays. Recall first that if F(t) and G(t) are formal power series with coefficients in a commutative ring R, with G(0) = 0, then the exponential Riordan array R[F, G] is the infinite lower-triangular matrix with entries

R[F, G] nk = n! k! [t n ] F(t)G(t) k ∈ R . (1.5)
If the ring R contains the rationals (as it will here), this says that the exponential generating function of the kth ] nk is nonvanishing only when n and k have the same parity; that is, only the even-even and odd-odd submatrices of R[F, G] are nonvanishing. In this situation we call R[F, G] a checkerboard exponential Riordan array. We will show that the output matrix O(P) is the even-even submatrix of a particular checkerboard exponential Riordan array. Clearly F must be the exponential generating function of the zeroth column of O(P), which by Proposition 1.2(a) are the even reduced Schett polynomials X 2n (x 2 , y 2 , z 2 ), alternated with zero entries. Hence we must have

column of R[F, G] is F(t)G(t) k /k!. Note also that if F is even and G is odd, then R[F, G
F(t) = ∞ ∑ n=0 X 2n (x 2 , y 2 , z 2 ) t 2n (2n)! = Dn(t; x, y, z) En(t; x, y, z) , (1.6)
where Dn and En are Dumont's hyperelliptic functions [6, section 3], and the final equality is [6, eq. (3.11)]. For G, it turns out that we will have G(t) = Sn(t; x, y, z), where Sn is another of Dumont's hyperelliptic functions. We will therefore prove:

Proposition 1.4 (Exponential generating function for the output matrix O(P)).

Let the matrix P be as in Proposition 1.2(a), and let F(t) and G(t) be as above. Then the output matrix O(P) is the even-even submatrix of the exponential Riordan array R[F, G]:

O(P) nk = (2n)! (2k)! [t 2n ] F(t)G(t) 2k . (1.7)
We can do something analogous for the output matrix O(Q) associated to the odd Schett polynomials, but this time it will be most convenient to find it as the odd-odd submatrix of a checkerboard exponential Riordan array. We will again have G(t) = Sn(t; x, y, z). Since the zeroth column of O(Q), which by Proposition 1.2(b) are the odd reduced Schett polynomials X 2n+1 (x 2 , y 2 , z 2 ), will now appear in the k = 1 column of R[F, G] in odd rows starting at n = 1, we must have

F(t)G(t) = ∞ ∑ n=0 X 2n+1 (x 2 , y 2 , z 2 ) t 2n+1 (2n + 1)! = Sn(t; x, y, z) Cn(t; x, y, z) , (1.8)
where Sn and Cn are Dumont's hyperelliptic functions, and the final equality is again [6, eq. (3.11)]. Therefore F(t) = Cn(t; x, y, z). We will then prove: Since the zeroth column of this submatrix is, by construction, the sequence of even (resp. odd) reduced Schett polynomials, this will also prove Proposition 1.2.

O(Q) nk = (2n + 1)! (2k + 1)! [t 2n+1 ] F(t)G(t) 2k+1 . ( 1
The production matrix of an exponential Riordan array R[F, G] is well known [START_REF] Sokal | Total positivity of some polynomial matrices that enumerate labeled trees and forests I: Forests of rooted labeled trees[END_REF]Theorem 2.19]

to be p nk = (n!/k!) (z n-k + k a n-k+1 ), where A(s) = ∑ ∞ n=0 a n s n and Z(s) = ∑ ∞ n=0 z n s n satisfy G ′ (t) = A(G(t)) , F ′ (t) F(t) = Z(G(t)) . (2.1)
Here we will give the analogous result for the even-even and odd-odd submatrices of a checkerboard exponential Riordan array. First we define some new series:

B(s) def = A(s) 2 , C(s) def = 2A(s)Z(s) , D(s) def = Z(s) 2 + A(s)Z ′ (s) . (2.2)
Since F is even and G is odd, it follows that A is even and Z is odd; then B and D are even, while C is odd, and we can write

B(t) = ∞ ∑ m=0 b m t 2m , C(t) = ∞ ∑ m=0 c m t 2m+1 , D(t) = ∞ ∑ m=0 d m t 2m . (2.3)
We then have:

Theorem 2.1 (Production matrices for submatrices of a checkerboard exponential Riordan array). Let R[F, G] be a checkerboard exponential Riordan array with invertible diagonal entries and F(0) = 1, and let R[F, G] ee and R[F, G] oo be its even-even and odd-odd submatrices.

Then:

(a) The production matrix of R[F, G] ee has entries

p nk = (2n)! (2k)! 2k(n + k) b n-k+1 + 2k c n-k + d n-k . (2.4) (b) The production matrix of R[F, G] oo has entries p nk = (2n + 1)! (2k + 1)! (2k + 1)(n + k + 1) b n-k+1 + (2k + 1) c n-k + d n-k . (2.5)
The proof will be based on the following simple but powerful result, which we discovered very recently and which to our surprise is apparently new; it is inspired by the integration-by-parts arguments employed by Stieltjes [START_REF] Stieltjes | Sur la réduction en fraction continue d'une série procédant selon les puissances descendantes d'une variable[END_REF], Rogers [START_REF] Rogers | On the representation of certain asymptotic series as convergent continued fractions[END_REF] and others [START_REF] Flajolet | Elliptic functions, continued fractions and doubled permutations[END_REF][START_REF] Milne | Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions[END_REF] to deduce classical J-fractions. Proposition 2.2 (Exponential-generating-function method for production matrices). Let A = (a nk ) n,k≥0 and P = (p nk ) n,k≥0 be row-finite matrices with entries in a commutative ring R containing the rationals, with a 0k = δ k0 . Fix integers r ≥ 1 and 0 ≤ s < r, and define the column exponential generating functions of A:

A [r,s] k (t) = ∞ ∑ n=0 a nk t rn+s (rn + s)! . (2.6)
Then the following are equivalent:

(a) For all k ≥ 0, d r dt r A [r,s] k (t) = ∞ ∑ n=0 p nk A [r,s] n (t) . (2.7) (b) A = O(P).
Proof. On both sides of (2.7), the only powers of t that can occur are t rm+s with m ≥ 0; for the left-hand side we have here used the hypothesis that 0 ≤ s < r. So take the coefficient of t rm+s /(rm + s)! on both sides of (2.7):

t rm+s (rm + s)! d r dt r A [r,s] k (t) = t r(m+1)+s (r(m + 1) + s)! A [r,s] k (t) = a m+1,k (2.8) while t rm+s (rm + s)! ∞ ∑ n=0 p nk A [r,s] n (t) = ∞ ∑ n=0 p nk a mn .
(2.9)

But a m+1,k = ∞ ∑ n=0
a mn p nk is precisely the recurrence stating that A = O(P). □ Note that, to prove that A = O(P), it suffices to verify the identities (2.7) for one pair (r, s). On the other hand, if A = O(P), then the identities (2.7) hold for all pairs (r, s) satisfying 0 ≤ s < r.

We remark also that (a) =⇒ (b) holds even when s ≥ r; but in this case the hypothesis (2.7) is unlikely to hold, because there are terms that will occur on the left-hand side (unless they happen to vanish) that cannot occur on the right-hand side, namely t rm+s with m < 0. We will now see an example of this. Sketch of proof of Theorem 2.1. We apply Proposition 2.2 with r = 2, taking s = 0 for part (a) and s = 1 for part (b). That is, we will compute the production matrix of the matrix with elements a nk = R[F, G] 2n+s,2k+s . Here we can take in principle any integer s ≥ 0, but we will see at the end why only s = 0, 1 give a valid final result.

For any k ∈ Z, let H k (t) be the exponential generating function of column 2k

+ s of R[F, G], H k (t) def = F(t) G(t) 2k+s (2k + s)! , ( 2.10)
where by convention we take H k (t) def = 0 if 2k + s < 0. A straightforward computation (about 1 page long) using (2.1)- (2.3) shows that

H ′′ k (t) = ∞ ∑ n=-1 p nk H n (t) (2.11)
where

p nk = (2n + s)! (2k + s)! (2k + s)(n + k + s) b n-k+1 + (2k + s) c n-k + d n-k . (2.12)
In general the term n = -1 can contribute to the sum (2.11) when k = 0, via b n-k+1 = b 0 (note that H -1 is nontrivial if s ≥ 2); but its coefficient is s(s -1) and hence vanishes for s ∈ {0, 1} (and in any case H -1 = 0 for s ∈ {0, 1}). Therefore, for s ∈ {0, 1} we can apply Proposition 2.2 with r = 2. □

We also have another proof of Theorem 2.1; we will publish it in the fuller version of this work.

Bishal Deb and Alan D. Sokal

We now apply Theorem 2.1 to the checkerboard exponential Riordan arrays defined in Propositions 1.4 and 1.5. From G(t) = Sn(t; x, y, z) and the differential equation satisfied by Sn [6, eq. (3.7)], we have immediately

A(s) = [(1 + x 2 s 2 )(1 + y 2 s 2 )(1 + z 2 s 2 )] 1/2 .
(2.13)

For Proposition 1.4 we have F(t) = Dn(t; x, y, z) En(t; x, y, z); using [6, eq. ( 3.3)/(3.4)] we get

Z(s) = s A(s) y 2 1 + y 2 s 2 + z 2 1 + z 2 s 2 .
(2.14)

For Proposition 1.5 we have F(t) = Cn(t; x, y, z); using [6, eq. ( 3.3)/(3.4)] we get

Z(s) = s A(s) x 2 1 + x 2 s 2 .
(2.15)

From this we compute, in each case, the series B, C, D and the coefficients b m , c m , d m ; this computation is straightforward and we leave it to the reader. Substituting these coefficients into Theorem 2.1 yields the production matrices of Proposition 1.2. We remark that when x = y and z = 0, these two checkerboard exponential Riordan arrays coincide; this array was found earlier in [2, eq. ( 6.24)].

Proof of Proposition 1.3

We begin by observing that the production matrices P and Q defined in Proposition 1.2 have a nice factorization. Let T(y, z) def = P(0, y, z) = Q(0, y, z) be the production matrix at x = 0: it is tridiagonal with matrix elements

T n,n+1 = 1 (3.1a) T n,n = (2n + 1) 2 (y 2 + z 2 ) (3.1b) T n,n-1 = 4n 2 (4n 2 -1)y 2 z 2 (3.1c)
And let L(x) be the lower-bidiagonal matrix with entries

L n,n = 1 (3.2a) L n,n-1 = 4n 2 x 2 (3.2b)
A simple computation shows: Proposition 3.1 (Factorization of the production matrices).

(a) P(x, y, z) = L(x) T(y, z).

(b) Q(x, y, z) = T(y, z) L(x).

Since a nonnegative bidiagonal matrix is totally positive, to prove Proposition 1.3 it suffices to show that T(y, z) is coefficientwise totally positive. This turns out to be surprisingly difficult; but it is contained within a tridiagonal special case of [1, Theorem C.1], which asserts the coefficientwise total positivity of a class of quadridiagonal lower-Hessenberg matrices, defined as follows. Let

P def = L 1 L 2 U + L 1 D 1 + L 2 D 2 (3.3)
where

L 1 = αI + ξ L , L 2 = βI + ηL (3.4) and
• L is the lower-bidiagonal matrix with the sequence a 0 , a 1 , . . . on the diagonal, the sequence b 1 , b 2 , . . . on the subdiagonal, and zeroes elsewhere;

• U is the upper-bidiagonal matrix with the sequence c 1 , c 2 , . . . on the superdiagonal, the sequence d 0 , d 1 , . . . on the diagonal, and zeroes elsewhere;

• D 1 is the diagonal matrix with entries e 0 , e 1 , . . . ;

• D 2 is the diagonal matrix with entries f 0 , f 1 , . . The quadridiagonal matrices P and Q of Proposition 1.2 are not (as far as we can tell) covered by Theorem 3.2, but the tridiagonal matrix T is. Taking d = 0 makes the matrix P tridiagonal; the further specializations

α = β = 1, ξ = y 2 , η = z 2 , a n = 0, b n = (2n)(2n + 1), c n = 1, e n = (2n + 1)z 2 , f n = (2n + 1)y 2 (3.5) then yield T.
Sketch of proof of Theorem 3.2. We first let Q = P| f=0 = L 1 (L 2 U + D 1 ). Next let p k , q k and ℓ k denote the k-th columns of the matrices P, Q and L 2 , respectively.

We use induction to prove that for any fixed pair of integers 0 ≤ k ≤ m + 1, the matrix (q 0 , q 1 , . . . , q k-1 , p k , . . . , p m ) is totally positive. We do this in the following steps:

• Step 1: The matrix Q is totally positive: this follows from the fact that bidiagonal matrices with non-negative entries are totally positive, together with the tridiagonal comparison theorem [START_REF] Sokal | Coefficientwise Hankel-total positivity[END_REF] [22, Proposition 3.1] [1, Proposition 2.6]. This establishes the base case k = m + 1 of our induction.

• Step 2: The matrix (q 0 , . . . , q k-1 , ℓ k ) is totally positive: When d = 0, this can easily be shown by direct consideration of the minors. The general case is significantly more difficult.

• Step 3: If the matrix (p k+1 , . . . , p m ) is totally positive, then so is (q 0 , . . . , q k-1 , ℓ k , p k+1 , . . . , p m ): We prove this by induction where the base case

k = m is Step 2. When k < m, we let t k+1 = p k+1 | b k+1 =0 f k+1 →ξb k+1 c k+1
and let (t n,k+1 ) n≥0 = t k+1 . Next let p k+1 be the same as t k+1 except that the entry t k,k+1 is made equal to 0. It is not difficult to show that the matrix ( p k+1 , p k+2 , . . . , p m ) is totally positive. Next, we notice that the matrix S = (q 0 , . . . , q k-1 , ℓ k , p k+1 , p k+2 , . . . , p m ) consists of two totally positive blocks which overlap in a single row. Using [11, p. 398], we get that S is totally positive. Finally, we obtain the desired result by right-multiplying S with the upper-bidiagonal matrix that has 1 on the diagonal, (α + ξa k )c k+1 in position (k, k + 1) and zeros elsewhere.

• Step 4 (Induction step): If the matrix (q 0 , q 1 , . . . , q k , p k+1 , . . . , p m ) is totally positive, then so is (q 0 , q 1 , . . . , q k-1 , p k , . . . , p m ): This follows from noticing p k = q k + f k ℓ k and then using the column-linearity of determinants.

The details of this proof can be found in [1, Appendix C]. □

Remarks

In a longer version of this work, we intend to treat the following generalizations: 1) We can define Schett polynomials in any number of variables, not just three. Fix an integer m ≥ 1, and define the Schett operator in m + 1 variables x 0 , x 1 , . . . , x m by

D m+1 def = m ∑ i=0 ∏ 0 ≤ j ≤ m j ̸ = i x j ∂ ∂x i . ( 4.1)
Then define the Schett polynomials in m + 1 variables initialized in x 0 by

X [m+1] n (x 0 , x 1 , . . . , x m ) def = (D m+1 ) n x 0 . (4.2)
These polynomials are clearly symmetric in x 1 , . . . , x m . Since x 0 plays a special role, we write x = x 0 and y = (y 1 , . . . , y m ) = (x 1 , . . . , x m ). It turns out that if we define a i = 1/y i and make an appropriate rescaling, then the Schett polynomial in m variables is obtained from the one in m + 1 variables by specializing to a m = 0. This allows us to take m → ∞ and define a symmetric-function generalization of the Schett polynomials.

For this generalization we have obtained analogues of Propositions 1.2, 1.4 and 1.5 for the case x = 0, but the coefficientwise total positivity of this production matrix is at present a conjecture. Furthermore, we have not yet been able to find a production matrix for the case x ̸ = 0.

2) Schett [17, Theorem III/1] showed that X n (1, 1, 1) = n!. It was therefore natural to interpret X n (x, y, z) as enumerating permutations of [n] def = {1, . . . , n} with respect to some bivariate statistic, and Dumont [START_REF] Dumont | A combinatorial interpretation for the Schett recurrence on the Jacobian elliptic functions[END_REF] identified the statistic: he showed that X n (x, y, z) =

x D n (x, y, z) for n even y D n (x, y, z) for n odd (4.3)

where

D n (x, y, z) def = ∑ σ∈S n (x 2
) cpeakodd(σ) (y 2 ) cpeakeven(σ) z cdrise(σ)+cdfall(σ)+fix(σ) ; (4.4) here these statistics count, respectively, odd cycle peaks, even cycle peaks, cycle double rises, cycle double falls, and fixed points in the permutation σ. (We remark that the y ↔ z symmetry is quite mysterious in this combinatorial interpretation.) A different combinatorial interpretation of the Schett polynomials, in terms of vertices of even and odd degree at even and odd levels of an increasing tree, was given by Lin and Ma [START_REF] Lin | A symmetry on weakly increasing trees and multiset Schett polynomials[END_REF]Theorem 1.6]. It will be interesting to find combinatorial interpretations for the higherorder Schett polynomials.

3) We can generalize (4.3)/(4.4) by introducing a factor λ cyc(σ) , where cyc(σ) denotes the number of cycles in the permutation σ. For these Schett polynomials with cycle-counting, we have found generalizations of (1.3)/(1.4) that empirically generate the correct zeroth-column sequences, but we have not yet been able to prove that they do so. On the other hand, we have proven the corresponding generalization of Proposition 1.3: these production matrices are totally positive, coefficientwise in x, y, z, λ, as a consequence of the d = 0 case of Theorem 3.2.

Proposition 1 . 5 (

 15 Exponential generating function for the output matrix O(Q)).Let the matrix Q be as in Proposition 1.2(b), and let F(t) and G(t) be as above. Then the output matrix O(Q) is the odd-odd submatrix of the exponential Riordan array R[F, G]:

5

 5 We shall prove Proposition 1.4 (resp. 1.5) by considering the checkerboard exponential Riordan array defined there and showing that the production matrix of its even-even (resp. odd-odd) submatrix is indeed what is written in Proposition 1.2(a) [resp.1.2(b)].

Theorem 3 . 2 .

 32 . ; and α, β, ξ, η, a = (a n ) n≥0 , b = (b n ) n≥1 , c = (c n ) n≥1 , d = (d n ) n≥0 , e = (e n ) n≥0 , f = ( f n ) n≥0 are all indeterminates. We then have: [1, Theorem C.1] The matrix P defined by (3.3)/(3.4) is totally positive, coefficientwise in the indeterminates α, β, ξ, η, a, b, c, d, e, f.

See[20, footnote 4] for many references.

See alsoFlajolet and Françon [8, Theorem 1] and Milne[START_REF] Milne | Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions, and Schur functions[END_REF] Theorems

3.2 and 3.11] for modern presentations of Rogers' elegant proof of the continued fraction.

Partially supported by the DIMERS project ANR-18-CE40-0033.