V Loisel

J.-A Zambaux

M Hadj-Achour

S Picchi

O Coindreau

R Meignen
email: renaud.meignen@irsn.fr

Oxidation during fuel-coolant interaction Advances and modeling

Keywords: FCI, Oxidation, Numerical Simulations

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

INTRODUCTION

During the progression of a severe accident with core melt in a nuclear reactor cooled with water, the relocation of the corium melt in the lower head may occur. The corium behavior in the lower head is currently the object of an R&D at the international level in the frame of the Ex-Vessel Retention (EVR) strategies, and in particular in the H2020 IVMR project [START_REF] Fichot | Some considerations to improve the methodology to assess In-Vessel Retention strategy for high-power reactors[END_REF]. It is expected that the liquid corium in the lower head will stratified in two or three layers with one being essentially oxidic, and one or two being essentially metallic. A most probable scenario is that the upper layer is formed with light metals with a temporarily quite high superheat. This layer would easily attack the vessel and break it through, resulting in the flow in the reactor cavity of this superheated metallic melt. A second possibility is the failure of the vessel by the bottom layer composed with heavy superheated metals. If the vessel comes to fail, and if the pit is flooded with water (a condition for the EVR strategies), the melt is then expected to flow out of the vessel with a metallic composition at first, inducing oxidation. Ex-vessel FCI is also an issue for the cases of BWR's with large (flooded) pit. Apart from a massive hydrogen release, there are three effects of oxidation on FCI. On the one hand, experimental evidences [START_REF] Cho | Experiments on interactions between zirconiumcontaining melt and water[END_REF] have shown that the potentially high amount of released chemical energy can contribute to increase the magnitude as well as the duration of the steam explosion. On the other hand, the hydrogen generation can substantially increase the void fraction during the premixing, which could limit to possibility to trigger a steam explosion. On a side note, the oxidation process is also responsible for the change in the material properties that can affect its coolability and solidification. Indeed, the strong explosions reported by Cho, Armstrong & Gunther could be obtained only with a fast triggering, before p.2 the melt jet solidifies. This behavior is interpreted as being essentially due to an increase of the solidification temperature bounds (solidus and liquidus) during the oxidation process.

Oxidation is also an issue regarding oxidic U-Zr-O melts. The U-Zr-O system is currently the field of efforts to clarify its behavior at high temperatures. Indeed, UO2/ZrO2 melts, as classically used for "prototypical" corium may slightly over-oxidize [START_REF] Mastromarino | Assessment of solid/liquid equilibria in the (U, Zr)O2+y system[END_REF]. This behavior could be noticed in the FARO experiments using UO2/ZrO2 melts and for which a production of hydrogen could be measured [START_REF] Magallon | Lessons learnt from FARO/TERMOS corium melt quenching experiments[END_REF]. For this system, the question of hydrogen production and energy release is not so important, but the change of composition affects the solidification properties and hence, should be taken into account in the models and calculations.

In the framework of nuclear safety, oxidation of metals in water and steam has been studied extensively for decades, resulting in an exhaustive database of reaction rates for metals at relatively low temperatures (i.e. at solid state). On the other hand, much less is known concerning temperatures above the melting point. First experimental works dealing with the hazards potentially induced by melt oxidation during the course of a Fuel-Coolant Interaction (FCI) date back to the 1950s and have evaluated the possibility of an explosive event under these conditions. In order to provide basic data, [START_REF] Bostrom | The high temperature oxidation of zirconium in water[END_REF][START_REF] Lemmon | Studies relating to the reaction between zirconium and water at high temperatures[END_REF][START_REF] Baker | Studies of metal-water reactions at high temperatures. III. Experimental and theoretical studies of the Zirconium-water reaction[END_REF] conducted experiments dedicated to the evaluation of oxidation rates of zirconium (or zircaloy) in both steam and water. Based on the results obtained, most of the theoretical models subsequently developed assumed that the kinetic of oxidation is governed either by vapor diffusion in the boundary layer, if the material remains fully melted, or by oxygen diffusion if the fuel is solid at the interface [START_REF] Corradini | Hydrogen generation during a core meltcoolant interaction[END_REF]) [START_REF] Young | Hydrogen generation during fuel/coolant interactions[END_REF] [START_REF] Lee | Development of a mechanistic model for hydrogen generation in fuel-coolant interactions[END_REF]. More recently, two sets of experiments were conducted by Cho, Armstrong & Gunther, (1988) with melts containing zirconium with steel or with zirconia at various ratios. As already explained, explosions took place in the triggered experiments only, otherwise rapid quenching and solidification was observed. The strengths of the explosions were directly linked to the amount of zirconium, which could be nearly totally oxidized during the explosions.

Interaction of aluminum with water has also been the subject of several works, in general motivated by the fact that aluminum is used in the core of various research and military nuclear reactors but also by the numerous accidents occurring in metal cast industries. The observations (e.g. [START_REF] Nelson | Steam explosions of single drops of pure and alloyed molten aluminum[END_REF] for a synthesis) of strong and fast explosions accompanied by strong and fast oxidation triggered a number of theories and papers. In particular, Epstein and co-workers attempted to explain the fact that surface oxidation could not produce a stable barrier to further oxidation at temperatures nearly down to the melting point. However, these works focused mostly on the explosion whereas only few data are available for non-explosive drops, to support the development oxidation kinetic models. [START_REF] Corradini | Hydrogen generation during a core meltcoolant interaction[END_REF] developed a model included in the TEXAS code to evaluate the kinetics of oxidation for three different configurations: coarse mixing, steam explosion and stratified melt quenching, without considering the impact of solidification. Later on, this model has been reworked by [START_REF] Young | Hydrogen generation during fuel/coolant interactions[END_REF] to correct mistakes, which caused a diminution of the calculated reaction rates. However, the corrected version gives much larger amounts of hydrogen than that of the experimental results, and the model no longer describes a significant difference between the different configurations. To overcome this issue, the mechanistic model of [START_REF] Lee | Development of a mechanistic model for hydrogen generation in fuel-coolant interactions[END_REF] accounts for the formation of an oxide coating. Although this improvement permitted a much better agreement with the experimental data the situation appears quite confusing on both understanding and modeling aspects. Indeed, the impact of the modeling depends primarily on the simulation of the FCI itself (drop size and local flow configuration), on which some confidence has been recognized only recently (Meignen et al, 2014). This present work aims at readdressing this question with the help of recent analyses and experiments, and proposing a new model of oxidation specifically designed for dealing with the various stages of FCI p.3 with the thermal-hydraulic code MC3D (Meignen et al., 2014) (Meignen, Raverdy, Picchi & Lamome, 2014). The verification of the model was confronted to the lack of precise experimental data for the oxidation phenomena itself in FCI conditions. To overcome this difficulty, a work of fine numerical simulations at the drop scale (quasi DNS) has also been engaged. The paper discusses the advances of the work, still limited regarding the necessary verification work against explosion experimental data.

PHYSICAL AND MATHEMATICAL MODELLING

Phenomenology

Fuel-Coolant Interaction

The contact between the melt and the water heralds the beginning of the fuel-coolant interaction. A recent status of FCI understanding and modeling capabilities is given by Meignen et al. (2014). It is recognized that, in most cases, the mixing phase starts with the melt fragmentation into coarse drops of millimetric sizes. This fragmentation is accompanied by vaporization, the intensity of which is mostly directed by the characteristic size of the fragmented melt. Also, oxidation would most likely occur for the fragmented or highly deformed parts of the melt. It is worth noting that the immediate neighborhood of the jet is mostly gaseous due to the significant amount of heat released during the quench. This is not necessarily the case in experiments but the recent simulations performed in the frame of the OECD SERENA-2 program (Meignen et al., 2014) indicates that, at reactor scale, i.e. for large jets (say larger than 10 cm), a large vapor film embeds the melt jet and a large fraction of the melt drops. The resulting molten drops are therefore created and remains for a substantial time in a vapor-rich environment before being convected into the water. Consequently, two configurations must be considered to track the time evolution of the melt oxidation all along the entire process (Fig. 1): when a drop is within an infinite medium of steam, and the film boiling configuration. It is nevertheless quite likely that, during premixing, the configuration of drop in gas is the most important regarding the effectiveness of oxidation. Indeed, the CCM experiments [START_REF] Spencer | Fragmentation and quench behavior of corium melt streams in water[END_REF], using a mixture of UO2/ZrO2 and stainless steel, indicated a much more effective oxidation in cases with saturated water, i.e. probably large void in mixture, than cases in subcooled water. In contrast, during a steam explosion event, the flow configuration changes drastically. A large part of the void collapses due to the pressurization. Although at large scale (i.e. reactor application) the void around the jet cannot completely collapse, it is likely that the configuration of drop in liquid is the most important to take into account.

p.4

Oxidation process

The consensus adopted in the literature describes the process of oxidation according to three steps, each one having its own characteristic time. (1) The first of them consists to supply the reactive surface with steam. This can only be done if the vapor diffuses through the hydrogen that is being generated at the surface. (2) The next step is the atomic dissociation of the water molecules, which releases both hydrogen atoms and energy. (3) The final stage of the oxidation process describes on one side the absorption and diffusion of the oxygen atoms inside the material's body, and on the other side the recombination of the hydrogen atoms to form H2.

It is well accepted that the second step is very fast is not a limiting process. It is also well accepted that H2O/H2 diffusion in gaseous boundary layers around the melt is a limiting process that must, in general, be accounted for. A first key issue is then to check at the third step to verify the ability of oxygen to be effectively rapidly diffused inside the melt. Other way, an oxidized layer may rapidly form and block the process. In such case, the oxide layer has very different physical and thermodynamical properties. In particular, it should induce in most cases the solidification of this layer. The solidification itself should in turn impose a faster decrease of the temperature in this layer, and so a slower oxidation rate.

During the premixing stage of the interaction, the typical length scale (R) of the fragmented melt is about 1 millimeter, and the typical time scale (τ) is about 1 s. The experiments involving oxidation for U-Zr-O melts [START_REF] Magallon | Lessons learnt from FARO/TERMOS corium melt quenching experiments[END_REF], [START_REF] Tyrpekl | Prototypic corium oxidation and hydrogen release[END_REF], U-Zr-SS-O melts [START_REF] Spencer | Fragmentation and quench behavior of corium melt streams in water[END_REF], or Zr-SS-O melts [START_REF] Cho | Experiments on interactions between zirconiumcontaining melt and water[END_REF] indicated that, during the premixing, the reaction was not necessarily completed but had at least a level of some % (up to about 30 % in saturated water). Thus, during this stage, a minimum diffusion coefficient (R²/ τ) of oxygen inside the melt of 10 -7 to 10 -6 m²/s may be required for partial or total oxidation. During the explosion, the typical length scale (R) of the fragmented melt during the premixing is less than 50 µm, and the typical time scale (τ) is about 1 ms. This implies that the minimum diffusion coefficient of oxygen in the melt for a complete oxidation may have to be as high as 10 -6 m²/s.

At the moment, we will restrict our attention to Zr-O and U-Zr-O systems, despite the fact that, for exvessel reactor applications, the case of metal mixtures U-Fe-Zr should be more appropriate.

Regarding U-Zr-O melts, the detailed analysis of [START_REF] Tyrpekl | Prototypic corium oxidation and hydrogen release[END_REF] of some KROTOS tests, some with explosion, some without, Fig. 2, indicated a final over-stoichiometry ((U-Zr)O2+y) rather homogenous, with y in between 0.11 and 0.17. This is consistent with recent thermodynamical studies of this system reported in [START_REF] Mastromarino | Assessment of solid/liquid equilibria in the (U, Zr)O2+y system[END_REF] indicating that the solidus and liquidus temperatures of initially stoichiometric U-Zr-O particles are notably reduced when the melt exposed in oxidizing environment, with a final over-stoichiometry of about 0.1. p.5 Fig. 2: Mass fraction for each size interval and corresponding mean over-stoichiometry in 3 KROTOS tests, from [START_REF] Tyrpekl | Prototypic corium oxidation and hydrogen release[END_REF] The characteristic necessary diffusion coefficients discussed above are quite higher than those usually reported in the literature. The diffusion of oxygen in UO2±x was analyzed in detail by [START_REF] Breitung | Oxygen self and chemical diffusion coefficients in UO2±x[END_REF] (see also [START_REF] Berthinier | Thermodynamic assessment of oxygen diffusion in non-stoichiometric UO2±x[END_REF]). For the considered temperatures (2500 -3000 K), he could propose a chemical-diffusion coefficient (not to be confused with self-diffusion) of the order of 10 -7 m²/s (self-diffusion coefficient of the order of 10 -9 m²/s). Recently, applying the molecular dynamics theory, [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transition in stoichiometric uranium dioxide[END_REF] found similar values for the oxygen self-diffusion in UO2 in liquid state.

Concerning the case of ZrO2-x, the situation seems a bit confusing since there are several available data for self-diffusion with rather large differences, depending of the diffusion mode. Yakub et al. also analyzed the case of liquid ZrO2-x and propose a law for self-diffusion. Our evaluation of the chemical diffusion, using Yakub's law and the NUCLEA database (e.g. [START_REF] Barrachin | New modelling of the U-O-Zr phase diagram in the hyper-stoichiometric region and consequences for the fuel rod liquefaction in oxidising conditions[END_REF]) yields, for the case x=0.1 (i.e. ZrO1.9) corresponding to the test KROTOS KS-5, also a value of the order of 10 -7 m²/s. Despite the abundant literature on Zr and zircaloy oxidation, experimental data relevant to oxygen diffusion in very high temperatures Zr could not be found. Extrapolations of diffusion data in zirconia in the range 1000-1500 K leads to chemical diffusion coefficients between 10 -8 m²/s and 10 -7 m²/s in the melting temperature range. Recently, the Molecular Dynamics method could be employed by (Mendelev & Bokstein, 2010) to produce a law for self-diffusion at high temperature, but still in solid phase, yielding similar values for the chemical diffusion.

Thus, for pure Zr, but also for (U-Zr)O2±x melts, although the values given above are compatible with a fast oxidation, an additional mechanism may be necessary to achieve oxidation as observed in the experiments. For zirconium, the strong release of energy might rapidly increase the temperature and thus increase the diffusion. However, an efficient mechanism in the liquid state is the movement of the fluid itself. The circulation alone in a spherical drop of fixed shape is not very efficient, although it may already multiply the diffusion by a factor 2 to 3. Much more effective are the oscillatory effects and local small scale turbulence. Several theories have been developed but require detailed knowledge of the local flow (see e.g. [START_REF] Turney | Air-water gas transfer and near-surface motions[END_REF]). Of course, such detailed knowledge of the flow is out of reach in the current available FCI models, thus at present, the choice is to introduce a law of the form: .

p.6

where F is a numerical function of temperature tending to 1 when the melt is supposed to be solid, and is the chemical diffusion coefficient (in the absence of fluid movement), obtained through thermodynamics analyses for the considered material.

At that point, it is important to discuss here another issue for FCI which is the solidification process. Some FCI models make the hypothesis that the solidification occurs mainly as crust growth at the surface of melt globules and MC3D also provides such model [START_REF] Uršič | Eulerian modelling of melt solidification impact during fuel-coolant interaction[END_REF] as an option. However, the mixing processes at liquid state previously discussed, due to surface stretching and local oscillations/turbulence, should also impact similarly the temperature diffusion. This is equivalent to multiply the conduction coefficient by one or two orders of magnitude so that, even oxides may behave as very conductive materials.

Oxidation and solidification are then strongly linked and a simplified modeling with homogenous oxidation grade and temperature inside the melt seems sufficiently accurate at first order.

Model of H2 generation

General modeling

We will now focus our attention on the modeling of oxidation and hydrogen generation with the hypothesis that, at least in the liquid state, the main limitation process is the diffusion in the gas mixture surrounding the melt globule, either in gaseous environment, either in film boiling conditions.

Let's consider the oxidation of x moles of a melted metal drop Me by y moles of steam leading to the creation of hydrogen and the production (or consumption) of heat:

• Me + • H O → Me O + • H .
(1)

In agreement with the assumption stating that the oxidation kinetic is constrained by the vapor-H2 interdiffusion characteristic time, the hydrogen quantity produced, providing the vapor arriving at the interface has been consumed by the reaction, can be written in terms of a Fick's law:

/ • • ! , (2)
where " stands for the generated mass of hydrogen per unit of area (kg•m -2 ) and is the hydrogen molar mass (i.e. 2×10 -3 kg•mol -1 ). The vapor-H2 inter-diffusion coefficient D (m 2 •s -1 ) has been experimentally determined by [START_REF] Furman | Metal-water reactions : V. the kinetics of metal-water reactions, low pressure studies[END_REF] as a function of the mixture temperature and pressure (expressed in atm):

/ 6.52 × 10 )* • + , -../ 0 12, (3)
The last term of (2) describes the gradient of hydrogen concentration (3 in mol•m -3 ) across the boundary layer surrounding the drop. Typical FCI conditions often involve a large temperature difference between the drops and the ambient fluid so that strong variations of the physical properties are expected across the boundary layer. Such conditions make the analytical determination of the temperature profile quite complex even if the weak mixture Prandtl number (Pr < 0.7) suggests that the velocity profile has a weak influence on that of the temperature. For that reason, this gradient of hydrogen concentration is p.7 approximated by the differential of hydrogen concentration expressed across the boundary layer whose expression depends on the drop configuration, such as:

d3 d5 ≈ 7 8 9 8 : ∆3 < = Sh @ (vapor), ( 4a
) ∆3 B (film boiling). ( 4b
)
The approximation in eq. 4a uses a diffusional length Lc/Sh that involves a Sherwood number to characterize the mass transfer around the drop. Traditionally, Lc is the diameter of the sphere or the cylinder. As to the film boiling case (eq. 4b), ∆3 is determined over the distance defined between the drop surface and the liquid-gas interface, i.e. the gas-vapor film thickness δ. However, the film thickness n is not constant around the drop and no analytical expression has been proposed up to now in such conditions. [START_REF] Epstein | Subcooled forced-convection film boiling in the forward stagnation region of a sphere or cylinder[END_REF] developed a way to evaluate this film thickness along with the heat transfers in the frame of a sphere or cylinder at high temperature submitted to a subcooled forced flow, the effects of oxidation not being accounted for. [START_REF] Epstein | Film boiling on a reactive surface[END_REF] later studied the implications of the hydrogen on the heat exchanges when generated in the surrounding film. Unfortunately, no explicit expression of the film thickness could be formally deduced. Therefore, the value of δ given by the Epstein-Hauser model will be used in a first place.

It may be noticed that these formulations do not account, once more, the effect of local oscillations and stretching of the surface. However, this may be achieved by the choice of adequate Sherwood numbers (or film equivalent thickness). The data are scarce and thus, at the moment, the modeling is based on the hypothesis of rigid drops.

The differential of hydrogen concentration mentioned above must be written accordingly to the drop configuration. At the gas-drop interface, we already have postulated that the incident steam is totally consumed by the reaction and then replaced by the hydrogen produced. Hence, the hydrogen partial pressure is equal to the total pressure p, and we have at the drop surface (r = 0):

3 C !D 0 E•+ F , ( 5
)
Ts being the surface temperature of the molten drop. At the other end of the thermal layer (r = δT), the hydrogen concentration resulting to its diffusion away from the drop is:

3 C !DG H 0 E•+ I . ( 6
)
This expression can only be valid for the vapor configuration. In the liquid configuration, the concentration of hydrogen is related to the conditions at the interface. These ones are not so clear. It is expected that an important production of vapor occurs at the interface, so one should be quite close to the saturation regarding total pressure. With such hypothesis, the hydrogen partial pressure can be neglected at the interface. However, this hypothesis needs further investigations.

Proposed formulations

Taking into account what has been previously stated, the formulation of the hydrogen production kinetic for a drop of diameter d subjected to a gaseous environment, can be written as follows:

p.8

JKL • • MN O1P Q • R 0 E•+ F - 0 E•+ I T (7)
The Sherwood number used in this model is usually derived from common empirical heat transfer correlations such as that of [START_REF] Ranz | Evaporation from drops[END_REF]by considering the analogy between conductionconvection heat transfers and mass diffusion around a given geometry:

Nu WX 2 + 0.6 • Re Z/ • Pr Z/] , (8)
Sh WX 2 + 0.6 • Re Z/ • Sc Z/] . (9)
In general, mass transfers around evaporating liquid droplets are best represented with a correction factor due to the mass flow of evaporation (e.g. [START_REF] Yuen | Heat-Transfer measurements of evaporating liquid droplets[END_REF]). Also, this correlation calls upon various physical parameters, strongly dependent of the temperature, that need to be correctly evaluated. To do so, in the absence of experimental data, the chosen path is to adjust the macroscopic model with numerical simulations performed with the MESO model, as described below in paragraph 3.2. A critical point is the choice for evaluating the diffusion coefficient, which changes strongly with the temperature along the boundary layer, eq (3).

On the other hand, in the film boiling regime, the formulation can be based on the film thickness as can be calculated from, e.g., [START_REF] Epstein | Subcooled forced-convection film boiling in the forward stagnation region of a sphere or cylinder[END_REF]:

_` a•b G • 0 E•+ F (10)
Alternatively, in a general way, one could use also a more generic formulation similar to (8), where the film thickness is replaced by d/Sh cd :

_` • • MN _Q • 0 E•+ F (11)
Obviously, oxidation may modify the film boiling characteristics. The use of the Epstein-Hauser correlation will be discussed later on.

Here also the objective is to compare macroscopic model and simulations. However, this task is not fully achieved and we will give here only hints of the progress of this work.

MODEL ADJUSTMENT AND VERIFICATION

For the case of droplet in gaseous environment, in the absence of data, it was decided to build and use a numerical simulation model. A comparison is then made between the results given by our subgrid model of oxidation (implemented in MC3D's premixing module PREMIX [START_REF] Meignen | The challenge of modeling fuel-coolant interaction: Part I -Premixing[END_REF]) and those extracted from the local scale simulations (run with MC3D's quasi-DNS module MESO). The MESO model is being improved in order to consider the film boiling configuration. However, only preliminary results will be given here. Furthermore, there exists some experimental data of oxidation rate, although quite imprecise, to adjust the model.

Description of the DNSs (MESO)

p.9 MESO is a quasi-DNS module of MC3D allowing local scale calculations. MC3D is a multiphase flow code based on eulerian modeling with a staggered regular grid (Cartesian or cylindrical), see [START_REF] Meignen | The challenge of modeling fuel-coolant interaction: Part I -Premixing[END_REF] for more general information. In the present case, two phases are considered:

-the melt, represented here as a fixed object, that can nevertheless represent conduction and diffusion processes ; -the gas : a mixture of vapor and hydrogen. In the present problem, the conduction and diffusion in the melt are not considered (set to very large values). Using a very large heat capacity allows also having a constant melt temperature. The ignoring of the melt temperature changes simply leads to consider a surface temperature driven problem. In reality, there might be a limitation of the heat transfer by conduction and adjustment of surface temperature. This does not modify the physics of the boundary layer as shown by the simulations themselves and the TREPAM experiments (no "history effect"). The adaptation of film boiling to boundary conditions is very fast. Similarly, for the oxidation process, the diffusion coefficient in the melt is set to a very large value so that the dynamics is limited by the diffusion in the gas boundary layer. The reality of this assumption was discussed previously and needs to be strengthened. In the MC3D global model, the actual diffusion (oxidation) is set to the minimum between diffusion in the melt and in the boundary layer, although modeling of diffusion in the melt is still under progress. The subsequent H2-H2O inter-diffusional transport mechanism taking place in the boundary layer is also accompanied by heat transports where both hydrogen and steam carry their own thermal energy to their new location. The mass diffusion flux between cells is written as

e f / • • d3 d5 • g kg/s
where / is the diffusion coefficient in eq (3). The associated energy flux is simply given by e k e f • l.

Reference results given by the direct numerical simulations are calculated on a fine 2D cylindrical grid (Fig. 3). The spherical drop is then in fact assimilated to a fixed cylinder situated in the center of a bidimensional transverse flow whose fluid properties are interpolated from steam-water tables as a function of pressure and enthalpy. As the heating object is here a cylinder, a deviation is expected when applied to a sphere. However, for the present purpose, due to the general high uncertainties, the difference is small and neglected.

A particular attention is paid to the surface close region where a refined meshing is necessary to capture the boundary layer. At least 10 meshes are required along the radial direction in the dynamic and thermal boundary layers.

p.10 In compliance with the model's assumptions described earlier, the simulations with MESO provide for the awareness of the rate-limiting H2-steam inter-diffusion mechanism.

Vapor Case

A sensitivity study was carried out in which the main parameters are varied over a broad range of values covering those usually encountered in FCIs (table II). All along the calculations, the mass of hydrogen produced is used to deduce the kinetics of oxidation.

The formulation (7) introduced in MC3D-PREMIX model was found, after some few tries, to give acceptable results compared to MESO provided that: -all properties are evaluated at the average temperature Tm, except:

-the fluid's density in the Reynolds number that is evaluated at the bulk temperature T∞, as proposed by [START_REF] Yuen | Heat-Transfer measurements of evaporating liquid droplets[END_REF]); -the diffusion coefficient which is evaluated at the surface temperature Td ; -the Sherwood number is adjusted to 50 % of the original formulation as

Sh mno 1 + 0.3 • Re Z/ • Sc Z/] (12)
Results deduced from the hydrogen productions calculated in the simulations are gathered in table II and plotted in Fig. 4. Calculations were all run up to the same time so that the data can be compared from one case to another one. A value of 100 % • s -1 would mean that the drop would be entirely oxidized in 1 s. Deviations reported in table II are on average of 4.6 ± 12.9%, which shows an acceptable agreement of the macroscopic model's results with the DNS.

p.11 The high grades reported in table II clearly indicates the effectiveness of oxidation in the vapor phase during premixing. The comparison among the various points is immediate only for cases with same drop size. The relatively weak impact of the surface (drop) temperature is noticed, in contrast with the impact of the pressure. It is then anticipated that a fine fragmentation under high pressure should lead to a very fast oxidation. In turn, fast oxidation, if accompanied by strong reaction heat (e.g. aluminum or zirconium) should enhance the explosion behavior. A process of "ignition" can then be envisaged. The velocity has also an important effect, which should enhance also this ignition effect. p.12 3.3. Film Boiling Case

Preliminary simulations with the macroscopic model of oxidation

The verification process of the macroscopic model in the film boiling configuration is a bit different from that in pure steam as a few experimental data exist in terms of oxidation kinetics. [START_REF] Lemmon | Studies relating to the reaction between zirconium and water at high temperatures[END_REF] conducted Zr drop experiments in a vertical water tank of controlled temperature. They deduced the rates of reaction from the measurements of the time tox during which an intense light is emitted by the drop while falling down and the extent of oxidation τox of the spherical debris collected after each run.

Obviously, a quite high uncertainty is attached to these measurements. Two runs (8 and 9) have been selected from the original experimental series to be numerically reproduced with PREMIX. The experimental conditions and results are given in table III. Very recently, the similar experiments were conducted in the MISTEE installation at KTH (not published yet, see [START_REF] Ma | MISTEE Investigation on Mechanisms of Steam Explosion during Severe Accidents[END_REF], see [START_REF] Zambaux | Study on thermal fragmentation characteristics of a superheated alumina droplet[END_REF] for more on MISTEE). The hydrogen production was based on the estimate of the bubble production, supposed to be rapidly at equilibrium, so with also a noticeable uncertainty. In conditions similar to the run 8 in table III, an oxidation rate of about 35 % was estimated, with however a faster kinetics of about 10%/s. The calculated time evolutions of oxidation along with the drop temperature have been plotted in Fig. 5for the two runs. The first observation that can be drawn from this figure is that predicted kinetics of reaction is of the order of magnitude but faster than that measured experimentally. However, the calculation is quite consistent with the most recent data. Mean drop temperatures are observed to increase until equilibrium is found with the heat losses resulting in a plateau. During this process, no significant variations in the kinetics of oxidation are noticed. It will be noticed that the drops in the MISTEE tests, reached the bottom of the tank rapidly, while remaining in a somewhat round shape, so, likely, almost solid. This indicates that the drop may have not reached the high temperature calculated in run 8, thus confirming the probably too fast oxidation rate.

p.13

DNSs of the film boiling configuration

In an attempt to adjust more precisely the kinetics of oxidation in the film boiling configuration, the following stage thus consists in performing DNSs with MESO, similarly to what was done for the gas configuration. Here, we will only give a brief status of the work as it is still in a preliminary phase. As stated previously, in the current MESO model, only one field is used to calculate both the liquid water (when the temperature is below saturation) and steam (when the temperature is above saturation). Around the saturation temperature, a mixture of liquid and vapor is considered at local equilibrium. Due to this single-fluid formulation, MESO simulations of film boiling are more easily carried out in high pressure conditions, where the differences in the fluid's physical properties between the liquid and vapor states are much less significant than those at low pressure. A 2-phase formulation is foreseen to be able to evaluate situations at low pressure. Nonetheless, liquid/liquid configuration is most important in explosion configuration, i.e. at high pressures. In a side work [START_REF] Meignen | Outcomes of the French ICE project on Fuel Coolant Interaction[END_REF]simulations of film boiling in the pressure range [5-100] MPa were done, indicating that film boiling under such conditions could be approximately reproduced by standard film boiling correlations as those integrated in the MC3D code, as well as results from TREPAM experiments [START_REF] Berthoud | Film boiling heat transfer around a very high temperature thin wire immersed into water at pressure from 1 to 210 bar: Experimental results and analysis[END_REF] . An example of calculations is given in Fig. 6. The MESO calculation compares well with the test experimental data and is in the range of the tested correlations. It may however be noticed from this figure that the Epstein-Hauser correlation used as first guess for the model over-predicts in general the heat transfer.

p.14 It is also clear that the oxidation should modify the transfer and film characteristics.

Only simulations in supercritical pressure conditions are here considered. In this domain, pseudo film boiling may be observed and the fluid does exhibit a high density pseudo-liquid state and low density pseudo-vapor state. An example of first performed simulations is given here below for a case at 100 MPa run with oxidation either enabled (case A1) or disabled (A2). The drop temperature (2150 K) is maintained fixed all along the simulations which are run for a typical explosion time of 3 ms. It is worth noting that in these simulations the reaction does not generate any heat and the cooling is neither taken into account (the drop has a fixed temperature). The inlet fluid temperature is 750 K, i.e. with a rather small but substantial water subcooling ( 54 K).

The fluid density fields are compared Fig. 7. The contours of hydrogen fraction are shown in Fig. 8. We find that, at the examined time scale here, a quasi-steady situation is rapidly reached. The equivalent volume of (gas) is not so different from A1 to A2. The film thicknesses measured at the stagnation point (δSP) and at the equator (δEQ) are gathered in table IV, compared with that given by the Epstein-Hauser correlation. In the absence of oxidation, the Epstein-Hauser correlation leads to a somewhat reduced film thickness, as already commented before. But, although this correlation is not dedicated to films comporting hydrogen, the order of magnitude of the calculated values remains quite satisfactory. It is also verified that the hydrogen boundary layer is embedded in the thermal one. Thus, along the pseudo-interface, the hydrogen fraction is effectively very small, as hypothesized in the model, eq. ( 10).

TREPAM exp. result p.15 Finally, one can compare the oxidation kinetics from the hypothesized formulation (10) with that obtained using MESO DNS, Fig. 9. Following the conclusions of previous section, the diffusivity might be computed at the surface temperature. This approximately doubles the oxidation rate compared to the standard assumption of using a mean temperature. It is seen that the MESO oxidation rate is in the same range, although notably smaller. These results are to be compared also with those reported in Fig. 5(at ambient pressure and spherical geometry). Although additional work is needed, in the three directions of DNS simulation, macro modeling and experiments, these results are encouraging and should lead to the derivation of an adjusted oxidation rate in near future. At the moment, the use of the formulation (10), with diffusion coefficient evaluated at a mean film temperature may be used for preliminary evaluations.

p.16

Fig. 1 :

 1 Fig. 1: Drop configuration studied in order to model oxidation

Fig. 3 :

 3 Fig. 3: Example of simulation domain used with MESO. The red represent the cylinder, coarsely meshed here. An inlet condition is applied at the bottom (large green line) and a pressure condition at the top (large orange line). The average inlet velocity is almost one-dimensional.

Fig. 4 :

 4 Fig. 4: Comparison between the kinetics of reaction given by the MESO DNS model and the eq (12) introduced in the MC3D-PREMIX application. Inset: magnification of the 0 -20% zone

Fig. 5 :

 5 Fig. 5: Temporal evolutions of the kinetics of oxidation and mean drop temperatures calculated with the macroscopic model and compared to the data by[START_REF] Lemmon | Studies relating to the reaction between zirconium and water at high temperatures[END_REF], blue diamonds, and[START_REF] Ma | MISTEE Investigation on Mechanisms of Steam Explosion during Severe Accidents[END_REF], orange diamond.

Fig. 6 :

 6 Fig.6: Comparison of the heat transfer calculated from the numerical simulations and the heat transfer given by three film boiling correlations and a result from the TREPAM test, at 24 MPa and high subcooling.

Fig. 7 :

 7 Fig. 7: Snapshots of the fluid density field around the drop for the two cases without oxidation (A1, top) and with (A2, bottom). Red is for the fluid high density zones (liquid water); Blue is for the low density regions (gas). The black line marks the contour of saturation temperature.

Fig. 8 :

 8 Fig. 8: Visualization of the 2D contours of hydrogen fraction and contour of saturation temperature (at total pressure) (simulation A2).

:

  Table IV: Measurement of the film at different locations around the drop. OE •Ž : film thickness at the upstream stagnation point; OE •• : film thickness evaluated with the Epstein-Hauser correlation; OE •' : film thickness at the drop equator. Run OE •Ž ss OE •• ss OE •' ss A1 0Considering a film composition of 50% steam and 50% hydrogen.

Fig. 9 :

 9 Fig. 9: Comparison of the hydrogen production from the preliminary proposal (10), considering two temperatures for the evaluation of diffusivity, with the MESO model result for the case A2.

Table II :

 II Test grid for the vapor configuration and kinetics of oxidation (in percentage of the total oxidation per second) deduced from the PREMIX and MESO calculations. The relative errors based on the results given by the MESO calculations are put in the last column.

	Run	D (mm)	q r s /t	u r v	w xyz	u { v	Varied parameter	Ox. Kinetics (% • t )} ) MESO PREMIX	Error (%)
	1	3	1	400	1	2500	Reference	10.1	11.2	+10.9
	2 3 4 5 6 7 8 9 10 11 12 13 14	1 10 3 3 3 3 3 3 3 3 3 3 3	1 1 0.1 5 10 20 1 1 1 1 1 1 1	400 400 400 400 400 400 1000 1500 460 550 600 400 400	1 1 1 1 1 1 1 1 10 50 100 1 1	2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2500 2200 3000	~ 1 mm ~ 10 mm € r 0.1 m/s € r 5 m/s € r 10 m/s € r 20 m/s r 1000 K r 1500 K ' 10 bar ' 50 bar ' 100 bar 2200 K 3000 K	72.1 1.3 7.6 16.4 22.4 31.7 8.1 7.9 23.4 46.1 65.2 9.0 11.8	73 1.5 5.4 18.8 24.1 31.5 9.6 9.1 23.5 44.3 60.5 10.7 12	+1.2 +15.4 -29 +14.6 +7.6 -0.6 +18.5 +15.2 +0.4 -3.9 -7.2 +18.9 +1.7

Table II :

 II Experimental conditions and results fromLemmon Jr. et al. [5] that have been used in the simulations. Extents of oxidation reported are those obtained by weight increase measurement. The drop diameter d is deduced from the mass md of the debris, supposing that they are spherical.

	Run 8	u r v 338.0	w xyz 1	u … v 2173	† ‡ˆ t 6.20	∆q r s/t 0.73	‰ … Š 0.59	{ ss 5.6	‹ ‡ˆ % 36.8
	9	366.5	1	2173	9.25	0.71	0.62	5.7	44.1

ACKNOWLEDGMENTS

This work is done in the frame of French RSNR-ICE research program on FCI. The authors acknowledge the sponsors of the ICE program, namely EDF, AREVA and the French Government through the Research National Agency (ANR). We also thank other participants in the RSNR-ICE program, from CEA and the University of Lorraine for their valuable comments on this work. Many thanks also to Severe Accident KTH team, particularly Sevostian Bechta and Weimin Ma, for valuable discussions and works on the MISTEE-Zr tests. We also acknowledge Marc Barrachin and Bruno Piar, IRSN, for their support.

CONCLUSIONS

In this work, a macroscopic model of oxidation based on the assumption that the H2-steam inter-diffusion is the rate-limiting mechanism is being derived for the FCI code MC3D for both premixing and explosion conditions. This model intends to describe the oxidation rate as long as the drop is liquid. In such case, it is likely that diffusion is very fast in the melt and thus the limiting process of oxidation is the diffusion in the gaseous boundary layer. In case of quenching and appearance of solidification, diffusion limitation in the melt may be considered thanks to available data. Two drop configurations are considered: the liquid/gas configuration, likely to be more representative for oxidation during premixing, and the liquid/liquid film boiling configuration, likely to be more representative for explosion. In order to verify/adjust the models, a quasi-DNS model has been built and is used to supplement the current lack of precise experimental data. The work is still in a preliminary status and first outcomes are given.

For the liquid/gas system, for which no data could be found, the DNS's allow a qualitative verification and quantitative adjustment of the proposed correlation, eq (12). The model already allows identifying a possible ignition/escalation effect during an explosion due to the strong impact of pressure and ambient velocity.

For the case of liquid/liquid configuration, one has preliminarily used some experimental data of zirconium oxidation. To complement the analysis, the work has been oriented towards the use of the DNS technic. Only preliminary results are shown here. They are nevertheless encouraging for the derivation of an adequate correlation. The next step is then related to the continuation of the DNS work to build a reliable correlation, before confronting to the available experimental data related to steam explosion.

p.17

Quite important efforts are however still needed to achieve the development of a full validated model of oxidation and its impact on FCI. In particular, over the course of this study, taking into account the variations of the material physical properties due to oxidation has become an important perspective. The melt energy as well as the entire oxidation process intimately depends on this feature. Solidification will be strongly affected. Thus, the final step will be to deduce the solidification mechanism from DNSs. We nevertheless encourage experimental work for both liquid/gas and liquid/liquid conditions. At that point, we can cite current efforts underway by both CEA and KTH institutes.

NOMENCLATURE

B

: film thickness (m) ∆' "

: enthalpy of reaction (J/mol) "

: density (kg/m 3 )


Satyajit Phadke

Roman Mysyk

Mérièm Anouti
email: meriem.anouti@univ-tours.fr

Effect of cation (Li + , Na + , K + , Rb + , Cs + ) in aqueous electrolyte on the electrochemical redox of Prussian Blue Analogue (PBA) cathodes

Keywords: Prussian blue, Cathode, Alkali cations, XRD, Batteries

Prussian blue analogue (PBA) material is a promising cathode for applications in Na-ion and K-ion batteries which can support high c-rates for charge and discharge. In this study, the material of composition [K 2 Cu II Fe II (CN) 6 ] was synthesized and its structural and electrochemical redox behaviour was investigated with 5 different alkali insertion cations (Li + , Na + , K + , Rb + , Cs + ). Galvanostatic measurements indicate that the redox potential strongly depends on the ionic radius of the inserted cation. The redox potential varies by ~ 400 mV between using Li + (0.79 Å) or Cs + (1.73 Å) in the electrolyte. The underlying modification of the Fe 2+ /Fe 3+ redox potential in PBA is proposed to be due to the weakening of the Fe-C bond in the material. This hypothesis is supported by XRD measurements which reveal that the lattice parameter of the de-intercalated host structure follows the same trend of monotonic increase with the cation size. The relatively minor volume changes accompanying the redox (1.2%-2.4%) allow the PBA to accommodate differently sized cations, although the structural hindrances are quite pronounced at high c-rates for the larger ones (Rb + and Cs + ). Cycle aging studies indicate that the minimum capacity fade rate is observed in case of K + and Rb + containing electrolyte. The peak intensity corresponding to the [220] crystallographic plane varies depending on the state of charge of PBA, since this plane contains the insertion cations. Owing to the sensitivity of the redox potential to the insertion cation coupled with the observed fast ion-exchange ability, the PBA material may find additional analytical applications such as ion sensing or filtration devices.

Introduction

Energy storage devices are gaining global importance owing to their increasing use in portable electronic devices, electric vehicles and stationary storage applications [1]. Additional high power capability (W kg -1 ) is desirable for specific cases such as acceleration and regenerative braking in electric vehicles, elevator backup systems and power reliability systems for aviation [2]. Traditionally, Li-ion batteries containing intercalation cathodes such as LiCoO 2 (LCO), LiNi 0.33 Mn 0.33 Co 0.33 O 2 (NMC) or LiFePO 4 (LFP) possess high energy density (120-180 Wh kg -1 ) but lack in terms of power density (W kg -1 ) [3,4]. Due to the diffusion limitations during intercalation and de-intercalation of Li + , the obtained capacity (mAh g -1 ) and nominal voltage (V) for these materials tends to reduce drastically at high crates.

Most of the intercalation electrode materials are also very cation specific. By contrast, Prussian blue analogue (PBA) materials with the chemical formula [A x M' a M" b (CN) 6 ] have been shown to reversibly intercalate a wide variety of cations [5][6][7]. Although, the theoretical gravimetric specific capacity is a little low for use as battery electrodes (60-100 mAh g -1 ), owing to their fast intercalation kinetics which are suitable for used as electrodes in hybrid supercapacitors [8][9][10]. The PBA materials have been shown to retain a large fraction of their theoretical capacity at scan rates as high as 50 C [9][10][11][12]. Other pseudocapacitive electrode materials in this category include inorganic materials such as MnO 2 , V 2 O 5 [13][14][15][16][17] and organic materials such as PAQS, PPy, Polyaniline [18][19][20][21][22].

PBAs are members of a family of multifunctional molecular materials with electronically active metal sublattices known for their unusual electronic, optical, photothermal and magnetic properties [23][24][25][26]. Margadonna et al. reported using synchrotron X-ray diffraction measurements that the PBA exhibits isotropic near zero thermal expansion (ZTE) in a large temperature range between 4.2 and 300 K [27]. Their high compositional flexibility combined with a host of interesting physical properties could be exploited to design multifunctional energy storage materials through the suitable selection of transition metal ions and/or interstitial units.

Recently there have been reports on the use of this material as a cathode in Na, K and Li-ion in batteries and capacitors [28][29][30][31]. However, to the best of our knowledge, an integrated structure-redox behavior study using different cations has not been reported. In this study, we investigate the effect of inserted cation (Li + , Na + , K + , Rb + , Cs + ) on the obtained capacity (mAh g -1 ), redox potential (V) and capacity fade rate (%/cycle) of the synthesized PBA material. The observed structural variations of the PBA host material from XRD measurements are used to explain the contrasting electrochemical behavior obtained in the presence of different cations.

2.Experimental

Material synthesis

Prussian blue analogue material with the following composition K 2 Cu II Fe II (CN) 6 was prepared by mixing equal volumes of 50 mM aqueous solution of K 4 Fe(CN) 6 and 100 mM aqueous solution of Cu(NO 3 ) 2 . The two solutions (120 mL each) were added dropwise to 60 mL of de-ionized water in a beaker. The de-ionized water was kept under magnetic stirring during the addition. The reaction between the two chemicals is fast and the PBA material instantaneously precipitates out. The precipitate was removed by vacuum filtration using Whatmann filter paper (GF/A). Following this, the precipitate was washed several times with water and then dried at 60 °C for 10 h. The as-synthesized powder is in the reduced (that is the discharged) state.

Electrode preparation

For preparing the electrodes, the PBA powder was dry and then mixed with Ketjen Black (KB) in a mortar and pestle. The powder mixture was added to a solution of N-methyl pyrrolidone (NMP) containing 5 wt% Polyvinylidene fluoride (PVDF). The final electrode composition was PBA:KB:PVDF (80:10:10) by weight. The obtained slurry was coated on graphite paper (Sigracet BC, SGL Carbon) and dried at 90 °C under vacuum for 3 h to remove the NMP.

Electrolytes

The electrolytes were prepared by dissolving the nitrates of the respective alkali cations (Li + , Na + , K + , Rb + , Cs + ) in de-ionized water to obtain a 1 M concentration. Using the same anion (nitrate) ensures that the only difference between the electrolytes is the cation. The pH of the electrolyte was adjusted to 2 by addition of nitric acid.

2.4.Electrochemical testing

For electrochemical testing, standard three electrode cells were assembled. The counter electrode was a commercial AC electrode while a standard Ag/AgCl reference electrode was used. The electrodes were immersed in a beaker containing the appropriate electrolyte. The electrochemical measurements were performed on a VMP-3 Biologic Potentiostat. The cycled electrodes were stopped, either in the charged state or the discharged state, removed from the electrolyte and washed thoroughly with water prior to XRD characterization.

2.5.XRD characterization

After drying, the as-prepared and charged and discharged electrodes were subjected to XRD under the following conditions. The wavelength of the Cu Kα radiation was λ = 1.5418 Å, while the angular resolution in 2θ scans was 0.02° over range of 10°-70°. Structural refinement of the obtained data was done using PowderCell software. Crystal structure visualizations were prepared using VESTA software.

3.Results and discussion

The crystal structure of the as-synthesized material has the formula K 2 Cu II Fe II (CN) 6 and belongs to the cubic space group Fm m. The dimension of the unit cell as calculated from XRD measurements on the PBA is a 0 = 9.9954 Å. Each unit cell contains 4 formula units of the molecule. The Fe 2+ ions occupy the corners and face centered positions of the unit cell as shown in Fig. 1. A C≡N group (not shown) is located between neighboring Cu and Fe ions to form the network of the host structure. The C≡N group is oriented in a way to form Fe-C and Cu-N bonds as shown in Fig. 1. In each unit cell, the eight K + cations occupy the Wyckoff position 8c. Out of these 8 cations, only 4 K + ions can be reversibly intercalated during the electrochemical reduction corresponding to the four Fe 2+ /Fe 3+ redox centers as shown in reactions (1) and (2). During oxidation, the Fe 2+ is re-oxidized to Fe 3+ and de-intercalation of the itinerant cation occurs as represented in Fig. 1. The calculated theoretical discharge capacity based on this reaction is 63 mAh g -1 . The obtained positions of the ions by refinement of the XRD data of the as-synthesized material are shown in Table 1. The five cations chosen for this investigation are Li + , Na + , K + , Rb + and Cs + . The radii of the ions as well as their diffusion coefficients obtained from the literature are shown in Table 2 [32,33]. It can be seen that the ionic radius of Li + is almost half of K + , while that of Cs + is approximately 30% higher.

Table 2. Ionic radii of cations of interest (A = Li + , Na + , K + , Rb + and Cs + ) in aqueous solutions, along with their diffusivity (25 °C) in aqueous electrolytes [32,33].

Electrochemical measurements

Redox potentials

The cyclic voltammetry at 1 mV s -1 performed on the PBA electrodes in electrolytes containing different cations is shown in Fig. 2(a). Based on the large variation in the shape the CVs and the redox potentials, it is obvious that the nature of inserted cation has a significant impact on the electronic environment of the Fe 2+ /Fe 3+ redox center. In case of Li + , a broad profile consisting of a multiple peaks is observed over a wide potential range of ~800 mV. In case of Na + , two distinct peaks can be observed during charge and discharge. By contrast, it is found that, for the larger cations (Rb + and Cs + ), there is only a single redox peak.

Table 3. Redox potentials (V vs. SHE) obtained from the oxidation and reduction peaks observed during cyclic voltammetry (1 mV s -1 ) of PBA. The reduction and oxidation peak potentials observed from cyclic voltammetry are summarized in Table 3. The redox potential is estimated by taking the average of a pair of the prominent oxidation and reduction peaks. It can be seen that the average redox potential increases with increasing cation size and is the highest for Cs + cation (0.95 V). By simply exchanging the cation in the electrolyte from Li + to Cs + , the redox potential shifts by almost ~400 mV. In fact, the cation dependence of redox potential has been reported previously by two groups separately working on Prussian Blue thin film electrodes of composition KFe II Fe III (CN) 6 in the 1980s [7,35]. However, no explanation was offered for the observation. Since we observe the same effect, in a material of different composition K 2 Cu II Fe II (CN) 6 , it appears that this is generally applicable to materials of hexametallocyanate family.

Cation E PA (V) E PC (V) E 1/2 (V
When the CV is conducted at a lower scan rate (0.1 mV s -1 ), the obtained scans are significantly modified as shown in Fig. 2(b). For all cations, there is an increase in electrochemical reversibility as observed from the lowering of (E PA -E PC ). In case of Rb + and Cs + , the peaks appear distinctly sharper indicating that diffusion of these cations in the PBA is more sluggish as compared to Na + and K + . As a result, allowing more time for the diffusion of cations by lowering the scan rate significantly influences the electrochemical behavior. In a previous work of Goodenough, it was shown that the potential of the Fe 2+ /Fe 3+ redox center can shift significantly based on the surrounding environment. In such materials the redox potential was found to shift by ~800 mV depending on the choice of the polyanions (SO 4 ) 2-, (PO 4 ) 3-, (SiO 4 ) 4-, (AsO 4 ) 3-or (WO 4 ) 3-[36]. It was shown that, as the Fe-O bond length increases, the Fe 2+ /Fe 3+ redox potential also increases [37,38]. More recently, it was predicted that by analogy in PBA materials where each Fe is bonded to 6 carbons, it is the nature of this Fe-C bond which would primarily affect the redox potential [39]. Based on the redox potentials obtained in this work, it appears that the Fe-C bond should be the longest in the material when Cs + cation is used. This in turn would translate to the increase of the redox potential as observed. By contrast, in case of Li + ion insertion, the redox potential observed is the lowest (0.58 V), which would imply that the Fe-C bond is the shortest. This hypothesis is supported by the XRD results which are discussed in a following section.

3.1.2.Voltage profiles and reversible capacity

The potential profiles obtained from galvanostatic cycling in presence of different cations (Li + , K + , Rb + and Cs + ) are shown in Fig. 3(a-d). It can be seen that irrespective of the cation in the electrolyte, the capacity on the first charge (1 C) is ~60 mAh g -1 and close to the theoretical capacity. However, during the first discharge, the capacity is lower and highly dependent on the cation. The extent of unrecovered capacity (shown in green) is larger for the larger cations namely Rb + (11 mAh g -1 ) and Cs + (24.4 mAh g -1 ). Additionally, the reduction in capacity (shown in red) by going to a higher c-rate (5 C) is also highest for these cations. This is indicative of diffusion constraints for the larger cations as inferred from the cyclic voltammetry.

The other distinction between cations is the flatness of the potential profiles. A flat profile during electrochemical reaction is observed when a two-phase system is encountered such as that occurring in case of LiFePO 4 or Li 4 Ti 5 O 12 [38,40,41]. In case of Li-Bi batteries also, a varying voltage is observed in the single phase region whereas a flat voltage when the cathode enters the two-phase region [42]. It is possible that the insertion of the larger cations (Rb + and Cs + ) causes significant structural distortion leading to a similar effect. On the other hand, the insertion of small cations such as Li + is easier to accommodate into the structure. The voltage profiles of Na + and K + exhibit intermediate behaviour; where during the initial portion (< 20%) of the discharge the voltage changes continuously whereas during the remaining part, the voltage remains relatively flat. These two regions are labelled (separated by a dotted line) in Fig. 3(b) taking K + as an example.

3.1.3.Capacity retention

The discharge capacity obtained for various itinerant cations at 5 C for the first 50 cycles is compared in Fig. 4. The charge and discharge voltage limits for the galvanostatic cycling were selected based on the cyclic voltammetry results shown earlier to ensure completion of the redox reaction. In Fig. 4 It can be seen that the initial capacity obtained in case of Na + (56.6 mAh g -1 ) and K + (55.1 mAh g -1 ) is the highest and close to the theoretical capacity. The initial capacities and the capacity fade rate (%/cycle) calculated by linear fitting of the data are summarized in Table 4.

For Li + , although the initial capacity is quite close to that of Na + and K + , the capacity fade rate was significantly higher than for all other cations (0.76 %/cycle). This is partly explained by the higher solubility of the PBA (Li + ion intercalated). In case of Li + , the electrolyte assumed the characteristic yellow color of ferrocyanide ion (Fe(CN) 6

4-

) after a few cycles indicating loss of material from the electrode. In case of all other cations, the electrolyte remained transparent until the completion of the test. For other cations such as Rb + and Cs + , although the capacity obtained is much lower, the capacity fade rates are still quite low (0.09 and 0.26 %/cycle respectively).

By comparing the discharge capacity obtained at 1 C and 5 C for all cations (Table 4), it is clear that the larger cations such as Rb + and especially Cs + face structural hindrances due to their large ionic radius. This becomes more evident at the higher c-rate, where Rb + (41 mAh g -1 ) and Cs + (28 mAh g -1 ) exhibit much lower capacity. The channel radius in the PBA lattice is approximately 1.6 Å, which is smaller than the ionic radius of Rb + (1.64 Å) and Cs + (1.73 Å) [6]. Other cations which are much smaller exhibit a very small (<4%) decrease in capacity on increasing the c-rate from 1 C to 5 C.

3.2.XRD measurements

To understand the accompanying structural changes, XRD measurements were conducted on the electrodes while in the fully charged (oxidized state), fully discharged (reduced state) and on the pristine material (Fig 5). The obtained XRD spectra clearly demonstrate three peaks belonging to the carbon current collector which are invariant (Fig 5a). These are indicated by gray dotted lines in Fig. 5(b). The remaining peaks belong to the active material in the electrode and are similar to those reported in literature [43][44][45]. These have been indexed accordingly.

Fig. 5 (a) Obtained XRD patterns for pristine PBA (blue), PBA in the charged/de-intercalated state (green) and the discharged/intercalated state (red). Pattern for uncoated carbon current collector is also shown (black). (b) Calculated unit cell parameters from the XRD measurements as a function of the insertion cation for the PBA material in the intercalated state (red) and the de-intercalated state (green). Also shown are the respective measured redox potentials

3.3.Cation nature and respiration of unit cell

As seen in Fig. 6, the peaks corresponding to PBA, shift towards higher 2θ upon discharge indicating that the unit cell contracts upon intercalation of ions. This observation was found to apply to insertion of all cations studied irrespective of their size. The structural contraction has been observed previously in other materials of the hexacyanometallate family [43].

The calculated unit cell dimensions for PBA under cycling with different cations are shown in Fig. 5(b) for the discharged and charged state respectively. It can be seen that in the charged (or de-intercalated) state, there is a strong dependence of the unit cell dimension and the choice of the itinerant cation. In case of using Cs + as the itinerant cation, the dimension is the largest (10.0697 Å) whereas in case of Li + , it is the smallest (10.0121 Å). According to the Equations ( 1) and ( 2) shown earlier, the composition of the material in the de-intercalated state should be the identical [KCu II Fe III (CN) 6 ] irrespective of the cation employed for the redox reaction. However, it is clear from the obtained XRD data that the PBA host material undergoes compositional changes resulting in an expansion of the unit cell.

One of the possibilities is cation exchange between the electrolyte and electrode as shown in the schematic and reaction in Fig. 7. Since the PBA electrodes are maintained in the electrolyte containing a large excess (~1000 times) of a particular cation, it is possible that exchange occurs between K + ions in the structure and the A + ions in the electrolyte. As a result, the modified composition would be closer to ACu II Fe III (CN) 6 where A = Li + , Na + , K + , Rb + or Cs + . This would explain the direct relationship between unit cell dimension and the ionic radius of the cation. Indeed, Prussian blue is known to be zeolitic in nature with the ability to readily exchange group I cations in aqueous solutions [6,[46][47][48]. This cationexchange ability is aided by the relatively small %ΔV during cycling which ranges between 2.42% and 1.13% in case of Cs + and Li + as shown in Fig. 5(b). These are quite small in comparison to those observed other cathode materials such as LiFePO 4 (6.81%) [38].

As a consequence, it can be seen that the PBA host material is not only exceptionally flexible to cation size but can also accommodate quite high c-rates with ease. For example the discharge capacity in case of Li + , Na + and K + ion only reduces by ~4% on increasing the current from 1 C to 5 C. The average dimension of the cubic particles of PBA is ~40 nm (400 Å) which provides a large electrode/electrolyte interface. Owing to the cubic crystal symmetry (6 faces of the cube can be used for cation exchange) the estimated diffusion distance would only be ~20 unit cells.

In the discharged state (intercalated) the PBA material lattice dimension does not exhibit any specific trend based on the cation size. The unit cell dimensions are mostly centered around the pristine material.

The two most prominent peaks in the XRD pattern correspond to the [200] and [220] crystallographic planes. A close up view of these peaks is shown in Fig. 6(a) for the charged and the discharged electrodes. The orientation of these planes in the unit cell and the atoms positioned on the planes are shown in Fig. 6(b). It can be seen that the insertion cation is located on the [220] plane but not on the [200] plane. Due to this we can clearly observe an increase in the intensity of the peak corresponding to [220] with the progression of discharge (intercalation). The peak height of the [200] plane is invariant. The graph shown here is that for K + but the same phenomenon was observed for all cations.

3.4.Structural interpretation of electrochemical redox potential

It can be seen that the increasing trend of the potential matches well with the increasing cation size (Fig. 7a-c) as well as the increasing lattice parameters in the charged state. This correlation further extends even to non-alkali cations as we found in case of ammonium ion during galvanostatic cycling. As shown in Fig. 7(b), the redox potential obtained (0.93 V vs. SHE) is slight lower compared to Cs + (0.95 V) in Fig. 3(d). The ionic radius of NH 4 + (1.54 Å) is also slightly lower than that of Cs + (1.73 Å) [33,47]. The observed cation ionic radius dependent expansion of the PBA host framework implies a larger stretching (and more weakening) of the Fe-C bond for a larger insertion cation. + and (c) Na + itinerant cation at 1C rate. Cations are inserted during discharge and vice versa as indicated This agrees well with the hypothesis presented in a previous section, wherein the Fe-C bond was predicted to be the largest for the case of Cs + and smallest for the case of Li + based on the measured redox potential. The cation present in the electrolyte is solely responsible for the variation in the redox potential of the host framework since the starting electrode material is the same in all cases. This also implies that the cation exchange process occurs (Fig. 8) quite rapidly in these materials and as soon as they are placed in contact with the electrolyte. In the galvanostatic experiments presented, the electrodes were placed in the electrolyte only few minutes prior to the start of cycling. However, in all cases the redox potentials were already modified starting from the first half cycle of the testing. Owing to the large dependence of the redox potential on the electrolyte cation and fast response time, this material may also find use in ion-sensing and filtration applications in addition to energy storage devices. around the itinerant cation. Dominant cation in the electrolyte (grey) depends on the electrolyte composition and could be Li + , Na + , K + , Rb + , Cs + which may be freely exchanged with the K + cation present in the as-synthesized material.

4.Conclusions

In this study, the intercalation behaviour of different cations into the PBA host cathode material is investigated. Although PBA can reversibly intercalate all tested alkali cations, the capacities obtained at 1 C and 5 C are dependent on the size of the cation and seem to be independent of the cation diffusivity in the electrolyte. Galvanostatic and cyclic voltammetry measurements indicate that the structural hindrance to diffusion of larger cations (Rb + and Cs + ) within the PBA structure results in a lower capacity. Based on aging studies it was found that K + and Na + are ideally suited for this cathode from a capacity perspective (>55 mAh g -1 at 5 C), whereas K + and Rb + cations exhibit the best capacity retention of 0.06 %/cycle and 0.09 %/cycle respectively. The redox potential was found to be strongly dependent on the size of the insertion cation. Structural investigations using XRD measurements indicate that the expansion of the unit cell and resulting lengthening of the Fe-C bond can explain the increase of the redox potential with cation size. The intensity of the peak corresponding to the plane [220] which contains the itinerant cation was found to be sensitive to the state of charge of the cathode. The sensitivity of the redox potential to the insertion cation coupled with the fast ion-exchange ability, renders the PBA material suitable for ion sensing device applications in addition to energy storage devices. Further work is required to better understand the role of the solvent in the intercalation mechanism.

Fig 1 .

 1 Fig 1. Crystal structure of K 2 CuFe(CN) 6 in the discharged (intercalated) and the charged (deintercalated) state. K + ions are shown in red and C≡N bonds are not shown for clarity. Fe and Cu lattice positions are labelled accordingly.

Fig. 2

 2 Fig. 2 Cyclic voltammetry at (a) 1 mV s -1 and (b) 0.1 mV s -1 on PBA under different itinerant cations

Fig. 3

 3 Fig. 3 Charge and discharge potential profile for the first cycle at 1 C and 5 C in electrolytes with different cations (a) Li + , (b) K + , (c) Rb + and (d) Cs + . The reduction in discharge capacity (mAh g -1 ) upon increasing the current from 1 C to 5 C is shown in red. The unrecoverable capacity of the first charge at 1 C is labelled in green.

  (a), the actual capacities are plotted as a function of the cycle number, whereas the normalized values are shown in Fig. 4(b).

Fig. 4

 4 Fig. 4 Discharge capacity expressed in mAh g -1 (a) and normalized discharge capacity (b) as functions of the cycle number for PBA under galvanostatic cycling (5 C) with different itinerant cations

Fig. 6

 6 Fig. 6 (a) Superimposed XRD pattern of PBA in the intercalated (discharged) and deintercalated (charged) state. Itinerant cation is K + . (b) Crystallographic planes [220] and [200] corresponding to two prominent peaks observed in the powder XRD pattern.

Fig. 7

 7 Fig. 7 Galvanostatic charge discharge curve for PBA with (a) Li + and (b) NH 4 + and (c) Na + itinerant cation at 1C rate. Cations are inserted during discharge and vice versa as indicated

Fig. 8

 8 Fig.8 Sub-section of the PBA unit cell showing the arrangement of Fe, Cu, C and N atoms around the itinerant cation. Dominant cation in the electrolyte (grey) depends on the electrolyte composition and could be Li + , Na + , K + , Rb + , Cs + which may be freely exchanged with the K + cation present in the as-synthesized material.

Fig. 1 Fig. 2 Fig. 3 Fig. 4

 1234 Fig. 1 Crystal structure of K2CuFe(CN)6 in the discharged (intercalated) and the charged (de-intercalated) state. K + ions are shown in red and C≡N bonds are not shown for clarity. Fe and Cu lattice positions are labelled accordingly.

Fig. 5 (

 5 Fig.5(a) Obtained XRD patterns for pristine PBA (blue), PBA in the charged/de-intercalated state (green) and in the discharged/intercalated state (red). Pattern for uncoated carbon current collector also shown (black). (b) Calculated unit cell parameters from the XRD measurements as a function of the insertion cation for the PBA material in the intercalated state (red) and the de-intercalated state (green). Also shown are the respective measured redox potentials.

17

 17

Fig. 6 Fig. 7

 67 Fig. 6 (a) Superimposed XRD pattern of PBA in the intercalated (discharged) and de-intercalated (charged) state. Itinerant cation is K + . (b) Crystallographic planes [220] and [200] corresponding to two prominent peaks observed in the powder XRD pattern.

Fig. 8

 8 Fig.8Sub-section of the PBA unit cell showing the arrangement of Fe, Cu, C and N atoms around the itinerant cation. Dominant cation in the electrolyte (grey) depends on the electrolyte composition and could be Li + , Na + , K + , Rb + , Cs + which may be freely exchanged with the K + cation present in the as-synthesized material.

Table 1 .

 1 Crystal structure data for K 2 CuFe(CN) 6 .

	Atom Site x	y	z	B iso	SOF
	Fe	4a	0.0000 0.0000 0.0000 3.5900 1.0000
	Cu	4b 0.5000 0.5000 0.5000 1.3600 1.0000
	K	8c	0.2500 0.2500 0.2500 17.0400 1.0000
	C	24e 0.7948 0.0000 0.0000 4.7900 1.0000
	N	24e 0.6860 0.0000 0.0000 3.1400 1.0000
	Space group (Fm m, a 0 = 9.9954 Å).

Table 4 .

 4 Initial

	discharge capacity (mAh g -1 ) obtained while using different itinerant cations
	at 5 C and 1 C rate of galvanostatic cycling along with calculated capacity fade rate
	(%/cycle).

Acknowledgments

The authors would like to thank "Le Studium Loire Valley Institute for Advanced Studies" for financial and logistical support to the researchers involved in this study and "Region Centre" for financial support under the "Lavoisier" program.

Conflicts of interest

There are no conflicts to declare.

TOC

The redox behaviour of Prussian blue analogue (PBA) material was investigated with 5 aqueous alkali nitrate electrolytes (Li + , Na + , K + , Rb + , Cs + ) showing crystallographic plane variation of PBA depending on the state of charge.