Didier T D D Le

X P P Lasseux

G Nguyen

N Vignoles

A Mano

Kuhn

D Lasseux

X P Nguyen

G Vignoles

N Mano

A Kuhn

Multi-scale modeling

Keywords:

Introduction

ABSTRACT

A multi-scale model of diffusion/reaction at play in a porous electrode is developed and solutions to the physico-electro-chemical coupled problem are provided. This represents a key step to progress in the optimization of new efficient and innovative micro-electro-devices that needs to be addressed from a chemical engineering point of view. The pore-scale model based on Fickian diffusion in the porous med ium and Nernstian layer and the electrochemical reaction governed by the Buttler-Volmer equation is upscaled using volume averaging to obtain a macroscopic model that describes the process on an effec tive equivalent medium. The validity and accuracy of the macroscopic model is successfully checked through the comparison with direct numerical simulations of the initial microscale model for amperom etry tests. Predictions obtained from the upscaled model on the current intensity versus the scanning potential during voltammetry reveal to be in very good agreement with experimental results reported in the literature. These results show the capability of the macroscopic model to analyze the behavior of the porous electrode. In particular , it provides an efficient tool to study the dependence of the current intensity on the microstructure of the porous material and on the electrochemical parameters with the perspective of optimizing the electrode efficiency.

Porous electrodes have become a privileged solution for minia turized and potentially in vivo implantable active and passive electro-analytical devices such as biosensors, bioactuators, batter ies, etc. thanks to their high specific surface area which allows effi cient in situ electrical energy production through the use of rather well-controlled redox couples. Indeed, for a given over-potential, such electrodes may provide electrical current at least one order of magnitude higher than classical flat electrodes of the same size [START_REF] Walcarius | Ordered porous thin films in electrochemical analysis[END_REF][START_REF] Szamocki | Tailored mesostructuring and biofunctionalization of gold for increased electroactivity[END_REF] making possible a reduction of the overall macroscopic size of electrodes down to a few tens of micrometers in thickness and a few millimeters in extent. For the past two decades, porous electrodes have often been synthesized by self-assembly and electrodeposition [START_REF] Reculusa | Synthesis of colloidal crystals of controllable thickness through the Langmuir-Blodgett technique[END_REF] using the Langmuir-Blodgett tech nique [START_REF] Blodgett | Films built by depositing successive monomolecular layers on a solid surface[END_REF]. These techniques allow to create a wide range of porous microstructures with tunable porosity as well as controllable architecture [START_REF] Karajic | Bottom-up generation of miniaturized coaxial double electrodes with tunable porosity[END_REF]. closure variable that maps V ( c A / onto c A (m) concentration of species X (X = A or B) (mol/m 3 ) bulk concentration of species A (mol/m 3 ) superficial average of c A (mol/m 3 ) intrinsic average of c A (mol/m 3 ) area average of c A (mol/m 2 ) concentration deviation (mol/m 3 ) dimensionless concentration averaged over the fluid do main molecular diffusion coefficient of species A (m 2 /s) effective diffusion tensor (m 2 /s) effective diffusion coefficient in the isotropic case (m 2 /s) diameter of spherical pore, pore connection window size (m) potential, standard potential (V) Faraday's constant (C/mol) initial concentration of species A (mol/m 3 ) concentration of species A at the interface A f e (mol/m 3 ) current per unit volume at the scale of the representa tive unit cell (A/m 3 ) total current intensity ..,>.,>?,.>.,,>-,. >-.., >-., -' /-.,,>-..,>,,> "�"..-"?v"'v",,•y"-/v"-. ,,,..,, ,. ,," .,." .,. "-,." .., ",."-r ,._ ,,,..,. .. -r"' .. ,,, ... ,. "' .,.",.." ..," ,,. "..,",, -'-,." ,,,." ,. '" ,.

. ,,. .,,..,,..,,..,,..,,. .,,. .,,. .,,..,,..,,. .,,. deposition of a colloidal template of self-assembled silica beads, with a diameter d, and a number of layers that can be adjusted, on a gold-coated glass substrate [START_REF] Reculusa | Synthesis of colloidal crystals of controllable thickness through the Langmuir-Blodgett technique[END_REF][START_REF] Szamocki | Tailored mesostructuring and biofunctionalization of gold for increased electroactivity[END_REF]) (see Fig. 1.1a). Secondly, sequential elec trodeposition of gold ( or other conducting materials like polypyrrol [START_REF] Heim | Engineering of complex macroporous materials through controlled electrodeposition in colloidal superstructures[END_REF]) and nickel layers is performed so that the limits between these layers, and hence their thickness, can be strictly controlled (Reculusa et al., 2011) (Fig. 1.1 b ). Finally, the nickel layer and the silica template are successively dissolved to obtain a two-electrode device (Fig. 1.1c and d). It should be noted that the purpose of the nickel layer is to form an electrical insulation between the two electrodes and the thickness of this layer may be typically of few beads diameters d,. The resulting porous struc ture is such that, due to the limitation of the electrodeposition pro cess in the interstice between touching beads, open connecting windows of characteristic size d e between two adjacent pores remain. Evidently, the mass and electron transfer within the por ous structure, as well as the reactive mechanism governing the whole process that conditions the resulting efficiency of the elec trode, are intimately related to the architecture of the porous material.

As indicated above, current available methods for electrode engineering allow for rather well-controlled micro-or nano structured porous materials with a given pore-size gradient and distribution to be achieved. However, the question of how an opti mal pore architecture can be designed to maximize the electrode power efficiency has been receiving an empirical answer so far. A rational approach is indeed crucial in order to clearly relate the current available at the macroscopic level to the microsctructure on the basis of a careful physical description of mass and electron transfer coupled to reaction mechanisms at play at the microscale (pore-scale) and their implication at the macroscale. So far, model ing has been often restricted to a 1D approach, making use, most of the time, of empirical macroscopic models [START_REF] Barnes | Voltammetry at porous electrodes: a theoretical study[END_REF][START_REF] Do | Mathematical modeling of a porous enzymatic electrode with direct electron transfer mechanism[END_REF]. A formal upscaled model, obtained from volume aver aging and describing the transport and reaction process in a porous electrode, taking into account three phases (solid, liquid and gas), was proposed two decades ago [START_REF] Vidts | Governing equations for transport in porous electrodes[END_REF]. However, there is no closure scheme available in this work so that effective parameters in the macroscopic equations can not be properly esti mated. In [START_REF] Ferguson | Nonequilibrium thermodynamics of porous electrodes[END_REF], correlations between the effective diffusivity (or conductivity) and the tortuosity were employed, based on Wiener or Hashin-Shtrikman limiting bounds, without any explicit dependence on the real geometry of the microstructure. In [START_REF] Ender | An extended homogenized porous electrode model for lithium-ion cell electrodes[END_REF], a homogenized porous electrode model for lithium/ion cell electrode was proposed where the effec tive conductivity was directly obtained from the experimental measurement.

Our purpose in the present study is to contribute to a physically sound multi-scale modeling of the coupled transfer and reactions in a simplified case where exchange of electrons directly occurs between the species in the fluid and the pore surfaces without any contribution of enzymes or redox mediator. As a proof of con cept, the reduction of 0 2 to H 2 0 2 has been chosen. In this context, at the pore-scale, mass transfer is essentially diffusive, while a cathodic oxygen reduction-like reaction can be considered making use of the Butler-Volmer model [START_REF] Butler | Studies in heterogeneous equilibria. part ii. The kinetic interpretation of the nernst theory of electromotive force[END_REF][START_REF] Butler | The mechanism of overvoltage and its relation to the combination of hydrogen atoms at metal electrodes[END_REF]. Start ing from the pore-scale initial boundary-value problem, the method of volume averaging [START_REF] Whitaker | The Method of Volume Averaging[END_REF] is employed in order to derive a closed macroscopic mass transfer equation con taining macroscopic effective coefficients together with the expression of the current available at the electrode. The effective mass transport coefficient is obtained from the solution of a clo sure problem that allows passing the microscale information up to the macroscopic level.

The article is organized as follows. The initial boundary value problem is formulated at the microscopic scale in Section 2, a numerical solution of which is provided on model porous struc tures in Section 3. With the aim of deriving a model that operates at the macroscale and that is much less demanding in terms of computational resources while keeping the essential information from the pore-scale, the upscaling is proposed in Section 4. This yields an effective model that allows the determination of the effective current available at the electrode. In Section 5, numerical solutions of the macroscopic upscaled model are reported for the different microstructures under consideration, characterized by their pore diameter d,, pore connection window size d e , porosity B t , specific surface area a v , and macroscopic geometries, defined by the electrode thickness L e i ee and fluid external diffusion layer (Nernstian layer) thickness L N . Successful comparisons with exper imental data are also reported in this section. Conclusive remarks are drawn in Section 6.

Microscopic pore-scale model

The porous electrode, immersed in the reactive solution, occu pies the spatial domain Q composed of the solid skeleton in Q, and the fluid phase in n 1 , the interface between the two being denoted by I, 1 = n 1 n Q, where redox reaction takes place ( Fig. 2.1 ). Here, we focus on the direct reduction reaction of oxygen at the cathode without any catalysis, so that the electrochemical process can be represented by [START_REF] Barnes | Voltammetry at porous electrodes: a theoretical study[END_REF] (1)

where species A and B may represent, for instance, oxygen and hydrogen peroxide respectively. We have denoted by r the position vector on I, 1 locating the sites where n electrons are captured by the acceptor A whereas k 0 is the electron transfer rate constant. Since the interest is laid upon reduction, it is sufficient to specify the mass balance for species A which diffuses from the external bulk fluid Q e in the Nernstian diffusion layer towards the electrode external surface, and then through the pore space n 1 towards the pore walls I, 1 (Fig. 2.1) where electrons transfer takes place. The Nernstian layer in the vicinity of the electrode's external surface represents the region where a significant concentration gradient occurs in the bulk fluid.

In practice, A is sufficiently diluted in the supporting electrolyte for mass transfer to be described as a Fickian process [START_REF] Fick | On liquid diffusion[END_REF]. The Initial Boundary Value Problem (JBVP) governing the diffusion reaction mechanism within the electrode at the pore-scale can hence be written as

Electrode/Bulk fluid interface ! _ ( A .... A i . ...... . 0 e ••••••••• 1 ••••••••••
8t = V • (DA VcA) inQ.f B.C.1 -n • DA VcA = -R red at I sf I.C. cA = h(r) rE0.t,t = 0 fEAfe,\lt (2a) (2b) (2c) (2d)
where cA = cA(r, t) and DA are the concentration at r and t and molecular diffusion coefficient of species A, respectively, while R red is the reduction reaction rate, and n the normal unit vector at the fluid solid interface pointing out of the fluid phase. In the expres sion of the external boundary condition B.C.2, Ate = Q f n 0. e is the entrance and/or exit boundary of the fluid phase O f from/into the external bulk fluid O e . In our case, the reacting molecule (oxygen) is not charged and is a dissolved species in the liquid in contact with the electrode, so, for the sake of simplicity, adsorption/desorption mechanisms are not taken into account in the present analysis.

The reduction reaction rate, R red , is related to the corresponding microscopic electrical current density, j , from Faraday's law [START_REF] Faraday | Experimental researches in electricity: third series[END_REF]

according to j Rred = nF (3)
Moreover, the relationship between the current density and the potential, E, is classically described by the Butler-Volmer equations, which, in the general case is expressed as [START_REF] Butler | Studies in heterogeneous equilibria. part ii. The kinetic interpretation of the nernst theory of electromotive force[END_REF][START_REF] Butler | The mechanism of overvoltage and its relation to the combination of hydrogen atoms at metal electrodes[END_REF] .

_ I F [ ( (1 -ex)nF(E -E °)) _ ( -exnF(E -E °)) ] J-n< 0 exp RT Ca exp RT C A (4)
In this expression, ex is the electron transfer coefficient ( ex takes val ues of about 0.5 ), E 0 the standard potential, Rand Fare the ideal gas and Faraday's constants respectively, T is the temperature and c 8 represents the concentration of species B resulting from the reduc tion of species A. In the particular case of simple oxygen reduction, the oxidation reaction can be completely ignored so that the first exponential term on the right hand side of Eq. ( 4) can be reasonably neglected. Under these circumstances, Eq. ( 4) can be simplified to the Tafel equation [START_REF] Tafel | Ober die Polarisation bei kathodischer Wasserstoffentwicklung ( concerning the polarisation during cathodic evolution of hydrogen)[END_REF] given by .

( -exnF(E -E 0 ) ) J = -nk 0 Fexp RT cA = -nk0FexAcA at I sf (5)
where exA is a potential dependent parameter which expression is ex = exp -anF(E-E") Hence

A IT . ' at I sf (6)
C BCC For a given structure, geometry and operating conditions, the IBVP defined in Eqs. ( 2) and ( 6) can be solved directly in order to finally compute the current available at the electrode given by I= -nkofo.A l cAclA .lrsf

Numerical solution of the microscopic model (7)

In practice, 9A(r, t) in Eq. ( 2d) is usually unknown at the inter face with the bulk fluid and the boundary condition B.C.2 shall be replaced by considering the Nernstian diffusion layer, O e , of thickness L N , out of which cA remains constant, i.e. cA = c�.

Direct numerical simulations (DNS) were carried out by solving the 3D microscopic pore-scale model described in Section 2. Sev eral arrangements representative of the porous electrode microstructure may be considered such as cubic (C), body centered cubic (BCC) and face-centered cubic (FCC) with periodic unit cells, of size £REV, represented in Fig. 3.1. Characteristics of these microstructures are imposed by the pore diameter d s and the connection window size d e between two adjacent pores from which the porosity and specific area can be determined as will be further discussed in Section 5.1. The FCC structure, which is the most compact one, was used in the pore-scale simulations as it realistically represents a homogeneous template as the one reported in Fig. 1.1 [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF] .

Simulations, which correspond to chronoamperometric 3D numerical experiments, were performed on a computational domain represented in Fig. 3.2. It is composed of the porous elec trode, between z = -L e i e c and z = 0, and the Nernstian fluid layer between z = 0 and z = L N where molecular diffusion, governed by Eq. ( 2a), takes place. The electrode is made of an array of 1 x 1 x �. • • unit cells in the x, y and z-directions respectively. Peri-tRfv odic boundary conditions are applied along x and y with the idea that the electrode is either of infinite extension in the x-y plane or has an annular shape with its mean radius much larger than its thickness (i.e. is quasi-plane) and its length much larger than fREv for the periodic boundary condition in the axial direction to be relevant. The additional boundary conditions are a zero diffu sive flux at the lower boundary of the electrode, z = -L e i e c, and con stant concentration, c�. at the upper boundary of the Nernstian layer, z = L N . The initial concentration is supposed to be uniform equal to c�. For completeness, periodic boundary conditions on both the con centration and flux in the x and y directions must be added to the above set of equations. The reference length £ REV used in the above dimensionless form is only a choice and does not necessarily corre spond to the relevant physical scaling which would rather impose d s as the reference length. This choice was made and will be kept for the 1 D macroscopic model in Section 5 in order to perform a direct comparison on concentration and current versus dimension less space and time variables.

•• •• •• •• •• • •• •• •• •• •• •• • L •• •• •• •• •• N •• •• •• •• •• •• •• • z-o • •• •• •• •• .. .. • •• •• •• •• •• •• •• •• •• •• •• •• •• •• • .. ..
To illustrate the solution, simulations were carried out with L; lec = 30 and LN = 20 µm (L� ""' 12). In practice, the value of LN is difficult to estimate accurately and the latter value is just a choice to illustrate the solution. Values of the other physical parameters used for the simulations are indicated in Table 3.1. These parame ters were chosen to be consistent with the experimental configura tion further considered in Section 5.3. In particular, the standard potential was taken as the one for the reduction represented in Eq. ( 1) relative to the Ag/ AgCl reference electrode. Since the stan dard potential for the reaction ( 1) relative to the standard hydro gen electrode (SHE) is 0.67 V whereas the potential difference of the Ag/AgCl electrode and SHE is 0.22 V, this leads to Ifl = 0.45 V in our case. In addition, it must be noted that the overall electron transfer results from a complex succession of elementary steps which detailed model is still a subject of debate, in particular for oxygen reduction [START_REF] Shinagawa | Insight on tafel slopes from a microkinetic analysis of aqueous electrocatalysis for energy conversion[END_REF]. In this latter case, there exist a pseudo-elementary rate determining step, involving a sin gle electron transfer, even though the overall reduction occurs with two electrons [START_REF] Zhang | PEM Fuel Cell Electrocatalysts and Catalyst Layers, chapter: Electrocatalytic Oxygen Reduction Reaction[END_REF]. For this reason, the value of n can be rather taken to be equal to 1 in the exponential term of the driving force in the Butler-Volmer Eq. ( 5). It should be recalled that the The software COMSOL Multiphysics was used to solve this prob lem. Careful attention was paid to the mesh convergence and, to fulfill this requirement in the configuration defined above, we used a physics-controlled mesh including extremely fine grid blocks composed of 1.3 10 7 tetrahedral elements in the overall domain represented in Fig. 3.2.

It should be noted that the value of k 0 can be extremely small (down to less than 10-9 emfs) when the electrode surface is not catalyzed by any enzymes or electrons are not transferred by a mediator [START_REF] Renslow | Oxygen reduction kinetics on graphite cathodes in sediment microbial fuel cells[END_REF]. Hence, in our case, the investi gated range of k 0 is from 10-7 cmf s to 10-9 cmf s.

Dimensionless concentration fields obtained from DNS in n.

0.8 >--",-----'�----+------+----� .. .. 0.6 1-------+--------l --'--.., � ., -+-. ------J .. .. .. .. .. .. 0. 4 1--�--+----�--'--------� ---1
ko = 10-9 cm/s ---ko = 10-8 cm/ s 0.2 e-----��+----7 In addition, the current available may be computed from the expression in Eq. ( 7). This numerical result will be shown and dis cussed in Section 5.2 when compared with the current obtained from the macroscopic model.

•••••• ko = 10-cm/s ... 50 ••• • •• •• • l • • •••••••••••r•••••••••••••
While DNS are able to provide detailed and accurate results, the important computational time and resources they require (typi cally 52 min for one simulation in the above-mentioned conditions on a Dell PowerEdge 430-2 processors Intel Xeon E5-2630v3) are such that they can not be reasonably envisaged if one is willing to carry out a thorough analysis of the macroscopic current-to microstructure relationship. Alternatively, an upscaling procedure can be applied to the above equations in order to obtain a macro scopic description of the coupled diffusion-reaction process in the porous medium and an expression of the current at the macroscale. This is the objective of the following section.

Macroscopic effective model

The IBVP under consideration given in Eqs. (2) corresponds to a diffusion with heterogeneous reaction, and resembles the one for which upscaling is detailed in [START_REF] Whitaker | The Method of Volume Averaging[END_REF] for a homogeneous porous medium using the volume averaging method. For the sake of brevity, the main result of the development is recalled with the necessary details reported in Appendix A.

The upscaling procedure starts with the definition of an averag ing domain V (of measure V and size r 0 ) including the solid and fluid domains V s and v 1 (of measure V s and v 1 respectively) and the solid fluid interface A s t ( of measure A st ) from which the poros ity, s 1 , and specific area, a v , can be obtained as In these equations, x is used to locate the centroid of the averaging volume while y = r -x locates any point within v 1 relative to x (see Fig. 4.1 ), and we shall drop this subscript, unless necessary. The derivation of the upscaled model is subject to a scale hierarchy defined by /i p « r 0 « L, /i p being the characteristic pore size and L the macroscopic size of the electrode. In the case of the configura tion of Fig. 3.2 for instance, £ p = d s , ro = £REV and L = L eie c• All through the article, this scale hierarchy is supposed to be satisfied. The upscaling procedure is carried out according to the four main steps reported in Appendix A and, under the constraints on the kinetic number Ki = ko�A r r « 1 and time scale �£ '' » 1, yields the following

A p macroscale diffusion/reaction equation 8(cA/ ( f) f f,f � = V • £1D eff • V (cA) -kolXAav (CA)
with the effective diffusion tensor D e .ff given by

D eff = DA (1 + i ( nbdA) f JA,t (14)
(15)

where b, having the dimension of a length, is the solution of the intrinsic closure problem given by ( see Appendix A for more details)

V 2 b = 0 in V 1 n • Vb = -n at A s t (b) = 0 b(r) = b(r + £;e;) (16a) (16b) (16c) (16d)
Eq. ( 14) operates at scale L once B t , av and D e .ff are computed over a representative unit cell of the structure under concern. When asso ciated to macroscopic boundary conditions, this equation may be solved on the effective medium yielding the field of (c A )f over Q. From this solution, the interest is now to obtain the available cur rent at the electrode. To do so, one can start with the expression of the current density at the scale of a representative unit cell of the porous medium, which, by making use of Eq. ( 5) is given by ( 17) where we have used the decomposition on cA given in Eq. (A.4). As shown in the development in Step 2 of Appendix A, the second area integral term on the right hand side of the above expression can be neglected whenever the constraint on the kinetic number ko�� 1 ' « 1 is satisfied. In addition, introducing the Taylor expansion of Eq. (A.10) into the remaining area integral term and using orders of magnitude estimates leads to (18)

The current per unit volume, i v , at the scale of the representative unit cell is hence given by .

_ (i) sJ A s f _ (i ' ) Iv -- V --av sf (19)
from which the total current available at the electrode is obtained as (20)

The accuracy of this result associated to the macroscopic model will be checked in Section 5.2 by a direct comparison with the current values computed from Eq. ( 5) in the DNS approach.

It should be noted that the upscaling approach does not make any assumption on the porous structure itself and can hence be applied to any complex structure provided length-scales separa tion is satisfied. In particular, it could be envisaged on real images obtained from tomography which may require a local geometrical modification of the structure to make it periodic over the REV. However, this modification would occur over a distance of typical length-scale /i p and would not significantly impact the values of the effective quantities. s_ Numerical simulations using the macroscale effective model In this section, numerical simulation results obtained with the macroscopic model detailed above are presented and comparisons are carried out both with results of DNS at the pore-scale and experimental data reported in [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF].

The porous electrode being now considered as an effective med ium, the problem to be solved, written in its dimensionless form, is given by noted that the reference length fREv is again a choice motivated by the comparison that will be performed with the 3D microscopic model. A more physcically relevant scaling in space would indeed rather be L e i e c• If, at the underlying pore-scale, periodicity is assumed in the x and y directions, as in Section 3, the problem to be solved is now 1D (in the z-direction) and the configuration is the one represented in Fig. 5.1. The dimensionless form of the clo sure problem yielding D e .ff is

V' 2 b' = o in Vt n • V*b' = -n at A sf (b') = 0 b'(r') = b'(r* +Ji;) with (22a) (22b) (22c) (22d) (23)
Relevant boundary conditions at the dividing surface between a fluid and a porous medium are a critical issue when modeling mass transfer coupled to reaction in a system like the one under consid eration here. In the present case, continuity of the concentration and flux expressed in Eqs. ( 21c) and ( 21d) at z• = 0 is adopted and the justification of such a choice is provided in Appendix B.

Effective coefficients

We shall first present results on the effective coefficients (porosity, specific area and effective diffusion coefficient) and examine their dependence upon the pore connection window size for different pore arrangements. As quoted before, in the absence of detailed microstructural experimental investigations, three pre liminary isotropic model structures may be considered, namely cubic (C), body-centered cubic (BCC) and face-centered cubic (FCC), the representative unit cells of which are depicted in Fig. 3.1. For these isotropic structures, D e .ff= 'D e ff l which, together with B f and a;= a v fREv, depend only on the relative pore connec- For the results presented in the next two sections, the FCC structure was considered and the problem in Eqs. (21) was solved using COMSOL Multiphysics. Chronoamperometry simulations were first carried out for comparison with ONS and voltammetry simulations were performed in a second step for comparison with experiments. The effect of the electrode thickness L e t e c, pore microstructure and electron transfer rate constant k 0 was investigated.

� 0.3 � 0.2 0.1 0 -C ---ECG ••••••FCC 5% . . . . 10% . . . . -••• • • . . ,•• .. ..
'-0 ••••••••••• •••••• •••••• ---ECG ...... pee •g 5 �=== =: -=--=--=-� C/l C/l C/l u .s 4 f----+-- C/l p u •••• ••• •••• ••• -------- ••• •• •• •• • Q 3� == =+ ===- ===t= =- ---==!=

Chronoamperometry simulations

Simulations of chronoamperometry using the 1O-macroscopic effective model (Eqs. ( 21)) were carried out in the same conditions as for 30 ONS at the microscale presented in Section 3, i.e. using parameters indicated in Table 3.1. These parameters are close to laboratory testing conditions reported in [START_REF] Karajic | Bottom-up generation of miniaturized coaxial double electrodes with tunable porosity[END_REF], [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF], [START_REF] Szamocki | Tailored mesostructuring and biofunctionalization of gold for increased electroactivity[END_REF]. For this microstruc ture, s 1 = 0.763, a�= 5.985 and s/D e tt /DA = 0.364. As in the 30 case, we took r; 1ec = 30 and L�"" 12 (LN = 20 µm). The 10 simula tions were performed with a mesh composed of 700 grid blocks, a spatial discretization that was checked to ensure mesh conver-gence of the solution. Special care was taken to account for the sharp concentration gradient present at z' = 0 and this was han dled by locally refining the mesh on both sides of this interface.

For comparison purposes, the field of c;, obtained from 30 ONS of Eqs. ( 8) was averaged on each FCC unit cell and then compared with (c;/, solution of the macroscopic effective model. In Fig. 5.4, dimensionless concentration profiles along the electrode obtained for k 0 = 10-s cm/s are represented for several dimensionless times ranging from 10 to 200.

Clearly, the agreement between the two approaches is excel lent, confirming that the macroscopic model correctly captures the diffusion/reaction mechanisms under consideration. It must be emphasized that the computational resources and time are con siderably smaller for simulations of the 1 O macroscopic model in comparison to 30 ONS. In the case under study the computational time is only 1s with the 1 O upscaled model, leading to a speed-up of roughly 3120.

To complete the validation, it is important to compare the total intensity available at the electrode estimated with the upscaled model and given by Eq. ( 20) to that in its original form in the 30 model expressed in Eq. ( 7). In Fig. 5.5, we have represented the 7)) and 1D macroscopic (Eq. ( 20)) models. ,.

.. .. .. .. ........ 3D-micromodel t* = 10 -•-3D-micromodel t* = 50 0.8 ----------- ----. -. . . . • • • • • 3D-micromodel t* = 100 -•••• 3D-micromodel t* = 200 . . . -� . . • . . •• •• •• • .. . . ---
--• • -" • •• -"-• •-• • -" .. , --+-- 0.2 f---+----+-- o�-�---�--�---�--�--� -25 -20 -15 -10 -5 0 z* lD-macromodel t* = 50 • • • • • • lD-macromodel t* = 100 -, -, -lD-macromodel t* = 200
h ----• ,;: i 20) relies.

0.6 0.2 -1i •••-/ r; l' - OA--•-= ••••=-----•-• -••••= •••= ••••�- 1 , , , 0
Evolutions of the concentration profiles within the electrode (cA/ and the Nernstian layer c;; obtained from the simulation of chronoamperometry of the 1 D macroscopic model are depicted in Fig. 5.6 for k 0 = 10-9 cm/s and k 0 = 10-7 cm/s, showing the con centration decrease with time due to diffusion and reaction in the electrode and diffusion from the Nernstian layer to the porous region. In particular, when k 0 is large, almost no concentration gra dient is present in the electrode as the mechanism is strongly dom inated by the electrochemical reaction process, the entire electrode operating in a mass transfer limited regime. Conversely, when k o is small (see Fig. 5.6a for k 0 = 10-9 cm/s) diffusion is significant and a concentration gradient persists in the electrode domain corre sponding to an intermediate regime. A stationary regime is also observed when the balance between diffusion and reaction is reached on the overall system. The equilibrium characteristic time strongly decreases when k 0 increases. Indeed, steady state is reached at t' '""' 3000 for k 0 = 10-9 cm/s and t' '""' 100 for k o = 10-7 cm/s.

The current density averaged on the entire electrode .l. r IJ) :f dV = __1__

1 was also computed according to the relationship l!.ll J!.l s I I,t in Eq. ( 18) and its evolution is represented in Fig. 5.7 for three values of k 0 , namely k 0 = 10-9 cm/s, k 0 = 1 o-s cm/s and k 0 = 10-7 cm/ s. For small values of k 0 (i.e. k 0 = 10-9 emfs), the average current density produced by the electrode is small and remains quasi con stant over time. For larger values of k 0 , a large current is produced at the early stage of the process and strongly decreases to reach a plateau at equilibrium. The influence of the microstructures on the electrical produc tion is illustrated in Fig. 5.8. Keeping all parameters the same, the current intensity per unit volume of the electrode rhf fn i v dV = jtij was computed for the three different pore struc tures C, BCC and FCC and two values of the electron transfer rate constant k 0 = 10-7 cm/s and k 0 = 10-9 cm/s. For both values of k 0 , the largest value of the current intensity is produced on the FCC structure, while the lowest is obtained on the C structure. This can be explained by the values of E t , a v and D e tf that are the largest for the FCC arrangement. For all three structures, the current inten sity decreases with time and reaches a plateau at equilibrium that is attained with almost the same characteristic time for the three structures under investigation. The amplitude of variation of the current intensity over time decreases strongly when k o increases and the rate of decrease is much stronger for a large value of k o .

Voltammetrycomparison with experimental data

The objective in this section is to compare our simulation of the 1D upscaled model with experimental data reported in [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF] in which the intensity of porous gold coated electrodes of different thicknesses was recorded during voltammetry. The porous electrode thicknesses correspond to 7, 27 and 80 half layers of silica templates made of beads of 1.2 µm in diameter and deposited onto a gold wire of 50 µm. Voltammetry was run under a potential scan from E = 0.2 to -0.4 V at the same rate rE = 5 mV/s. The electrode was immersed in a 0.05 M H2S0 4 aque ous solution saturated with oxygen. The number of electron trans ferred in that case corresponds to n = 2.

The diffusion coefficient of oxygen in a 0.05 M H 2 S0 4 solution can be taken as TJA = 2. 10-9 m 2 /s (see [START_REF] Bala | Limiting diffusional currents of oxygen reduction in 0.5-11 m sulphuric acid[END_REF] while Nernstian layer thicknesses L N = 65 µm for bare wire, 27 and 80 half-layers and L N = 110 µm for 7 half-layers were adopted. It must be noted that the Nernstian layer thickness should be considered as a time dependent parameter [START_REF] Molina | On the meaning of the diffusion layer thickness for slow electrode reactions[END_REF]. For the sake of simplic ity, this dependence was disregarded as a first approach in the pre sent work. Two other coefficients need to be chosen, namely the electron transfer rate constant, k 0 , and the charge transfer coeffi cient, rx, for which no values are available so far. Depending on the material coating the pores and the texture of the surface coat ing, k 0 can vary over several orders of magnitude Oirkovsky et al., 2010;[START_REF] Wang | A kinetic study of oxygen reduction reaction and characterization on electrodeposited gold nanoparticles of diameter between 17 nm and 40 nm in 0.5 m sulfuric acid[END_REF]. Again, the value of k 0 may be very small for a redox reaction without catalysis [START_REF] Renslow | Oxygen reduction kinetics on graphite cathodes in sediment microbial fuel cells[END_REF]. As a consequence the values of k 0 and rx were adjusted to achieve the best fit with the intensity curve corresponding to the 27 half layers electrode thickness of Fig. 5.2 in [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF]. The values k 0 = 10-9 cm/s and rx = 0.66 for a gold electrode yielded the best fit. They were kept for all the simulations performed with the other electrode thicknesses (7 and 80 half-layers). For the whole range of the scanning potential investigated in the simula tion of voltammetry, the value of the kinetic number Ki = ko;: d , remains between 3.4 10-6 and 0.09 and this ensures that the esti mation of the current intensity using Eq. ( 20) is valid.

I I \ I '.,, I ' I \ I \ I \ 10 --,' '. ----+-- ' ' �0.2 -0.1 0 --Bare wire ---7HL ••••••27HL -•-•• SOHL 0.1 Potential vs. EA g /A g Cl (V) 0.2
In Fig. 5.9, we have represented the intensity per electrode unit length versus the scanning potential obtained from our numerical simulation together with the experimental results of Fig. 5.2 in [START_REF] Reculusa | Design of catalytically active cylindrical and macroporous gold microelectrodes[END_REF]. The agreement of our simulation results with measurements on the electrodes made of 7 and 80 half layers is quite satisfactory. The case of the bare-wire is quite differ ent from the others as no porous medium is present and the gold layer may lead to a different value of IX. The simulation in that case was carried out using Eq. (Sa) in the Nerrnstian diffusion layer with the boundary conditions B.C.1 in Eq. (Sb) applied at z' = 0 and B. C.3 in Eq. (Sd) together with initial condition in Eq. (Se) for 0 < z' < r;. Indeed, a very good fit was obtained by taking IX= 0.61 in that case as reported in Fig. 5.9. It must be observed that, for a given potential, the absolute intensity increases with the porous thickness. To better evaluate the fit, the rela tive error between the numerical results and experimental data on the current per unit length of the electrode as represented in Fig. 5.9 are reported in Fig. 5.10. The reference value of the current was taken as its maximum experimental value for each electrode, i.e. at a potential of -0.2 V. The maximum error is about 14% for the 7 half-layers electrode at a potential of -0.1 V while the error is around 5% on average for the whole dataset.

Conclusions

A multiscale approach to model porous electrodes used in active or passive electro-analytical devices was developed in this work in the context of reduction reaction of oxygen at the cathode in the absence of catalysis. Within this context, the pore-scale model consists in a coupled diffusion-electrochemical reaction which can be solved using DNS at the cost of very significant com putational resources. With the aim of developing a strategy to ana lyze the relationships between the porous medium microstructure and the macroscopic properties(i.e. the current intensity available at the electrode), an upscaled model operating at the scale of an effective equivalent medium was derived by upscaling the pore scale model with the aid of the volume averaging method. The effective diffusion tensor present in this model can be simply determined from the solution of an intrinsic closure problem on a representative elementary volume of the microsctructure. This upscaled model contains all the essential information from the pore-scale and is far less computationally demanding. Its validity and accuracy was proven through comparative simulations of amperometry at both the pore-scale and macroscale showing an excellent agreement between the two approaches on the concen tration profiles and macroscopic current intensity available at the electrode. The speed-up of the computational procedure achieved by using the effective macroscopic model is roughly 3100 com pared to DNS. Various porous model structures (C, BCC, FCC) and values of the parameters ( electron transfer rate constant) were investigated as illustrative examples.

Simulation of voltammetry tests performed with the upscaled model were compared to intensity versus scanning potential experimental data reported in the literature. Parameters were cho sen as close as possible to the corresponding experimental config uration except the Nernstian layer thickness and the electron transfer coefficient that are unknown in the experimental approach and need to be adjusted. A very good agreement was achieved between predictions from the upscaled model and the experimental results confirming the relevance of this model.

Further work is necessary to complete the modeling of the elec trochemical process beyond the simple case of oxygen reduction in order to analyze more complex situations involving enzymes as a catalyst in the case of Direct Electron Transfer or Mediated Electron Transfer for which a mediator species shuttles electrons between enzymes' active centers and the electrode internal fluid-solid interfaces.

The high potential capability of the multiscale approach devel oped in the present work opens the way to investigate the optimal design of the porous architecture in order to improve the electrode effectiveness and this represents a key step to progress in the development of these innovative micro-electrochemical devices that needs to be addressed from the point of view of chemical engineering.

  interface in the averaging volume measure of A, 1 (m 2 ) interface between the fluid phase in the electrode and the external bulk fluid specific area ( m-1 )

  (A) unit tensor solid/fluid interface current density (A/m 2 ) electron transfer rate constant ( cm/s) effective reaction rate coefficient ( m/ s) pore-scale kinetic number, Ki= ko�,r, electrode thickness (m) , Nernstian layer thickness (m) periodic lattice vector in the ith-direction (m) characteristic pore size ( R p = d,) ( m) size of the periodic representative unit cell (m)

  Fig. 1.1. Two-coaxial electrochemical cell synthesis (from Karajic et al., 2015). Top: schematic representation of the different steps. Bottom: Corresponding cross-sectional SEM images of the material.

Fig. 2

 2 Fig. 2.1. Pore-scale configuration with the solid and fluid domains, their interface and a diffusive path of species A.

  oCA T.D.Le et al./Chemical Engineering Science 173 (2017) [153][154][155][156][157][158][159][160][161][162][163][164][165][166][167]

e2

  Fig. 3.1. Periodic unit cells of the three arrangements considered for the porous electrode microstructures: C, BCC and FCC.

  Fig. 3.2. 3D domain for the DNS.

F�LID

  Relative pore connection window size dcfds 15% Size of the periodic unit cell £REV 1.678 �LID value of L; iec represents the number of unit cells composing the electrode in the z-direction.

Fig

  Fig. 3.3. Dimensionless concentration fields c;; at t' � 10,100,200 for k0 � 10-3 cm/s.

  Fig. 3.4. Dimensionless dynamics of the averaged dimensionless concentration c;.; for three different values of k0•

  Three different averaging operators are used to carry out the upscaling, namely the superficial, intrinsic and area averages respectively

Fig. 4

 4 Fig. 4.1. Averaging domain for a two-phase system.

  Fig. 5.1. Configuration of the 1D domain including the electrode (!1) and Nernstian layer (!1,).

  ===---==--==I =---== 5% 10% 15% 20% Pore connection window size d e / d 8 a) tion window size d c /d 5 • The effective diffusion coefficient 'D e ff was computed by solving the closure problem in Eqs. (16) (or (22)) on the corresponding periodic unit cell of each structure using COMSOL Multiphysics. The porosity and dimensionless specific area are represented versus d c !d s in Fig. 5.2 for O ,s:; d c !d s ,s;; 20%. As expected, B J increases with d c !d s while a; decreases. The variations of both coefficients remain small over the investigated range of d c ld s as they are less than 7% for the FCC structure for which porosity and specific area are the largest due to its optimal arrangement. In Fig. 5.3, we have reported the normalized effective diffusion coefficient B t 'D e ff /'DA versus d c !d s for the three different types of sphere packing under consideration. For each of them, the effective diffusion coefficient increases almost linearly with the pore con nection window size. The FCC structure yields the largest values of'D e ff, the simple cubic packing providing the smallest while inter mediate values are obtained for the BCC arrangement.

15%

  Fig. 5.3. Normalized effective diffusion coefficient e/D,JJ /DA for the three model structures of Fig. 3.1 versus dc/d,.

  u

20%

  Fig. 5.2. Dependence of (a) the porosity and (b) normalized specific area on the pore connection window size for the three model structures C, BCC and FCC (see Fig. 3.1 ).

  *' , ,.......................... .... .... --0.6 f----+----+---t------t----+r-"" • "------l --lD-macromodel t* = 10 __ ... ,,..,.. , ..,."-

Fig. 5 . 4 .

 54 Fig. 5.4. Dimensionless average concentration profi les along the porous electrode obtained from 3D DNS and simulation of the 1 D macroscopic model for four different dimensionles times. k0 � 10-s cm/s.

  fig. 5.6. Dimensionless concentration profi les in both the electrode ( -30,;; z•,;; 0) and Nernstian layer (0,;; z',;; 12) obtained from simulations of 1D macroscale chronoamperometry with (a) ko = 10 9 cm/s and (b) k0 = 10 7 cm/s.

  Fig. 5.7. Evolution of the average current density on the electrode for three values of ko.

  Fig. 5.8. Evolution of the current intensity per unit volume of the electrode for the three model structures C. BCC and FCC (see Fig. 3.1) and for (a) k0 = 10 7 cm/s, (b) ko = 10-9 cm/s.

Fig. 5 . 9 .

 59 Fig. 5.9. Intensity versus scanning potential obtained from voltammetry numerical simulations using the 1D upscaled model. Comparison with voltammetry experimental data reported in Reculusa et al. (2011 ).

Fig. 5 .

 5 Fig. 5.10. Relative error between numerical results and experimental data.

Table 3 .1

 3 Parameters used in the simulations.

	Parameter	Symbol	Value	Unit
	Ideal gas constant

R 8.314 J/(molK) Faraday's constant

Acknowledgement

This work was supported by the LabEx AMADEus (ANR-10-LABX-42) within IdEx Bordeaux (ANR-10-IDEX-03-02), i.e. the "Investissements d'Avenir Programme" of the French government managed by the Agence Nationale de la Recherche.

Appendix A. Volume averaging and derivation of the upscaled model

In this Appendix, the upscaling procedure to derive the macro scopic diffusion/reaction equation by volume averaging the micro scopic IBVP is reported. Volume averaging is applied according to the following four main steps. More complete details can be found in Chap. 1 of [START_REF] Whitaker | The Method of Volume Averaging[END_REF] (see also [START_REF] Valdes-Parada | On diffusion, dispersion and reaction in porous media[END_REF]. Again, we use c A to denote c A (r, t).

A.1. Step 1: Application of the superficial average operator

The superficial average operator is applied to the microscale IBVP on c A and, with the purpose of deriving a model involving (c A ) (or (c A /) only, time and spatial derivation must be inter changed with volume averaging. This is achieved by using the gen eral transport theorem, which, since the averaging volume is fixed in time and the porous medium is rigid, reduces to

and the spatial averaging theorem (or Leibniz [START_REF] Truesdell | The Classical Field Theories[END_REF], [START_REF] Whitaker | The Method of Volume Averaging[END_REF] rule) given by

11•

together with a straightforward similar form for the divergence operator. When this theorem is employed twice and by making use of the boundary condition at A s t (see Eq. (Sb)), one arrives at the following average form

where the intrinsic average concentration was used instead of the superficial average.

A.2. Step 2: Decomposition of c A and simplifications due to length-scale constraints

The average Eq. (A.3) contains both average and point-wise con centrations. To remove the latter, the spatial decomposition (A.4) is introduced [START_REF] Gray | A derivation of the equations for multi-phase transport[END_REF], where c A is the deviation of concentra tion which fluctuates at a typical length-scale C p while (c A / experi ences significant variations at the scale L. It should be noted that a consequence of this decomposition is (cA) = 0. When this decompo sition is inserted into Eq. (A.3), one gets

Let us now focus attention onto the area integral term i J� n(c A / dA on the right hand side of Eq. (A.5). This term must

,f be evaluated at the centroid x of the averaging volume V and requires first the evaluation of (c A / at any point x +yon A, 1 con tained in V, making this term non-local. A Taylor expansion given by (cA/ lx+y = (cA/ lx + Y • V(c A/lx + ½yy: VV(c A/lx +... (A.6) may now be employed and, when introduced back into Eq. (A.5) together with the orders of magnitude estimates to identify the sig nificant terms, it is not hard to deduce that Eq. (A.5) can be simpli fied to (see [START_REF] Whitaker | The Method of Volume Averaging[END_REF] for more details)

. a(cA/

where we have introduced the decomposition of the concentration into the reactive term. In addition, it can be shown that, whenever the pore-scale kinetic number Ki = k0;; 1 r satisfies Ki« 1 (A.8)

we have c A « (c A / at A, 1 (see [START_REF] Whitaker | The Method of Volume Averaging[END_REF]. Under these circum stances, Eq. (A.7) can be simplified to the following form

A close attention needs to be dedicated to the last area integral term on the right hand side of this equation since it is non-local. Indeed, when the Taylor expansion of Eq. (A.6) is introduced in this integral, one has (A.10)

By estimating the orders of magnitude of the integral terms, it can be proven that ((cA/), 1 "" (cA/ [START_REF] Whitaker | The Method of Volume Averaging[END_REF]. As a consequence the average Eq. (A.9) can be simplified to the following form

The boundary condition at the solid-fluid interface can be written as the idea that a v = 0 ( e; 1 ). This indicates that the macroscopic diffu sion/reaction equation may be written as

Appendix B. Boundary condtions at the dividing surface between the electrode and the fluid

This appendix is dedicated to a justification of the relevance of boundary conditions in Eqs. ( 21c) and (21d) indicating that both the concentration and flux can be considered as continuous at the dividing surface between the electrode and the fluid. For the sake of simplicity, the discussion is carried out on the dimensional version of the boundary conditions.

A formal derivation of physically justified boundary conditions for mass transfer coupled to reaction between a porous medium and a fluid was proposed in the literature [START_REF] Valdes-Parada | Diffusive mass transfer between a microporous medium and an homogeneous fluid: Jump boundary conditions[END_REF] leading to a jump of the flux at the dividing surface given by To be more specific, a quantitative analysis can be made taking the particular FCC structure under consideration that best repre sents the real configuration. For this structure, 1:: It can be easily shown that the right hand side of (B.9) is mini mum when 1 = 0 (i.e. in the limiting case of touching spheres)

yielding Ki « 0.24 as a safe constraint. Clearly, for all cases under consideration in the present work, this constraint is satisfied, justi fying to employ continuity of the flux (and of the concentration) as boundary conditions between the porous electrode and the fluid.