Rémi Deterre

Pierre Mousseau

Alain Sarda

Julien Launay

Julien Launay Thermal

Thermal control and Energy balance in polymer processing

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

INTRODUCTION

Improving the energy efficiency of tooling is a challenge for rubber industries today in terms of energy consumption, productivity and product quality. The design of the tools evolves towards the reduction of the heated mass and the thermal insulation of the molds [(The Society of the Plastics Industry and Energy, 2005), (Smithers, 2007), (Project, 2007), (Report, 1998)]. In this work, we are interested in controlling and mastering the thermal regulation of a tool used in the shaping of the elastomeric materials. The aim is to propose a new approach of a tool design for the improvement of heat transfers and better control in parts quality. We present the design and instrumentation of the experimental set-up. Experimental measurements allowed us to understand the thermal of the tool and to show the thermal heterogeneities on the surface of the mold and in the various cavities. Injection molding tests of the rubber and a thermal balance on the energy consumption of the tool were carried out.

EXPERIMENTAL SET-UP

Rubber injection molding process

The Figure 1 describes the injection device used for the samples molding. We used a vertical rubber injection molding machine (REP V39). The mold is fixed on two heating plates. The maximum clamp force is 110 T. The rubber is introduced into the storage jar at 70 °C. In a second stage, the injection piston pushes the polymer inside the hot mold regulated at 180 °C.

Rubber injection molding process

The Figure 2 describes the molded parts.

Thermal control and Energy balance in polymer processing R. Deterre, P. Mousseau, A. Sarda

GEPEA -OSE Université de Nantes

J. Launay

Capacités Université de Nantes ABSTRACT: The rubber industry is facing a lack of thermal control during rubber processing. In the process of injection molding of elastomers, improving the energy efficiency of the tools is a current challenge for industry in terms of energy consumption, productivity and product quality. Experimental measurements allow us to understand the thermal of the tool and to show the thermal heterogeneities on the surface of the mold and in the various cavities. Tests of injection molding of the rubber and a thermal balance on the energy consumption of the tool are carried out. In the rubber industry, 20% of the energy consumed by capital goods comes from heating processes; more than 50% of heat losses are linked to insufficient control and thermal insulation of Molds. The design of the tooling evolves in particular towards the reduction of the heated mass and the thermal insulation of the molds. The rubber compound is based on Ethylene-Propylene-Diene Monomer matrix. The compound has a density of 1070 kg /m3, a thermal conductivity of 0.298 W/m /K and a constant heat mass of 1490 J/kg/K. we used a RPA (Rubber Process Analyzer) in order to measure the kinetics curves of the rubber compound vulcanization.

Mold and thermal instrumentation

The mold is equipped with thermal and pressure sensors (cf. Figure 3). The thermal device is able to measure the rubber temperature evolution in the channels and in the various cavities (Fekiri, et al., 2017).

THERMO-KINETIC MODEL OF MOLDING

The model geometry used for the numerical curing simulation is represented in Figure 4. Mould temperature regulation is imposed on the boundaries Γ1 and Γ5. Ωm represents the metal boundaries of the mould, Ωis represents the insulating guarding element and Ωc represents the rubber part. T+ represents the mould temperature that is to be prescribed for the optimized curing cycle. During the phase (1) in Figure 2, heat transfer in the rubber part is described by equations ( 1),( 2) and ( 3) with a source term related to the enthalpy of the vulcanization reaction.

  c c c c c c T Cp T T H tt                  (1)
The boundary conditions of phase II are described by equations ( 8) and ( 9).

    T t t T t t      
(2)

   6 7 8 c c c ext T T T h T T T z r z                          (3)
We used a modification of Isayev's model (induction + vulcanization) by adding a reversion model (equation 4-11), according to Coran's model (Coran, 1964).

( ) . ( ) ( ). ( , ) ( ). ( )

c W t F T W t G T W t H T t              (4) 0 1 () 1 1 i i i if t Wt if t       (5)
The reduced induction time i t is given in equation ( 12):

  0 (T t, r, z ) t i i dt t t   (6)
With:

( / ) 0

B TT i t t e  (7)
The second term of equation ( 4), F, is defined as follows:

1/ 1/ 1/ ( , ) (1 )
n n n n n F T nk      (8)
With:

( / ) 0 e E RT kk   (9)
The last terms (4) describe the reversion reaction.

  The temperature measurements used to validate the thermal model were recorded during various moulding cycles. The comparison between experi-mental measurements and our numerical model solution shows a good agreement. The comparison between experimental vulcanization values and the calculated ones showed, also, a good agreement.

THERMAL BEHAVIOR OF RUBBER AND

MOLD We investigate the temperature evolution depending from the part and the mold configurations.

Viscous heating during filling

The Figure 5 reports the surface temperature measured in the channel during the filling stage. We observe a temperature rise more than 30°C of the rubber during the filling stage. The magnitude of this value is 5 time higher than the average temperature rise corresponding to an adiabatic calculation (Agassant, et al., 2017). This is due to the high temperature gradient developed at the runner wall.

Effect of the rubber part

We used the thermal model developed previously, in order to simulate various industrial case studies. The following one concerns a rubber boots. The Figure 6 reports the temperature gradient in a mold section during a whole cycle. We detect a big difference between the mold and the metal insert. This is due to the insulation effect due to the rubber presence in the cavity. Such effect induces thermal heterogeneity, especially for the metal inserts in the mold. Such a model is often use as a decision support tool in order to choose the best thermal conditions for the production setup.

Effect of the part geometry

The following case study shows the molding of various part geometries in the same mold. The Figure 7 reports the temperature gradient in a mold section. The Figure 8 shows the temperature versus time of various parts molded in the same mold. We observe the big influence of the part geometry especially the part thickness effect. Nevertheless, such a model allows to predict and eventually to show the possibility to make various parts in the same mold. In another way, this model was used in order to mold various rubber compounds (with various cure kinetics) in various cavities in the same mold during the same molding cycle.

Effect of the number of parts

We studied the effect of the part number molded in the same mold. The Figure 9 shows the temperature and the cure evolution versus time in a multicavity mold. The curves compare the temperature and the state of cure of the central part with the peripheral one.

Thermal regulation temperature = 180°C Such analysis is necessary to determine the maximum number of parts and eventually to the thermal improvements necessary to undertake (e.g. thermal insulation).

EFFECT OF PROCESS PARAMETERS

We used the coupled rubber part and mold simulation in order to investigate the effect of various process parameters.

Thermal cycling

We measured the temperature evolution at various places in the mold versus time, in the previous industrial case study. The Figure 12 shows the various temperatures curves during many cycles with various cycle times. We observe the unstationary thermal behavior of the mold. Such behavior should explain the effect of cycling change on the part quality.

Thermal regulation

The temperature measurements have been done in the same industrial case but with various thermal regulation conditions.

The Figure 11 shows effect of a classical regulation system (e.g. PID) on the temperature versus time evolution. We observe the temperature overshoot due to the thermal regulation effect.

A realistic modeling of the thermal regulation reveals a 15K thermal gradient along an industrial thermal resistance. Such a gradient leads to a 5K gradient at the mold surface. Moreover, we simulate the effect of various regulation types. In the previous industrial case, a central regulation point leads to 180°C in the center and 160°C on the sides. In the same case, a 3 points regulation system results in 180° in the center and 180°C+-1°C on the sides.

ENERGY BALANCE IN RUBBER MOLDING

The validated thermal model allows to calculate the energy balance during a molding cycle. The Figure 12 illustrates the energy losses around a mold section. The energy balance corresponding to this case indicates the following repartition of the global energy spent during the whole molding cycle: 66% loss towards the press plates, 14% convection during mold opening, and 12% convection during the mold close phase. Such balance reveals also that less than 10% of the total energy is devoted to the vulcanization process.

CONCLUSION

The design of the tooling evolves in particular towards the reduction of the heated mass and the thermal insulation of the molds. Experimental measurements allow us to understand the thermal of the tool and to show the thermal heterogeneities on the surface of the mold and in the various cavities. Tests of injection molding of the rubber and a thermal balance on the energy consumption reveals that 20% of the energy consumed by capital goods comes from heating processes. More than 50% of heat losses are linked to inadequate control and thermal insulation of molds. A. Coran (1964)

Figure 1 :

 1 Figure 1: Injection molding press and mold.

Figure 2 :

 2 Figure 2: rubber parts and runners used in the experimental setup.

Figure 3 :

 3 Figure 3: view of the mold cavities equipped with sensors.

Figure 4 :

 4 Figure 4: Schematic diagram of the model geometry.

Figure 5 :

 5 Figure 5: viscous heating measurements during filling stage.

Figure 6 :

 6 Figure 6: temperature and state of cure field during the molding of a boots.

Figure 7 :

 7 Figure 7: temperature field in the cross section of a mold during the molding cycle of various parts in the same mold.

Figure 8 :

 8 Figure 8: viscous heating measurement during filling stage.

Figure 9 :

 9 Figure 9: temperature and cure evolution during the molding of numerous parts.

Figure 10 :

 10 Figure 10: effect the cycle time variation on the mold temperature (at various places in the runners and the cavities).

Figure 11 :

 11 Figure 11: effect of the thermal regulation on the mold temperature at various places in the runners and the cavities.

Figure 12 :

 12 Figure 12: energy balance during molding cycle.

  . Vulcanization. Part VI. A model and treatment for scorch delay kinetics. Rubber Chemistry and Technology, 37: 689-697. J.-F. Agassant, P. Avenas, P. J. Carreau, B. Vergnes and M. Vincent (2017). Polymer processing: principles and modeling. Carl Hanser Verlag GmbH Co KG. Report, United Nations Industrial Development Organization (UNIDO). Ministry of International Trade and Industry (MITI), Japan, Hanoi, 1998. The Society of the Plastics Industry, U.S.D.o. Energy, in: I. The Society of the Plastics Industry (Ed.), U.S. Department of Energy, Washington, DC, 2005. N. Fekiri, C. Canto, Y. Madec, P. Mousseau, C. Plot and A. Sarda (2017). Design and thermal analysis of a mold used in the injection of elastomers. AIP Con-ference Proceedings, 1896: 120009.

183,4°C 174,4°C 165,4°C 147,6°C 111,8°C 93,8°C 156,5°C 138,6°C 129,7°C 120,7°C 102,8°C Time (s) Temperature (°C)


Anh-Minh Nguyen

Mounib Bahri

Sébastien Dreyfuss

Simona Moldovan

Antoine Miche

Christophe Méthivier

Ovidiu Ersen

Nicolas Mézailles
email: mezailles@chimie.ups-tlse.fr

Sophie Carenco
email: sophie.carenco@sorbonne-universite.fr

Bimetallic Phosphide (Ni,Cu) 2 P Nanoparticles by Inward Phosphorus Migration and Outward Copper Migration

come

Introduction

Metal phosphides (M x P y ) have a large variety of compositions and a wide range of applications in catalysis, 1-3 magnetism, [START_REF] Gregg | Controlled Synthesis of MnP Nanorods: Effect of Shape Anisotropy on Magnetization[END_REF][START_REF] Imai | Static and Dynamical Magnetic Properties of the Itinerant Ferromagnet LaCo2P2[END_REF][START_REF] Balli | Magnetocaloric Effect in Ternary Metal Phosphides (Fe1-XNix)2P[END_REF] lithium-ion batteries, [START_REF] Gillot | Electrochemical Reactivity and Design of NiP2 Negative Electrodes for Secondary Li-Ion Batteries[END_REF][START_REF] Boyanov | Nanostructured Transition Metal Phosphide as Negative Electrode for Lithium-Ion Batteries[END_REF][START_REF] Boyanov | P-Redox Mechanism at the Origin of the High Lithium Storage in NiP 2-Based Batteries[END_REF] as semi-conductors 10- 12 and for superconductivity. [START_REF] Mine | Nickel-Based Phosphide Superconductor with Infinite-Layer Structure, BaNi2P2[END_REF] At the nanoscale, they have also attracted great interest due to their promising performances especially in electrocatalysis [START_REF] Shi | Recent Advances in Transition Metal Phosphide Nanomaterials: Synthesis and Applications in Hydrogen Evolution Reaction[END_REF][START_REF] Popczun | Nanostructured Nickel Phosphide as an Electrocatalyst for the Hydrogen Evolution Reaction[END_REF] and photoluminescence [START_REF] Micic | Synthesis and Characterization of InP, GaP, and GaInP2 Quantum Dots[END_REF][START_REF] Wang | Highly Polarized Photoluminescence and Photodetection from Single Indium Phosphide Nanowires[END_REF][START_REF] Tamang | Chemistry of InP Nanocrystal Syntheses[END_REF] among other applications. While chemists have been initially working on synthesis of monometallic phosphide nanoparticles [START_REF] Tamang | Chemistry of InP Nanocrystal Syntheses[END_REF][START_REF] Henkes | Trioctylphosphine: A General Phosphorus Source for the Low-Temperature Conversion of Metals into Metal Phosphides[END_REF][START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF] , recent efforts have focused on the controlled synthesis of bimetallic phosphide nanoparticles, which show enhanced properties. [START_REF] Mutinda | Synthesis and Characterization of Co2-XRhxP Nanoparticles and Their Catalytic Activity towards the Oxygen Evolution Reaction[END_REF][START_REF] Li | Efficient Water Oxidation Using CoMnP Nanoparticles[END_REF][START_REF] Hitihami-Mudiyanselage | Synthesis and Characterization of Discrete FexNi2-XP Nanocrystals (0 < x < 2): Compositional Effects on Magnetic Properties[END_REF][START_REF] Liyanage | Simultaneous Control of Composition, Size, and Morphology in Discrete Ni2-XCoxP Nanoparticles[END_REF] Most reported studies relied on the non-quantitative decomposition of tri-n-octylphosphine (TOP), triphenylphosphine (PPh 3 ) or triphenylphosphite (TPP) in the presence of metal complexes or salts, thus forming metal phosphide nanoparticles in one step or two steps. [START_REF] Liu | Colloidal Ni2-: XCoxP Nanocrystals for the Hydrogen Evolution Reaction[END_REF] To synthesize monometallic phosphides, insertion of phosphorus into preformed nanoparticles was also investigated. This approach allows an excellent control over size, shape and crystalline phase because it preserves the morphology and size-distribution of the starting metal nanoparticles. [START_REF] Henkes | Template-Assisted Synthesis of Shape-Controlled Rh2P Nanocrystals[END_REF] Nevertheless, this insertion route was rarely used on bimetallic nanoparticles [START_REF] Duan | Au/Ni12P5 Core/Shell Nanocrystals from Bimetallic Heterostructures: In Situ Synthesis, Evolution and Supercapacitor Properties[END_REF] despite the large number of candidates available. [START_REF] Gilroy | Bimetallic Nanocrystals: Syntheses, Properties, and Applications[END_REF] Indeed, synthesis of bimetallic nanoparticles, such as Au-Cu, Pt-Pd, Pt-Ni and Cu-Ni [START_REF] Yamauchi | Magnetic Cu-Ni (Core-Shell) Nanoparticles in a One-Pot Reaction under Microwave Irradiation[END_REF][START_REF] Zhang | Near-Monodisperse Ni-Cu Bimetallic Nanocrystals of Variable Composition: Controlled Synthesis and Catalytic Activity for H 2 Generation[END_REF][START_REF] Bonet | Synthesis and Characterization of Bimetallic Ni-Cu Particles[END_REF][START_REF] Liu | Triphenyl Phosphite as the Phosphorus Source for the Scalable and Cost-Effective Production of Transition Metal Phosphides[END_REF] , with tunable composition and precise atomic order control is a mature field.

Copper phosphide and nickel phosphide nanoparticles show good performance in (electro)catalytic processes. [START_REF] Popczun | Nanostructured Nickel Phosphide as an Electrocatalyst for the Hydrogen Evolution Reaction[END_REF][START_REF] Wang | MOF-Derived Bifunctional Cu3P Nanoparticles Coated by a N,P-Codoped Carbon Shell for Hydrogen Evolution and Oxygen Reduction[END_REF][START_REF] Peng | Nanostructured Ni2P as a Robust Catalyst for the Hydrolytic Dehydrogenation of Ammonia-Borane[END_REF] While bulk mixed phosphides of copper and nickel (Ni,Cu) 3 P and (Ni,Cu) 2 P have been prepared by Nowotny et al. and Adlwarth et al., [START_REF] Nowotny | Ein Beitrag Zur Kenntnis Ternärer Phosphorlegierungen[END_REF][START_REF] Adlwarth | Crystallization of Amorphous Cu-Ni-P Alloys[END_REF] nanoscaled Cu-Ni-P have been synthesized by solution-based route [START_REF] Zhao | Composition-Dependent Morphology of Bi-and Trimetallic Phosphides: Construction of Amorphous Pd-Cu-Ni-P Nanoparticles as a Selective and Versatile Catalyst[END_REF] or by electrical wire explosion. [START_REF] Kim | Cu-Ni-P Alloy Nanoparticles Prepared by Electrical Wire Explosion[END_REF] In these last two examples, the nanoparticles crystallized in cubic phases, isostructural to metallic Cu and Ni, and the latter showed low phosphorus content (≤ 8.3 at.%) as well as inhomogeneous composition of individual particles.

There is no report so far on Ni-Cu-P nanoscaled phosphides presenting a crystal structure indexed as a metal phosphide. Hence, a strategy that involves insertion of phosphorus into pre-formed Ni-Cu nanoparticles does seem particularly appealing to target Ni-Cu-P nanoparticles by colloidal synthesis.

For this purpose, white phosphorus (P 4 ) shows several interesting features as a phosphorus donor. It acts as a stoichiometric P-donor for both nickel and copper, forming crystalline Ni 2 P and Cu 3 P nanoparticles. [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF][START_REF] Carenco | Magnetic Core-Shell Nanoparticles from Nanoscale-Induced Phase Segregation[END_REF] No by-product is produced during the reaction and phosphorus insertion started at temperature as low as 100 °C.

Furthermore, it was shown to be a stoichiometric donor in other cases such as Pd 5 P 2 , [START_REF] Carenco | Structural Transitions at the Nanoscale: The Example of Palladium Phosphides Synthesized from White Phosphorus[END_REF] PdP 2 , [START_REF] Carenco | Structural Transitions at the Nanoscale: The Example of Palladium Phosphides Synthesized from White Phosphorus[END_REF] FeP [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF] or InP. [START_REF] Carenco | White Phosphorus and Metal Nanoparticles: A Versatile Route to Metal Phosphide Nanoparticles[END_REF] Interestingly, the phosphorus insertion occurred through amorphization of the nanoparticles for nickel and palladium, but not for copper.

Herein, we propose a one-step colloidal route for Cu 1-x Ni x nanoparticles. We investigated the reaction of core-shell Cu 0.25 Ni 0.75 nanoparticles with white phosphorus, for the production of crystalline Ni-Cu-P nanoparticles. The stoichiometry was selected to target full phosphidation of Ni into Ni 2 P. This study provides insights on three mechanistic aspects: (i) preference of phosphorus towards one of the metals, (ii) amorphization and recrystallization during insertion of phosphorus and (iii) outward migration of copper vs. inward migration of phosphorus.

Results and discussion

Synthesis and characterization of bimetallic Cu 1-x Ni x nanoparticles.

In order to prepare bimetallic Cu 1-x Ni x nanoparticles (x = 0.33, 0.50, 0.67, 0.75), we adapted a synthetic protocol designed for pure nickel nanoparticles. [START_REF] Carenco | Controlled Design of Size-Tunable Monodisperse Nickel Nanoparticles[END_REF] Cu(acac) 2 and Ni(acac) 2 were reduced at 220 °C by oleylamine, in the presence of tri-n-octylphosphine (TOP) that served as a stabilizing agent (see experimental section). The reactions were carried out under inert atmosphere to yield colloidal solutions. Black powders of Cu 1-

x Ni x nanoparticles were collected by centrifugation and washing in air. Inductively coupled plasma with atomic emission spectroscopy (ICP-AES) showed that the final compositions are in good accordance with the expected ones (ESI Table S1).

X-ray diffraction (XRD) pattern of Cu 0.25 Ni 0.75 nanoparticles is presented in Figure 1.

The nanoparticles crystallized in the same fcc phase as metallic copper and nickel. However, their diffraction peaks were shifted to larger angles compared to Cu and smaller angles compared to Ni (Figure 1a and ESI Figure S1). The (111) peaks were fitted by two pseudo-Voigt curves to identify two components at 43.6 ° and 44.4 ° (Figure 1b). The former was attributed to Cu-rich crystalline domains and the latter to Ni-rich ones. The presence of two components indicated that copper and nickel do not form a single-phase solid solution. XRD patterns of Cu 1-x Ni x nanoparticles (x = 0.25, 0.50, 0.67) synthesized with other metal ratios show gradual shifts of all three peaks (111), ( 200) and (220) to larger angles with increasing nickel content (ESI Figure S1).

Cu-rich and Ni-rich phases could be distinguished from the (111) peak of all Cu 1-x Ni x samples. The two components were resolved for x ≥ 0.67. According to the Scherrer equation, the Ni-rich and Cu-rich crystallite domains were respectively 8 nm and 11 nm large, in average. Bright-field transmission electron microscopy (TEM) images of Cu 0.25 Ni 0.75 showed quasi-spherical nanoparticles with average diameter of 40 nm (Figure 1c). This diameter was larger than the average crystallite size, suggesting that the nanoparticles were polycrystalline. Accordingly, high resolution scanning transmission electron microscopy (HR-STEM) revealed several crystalline domains of ca. 10 nm within one particle (ESI Figure S2). TEM images of Cu 0.75 Ni 0.25 , Cu 0.50 Ni 0.50 and Cu 0.33 Ni 0.67 nanoparticles are shown in ESI Figure S3. The size distribution of the nanoparticles became broader with increasing copper content. This is likely due to the increasing copper leaching from copper-rich nanoparticles in the presence of TOP, [START_REF] Frogneux | Surprisingly High Sensitivity of Copper Nanoparticles toward Coordinating Ligands: Consequences for the Hydride Reduction of Benzaldehyde[END_REF] which fastened the defocusing size ripening of the nanoparticles.

Energy dispersive spectroscopy (EDS) in STEM showed a core-shell structure for the Cu 0.25 Ni 0.75 nanoparticles (Figure 1). This partial segregation is compatible with the XRD pattern which indicated two types of crystallites. The core was rich in Cu while the shell was rich in Ni. Based on EDS maps performed on the TEM, apparent ratios could be estimated, although the measurement in the center of the nanoparticle also took into account the shell that is crossed twice by the electron beam. Uncertainty of all the values discussed below is in the order of 10 %. The apparent composition was Cu 0.07 Ni 0.93 in the shell and Cu 0.46 Ni 0.54 in the core. STEM-EDS maps of Cu 1-x Ni x nanoparticles with others metal ratios showed core-shell morphology as well (ESI Figure S4). In particular, Cu 0.33 Ni 0.67 nanoparticles with similar size to Cu 0.25 Ni 0.75 , showed an apparent composition of Cu 0.09 Ni 0.91 in the shell and Cu 0.57 Ni 0.43 in the core.

As expected, the shell was thinner than that of Cu 0.25 Ni 0.75 nanoparticles i.e. 7 nm compared to 10 nm.

The core-shell structure was favored over an alloy or Janus structure, in agreement with the syntheses previously developed in the literature for other copper-nickel nanoparticles. [START_REF] Yamauchi | Magnetic Cu-Ni (Core-Shell) Nanoparticles in a One-Pot Reaction under Microwave Irradiation[END_REF][START_REF] Zhang | Near-Monodisperse Ni-Cu Bimetallic Nanocrystals of Variable Composition: Controlled Synthesis and Catalytic Activity for H 2 Generation[END_REF][START_REF] Bonet | Synthesis and Characterization of Bimetallic Ni-Cu Particles[END_REF] However, our results differ from the work of Liu et al. who obtained alloys using a similar approach but at a lower temperature i.e. 180 °C and with longer reaction time of 4 h. [START_REF] Liu | Facile Synthesis of Cu/Ni Alloy Nanospheres with Tunable Size and Elemental Ratio[END_REF] In our reaction, the solution readily turned dark at ca. 180 °C. We know from our previous work that the reduction of Ni(acac) 2 starts at 213 °C. [START_REF] Carenco | Controlled Design of Size-Tunable Monodisperse Nickel Nanoparticles[END_REF][START_REF] Carenco | Revisiting the Molecular Roots of a Ubiquitously Successful Synthesis: Nickel(0) Nanoparticles by Reduction of [Ni(Acetylacetonate) 2[END_REF] Hence, we propose that Cu nuclei form at 180 °C and that reduction of nickel at a higher temperature is responsible for the partially segregated morphology. On the powder of nanoparticles isolated in air, X-ray photoelectron spectroscopy (XPS) of the P 2p region showed the presence of phosphorus species on the surface of the nanoparticles. Each component was represented by 2p 3/2 and 2p 1/2 doublet peaks with a spin-orbit splitting of 0.84 eV. In the following, we will only indicate the binding energy (BE) of the 2p 3/2 component. Three components were identified at 129.4, 132.6 and 133.0 eV (ESI Figure S5b). The component at 129.4 eV, representing 13% of the total phosphorus content detected at the surface, was attributed to nickel phosphide. [START_REF] Nemoshalenko | Investigation of the Atomic Charges in Iron, Cobalt and Nickel Phosphide[END_REF] This phosphide came from a partial decomposition of TOP and contaminates the surface, in agreement with previous reports. [START_REF] Carenco | The Birth of Nickel Phosphide Catalysts: Monitoring Phosphorus Insertion into Nickel[END_REF][START_REF] Moreau | Defining Crystalline/Amorphous Phases of Nanoparticles through X-Ray Absorption Spectroscopy and X-Ray Diffraction: The Case of Nickel Phosphide[END_REF][START_REF] Moreau | Unintended Phosphorus Doping of Nickel Nanoparticles during Synthesis with TOP: A Discovery through Structural Analysis[END_REF] The component at 132.6 eV was attributed to phosphine oxide, [START_REF] Yerushalmi | Phosphine Oxide Monolayers on SiO2 Surfaces[END_REF][START_REF] Carenco | Synthesis and Structural Evolution of Nickel-Cobalt Nanoparticles under H2 and CO2[END_REF] originating from oxidation of TOP into tri-n-octylphosphine oxide (TOPO) during the post-reaction treatment. Lastly, the component at 133.0 eV was attributed to phosphates, from partial phosphide oxidation. [START_REF] Franke | Auger Parameters and Relaxation Energies of Phosphorus in Solid Compounds[END_REF] On the metals side, Ni 2p and Cu 2p were measured (ESI Figure S5c andd). Partially oxidized nickel was detected on the surface while no signal of oxidized Cu was detected in the Cu 2p region, in agreement with the proposed structure of a nickel thick shell. Stoichiometric reaction of white phosphorus on copper and nickel nanoparticles was previously reported. [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF][START_REF] Carenco | Magnetic Core-Shell Nanoparticles from Nanoscale-Induced Phase Segregation[END_REF] Using stoichiometric amounts of P 4 , crystalline Cu 3 P was formed at 100 °C, while Ni 2 P only partially crystallized at 150 °C and fully crystallized at 220 °C. Using sub-stoichiometric amounts of P 4 , Ni 3 P did not crystallize. Rather, nanoparticles with Ni 2 P core and metallic Ni shell were obtained. In the case of copper, faster crystallization of the phosphide phase led to nanoparticles with the opposite structure: metallic Cu core and Cu 3 P shell.

Phosphorus insertion into

In this work, core-shell Cu 0.25 Ni 0.75 nanoparticles were reacted with P 4 in substoichiometric amount compared to the fully phosphidized Cu 3 P-Ni 2 P. The chosen stoichiometry of the final product was Cu 0.25 Ni 0.75 P 0.375 , corresponding to hypothetical "Cu-Ni 2 P". Two borderline scenarios were envisioned (Figure 2): either a diffusion of phosphorus limited to the shell, yielding a metallic core and a phosphide shell, or a diffusion of phosphorus to the core, yielding a bimetallic phosphide alloy. The outcome would depend on the relative kinetics of crystallization of both phosphide phases and the preference of phosphorus towards one of the metals.

(i) Preference of phosphorus towards one of the metals

To obtain a fully crystallized product, as-synthesized Cu 0.25 Ni 0.75 nanoparticles were reacted directly after synthesis (no further purification) with a solution of P 4 in toluene at 250 °C for 2 h. Toluene was evaporated in vacuo before heating to avoid reflux at the reaction temperature. [START_REF] Peng | Nanostructured Ni2P as a Robust Catalyst for the Hydrolytic Dehydrogenation of Ammonia-Borane[END_REF] P NMR spectrum of the supernatant after centrifugation shows that P 4 was completely consumed (ESI Figure S7b). TOP remained from the first step of the synthesis and gave a signal at -31.9 ppm (see inset). Two phosphorus species at oxidation degree +V were detected at 43.2 and 43.1 ppm, one of them likely corresponding to TOPO and the other unidentified. Chemical composition of the powder was evaluated by EDS. The measured composition Cu 0.22 Ni 0.78 P 0.34 was comparable to the expected one, indicating a quantitative reaction between the coppernickel nanoparticles and P 4 .

XRD patterns of the resulting nanoparticles showed a phase isostructural to hexagonal Ni 2 P as well as fcc metallic Cu (Figure 3). Diffraction peaks of the phosphide phase were shifted from those of the reported Ni 2 P (JCPDS 74-1385). Peaks of (hkl) (l ≠ 0) planes were shifted towards wider angles while those of (hkl) (l = 0) planes were shifted towards smaller angles, suggesting a distortion of the unit cell. We therefore propose the formation of a mixed phosphide (Ni,Cu) 2 P phase, with a partial substitution of copper to nickel in the unit cell resulting in the structural distortion.

To further substantiate this proposal, we went back to the few articles discussing these structural parameters. Distortion of the unit cell is often discussed using the ratio c/a of lattice parameters. [START_REF] Dera | Structure and Behavior of the Barringerite Ni End-Member, Ni2P, at Deep Earth Conditions and Implications for Natural Fe-Ni Phosphides in Planetary Cores[END_REF] Nowotny et al. reported that macroscopic (Ni,Cu) 2 P phase can form for Cu:Ni ratios up to ca. 25%. [START_REF] Nowotny | Ein Beitrag Zur Kenntnis Ternärer Phosphorlegierungen[END_REF] Measurement uncertainty of our XRD data is around 0.001 nm for distances (this value depends on the angle, see Supporting Information page S15). At 25% Cu, lattice constants showed slightly deviated values of a = 0.584 nm and c = 0.340 nm from those of pure Ni 2 P phase (a = 0.586 nm and c = 0.338 nm). The resulting c/a ratio was 0.582 vs. 0.577 for pure Ni 2 P. In our case, profile matching of the XRD pattern showed lattice constants of a = 0.591 nm and c = 0.333 nm, giving a c/a ratio of 0.563. This indicated a more significant distortion at the nanoscale. Interestingly, as the c/a ratio decreased from that of pure bulk Ni 2 P phase, the distortion was also not of the same direction as in the bulk mixed phase (where the c/a ratio increases). We calculated the c/a ratio in nanoscale (Ni,Fe) 2 P or (Ni,Co) 2 P reported by another group, and it showed increasing values when compared to the pure hexagonal Ni 2 P phase (ESI Table S3). [START_REF] Hitihami-Mudiyanselage | Synthesis and Characterization of Discrete FexNi2-XP Nanocrystals (0 < x < 2): Compositional Effects on Magnetic Properties[END_REF][START_REF] Liyanage | Simultaneous Control of Composition, Size, and Morphology in Discrete Ni2-XCoxP Nanoparticles[END_REF] To the best of our knowledge, the (Ni,Cu) 2 P distortion reported in this article is the first example of decreasing c/a ratio for mixed phosphide phase. At this point, detection of metallic copper and of a bimetallic phosphide phase suggested a reaction closer to scenario (b) (Figure 2). This reasoning was shattered by TEM images that show a majority of hollow nanoparticles (Figure 4a). These hollow nanoparticles were monocrystalline (ESI Figure S8) whereas the starting nanoparticles were polycrystalline. This indicated that the nanoparticles were amorphized at an intermediate step, which will be studied in the next section. Lattice fringes with spacing of 0.51 nm were observed, corresponding to (100) crystal planes of the hexagonal Ni 2 P phase (Figure 4b). STEM-EDS showed that the hollow phosphide nanoparticles contained both Cu, Ni and phosphorus, in agreement with the proposed mixed phosphide (Ni,Cu) 2 P phase (Figure 4e-h). This was also confirmed by elemental profiles of a delineated zone across a hollow nanoparticle (ESI Figure S6). Alongside the hollow phosphide nanoparticles, higher contrast zones were also observed, either inside the shells or on top of them (Figure 4a,c,d). EDS elemental maps indicate that they did not contain phosphorus (Figure 4g), but only Cu. They were therefore the metallic copper crystallites determined in the XRD pattern. XPS of the nanoparticles powder in the Cu 2p region (ESI Figure S5d) showed the presence of metallic and oxidized Cu, confirming EDS observations. The lower B.E. of the Cu 2p peak could also partially arise from the copper phosphide, as the reported compound show similar binding energies for Cu (0) (932.7 eV), Cu 3 P (932.8 eV) [START_REF] Chen | Synthesis of Copper Phosphide Nanotube Arrays as Electrodes for Asymmetric Supercapacitors[END_REF] and CuP 2 (between 932.2 and 932.4 eV). [START_REF] Nefedov | A Study by XPS and XRS of the Participation in Chemical Bonding of the 3d Electrons of Copper, Zinc and Gallium[END_REF] In the Ni 2p 3/2 region, peaks of oxidized nickel were present at 855.2 and 855.9 eV. [START_REF] Biesinger | X-Ray Photoelectron Spectroscopic Chemical State Quantification of Mixed Nickel Metal, Oxide and Hydroxide Systems[END_REF] Reduced nickel was also detected at 852.9 eV (ESI Figure S5c). It was slightly shifted to higher binding energy (B.E.) than bimetallic copper-nickel nanoparticles whose main peak was at 852.3 eV. Unfortunately, Ni (0) and Ni 2 P are hardly distinguished by XPS.

Previous study by Li et al. attributed them the same B.E. of 853.1 eV, [START_REF] Li | XPS Studies on Surface Electronic Characteristics of Ni-B and Ni-P Amorphous Alloy and Its Correlation to Their Catalytic Properties[END_REF] while others tabulate Ni (0) at 852.7 eV and Ni 2 P at 852.9 eV, only 0.2 eV apart. [START_REF] Blanchard | X-Ray Photoelectron and Absorption Spectroscopy of Metal-Rich Phosphides M2P and M3P (M = Cr-Ni)[END_REF] XPS relative peak area indicated an apparent surface coverage of copper vs. nickel of one third of copper for two third of nickel, much richer in copper than what was found on the nanoparticles prior to phosphidation. This indicated that most of the copper migrated to the surface or to the outside of the nanoparticles, forming aggregates on their own, in agreement with STEM-EDS analysis. Besides, the apparent ratio of surface phosphorus vs. the sum of the metals was ca 60%, while it was only 30% prior to the reaction with white phosphorus. Oxidized phosphorus species, similar to these of the starting nanoparticles, were still detected in the form of TOPO and phosphates at 132.8 and 133.2 eV respectively. P 2p indicated the presence of metal phosphide at 129.5 eV, [START_REF] Nemoshalenko | Investigation of the Atomic Charges in Iron, Cobalt and Nickel Phosphide[END_REF] representing 33% of the surface phosphorus (ESI Figure S5b). A minor component was also detected at 130.2 eV, representing 14%. This was assigned to a different reduced phosphorus species as the B.E. value is close to that of metal phosphide. Altogether, XPS confirmed the formation of surface metal phosphide on the nanoparticles and the presence of metallic copper on the surface of the nanoparticles and/or as separate nanoparticles.

The formation of hollow (Ni,Cu) 2 P nanoparticles and metallic Cu nanoparticles suggested an outward migration of copper from the core to the shell and even outside of the starting nanoparticles during the reaction with P 4 , a scenario that was not initially envisioned. Moreover, phosphorus showed a preference towards nickel as shown by STEM-EDS and XRD, possibly due to the initial core-shell morphology where nickel was in contact with the reaction medium.

(ii) Amorphization and recrystallization during insertion of phosphorus

As mentioned above, bimetallic copper-nickel nanoparticles likely underwent amorphization during phosphorus insertion. This hypothesis is made based on previous observations on the phosphidation of crystalline nickel nanoparticles by white phosphorus, for which we demonstrated the formation of an amorphous intermediate even when a low stoichiometry of phosphorus was reacted (down to 0.25 P equiv. vs.

Ni). [START_REF] Carenco | Magnetic Core-Shell Nanoparticles from Nanoscale-Induced Phase Segregation[END_REF] This was expected because there is no topotactic transformation possible from Ni fcc to hexagonal Ni 2 P. [START_REF] Carenco | Designing Nanoparticles and Nanoalloys with Controlled Surface and Reactivity[END_REF] A fully amorphous nickel phosphide compound, showing a broad peak from 40 to 50 ° in XRD, was isolated at low temperature (90 °C). [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF] In the present case, nickel is exposed at the surface of the nanoparticles and has to react first with P 4 .It would thus make sense that the first step of the reaction was the formation of a nickel phosphide amorphous shell.

In order to investigate the reaction mechanism, including the surprising and unprecedented outward diffusion of copper, a second reaction was performed on the Cu 0.25 Ni 0.75 nanoparticles, with the same stoichiometry of P 4 , but at a lower temperature of 150 °C. [START_REF] Peng | Nanostructured Ni2P as a Robust Catalyst for the Hydrolytic Dehydrogenation of Ammonia-Borane[END_REF] P NMR spectrum of the supernatant after centrifugation showed again that P 4 was entirely consumed, suggesting a complete conversion (ESI Figure S7c). Indeed, only TOP and TOPO were detected at -31.2 and 44.2 ppm. EDS indicates a chemical composition of Cu 0.23 Ni 0.77 P 0.43 , comparable to these of the product formed at 250 °C if one takes into account the experimental error of EDS (ca. 5%) and the fact that the phosphorus-containing ligands (of variable amount from sample to sample) contribute to the overall phosphorus amount.

The isolated powder was partially amorphous as shown by XRD (Figure 5a) through a broad feature from 40 to 50 °. This feature may correspond to amorphous nickel phosphide [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF][START_REF] Carenco | Magnetic Core-Shell Nanoparticles from Nanoscale-Induced Phase Segregation[END_REF] but also to amorphous copper phosphide or copper-nickel phosphide. The diffractogram also presented diffraction peaks of hexagonal Ni 2 P and fcc metallic Cu.

This confirmed a two-step process with first, an amorphization driven by phosphorus insertion and/or copper migration, and second, a recrystallization. Interestingly, no shift was observed for the diffraction peaks of the Ni 2 P phase: pure Ni 2 P phase seems to crystallize first, before evolving towards mixed (Ni,Cu) 2 P phase as a consequence of copper outward migration.

In order to confirm this, the obtained partially amorphous nanoparticles were redispersed in oleylamine and annealed at 250 °C for 2 h. As expected, XRD pattern of the annealed product showed crystalline (Ni,Cu) 2 P alongside fcc metallic copper, but no broad signal resulting from an amorphous phase. However, diffraction peaks of the mixed phosphide phase were more shifted compared with these of Ni 2 P, than in those of the nanoparticles formed at 250 °C. In fact, the c/a ratio decreases to 0.559. TEM of the sample synthesized at 150 °C showed full and partially hollow nanoparticles, alongside hollow nanoparticles (Figure 5b). This suggested that outward migration of copper already started and was still undergoing at this temperature. Surprisingly, no significant morphology change was observed in the annealed sample (Figure 5c): copper crystallites were not observed outside the nanoparticles. Isolating the nanoparticles after the first reaction step at 150 °C seems to deactivate the outward migration of copper. This isolation step removed most of the TOP from the reaction medium: initially 0.8 equiv. vs. the metals, which represents a large excess vs. the surface metal atoms. We recently showed that alkylphosphines are responsible for strong leaching of copper nanoparticles to solution. [START_REF] Frogneux | Surprisingly High Sensitivity of Copper Nanoparticles toward Coordinating Ligands: Consequences for the Hydride Reduction of Benzaldehyde[END_REF] Hence, we propose that, for the one-step reaction described in the previous section, the presence of TOP in the reaction medium was a significant driving force for the outward migration of copper by displacing the equilibrium in the favor of soluble copper species. These species were then able to redeposit on the metal phosphide nanoparticles, as will be discussed in detail in the last section of the article. STEM-EDS elemental maps of the 150 °C sample showed that the nanoparticles react inhomogeneously with P 4 (Figure 6), providing an unexpected opportunity to catch snapshots of the reaction steps on the same TEM grid. Four different stages of phosphorus insertion could be distinguished, numbered intermediates (1) to (4) (Figure 6f-g). As shown in phosphorus STEM-EDS map (Figure 6e), for intermediate (1), phosphorus insertion was incomplete and stopped at the surface of the nanoparticle, whereas, for intermediates (2) to (4), phosphorus completely occupied the shell of the nanoparticles. Phosphorus diffusion always stopped at the shell and never continued into the core. This confirmed our previous conclusion on the preference of phosphorus towards nickel rather than copper in this system.

The copper map (Figure 6c) showed that the core of intermediates (1) and (2) were still intact while intermediates (3) and (4) were partly and fully hollow, respectively.

Interestingly, the shell of intermediates (2) to (4) always contained the three elements, suggesting the presence of amorphous bimetallic phosphide, detected as a broad amorphous species by XRD below the crystalline Ni 2 P (Figure 5a). The amorphous nature of these intermediates was confirmed by Selected Area Electronic Diffraction (SAED) (ESI Figure S9). This indicated that there are two competing reaction pathways: a first where crystalline Ni 2 P formed first then evolved towards mixed (Ni,Cu) 2 P through copper migration and a second where copper migration occurred first to form amorphous mixed phosphide that crystallized afterward, at 250 °C.

XPS of the non-annealed sample in the Cu 2p and Ni 2p regions showed again the presence of both reduced and oxidized species (ESI Figure S5). While comparing with the sample formed at 250 °C, the apparent copper content measured at the surface increased from 14% to 25% and the nickel content decreases from 27% to 14%. This is likely due to the fact that outward migration of copper was still undergoing. We detected both remaining copper inside the shell or small copper species outside. P 2p region of the non-annealed was also similar to that of the sample formed at 250 °C (ESI Figure S5).

(iii) Outward migration of copper vs. inward migration of phosphorus

In order to determine the fate of copper that migrated outward the nanoparticles, and potentially/partially outside, during the reaction with white phosphorus at 250 °C, TEM images of aliquots from the reaction mixture were taken at different reaction times (Figure 7). As expected, the nanoparticles were still not hollow at room temperature (Figure 7a). The majority of the nanoparticles was already hollow at 180 °C (Figure 7b), consistently with the TEM of nanoparticles prepared at 150 °C shown in the previous section. When the temperature reached 250 °C, small nanoparticles of diameter of around 4 nm were detected outside of the hollow nanoparticles (Figure 7c). After 30 min at this temperature, larger particles were found (Figure 7d), consistent with our observation of copper nanoparticles by STEM-EDS in the final product (Figure 4).

Outward migration of copper eventually resulted in nucleation and growth of copper nanoparticles outside of hollow phosphide nanoparticles. The migration of copper outside the nanoparticles was further confirmed using in situ STEM-EDS mapping (Figure 8). As discussed above, the sample formed after reaction with white phosphorus at 150 °C presented several intermediates of phosphorus insertion (Figure 6). Thus, we heated this sample under argon (1 bar) from room temperature up to 250 °C in a TEM gas cell. This experiment was slightly different than heating in solution, but since copper easily migrates on the silicon nitride membrane upon heating, we considered it a worthwhile comparison experiment. As expected, STEM images showed that the thickness of the core decreases from 18 nm to 6 nm between 100 °C (Figure 8a,c) and 250 °C (Figure 8b,d). Moreover, STEM-EDS elemental maps showed the copper migrating to the outside of the nanoparticles and redepositing nearby, from Figure 8e to Figure 8f. We showed that the inward migration of phosphorus and outward migration of copper already started at 150 °C when the core-shell copper-nickel nanoparticles were reacted with white phosphorus. However, between 150 and 250 °C, it was shown that TOP could also act as a phosphorus source for insertion into metal. [START_REF] Carenco | The Birth of Nickel Phosphide Catalysts: Monitoring Phosphorus Insertion into Nickel[END_REF][START_REF] Moreau | Defining Crystalline/Amorphous Phases of Nanoparticles through X-Ray Absorption Spectroscopy and X-Ray Diffraction: The Case of Nickel Phosphide[END_REF][START_REF] Moreau | Unintended Phosphorus Doping of Nickel Nanoparticles during Synthesis with TOP: A Discovery through Structural Analysis[END_REF] A control reaction was thus carried out at 180 °C in the same fashion without adding white phosphorus.

The only phosphorus source for insertion in this case was the TOP remaining from the first step of the bimetallic nanoparticle synthesis. No change in size, shape, crystal structure, chemical composition as well as apparent copper content measured on the surface of the nanoparticles was observed after the heating (ESI Figure S10), confirming that P 4 , as a more reactive P-donor, is required to activate outward migration of copper at 180 °C.

We then carried out a second control reaction at 250 °C. This time, XRD pattern of the sample showed pure orthorhombic phosphide Ni 12 P 5 and fcc metallic copper phases (ESI Figure S11), indicating phosphorus insertion. Chemical composition measured by EDS confirmed enrichment in phosphorus (ESI Figure S11). Interestingly, TEM images and STEM-EDS mapping showed full and faceted nanoparticles alongside only a few hollow nanoparticles (ESI Figure S11 and Figure S12). No copper was detected outside of the nanoparticles. XPS only showed very small apparent Cu-content (3 at.% of total Cu, Ni and P) on the surface (ESI Figure S11). The high-temperature degradation of TOP seems to inhibit copper migration, directly resulting in a crystalline phosphide shell.

Discussion

(i) Regarding the preferred formation of nickel-rich phosphide over copper-rich phosphide.

Overall, the above paragraphs describe the formation of a nickel-rich NiCuP alloy.

While some of the copper domains are not phosphidized, there is no remaining nickel domain at the end of the reaction. This suggests a preference of phosphorus for the nickel rather than for the copper.

Two aspects can be considered:

(1) The reaction path. Due to the core-shell morphology of the starting nanoparticles, phosphorus reacts first with nickel. The nickel phosphide, once formed, traps a significant amount of the available phosphorus species. This should however be reconciled with the fact that copper is able to diffuse through the nickel-rich phosphide shell. Kinetic control cannot be solely responsible for the observed result.

(2) The stability of the final phases. Data are not available on the ternary alloys but they are for the binary ones: the enthalpy of formation of Ni 2 P at 298 K (from -190 to -136 kJ/mol depending on the studies) is more negative than these of Cu 3 P (from -52 to -135 kJ/mol). [START_REF] Schlesinger | The Thermodynamic Properties of Phosphorus and Solid Binary Phosphides[END_REF] At the nanoscale and using the most reactive phosphorus source available (P 4 ), we can also note that nickel nanoparticles fully convert in Ni 2 P nanoparticles at a lower temperature (150 °C) than these required to convert Cu nanoparticles in Cu 3 P ones (250 °C or at least above 220 °C). [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF] These two arguments suggest a higher stability of nickel-rich phosphide over copper-rich ones.

To further advance this discussion, it would be interesting to perform the reaction of P 4 with a nickel-copper alloy or with an inverted core-shell (nickel inside, copper outside) and to measure the enthalpy of formation of the ternary alloys. Both go beyond the scope of this study. Also, a more systematic phosphidation study (preferably using a stoichiometric P donor) on other bimetallic nanoparticles (eg. NiPd) would be interesting, in order to eventually propose a general rule or law regarding the selectivity of the phosphidation reaction.

(ii) Regarding the formation of hollow nanostructures.

The formation of hollow nanostructures was previously observed for a number of materials, including metal oxides (as a result of oxidation reaction from O 2 ), [START_REF] Carenco | Synthesis and Structural Evolution of Nickel-Cobalt Nanoparticles under H2 and CO2[END_REF][START_REF] Varón | Spontaneous Formation of Hollow Cobalt Oxide Nanoparticles by the Kirkendall Effect at Room Temperature at the Water-Air Interface[END_REF] metal sulfides (due to sulfidation, eg. from S 8 ) [START_REF] Cabot | Sulfidation of Cadmium at the Nanoscale[END_REF] and phosphides (due to phosphidation, eg. from TOP [START_REF] Henkes | Template-Assisted Synthesis of Shape-Controlled Rh2P Nanocrystals[END_REF] or from P 4 ). [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF][START_REF] Carenco | 25th Anniversary Article: Exploring Nanoscaled Matter from Speciation to Phase Diagrams: Metal Phosphide Nanoparticles as a Case of Study[END_REF] The holes result from the coalescence of vacancies in the center of the nanoparticles. The vacancies are generated as the inner element (typically, the metal) diffuses outward faster than the outer element (typically, sulfur, oxygen, phosphorus) diffuses inward. It is often called "nano-Kirkendall" effect due to the phenomenological similitude with the original Kirkendall effect, which is the appearance of vacancies at the interface of two metals A and B, as a consequence of unbalanced diffusion coefficients of A toward B vs. B toward A. [START_REF] Smigelskas | Zinc Diffusion in Alpha Brass[END_REF] In the present case, the closest phenomenon related to a Kirkendall effect would have been the formation of vacancies at the copper-nickel interface due to unbalanced nickel and copper diffusion rates. Such phenomenon could be expected at a high enough temperature because the nickel diffuses slower in copper than the copper in the nickel, as well as in copper-nickel alloys. [START_REF] Porter | Phase Transformations in Metals and Alloys[END_REF] Here, the situation is different: the copper outward diffusion is a consequence of the reaction with an external non-metallic element: phosphorus. For clarity, we will hence avoid the term "nanoscale Kirkendall effect".

Under fairly similar reaction conditions, the phosphidation of single-metal nanoparticles of nickel or copper did not yield hollow nanoparticles in our previous studies. [START_REF] Carenco | Metal-Dependent Interplay between Crystallization and Phosphorus Diffusion during the Synthesis of Metal Phosphide Nanoparticles[END_REF] It should be noted that the study on the copper nanoparticles was performed in the absence of phosphine ligands. Based on these observations, we propose that:

(1) Nickel-rich NiCuP alloy allows the diffusion of Cu from the inside to the outside. This is especially well observed in situ (Figure 8).

(2) The formation of copper-phosphine species likely acts as a driving force for the copper migrating outside of the nanoparticles, as was also observed by Robinson and coworkers on copper sulfide nanoparticles. 63 Such copper-phosphine species were partly described in a previous work as complexes such as Cu(PR 3 ) 2 and clusters. [START_REF] Frogneux | Surprisingly High Sensitivity of Copper Nanoparticles toward Coordinating Ligands: Consequences for the Hydride Reduction of Benzaldehyde[END_REF]

Conclusions

In this work, we studied the reaction between core-shell copper-nickel nanoparticles (Cu:Ni in 1:3 ratio) and white phosphorus. Surprisingly, we obtained hollow Ni-rich phosphide nanoparticles and Cu nanoparticles, suggesting a preference of phosphorus towards nickel rather than copper in this particular case. The phosphide nanoparticles were shown to crystallize in a mixed (Ni,Cu) 2 P phase isostructural to Ni 2 P for the first time. Partial substitution of Ni atoms by Cu atoms in the structure leads to a distortion of the unit cell. The lattice constant ratio c/a decreases from 0.577 (pure Ni 2 P) to 0.563, in contrast with other reported Ni-containing bimetallic phosphide nanoparticles. [START_REF] Hitihami-Mudiyanselage | Synthesis and Characterization of Discrete FexNi2-XP Nanocrystals (0 < x < 2): Compositional Effects on Magnetic Properties[END_REF][START_REF] Liyanage | Simultaneous Control of Composition, Size, and Morphology in Discrete Ni2-XCoxP Nanoparticles[END_REF] From a mechanistic point of view, we showed that phosphorus insertion into the bimetallic nanoparticles triggers amorphization of the latter as well as outward migration of copper to leave a void in the core (Figure 9). During the outward migration of copper, two competing reaction pathways can occur. Either an amorphous intermediate of (Ni,Cu) 2 P forms first and crystallizes into mixed phosphide and/or pure Ni 2 P forms first and evolves towards (Ni,Cu) 2 P. Lastly, the copper that migrates outside the nanoparticles crystallizes into metallic fcc Cu nanoparticles. The low-temperature reactivity of white phosphorus promotes outward migration of copper while in the case of high-temperature degradation of TOP, the core-shell nanostructure of the starting bimetallic nanoparticles is mostly preserved. The resulting relative kinetics between copper migration and crystallization of the phosphide allows formation of a mixed phosphide (Ni,Cu) 2 P phase which is not observed when TOP is used as a sole source of P. The mixed phosphide nanoparticles were monocrystalline and faceted to expose the (001) surface, which is known to be active in HER, [START_REF] Popczun | Nanostructured Nickel Phosphide as an Electrocatalyst for the Hydrogen Evolution Reaction[END_REF] opening an avenue for further use of these unique structures as electrocatalysts.

This study of the phosphorus insertion into bimetallic nanoparticles using white phosphorus underlines the competing processes occurring during the reaction. These results from the interaction of stabilizing ligands with copper, as well as phosphorus migration and phosphide phase crystallization. Using a combination of local and ensemble analytic techniques, a complete rationalization of these processes is presented for the first time, in the case of bimetallic copper-nickel nanoparticles. This study paves the way for mechanistic studies of other bimetallic phosphide nanoparticles syntheses. Electrocatalytic activity of this new type of bimetallic metal phosphide is currently being assessed in our laboratories.

Experimental section

Chemicals. Oleylamine (OAm; technical grade, 70%) was purchased from Sigma-Aldrich. Tri-n-octylphosphine (TOP; 97%), copper (II) acetylacetonate (Cu(acac) 2 ; 98+%) and nickel (II) acetylacetonate (Ni(acac) 2 ; anhydrous, min. 95%) were purchased from Strem Chemicals and stored in the glovebox. All chemicals described above were used without further purification. White phosphorus (P 4 ) was taken from the laboratory's own chemical stockroom and was dissolved in toluene. A typical P 4 solution has a concentration around 0.4 mol/L in P. Safety note: White phosphorus is stable in water but highly flammable and toxic if swallowed or inhaled. It is incompatible with oxidizing agents, reducing agents and bases. It is light and heat sensitive and should be handled accordingly.

Synthesis of Cu 1-x Ni

x Nanoparticles. Bimetallic copper-nickel nanoparticles were prepared via a solvothermal reaction under either purified argon or nitrogen atmosphere using standard air-free techniques with Schlenk line. In a typical synthesis of Cu 0.25 Ni 0.75 , Cu(acac) 2 (0.51 g, 1.95 mmol, 0.25 equiv.), Ni(acac) 2 (1.50 g, 5.85 mmol, 0.75 equiv.) and 2.80 mL of TOP (6.24 mmol, 0.8 equiv.) were added to 20.8 g of OAm (78.0 mmol, 10 equiv.) in a 100 mL three-necked flask at room temperature. For nanoparticles with other Cu/Ni ratios, the quantity of each metal precursor was changed in a way that keeps the total amount at 7.80 mmol. The mixture was degassed under vacuum for 5 min then heated to 220 °C, giving quickly a black solution. The solution was stirred at this temperature for 2 h under inert atmosphere. After heating, it was left to cool to room temperature. The nanoparticles were isolated using acetone and were washed at least three times using a THF/ethanol (1/4) mixture to remove the remaining reagents and organic matter. 465 mg of black powder were obtained, which corresponds to a 100 % yield of Cu 0.25 Ni 0.75 (this calculation neglects the weight of organic ligands).

Reaction of Cu 0.25 Ni 0.75 Nanoparticles with White Phosphorus. The prepared bimetallic copper-nickel nanoparticles were not isolated before the reaction with white phosphorus. The crude solution from the first step was used as such. For a Cu 0.25 Ni 0.75 P 0.375 stoichiometry, 8.6 mL of a P 4 solution in toluene at 0.34 M in P (2.9 mmol, 0.375 equiv.) were added to the crude reaction mixture of the synthesis of Cu 0.25 Ni 0.75 . The mixture was degassed under vacuum for 30 min to eliminate the toluene. It was heated to 250 °C for 2 h under inert atmosphere then was left to cool to room temperature. The nanoparticles were isolated using the same method as for bimetallic nanoparticles. Supernatant from the first centrifugation were collected to analyze with [START_REF] Peng | Nanostructured Ni2P as a Robust Catalyst for the Hydrolytic Dehydrogenation of Ammonia-Borane[END_REF]

Scanning Transmission Electron Microscopy, elemental mapping by Energy

Dispersive Spectroscopy and in situ experiments. STEM bright-field and HAADF, STEM-EDS and in situ observations were performed using a spherical aberration corrected JEOL 2100 F operating at 200 kV. EDS mapping was performed using JEOL Silicon Drift Detector (DrySD60GV: sensor size 60 mm 2 ) with a solid angle of approximately 0.5 srad. For the acquisition of EDS maps, molybdenum grids were used to prevent the unsuited contribution of the traditional copper TEM grids to the Cu signal originating from the nanoparticles. In situ STEM annealing of Cu 0.25 Ni 0.75 P 0.375 nanoparticles was carried out using the Atmosphere device purchased from Protochips Inc. (Raleigh, NC, USA). [START_REF] Allard | Novel MEMS-Based Gas-Cell/Heating Specimen Holder Provides Advanced Imaging Capabilities for In Situ Reaction Studies[END_REF] The sample was placed between two micro-electricalmechanical systems (MEMS)-based closed cell (SiN windows). The temperature and the gas flow in the cell were controlled by the heating holder and a gas delivery manifold. The samples were dispersed in n-hexane and drop-casted on the SiN membrane. The in situ experiments were done at atmospheric pressure under Ar. Four temperatures were chosen for the observations: 100, 150, 200 and 250 °C. At each temperature, the ramp duration was 20 min and the heating rate was 5 °C/min. Energy Dispersive X-ray Spectroscopy. A small amount of powder was deposited on carbon adhesive tape on a scanning electron microscope sample holder. EDS analyses were performed on a SEM HITACHI S-3400N instrument at 20 kV. Cobalt was used as calibrator and the analyses were performed on at least three different zones on the samples to calculate an average chemical composition.

Nuclear Magnetic Resonance Spectroscopy. [START_REF] Peng | Nanostructured Ni2P as a Robust Catalyst for the Hydrolytic Dehydrogenation of Ammonia-Borane[END_REF] P liquid NMR spectra were recorded on a Bruker AC-300 SY spectrometer. Chemical shifts are relative to an 85% H 3 PO 4 external reference. P 4 was detected as a singlet at -521 ppm.

X-ray Photoelectron Spectroscopy. Spectra were collected on a Omicron Argus X-ray photoelectron spectrometer, using a monochromated Al Kα (hυ = 1486.6 eV) radiation source having a 300 W electron beam power. The samples were analyzed under ultrahigh vacuum conditions (10 -8 Pa). After recording a broad range spectrum (pass energy 100 eV), high resolution spectra were recorded for the C 1s, Cu 2p, Ni 2p, P 2p core XPS levels (pass energy 200 eV). The binding energies were calibrated with respect to the C 1s peak at 284.8 eV. Spectrum processing was carried out using the CasaXPS software package.

Table of content entry

Phosphidation of core-shell CuNi nanoparticles yields (Ni,Cu) 2 P alloy but also unexpected outward migration of copper outside the nanoparticles.

Figure 1 :

 1 Figure 1: a) XRD patterns of Cu, Cu 0.25 Ni 0.75 and Ni nanoparticles. b) Deconvolution of the (111) peak of Cu 0.25 Ni 0.75 nanoparticles with its copper-rich component at 43.6° (orange, corresponding to the stoichiometry Cu 0.7 Ni 0.3 based on Vegard's law) and nickel-rich component at 44.4° (green, corresponding to the stoichiometry Cu 0.1 Ni 0.9 based on Vegard's law). The dotted lines are a guide to the eye for pure copper (orange) and nickel (green). c)

  core-shell Cu 0.25 Ni 0.75 nanoparticles.

Figure 2 :

 2 Figure 2: Two borderline scenarios of phosphorus insertion into core-shell Cu-Ni nanoparticle.

Figure 3 :

 3 Figure 3: XRD patterns of Cu 0.25 Ni 0.75 nanoparticles after reaction with P 4 at 250 °C. Reference patterns of Ni 2 P (JCPDS 74-1385) and Cu (JCPDS 04-0836) are in blue and orange. The blue and red arrows are a guide to the eye for visually remarkable shifts of diffraction peaks towards

Figure 4 :

 4 Figure 4: a) TEM image of Cu 0.25 Ni 0.75 nanoparticles after reaction with P 4 at 250 °C. b) TEM image of a hollow nanoparticle. Inset: FFT of the image. c) STEM bright-field and d) STEM-

Figure 5 :

 5 Figure 5: a) XRD patterns of the nanoparticles formed after reaction with P 4 at 150 °C before and after annealing in oleylamine at 250 °C for 2 h. Reference patterns of Ni 2 P (JCPDS 74-1385) and Cu (JCPDS 04-0836) are in blue and orange. The blue and red arrows are a guide

Figure 6 :

 6 Figure 6: a) STEM bright-field and b) STEM-HAADF images of the nanoparticles after reaction

Figure 7 :

 7 Figure 7: TEM images of the nanoparticles during the reaction with P 4 at 250 °C. The sample was taken at a) room temperature, b) 180 °C during the heating ramp, c) when the temperature

21

 21

Figure 8 :

 8 Figure 8: In situ annealing of the nanoparticles formed after reaction with P 4 at 150 °C. STEM bright-field images of the nanoparticles heated in situ under argon at a) 100 °C and b) 250 °C. c)

Figure 9 :

 9 Figure 9: Formation mechanism of hollow mixed phosphide (Ni,Cu) 2 P nanoparticles. The asterisk indicates two possible and competing pathways.

  P NMR. Syntheses with different quantities of P 4 and temperature were carried out in the same fashion. Control Reaction of Cu 0.25 Ni 0.75 Nanoparticles with White Phosphorus. The reaction medium of a typical Cu 0.25 Ni 0.75 nanoparticles synthesis was left to cool to room temperature. It was then heated to 250 °C for 2 h under inert atmosphere. The work-up procedure was done in the same fashion as in the actual reactions with white phosphorus. Annealing of Cu 0.25 Ni 0.75 P 0.375 Nanoparticles. Previously isolated Cu 0.25 Ni 0.75 P 0.375 nanoparticles were re-dispersed in 15 g of OAm. The mixture was degassed under vacuum at 50 °C for 15 min then heated to 250 °C. The black solution was stirred at this temperature for 2 h under inert atmosphere. The nanoparticles were isolated using the same method as for bimetallic nanoparticles. X-ray Diffraction. Powder XRD measurements were performed with a Bruker D8 Xray diffractometer operating in the reflection mode at Cu Kα radiation with 40 kV beam voltage and 40 mA beam current. Typical diffraction patterns were collected with steps of 0.05° and a scanning speed of 2.5-5 s/point. For easier phase identification, the background was subtracted using EVA software. Profile matching of diffraction patterns was carried out using the Thompson-Cox-Hastings pseudo-Voigt function model with Fullprof program. Conventional Transmission Electron Microscopy. TEM were recorded on a TECHNAI G2 Spirit microscope operating at 120 kV. Samples were prepared by evaporating a drop of n-hexane diluted nanoparticles suspension on a carbon-coated copper grid.

Acknowledgements

Sorbonne Université, CNRS, and the Collège de France are acknowledged for financial support. Labex MATISSE is acknowledged for funding the master thesis (ANR-11-IDEX-0004-02). The authors acknowledge financial support from the CNRS-CEA "METSA" French network (FR CNRS 3507) for the TEM experiments. We thank Isabelle Génois and Mohamed Selmane for the help with EDS measurements and profile matching of XRD patterns.

Conflict of interest

There is no conflict to declare.

Supporting Information

Supporting information available: complementary data on the nanoparticles before and after reaction (XRD, HR-STEM, STEM-EDS, XPS, ICP-AES, NMR). Figures 3 and5a redrawn in logarithmic scales. This material is available free of charge via the internet at http://pubs.acs.org.


M R Petryk
email: mashapetryk@gmail.com

A Khimich
email: khimich505@gmail.com

M M Petryk

J Fraissard
email: jacques.fraissard@upmc.fr

Experimental and Computer simulation studies of dehydration on microporous adsorbent of natural gas used as motor fuel

Keywords: Natural gas dehydration, diffusion of adsorbed gas, adsorption and desorption of gases, modelling, Heaviside's operational method, Laplace integral transform

An experimental and theoretical study of the dehydration of natural gas using microporous silica beds for motor fuel technology in extreme winter climates is described. Analytical solutions to the problem of non-isothermal adsorption and desorption are based on Heaviside's operational method and Laplace integral transform, but the development of calculations is quite original. Experimental and modeling distributions of moisture and temperatures of gas at the inlet and outlet of the silica beds for each adsorption -desorption phase at different times are presented. The distribution of moisture within the beds for the full dehydration -regeneration cycle is determined.

Introduction

The main anthropogenic sources of atmospheric pollutants are the processes by which energy is generated for transport and industry. It has been demonstrated that the transport sector is the emission source that contributes the most to global warming at present, and it will probably remain so in the immediate future [START_REF] Unger | Attribution of climate forcing to economic sectors[END_REF].

Natural gas is an important source of primary energy. Its use as a motor fuel for transport and other sectors of industry saves liquid petroleum products, significantly improves the atmosphere of cities and slows the process of global warming. However, the reliability of the exploitation of process equipments, including motors of vehicles that run on natural gas, is largely dependent on the quality of gas used as fuel. This must be in accordance with European Union specifications governing the security and safety of such technical equipment's [START_REF]Euro 5 and Euro 6 standards: reduction of pollutant emissions from light vehicles[END_REF].

Under normal production conditions, natural gas contains water vapor. Therefore, for the reliable exploitation of vehicles in winter at temperatures down to -30 o C it is accepted that the humidity content of the gas must not exceed 0.009 g/m 3 , corresponding to a dew point of -60 o C under normal conditions [START_REF] Farag | Natural gas dehydration by desiccant materials[END_REF].

Extensive literature is available on common gas dehydration systems including solid and liquid desiccant and refrigeration-based systems [START_REF] Gandhidasan | Dehydration of natural gas using solid desiccants[END_REF][START_REF] Karimi | Selective dehydration of high-pressure natural gas using supersonic nozzles[END_REF][START_REF]Gas Processors Suppliers Association[END_REF][START_REF] Bahadori | Simple correlation accurately predicts densities of glycol solutions[END_REF]. Netusil and Pavel [START_REF] Netusil | Comparison of three methods for natural gas dehydration[END_REF] have compared the different methods for natural gas dehydration that are widely applied in industry, according to their energy requirements and suitability for use. They concluded that adsorption is preferred when a very low T dew of natural gas is required. Currently, adsorption-based processes rely on a technology which uses high temperatures for the regeneration of the adsorbent [START_REF] Farag | Natural gas dehydration by desiccant materials[END_REF][START_REF] Puertolas | Modelling the heat and mass transfers of propane onto a ZSM-5 zeolite[END_REF][START_REF] Petryk | The gas dehydration block of NGV filling stations BCV 250[END_REF]. For example, molecular sieves require large amounts of energy to heat them to the regeneration temperature (over 350 °C), and then to cool them to the adsorption temperature ( 25 °C), increasing the cost of gas used as motor fuel. Lower operating costs for drying gas are possible with short adsorption-desorption cycles and low-temperature regeneration of the adsorbent. In our plant, a microporous silica gel is used because of its ability to provide extremely low dew points. But the main purpose of this study is not to confirm the effectiveness of the adsorption-desorption technology for the dehydration of natural gas and to show that our installation is very efficient. For the first time, it has been possible to specify all the processes along the columns during non-isothermal adsorption and desorption. For the modeling we have used the Heaviside operational method and the Laplace transform, but the development of calculations is quite original.

Experimental system for gas dehydration and adsorbent regeneration

The scheme of the plant is presented in Figure 1. The two columns have the dimensions: height: 1.65 m, diameter: 0.8 m. They are filled with microporous silica gel KSMH (Ukraine) with the following characteristics: -density: 720 g/dm 3 ; -specific surface area: 450-600 m 2 /g; pore diameter: 5*10 -9 -8*10 -9 m; pore volume: 0.6-0.8 ml/g; radius of solid particles: 1.4-3 mm.

The natural gas to be dehydrated is at a temperature of 10-12 ° C and a pressure of 25 MPa; it cools the column (for example A1 in the figure) at the same time as water is adsorbed. On leaving A1 the water concentration of the gas should not be greater than 0.009 g/m 3 . Part of the dehydrated gas is heated in the counter-flow heat exchanger and sent to regenerate the adsorbent (A2 in the figure) at a pressure of 1.0-1.2 MPa and temperature of 125-130 °C. It thus warms the second column and eliminates the desorbed water. The two columns of silica (A1, A2) are alternately in the "dehydration" or "regeneration" mode. The experimental conditions of the system are summarized in Table 1. The study was carried out during four-hour cycles of dehydration and regeneration.

Mathematical model of non-isothermic adsorption and desorption in microporous solids

The modeling of the kinetics of gas dehydration processes on a microporous adsorbent and the corresponding regeneration is based on our approach [START_REF] Sergienko | Mathematical Modelling of Diffusion Process in Microporous Media (Numerical analysis and application[END_REF] using a mathematical model that includes mass balance and heat in ref. 12 and 13. The meaning of the terms is given in the Nomenclature.

,

er g ads c t z a t z c c uD t t z z T t z T a T H uh Q T t z t z a c Ka z t t                                (1) Initial conditions: a) adsorption: b) desorption: ( , ) | 0 to c t z   , 0 ( , ) | to c t z c   , (2) 0
( , ) | to T t z T   , (3)
Boundary conditions: a) adsorption: b) desorption:

( , ) | in zo c t z c   , 0 ( , ) | ( ) in z c t z c t   , ( 4
) ( , ) | 0 z c t z z     , ( , ) | 0 z c t z z     , (5) 0
( , ) | ( ) z in T t z T t   , ( , ) | 0 z T t z z     . ( 6
)
During adsorption C in is roughly constant for z = 0 (top of the columns). We use the same symbol C 0 for adsorption and desorption because the calculation methodology is the same for both.

For the model we choose columns of unlimited height, since it is not known in advance what the concentration of moisture in the gas at the exit of the column of regeneration will be. However, if an infinite length is assumed, the concentration will decrease from the beginning to the end, where it will be equal to 0. If we apply this condition for a limited height of the columns at the end of which the gas leaves, the calculation will not be completely correct.

The solution of the system (1)-( 7) was obtained using Heaviside's operational method and is presented in the Appendix [START_REF] Petryk | Mathematical Modeling of Nonlinear Non-isothermic Process of Diffusion and Adsorption in Compressed Layer of Adsorbent. Integral Transformations and Application in Boundary Problems[END_REF]14,[START_REF] Lavrentiev | Methods of theory of functions of a complex variable[END_REF].

Let us point out that for a first approach to this problem, given the complexity of the calculations, the authors have used some simplifications; for example, the co-adsorption of methane is neglected; the Henry's law constant (K) was calculated for the temperature T =40 C according to the reference [START_REF] Do | Adsorption Analysis: Equilibria and Kinetics[END_REF] and supposed to be constant K = 300; the isosteric heat of adsorption for the water isotherm was also assumed to be constant Q ads =2200 kJ/kg [START_REF] Do | Adsorption Analysis: Equilibria and Kinetics[END_REF][START_REF] Keltsev | The fundamentals of the adsorption technique[END_REF]. The total mass transfer coefficient β = 0.85-0.95 s-1. was calculated as the inverse of the total resistance in the system, which includes mainly the external and internal mass transfer resistance [START_REF] Aérov | Hydraulic and Thermal Principles of the Operation of Installations Incorporating Stationary and Fluidized Granular Beds[END_REF]:

int iintra 1 1 1 er    .

Analysis of experiment and simulation

Moisture concentrations of the gas that enters the drying adsorbent column (adsorption phase) during a dehydration cycle are presented in Figure 2. These variations are due to the minor fluctuations of the gas pressure in the network. The corresponding average moisture at the inlet, 0.175 g/m 3 , is used for the simulations. Figure 3 displays the moisture variations against time of dehydrated gas at the outlet of the adsorbent column. The average concentration (used for simulation) does not exceed 0.003 g/m 3 , which correspond to the technical requirements of the quality of gas for its further use as a motor fuel [START_REF] Petryk | The gas dehydration block of NGV filling stations BCV 250[END_REF].

Adsorbent regeneration (desorption phase) is carried out with a small fraction of the dehydrated gas (10% in our experiments) heated at 125-130 °С. Figure 4 shows the variation of the temperature near the inlet T in (T 1 ,T 2 ) and outlet T out (T 3 ,T 4 ) of the column during the desorption phase. The variation of the moisture of the gas at the outlet of the column of regeneration (Fig. 5) shows the kinetics of the internal desorption of the moisture during 240 minutes.

Figure 4 and 5 show that the regeneration process is characterized by two specific periods: the heating of the adsorbent, corresponding roughly to the first 90 minutes (Fig. 4, curves T 1 and T 3 , Fig. 5, curve w 1 ) and the regeneration period (Fig. 4, curves T 2 and T 4 , Fig. 5, curve w 2 ). At the beginning of the regeneration, during bed heating, desorption of water is low (moisture content of the regenerated gas is 0.5-0.9 g/m 3 ), and the average temperature of the gas at the outlet of the column reaches 30-50 °C. Further there is a higher and stable temperature T in  120-125 °C and T out increases continuously. After about 90 minutes there is a sharp increase in the desorption with a maximum around 120 minutes (Fig. 5, point A = 3.75 g/m 3 ) due to the elimination of the water very slightly physisorbed on the adsorbent. Then the average moisture concentration of the gas decreases sharply and is 0.50-0.2 g/m 3 after about 200 minutes. At the end of the regeneration cycle (240 minutes) this concentration is negligible (less than 0.005 g/m 3 ). Figure 6 shows the amount of adsorbed and desorbed water during the four-hour "dryingregeneration" cycle. During this time, 1089 g of water are adsorbed and 1067 g are eliminated, so 22 g of moisture are still left in the adsorbent, i.e. about 2% of adsorbed moisture. This very small difference could correspond either to experimental errors or to the formation of few stable OH groups at the highest temperature.

Figure 7 shows the adsorption isotherms for the entire water concentration range in the column calculated using the Langmuir equation of the adsorption equilibrium [START_REF] Do | Adsorption Analysis: Equilibria and Kinetics[END_REF][START_REF] Ruthven | Principles of Adsorption and Adsorption Processes[END_REF] Remarks on the approximations: -The Henry law constant K is assumed independent of T and p. In a more precise study we will try to take into account the variarion of K with the pressure using for example the data of Do [START_REF] Do | Adsorption Analysis: Equilibria and Kinetics[END_REF] and Karger et al [START_REF] Kärger | Diffusion in Nanoporous Materials[END_REF] for the adsorption of water on silica, in a form of the type 0 K k p  which will take into account the influence of pressure differently for adsorption (p = 25 MPa) and desorption (p = 1.2 MPa), where .

-In this model, the pressure drop in the column is not taken into account. The next model will be improved by the pressure change equation.

-The isosteric heat of adsorption for the water isotherm is assumed constant. In the proposed new nonlinear model, the temperature dependence will be implemented directly in the Langmuir nonlinear adsorption equilibrium, which should improve the simulation results.

-The influence of the adsorption of methane is not taken into account. It is certain that there must be some amount of methane adsorbed at this high pressure. The literature data, mainly on the adsorption of CH 4 on activated carbon (energy storage problem), all show that the presence of water partially inhibits the adsorption of methane, even at high CH 4 pressures; only in the case of very humid systems can there be formation of methane hydrates at very high pressure [START_REF] Zhou | Effect of moisture in microporous activated carbon on the adsorption of methane[END_REF][START_REF] Krooss | High-pressure methane and carbon dioxide adsorption on dry and moisture-equilibrated Pennsylvanian coals[END_REF]. This is why the lack of experimental data on our installation made any simulation impossible. The problem is all the more complex as the adsorption of the methane should be dependent on the corresponding concentration of adsorbed water at each point of the column and at each instant.

The small difference between the theoretical and experimental results could be due to the fact that the experimental data take into account the whole environment, and therefore the presence of adsorbed methane, which is not the case of simulation. This could mean that the concentration of adsorbed methane is low.

Conclusion

Our experimental and theoretical studies have confirmed the effectiveness of the "adsorptiondesorption" technology for the dehydration of natural gas used as fuel in extreme winter climates. After each drying cycle, the residual moisture content of the gas is lower than the maximum allowable value of 0.009 g / m3. But this was not the main purpose of our study. The solution of the proposed mathematical model for the gas drying process using the adsorption of water on porous solid and its regeneration is based on an original algorithm using the Heaviside operational method and the Laplace transform. The development of calculations is quite original. The result allows instantaneous knowledge along the columns, during the non-isothermal adsorption and desorption, of: -the moisture of the gas phase; -the water concentration of the adsorbed phase; -the temperature of the gas stream. This original mathematical treatment can serve as a model for many applications relating to this type of process, such as for example the purification of confined atmospheres.

Nomenclature

c -concentration of moisture of the gas phase in the column; a -concentration of moisture adsorbed in the solid phase; T -temperature of gas phase flow, °C; u -velocity of gas phase flow, m/s 2 ; int er D -effective longitudinal diffusion coefficient;  -coefficient of thermal diffusion along the columns; g h -gas heat capacity; ads Q -heat of adsorption, kJ/kg; μmolecular mass of adsorbat, kg/mole; H -total heat capacity of the adsorbent and gas;  -total mass transfert coefficient, s -1 ; β intercoefficient of inter mass transfer, s -1 ; β intra -coefficient of intra mass transfer, s -1 ; z -distance from the top of the bed for mathematical simulation, m; Z -dimensionless coordinate = abscissa z/height of the column. in -index of parameter names (concentration, temperature) in the inlet of the column.

,

pt pt pt c p z c t z dt L c a p Z a t z dt L a T p z T t z dt L T               (A.7)
where p is a complex-value parameter of the Laplace transformation.

b) then the solutions, c and T, are obtained by using the Laplace inverse transform of the functions c  and T  with respect to the variable p.

Applying the Laplace transformation to eq. (A.1)-(A.6) one obtains: The solution of eq. (A.13) with conditions (A.10), (A.11) is [START_REF] Sergienko | Mathematical Modelling of Diffusion Process in Microporous Media (Numerical analysis and application[END_REF]14,[START_REF] Petryk | Liquid Flowing from Porous particles During the Pressing of Biological Materials[END_REF]: where    

2 0 int 2 2 2 0 2 ( , ) ( , ) , ( ( , ) ) 0,
er g ads dd pc p z c pa p z u c D c dz dz dT H pT p z T uh T Q pa T T z dz                         (A.8)
    1 () * 2 0 int 0 1 , , ,
1/2 22 1 1 1 1 1 , Re 0 4 p u q p        ,   ,, pz  E -Cauchy function [11, 15]               1 1 1 2 1
                        . (A.17)
Calculating the Laplace originals in eq. (A.17), we obtain:

    1 1 1 1 () 1 1 1 0 2 0 0 () 1 1 2 0 1 1 , ( ) 1 1 1 1 ( 1) u pz z in u pz z c K e c t z c L L c c e L K p K p K p e c L L e p K p                                                    , (A.18)
where the Laplace original which describe the of moisture in the gas phase and adsorbed on the solid:

        0 1 1 int 1 0 2 () 2
                           W (A.19) where   1/2 1/2 2 2 2 2 1 2 1 1,2 ( ) ( ) ( ) ( ) 2                ,   22
             1 int int 1 2 0 00 1 2 0 0 ,, 11
, for the desorption phase, eq. (A.20) will be: Similarly, the analytical solution of (A.1)-(A.6) which describes the temperature distribution in the adsorbent is [START_REF] Petryk | Mathematical Modeling of Nonlinear Non-isothermic Process of Diffusion and Adsorption in Compressed Layer of Adsorbent. Integral Transformations and Application in Boundary Problems[END_REF][START_REF] Lavrentiev | Methods of theory of functions of a complex variable[END_REF]:

er er u z Kt D in c u t z K t s D c c K c t z c e c c e t z KK c e e s z ds                W W (A.20)       0 ,, t
             int int int 1 2 2 0 0 0 0 0 1 2 0 0 0 , , ( ) , 11 ,
                  0 0 0 0 0 0 0 , z 0

Figure 1 .

 1 Figure 1. Schema of experimental plant with water adsorption in A1 and regeneration in A2.

Figure 2 .

 2 Figure 2. Experimental concentration of gas moisture at the inlet of the adsorbent column during the adsorption phase

Figure 3 .

 3 Figure 3. Moisture content of dehydrated gas at the outlet of the adsorbent column.( blue square pointsexperiment; dotted red line -model).

Figure 4 .

 4 Figure 4. The temperature of gas near the inlet T in (T 1 ,T 2 ) and at the outlet T out ( T 3 ,T 4 ) of the column of regeneration (squares -experiment, dotted line -model) .

Figure 5 .

 5 Figure 5. Moisture of the gas at the outlet of the column during the regeneration of the adsorbent during 240 minutes (square red points -experiment, dotted blue line -model)

Figure 6 .

 6 Figure 6. Variation of the amount of adsorbed and desorbed water during the four-hour cycle (red diagram -regeneration, blue -drying)

Figure 7 .

 7 Figure 7. The isotherms for the entire range of moisture concentration in the column.

  heat loss through the wall of the adsorbent; R -radius of solid particles of the adsorbent, m ; h  -heat transfer coefficient; K -Henry's constant;

Table 1 :

 1 Experimental conditions

	Environment temperature, °С	-5 to +3
	Gas entering for dehydration:
	-Consumption, m 3 /h	1600
	-Pressure, MPa	25.0
	-Temperature, °C	10-12
	-Moisture content, g/m 3	0.16-0.18
	Regeneration of the adsorbent:
	-Consumption (10-11% of gas consumption for drying), m 3 /h	155-160
	-Pressure, MPa	1.0-1.2
	-Temperature, °C -Moisture content, g/m 3	125-130 0.00035-0.0017

  :

	Here	ads HQ 	ads	/		-energy of activation, kJ/mole; R -gas constant, kJ/(mole.K);,	full a -the
	concentration (amount) of adsorbate at saturation of the micropores and intercrystallite space of the
	adsorbent, kg/kg; p -pressure;	' 0 k -the equilibrium constant.
	a	full a		1	' 0 kp ' 0 exp exp ads ads H RT H kp RT          	.

Acknowledgements

This work was supported by the Ternopil Ivan Puluj National Technical University and the Academy of Sciences of Ukraine.

Appendix

Methodology of analytical solutions of the adsorption-desorption model

The mathematical model of adsorption and desorption cycle of drying gas process and adsorbent regeneration, including mass balance and temperature [START_REF] Sergienko | Mathematical Modelling of Diffusion Process in Microporous Media (Numerical analysis and application[END_REF][START_REF] Petryk | Mathematical Modeling of Nonlinear Non-isothermic Process of Diffusion and Adsorption in Compressed Layer of Adsorbent. Integral Transformations and Application in Boundary Problems[END_REF] is given by:

,

We use the same symbol c 0 for the adsorption and desorption because the methodology of calculation is the same for both.

First of all we obtain the analytical solution of the eq. To find the analytical solution of the problem (A.1)-(A.6) where the functions c and T depend on the concentration, the time and coordinate z, we use Heaviside's operational method [14,[START_REF] Lavrentiev | Methods of theory of functions of a complex variable[END_REF]. Solutions c and T are obtained according to the following procedure: a) first the Laplace transforms of the functions c and T are determined: