Vincent Guillemot
email: vincent.guillemot@pasteur.fr

Julie Le Borgne

Arnaud Gloaguen

Arthur Tenenhaus

Gilbert Saporta

Sylvie Chollet

Derek Beaton

Hervé Abdi

Sparse Multiple Correspondence Analysis

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

Résumé. L'Analyse des Correspondances Multiples (ACM) est la méthode de choix pour l'analyse des données catégorielles multivariées. Basée sur la décomposition en valeurs singulières (SVD), l'ACM bénéficie naturellement des extensions de cette dernière, dont celles qui permettent de réaliser des analyses parcimonieuses. L'algorithme permettant de réaliser l'ACM parcimonieuse nécessite deux propriétés particulières additionnelles: l'inclusion des matrices de métriques (masses et poids) caractéristiques de l'ACM, et la possibilité de sélectionner des groupes entiers de variables (un groupe étant constitué du codage disjonctif complet d'une variable catégorielle). Nous proposons un algorithme pour l'ACM parcimonieuse basé sur la décomposition en valeurs singulières généralisée et une projection sur la boule `1,2 . Nous illustrons notre méthode avec les résultats d'une enquête par questionnaires sur les connaissances alimentaires et la perception de fromages.

Mots-clés. Parcimonie, Analyse Multivariée, Analyse des Correspondances Multiples

Abstract. Multiple Correspondence Analysis (MCA) is the method of choice for the multivariate analysis of categorical data. In MCA each qualitative variable is represented by a group of binary variables (with a coding scheme called "complete disjunctive coding") and each binary variable has a weight inversely proportional to its frequency. The data matrix concatenates all these binary variables, and once normalized and centered this data matrix is analyzed with a generalized singular value decomposition (GSVD) that incorporates the variable weights as constraints (or "metric"). The GSVD is, of course, based on the plain SVD and so MCA can be sparsified by extending algorithms designed to sparsify the SVD. To do so requires two additional features: to include weights and to be able to sparsify entire groups of variables at once. Another important feature of such a sparsification should be to preserve the orthogonality of the components, Here, we integrate all these constraints by using an exact projection scheme onto the intersection of subspaces (i.e., balls) where each ball represents a specific type of constraints. We illustrate our procedure with the data from a questionnaire survey on the perception of cheese in two French cities.

Keywords. Sparsity, Multivariate Analysis, Multiple Correspondence Analysis 1 a

Akin to principal component analysis (PCA), Multiple Correspondence Analysis (MCA, see for reviews [START_REF] Abdi | Multiple Correspondence Analysis[END_REF][START_REF] Roux | Multiple Correspondence Analysis.S a g e , T h o u s a n d[END_REF][START_REF] Saporta | Probabilités, Analyse des Données et Statistique.T e c h n i p[END_REF]) is a multivariate analysis method that analyzes the structure of a set of I observations described by K qualitative variables each comprising J k modalities. In MCA, the response of an observation to a qualitative variable is represented by a binary vector. The I by J data matrix to analyze (denoted Y), called the disjunctive table, is the concatenation of K matrices each with I observations and J k columns. In MCA, the matrix actually analyzed is the centered probability matrix denoted X,a n dc o m p u t e da s

X = Y ⇥ (IK) -1 -rc > with r = Y1 ⇥ (IK) -1 and c = Y > 1 ⇥ (IK) -1 .
(

) 1
MCA is obtained from the generalized singular value decomposition (GSVD) of X as

X = P∆Q > with P > MP = Q > WQ = I where M =diag{r} and W =diag{c} (2)
with the diag operator transforming a vector into a diagonal matrix. In MCA, the interpretation of the dimensions is greatly facilitated when variables have either a very large or a very small (rather than a medium) contribution to a given dimension. In standard PCA such a pattern is obtained by rotation of the dimensions (e.g., with VARIMAX) or by sparsification (see [START_REF] Guillemot | A constrained singular value decomposition method that integrates sparsity and orthogonality[END_REF]). To extend sparsification to MCA (for previous work on Sparse MCA see [START_REF] Bernard | Sparse principal component analysis for multiblock data and its extension to sparse multiple correspondence analysis[END_REF][START_REF] Mori | Sparse Multiple Correspondence Analysis[END_REF]), the procedure needs to rely on extensions of sparsification that can incorporate 1) the constraints imposed by the matrices M and W and 2) a group constraint which imposes that the entire block of columns representing a variable is is selected or discarded by the sparsification procedure. In addition, as for standard sparsification of PCA, the interpretation of the results is improved when the sparsified dimensions are pairwise orthogonal, but previous sparsification methods for MCA do not implement the orthogonality constraint and sparsify only single variables. In this paper we present a generalization of the sparsified SVD (the SGSVD) that implements all the aforementioned constraints. We first present the method as an optimization problem, derive the algorithm for this optimization problem, and illustrate this new method with the analysis of aq u e s t i o n n a i r eo nf oodp r e f e r e n c e .

Method

The algorithm that we propose solves the following optimization problem arg min

P,∆,Q 1 2 X -P∆Q > 2 2 with ( P > MP = I Q > WQ = I ,a n d8`=1,...,R ( kp `kG  c 1,k q `kG  c 2,` ( 3)
where k.k 2 is the Euclidean norm, c 1,`a nd c 2,`a re positive scalars controlling the sparsity constraint for the pseudo-singular vectors, and where the group norm is defined as:

kxk G = P G g=1 kx ιg k 2 .
In other words, it is the `1-norm of the vector containing the `2-norm of the sub-vectors defined by the groups. The `1,2 -ball associated with this norm is noted B 1,2 (•). Apply weights and masses to X; for `=1,...,R do p (0) and q (0) are randomly initialized;

δ (0) ← 0; δ (1) ← p (0)> Xq (0) ; s ← 0; while δ (s+1) -δ (s) ≥ " do p (s+1) ← proj(Xq (s) , B 1,2 (c 1,`) \ B 2 (1) \ P ? ); q (s+1) ← proj(X > p (s+1) , B 1,2 (c 2,`) \ B 2 (1) \ Q ? ); δ (s+1) ← p (s+1)> Xq (s+1) ; s ← s +1 ; end δ `← δ (s+1) ; P ← vec P, p (s+1) ; Q ← vec Q, q (
s+1) ; end Apply inverse weights and masses to P and Q; Algorithm 1: General algorithm of the sparse MCA.

The sparse MCA algorithm

The sparse MCA algorithm is presented in Algorithm 1. It is essentially an alternate projection algorithm. Its key component is the projection onto the intersection between the ball defined by the group constraint, the `2-ball, and the space orthogonal to the already estimated singular triplets (left or right).

To achieve the pro jections on B 1,2 (c 1,`) \ B 2 (1) \ P ? and B 1,2 (c 2,`) \ B 2 (1) \ Q ? ,w e perform a Projection Onto Convex Sets (POCS) [START_REF]Comb ettes. The foundations of set theoretic estimation[END_REF] with two components: the projection onto the intersection of the group ball and the `2-ball, and the projection onto the orthogonal spaces defined by the already estimated pseudo-singular vectors. We detail the first projection in the next section.

Projection onto the intersection of the group and the `2-balls

Here, we present a fast and exact algorithm for the projection of x,afi x e dv e c t o ro fR n that comprises K non-overlapping groups, onto the intersection of an `1,2 -ball of radius c and the `2-ball of radius 1. This generalizes the projection onto B 1 (c) \ B 2 (1) [START_REF] Gloaguen | An efficient algorithm to satisfy `1 and `2 constraints[END_REF].

⇤ is such that k prox λ ⇤ kk G (x)k G = c if and only if k prox λ ⇤ kk 1 (v)k 1 = c,
where prox f is the proximal operator of a convex function f .R e c a l lt h a tt h ep r o j e c t i o n onto a ball is intimately linked to the proximal operator of the ball's norm. So, projecting [START_REF] Abdi | Multiple Correspondence Analysis[END_REF], which can be achieved with the algorithm presented in [START_REF] Gloaguen | An efficient algorithm to satisfy `1 and `2 constraints[END_REF].

x onto B 1,2 (c) \ B 2 (1) is equivalent to projecting v onto B 1 (c) \ B 2

Results

We analyzed the answers to two sets of questions of a survey on cheese answered by a sample of French participants from the two French cities of Angers and Lille. The 8q u e s t i o n sf r o mt h efi r s ts e te v a l u a t ek n o w l e d g ew i t ha n s w e r sc o d e da se i t h e rc o r r e c t or incorrect. The 23 questions from the second set evaluate the behaviors, opinions, or attitudes of the respondents toward cheese that are either farm-made or industrial. These questions are answered with a 4 point Likert scale (from 1 meaning "I totally agree" to 4m e a n i n g" It o t a l l yd i s a g r e e " ) . I na d d i t i o nw eh a v ei n f o r m a t i o na bo u tt h er e s po n d e n t s : Sex, Age (coded in 4 categories), and the city where they live (Angers or Lille).

AregularMCAwasappliedtothisdata,aswellasasparseMCAwiththeconstraints that each dimension be based on only one group of variables (i.e., a unique categorical variable) and only one group of observations (i.e., city of origin).

The Scree-plots on Figures 1a and1b show that the first dimension of the regular MCA captures most of the variability of the data, whereas for the sparse MCA the pseudoeigenvalues (normalized by the total inertia) are almost all equal. The results of the sparse MCA (see Figures 2c,2d, 2e, and 2f) reveal that: Age is an important component on the structure of the Angers group, followed by "knowledge" and gender whereas the second city of origin (Lille) is associated with the fourth dimension.

Conclusion

We developed a sparse version of the MCA that incorporates into the GSVD, the group constraints imposed on the different modalities of a qualitative variable, and illustrated with real data that this new approach can simplify the interpretation of the factorial dimensions as well as reveal deeper insights. Future directions will include taking into account a hierarchical structure of either variables or observations such as overlapping groups of grouped variables as can be found, for example, for SNP data structured into pathways.

  , ", R Result: Sparse MCA of X Define P = 0; Define Q = 0;

A

  Denote by G the set of the indices defining the groups: G = {◆ k ,k =1 ,...,K},w h e r e ◆ k indicates the variables contained in group k and K is the number of groups. Let v be the vector containing all the `2-norms of the sub-vectors x ι k , k =1 ,...,K.T h er e a l and positive value λ

  (a) MCA scree-plot. (b) Sparse MCA scree plot.

Figure 1 :

 1 Figure 1: Scree-plots of the regular (left) and sparse (right) MCA.

Dimension 1 .

 1 λ = 0.071. τ = 98% Dimension 2. λ =

Figure 2 :

 2 Figure 2: Variable and observation maps for MCA and sparse MCA. Only Dimensions 1 and 2 are shown for the regular MCA. For the sparse MCA, we show Dimensions 1 and 2( m i d d l efi g u r e s )a n dD i m e n s i o n s3a n d4( l o w e rfi g u r e s ) .

sciencesconf.org:jds2020:319875


Thomas Sanchez

Simon Gillet

Viacheslav Shkirskiy

Vincent Vivier

Jaques Echouard

Jolanta Światowska

Polina Volovitch

New experimental approach for intelligent screening of buried metal/oxide/polymer interfaces via local electrochemistry: example of undamaged model epoxy-coated Zn alloys

Keywords: Local electrochemical impedance mapping (LEIM), Underpaint degradation, Zn alloys, conversion coatings, AC-DC-AC Highlights

teaching and research institutions in France or abroad, or from public or private research centers.   New experimental approach for intelligent screening of buried metal/oxide/polymer interfaces via local electrochemistry: example of undamaged model epoxy-coated Zn alloys

Introduction

Polymer coatings have been used for centuries for corrosion protection of metals and alloys due to their good barrier properties limiting water and oxygen access to the metallic substrates [1,[START_REF] Walter | A critical review of the protection of metals by paints[END_REF]. Barrier protection mechanisms of organic coatings have been extensively studied and the formulations of hydrophobic binders, as well as the addition of pigments, have been optimized to reduce the permeability of the coating to water and oxygen [START_REF] Montermor | Functional and smart coatings for corrosion protection: A review of recent advances[END_REF]. However, as water reaches the interface, underpaint reactivity can develop. Localized reactive spots under the polymer coating lead to the metal/oxide/polymer interface disbondment and delamination of the polymer coating from the metallic substrate. This delamination of the coating can accelerate the system degradation and must be considered with caution [START_REF] Funke | Towards environmentally acceptable corrosion protection by organic coating[END_REF][START_REF] Funke | Problems and progress in organic coatings science and technology[END_REF].

In order to ensure a good adherence of the polymer and improve corrosion resistance, the metallic interface is usually modified by conversion coatings [START_REF] Funke | The role of adhesion in corrosion protection by organic coatings[END_REF]. These coatings are thin oxidized layers formed via chemical and/or electrochemical process at the metal surface [7,[START_REF] Makhlouf | Current and advanced coating technologies for industrial applications[END_REF],9]. Among conversion coatings, the most used in the last decades were chromate-based conversion coatings [START_REF] Zhao | Effects of chromate and chromate conversion coatings on corrosion of aluminum alloy 2024-T3[END_REF]. Their efficiency is supposed to be due to the combination of excellent anti-corrosion properties guaranteed by self-healing ability and good adhesion of chromates to polymer coating [11][START_REF] Jeffcoate | Chromate in conversion coatings: a XANES study of its concentration and mobility[END_REF]. However, chromates were proved to be genotoxic and carcinogen [START_REF] Epa | Principles of Environmental Impact Assessment Review: Appendix A: Environmental Impact Assessment Checklist[END_REF] and since 2006, European legislation tries to tackle hexavalent chromium via Registration, Evaluation, Authorization and Restriction of Chemicals (REACH) and Restriction of the use of certain Hazardous Substances (RoHS) [START_REF]European Commission regulation[END_REF]. According to these legislations, the application of Cr(VI) compounds should be banned in Europe by 2022 even though some industries got several years of extension to find replacement solutions. Therefore, the development and the optimization of new chromate-free conversion coating is strongly required [15-,16,17], whereas Cr(III)-based conversion coatings have been considered as a possible replacement [START_REF] Berger | A comparative study of the corrosion protective properties of chromium and chromium free passivation methods[END_REF][START_REF] Zhang | Comparison of the morphology and corrosion performance of Cr(VI)-and Cr(III)-based conversion coatings on zinc[END_REF].

In order to optimize the new generation of conversion coatings in a reasonable time scale, and to better understand their performances, it is necessary to develop new analytical tools to access the metal/oxide/polymer interface stability. To detect and localize the interfacial disbondment or deterioration, Scanning Kelvin Probe (SKP) is the most well-known and precise tool, used for both, blisters or delamination front detection and the measurement of the kinetics of polymer delamination front propagation in atmospheric corrosion [START_REF] Nazarov | Application of Scanning Kelvin Probe in the Study of Protective Paints[END_REF][START_REF] Stratman | A new technique able to measure directly the delamination of organic polymer films[END_REF]. SKP showed that delamination of a polymer with weak adhesion to zinc is mainly cathodically driven [START_REF] Fürbeth | The delamination of polymeric coatings from electrogalvanized steel -a mechanistic approach. Part 3: delamination kinetics and influence of CO2[END_REF]. In immersion conditions, electrochemical impedance spectroscopy (EIS) is the most used technique to detect interfacial disbondment from an artificially introduced defect as well as to monitor the polymer coating degradation [START_REF] Amirudin | Application of electrochemical impedance spectroscopy to study the degradation of polymer-coated metals[END_REF][START_REF] Murray | Electrochemical test methods for evaluating organic coatings on metals: an update. Part III: Multiple test parameter measurements[END_REF][START_REF] Bonora | Electrochemical impedance spectroscopy as a tool for investigating underpaint corrosion[END_REF].Different approaches have been developed to quantify the global degradation phenomena in painted systems: water uptake, diffusion of soluble ions, evolution of the total delaminated area etc. [START_REF] Ogle | An electrochemical study of the delamination of polymer coatings on galvanized steel[END_REF][START_REF] Deflorian | An EIS study of ion diffusion through organic coatings[END_REF][START_REF] Akbarinezhad | Another approach for ranking and evaluating organic paint coatings via electrochemical impedance spectroscopy[END_REF]. To localize the disbondment at polymer-metal interface, Local Electrochemical Impedance Mapping (LEIM) [START_REF] Jorcin | Delaminated areas beneath organic coating: A local electrochemical impedance approach[END_REF][START_REF] Shkirskiy | Development of anodic and cathodic blisters at a model Zn/epoxy interface studied using local electrochemical impedance[END_REF] and Scanning Vibrating Electrode (SVET) [START_REF] Santana | Sensing electrochemical activity in polymer-coated metals during the early stages of coating degradation by means of the scanning vibrating electrode technique[END_REF] can both detect the electrochemical reactivity in delaminated areas and distinguish the anodic and the cathodic reactive zones. However, contrary to LEIM, SVET cannot detect the underpaint reactivity in the absence of a defect in the paint. In addition, a quantitative LEIM was recently proposed to study the disbondment front propagation kinetics [START_REF] Shkirskiy | Development of quantitative Local Electrochemical Impedance Mapping: an efficient tool for the evaluation of delamination kinetics[END_REF] during immersion. However, it should be mentioned that most of previously reported studies on local degradation of metal/oxide/polymer interfaces using local electrochemical techniques concerned the interface delamination starting from an artificial defect, intentionally introduced at the metal/oxide/polymer interface or in the coating prior to the aging process.

For an undamaged coated surface, in addition to long term immersion, AC-DC-AC accelerated electrochemical degradation procedure via alternating cathodic polarization (DC) and EIS measurement at open circuit potential (AC) [START_REF] Allahar | Understanding ac-dc-ac accelerated test results[END_REF] is often used. Depending on the potential of the cathodic polarization during AC-DC-AC procedure, it can either be used to assess the polymer stability [START_REF] Suay | Rapid assessment of automotive epoxy primers by electrochemical techniques[END_REF][START_REF] Bethencourt | Lifetime prediction of waterborne acrylic paints with the AC-DC-AC method[END_REF], or to study the underpaint reactivity and delamination [START_REF] Hollaender | Rapid assessment of food/package interactions by electrochemical impedance spectroscopy (EIS)[END_REF]. The application of cathodic potential promotes the initiation and propagation of the metal/oxide/polymer interface degradation [START_REF] Montoya | A cathodic delamination study of coatings with and without mechanical defects[END_REF]. Under polarization lower than -1 V/SHE, both the oxygen reduction (Eq. 1) and the water reduction (Eq. 2) can occur [START_REF] Leidheiser | Cathodic delamination of polybutadiene from steel: a review[END_REF] thus leading to a pH increase at the buried metal/oxide/polymer interface. Using pH sensitive indicator like phenolphthalein, these cathodic reactive areas can be evidenced by in situ Time Lapse Microscopy (TLM) of the sample surface [START_REF] Zhang | Corrosion-Sensing Behavior of an Acrylic-Based Coating System[END_REF][START_REF] Maia | Active sensing coating for early detection of corrosion processes[END_REF]. (2) In the case of large cathodic over-potentials, high local pH, peroxides and free radicals can also damage the polymer matrix; the polarization conditions should then be chosen carefully for each system and should take into account the chemistry of the organic coating.

Epoxy resins are among the most used primers for Zn alloys because of their hydrophobicity and their ability to incorporate fillers and pigments for corrosion inhibition, anti-fouling, self-healing, etc. [START_REF] Howarth | The use of water-based epoxies for anti-corrosive primers[END_REF]. Due to the high impedance of thick commercial epoxy coatings and the presence of numerous additives, in situ study of the underpaint reactivity in such a system by means of electrochemical tests, vibrational spectroscopy and/or optical microscopy is often difficult and the results interpretation can be controversial. In order to understand the underpaint reactivity, the use of a model polymer seems to be a promising approach [START_REF] Qhach | Etude de la durabilité d'un primaire epoxy anticorrosion: rôle de l'interphase polymère/métal et conséquence sur l'adhérence[END_REF]. In the literature, several weakly adherent coatings, like polyvinyl butyral (PVB) or polyvinyl alcohol (PVA), were used in SKP studies of underpaint reactivity [START_REF] Fürbeth | The delamination of polymeric coatings from electrogalvanized steel -a mechanistic approach. Part 1: delamination from a defect with intact zinc layer[END_REF][START_REF] Iqbal | Study of polymer coating delamination kinetics on zinc modified with zinc oxide of different morphologies[END_REF]. However, they do not sustain long term immersion because of their high solubility in water and weak adhesion to metallic substrates [START_REF] Mallapragada | Dissolution mechanism of semicrystalline poly(vinyl alcohol) in water[END_REF]. For stability evaluations in long immersion and aggressive aging tests, some groups performed experiment using thick epoxy polymers [START_REF] Jorcin | Delaminated areas beneath organic coating: A local electrochemical impedance approach[END_REF][START_REF] Nazarov | Assessment of steel corrosion and deadhesion of epoxy barrier paint by scanning Kelvin probe[END_REF]. Nonetheless, the measurements were either made in atmospheric conditions or required to introduce a well localized artificial defect in the coating or at the interface prior to the degradation procedures. The latter can influence the degradation mechanisms and is not appropriate if the initial degradation is of interest.

Electrochemical impedance studies demonstrated that polymer also degrades in electrochemical tests (see for instance [START_REF] Zhang | Studies of water transport behavior and impedance models of epoxycoated metals in NaCl solution by EIS[END_REF][START_REF] Olivier | Use of Electrochemical Impedance Spectroscopy (EIS) for the evaluation of Electrocoatings performances[END_REF]); however, to our knowledge, a combined approach, taking into account the possible evolution of all the components of metal/oxide/polymer interfaces is still missing. It seems hence to be timely to develop an experimental approach able to consider the initiation and propagation of the underpaint reactivity and dissociate the intrinsic interface response from the possible polymer evolution under electrochemical testing of buried metal/oxide/polymer interface. The later requires the development of a model thin epoxy coating with a good adhesion to metal interfaces and characterized stability [START_REF] Gu | Linking Accelerated Laboratory Test with Outdoor Performance Results for a Model Epoxy Coating System[END_REF], [START_REF] Rouw | Model epoxy powder coatings and their adhesion to steel[END_REF]. The model polymer should be homogeneous in terms of chemical composition and thickness, thin enough for in situ electrochemical impedance measurements, transparent in the visible light for an optical survey of the substrate/polymer interface and be stable to sustain AC-DC-AC procedure.

In this work, a new methodological approach is proposed for in situ accelerated mechanistic study of the underpaint stability of zinc alloy/epoxy interfaces in immersion condition without introducing an initial defect in the coated system. The methodology includes a specific aging procedure to accelerate the interface degradation and the use of an optimized model epoxy polymer, meeting the requirements listed above. The validation is made for a zinc alloy with Cr(III) conversion coatings designed for epoxy painting.

Materials and methods

Materials

Substrates preparation

Two substrate materials were used: hot dip galvanized zinc and zinc alloy coated steel. Zn coated (hot dip galvanized, 0.2 wt. % of Al in the coating) plates of low carbon steel were used to test the developed epoxy coatings, because the behavior of this material is well known and the response of the epoxy can be easily distinguished from the response of Zn substrate. Prior to the application of organic coating, the Zn samples were rinsed with Milli-Q water and degreased in 1 M NaOH solution for 30 s to remove the oxide layer and then rinsed with Milli-Q water and absolute ethanol before being dried with compressed air.

Zinc-nickel alloy coating was used as the substrate for the conversion coating study. The alloy was prepared by a following procedure. Steel plates of 1015 cm² were degreased and grit-blasted. Electrodeposition of Zn alloy was performed in alkaline 12-16 % ZnNi bath for 45 minutes at applied current of 3 A/dm². After electrodeposition, the samples were rinsed with deionized water for 10 min and dipped in trivalent chromium passivation bath, then rinsed for 3 min in deionized water and dried with compressed air. Relief-baking of the samples was performed at around 200 °C for 23 h to alleviate hydrogen embrittlement. X-ray fluorescence (XRF) measurement was performed by Fisherscope X-Ray XDAL and showed 18 ± 2 µm in thickness and 14 ± 2 % in nickel composition. X-ray diffraction (XRD) measurement was performed by Empyrean Panalytical and showed only the presence of the gamma phase (Zn11Ni2). The Cr(III) content at the top surface was found to be between 5 and 9 at. % from the XPS measurement.

In order to reduce adhesion of polymer coating to the substrate, conversion coating was modified for some samples. For this, the samples of Zn alloy with conversion coating were immersed in 0.1 M NaOH for 5 s then rinsed with milli-Q water and dried with compressed air before application of polymer coating. Zn alloy samples with modified conversion coating are referred as "samples of type 1" while the Zn samples with non-modified conversion coating are referred as "sample of type 2" below. XPS characterization of the surface could not evidence a chemical composition difference between the two types of coating within a 2 % accuracy, which is consistent with the accuracy of the XPS measurement on rough surfaces.

For the model epoxy coating and degradation tests, 55 cm² samples were cut from the central part of each 1510 cm² plate to prevent the edge effects that can be observed in electrodeposition process. Prior to the polymer coating, the samples were stored in a desiccator.

Epoxy coating

The model epoxy formulation was a DiGlycidyl Ether of Bisphenol-A prepolymer and TriEthylene TetrAmine hardener (DGEBA-TETA) used in (1:1) ratio. One should note that Bisphenol-A is currently listed in the Candidate List of substances of very high concern, however in this work it is not proposed for industrial application but for methodological demonstration. If the method is applied at large scale, alternative epoxy coating could be formulated. For viscosity requirements, the preparation was diluted in 3 mL of ethanol (analytical grade) for 4 g of mixture. Prior to spin-coating procedure, the preparation was placed in a hermetic beaker and mixed at 300 rpm for 20 minutes then left to stand for 15 minutes in order to remove air bubbles.

Just before the coating application, the surface of the substrate was rinsed with absolute ethanol to improve the surface's wettability. The polymer coating was deposited by spin coating process using a POLOS SP15 spin-coater. Once the sample was placed and held in the spin-coater, the resin was poured in large excess and spread on the whole surface, the spin coating was performed twice with a 15 s relaxation time in between with following parameters: 3000 RPM rotary speed and 500 RPM/s linear acceleration for 35 s. After spin-coating, the samples were cured at 50 ± 5 °C for 14-16 h for curing and desolvatation [START_REF] Dang | Effects of curing program on mechanical behavior and water absorption of DGEBA/TETA epoxy network[END_REF].

With this procedure, the obtained coating's thickness controlled by optical microscopy (see section 2.2.2. for details of optical measurement) was ca. 10 ± 1 µm.

The samples were stored in a desiccator between the coating application and the degradation test. Degradation tests were made in the following week.

The chemical bonds formed after curing for stoichiometric formulation were reported in literature [START_REF] Moraes D'almeida | Ageing of the DGEBA/TETA Epoxy System with off-Stoichiometric Compositions[END_REF] and are schematically presented in Fig. 1 [53].

<Fig. 1> For the tests eight 55 cm² samples of each type (type 1 and type 2 described in section 2.1) were coated with model epoxy polymer and tested with the adapted AC-DC-AC procedure as detailed hereafter (see section 2.3.2). For comparison, additional four samples (size 1015 cm², two of each type) were also coated with a 60 µm thick layer of a commercial epoxy primer paint. The last four samples were used in 1000 h immersion test and AC-DC-AC degradation tests. These panels are named hereafter as "industrial epoxy coating". gTo distinguish the same substrates covered with two different coating, the letter A is added to the name/type of the sample with the industrial thick epoxy coated samples and the letter B for the samples with model epoxy. The combinations of studied substrates and coatings are summarized in Table 1. <Table 1>

Polymer characterization

No specific drying procedure was used before ex situ measurement except the electrolyte removal from the surface by compressed air. The evolution of the hydrophobic properties of the polymer were followed by static drop contact angle measurements using a Dataphysics OCA 15EC Package 2 contact angle automatic system with SCA 20 software module using a drop of 25 µL Quartex water deposited from 1 cm height. Each measurement was repeated 3 times with 3 drops and 3 different locations on each sample.

The chemical evolution of the polymer matrix was studied by Attenuated Total Reflectance Infrared Spectroscopy (ATR-IR) with Bruker Tensor 27 equipped with a Pyke MIRacle TM single reflection ATR. ATR was used in a standard configuration with the coating directly pushed against the crystal. Each IR spectrum was recorded from 4000 to 400 cm -1 with a resolution of 2 cm -1 and integrated over 512 cycles, at least two spectra were measured for each sample at two different locations. Before every ATR measurement, the crystal was cleaned with absolute ethanol and a background measurement was performed for the cleaned crystal. Before the measurement, a spectrum of a reference coated sample stored in a desiccator was recorded. Thus, the reproducibility of the measurement could be verified so that normalization of the IR spectra of degraded samples was unnecessary. This protocol was proposed since during degradation of the polymer coating no constant peak could be defined in the ATR-IR spectra. The peaks positions of the characteristic functional groups in DGEBA-TETA epoxy resin followed in this study [START_REF] Charlesby | Crosslinking and degradation of polymers[END_REF][START_REF] Morsch | The degradation mechanism of an epoxy-phenolic can coating[END_REF] are presented in Table 2.

<Table 2> Optical characterization of the samples was performed using a Keyence VHX-5000 3D microscope. Due to its short depth of focus (< 1 µm), it was possible to measure the thickness of the model epoxy by focusing alternatively on the non-coated substrate and on the top of the transparent coating using the reflection of the confocal light source. It was also used to observe the presence of corrosion products under the coating, as well as the deformation of the top surface of the coating in 3D-composition mode.

Degradation procedures

Three different degradation procedures were used. One sample was also kept in a desiccator after curing to obtain the evolution of a reference coating.

1. A humid chamber was used the first stability verifications of the model epoxy. Two samples were kept above 90 % relative humidity at room temperature. For contact angle and ATR-IR measurements, every hour they were taken out, rinsed with Milli-Q water and dried with compressed air before placed back in the humid chamber.

2. An immersion was used to verify immersion stability of the model epoxy. Four samples were immersed in 25 mL solution of 0.01 M NaCl with 1 wt. % of phenolphthalein in a sealed glass cell of 4.4 cm in diameter at room temperature. According to the literature, the influence of the phenolphthalein on the cathodic reactivity of polymer/zinc alloy interface can be neglected 32 . To perform ex situ measurements during immersion, samples were rinsed with milli-Q water and dried with compressed air before measurements and placed back to the immersion solution in less than 5 min. Immersion electrolyte was kept in a beaker during ex situ characterization and poured into the electrochemical cell to continue immersion test.

3. An AC-DC-AC test was applied to study the metal/oxide/polymer interface stability. For this test an electrochemical cathodic polarization cycling was performed on 5 coated samples. The samples were immersed for 1 hour in 25 mL of 0.01 M NaCl with 1 wt. % of phenolphthalein at room temperature to reach a stable value of open circuit potential (OCP). After the first EIS measurement, AC-DC-AC cycling was performed as presented in Fig. 2. It consisted of 15 minutes of cathodic polarization at -1.5 V/Ag/AgCl followed by 15 min immersion at OCP and a remaining period of 30 min for different measurements at the OCP. The procedure lasted 7 cycles for a total duration of 8.5 hours. After the cycling and before ex situ characterizations the sample was rinsed with milli-Q water and dried with compressed air. <Fig.2>

Electrochemical characterization and Time Lapse Microscopy

Electrochemical impedance spectroscopy (EIS) was performed using a Solartron Modulab potentiostat with ECS software interface. The three-electrode electrochemical cell consisted of a silver/silver-chloride (Ag/AgCl) reference electrode (RE), a 6.6 mm in diameter graphite rod as counter electrode (CE) and the coated sample as working electrode (WE). The cell was placed in a Faraday cage to avoid external interferences. EIS spectra were carried out at the OCP from 100 kHz to 0.1 Hz with 6 points per frequency decade and 10 mVRMS sinewave signal. Evolution of the low frequency (0.1 Hz) impedance modulus was used for tracking the stability of the metal/oxide/polymer interface.

Separately, local electrochemical impedance mapping (LEIM) was performed on three additional samples under AC-DC-AC tests. The electrochemical setup is described in detail in [START_REF] Shkirskiy | Development of anodic and cathodic blisters at a model Zn/epoxy interface studied using local electrochemical impedance[END_REF]. The local probe consisted of two Ag wires of 150 µm in diameter sealed in twin-capillaries using 3M glue. The distance between the two wires was 400 µm. The capillaries were bended so that the Ag electrodes were perpendicular to the surface of the sample, and thus, only the normal contribution of the current density was monitored. AgCl was deposited on both Ag electrodes of the probe prior to LEIS experiment by 5 min potentiostatic oxidation at 0.4 V/Ag/AgCl in 0.5 M NaCl solution. OCP and EIS measurements of the Ag/AgCl surfaces of the probe were performed during its immersion in 0.01 M NaCl solution at different times between 5 min and 10 h after the preparation of the probe to verify its stability. A home-made setup using Solartron Modulab software with auxiliary input and a 3-axis stepper motor-controlled by Sensolytics was used for LEIM measurements. The working electrode was a 2 cm diameter sample's surface. The counter electrode was a 3 cm diameter toroid of platinum wire placed above the surface of working electrode to guarantee symmetrical distribution of the current lines. During the measurement, the probe to sample distance was fixed at 100 ± 25 µm. For electrochemical impedance mapping, the potential oscillation was set to 15 mVRMS and 100 Hz; the X-Y maps consisted of a square grid of 3131 points distant of 100 µm to cover a 33 mm² area, the mapping was performed in 18 min.

In situ time lapse microscopy (TLM) using an HD USB microscope with 2 MP CMOS detector and a 4000 K white LED enlightenment was set up in the electrochemical cells for both AC-DC-AC and local electrochemistry tests. The objective to surface distance was set to 4 cm for a magnification of 80, leading to a 100 µm² pixel size. A picture was taken every minute during immersion. Local underpaint cathodic reactivity was detected from the color change of phenolphthalein, turning from colorless to pink color in the pH range of 8.2 -10. The increase of pH due to cathodic OH -release (reactions 1-2) is relevant for both, immersion and cathodic polarization. Percolation of the polymer and opening of pores could also be observed by TLM during immersion test. During cathodic polarization in the AC-DC-AC test, the localized hydrogen evolution can be monitored.

The experiments were made in a closed system in order to limit evaporation and keep constant electrolyte level (marked in the cell). This was achieved with the use of parafilm grafting tape, sealing the electrochemical cell and the parts of the setup (objective etc.). The electrolyte level visually stayed constant during 7.5 hours of the testing procedure.

Degradation procedures and characterizations used for the evaluation of the samples of different types are listed in Table 1.

Results

Stability of the epoxy coating on Zn substrate during immersion

Fig. 3.a shows typical ATR-IR spectra of the epoxy coated Zn samples at different times of immersion. The peaks attribution is shown in Table 2. The peaks referred as 6 and 1 correspond to characteristic peaks of OH -groups for scissors and stretching vibrations. The scissors vibration at 1644 cm -1 is related to the water molecule and is referred as "free OH" while the stretching vibration at 3400 cm -1 is related to all the OH groups and is referred as "total OH" [START_REF] Morsch | Water transport in an epoxy-phenolic coating[END_REF][START_REF] Musto | The transport of water in a tetrafunctional epoxy resin by near-infrared Fourier transform spectroscopy[END_REF]. The evolution of the peak's intensities averaged for the four immersed samples and the standard deviation are presented in Fig. 3.c. Both intensities increase and follow the same exponential trend (Eq. (3)) for the first 65 h. This trend is consistent with the water uptake of the coating, which seems to reach the saturation level from 20 to 25 hours of immersion: both peaks intensities associated with OH become stable. After 65 to 77 h of immersion, the intensities of these peaks change suddenly: the peak 1 corresponding to the "total OH" increases, whereas the peak 6 associated with the "free OH" decreases. This change in intensity is attributed to chemical evolution within the epoxy polymer coating occurring at around 70 h of immersion in a 0.01 M NaCl solution. The experimentally observed evolution of the OH-group peaks intensities with time could be fitted with the empirical equation (3) for both OH peaks.

𝐴(𝑡) = 𝑨 sat -𝐴 1 × 𝑒 -𝑏×𝑡 , (3)
In equation (3), t is the immersion time, A(t) is the peak intensity at time t, 𝑨 sat is the peak intensity at saturation, A 1 is the pre-exponential factor, and b is the exponential time constant obtained from the fitting. The value b = 0.16 ± 0.01 s -1 was the same for both "free OH" and "total OH" peaks evolution. Fig. 3.d shows the average evolution of the intensities of the peaks attributed to the typical organic bonds in DGEBA-TETA epoxy coating 2, 3, 4 and 5 in Table 2 for the four samples during immersion. For clarity, standard deviation of the measurement is not shown in the figure; however, the tendency was similar for all four samples. Their intensities are stable for the first 60 hours of immersion; therefore, it can be considered that only water uptake occurred during this period. After 60 hours, all the peaks' intensities attributed to the typical organic bonds in epoxy matrix (see Table 2) decrease strongly. This intensity drop coincides with the change in behavior of the two peaks corresponding to OH positions. These trends confirm that the model DGEBA-TETA epoxy coating starts to degrade after around 70 hours of immersion in a 0.01 M NaCl solution at room temperature.

<Fig. 3>

In addition to ATR-IR spectra, the evolution of the contact angle is shown in Fig. 4.a. The contact angle of the reference sample stored in humid air was also measured for comparison. Under both conditions, immersion and humid air, the contact angle drops from 75° to around 70° during the first hour that evidences a first decrease in hydrophobicity. This can be attributed to the hydration of the top surface of the epoxy coating. Until the 70 th hour, the contact angle remains stable (70° for all samples). After 70 hours of immersion, the contact angle decreases down to 60° and the values become more scattered depending on the analyzed areas. The increase in hydrophilicity of the coating corroborates with ATR-IR results: a first phase of water accumulation within the firsts hours, a saturation of water and stability domain of the system up to 50 hours and after 60-70 hours, the beginning of model epoxy matrix degradation.

<Fig.4>

In situ optical TLM images (Fig. 5 a) show no change of the surface up to 50 hours of immersion. Between 50 and 75 hours of immersion (Figs. 5b, d, ande) localized porosity appears and formation of corrosion products becomes visible around some open pores. A sudden formation of corrosion products with the opening pores could indicate the onset of Zn dissolution under the coating prior to the coating degradation and the decrease in the cross linking. This initial underpaint reactivity can be possible due to porosity of the polymer coating. Fig. 5.c and f show the coating analysis in 3D microscope after 10 hours and 75 hours of immersion consequently. The images are reconstructed using a high-resolution 3Dcomposition over 20 optical images taken over 9 µm depth range. The colors are attributed to light reflection, thus corresponding to the deformation of the coating due to the growth of corrosion products underneath the coating or by swelling. It can be seen that the coating surface is significantly deformed after the immersion test (Fig. 5.c) if compared to non-deformed state (Fig. 5. f), which can be attributed to the swelling of the polymer and also is coherent with the ATR IR observed water content increase. The coating deformation is measured to be ca. 8 µm protruding above the initial coating surface.

<Fig. 5> Typical electrochemical impedance spectra of the Zn coated samples after 1 and 72 hours of immersion are presented in Fig. 6.a. The EIS spectrum after 1 h of immersion shows two separate depressed semi-circles attributed to the polymer coating in the high frequency domain (1000 -5 Hz) and the double layer at the metallic interface in the lower frequency domain (5-0.1 Hz) as reported in literature 24 . After 1 h of immersion the presence of second semi-arc at low frequencies clearly shows the existence of a double layer capacitance with a value of ca. 1 µF cm -2 . It clearly indicates that the polymer coating is highly porous so that water can reach the metallic substrate during this period of immersion. Moreover, it evidences also that the metallic surface is not entirely covered by the model epoxy coating, i.e. there are some zones with pre-existing disbondment at the metal/oxide/polymer coating interface. The Nyquist plot after 72 hours of immersion shows a single depressed semi-circle attributed to the response of the delaminated metallic surface. However, the contribution from the polymer coating and and the corrosion product layer cannot be completely excluded at this time. A thorough data analysis with the fitting was attempted but no simple equivalent circuit was found to fit correctly this single time constant. Various parameter combinations could give valid fitting without proper physical meaning.

The evolution of the electrochemical impedance modulus at 0.1 Hz during the immersion is presented in Fig. 6.b. The impedance value of ca. 30 k cm 2 remains stable with an abrupt drop at 50 hours to a steady value of ca. 5 k cm 2 . The decrease of electrochemical impedance modulus can be explained as a loss of adherence and the degradation at the paint/metal interface due to the formation of pores in the coating. It should be noted that the polymer composition remains stable during 50 hours as confirmed from ATR-IR and TLM observations (vide supra).

<Fig. 6>

Stability of the epoxy coating on Zn substrate under AC-DC-AC cycling

In order to prove the applicability of the proposed model epoxy coating for a fast-electrochemical assessment of epoxy paint/metal interface, the evolution of the model epoxy coating was surveyed during the AC-DC-AC test.

The ATR-IR spectra on Fig. 3.b show no variation in the peak intensities related to the characteristic groups of the epoxy matrix except the peak 5 corresponding to the C-H bond, while both peaks corresponding to "free OH" and "total OH", see Table 2, increase in intensity which is relevant to water uptake. In addition, the hydrophobicity measured by contact angle remains stable throughout the 6 polarization cycles, as presented in Fig. 4.b. The results are coherent with water uptake in the polymer pores without polymer degradation. As for the immersion test, electrochemical impedance at low frequency, shown in Fig. 6.c, is also stable for 4 cycles and decreases after the 5 th cycle.

It can be then concluded, that the proposed model epoxy coating can sustain up to 50 hours of immersion or 7 cathodic polarization cycles without degradation. In addition, with both degradation procedures, the polymer did not show any sign of degradation until the disbondment occurred at the metal/oxide/polymer interface [START_REF] Hollaender | Application of impedance methods for corrosion protecting properties of coated substrates in metal packaging[END_REF].

Zn alloy/oxide/model epoxy interface evolution under modified AC-DC-AC test

The developed epoxy coating was applied on the Zn alloy with two types of conversion coating defined as described in the experimental part (section 2.1.1: damaged and undamaged samples corresponding to sample 1 and sample 2, respectively). The modified AC-DC-AC procedure was used to compare the electrochemical stability of zinc alloy/conversion coating/polymer interfaces in these two systems.

In the first step, the behavior of two types of samples without application of the polymer was compared. After one hour of immersion in a 0.01 M NaCl solution, the OCP of both types of samples was around -0.56 V vs. Ag/AgCl. Furthermore, the samples with both types of conversion coating show a similar EIS response (Fig. 7.a). A slight difference in the roughness of the Zn alloy substrate can change the angle in the CPE used for the fitting [START_REF] Viera | Surface roughness influence on CPE parameters in electrolytic cells[END_REF], and can explain the slight differences in the shape of the Nyquist plot.

<Fig. 7> In contrast, once the samples are coated with the model epoxy coating (referred as type 1B and type 2B in Table 1), the electrochemical behavior of the two systems is quite different (Fig. 7.b). The steep decrease in the low frequency electrochemical impedance modulus (|Z| 0.1 Hz) from the 3 rd cycle (Fig.

7.c)

shows an important increase of the system reactivity in the case of the samples of type 1, while the impedance modulus value remains almost the same for the samples of type 2 during all the 7 cycles. For comparison, in the Zn samples without conversion coating (Fig. 6), the degradation of the metal/oxide/polymer interface was visible from the sharp decrease of electrochemical impedance modulus. Thus, the difference in the evolution of the EIS of Zn alloy with conversion coatings of type 1 and 2 (sample type 1B and 2B) with cycling reveals a lower adherence of the model paint to the substrate in the case of type 1 samples than in the case of type 2 substrates.

The in situ microscopy images of both samples after 3 and 5 degradation cycles are presented in Fig. 8. The sample of type 1B (Fig. 8 left) shows a more significant cathodic hydrogen evolution (Eq. 2) than the sample of type 2B as it can be evidenced by the bubbles present on the surface. In addition, the spreading of pink-colored areas on the surface of type 1B sample indicates the presence of OH -ions (Eq. 1 and Eq. 2) inside and over the coating because of the phenolphthalein indicator in the electrolyte. Although the oxygen reduction reaction can occur during the modified AC-DC-AC degradation cycling for both types of samples, no OH -were detected by the indicator on the sample of type 2B (Fig. 8

right).

<Fig. 8> Thus, in situ TLM confirms EIS results under modified AC-DC-AC procedure: samples of type 1 show a much weaker bonding at the metal/oxide/polymer interface than the samples of type 2. The difference in the surface properties of the sample type 1 and type 2 was not observed by EIS before epoxy coating application, however the proposed AC-DC-AC procedure applied on the epoxy coated samples in combination with EIS and TLM observations allowed easily to evidence a difference between these two types of samples. 9 a. While TLM shows no visual degradation (no pores in the polymer, no corrosion products, no pink color indicating high pH) until at least 3 cycles, LEIM clearly demonstrates inhomogeneous distribution of the local admittance, revealing localized damage at the metal/oxide/polymer interface. Indeed, the zones indicated by red color in Fig. 9 b have the values of admittance about one order of magnitude higher compared with the other parts of the scanned surface.

Spatial distribution of underpaint reactivity

The measured admittance values up to 0.5 2 S cm -2 , meaning the current is in the mA range, ensure that the developed epoxy coating allows a proper measurement of the underpaint reactivity. One should note that even if Fig. 9 illustrates only one example of the zone with LEIM detected inhomogeneous reactivity, strongly inhomogeneous degradation was detected in multiple locations for the samples of Type 1. For comparison, in the same testing procedure, the samples of Type 2 did not show degradation neither by TLM (Fig. 8 b) nor by LEIM (not shown).

<Fig. 9> In order to verify that the interface evolution under the model thin epoxy polymer is coherent with the degradation expected under a thick strong industrial epoxy polymer, and in order to verify if the difference between the conversion coatings of type 1 and type 2 can be detected by LEIM under such a thick layer, additional tests were made with the samples 1A and 2A, coated by industrial paint.

The samples of type 2A with thick coatings did not evolve neither in TLM nor in LEIM during 10 AC-DE-AC cycles and are not shown in the figures. The samples of type 1A coated with the same industrial paint formed blisters during cycling. Fig. 9 c shows a typical in situ TLM image of a sample of type 1, coated with industrial epoxy coating after 10 AC-DC-AC cycles. Blisters are clearly visible on the surface, indicating localized degradation of the metal/oxide/polymer interface in their vicinity. The black square on the TLM image represents the 44 mm² area scanned by LEIM. The LEIM measured at 100 Hz is presented in Fig. 9 d. At the same time, the LEIM map shows a homogenous distribution of admittance values around 0.2 S cm -2 . Although the presence of degraded area on the sample detected by TLM, the high impedance of the applied epoxy primer masks the electrochemical response from the metal/oxide/polymer interface in the case of our small local impedance probe.

Finally, several samples of type 1A and 2A with industrial coating were immersed for 1000 hours in the demineralized water. After the test, the samples of type 1A were blistered while the samples of type 2A did not. To conclude, the LEIM experiments with model epoxy clearly distinguished two types of conversion coatings, evidenced the localized character of degradation and gave the relative stability order similar to the order obtained in immersion tests and AC-DC-AC tests of the same substrates coated by industrial coating.

Discussion

Speculative degradation mechanism of model epoxy

Three steps of evolution were observed for the model zinc/epoxy interface with developed DGEBA-TETA epoxy polymer exposed 0.01M NaCl electrolyte, as demonstrated by combined EIS, ATR-IR and contact angle measurements: -A saturation of the model epoxy coating by water reaching the metal/oxide/polymer interface during the first hours; -A stable behavior of the polymer coating until the polymer disbondment at the metal/oxide/polymer interface at around 50 hours of immersion; -A polymer coating degradation after 70 h of immersion. In situ TLM demonstrated the emergence and the opening of pores, the percolation of the epoxy coating, which can be attributed to a decrease of the cross-linking of the polymer. The evolution of the ATR-IR peaks corresponding to characteristics bonds of epoxy polymer also evidences that the organic bonds were most likely broken during the polymer's degradation (Fig. 1.b). Our results also demonstrated that the degradation of both the metal/oxide/polymer interface and the epoxy coating was enhanced by the cathodic polarization cycling of the coated samples. During AC-DC-AC aging procedure, hydroxides ions were detected by TLM, which is in agreement with Eq. 1.

The degradation mechanism of the polymer coating seems hence to be driven by the local pH increase. Consequently, with a stronger metal/oxide/polymer interfacial bonding the cathodic current under polarization cycling would be reduced leading to a less significant increase of pH under the paint. Hence, the developed model epoxy coating could sustain more severe AC-DC-AC degradation test, such as lower potential or increased number of cycles, for the systems with optimized conversion coating.

Leached by the substrate anodic dissolution metallic cations or possible presence of intermediate peroxides formed by the cathodic reactions reported in Refs. [START_REF] Gervasio | Determination of the oxygen reduction products on ASTM A516 steel during cathodic protection[END_REF][START_REF] Leidheiser | Some Substrate and Environmental Influences on the Cathodic Delamination of Organic Coatings[END_REF] could also accelerate polymer degradation, however our work is not able to confirm or disclaim their production and their role in the degradation mechanism.

Applicability of the modified AC-DC-AC protocol to discriminate conversion coatings

The combination of local electrochemical impedance and in situ Time Lapse Microscopy evidenced that the methodology using DGEBA-TETA polymer coating and modified AC-DC-AC procedure permitted to detect localized metal/oxide/polymer interface degradation. A speculative schema of the metal/oxide/polymer interface degradation process induced by the test is presented in Fig. 10. Once water and aggressive species diffuse through the polymer and reach the metal/polymer interface, deadhesion is stronger for conversion coating of type 1 samples than for type 2 samples. More intensive cathodic reaction leads to a quicker evolution of the impedance modulus with time for type 1 samples than for type 2 samples. Finally, hydroxides generated at the interface, diffuse through the coating and degrade the polymer, which arrives quicker for the type 1 sample than for type 2 sample.

<Fig. 10>

The approach allowed to discriminate two conversion coatings on Zn alloys as a function of their ability to form a stable metal/epoxy polymer interface. The proposed methodology is of great interest, because the results can be obtained in approximately 24 hours (1 hour for the coating preparation, 14 hours for curing, 9 hours for the measurements), in comparison to the commonly used 1000 hours of immersion. Thus, this methodology can be used as a fast and smart screening test of conversion coatings on Zn alloys for epoxy primers compatibility. In addition, it could give more comprehensive insights into the degradation mechanism of the metal/oxide/polymer interface such as its sensitivity to cathodic delamination. The combination of TLM and LEIM can indicate the heterogeneous or homogeneous nature of the metal/oxide/polymer disbandment at different scales and with a spatial resolution of approximately 100 µm.

The proposed methodology evaluates the intrinsic stability of conversion coating in confined zones under epoxy polymer and discriminates coating with close chemistry. Such an approach however cannot take into account possible variations of conversion layer stability related to interactions between the conversion coating and an industrial epoxy polymer, containing numerous additives. Comparison of conversion coatings with very different chemistry and hence probably different degradation mechanisms can be also erroneous without additional characterizations. Though, chemical characterization of the interface and of the effect of the polymer chemistry on the stability of the conversion layer with several model polymers could nicely complete the proposed approach.

Physical interpretation of LEIM on undamaged coated samples

Although LEIM is a common technique in the investigation of metal/oxide/polymer interface, most published studies demonstrate that pre-degraded polymer coating with either scarification and/or drilling of the coating was used to initiate the delamination process [START_REF] Jorcin | Delaminated areas beneath organic coating: A local electrochemical impedance approach[END_REF]. Other groups have studied blistering mechanism, i.e. a destabilization of buried metal/paint interface via LEIM without the generation of a localized defect in the coating. In most cases, the reactivity was enhanced at metal/oxide/polymer interface by using either NaCl salt powder or a drop of NaCl solution at the defined area of the metal substrate prior to the polymer coating deposition [START_REF] Zhou | Localized electrochemical impedance spectroscopy for studying the degradation of organic coatings[END_REF], [START_REF] Williams | The mechanism of group (I) chloride initiated filiform corrosion on iron[END_REF]. Thus, in these experiments a detection of the defect was foreseen. In our work, in order to initiate the metal/oxide/polymer interface reactivity, the conversion coating was slightly degraded then the samples were cleaned prior to paint application so that the loss of stability of the buried metal/oxide/polymer interface was only originating from the conversion coating quality. According to our knowledge, this approach with the unknown random distribution of surface defects before measurements, corresponding to the potentially weak interfacial bonding, was used for the first time.

On the other hand, one could contest the accurate contribution of the underpaint reactivity detected by LEIM in this work. Indeed, this technique is often used to detect defects within the organic coating itself more than at the hidden metal/oxide/polymer interface [START_REF] Upadhyay | Localized electrochemical characterization of organic coatings: A brief review[END_REF]. One could then wonder whether the active sites detected by LEIM were related to interfacial reactivity or to intrinsic defects in the polymer like a localized higher porosity of the model coating [START_REF] Taylor | Incentives for using local electrochemical impedance methods in the investigation of organic coatings[END_REF] or locally thinner polymer [START_REF] Darowicki | Assessment of organic coating degradation via local impedance imaging[END_REF] as described in the literature. Indeed, the existence of local high porosity in the polymer coating could induce a higher diffusion of oxygen under cathodic polarization and therefore, pH increase leading to the locally accelerated disbondment. A corollary hypothesis would be that these intrinsic defects in the polymer coating could result from the presence of a locally damaged conversion coating underneath: during the paint application and curing, the constraint relaxation within the polymer coating could lead to a segregation of intrinsic defects in the vicinity of a locally weaker metal/oxide/polymer interface. However, as described by Taylor et al. [START_REF] Taylor | Incentives for using local electrochemical impedance methods in the investigation of organic coatings[END_REF] the presence of underpaint reactivity and, therefore, the corrosion products in the pores leads to an enhancement of the barrier effect, thus, the increase of impedance modulus of the system. It was pointed out, that the deposition of zinc corrosion products occurs very shortly after the pore opening in the coating, indicating that the underpaint metal already reacts.

The question of the initiation of the local reactivity remains uncertain, both metal/oxide/polymer disbondment and intrinsic defect in the polymer could be the origin of the local reactivity, and might also be linked one to another. Still, this behavior was only evidenced on the samples of type 1, i.e. with supposedly weaker metal/oxide/polymer interface.

Conclusions

A new methodology using a model epoxy polymer and adapted to an AC-DC-AC accelerated degradation procedure was proposed to study and to understand stability of buried metal/oxide/polymer interface. It allowed to discriminate conversion coatings on zinc alloys in terms of ability to generate a stable metal/oxide/polymer interface in epoxy coated systems. The relative stability of two substrates with model thin coating, evaluated in the developed electrochemical test, correlated with the observations obtained for the same substrates coated with thick commercial epoxy primer after 1000 h of immersion. The proposed testing procedure lasts less than 24 h compared with a classical immersion test, making it interesting for the fast and smart ranking of the quality of conversion coatings.

In addition, the combination of in situ optical imaging and local electrochemical impedance in the accelerated test allowed to identify the heterogeneous character of interfacial disbondment on a large surface for which the possible locations from which the degradation can initiate are initially unknown and randomly distributed. This allows to obtain reliable results without preliminary introducing of an artificial defect as usually reported in the literature. The latter allows to have the true response of the sample protected by the coating without having to ask the question of the driving role that the artificial defect could have on the degradation of the coating.

Figure captions

Fig. 9 a

 9 Fig. 9 a presents an example of in situ image of a sample of type 1B, coated with the developed epoxy coating. The picture was taken after the relaxation step of the 4 th AC-DC-AC cycle, just before the LEIM measurement. Localized hydrogen evolution as well as slightly pink areas corresponding to oxygen reduction reaction sites are clearly visible in the figure. Fig. 9 b shows the LEIM map of the modulus of the local electrochemical admittance, i.e. the inverse of impedance, recorded at the 33 mm² area delimited by black square on the TLM image in Fig. 9 a.While TLM shows no visual degradation (no pores in the polymer, no corrosion products, no pink color indicating high pH) until at least 3 cycles, LEIM clearly demonstrates inhomogeneous distribution of the local admittance, revealing localized damage at the metal/oxide/polymer interface. Indeed, the zones indicated by red color in Fig.9b have the values of admittance about one order of magnitude higher compared with the other parts of the scanned surface.The measured admittance values up to 0.5 2 S cm -2 , meaning the current is in the mA range, ensure that the developed epoxy coating allows a proper measurement of the underpaint reactivity. One should note that even if Fig.9illustrates only one example of the zone with LEIM detected inhomogeneous reactivity, strongly inhomogeneous degradation was detected in multiple locations for the samples of Type 1. For comparison, in the same testing procedure, the samples of Type 2 did not show degradation neither by TLM (Fig.8 b) nor by LEIM (not shown).<Fig. 9> In order to verify that the interface evolution under the model thin epoxy polymer is coherent with the degradation expected under a thick strong industrial epoxy polymer, and in order to verify if the

Fig. 1 .

 1 Fig. 1. Schematic representation of (a) chemical bonding in cured DGEBA-TETA epoxy resin (adapted from Bernassau et al. [53]) and (b) epoxy degradation induced by interfacial pH increase.

Fig. 2 .

 2 Fig. 2. Schematic description of the AC-DC-AC procedure cycle used in the study.

Fig. 3 .

 3 Fig. 3. ATR-IR spectra (a, b) and evolution of peaks intensities with time (c, d) of the epoxy polymer over Zn HDG in the low wavelength region. (a) after 100 h of immersion and (b) after 7 AC-DC-AC cycles compared with the initial spectra of as coated epoxy polymer as indicated; (c) average evolution of the OH peaks intensities corresponding to "free -OH" and "total -OH" as a function of immersion time (see text for definition) and (d) average evolution of epoxy characteristic peaks (Table 1) intensities as a function of immersion time. Dashed line in (c) corresponds to the exponential fitting with Eq. 3 of the peaks intensities (A) vs immersion time (t).

Fig. 4 . 5 .

 45 Fig. 4. Evolution of average contact angle values on the model epoxy surface (a) in humid atmosphere (over 95%) and immersion in 10 mM NaCl water as indicated, (b) in modified AC-DC-AC ageing test. Fig. 5. Examples of optical images of the epoxy coating on Zn after (a) 50 h; (b) 60; (d) 70 and (e) 75 h of immersion in a 10 mM NaCl electrolyte by in situ Time Lapse Microscopy. Images (c) and (d) represent 3D image composition in initial state (10 h of immersion) and after 75 h of immersion consequently. Strong deformation after immersion can be seen (see text for details).

Fig. 6 .Fig. 7 .

 67 Fig. 6. Typical EIS spectra (a) and evolution of average low frequency electrochemical impedance modulus (b, c) of model epoxy coated Zn at different immersion times (example after 1 and 72 h of immersion time) in a 10 mM NaCl aqueous solution. (a, b) during immersion and (c) under modified AC-DC-AC degradation procedure Fig. 7. Typical EIS response of Zn alloy with two types of conversion coatings (Type 1 and Type 2 substrates in Tab. 1) in a 10 mM NaCl solution over modified AC-DC-AC aging procedure for (a) uncoated substrates and (b, c) substrates with model epoxy coating. Figure (c) presents average evolution of the low frequency electrochemical impedance modulus for the samples shown in (b).

Fig. 8 . 9 .

 89 Fig. 8. In situ Time Lapse Microscopy observation during OCP measurement after 3 and 5 AC-DC-AC test cycles for epoxy coated Zn alloy samples of type 1B and type 2B as indicated. Fig. 9. in situ TLM (a, c) and LEIM maps (b, d) of the Zn alloy sample of type 1B (Type 1 substrate covered with model epoxy) after 4 AC-DC cycles (a, b) and type 1A (Type 1 substrate coated with strong epoxy primer) after 10 AC-DC cycle (c, d). The areas mapped by LEIM are delimited in TLM images by squares.

Fig. 10 .

 10 Fig. 10. Schematic description of the metal/polymer interface degradation in the proposed accelerated test. During immersion, water and aggressive species diffuse through the polymer and reach the metal/polymer interface (a). In function of weak (b) or strong (c) stability of the metal/oxide/polymer interface, the underpaint reactivity under polarization leads to different intensity of cathodic reaction and to different EIS response (Type 1 or Type 2) respectively. Finally, hydroxides generated at the interface diffuse through the coating and degrade the polymer (d).

  Figure 5 a. b. c.

Tables

 Table 1:Studied samples (substrate and polymer types, used name, number of studied samples N), applied degradation tests and characterizations methods.

Substrate