Yaqiong Cao

Hongpeng Zheng a

Junlei Tang
email: tangjunlei@126.com

Bing Lin

Yingying Wang

Jinkai Yuan

Hongpeng Zheng

Photoconversion and Photoelectron Storage Coatings Based on PANI, BiVO 4 and Graphene

Keywords: BiVO4, polyaniline, photoelectron storage, photocathodic protection

Based on one of the most abundant resources on earth-sunlight, photocathodic protection (PCP) coating is a photoelectron-saving and intelligent anti-corrosion measure.

Herein, we created a photoconversion and photoelectron storage coating composed of epoxy resin matrix, graphene, and sunlight-responsive polyaniline@bismuth vanadate nanocomposites (PANI@BiVO4). The PANI@BiVO4 nanocomposites were evenly distributed into the 3D network formed by graphene flakes. Such a unique design allows for storing the generated electrons, from the photoelectric conversion of BiVO4 upon light exposure, in the layer of PANI via electrochemical reactions. The stored electrons can be released along the 3D conductive graphene network to the metal interface even in dark conditions for more than 12 hours.

Introduction

Based on one of the most abundant resources on earth-sunlight, photocathodic protection (PCP) coating is an energy-saving and versatile corrosion protection coating. Today, the practical applications of PCP coatings pose multiple technical challenges. The first challenge relies on the efficient conversion of light to electrons. Photoelectric conversion can be achieved using semiconductors with a small band gap. BiVO4 has a stronger response to sunlight [1][2][3],

showing promising values in the PCP coating. The second challenge, which is the most difficult to tackle, is to ensure a continuous supply of electrons. The combination of photoelectron storage materials and photoelectric materials can be an effective strategy to meet the end [4,5].

Conductive polymers are flexible and have excellent redox properties and interfacial compatibility with resin matrices, such as polyaniline (PANI) [6]. Electrons can be stored in these active materials by an electrochemical reduction reaction under light and released via a peroxidation process in the dark. In addition, under light irradiation, PANI can be stimulated to produce enriched oxygen vacancies, which contribute to negative electron storage [7].

Current technologies to ensure a continuous supply of electrons require constant light radiation. However, the practical scenario in the real situation is that a coating serves in the alltime domain (light+dark). Under cloudy or dark conditions, photoelectric conversion can be significantly weakened or even stopped.

In this work, to realize the all-time domain protection of metals, we designed a solar photoelectron powered photoelectron storge coating. BiVO4 particles were chosen as the photoelectric converting material. Their surfaces were coated with PANI to store the generated electrons. These generated electrons were transported along the conductive Gr network to reach the stainless steel interface. Such unique photoelectron storage coating is expected to induce a cathodic protection on metals in the dark. We designed the simulated actual illumination cycle (2 hours light + 12 hours dark) to investigate the potential change of photoelectron storge coating in the all-time domain service period. Photoelectrochemical and structural characterizations were performed to study the photoelectric conversion, electron storage and photocathodic protection of the photoelectron storge coatings.

Results and Discussion

Morphology and photocathodic protection of the PANI@BiVO4-Gr composite coating

The functional groups of the prepared coating were characterized to clarify the successful preparation of the composite coating. The synthesis process made PANI into a doped state with redox performance [8], and the BiVO4 remained a light response performance without damaging its chemical composition (Figs. (S2,S3)).

The cross-sectional and surface morphologies of the PANI@BiVO4-Gr composite coatings were shown in Figs. (1, S4). Compared to other ratios, when the n (PANI): n (BiVO4) was 1:2 (Fig. 1c), the distribution of Gr sheets was very uniform, and the EDS spectra showed that the PANI@BiVO4 nanocomposites were evenly dispersed inside the P1B2 coating. The SEM images showed the cross-sectional morphologies of the coating on a two-dimensional plane, and Gr sheets were uniformly distributed at the cross-section of the coatings, indicating the Gr sheets inside the P1B2 coating formed a uniform three-dimensional network. The uniformly dispersed PANI@BiVO4 nanocomposites exerted a steric effect to prevent Gr sheets agglomeration (Fig. S5), which promoted the formation of a three-dimensional network structure. As shown in Fig. 2a, compared with EP coating, the absorption intensities of P1B2 and P1B3 coatings in the visible region were significantly improved and the absorption edge showed a redshift. The calculation results of the bandgap energy (Eg) of the coatings (Fig. S6) further demonstrated that the more efficient electronic interaction between PANI, BiVO4, and Gr led to the expansion of the light absorption range [9]. Fig. 2(b-d) showed the polarization curves of PANI@BiVO4-Gr composite coatings coated 304L SS in 2h light-12h dark test (3.5 wt.% NaCl). The EOCP of coated samples after 2h of illumination decreased to about -0.6 VSCE (Fig. 2c). Although the P1B0.5 and P1B1 coatings did not show an obvious peak in the UV-visible diffusive reflection absorption spectra due to the low content of BiVO4, the addition of proper amount of high conductivity Gr and the gradual formation of three-dimensional conductive network enabled the transmission of electrons from composite coating to stanless steel substrate in NaCl solution (Fig. S7), thus showing a negative shift of potential of test sample under illumination. The iTafel was approximately 3 orders of magnitude higher than that before the 2h illumination (Table S1), indicating the cathodic effect was significantly enhanced by the illumination, which meant the composite coating could generate a large number of photoelectrons under visible light illumination. The potential of the P1B2 coating returned to the starting position after 18.5h of light off (Fig. S8). Combined with these results, for P1B2 coating, it can be seeing the photoelectrons stored in the coating were not completely released in the 12h natural discharge. Therefore, the photocathodic protection of the P1B2 coating in the dark was longer than 12h.

Photoelectron storage mechanism of PANI@BiVO4-Gr composite coating

With the increase of BiVO4 content, more photoelectrons were captured and stored (Fig. S9).

To further clarify the reasons for composite coatings showing different photoelectron storage behaviors, the XPS spectra of composite coatings while before 2h illumination and after Light 2h-Dark12h cycle were measured.

The intrinsic oxidation state of PANI can range from the fully oxidized pernigraniline (PN, y = 0), through half-oxidized emeraldine (EM, y = 0.5), to fully reduced leucoemeraldine (LE, y = 1) [10]. Through Gaussian fitting, N1s XPS spectra can be separated into four peaks. The peaks corresponding to quinoid di-imine nitrogen (-N=) and benzenoid di-amine nitrogen (-NH-) are centered at 398.6 and 399.5 eV, respectively. The nitrogen contributions at 400.5 and 401.7 eV can be ascribed to positively charged nitrogen (N+): oxidized amine and protonated imine, respectively [11]. Where the -N=, -NH + -and -NH + = groups originated from the oxidized state of PANI and the -NH-structure originated from the reduced state [12]. As shown in Table 1, before 2h illumination, the PANI in four photoelectron storge coatings was closer to conductive emeraldine (EM, y= 0.5). After Light 2h-Dark 12h, there was a increase in the percentage of reduced N component (-NH-), accompanied by the decrease in oxidized N components (-NH + = and -N=), which was due to the reduction of emeraldine PANI [13].

The structure of EM had a framework of linearly repeating "quinone and benzene" groups.

It intentionally introduced holes to maximize charging capacity by removing two electrons from alternating nitrogen atoms [14]. Owing to the PANI with charge-transfer excitation-like transition from the HOMO to the LUMO, which made the excited photogenerated electrons transfer, thereby producing more electron-hole pairs [15], and acting as the photoelectron storage units in composite coatings. The P1B2 coating possessed the most electron-hole pairs (Table 1), which proved that it exhibited optimal photoelectron storage performance. Fig. 3b showed the cyclic voltammetry of composite coatings, with two pairs of obvious redox peaks appearing in all curves. The current peak of anodic peak and cathodic peak were assigned to oxidation reaction and reduction reaction, respectively [16]. From the curve, after Light 2h, the current peak of the anodic peak was much larger than Dark. After Light 2h-Dark 12h, the current peak at the anodic peak was smaller than that after Light 2h, while the current peak at the cathodic peak increased. According to the XPS results of the coating, redox occured in the composite coating, and the redox speed gradually increased with the complete construction of the three-dimensional conductive network while the charge transfer resistance decreased. Especially, the difference between the CV curvilinear integral area of P1B2 coating after Light 2h and after Light 2h-Dark 12h was the smallest, which again proved that P1B2 coating had the best electronic storage performance.

Electron migration mechanism of PANI@BiVO4-Gr composite coating

The electron migration mechanism of PANI@BiVO4-Gr composite coating was shown in Fig. 4. Since the contact between Gr and BiVO4 appeared to be ohmic in nature, making the electrons shuttle back and forth between them. At the junction, because of high aspect ratio and electron mobility of Gr, photoelectrons tended to move away from BiVO4. BiVO4 and PANI in PANI@BiVO4-Gr composite coating might form a Z-type heterojunction [17], that was, holes on the valance band (VB) of BiVO4 were transferred to the HOMO of PANI, while electrons from the LUMO of PANI were transferred to the conduction band (CB) of BiVO4.

When the PANI@BiVO4-Gr composite coating was exposed to direct solar light irradiation, partial photogenerated electrons from BiVO4 CB quickly transferred to 304L SS via the high mobility Gr transport bridge. The another part electrons recombined with the holes in the VB of PANI. This caused a hole accumulation in the VB of BiVO4. These holes could not move towards PANI VB due to very low hole mobility of Gr. This led to effective separation of electrons and holes in BiVO4 and partial accumulation of electrons on PANI CB. This allowed photoelectrons to storage in the PANI@BiVO4-Gr composite coating and migration to 304L SS in the dark. Hence, the PANI@BiVO4-Gr composite coating provided sustainable cathodic protection for 304L SS.

Conclusions

The PANI@BiVO4-Gr composite coating was prepared. PANI@BiVO4-Gr composite coating provided cathodic protection on 304L stainless steel. The optimal addition ratio of PANI and BiVO4 was n (PANI): n (BiVO4) =1:2. The abundant PANI@BiVO4 nanocomposites and the construction of a three-dimensional conductive network in the composite coating widened the light wave response range and absorption intensity. Stimulated by visible light, the PANI@BiVO4 nanocomposites inside the composite coatings transformed and stored abundant photoelectrons, which made the corrosion potential of the coating move negatively. The photoelectrons were transmitted along the conductive Gr network to the 304L SS surface, providing cathodic protection. Under dark conditions, the composite coating showed excellent long-acting cathodic protection with a duration period of more than 12 hours. thermosetting resins. Due to its good flowability and convenient molding, it played an important role in various industries such as corrosion protection, aerospace, machinery, electronics, etc. Its molecular structure had the following characteristics (Fig. S1a): the two ends of the macromolecule were epoxy groups made the resin reactive, and the cured product also had strong adhesion and cohesion [1]. There were many ether bonds on the main chain of macromolecules, and the ether bonds and C-C bonds gave the macromolecules flexibility. The light group and ether bond were polar groups that could improve their wettability and adhesion. There were many methylene groups, benzene rings, and isopropyl groups on the main chain of macromolecules, and isopropyl groups gave the macromolecules a certain degree of rigidity. The benzene rings endowed polymers with heat resistance and rigidity. The bond energy of the -C-O-bond was high, which could improve their alkali resistance. But after curing, it presented a threedimensional cross-linked network rigid structure, with low-temperature brittleness.

Polyamide had long fatty acid carbon chains and amino groups in its structure (Fig. S1b). When used as a curing agent and cured with epoxy resin, it can enhance the toughness and adhesion of the cured coating.

For the PANI@BiVO4-Gr composite coating, it had better physical properties such as adhesion, flexibility and rigidity. The dosage combination of E-44 and polyamide was more flexible and convenient to use compared to aliphatic and aromatic amines cured epoxy resins [2]. Currently, the combination of E-44 and polyamide 651 is widely used in anti-corrosion coatings [3][4][5][6]. Most of all, the epoxy matrix with excellent transparency allowed a large amount of light to penetrate the coating and be captured by BiVO4, which was used for photocathode protection.

Preparation and Characterization

2.1 Preparation of PANI@BiVO4 nanocomposite. PANI@BiVO4 nanocomposite was prepared by in situ oxidative chemical polymerization method. First, aniline (1mL) was dropwise added to the HCl solution (150mL, 1 mol/L) with magnetic stirring for 5min, then BiVO4 (1.74g, n(PANI): n(BiVO4) =1:0.5) was added and stirred for 30min in an ice-water bath. Second, APS (2.40g) was dropwise added to HCl solution (50mL, 1 mol/L) with magnetic stirring for 5min, then stirred magnetically for 4h in an icewater bath. Afterward, the reaction solution was filtered and washed repeatedly with deionized water and ethanol until the filtrate became colorless and transparent. Finally, the collected product was dried in an oven at 60℃ for 2h to obtain PANI@BiVO4 nanocomposites. The preparation of PANI@BiVO4 nanocomposites with n (PANI): n (BiVO4) =1:1, 1:2, and 1:3 followed the same protocol aforementioned.

2.2 Preparation of PANI@BiVO4 -Gr composite coating. Firstly, the 304L SS sheets were polished with 1000# sandpaper, rinsed repeatedly with deionized water and ethanol, and placed in a dry dish for later use. EP (18.00g) was dissolved in n-butanol (8.00g). PANI@BiVO4 nanocomposite (6.00g) and Gr (0.46g) were mixed, then added to the coating resin with magnetic stirring for 1h. Secondly, add the curing agent (mass ratio of EP: curing agent=5:2) to the above mixture and magnetic stirring for 30min to obtain the PANI@BiVO4-Gr composite coating. Finally, a uniform layer of PANI@BiVO4-Gr composite coating was brushed on the surface of pre-treated 304L

SS sheets, then it was dried in an oven for 6h (40℃) to obtain a PANI@BiVO4-Gr composite coating sample. The prepared coating films were about 200±10 μm thick.

The preparation of PANI@BiVO4-Gr composite coating with n (PANI): n (BiVO4) =1:0.5, 1:1, 1:2, and 1:3 followed the same protocol, hereafter named P1B0.5, P1B1, P1B2, and P1B3 coatings, respectively.

Characterization of PANI@BiVO4

-Gr composite coating. The morphologies of the BiVO4, PANI@BiVO4 nanocomposites and PANI@BiVO4-Gr composite coatings were investigated by scanning electron microscopy (SEM, ZEISS GeminiSEM 300, Germany) and transmission electron microscope (TEM, FEI Tecnai F20, USA). A Fourier transform infrared spectrometer (FT-IR, Nicolet6700, USA) and ultravioletvisible absorption spectra (UV-vis, PE lambda 750, USA) were used to study the change in the functional group and behavior in response to visible light with common schema by using BaSO4 as a white standard, respectively. X-ray photoelectron spectroscopy (XPS) was used to obtain the N 1s XPS spectra of photoelectron storge coatings.

Photoelectrochemical properties were measured on a coated sample in a conventional three-electrode cell using a CS350M workstation (Wuhan Corrtest, China). The working electrode, counter electrode, and reference electrode were the coated sample, a platinum plat, and a saturated calomel reference electrode (SCE), respectively. The electrolyte was 3.5wt.% NaCl solution. A 300 W full spectrum sunlight simulation test lamp (ULTRA VITALUX 300W 230V E27, Germany) was used as the light source.

Results and Discussion

Structures and morphologies of PANI@BiVO4-Gr composite coatings

Fig. S2 showed the FT-IR spectra of Gr, PANI, PANI@BiVO4 nanocomposites and PANI@BiVO4-Gr composite coating. In FT-IR spectra of PANI and PANI@BiVO4 nanocomposites, bands at 1292 cm -1 were linked to C-N stretching vibration with aromatic conjugation [9,10]. A broad band at 1103 cm -1 was attributed to C=N stretching vibrations of -N= quinoid(Q) =N-group [9]. The band centered at 790 cm -1 was the out-of-plane bending vibration of C-H group in 1,4-disubstituted benzene rings [11]. The band near 1478 cm -1 belonged to C=C stretching deformation of -Nbenzenoid(B)-N-group. Above bands suggested that conducting PANI was in the emeraldine form, which was also a symbol of polaron delocalization [9,10]. Similar characteristic peaks also appeared in the PANI@BiVO4-Gr composite coating, but the position of the peak was shifted, which was mainly due to the bond interaction between Gr and the PANI@BiVO4 nanocomposites [12].

Light absorption performance and photoelectric conversion performance of PANI@BiVO4-Gr composite coating

The optical band gap of P1B2 coating and P1B3 coating were calculated by the Tauc plots to be 2.22 eV and 2.41 eV, respectively (Fig. S6). Appropriate ratio of PANI and BiVO4 resulted in overall band gap reduction. The redshifts in the absorption edge of the coatings (Fig. 2) occurred due to the effective formation of a heterojunction, which would have contributed to enhanced the light absorption properties of the PANI@BiVO4-Gr composite coatings. During the light period, the PANI@BiVO4-Gr coating generated photoelectrons, and the current density rose rapidly. For P1B0.5 and P1B1 coatings, the added current density value caused by the light was lower than that of P1B2 and P1B3 coatings. That means the P1B2 and P1B3 coatings converted more photoelectrons than that of P1B0.5 and P1B1 coatings. When the visible light was OFF, the current density of the coated sample negatively moved and gradually stabilized, and was higher than the initial current density. The reason for the negative movement was due to the PANI@BiVO4-Gr coating releasing electrons, which was caused by the photochemical reaction. And, as the electron storage material, the PANI continued to release the electrons, so the current density of the PANI@BiVO4-Gr coated sample was gradually stable and higher than the initial current density. The current density of P1B2 coating was relatively stable between the two light-dark cycles. Studies suggest that under light irradiation, PANI was stimulated to produce enriched oxygen vacancies to store the electrons [13].

In the PANI@BiVO4 nanocomposites, BiVO4 could exhibit an excellent charge separation efficiency owing to the presence of enriched oxygen vacancies which acted as shallow donors by PANI [14]. Therefore, the optimum photoelectron conversion and storage could be achieved by the polymerization of PANI and BiVO4 with an appropriate ratio, which was proved to be n (PANI): n (BiVO4) =1:2. Also, combined with the results in Fig. S8, the above results showed that the appropriate addition of BiVO4 improves the stability of charge-discharge cycle of the PANI@BiVO4-Gr coatings, which was very important for long-term application.

Photocathode protection performance and energy storage performance of PANI@BiVO4-Gr composite coating

Fig. S8 showed the potential of the prepared PANI@BiVO4-Gr composite coating in the absence and presence of visible light illumination in NaCl solution at open circuit potential. When the light illumination was ON, the corrosion potential of each PANI@BiVO4-Gr coating became negative quickly. After 2h of light illumination, the potentials of P1B0.5 coating (Fig. S8a), P1B1 coating (Fig. S8b), P1B2 coating (Fig. S8c) and P1B3 coating (Fig. S8d) were -0.552V, -0.660V, -0.697V and -0.736V, respectively. This meant that P1B2 and P1B3 coatings had converted and stored more photoelectrons than that of P1B0.5 and P1B1 coatings.

When the light illumination was OFF, the corrosion potential of each PANI@BiVO4-Gr coating shifted in the positive direction. It was proved that the PANI@BiVO4-Gr coating had the capability of releasing electrons. The potential of P1B0.5 and P1B1 coatings did not return to the starting position after discharging. In contrast, P1B2 and P1B3 coatings returned to their starting positions. Combined with Fig. 1, it could be inferred that P1B2 and P1B3 coatings had a more uniform conductive network of Gr, so the electron transfer was more efficient. As seen in Figs. S8 (c,d), the potential of the P1B2 coating returned to the starting position after 18.5h of light off, while the P1B3 coating returned to the starting position after 7.5h of light off. This indicates that the discharge time of P1B2 coating was longer, which meant that it can perform longer cathodic protection. The discharge curves at 10μA/cm 2 of PANI@BiVO4-Gr composite coating, P1B2 coating without Gr and PANI@BiVO4-CNT composite coatings were shown in Fig. S9a. Since the potential of each coating was different after 2h of light charging (Fig. S8), the discharge potential window was different for each coating. For PANI@BiVO4-Gr composite coating, the discharge curves of P1B1, P1B2, and P1B3 coatings were more stable than that of P1B0.5 coating. The reason for this difference was that the increase in PANI@BiVO4 nanocomposite content led to more electron storage units, and the three-dimensional network constructed by Gr was more uniform, both of which resulted in more stable electron transport.

To further analyze the constant discharge curves of the composite coatings, the areaspecific capacitance of each coating under illumination for 2h was evaluated, and the results were shown in Fig. S9b. The energy storage behavior was revealed by the areaspecific capacitance of the coating, which was calculated from the composite coating at a discharge of 10μA/cm 2 after 2 hours of illumination by the following formula [15]:

𝐶 𝐴 = 𝐼×∆𝑡 ∆𝑉×𝑆 (1)
where CA was the area-specific capacitance, I was the constant current, ∆t was the discharge time, ∆V was the potential difference, and S was the working area of the sample.

The area-specific capacitance of the P1B0.5, P1B1, P1B2 and P1B3 coatings after 2h light was 17.126 mF/cm 2 , 26.036mF/cm 2 , 33.211 mF/cm 2 and 71.328 mF/cm 2 , respectively. With the increase of BiVO4, more photoelectrons would be captured and stored. These stored electrons would be used for photocathode protection of the PANI@BiVO4-Gr composite coating for 304L SS. It was worth noting that even though P1B3 coating had the largest area-specific capacity, the agglomeration of PANI@BiVO4 nanocomposites adversely affected the energy storage behavior of PANI@BiVO4-Gr coating and the construction of Gr conductive network when combined with the cross-section morphology in Fig. 1.

Combined with the above results, the cross-section morphology of the coating, and the electrochemical test results, it could be seen that more PANI@BiVO4 nanocomposites were conducive to the conversion and storage of more photoelectrons, and a more uniform conductive three-dimensional network was conducive to the efficient transfer of electrons, which together promoted the long-term photocathodic protection of the PANI@BiVO4-Gr composite coating.

Moreover, in order to clarify the effect of Gr addition on the energy storage performance of the composite coating, the same test was carried out on the P1B2 coating without Gr (P1B2 no Gr) and PANI@BiVO4-CNT composite coating (P1B2C1.5). For the P1B2 coating, with or without the addition of carbon materials, the discharge curve exhibited nonlinearity, which was due to the pseudo capacitive characteristics of PANI. The internal resistance of composite coatings exhibited a rapidly changing voltage region at the beginning of the discharge curve during constant current discharge, which can be represented by the change in the magnitude of the internal resistance of the coating using "IR drop". Due to differences in the structure and composition of the coating, the performance of "IR drop" varied at the same discharge current density. When Gr or CNT was added to the composite coating, the conductivity of the composite coating can be improved, manifested as a decrease in "IR drop", while the highest value was achieved when no carbon materials were added.

Further evaluated the area specific capacitance of P1B2 no Gr coating and P1B2C1.5

coating. The area specific capacity of the P1B2 no Gr coating and P1B2C1.5 coating after 2h light illumination was 22.888 mF/cm 2 and 38.193 mF/cm 2 , respectively, 0.69 and 1.15 times that of the P1B2 coating.

Combined with the cross-section morphology of the P1B2 coating (Fig. 1c), PANI@BiVO4-Gr composite coating without PANI@BiVO4 nanocomposites (Fig. S5), and P1B2C1.5 coating (Fig. S10), it could be seen that the uniformly dispersed PANI@BiVO4 nanocomposites exerted a steric effect to prevent Gr sheets agglomeration, which promoted the formation of a three-dimensional network structure.

Highly conductive Gr in the coating acted as a transport bridge for photogenerated electrons. Therefore, the addition of Gr enabled a high area specific capacitance to be maintained during stable charging and discharging, which realized the storage and transmission of photoelectrons in the PANI@BiVO4-Gr composite coating.

Physical properties of PANI@BiVO4-Gr composite coating

The physical properties of PANI@BiVO4-CNT composite coatings were the basis for studying their protective properties, therefore, the basic properties like adhesion, tensile strength, impact resistance of the PANI@BiVO4-Gr composite coatings had been characterized.

On the basis of having an excellent resin matrix, the addition of appropriate amounts of PANI@BiVO4 nanoparticles and Gr made the coating denser. The adhesion of P1B2 and P1B3 coatings was 5A grade (Fig. S11). P1B2 and P1B3 coatings did not crack after impact (Fig. S13). The tensile strength of P1B2 coating was 1.27 times that of P1B0.5 coating (Fig. S12). PANI@BiVO4 nanoparticles and Gr could improve the ability to resist deformation by limiting the movement of molecular chains, and delay the damage of the coating by deflecting the crack propagation direction and dissipating the energy at the crack tip [16].

Corrosion resistance of PANI@BiVO4-Gr composite coating

Based on the results of the above tests, the weather resistance and corrosion resistance of the PANI@BiVO4-Gr composite coatings were further evaluated.

The WCA of P1B0.5, P1B1, P1B2, and P1B3 coatings was 80.49 °, 94.86 °, 105.89 °, and 100.94 °, respectively (Fig. S14a). After immersion for 30 days, the water absorption rate of P1B0.5 coating was 1.65 times that of P1B2 coating (Fig. S14b). No corrosion products were generated on the surface of the all coatings after 14d of salt spray test (Fig. S15). The PANI@BiVO4-Gr composite coatings exhibited good anticorrosion effect. The interface bonding formed by the good dispersion of PANI@BiVO4 nanoparticles and Gr sheets in the coating (Fig. 1) was beneficial for the hydrophobicity of the PANI@BiVO4-Gr composite coatings. The three-dimensional network structure formed by the Gr sheet and the good compatibility with the matrix inhibited the invasion of corrosive media and internal pore expansion of epoxy resin under long-term immersion, allowing the coating to maintain longer stability [17].

Corrosion resistance of damaged PANI@BiVO4-Gr composite coating

The coating was prone to damage during service, resulting in direct exposure of the protected metal to corrosive environmental media. To investigate whether the damaged PANI@BiVO4-Gr composite coating could still protect 304L SS, damaged areas of 1%, 5%, and 10% relative to the electrochemical working area were prefabricated on the surface of the PANI@BiVO4-Gr composite coatings (EP coating as a comparison), which then were immersed in 3.5 wt.% NaCl solution (45℃) to measure the corrosion potential.

The simulated actual illumination cycle (2 hours light + 12 hours dark) was designed to investigate the photocathodic protection of PANI@BiVO4-Gr composite coating in the all-time domain service period in 3.5 wt.% NaCl solution (45 ℃). Fig. S17 showed the self-corrosion potential of 304L SS, EP coatings and PANI@BiVO4-Gr composite coatings with 1%, 5% and 10% damaged area after being immersed in 28 light-dark cycles, and the self-corrosion potential of samples immersed in the dark for 384h as the comparison. As present in Fig. S17f, under the test conditions of light-dark cycles and pure darkness, the corrosion potential of 304L SS decreased throughout the whole process with little difference in potential value. Light or darkness had no significant effect on the change of self-corrosion potential of 304L SS. For damaged EP coatings, similar patterns were observed.

Fig. S17. The self-corrosion potential of 304L SS, EP coatings and PANI@BiVO4-Gr composite coatings with 1%, 5% and 10% damaged area after being immersed in 28 light-dark cycles (immersed in the dark for 384h as the comparison) in NaCl solution.

Under the light-dark cycle condition, seen in Figs. S17 (a, b, c, andd), the corrosion potential of the four PANI@BiVO4-Gr coatings showed a significant negative shift. It showed that BiVO4 in the coating system captured photoelectrons and the photoelectric response of the coating caused a negative shift in the potential of the coating. At this time, the PANI@BiVO4-Gr coating began to play the role of photocathode protection.

After 2h of light illumination, it entered the dark, and the potential of the coating moved in the positive direction, indicating that the PANI@BiVO4-Gr coating released the electrons and continued to provide cathodic protection for the 304L SS. Among them, the first cycle of 2h of light made the potential of P1B0.5-1%, P1B1-1%, P1B2-1%, and P1B3-1% coatings negative to -0.538V, -0.66V, -0.661V, and -0.748V, respectively. After 12h in the dark, the potential of P1B0.5-1%, P1B1-1%, P1B2-1%, and P1B3-1% coatings was positively shifted to -0.269V, -0.396V, -0.412V, and -0.19V, respectively. The four damaged PANI@BiVO4-Gr coatings showed a good photoelectric response and could provide cathodic protection to 304L SS in the dark for more than 12h. The corrosion potential of the four PANI@BiVO4-Gr coatings showed a stable stage and a declining stage. The longer the stable stage was, the better the cathodic protection performance of the PANI@BiVO4-Gr coating system could be. The P1B2-1% coating displayed the longest stable stage (16th cycle), while the P1B0.5-1%, P1B1-1%, and P1B3-1% coatings showed stable stages of the 8th cycle, 12th cycle, and 12th cycle, respectively. When the damaged area reached 5% and 10%, the PANI@BiVO4-Gr coating also showed a certain cathodic protection effect under the condition of the light-dark cycle, as shown in Figs. S17. But after 2 hours of illumination, the potential of the coating was less than 1% of the damaged area PANI@BiVO4-Gr coating. This indicated that the decrease in the effective working area of the coating resulted in a decrease in the amounts of photoelectrons that the coating could capture, and the weakened photoelectric response performance of the coating resulted in a shortened photocathodic protection time for 304L SS. To further analyze the protection of the damaged PANI@BiVO4-Gr composite coating on 304L SS, SEM was used to characterize the corrosion morphologies of the uncoated 304L SS substrate, the metal substrate beneath the damaged EP coatings and PANI@BiVO4-Gr composite coatings after 384h of total darkness and 28 cycles of light-dark conditions. The SEM images of the metal surfaces coated with PANI@BiVO4-Gr coatings with 1%, 5% and 10% damaged area were shown in Fig.

S19

. Image J was used to make statistics on the proportion of stable pits and the area of the largest pits. The results were shown in Table S3. As seen in Figs. S19, after immersing in pure dark condition and light-dark cycle condition, the surface of 304L SS showed serious corrosion. The proportion of stable pits was 1.923% and 2.244% under pure dark and cyclic conditions, respectively, and the maximum pit area was 11.713 μm 2 and 9.9 μm 2 , respectively. For uncoated 304L SS, the Cl -in the solution caused pit corrosion to the passive film [START_REF] Asduzzaman | Effects of concentration of sodium chloride solution on the pitting corrosion behavior of AISI-304L austenitic stainless steel[END_REF]. After applying EP coating on the surface of 304L SS, the pitting corrosion at the damaged coating area improved. In dark conditions, the proportion of stable pits and maximum pit area of 304L SS surface were 7.55 and 52.91 times that of EP-10%, respectively. Under the conditions of 28 cycles of light-dark, the proportion of stable pits and maximum pit area of 304L SS surface were 12.88 and 40.81 times that of EP-10%, respectively. The above results indicated that the presence or absence of light has no significant effect on the EP coating.

Under the light-dark cycle condition, the 304L SS substrates beneath the four damaged PANI@BiVO4-Gr coatings displayed much less corrosion than that under pure dark condition. The proportion of stable pits on P1B0.5-1%, P1B1-1%, P1B2-1% and P1B3-1% was 0.492%, 0. 288%, 0.004% and 0.011%, respectively. The maximum pit area was 0.124 μm 2 ,0.119 μm 2 ,0.011 μm 2 and 0.153 μm 2 , respectively. The above values were all smaller than the results of the PANI@BiVO4-Gr-5%/ PANI@BiVO4-Gr-10% coatings. P1B2 coating showed the optimal cathodic protection effect.

Combining Fig. S17 and cross-sectional morphology characterization (Fig. 1), PANI@BiVO4 nanocomposite materials not only add units for converting and storing photoelectrons, but also help to form a uniform conductive Gr network, which could transfer abundant photoelectrons to the metal interface through the network. Even if the coating was damaged, it could still effectively perform cathodic protection performance.

Electron migration mechanism of PANI@BiVO4-Gr composite coating

The Mott-Schottky plots of PANI@BiVO4-Gr composite coatings were constructed as shown in Fig. S20. According to previous reports, the positive slope of the tangent line indicated that BiVO4 was a n-type semiconductor [START_REF] Gao | Fabrication of 3D lignosulfonate composited sponges impregnated by BiVO4/polyaniline/Ag ternary photocatalyst for synergistic adsorption-photodegradation of fluoroquinolones in water[END_REF]. While PANI could switch from a n-type to p-type semiconductor on its own by varying potentials [START_REF] Seger | An n-Type to p-Type Switchable Photoelectrode Assembled from Alternating Exfoliated Titania Nanosheets and Polyaniline Layers[END_REF]. The intersection of the tangent line with the X-axis represented the flat band potential (Efb) [START_REF] Zou | Construction of all-solidstate Z-scheme 2D BiVO4/Ag/CdS composites with robust photoactivity and stability[END_REF]. The P1B1, P1B2 and P1B3 coatings exhibited a positive slope, while P1B0.5 coating exhibited a negative slope, indicating that the foregoing three coatings were made of n-type semiconductors, while the latter was made of p-type semiconductor.

Fig. 1 .

 1 Fig. 1. Cross-sectional SEM images of PANI@BiVO4-Gr composite coatings (a) P1B0.5 coating, (b) P1B1 coating, (c) P1B2 coating, (d) P1B3 coating.

Fig. 2 .

 2 Fig. 2. (a) UV-visible diffusive reflection absorption spectra and the polarization curve of the PANI@BiVO4-Gr composite coating (b) before the illumination, (c) after the 2h illumination, (d) after the Light 2h-Dark 12h in NaCl solution.

Fig. 3 .

 3 Fig. 3. (a) N 1s XPS spectra of photoelectron storge coatings before 2h illumination and after Light 2h-Dark12h. (b) cyclic voltammogram of photoelectron storge coatings at a scan rate of 100 mV/s.

Fig. 4 .

 4 Fig. 4. Schematic diagram illustrating the mechanistic aspect of PANI@BiVO 4 -Gr composite coating.

Fig. S1 .

 S1 Fig. S1. Molecular Structure of E-44 epoxy resin (a) and polyamide (b)[7,8]

Fig. S2 .

 S2 Fig. S2. FT-IR spectra of Gr, PANI, PANI@BiVO4 nanocomposites and PANI@BiVO4-Gr

Fig. S3 .

 S3 Fig. S3. The SEM and TEM images of (a, c) BiVO4 and (b, d) PANI@BiVO4 nanocomposites.

Fig. S4 .

 S4 Fig. S4. The surface SEM images of PANI@BiVO4-Gr composite coatings (a) P1B0.5 coating,

Fig. S5 .

 S5 Fig. S5. The cross-sectional SEM image of the coating without adding PANI@BiVO4

Fig. S6 .

 S6 Fig. S6. Plots of the transformed Kubelka-Munk function versus the energy of light.

Fig. S7

 S7 Fig.S7showed the I-t curves of the prepared PANI@BiVO4-Gr composite coating

Fig. S7 .

 S7 Fig. S7. The I-t curves of the prepared PANI@BiVO4-Gr composite coating in the absence and

Fig. S8 .

 S8 Fig. S8. The potential of the prepared PANI@BiVO4-Gr composite coating in the absence and

Fig. S9 .

 S9 Fig. S9. (a) The constant current discharge curve and (b) The area-specific capacitance of

Fig. S10 .

 S10 Fig. S10. Cross-sectional SEM image of PANI@BiVO4-CNT composite coating (P1B2C1.5).

Fig. S11 .

 S11 Fig. S11. Adhesion of PANI@BiVO4-Gr composite coatings.

Fig. S13 .

 S13 Fig. S13. The results of impact resistance test of PANI@BiVO4-Gr composite coatings. Test

Fig. S14 .

 S14 Fig. S14. Water contact angle and water absorption rate of PANI@BiVO4-Gr composite coatings.

Fig. S15 .

 S15 Fig. S15. PANI@BiVO4-Gr composite coatings salt spray testing.

  Fig.S16showed a schematic diagram of the light + dark test cycles and the light + dark cycles test apparatus. 28 cycles of light + dark test was designed to verify the cyclability of charge and discharge of PANI@BiVO4-Gr composite coating. The corrosion potential of the damaged PANI@BiVO4-Gr composite coating was measured using a High Precision Digital Multimeter (± 0.001V, Agilent 34410A, USA). A 300 W full spectrum sunlight simulation test lamp (ULTRA VITALUX 300W 230V E27, Germany) was used as the light source.

Fig. S16 .

 S16 Fig. S16. (a) Schematic diagram of the light + dark test cycles and (b) light + dark cycles test

Fig. S18 .

 S18 Fig. S18. Macro morphology of 304L SS and the P1B2-10% coating after immersed in 28 light-

Fig. S19 .

 S19 Fig. S19. SEM images of 304L SS, EP coatings and PANI@BiVO4-Gr composite coatings with

  When two substances contacted each other, the electrons at the Fermi level (Ef) of PANI would migrate to the Fermi level of BiVO4 due to the work function of PANI was lower than that of BiVO4 until their Fermi level equality. The flat band potential of P1B0.5, P1B1, P1B2 and P1B3 coatings was determined from the intercept of linear region (C -2 = 0) of Mott-Schottky plot and was converted to normalized hydrogen electrode scale (NHE). The flat band potential of P1B0.5, P1B1, P1B2 and P1B3 coatings was determined to be 1.99 eV, 0.15eV, -0.79 eV and -0.65 eV, respectively w.r.t. NHE. The flat band potential of BiVO4 shifted upward after recombining with an appropriate amount of PANI and adding Gr[START_REF] Lakhera | Design of a highly efficient ternary AgI/rGO/BiVO4 nanocomposite and its direct solar light induced photocatalytic activity[END_REF]. Generally, the flat-band potential of the n-type semiconductor could be regarded as conduction band potential[START_REF] Li | Ingenious preparation of "layered-closed" TiO2-BiVO4-CdS film and its highly stable and sensitive photoelectrochemical cathodic protection performance[END_REF]. The negative shift of the potential of the prepared PANI@BiVO4-Gr composite coating in the presence of visible light illumination in NaCl solution at open circuit potential (Fig.S8) could be assigned to the change of the negative shift of conduction band potential. In other words, the essence of PANI@BiVO4-Gr composite coating cathodic protection lied in achieving the potential balance between the coating and protected metal (304L SS) through electron transfer. The negative shift of conduction band potential increased the driving force for electron transport, thereby realizing excellent photocathode protection performance.

Table 1 .

 1 Relative concentrations of nitrogen groups resulting from the fitting of Gaussian components to

		the N 1s photoelectron spectra.
	Sample	Relative concentrations of nitrogen groups (at.%) -NH--NH + = -N= -NH + -
	P1B0.5-Dark	50.25	13.57	26.63	9.55
	P1B1-Dark	43.48	13.91	33.04	9.57
	P1B2-Dark	40.65	10.98	40.24	8.13
	P1B3-Dark	44.84	21.08	27.80	6.28
	P1B0.5-Light 2h-Dark 12h	55.25	16.57	17.13	11.05
	P1B1-Light 2h-Dark 12h	53.76	14.52	20.43	11.29
	P1B2-Light 2h-Dark 12h	57.14	9.71	25.14	8.00
	P1B3-Light 2h-Dark 12h	55.25	25.41	14.92	4.42

Table S1 .

 S1 Electrochemical parameters of polarization curve of the 304L SS and PANI@BiVO 4 -Gr

				composite coating.
	Sample	Before illumination ITafel (A/cm 2 ) EOCP (V) ITafel (A/cm 2 ) EOCP (V) ITafel (A/cm 2 ) After illumination 2h After light 2h -dark 12h EOCP (V)
	P1B0.5	9.298×10 -8	-0.143	1.059×10 -4	-0.585	1.346×10 -7	-0.171
	P1B1	1.340×10 -7	-0.181	1.200×10 -4	-0.603	2.271×10 -6	-0.305
	P1B2	3.745×10 -7	-0.021	3.248×10 -4	-0.658	1.005×10 -6	-0.203
	P1B3	3.058×10 -7	-0.091	3.721×10 -4	-0.665	3.101×10 -7	-0.121
	304L SS	3.044×10 -8	-0.215	2.891×10 -8	-0.214	2.747×10 -8	-0.221

Table S3 .

 S3 Statistics of the proportion of stable pits and the area of the largest pit.

		P1B1-10%	0.763	0.507
		P1B2-1%	0.123	0.015
		P1B2-5%	0.171	0.015
		P1B2-10%	0.180	0.025
		P1B3-1%	0.180	0.014
		P1B3-5%	0.281	0.196
		P1B3-10%	0.968	0.126
		EP-1%	0.379	0.615
		EP-5%	1.199	0.820
		EP-10%	0.886	0.350
		304L SS	7.097	18.119
		P1B0.5-1%	0.285	0.025
		P1B0.5-5%	0.478	0.057
		P1B0.5-10%	0.384	0.046
		P1B1-1%	0.078	0.001
		P1B1-5%	0.204	0.016
		P1B1-10%	0.425	0.105
	In 28 light-	P1B2-1%	0.169	0.001
	dark cycles	P1B2-5%	0.116	0.009
		P1B2-10%	0.149	0.013
		P1B3-1%	0.168	0.005
		P1B3-5%	0.147	0.005
		P1B3-10%	0.232	0.039
		EP-1%	0.577	0.372
		EP-5%	0.523	0.236
		EP-10%	0.551	0.444
	Condition	Sample	The proportion of stable pits (at.%)	The area of the largest pit (μm 2 )
		304L SS	6.689	18.520
		P1B0.5-1%	0.333	0.160
	Pure dark	P1B0.5-5%	0.772	0.252
	384h	P1B0.5-10%	0.639	0.199
		P1B1-1%	0.633	0.208
		P1B1-5%	0.747	0.224

Acknowledgments

This study was supported by the Special project for the Sichuan Science and Technology Program under the Grant Nos. 2021ZDZX0002, 2021ZYCD003; National Natural Science Foundation of China, Nos. 52101076, 52201088.

Appendix A. Supplementary data

The detailed experimental section and auxiliary figures are presented in the Supplementary Material.

Supplementary Materials


Tien Dung Le

Christian Moyne
email: christian.moyne@univ-lorraine.fr

Mohamed Khaled Bourbatache
email: mohamed-khaled.bourbatache@insa-rennes.fr

Olivier Millet
email: olivier.millet@univ-lr.fr

Upscaled model for the diusion/heterogeneous reaction in porous media: boundary layer problem

Keywords: Homogenization, Diusion/reaction problem, Spectral approach, Boundary layer problem

Multi-species diusion/heterogeneous reaction coupled problem in porous media involving a boundary layer problem is studied. The spectral approach developed in our previous work allows to derive a macroscopic model in the high Damköhler number regime combining homogenization technique and spectral approach. Such a homogenized model cannot describe the perturbation at the external boundary where the chemical equilibrium is not necessarily satised. We construct herein a modied model involving additional variables, which are rapidly-decaying functions, to capture the complex physics in the boundary layer. Numerical simulations underline the accuracy of the proposed correction in both steady and transient states.

Introduction

Reactive transport of multi-species in porous media, which is of major interest for many applications in numerous areas such as electrochemical systems, agronomy, geology, etc., has received considerable attention from the modeling point of view. Macroscopic models involving eective coecients can be rigorously derived from upscaling approaches [START_REF] Allaire | Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media[END_REF][START_REF] Bloch | Upscaling of diusion-reaction phenomena by homogenisation technique: Possible appearance of morphogenesis[END_REF][START_REF] Battiato | Applicability regimes for macroscopic models of reactive transport in porous media[END_REF][START_REF] Boso | Homogenizability conditions for multicomponent reactive transport[END_REF][START_REF] Lugo-Méndez | Upscaling diusion and nonlinear reactive mass transport in homogeneous porous media[END_REF][START_REF] Tartakovsky | Hybrid simulations of reactiondiusion systems in porous media[END_REF][START_REF] Qiu | Upscaling multicomponent transport in porous media with a linear reversible heterogeneous reaction[END_REF][START_REF] Valdés-Parada | On diusion, dispersion and reaction in porous media[END_REF][START_REF] Valdés-Parada | Diusion and heterogeneous reaction in porous media: The macroscale model revisited[END_REF]. A challenging case refers to the predominant reaction situation when the medium exhibits a complex behavior and the eective coecients in the macroscopic equations depend on reaction rates [START_REF] Allaire | Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media[END_REF][START_REF] Battiato | Applicability regimes for macroscopic models of reactive transport in porous media[END_REF][START_REF] Allaire | Homogenization of a convectiondiusion model with reaction in a porous medium[END_REF]. More sophisticated models are required to accurately describe the non-equilibrium state at very short times that are of the order of the characteristic reaction time [START_REF] Bourbatache | Upscaling diusionreaction in porous media[END_REF][START_REF] Mauri | Dispersion, convection, and reaction in porous media[END_REF]. In spite of the diculty, macroscopic models were derived by using the periodic homogenization technique and a spectral approach in our previous work [START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF][START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF][START_REF] Bourbatache | Homogenized model for diusion and heterogeneous reaction in porous media: numerical study and validation[END_REF]. However, the homogenization procedure imposes a spatial periodicity condition at the boundary of the unit cell. In this framework, the physics inside the porous domain can be accurately captured by the macroscopic models while at the outer surface of the domain, periodicity conditions cannot be respected. This leads to a boundary layer problem depending on the boundary condition type.

The boundary layer problems have been widely studied for elastic periodic composites [START_REF] Dumontet | Study of a boundary layer problem in elastic composite materials[END_REF][START_REF] Koley | Study of boundary layer eects at ply interfaces of laminated composites using homogenization theory[END_REF], for elastic beams [START_REF] Buannic | Higher-order eective modelling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution[END_REF], for elastic shells [START_REF] Destuynder | Sur une justication des modèles de plaques et de coques par les méthodes asymptotiques[END_REF] or in a more general way in [START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF][START_REF] Amar | On the exponential decay for boundary layer[END_REF]. Matine et al. [START_REF] Matine | Modeling of thermophysical properties in heterogeneous periodic media according to a multi-scale approach: Eective conductivity tensor and edge eects[END_REF][START_REF] Matine | Transient heat conduction within periodic heterogeneous media: A space-time homogenization approach[END_REF] proposed to extend this approach for thermal problem in periodic structures, to take into account the edge eects. However, to the best of our knowledge there is practically no work in the literature addressing the boundary layer problem for the coupling of predominating reaction and diusion of multi-species in porous media.

In a sequence of papers, the coupling of diusion/reaction of two species with dierent molecular diusion coecients has been studied combining periodic homogenization technique with a spectral approach [START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF][START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF][START_REF] Bourbatache | Homogenized model for diusion and heterogeneous reaction in porous media: numerical study and validation[END_REF][START_REF] Bourbatache | Limits of classical homogenization procedure for coupled diusion-heterogeneous reaction processes in porous media[END_REF]. Concentrations of the initial pore-scale problem are expanded into a series development related to a spectral problem dened in a unit cell, yielding a new local problem to be homogenized. This approach is able to capture the complex coupling at short times when the chemical equilibrium is not reached. However, such a macroscopic model cannot provide information about the boundary layer that develops in the vicinity of the outer boundary due to the boundary conditions. In this work, corrections are added to the macroscopic model to take into account the boundary conditions at the outer boundary of the domain. In steady state, terms of the rst order need to be corrected by introducing new variables whereas in transient regime, corrections at lower order are added to capture the perturbation in the boundary layer at short times. Such a modication implies new closure variables that must be solved over several unit cells from the edge of the domain. For larger distances, these decreasing functions vanish.

The paper is organized as follows. In Section 2, the main result of the upscaling procedure based on the spectral approach and homogenization technique for high Damköhler number is recalled.

Macroscopic model for high Damköhler number: a recall

In this section, the main steps and results of the homogenization procedure to derive the macroscopic equations for the diusion-reaction problem, developed in a previous work [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF] are recalled.

To simplify the notation, the superscript * indicating dimensional quantities in the preceding reference is omitted.

A porous medium occupying a macroscopic domain Ω with a characteristic length L, is composed of an immobile uid phase Ω f and of a rigid solid phase Ω s with a solid-uid interface Γ f s . The medium is constituted of the repetition of a periodic elementary cell Y = Y f ∪ Y s of characteristic length l, composed of the uid phase Y f and the solid phase Y s together with the solid-uid interface ∂Y f s . The boundary ∂Y = ∂Y f s ∪ ∂Y e is constituted of the uid-solid interface ∂Y f s assumed to be impervious and of the external interface ∂Y e = ∂Y se ∪ ∂Y f e separating two juxtaposed elementary cells. The macroscopic and microscopic spatial coordinates are x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ) respectively. The scale separation (l ≪ L) allows the introduction of the small parameter ε = l/L. Let c 1 and c 2 be the concentrations of species A and B respectively. The transport is ruled by a Fickian process. On the uid-solid interface Γ f s , linear chemical reactions exchange species A and B. The microscopic diusion/reaction equations at the pore-scale are written as

                 ∂c 1 ∂t -∇ • (D 1 ∇c 1 ) = 0 in Ω f ∂c 2 ∂t -∇ • (D 2 ∇c 2 ) = 0 in Ω f -D 1 ∇c 1 • n f s = k 1 c 1 -k 2 c 2 at Γ f s -D 2 ∇c 2 • n f s = k 2 c 2 -k 1 c 1 at Γ f s (1)
where

D
               -∇ • (D 1 ∇ψ 1,n ) = λ n ψ 1,n in Y f -∇ • (D 2 ∇ψ 2,n ) = λ n ψ 2,n in Y f -D 1 ∇ψ 1,n • n f s = k 1 ψ 1,n -k 2 ψ 2,n at ∂Y f s -D 2 ∇ψ 2,n • n f s = k 2 ψ 2,n -k 1 ψ 1,n at ∂Y f s (2)
where ψ 1,n (y) and ψ 2,n (y) represent the succesive eigenfunctions sharing the same positive eigenvalue λ n ordered in ascending order with n ∈ N. The eigenfunctions ψ 1,n (y) and ψ 2,n (y) are dened within a same multiplicative constant, which can be determined according to

k 1 ψ 2 1,n f + k 2 ψ 2 2,n f = k 1 + k 2 (3)
where ⟨ ⟩ f denotes the volume average over the uid phase.

For n = 0, λ 0 = 0 and the two eigenfunctions are constant satisfying the condition k 1 ψ 1,0k 2 ψ 2,0 = 0. They are given by

ψ 1,0 = k 2 k 1 , ψ 2,0 = k 1 k 2 (4)
Finally note that for n ≥ 1

⟨ψ 1,n ⟩ f + ⟨ψ 2,n ⟩ f = 0 (5)
resulting from the compatibility condition of Eqs. [START_REF] Bloch | Upscaling of diusion-reaction phenomena by homogenisation technique: Possible appearance of morphogenesis[END_REF].

c i (i ∈ {1, 2}) are then sought in an expansion related to the spectral problem (2) as follows

c i (t, x, y) = ∞ n=0 ψ i,n (y) exp (-λ n t) v i,n (t, x, y) (6)
with new variables v i,n depending on time and position.

Let dene the macroscopic Damköhler number Da L as the ratio of the macroscopic diusion time to the reaction one

Da L = k r L D r (7)
where k r and D r denote reference reaction rate and diusion coecient. In [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF], the Damköhler number appears naturally through the dimensional analysis which has been skipped in this summary J o u r n a l P r e -p r o o f

Journal Pre-proof of the results. The reference time is chosen as the macroscopic diusion time t r = L 2 /D r . The reader can refer to [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF] for more details on the procedure.

It has been shown in [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF] that for small Damköhler number of order O(ε), a classical homogenization procedure can correctly predict the macroscopic laws. However, for higher Damköhler number of order O(ε 0 ) or O(ε -1 ), special development is needed to precisely predict the behavior of diusion/reaction mechanism for short time [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF]. It should be noted that a slight dierence between the two cases O(ε 0 ) and O(ε -1 ) comes from the closure problem in solid/uid interface condition. In this work, we deal with the most interesting case of high Damköhler number with Da L = O(ε -1 ). By inserting [START_REF] Tartakovsky | Hybrid simulations of reactiondiusion systems in porous media[END_REF] in the initial problem [START_REF] Allaire | Two-scale expansion with drift approach to the Taylor dispersion for reactive transport through porous media[END_REF] and making use of the denition [START_REF] Qiu | Upscaling multicomponent transport in porous media with a linear reversible heterogeneous reaction[END_REF], performing a dimensional analysis, the ε-microscopic model for v i,n in dimensional space reads as

                   ψ 2 1,n ∂v (ε) 1,n ∂t = ∇ • D 1,n ∇v (ε) 1,n in Y f ψ 2 2,n ∂v (ε) 2,n ∂t = ∇ • D 2,n ∇v (ε) 2,n -D 1,n ∇v (ε) 1,n • n f s = ε -1 k 2 ψ 1,n ψ 2,n (v (ε) 1,n -v (ε) 2,n ) at ∂Y f s -D 2,n ∇v (ε) 2,n • n f s = ε -1 k 1 ψ 1,n ψ 2,n (v (ε) 2,n -v (ε) 1,n ) (8) with D 1,n = D 1 ψ 2 1,n and D 2,n = D 2 ψ 2 2,n
. By collecting the terms in the dierent powers of ε, one leads to the following results.

• Slow variables: At the leading order, the solution of v is v

(0) 1,n (t, x, y) = v (0) 2,n (t, x, y) = v (0)
n (t, x).

• Fluctuation: the solution for v

n depending on t and

x 1 v (1) 1,n = χ 1,n (y) • ∇ x v (0) n (t, x) + v (1)
n (t, x) v (1) 2,n = χ 2,n (y) • ∇ x v (0) n (t, x) + v (1) n (t, x) (9)
J o u r n a l P r e -p r o o f Journal Pre-proof where the vectors χ 1,n and χ 2,n satisfy the closure problem2

               0 = ∇ y • D 1,n I + ∇ y χ 1,n in Y f 0 = ∇ y • D 2,n I + ∇ y χ 2,n -D 1,n n f s • I + ∇ y χ 1,n = k 2 ψ 1,n ψ 2,n χ 1,n -χ 2,n at ∂Y f s -D 2,n n f s • I + ∇ y χ 2,n = k 1 ψ 1,n ψ 2,n χ 2,n -χ 1,n (10)
The particular case n = 0 corresponds to the eigenvalue λ 0 = 0 and to the eigenfunctions (4).

In this case χ 1,0 ≡ χ 2,0 ≡ χ and the local problem [START_REF] Allaire | Homogenization of a convectiondiusion model with reaction in a porous medium[END_REF] reduces to the standard closure problem

   ∇ 2 yy χ = 0 in Y f (I + ∇ y χ) • n f s = 0 at ∂Y f s (11) This leads to v (1) 1,0 = v (1) 2,0 = v (1) 0 = χ(y) • ∇ x v (0) 0 (t, x) + v(1) 0 (t, x) (12)
• Macroscopic equation: the macroscopic problem for v

n is derived as

∂v (0) n ∂t = ∇ x • D v,n • ∇ x v (0) n ( 13
)
where the eective diusion tensor D v,n is dened by

D v,n = k 1 k 1 + k 2 D 1,n I + ∇ y χ 1,n T f + k 2 k 1 + k 2 D 2,n I + ∇ y χ 2,n T f (14)
For n = 0, the eective tensor is given by

D v,0 = k 1 D 2 + k 2 D 1 k 1 + k 2 ⟨I + (∇ y χ) T ⟩ f (15)
• Macroscopic equations for the concentrations: As discussed in [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF], only the two rst eigenvalues λ 0 = 0 and λ 1 with their corresponding eigenfunctions are considered. In the asymptotic development ( 6), the higher order terms can be ignored due to the exponential decay in time. The averaged concentrations at the leading order are given by

⟨c (0) 1 ⟩ f = ψ 1,0 v (0) 0 + ⟨ψ 1,1 ⟩ f exp(-λ 1 t) v (0) 1 ⟨c (0) 2 ⟩ f = ψ 2,0 v (0) 0 + ⟨ψ 2,1 ⟩ f exp(-λ 1 t) v (0) 1 (16)
Omitting for the sake of simplicity the subscript (0) related to the scale order, using [START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF] for n = 0

and n = 1, and considering the time derivative of ( 16) and the compatibility condition [START_REF] Lugo-Méndez | Upscaling diusion and nonlinear reactive mass transport in homogeneous porous media[END_REF] for n = 1 result in the following mass conservation equations at the leading order 3 :

                       ∂⟨c 1 ⟩ f ∂t -∇ x • k 1 D v,1 + k 2 D v,0 k 1 + k 2 • ∇ x ⟨c 1 ⟩ f -∇ x • k 2 (D v,0 -D v,1 ) k 1 + k 2 • ∇ x ⟨c 2 ⟩ f + λ 1 k 1 ⟨c 1 ⟩ f -k 2 ⟨c 2 ⟩ f k 1 + k 2 = 0 ∂⟨c 2 ⟩ f ∂t -∇ x • k 1 (D v,0 -D v,1 ) k 1 + k 2 • ∇ x ⟨c 1 ⟩ f -∇ x • k 2 D v,1 + k 1 D v,0 k 1 + k 2 • ∇ x ⟨c 2 ⟩ f -λ 1 k 1 ⟨c 1 ⟩ f -k 2 ⟨c 2 ⟩ f k 1 + k 2 = 0 (17)
Given the eective coecient tensors D v,0 and D v,1 , and the initial and boundary conditions, the coupled diusion-reaction Eqs. ( 17) can be solved to compute the concentration proles.

Boundary layer problem

The macroscopic model based on the homogenization technique relies on the periodic condition hypothesis. As a consequence, it cannot capture the boundary layer problem developed in the vicinity of the edge of the domain when an outer boundary condition of Dirichlet type is imposed.

In this section, corrections are made to precisely take into account this boundary layer in the upscaled problem.

Let consider a parallelepiped rectangular porous medium with macro and microscopic coordinates {x 1 , x 2 , x 3 } and {y 1 , y 2 , y 3 } respectively 4 . At the inlet boundary x 1 = 0 (and y 1 = 0), 3 Eqs. ( 17) are dimensional equations corresponding to Eqs. (49) of [START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF], where the stars have been omitted. 4 It should be noted that the semi-innite

domain x1 ∈ [0, ∞], x2, x3 ∈ [-∞, ∞] can
c 1 = k 2 k 1 v (0) 0 + v (1) 0 + ψ 1,1 (y) exp(-λ 1 t) v (0) 1 + v (1) 1,1 c 2 = k 1 k 2 v (0) 0 + v (1) 0 + ψ 2,1 (y) exp(-λ 1 t) v (0) 1 + v (1) 2,1 (18)
It should be noted that the formal parameter ε can be omitted for the calculation of the concentrations. This is a quasi-exact solution in the domain, except for the boundary layer in the vicinity of the interface where boundary conditions should be imposed for the variable v (0) 0 in order to solve the problem [START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF]. Two distinct problems related to the permanent and transient regimes need to be considered.

Steady state

In the steady state when t → ∞, only the rst order variable v

c 1 = k 2 k 1 v (0) 0 + v (1) 0 + v (1) 0,BL + u (0) 1BL c 2 = k 1 k 2 v (0) 0 + v (1) 0 + v (1) 0,BL + u (0) 2BL (19)
It should be underlined that the boundary layer problem needs to be solved in a reduced uid domain Y † f (with a corresponding solid/uid interface ∂Y † f s ) composed of the innite repetition of the unit cell Y in the direction Oy 1 perpendicular to the edge.

J o u r n a l P r e -p r o o f

Journal Pre-proof

3.1.1. Order O(ε 0 )
Inserting [START_REF] Destuynder | Sur une justication des modèles de plaques et de coques par les méthodes asymptotiques[END_REF] in the initial problem (1) in steady state, the problem for u

               0 = ∇ y • D 1 ∇ y u (0) 1BL in Y † f 0 = ∇ y • D 2 ∇ y u (0) 2BL -D 1 ∇ y u (0) 1BL • n f s = k 1 u (0) 1BL -k 2 u (0) 2BL on ∂Y † f s -D 2 ∇ y u (0) 2BL • n f s = k 2 u (0) 2BL -k 1 u (0) 1BL (20) u (0)
1BL and u

(0)
2BL are periodic in y 2 and y 3 and satisfy the following boundary conditions

y 1 = 0 c 1D = k 2 k 1 v (0) 0D + u (0) 1BL (y 1 = 0) c 2D = k 1 k 2 v (0) 0D + u (0) 2BL (y 1 = 0) y 1 → ∞ u (0) 1BL (y 1 ) → 0 u (0) 2BL (y 1 ) → 0 (21)
where v

s (0) BL = D 1 u (0) 1BL + D 2 u (0) 2BL d (0) BL = k 1 u (0) 1BL -k 2 u (0) 2BL (22)
Inserting the above denitions in [START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF], the problems for s BL are separate and given by

                 0 = ∆ yy s (0) BL in Y † f 0 = ∆ yy d (0) BL -∇ y s (0) BL • n f s = 0 on ∂Y † f s -∇ y d (0) BL • n f s = k 1 D 1 + k 2 D 2 d (0) BL (23) s (0)
BL and d

BL are also periodic in y 2 and y 3 . In addition, by using the denitions [START_REF] Matine | Modeling of thermophysical properties in heterogeneous periodic media according to a multi-scale approach: Eective conductivity tensor and edge eects[END_REF] of s

BL and

d (0)
BL in [START_REF] Amar | On the exponential decay for boundary layer[END_REF], one obtains the following boundary conditions

y 1 = 0 s (0) BL = D 1 c 1D + D 2 c 2D - k 1 D 2 + k 2 D 1 √ k 1 k 2 v (0) 0D d (0) BL = d D = k 1 c 1D -k 2 c 2D y 1 → ∞ s (0) BL (y 1 ) → 0 d (0) BL (y 1 ) → 0 (24)
J o u r n a l P r e -p r o o f

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In steady state, the solution of the problem [START_REF] Matine | Transient heat conduction within periodic heterogeneous media: A space-time homogenization approach[END_REF] for s

(0)
BL is a constant equal to 0 due to the boundary condition ( 24) at y 1 → ∞. From ( 24), the Dirichlet condition for v (0) 0 at y 1 = 0 reads as

v (0) 0D = k 1 k 2 D 1 c 1D + D 2 c 2D k 1 D 2 + k 2 D 1 (25)
so that at the leading order, the boundary values of the concentrations imposed at x 1 = 0 to calculate the solution of the inner problem [START_REF] Koley | Study of boundary layer eects at ply interfaces of laminated composites using homogenization theory[END_REF] are given by

       c (0) 1D = k 2 k 1 v (0) 0D = k 2 D 1 c 1D + D 2 c 2D k 1 D 2 + k 2 D 1 c (0) 2D = k 1 k 2 v (0) 0D = k 1 D 1 c 1D + D 2 c 2D k 1 D 2 + k 2 D 1 (26)
Inserting ( 25) into [START_REF] Amar | On the exponential decay for boundary layer[END_REF] gives the boundary values for u

1BL and u

(0) 2BL      u (0) 1BL( y 1 = 0) = D 2 k 1 c 1D -k 2 c 2D k 1 D 2 + k 2 D 1 u (0) 2BL( y 1 = 0) = D 1 k 2 c 2D -k 1 c 1D k 1 D 2 + k 2 D 1 (27)
It should be noted that the problem for d

BL in ( 23) and ( 24) is an exponential decay problem-type involving a thin boundary layer aected by this correction.

To summarize, in steady state, at the order O(ε 0 ), to capture the boundary layer problem when the Dirichlet conditions for the concentrations do not verify the equilibrium condition, the problems for v (0) 0 (Eq. ( 13) for n = 0) and for u This correction comes from the fact that in the concentrations given by ( 18), the term v

(1) 0 is not uniform in space at the boundary y 1 = 0. Similarly to the expression (12) for v

(1) 0 , the boundary correction v

(1)

0,BL is sought in the form v (1) 0,BL = ω • ∇ x v (0) 0 (28)
The vector ω periodic in y 2 and y 3 must satisfy the closure

∇ 2 yy ω = 0 in Y † f -∇ y ω • n = 0 on ∂Y † f s (29)
J o u r n a l P r e -p r o o f

Journal Pre-proof

At the boundary y 1 = 0 in order to satisfy the boundary condition v

(1)

0 + v (1)
0,BL = 0, we impose

ω + χ = 0 ( 30
)
where χ is the solution of the closure problem [START_REF] Bourbatache | Upscaling diusionreaction in porous media[END_REF]. In the direction y 1 of unit vector e 1 , for

y 1 → ∞, a homogeneous Neuman condition is imposed ∇ y ω • e 1 = 0 (31)
The vector ω will reach a constant value at large value of y 1 resulting from the averaging of the ω = -χ values at the interface y 1 = 0. As this constant value must be 0, the indetermination of the χ problem ( 11) is exploited subtracting this value from ω 5 .

In the particular case of a symmetrical unit cell, in the frame of the symmetry axes centered on the cell, imposing a null volume average ensures the unicity of χ where the components χ i are odd relative to y i (and even relative to the other coordinates y j̸ =i ). Therefore, if the external surface of the medium coincides with the surface of the unit cell Y of normal e 1 , χ 1 is null due to the combination of oddness and periodicity. Consequently, ω 1 = 0 on the surface y 1 = 0 of Y † . Hence

ω 1 is identically null in Y † f . In conclusion, if the gradient ∇ x v (0) 0 is parallel to e 1 , the correction v (1) 0,BL is null over Y † f .

Transient state

In the transient regime, the terms at order O(ε) in ( 18) must be corrected to better describe the complex unsteady non-equilibrium eect in the boundary layer. It is important to note that due to the large Damköhler number, the physical coupling is aected by the boundary condition only for short times. Beyond this time, the steady state is established. The O(ε 0 )-corrective terms for the two concentrations u

1,1BL and v

2,1BL into the expression of the concentrations [START_REF] Buannic | Higher-order eective modelling of periodic heterogeneous beams. II. Derivation of the proper boundary conditions for the interior asymptotic solution[END_REF] as 5 We recall that the boundary conditions are imposed for c1 and c2 at the main order, that implies according to

c 1 = k 2 k 1 v (0) 0 + v (1) 0 + v (1) 0,BL + u (0) 1BL + ψ 1,1 (y) exp(-λ 1 t) v (0) 1 + v (1) 1,1 + v (1) 1,1BL c 2 = k 1 k 2 v (0) 0 + v (1) 0 + v (1) 0,BL + u (0) 2BL + ψ 2,1 (y) exp(-λ 1 t) v (0) 1 + v (1) 2,1 + v (1) 2,1BL (32)
(19) that v (1) 0 + v (1)
0,BL = 0 at order O (ε).

2BL depending on y then reads as

                   ∂u (0) 1BL ∂τ = ∇ y • D 1 ∇ y u (0) 1BL in Y † f ∂u (0) 2BL ∂τ = ∇ y • D 2 ∇ y u (0) 2BL -D 1 ∇ y u (0) 1BL • n f s = k 1 u (0) 1BL -k 2 u (0) 2BL on ∂Y † f s -D 2 ∇ y u (0) 2BL • n f s = k 2 u (0) 2BL -k 1 u (0) 1BL (33)
where u

1BL and u

2BL are periodic in y 2 and y 3 and satisfy the boundary conditions

y 1 = 0 c 1D = k 2 k 1 v (0) 0D + u (0) 1BL(y1=0) + ψ 1,1 (y 1 =0) exp(-λ 1 t)v (0) 1 (t, x) (y1=0) c 2D = k 1 k 2 v (0) 0D + u (0) 2BL(y1=0) + ψ 2,1 (y 1 =0) exp(-λ 1 t)v (0) 1 (t, x) (y1=0) y 1 → ∞ u (0) 1BL (y 1 ) → 0 u (0) 2BL (y 1 ) → 0 t = 0 u (0) 1BL = u (0) 2BL = 0 (34)
To solve the boundary problem and the upscaled model, the initial and boundary conditions for v

(0) 0 , v (0) 1 , u (0)
1BL and u

2BL must be specied. The boundary values of v

(0) 0 , u (0)
1BL and u (0) 2BL

should verify the steady state being given by ( 25) and ( 27). The boundary condition for v (0) 1 can be legitimately adopted as v (0) 1 (y 1 = 0) = 0.

For the initial condition, owing to the compatibility condition [START_REF] Lugo-Méndez | Upscaling diusion and nonlinear reactive mass transport in homogeneous porous media[END_REF] for n = 1, from ( 16) we obtain

v (0) 0 (t = 0, x) = √ k 1 k 2 k 1 + k 2 ⟨c (0) 1 ⟩ f (t = 0, x) + ⟨c (0) 2 ⟩ f (t = 0, x) v (0) 1 (t = 0, x) = k 1 ⟨c (0) 1 ⟩ f (t = 0, x) -k 2 ⟨c (0) 2 ⟩ f (t = 0, x) ⟨ψ 1,1 ⟩ f (k 1 + k 2 ) (35)
It is legitimate to adopt that for τ = 0, no correction is needed leading to u

(0) 1BL (τ = 0, y) = u (0) 2BL (τ = 0, y) = 0. 3.2.2. Order O(ε 1 )
We now construct the closure problem for the O(ε 1 )-corrective terms. First note that the steady state problem for v 2,1 given by Eq. ( 8) written in y coordinates

as 6                    ψ 2 1,1 ∂v (1) 1,1BL ∂τ = ∇ y • D 1,1 ∇ y v (1) 1,1BL in Y † f ψ 2 2,1 ∂v (1) 2,1BL ∂τ = ∇ y • D 2,1 ∇ y v (1) 2,1BL -D 1,1 ∇ y v (1) 1,1BL • n f s = k 2 ψ 1,1 ψ 2,1 (v (1) 1,1BL -v (1) 2,1BL ) on ∂Y † f s -D 2,1 ∇ y v (1) 2,1BL • n f s = k 1 ψ 1,1 ψ 2,1 (v (1) 2,1BL -v (1) 1,1BL ) (36)
Similar to the solution (9) for v

(1)
1,1 and v

(1)
2,1 , the solution of v

(1)
1,1BL and v

(1) 2,1BL is sought in the form v (1) 1,1BL = ζ 1 (y, τ ) • ∇ x v (0) 1 (x, t) v (1) 2,1BL = ζ 2 (y, τ ) • ∇ x v (0) 1 (x, t) (37)
Inserting (37) into (36), noting that the slow variable v

                 ψ 2 1,1 ∂ζ 1 ∂τ = ∇ y • D 1,1 ∇ y ζ 1 in Y † f ψ 2 2,1 ∂ζ 2 ∂τ = ∇ y • D 2,1 ∇ y ζ 2 -D 1,1 n f s • ∇ y ζ 1 = k 2 ψ 1,1 ψ 2,1 (ζ 1 -ζ 2 ) on ∂Y † f s -D 2,1 n f s • ∇ y ζ 2 = k 1 ψ 1,1 ψ 2,1 (ζ 2 -ζ 1 ) (38)
complemented by the boundary condition at y 1 = 0

χ 1,1 + ζ 1 = 0 χ 2,1 + ζ 2 = 0 (39)
where χ 1,1 and χ 2,1 are solution of [START_REF] Allaire | Homogenization of a convectiondiusion model with reaction in a porous medium[END_REF] for n = 1. This condition ensures that at the boundary, the terms at order O(ε) vanish so that only the terms at O(ε 0 ) are considered for the boundary values.

Given the solutions of the rst order variables v

1,1BL and v The values of the physical parameters used are shown in Table 1. Note that the microscopic Damköhler numbers that constrain the values of the reaction rates k 1 and k 2 are now dened as

Da 1 = k 1 l D 1 and Da 2 = k 2 l D 2 = Da 1 α β (40)
with the ratios α = k 2 /k 1 and β = D 2 /D 1 .

The numerical study is carried out using COMSOL Multiphysics software based on the nite element method.

Numerical results in steady state

In steady state as t → ∞, the corrections for the concentrations c 1 and c 2 are given by [START_REF] Destuynder | Sur une justication des modèles de plaques et de coques par les méthodes asymptotiques[END_REF] with additional variables u

1BL and u

2BL (recall that as mentioned in 3.1.2 for the symmetric cell considered here the corrective term v

(1) 0,BL is identically zero). First, the macroscopic variable v

(0) 0 , l N φ D 1 D 2 β Da 1 k 1 α k 2 Da 2 c 1D c 2D 0.01 20 0.8 1 βD 1 2 100 Da 1 D 1 /l 2 αk 1 Da 1 α/β 1 c 1D Table 1: Parameters used in simulations.
solution of the homogenized diusion problem [START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF] in steady state for n = 0 with the homogenized diusion tensor given by [START_REF] Bourbatache | Homogenized model for diusion and heterogeneous reaction in porous media: numerical study and validation[END_REF], is computed in the eective medium. A Dirichlet boundary condition ( 25) is applied at the inlet whereas v

0 = χ • ∇ x v (0) 0 ( 41
)
where χ is solution of the closure problem (11) solved numerically on the unit cell.

Finally, the correction variables u

1BL and u

2BL , which are solutions of the local problem [START_REF] Allaire | Boundary layer tails in periodic homogenization[END_REF] with the boundary conditions at the inlet given by ( 27) and a zero-value condition far from the inlet, are solved in the microscopic pore-geometry for at least several unit cells from the inlet due to the exponential-decay behavior of these functions. This point will be proved in the numerical results in the sequel.

Let dene the y 2 -average operator of a function f as Given the solutions for v

⟨f ⟩ y2 = 1 l l 0 f dy 2 (42)
(0) 0 , v (1)
0 , u

1BL and u

2BL , knowing that v

(1) 0,BL ≡ 0, the y 2 -averaged concentrations ⟨c i ⟩ y2 of the corrected model can be computed from [START_REF] Destuynder | Sur une justication des modèles de plaques et de coques par les méthodes asymptotiques[END_REF]. In order to compare these results with the solution of the original model, the non-corrected homogenized model (HM) given by the system ( 17) is solved in the eective porous media. To do that, we rst compute the eigenvalue λ 1 and the eigenfunctions ψ 1,1 and ψ 2,1 from the spectral problem (2) for n = 1. Given the eigenfunctions, the closure problem (10) for χ 1,1 and χ 2,1 is solved in the unit cell to compute the eective coecient D v,1 from ( 14) for n = 1. Moreover, the closure problem for χ in [START_REF] Bourbatache | Upscaling diusionreaction in porous media[END_REF] is purely geometric and is solved in the unit cell to give the eective tensor D v,0 from (15).

To validate the proposed corrections in steady state, a direct numerical simulation of the porescale model (PSM) given in Eq. ( 1) is performed with the pore-scale geometry of Figure 1 to obtain the concentration elds which are averaged over the y 2 direction. Figure 3

c i y2 k2 k1 (v (0) 0 + v (1) 0 ) + u (0) 1BL y2 k1 k2 (v (0) 0 + v (1) 0 ) + u (0) 2BL y2 c1 y2 HM c2 y2 HM c1 y2 PSM c2 y2 PSM k2 k1 (v (0) 0 + v (1) 0 ) + u (0) 1BL y2 k1 k2 (v (0) 0 + v (1) 0 ) + u (0) 2BL y2 c1 y2 HM c2 y2 HM c1 y2 PSM c2 y2 PSM
1,1 and v

2,1 can be computed from the solution (9).

The rst-order corrections u (0) in the general case and vanishes for symmetric elementary cells, which is the case here. Far from the boundary, only the rst-order variable v (0) 0 diers signicantly from zero. In Figure 7b, the terms corresponding to n = 1 are plotted. The uctuations are due to the periodic solution of the eigenfunction ψ 1,1 . Due to exponential decay, the terms disappear over time. We observe that the time-dependent correction terms at this order are very small and can be neglected. The boundary layer problem for the coupled diusion/reaction process for high Damköhler number in porous media has been studied. The macroscopic mass conservation law derived from the homogenization technique and based on a spectral approach represents a quasi-exact solution within the domain but fails to describe the physics in a thin layer near the surface when a nonequilibrium Dirichlet boundary condition is imposed. In this context, corrective terms have to be added to the concentrations for both steady state and transient regime.

In steady state, two corrective functions are introduced to adjust the rst order variable corresponding to the null eigenvalue to capture the non-equilibrium state in the boundary layer. These terms decay exponentially from the boundary and need to be solved only on a few unit cells adjacent to the boundary. Numerical simulations prove that this correction is in excellent agreement with the direct numerical simulation of the pore-scale model.

In transient regime, the O(ε)-variables need to be corrected with additional terms whose solutions involve new closure problems. Numerical results show a small contribution of these terms.

Section 3

 3 is devoted to the development of the modied model in order to address the boundary layer problem in both steady and transient regimes. Numerical simulations are performed in Section 4 to validate the proposed model and to underline the importance of the correction terms. Conclusions are drawn in Section 5.

  1 and D 2 denote the diusion coecients of A and B respectively, k 1 and k 2 the reaction rates. These coecients have constant values during the process. n f s is the normal unit vector at the solid/uid interface pointing out of the uid phase. The microscopic problem is completed by the initial conditions for given values of c 1 (t = 0) and c 2 (t = 0) and the boundary conditions at the outer edges of the domain. In this work, we only consider the Dirichlet conditions in which the concentrations are given at the edges. The initial model is transformed into a new problem associated with the following periodic spectral problem dened on the periodic unit cell Y

  are sought in the following form, to within one additive constant v

  be reduced to a parallelepiped when the periodic conditions in x2 and x3 directions are applied and a nite region in the vicinity of the boundary is considered for the boundary layer problem J o u r n a l P r e -p r o o f Journal Pre-proof Dirichlet conditions for the concentrations, c 1 = c 1D and c 2 = c 2D are imposed. The most interesting case is when k 1 c 1D ̸ = k 2 c 2D corresponding to a non-equilibrium situation. This leads to a thin boundary layer for the concentration proles in the vicinity of the interface x 1 = 0. Our main objective is to incorporate this layer in the upscaled model. To accomplish this task, we rst consider the development (6) with the two rst orders in ε and the two rst eigenvalues together with the corresponding eigenfunctions:

  related to the zero eigenvalue, play a role in the boundary layer. To take into account the boundary layer, the expansion of the concentrations needs to be improved in the sense of [20]. Considering Dirichlet conditions, since two boundary values c 1D and c 2D are given for the concentrations and only one variable v (0) 0 is involved at the interface, it is necessary to introduce new boundary variables u (0) 1BL (x, y) and u (0) 2BL (x, y) at order O ε 0 and v (1) 0,BL at order O ε 1 as

  boundary layer at order O(ε -2 ) in volume and O(ε -1 ) at the interface reads as

  is the unknown Dirichlet condition value of v (0) 0 at y 1 = 0. The problem (20) can be transformed by introducing the auxiliary variables

2 .

 2 given by[START_REF] Bird | Transport Phenomena[END_REF] and (27) respectively. The concentrations c 1 and c 2 must be adjusted by a thin boundary layer involving u Order O(ε 1 )

2 ,

 2 BL are now transient. We also introduce the O(ε)-corrective terms v

3. 2 . 1 .

 21 Order O(ε 0 ) Let dene the characteristic short time τ related to the macroscopic time t by τ = ε -2 t. The transient problem for u

1

 1 corresponding to the rst non zero eigenvalue must be a transient problem. At short J o u r n a l P r e -p r o o f Journal Pre-proof times, inserting (32) in the initial problem (1) for c 1 and c 2 , the transient problem of v has the same form as the problem of v

  , t) is independent on the short characteristic time τ , gives rise to the local unsteady problem of the vectors ζ 1 and ζ 2

  solving the closure problems (29) and (38), the O(ε 1 )-correction terms v

( 1 ) 2 ,Figure 1 :

 121 Figure 1: Porous medium and homogenized medium used in the numerical simulation.

  is imposed at the outlet. If the concentrations values imposed at the interface are c 1D = c 2D = 1, the imposed value for v 1 = 0) = 0.75. Secondly, given the eld of v

Figure 2 (

 2 Figure 2(a) displays the variation of the y 2 -averaged variables u 0 1BL y2 and u 0 2BL y2 with respect to the position x 1 /L. We can observe that the correction u 0 1BL (respectively u 0 2BL ) decreases (respectively increases) rapidly and decays towards zero after several unit cells in the vicinity of the inlet. Thus, this correction only aects a thin layer (boundary layer) from the edge whose thickness depends on the Damköhler numbers. The variation of the dierence d 0 BL y2 with respect to x 1 /L is plotted in Figure 2(b). It is clearly observed that a non-equilibrium state where d 0 BL y2 ̸ = 0 is established in a thin boundary layer.

Figure 2 :

 2 Figure 2: Variation of (a): u 0 1BL y 2 and u 0 2BL y 2 and (b): d 0 BL y 2 versus x1/L for Da1 = 100.

displays the y 2

 2 -averaged concentration proles of ⟨c 1 ⟩ y2 and ⟨c 2 ⟩ y2 obtained from the corrected model (continuous line), the original HM without correction (dotted line) and the PSM (line-marker). An excellent agreement between the corrected model and the PSM is observed, which satisfactorily validates the proposed model. The original HM solution fails to capture the complex physics in the boundary layer and leads to an inaccurate prediction of the concentration elds in the entire domain. We can also observe that ⟨c 1 ⟩ y2 (respectively ⟨c 2 ⟩ y2 ) increases (respectively decreases) quickly in the boundary layer to attain a transition point delimiting the boundary zone.

Figure 3 :

 3 Figure 3: y2-averaged concentrations versus x1/L for Da1 = 100 obtained by the corrected model (line), the original HM (dotted line) and the PSM (line-marker).

4. 2 .1 1 ,

 21 Numerical results in transient regimeThe corrected solution for the concentrations involving additional variables at order O(ε 1 ) in transient regime is given by Eq. (32). To compute the concentrations, rst we solve the transient problems for the rst-order variables v given by[START_REF] Bourbatache | Upscaling coupled heterogeneous diusion reaction equations in porous media[END_REF] with the initial conditions (35). In addition, the Dirichlet boundary condition for v (0) 0[START_REF] Bird | Transport Phenomena[END_REF] is imposed at the inlet and v the zero value is imposed on both boundaries. Given the elds of v

Figure 4 :

 4 Figure 4: Concentration variation of (a) ⟨c1⟩y 2 and (b) ⟨c2⟩y 2 with time for dierent positions x1 = [l/4, 2l, 3l],obtained from the corrected model (dashed line-marker), PSM (line) and the original HM (dotted line-marker).

Figures 5 and 6 display the y 2 -Figure 5 : 20 J

 2520 Figures 5 and 6 display the y 2 -averaged concentration proles of ⟨c 1 ⟩ y2 and ⟨c 2 ⟩ y2 with respect to x 1 /L obtained from the three models at dierent times t. Small uctuations in the concentrations obtained by the corrected model are observed at the beginning of the process due to the periodic variation of the eigenfunctions, which vanish with time. Again, the corrected model is in very good agreement with the PSM while the original HM fails to reproduce the boundary layer behavior correctly. As a result, this error propagates with time away from the boundary, leading to inaccurate prediction of concentration proles.

Figure 6 : 21 J

 621 Figure 6: Concentration ⟨c2⟩y 2 versus x1/L for dierent times. 21 J o u r n a l P r e -p r o o f

Figure 7 :

 7 Figure 7: Corrective term proles for dierent times.

This necessary condition comes from the right hand side of the associated boundary conditions in Eqs.[START_REF] Valdés-Parada | On diusion, dispersion and reaction in porous media[END_REF]

Here, rather than using the tensorial denition in[START_REF] Le | A spectral approach for homogenization of diusion and heterogeneous reaction in porous media[END_REF] in which a transpose operator is needed, the index notation for the gradient of a vector dened in[START_REF] Bird | Transport Phenomena[END_REF] is adopted.

Starting again from Eq. (8) with the scaling τ = ε -2 t on time, we would obtain (36) at order O (ε).

J o u r n a l P r e -p r o o f

J o u r n a l P r e -p r o o f

Journal Pre-proof

In order to validate the proposed corrected model, a direct numerical simulation of the PSM (Eq. 1) is numerically solved in the pore-scale geometry in transient regime. Initial values of the concentrations are imposed as c 1 (t = 0) = c 2 (t = 0) = 0.1, which corresponds to a non-equilibrium initial state. Parameters used in the simulation are given in Table 1. Moreover, macroscopic equations without correction [START_REF] Koley | Study of boundary layer eects at ply interfaces of laminated composites using homogenization theory[END_REF] are also numerically solved in the same geometry.

Figure 4 shows the variation of the y 2 -averaged concentrations ⟨c 1 ⟩ y2 and ⟨c 2 ⟩ y2 with time at dierent positions x 1 = [l/4, 2l, 3l], obtained from the corrected model, the original homogenized model (HM) and the direct numerical simulation of the pore-scale model (PSM). Excellent agreement is obtained between the PSM and the corrected model for the three observation points located at dierent x 1 positions, which satisfactorily validates the proposed transient model. At very short times when the local chemical reaction dominates the transport mechanism and the boundary layer is located very close to the surface, all three models predict the same results at the observation points. However, for longer times, when the boundary layer propagates through the domain, a dierence between the original HM and the PSM appears and this is all the earlier as the position of the point is close to the surface at the left boundary.

J o u r n a l P r e -p r o o f

Journal Pre-proof -Boundary layer problem of diffusion/reaction process in porous media is studied -Macro-model derived from homogenization technique and spectral approach is recalled -Corrective terms are introduced to modify the concentration's expansion -These terms decay rapidly from the boundary and are solved on several unit cells -Numerical simulations highlight the role of the corrective terms in boundary layer

Declaration of interests

The authors declare that they have no known competng fnancial interests or personal relatonships ☒ that could have appeared to influence the work reported in this paper.

The authors declare the following fnancial interests/personal relatonships which may be considered ☐ as potental competng interests:


Monica Ashokumar

Jean-Luc Schwartz

Takayuki Ito

Modulation of speech perception due to somatosensory inputs in individuals with hearing-impairment

HAL is

INTRODUCTION•➢•

  Orofacial somatosensory inputs alter speech perception. • This somatosensory effect could be acquired during development of speech production. • In hearing-impaired individuals, speech production acquisition could be dependent on their hearing profile and this could intervene in the development of auditory-somatosensory effect. • This study assessed whether and how the somatosensory effect in speech perception can be induced in individuals with hearing impairment. METHODS Hearing threshold in noise • To detect three digits in noise RESULTS ➢ Somatosensory effect in speech perception • Vowel identification test using /e/-/ø/ continuum. • Facial skin-stretch in backward direction. French native speakers Hearing-impaired: 21 Normal hearing: 25 G1 : Relatively low hearing threshold (-7 to -1 dB SNR). Identify vowels correctly G2 : High hearing threshold (≥ 0 dB SNR). Unable to identify vowels correctly, (a) elderly (>55 years) (b) long duration of hearing device usage (>18 years) ➢ Hearing profile • Questionnaire