Benoit Goussen
email: benoit.goussen@gmail.com

Rémy Beaudouin

Morgan Dutilleul

Adeline Buisset-Goussen

Jean-Marc Bonzom

Alexandre R R Péry

Energy-based modelling to assess effects of chemicals on Caenorhabditis elegans: a case study on uranium

Keywords:

corrections, structural formatting, and other quality control mechanisms may not be reflected in this document.

Introduction

The evaluation of the long term effects of pollutants on the environment is required for an effective protection of the ecosystems. According to [START_REF] Yeates | Nematode diversity in agroecosystems[END_REF], a useful bioindicator of ecosystemic stress should be the nematodes since they are abundant key decomposers in soil ecosystem. Within this phylum, a powerful animal model for measuring the effects of pollutant is the ubiquitous free-living nematode Caenorhabditis elegans. It measures 250 µm at hatching and up to 1.4 mm at adult stage. It has several advantages for ecotoxicological assessment such as (i) a short life cycle (C. elegans breeds in three days after hatching at 20 • C), (ii) a short life span (21 days at 20 • C), and (iii) a high fecundity [START_REF] Byerly | The life cycle of the nematode Caenorhabditis elegans : I. Wild-type growth and reproduction[END_REF]. C. elegans allows the assessment of life-history traits such as length, reproduction, or survival at various time steps during an experimentation. Therefore, this nematode is widely used to produce data for the assessment of chemical effects on the environment [START_REF] Boyd | A high-throughput method for assessing chemical toxicity using a Caenorhabditis elegans reproduction assay[END_REF]Dutilleul et al., 2013;Goussen et al., 2013;[START_REF] Smith | A Discrete Time Model for the Analysis of Medium-Throughput C. elegans Growth Data[END_REF][START_REF] Swain | Linking toxicant physiological mode of action with induced gene expression changes in Caenorhabditis elegans[END_REF].

The assessment of (eco)toxicological data through models based on the Dynamic Energy Budget (DEB) theory (Billoir et al., 2008b;Kooijman and Bedaux, 1996), known as DEBtox models is relevant to assess the effects of toxic compounds on organisms. Indeed, this method allows the simultaneous modelling of multiple endpoints (growth and reproduction) and the parameters estimated are independent of the exposure time (Billoir et al., 2008b;[START_REF] Jager | Simultaneous Modeling of Multiple End Points in Life-Cycle Toxicity Tests[END_REF]. Such models can be used to assess physiological modifications (e.g. changes in growth rate, maximal length, reproduction rate, cumulated reproduction) (Augustine et al., 2012a;Billoir et al., 2008a;[START_REF] Massarin | Biology-Based Modeling To Analyze Uranium Toxicity Data on Daphnia magna in a Multigeneration Study[END_REF], or differential life strategy responses [START_REF] Álvarez | Responses to stress of Caenorhabditis elegans populations with different reproductive strategies[END_REF] of an organism exposed to a pollutant. DEB modelling can also be adapted to other types of analysis. Thereby, individual based model (IBM) based on DEB modelling can permit to analyse population level data, especially multi-generation studies [START_REF] Beaudouin | Individual-based model of Chironomus riparius population dynamics over several generations to explore adaptation following exposure to uraniumspiked sediments[END_REF][START_REF] Martin | Extrapolating ecotoxicological effects from individuals to populations: a generic approach based on Dynamic Energy Budget theory and individualbased modeling[END_REF].

Uranium, a pollutant of growing interest (OECD, International Atomic Energy Agency (IAEA), 2012), is a radioactive heavy metal that can be found naturally in the environment. It is both a chemical and a radiological toxic but its chemical toxicity is considered to be dominant [START_REF] Sheppard | Derivation of ecotoxicity thresholds for uranium[END_REF][START_REF] Zeman | Effects of waterborne uranium on survival, growth, reproduction and physiological processes of the freshwater cladoceran Daphnia magna[END_REF].

We recently produced multi-generation data for nematodes exposed to uranium (Goussen et al., 2013). These data will be analysed with a DEBtox model for nematodes. Prior to this, we had to propose a relevant and robust DEBtox modelling approach for nematodes. This is the aim of this paper. Jager et al. (2005) presented a DEB model applied to the nematode. This model has been adapted to the assessment of ecotoxicological data (resulting in a so called DEBtox model) and used several times (Alda Ál-varez et al., 2006, 2005;[START_REF] Swain | Linking toxicant physiological mode of action with induced gene expression changes in Caenorhabditis elegans[END_REF]. In the present study, we proposed an augmented version of the DEBtox equations for nematodes by Jager et al. (2005) based on biological characteristics that were not covered. We evaluated both nematode DEBtox models to analyse data on the effects of the uranium on the free living nematode Caenorhabditis elegans. We also assessed the output robustness and sensitivity to parameters.

Materials and Methods

Experimental data were extracted from (Goussen et al., 2013). Extended description is presented in supplementary information.

Model description

The Dynamic Energy Budget (DEB) theory (Kooijman, 2010) is based on a mathematical description of the uptake and use of energy within an organism. The general DEB framework generally predicts that individuals' growth follows a von Bertalanffy growth curve provided feeding is ad libitum or at constant density [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF] and the growth is isomorphic (i.e. the organism shape do not change with growth). However as demonstrated by [START_REF] Byerly | The life cycle of the nematode Caenorhabditis elegans : I. Wild-type growth and reproduction[END_REF], C. elegans growth curve deviates from the expected von Bertalanffy curve. According to [START_REF] Knight | A novel mode of ecdysozoan growth in Caenorhabditis elegans[END_REF], the C. elegans buccal cavity grows in a stepwise manner at each moult whereas the rest of the body growth is linear between the moults. [START_REF] Knight | A novel mode of ecdysozoan growth in Caenorhabditis elegans[END_REF] assumed that this process induces a food limitation thus resulting in an initial growth slower than the later growth. Jager et al. (2005) dealt with this particularity assuming the C. elegans ingestion rate follows a size-dependent feeding limitation function and incorporated this in their nematode DEB model. These authors also assumed an immediate stop of the reproduction when the sperm runs out. Nevertheless, experimental data do not show an immediate stop of reproduction, but rather a more smooth transition (see for example Byerly et al. (1976) Fig. 6 and8 and[START_REF] Muschiol | Life cycle and population growth rate of Caenorhabditis elegans studied by a new method[END_REF] Fig. 2). We here propose two modifications of this set of equations in order to take into account some known biological specificities of the nematode. (i) The C. elegans male gamete limitation induces a progressive stop of the egg laying [START_REF] Maupas | Modes et formes de reproduction des nématodes[END_REF][START_REF] Ward | Fertilization and sperm competition in the nematode Caenorhabditis elegans[END_REF]. (ii) We also modified the size-dependent feeding limitation function by adding a new parameter which allows the nematode to access food whatever its length. Indeed, the approach from Jager et al. (2005) assumed that a small nematode would never be able to access enough food to growth if food is just sub-optimal (see section results in this paper). With our approach, we assumed that this nematode will be able to access to a sufficient small quantity of food, such as small bacteria (bacteria is the main food source of C. elegans [START_REF] Abada | C. elegans behavior of preference choice on bacterial food[END_REF], to initiate growth. The set of equations in our DEBtox model for nematodes is thus:

s f (l) = α 1 -1 + l 3 f l 3 -1 (1a) dl dt = rB [(1 -s f ) f -l] (1b) dR dt = RM 1 -l 3 p g + l g + (1 -s f ) f (1 -s f ) f l 2 -l 3 p RMax -R RMax (1c)
with s f the size-dependent ingestion limitation function, (1α) (-) the proportion of food available whatever the length, r B (h -1 ) the von Bertalanffy growth rate, f (-) the actual ingestion rate divided by the maximal ingestion rate for a body size, R (#) the cumulated reproduction, R Max (#) the maximal cumulated reproduction, R M (# h -1 ) the maximum reproduction rate, and g (-) the investment ratio. L (µm) represents the body length at time t, L 0 (µm) the body length at birth (i.e. the start of feeding in the DEB framework), L f (µm) the body length at which the ingestion rate is half the maximum ingestion rate, and L p (µm) the body length at puberty. All these length data are scaled by the maximal length L inf (µm) resulting in scaled parameters l, l 0 , l f , and l p (-).

The internal concentration of the pollutant scaled by the ratio between the uptake rate and the elimination rate (which can be regarded as the bioconcentration factor; [START_REF] Péry | Analysis of bioassays with time-varying concentrations[END_REF], c q , has been calculated as described by Kooijman and Bedaux (1996). These authors described a one-compartment toxicokinetic model with growth dilution, chemical exchanges depending on a surface area:volume ratio, and an elimination rate depending on food availability. This equation has been modified with the s f function resulting in the following equation:

dcq dt = C ke (1 -s f ) f l -cq ke (1 -s f ) f l + d ln l 3 dt (2)
where k e (h -1 ) is the elimination rate and C (mM) the real exposure concentration.

The toxic stress function, s, was calculated as s(c q ) = max [0, b (c q -N EC)] with b (mM -1 ) the slope of the effect and N EC (mM) the no-effect concentration.

Combining Equations 1 with the DEBtox equations rederived from the original ones by Billoir et al. (2008b), we obtained the set of equations presented in Table 1.

According to a previous study, the uranium negatively affects both the growth and the reproduction of C. elegans at comparable exposure concentrations (Goussen et al., 2013). As a consequence, the modes of action that may apply to uranium on nematodes are the growth, assimilation, or maintenance models.

Estimation of the parameters

Parameters estimations were performed using the statistical computing software R 2.15 (R Core Team, 2012) and the JAGS 3.3.0 (Just Another Gibbs Sampler) software, which is a program for the statistical analysis of Bayesian hierarchical models by Markov Chain Monte Carlo (MCMC) [START_REF] Plummer | JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling[END_REF]. The initial parameters' distributions were chosen according to the literature (Jager et al., 2005) (Table 2). We chose positive truncated normal distribution for biological parameters and we applied a coefficient of variation of 20% to these values. Parameters for which less information was available were distributed according to a non-informative uniform distribution. The different modes of action models were compared using the Deviance Information Criterion (DIC) which is suitable for Bayesian calibration comparison [START_REF] Spiegelhalter | Bayesian measures of model complexity and fit[END_REF]. The DIC is a generalisation of the Akaike's Information Criterion (AIC). As for the AIC, the lower DIC points out the model that will make the best short-term predictions and a DIC difference of less than five indicates that the models show similar fits [START_REF] Spiegelhalter | WinBUGS User Manual -Version 1.4[END_REF]. The DIC is a penalized deviance statistics as it includes penalty as a function of the effective number of parameters of the model [START_REF] Spiegelhalter | Bayesian measures of model complexity and fit[END_REF].

Sensitivity and uncertainty analysis

An uncertainty analysis was performed on the model parameters. Parameters' distributions were the same as for the estimation of the parameters apart for the use of triangle distributions instead of normal distributions. The triangle distribution permits a larger exploration of the space of parameters values, because extreme values have higher probability, resulting in a more robust model analysis. The experimental design was built using a Monte Carlo approach. The variances of the model output were estimated by Monte Carlo integrations. Two independent input sample matrices (n × p) were used to compute the Monte Carlo integration. The sample size (n) was of 30,000 and the number of parameters (p) is indicated in the Table S2 for each model.

We also performed a sensitivity analysis on the parameters using a variance-based Sobol method (scheme by [START_REF] Saltelli | Making best use of model evaluations to compute sensitivity indices[END_REF]). The sensitivity analysis allows the identification of the main input contributors to the variation of the model outcome. Thus the sensitivity analysis indicates the parameters on which a particular emphasis must be paid. The Sobol's method is a global and model independent sensitivity analysis method that quantifies the amount of variance that each parameter contributes to the unconditional variance of the model output. The parameters' values, the distribution, and the experimental design were the same as for the uncertainty analysis. We then calculated the first order (Si; variance contribution of one parameter to the total model variance) and total order (STi; result of the main effect of one parameter and

dl dt = rB (1 -s f ) f + g g + [(1 -s f ) f (1 -s (cq))] [(1 -s f ) f (1 -s (sq)) -l] (3a) dR dt = RM 1 -l 3 p g + l g + (1 -s f ) f (1 -s (cq)) (1 -s f ) f (1 -s (cq)) l 2 -l 3 p RMax -R RMax (3b)
Maintenance model:

dl dt = rB [(1 -s f ) f -(1 + s (cq)) l] (4a) dR dt = RM 1 -l 3 p (1 + s (cq)) (1 -s f ) f l 2 g (1 + s (cq)) -1 + l g + (1 -s f ) f -l 3 p RMax -R RMax (4b)
Growth model:

dl dt = rB (1 -s f ) f + g (1 -s f ) f + g (1 + s (cq)) [(1 -s f ) f -l] (5a) dR dt = RM 1 -l 3 p (1 -s f ) f l 2 g (1 + s (cq)) + l g (1 + s (cq)) + (1 -s f ) f -l 3 p RMax -R RMax (5b)
Hazard model:

dl dt = rB [(1 -s f ) f -l] (6a) dR dt = RM 1 -l 3 p (1 -s f ) f l 2 g + l g + (1 -s f ) f -l 3 p exp -s(cq ) RMax -R RMax (6b)
Costs model:

dl dt = rB [(1 -s f ) f -l] (7a) dR dt = RM 1 -l 3 p (1 -s f ) f l 2 g + l g + (1 -s f ) f -l 3 p (1 + s (cq)) -1 RMax -R RMax (7b)
of all its interactions with the other parameters) Sobol index. Calculations were performed thanks to the function "sobol2002" from the sensitivity R-package [START_REF] Pujol | sensitivity: Sensitivity Analysis[END_REF] with the statistical computing software R 2.15 (R Core Team, 2012).

Results

Control data fit accuracy

Our model (Eq. 1) provided a relevant fit of both the growth and the reproduction data for the control conditions (Fig 1A and B). The parameters we obtained by fitting the control data set are presented in Table 2.

Both our model and the model proposed by Jager et al. (2005) present a similar fit accuracy (∆DIC < 5, see Table S2).

Uncertainty analysis performed on the model proposed by Jager et al. (2005) showed that this model does not allow growth of hatchlings at suboptimal food availabilities. Indeed, more than 50% of the simulated cases gave a non growing and therefore a non reproductive nematode (Fig. S1A andS2A). In contrast, our model allows the hatchlings nematodes to grow at much lower food levels (Eq. 1) (Fig. S1B andS2B). Model exploration also showed that the model we used is robust for the parameters estimations. Indeed, as shown by the Figures S1C andD, our model allows to test a larger range of values leading to a significant growth, while performing the parameters estimations, compared to the initial model proposed by Jager et al. (2005). Sensitivity analysis (Fig. 2 andS3) performed on our assimilation model showed that the main impact contributors to the growth model (Eq. 3a) output variations are parameters α (-) and the exposure concentration, followed by parameters b (mM -1 ), f (-), L f (µm), and N EC (mM). Although L f (µm) has a non negligible STi, its Si remains negligible. Parameters R Max (#), L p (µm), R M (# h -1 ), and g (-) did not influence the growth model output. Regarding the reproduction model (Eq. 3b), parameters f (-) and the exposure concentration are the main contributors. These parameters are followed by parameters b (mM -1 ), α (-), L inf (µm), L p (µm), and N EC (mM). Parameters L 0 (µm), R M (# h -1 ), R Max (#), and g (-) did not influence the reproduction model output.

Mode of action and sensitivity analysis

Parameter estimations

Parameter estimations for the assimilation mode of action (Eq. 3) are presented in Table 2 and Figure S4. The distributions of the parameters estimated with the control data only were close to the distributions of the parameters estimated with all the data. Indeed, the means of the parameters estimates were within the confidence interval of each other. According to these estimations, the individuals were mostly fed ad libitum (f = 0.97). The estimated no-effect concentration (N EC (mM)) was of 0.42 ± 0.06 mM and the elimination rate k e (h -1 ) was of 0.16 ± 0.11 h -1 indicating that under continuous exposure, internal concentration steady state should be reached within one day.

The slope of the effect (b (mM -1 )) was estimated at 0.31 ± 0.03. The slope of the effect (b (mM -1 )) and the no-effect concentration (N EC (mM)) were slightly correlated (correlation coefficient of 0.17). The newly introduced parameter α (-) was estimated at 0.88±0.07 for the control data set and at 0.86 ± 0.06 for all the data.

As shown in Figure 3, uranium affects both growth (especially the individuals' maximum length) and reproduction (both the onset of the reproduction and the maximum cumulated reproduction).

Discussion

In this study, the effect of uranium on growth and reproduction of the nematode C. elegans has been assessed using a bioenergetic approach.

We have shown that our model presents a relevant fit for C. elegans growth and reproduction. The model we used presents a similar fit quality to the model proposed by Jager et al. (2005) (∆DIC < 5). Nevertheless, our model has been shown to be more robust and to better account for the end of the reproductive period. Our model was more flexible, allowed a larger variation around the optimal values, and therefore, could handle a larger range of biological possibilities. Indeed, as shown in Figure S1C, the previous model did not allow growth for organisms slightly limited (f ≤ 0.9) around the L f (µm) optimal values presented by Jager et al. (2005) (≈ 444 µm). Nevertheless, as demonstrated by [START_REF] Palgunow | Dietary restriction during development enlarges intestinal and hypodermal lipid droplets in Caenorhabditis elegans[END_REF], C. elegans can handle a large variation of feeding conditions. These authors submitted wild-type N2 nematodes to a so-called "developmental-Dietary Restriction" (dDR) experiment (organisms fed by 250 µL E. coli, OD 600 ranging from 0.3 to 6, dDR starting from egg stage) and did not observed any arrest of growth (the authors observed a reduction of 74% of body length at adult stage for the most stringent dDR condition). According to Figure S1D, the new formulation allowed organisms growth for a larger range of dietary restriction. The new formulation of the reproduction is a descriptive model that also takes into account (i) the fact that sperm cells are limited since they are produced before oocytes during the early stage of development of C. elegans (mainly during the beginning of stage L4) [START_REF] Hubbard | Introduction to the germ line (september 1[END_REF][START_REF] Lints | Reproductive system, germ line[END_REF] and (ii) the slow decrease of the laying at the end of the reproductive period [START_REF] Byerly | The life cycle of the nematode Caenorhabditis elegans : I. Wild-type growth and reproduction[END_REF][START_REF] Muschiol | Life cycle and population growth rate of Caenorhabditis elegans studied by a new method[END_REF]. A mechanistic understanding of this new formulation of the reproduction may be as follows: if it is assumed that an egg-laying involves an ejaculate of several male gamete, this may lead to a decrease of the density of sperm as the male gamete reserve decreases. Further experiments would be required to confirm this hypothesis. It should be necessary to be able to measure sperm density.

Our results show that uranium affects both the reproduction and the growth of C. elegans. Our modelling results suggest that the mode of action of uranium is an effect on the assimilation of energy from food. Indeed, the Deviance Information Criterion we obtained for the assimilation model was much better than for all other modes of action. This result is consistent with other experimental and modelling studies which tackled the mode of action of uranium. Augustine et al. (2012a) showed that the mode of action of uranium on Danio rerio could either be assimilation or maintenance. A complementary study reveals that uranium induced an increase of damage in gut cell mitochondria, a loss of the gut wall architecture, the presence of large necrotic zones, and an overall decrease in gut bacteria (Augustine et al., 2012b). As these results reveal negative impacts on organs linked with the assimilation of food, they are in accordance with effects on the assimilation of energy from food. Similarly, DEBtox analysis of the uranium toxicity on Daphnia magna exposed during three generations showed that for the two first generations, the assimilation and the maintenance captured effects on growth and reproduction equally well, but for the last generation, the assimilation mode of action better captured the uranium effects [START_REF] Massarin | Biology-Based Modeling To Analyze Uranium Toxicity Data on Daphnia magna in a Multigeneration Study[END_REF]. The authors concluded in favour of the assimilation mode of action based on this last result and on histological evidences. These results were indeed supported by measurements demonstrating that uranium induced structural damages on intestinal epithelium [START_REF] Massarin | Biology-Based Modeling To Analyze Uranium Toxicity Data on Daphnia magna in a Multigeneration Study[END_REF][START_REF] Zeman | Toxicité d'un mélange binaire sur la daphnie Daphnia magna: Etude des effets biologiques de l'uranium et du sélénium seuls et en mélange[END_REF], a significant decrease of the ingestion rate expressed per daphnid [START_REF] Zeman | Effects of waterborne uranium on survival, growth, reproduction and physiological processes of the freshwater cladoceran Daphnia magna[END_REF], and a significant reduction in carbon ingestion and assimilation rates [START_REF] Massarin | Effects of chronic uranium exposure on life history and physiology of Daphnia magna over three successive generations[END_REF]. The same kind of histological damages to the digestive tract was also reported on the earthworm Eisenia fetida exposed to soil contaminated with uranium [START_REF] Giovanetti | Bioaccumulation and biological effects in the earthworm Eisenia fetida exposed to natural and depleted uranium[END_REF].

Our conclusions relative to the mode of action are based on "goodness of fit" and on information from the literature. Using "goodness of fit" only would have required complementary experiments. Indeed, the outcome of the comparison between different modes of action may be biased by experimental variability. Another important point is that, in DEBtox modelling, the level of effects is assumed to be linearly dependent on the internal concentration, once the NEC is exceeded. This is a strong assumption to account for the actual shape of the dose response with a minimum set of parameters to estimate. This may bias the comparison between different modes of action, if the actual dose response is far from linear. Thereby, previous studies assessing modes of action through DEBtox models proposed complementary measures to a standard test such as food limitation studies or histological observations [START_REF] Massarin | Biology-Based Modeling To Analyze Uranium Toxicity Data on Daphnia magna in a Multigeneration Study[END_REF].

Even though we obtained a biologically pertinent model, it did not fit very well the maximum egg laying under stress conditions. A better fit may be obtained using a double effect (both direct and indirect effect) on the C. elegans reproduction (Table S1). Thereby a hazard additional effect (Eq. S1) or a cost additional effect (Eq. S2) on the male gametes could be considered in addition with the assimilation model (Eq. 3).

Applied to our data, Eq. S1 gave a DIC of 4,398 and Eq. S2 a DIC of 4,399. Although these DIC are lower than the DIC we obtained in Table S2, to conclude with certainty, and retain this model, further experimental analysis are required to assess and validate this double effect. It should be interesting to assess the evolution of the number of male gamete produced under uranium exposure through, for example, the use of DAPI staining and counting spermatid nuclei located within the gonad as described by [START_REF] Cutter | Sperm-limited fecundity in nematodes: How many sperm are enough?[END_REF].

The sensitivity analysis we performed showed that the parameters related to food assimilation such as f (-) and α (-) greatly influence the model output. Indeed, according to the DEB framework, food assimilation is the main organism function for the acquisition of energy, and therefore, this function drives all other organism functions. Thereby we believe that the food distribution must be constant throughout the replicates and conditions in toxicity test experiments. Moreover, the food linked parameters should be re-estimated during the data analysis. The impacts of the variations of exposure concentration on the model output were also assessed thanks to the sensitivity analysis. As this parameter is one of the most influential parameters on the output variability of the model, we recommend the careful analysis of the actual concentration present in the medium.

Conclusion

Our study proposed a robust and consistent DEB model for the nematode. Our results supported the hypothesis that uranium affects the assimilation of energy from food in many species. We also confirmed the importance of accurately assessing the actual exposure concentration of contaminant and of food while analysing (toxicity) data through modelling. We highlight some points of attention when using modelling for assessing (eco)toxicological data such as the importance of supporting the modelling results by comple-mentary studies. The model developed and evaluated in the present study appears suitable to be used in the analysis of multi-generational datasets such as the one we recently produced (Goussen et al., 2013) in order to assess the evolution of individuals exposed to long term stress.

Acknowledgement

We are especially grateful to Florian Parisot for experimental help, Catherine Lecomte for discussions and suggestion on this project, Audrey Sternalski for discussion and punctual help, Virginie Camilleri for technical assistance with the ICP-AES measurements, and Cleo Tebby for linguistic corrections and discussion. We also thank Henrique Teotónio for providing us with his base population and for comments. We also thanks two anonymous reviewers who helped us to greatly improve the manuscript. This work was part of the Envirhom-Eco research program supported by the French Institute for Radioprotection and Nuclear Safety (IRSN) and the 190 program supported by the French Ministry of Ecology.

A Supplementary material A.1 Test organism

The C. elegans population used in this study is the EEV-A 0 population created by [START_REF] Teotónio | Evolution of outcrossing in experimental populations of Caenorhabditis elegans[END_REF] from a mixture of 16 wild isolates. This androdioecious population was derived through a funnel cross strategy. Briefly, two-isolate hybrids were obtained by crossing, in a pair-wise fashion, each of the wild isolates. Then, four-isolate hybrids were obtained by inter crossing in a pair-wise fashion the two-isolate hybrids. Hybridizations continued until the 16-isolates hybrids were created. The population was then maintained over 140 generations and no significant loss of genetic diversity was observed by the authors after the recombinationselection equilibrium was mostly reached.

Prior to exposure, nematode populations were maintained at 20 • C, 80% RH in 9 cm Petri dishes filled with modified nematode growth medium (NGM; use of HEPES buffer instead of potassium phosphate buffer (Dutilleul et al., 2013;Goussen et al., 2013)) seeded with Escherichia coli strain OP50 [START_REF] Brenner | Genetics of Caenorhabditis elegans[END_REF][START_REF] Stiernagle | Maintenance of C. elegans[END_REF]. E. coli OP50 cultures were grown overnight in L-Broth rich medium at 37 • C. Then cultures (OD 600 = 3) were washed twice with a 5 g L -1 NaCl solution in order to remove LB medium since it contains phosphate. Petri dishes were seeded with 1 mL of a 20:1 mixture and left overnight to allow the bacterial culture to dry. Petri dishes were then exposed to UV doses (Bio-Link Crosslinker, λ = 254 nm; intensity = 200 µwatt m -2 ) for 15 minutes to stop bacterial growth and to avoid uncontrolled heterogeneity in food availability between Petri dishes. To ensure nematodes were fed ad libitum they were transferred into new dishes every three days. To do so, nematodes were washed off the Petri dishes with a M9-modified solution (use of HEPES buffer instead of potassium phosphate buffer). Nematodes picked up were pooled in a 15 mL falcon tube and the number of individuals in a tube was estimated based on three sample drops of 5 µL [START_REF] Teotónio | Evolution of outcrossing in experimental populations of Caenorhabditis elegans[END_REF]. The volume corresponding to 500 individuals was then transferred into new Petri dishes.

A.2 Experimental conditions

Experimental data were extracted from a previous study (Goussen et al., 2013). Briefly, individuals from the maintained population were exposed to a range of seven nominal concentrations of uranium: 0 (control), 0.1, 0.3, 0.5, 0.9, 1.1, and 1.2 mM U. For this purpose, 12-well tissue-plates were used. Plates were filled with modified NGM. The uranium stock solution was obtained by dilution of uranyl nitrate (UO 2 (NO 3 ) 2 ,6H 2 O, Sigma-Aldrich, France). Uranium solution was added to the modified NGM just before flowing the plates. To measure real exposure concentration, 100 µL of NGM samples were collected for each treatment and stored at 4 • C. The samples were digested with a combina-tion of 1 mL HNO 3 and 1 mL H 2 O 2 at 90 • C prior to measurement with ICP-AES (Optima 4300 DV, Perkin-Elmer, USA; detection limit 0.04 µM). Around 60 gravid hermaphrodites were randomly picked up from the maintained population dishes, placed on new plates, and allowed to lay for 90 minutes. Eggs were then pooled and randomly deposited in the 12-well tissue-plates. At least 12 wells were used for each treatment (one egg per well). Survival, growth, and egg laying were monitored individually for six days. Survival was measured by stimulating each worm with a platinum transfer pick. A nematode was scored as dead if no head or body movement was triggered by three repeated stimulations [START_REF] Sutphin | Measuring Caenorhabditis elegans Life Span on Solid Media[END_REF][START_REF] Swain | elegans metallothioneins: New insights into the phenotypic effects of cadmium toxicosis[END_REF]. Nematodes were photographed daily using a stereomicroscope (ZEISS SteREO Discovery V20, ×240 and ×160 magnification respectively for juveniles and adults) coupled with a computerconnected camera (Nikon D5000). Body length was measured using the ImageJ software [START_REF] Rasband | ImageJ[END_REF] and a micrometer scale measure. Egg laying was recorded by visual scoring. The few worms that were lost by crawling off the plates or desiccating on the sides of the plates were removed from the data.

A.3 Model description

According to the DEB theory, energy is taken up from food, assimilated, and stored into reserves. This energy is then dispatched between three main processes: (i) maintenance, (ii) growth, and (iii) reproduction. Models based on the DEB theory allowing the analyse of ecotoxicological data (i.e. DEBtox models) have been built by Kooijman and Bedaux (Kooijman and Bedaux, 1996a,b,c) and corrected by Billoir et al. (2008). DEBtox models describe the perturbation of energy management when an organism is exposed to a pollutant. This kind of model assumes that the effects appear when the internal concentration exceeds a threshold called the no-effect concentration (NEC). The DEBtox models describe five modes of action of chemicals on physiological processes. Two of these have a direct effect on the reproduction and the three others have indirect effects on the reproduction in conjunction with effects on growth. The two first are the cost model (overhead costs for the creation of eggs) and the hazard model (mortality during oogenesis), and the three others are the assimilation model (decrease of the assimilation of energy from food), the maintenance model (increase of maintenance costs), and the growth model (overhead costs for the creation of growth units).

A.4 Estimation of the parameters

For each effect model, we performed an adaptive phase of 10,000 iterations. We then performed another 90,000 iterations. These iterations were performed on three independent MCMC chains. Chains' convergences were assessed using the [START_REF] Gelman | Inference from Iterative Simulation Using Multiple Sequences[END_REF] convergence statistic modified by [START_REF] Brooks | General Methods for Monitoring Convergence of Iterative Simulations[END_REF]. For the few estimations for which the convergence was not fully reached during this first run, we performed an adaptive phase of 15,000 iterations followed by 135,000 iterations on five independent MCMC chains. Model fit quality was checked by plotting the fits for both the growth and the reproduction data. This assessment was performed using the mean and confidence interval of the 1,000 calculations corresponding to the 1,000 last iterations of the estimation process.

Table S1: C. elegans DEBtox reproduction equations with two effects

Hazard additional effect:

dR dt = RM 1 -l 3 p g + l g + (1 -s f ) f (1 -s (cq)) (1 -s f ) f (1 -s (cq)) l 2 -l 3 p RMax exp -s(cq) -R RMax exp -s(cq ) (S1)
Cost additional effect:

dR dt = RM 1 -l 3 p g + l g + (1 -s f ) f (1 -s (cq)) (1 -s f ) f (1 -s (cq)) l 2 -l 3 p RMax (1 + s (cq)) -1 -R RMax (1 + s (cq)) -1 (

Figure 1 :

 1 Figure 1: C. elegans length (A) and cumulated reproduction (B) in relation to age for the control conditions. Age is indicated as hour since hatching. Observed data are represented by points, the model fits (Eq. 1) are represented by the regression line. Dashed lines represent 95th percentiles of the posterior distribution. Dotted lines represent the model fitted with the equations of Jager et al. (2005).

BFigure 2 :

 2 Figure 2: Sensitivity analysis for the assimilation model (Eq. 3). Mean of the sensitivity indices at the different times for the growth equation (A) and reproduction equation (B). Total order indices (STi) are presented in dark grey and first order indices (Si) in light grey. Parameters were ordered according to the total order Sobol sensitivity indices.

FFigure 3 :

 3 Figure 3: Growth and reproduction results for the assimilation model (Eq 3). Panels A and B represent the results for a concentration below the no-effect concentration (N EC (mM)) for respectively the growth and the reproduction. Panels C, D (two lowest), E, and F (two highest) represent the results for the concentrations over the non effect concentration (N EC (mM)) for the growth and the reproduction. Model fits are represented by regression lines. Data are represented by symbols. Symbols have been slightly shifted in time to facilitate graphics' reading.

Figure S2 :Figure S3 :Figure S4 :

 S2S3S4 Figure S2: Uncertainty analysis for reproduction. Panel A represent the uncertainty analysis of the model proposed by Jager et al. (2005) and panel B represent the uncertainty analysis of the model we used (Eq. 1). The solid line represent the median of the model output. Dashed lines represent the 95% quantiles of the model output.

Table 1 :

 1 C. elegans DEBtox equations

	Assimilation model:

  by over 32 points than all other modes of action DIC. In the same way as for the control model, theJager et al. (2005) model and our model have a non distinguishable quality of fit (∆DIC < 5) for the assimilation model.

Table

S2

presents the DIC comparison between the model by

Jager et al. (2005)

and our model for all the modes of action. The best mode of action is the assimilation model. Indeed, the DIC of the assimilation model is lower

Table 2 :

 2 Prior and posterior (mean±SD) distributions of the model parameters using the assimilation model (Eq. 3)Prior and posterior (mean±SD) distributions of the model parameters using the assimilation model (Eq. 3) with the control data only or with all data (Full data set). The normal distribution of mean µ and standard deviation σ is noted

	Parameter	Prior	Control data set	Full data set
			(mean±SD)	(mean±SD)
	α (-)	U [0, 1]	0.88 ± 0.07	0.86 ± 0.06
	f (-)	U [0, 1]	0.97 ± 0.02	0.97 ± 0.03
	L 0 (µm)	N (282, 56) I ]0, +∞]	219 ± 43.53	165 ± 37.37
	L f (µm) L p (µm)	N (444, 89) I ]0, +∞] N (1020, 204) I ]0, +∞]	418 ± 50.96 764 ± 87.18	364 ± 47.35 818 ± 57.64
	L inf (µm)	N (1540, 308) I ]0, +∞]	1414 ± 46.32	1383 ± 41.14
	r B (h -1 ) g (-)	N (0.039, 7.8 10 -3 ) I ]0, +∞] U [0, 20]	0.034 ± 0.005 10 ± 5.48	0.031 ± 0.003 12 ± 4.88
	R M (# h -1 )	N (12.67, 2.53) I ]0, +∞]	15.8 ± 2.02	15.1 ± 1.90
	R Max (#)	N (265, 53) I ]0, +∞]	312 ± 27.42	249 ± 22.13
	k e (h -1 )	U [0, 1]	-	0.16 ± 0.11
	b (mM -1 )	U [0, 5]	-	0.31 ± 0.03
	N EC (mM)	U [0, 2]	-	0.42 ± 0.06

N (µ, σ), and the uniform distribution between a and b is noted U [a, b]. I ]c, d] denotes an interval censoring between c (excluded) and d

Table S2 :

 S2 Comparison of the fit quality between modelsComparison between the model fromJager et al. (2005) and our model for all the modes of action. The comparison is based on the log-likelihood and the Deviance Information Criterion (DIC)

		Mode of action	Number of parameters	Log-likelihood	DIC
	Jager et al. (2005)	Control	9	-195.5	399
		Assimilation	12	-2,198.5	4,408
		Maintenance	12	-2,227.5	4,485
		Growth	12	-2,216.5	4,451
	Present study	Control	10	-196.5	401
		Assimilation	13	-2,197.0	4,405
		Maintenance	13	-2,244.5	4,687
		Growth	13	-2,213.0	4,437

A.5 Supplementary equations

A.6 Supplementary Figures


Marco Musiani

Mark E Orazem

Nadine Pébère
email: nadine.pebere@ensiacet.fr

Bernard Tribollet

Vincent Vivier

Determination of resistivity profiles in anti-corrosion coatings from constant-phase-element parameters

Keywords: CPE, electrolyte uptake, impedance modeling, pores

Introduction

Electrochemical impedance spectroscopy is a popular technique for the assessment of the corrosion protection performance of organic coatings. The analysis of impedance data is often based on the equivalent circuit proposed by Beaunier et al. [START_REF] Beaunier | Etude électrochimique et par microscopie électronique a balayage du fer recouvert de peinture[END_REF] in which the coating capacity and the double layer capacity are replaced by constant phase elements (CPE), the impedance of which is Z = 1/(j)  Q. Substitution of CPEs for capacities may strongly improve the quality of the fitting between the experimental data and the equivalent circuit, but creates ambiguities in the physical interpretation of the results, since Q cannot be simply identified with the coating capacity [START_REF] Orazem | Dielectric Properties of Materials Showing Constant-Phase-Element (CPE) Impedance Response[END_REF] and cannot be directly used to calculate the water uptake in the coating through the Brasher and Kingsbury [START_REF] Brasher | Electrical measurements in the study of immersed paint coatings on metal. I. Comparison between capacitance and gravimetric methods of estimating water-uptake[END_REF] approach. Various formulas have been proposed for the calculation of the effective capacity from CPE parameters [START_REF] Brug | The analysis of electrode impedances complicated by the presence of a constant phase element[END_REF][START_REF] Hsu | Concerning the conversion of the constant phase element parameter Y-0 into a capacitance[END_REF], but their application requires knowledge of the physical origin of the CPE behavior [START_REF] Orazem | Dielectric Properties of Materials Showing Constant-Phase-Element (CPE) Impedance Response[END_REF][START_REF] Hirschorn | Determination of effective capacitance and film thickness from constant-phase-element parameters[END_REF].

In a recent paper [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], our group has shown that the CPE behavior observed in the impedance of a coated system may be explained by an inhomogeneous electrolyte uptake if (i) the electrolyte volume fraction decreases from the coating/electrolyte interface to the metal/coating interface according to a power-law and

(ii) the resistivity and permittivity of the system are calculated using effective medium theory formulas corresponding to a parallel combination of coating material and electrolyte. This model, henceforth briefly called "power-law model" was compared to experimental data obtained with a hybrid sol-gel coating exposed to a 0.5 M NaCl solution. It was concluded that, as a result of the progressive electrolyte uptake for increasing exposure time, the resistivity of the coating markedly decreased (after one week, the decrease at the coating/electrolyte interface attained ca. 6 orders of magnitude, with respect to the dry coating), while its permittivity remained practically unchanged. Furthermore, it was shown that the number and/or section of the through pores, i.e. pores as deep as the whole coating thickness, directly connecting the electrolyte with the metal underneath the coating through a low-resistance path, increased with exposure time. These results allowed the extension of a previously derived model [START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 1. Theory[END_REF][START_REF] Musiani | Constant-Phase-Element Behavior Caused by Coupled Resistivity and Permittivity Distributions in Films[END_REF] applied so far to passive oxides and human skin [START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 2. Applications[END_REF],

to anti-corrosion coatings.

The present paper aims at extending the investigation described in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], focusing mainly on three issues: (i) the effect of the assumptions made on coating and electrolyte resistivity on the regressed parameters; (ii) the analysis of the relative amounts of electrolyte within through and short pores; (iii) the analysis of experimental data relevant to other coating/electrolyte combinations, based on the power-law model, and a discussion of the different behavior of different systems.

Experimental

A description of the coatings, their method of preparation, and the electrochemical techniques used for their characterization are presented in this section.

Hybrid sol-gel coatings

Two coatings were prepared. For both systems, the organosiloxane sol was prepared by using 3glycidoxypropylethoxysilane (GLYEO), tetraethylorthosilicate (TEOS) and butanol. A first solution was made by adding a polyaminoamide (PAA) to the sol, with a GLYEO/TEOS/PAA mass ratio of 3:1:2. The solution was prepared with an excess of organosilane, therefore the resulting coating was called O. The process takes advantage of the two moieties of the organosilane molecule (GLYEO). The organic part of the GLYEO reacts with the amine functions of the PAA to develop the organic part of the hybrid coating, while the silane part undergoes the usual hydrolysis and condensation steps of sol-gel synthesis, to form a silicon-based network.

TEOS was added to increase the coating density. In the second solution, an epoxy compound was added to the first solution, with a GLYEO/TEOS/PAA/Epoxy ratio of 1:1:10:8. Due to the presence of an epoxy in the solution, the resulting coating was called sample E. The addition of the epoxy compound in the liquid sol-gel resin required a different procedure for mixing together the compounds. For this reason, sample E is not comparable, from a chemical point of view, with sample O. The sol-gel films did not contain any pigments or fillers.

The coatings were deposited onto a 2024 T3 aluminium alloy currently used in the aerospace industry. The specimens consisted of 125 mm × 80 mm × 1.6 mm plates machined from rolled plate. Before painting, the samples were degreased at 60°C (pH = 9) for 15 min, rinsed twice with distilled water, then etched in an acid bath at 52 °C for 10 min, and rinsed again with distilled water. The liquid paints were applied by air spraying and cured at 100°C for 1 h. The coatings thickness was measured at several locations from the observation of the cross-sections of the samples by SEM and found to have an average value of 20 ± 2 µm.

Electrochemical impedance measurements

A three-electrode electrochemical cell was used in which the coated specimen served as the working electrode.

A cylindrical Plexiglas tube was assembled on top of the coated sample, exposing a surface area of 24 cm². A large platinum sheet was used as counter-electrode. A saturated mercury/mercurous sulphate reference electrode was used for electrolytes containing Na 2 SO 4 , and a saturated calomel reference electrode was used in electrolytes containing NaCl. The electrochemical cell was open to air and was kept at room temperature with an average value of 17°C and which may have undergone fluctuations by ±2°C. Following our previous work, the resulting variations in the dielectric constant of water were unlikely to have significant effects on the experimental impedance response [START_REF] Musiani | Constant-Phase-Element Behavior Caused by Coupled Resistivity and Permittivity Distributions in Films[END_REF]. Electrochemical impedance measurements were carried out using a Biologic VSP apparatus. The impedance diagrams were obtained under potentiostatic conditions at the corrosion potential over a frequency range of 60 kHz to 10 mHz with 6 points per decade, using a 20 mV peakto-peak sinusoidal voltage. The electrochemical behaviour of the coatings was characterized in either 0.5 M Na 2 SO 4 or 0.5 M NaCl, for exposure times ranging from 2 h to 168 h. The Na 2 SO 4 solution was used to achieve a better knowledge of the intrinsic properties of the coatings because the evolution of the system was slower than in NaCl solution. The results concerning sample E in the NaCl solution were previously reported in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF].

Model for data analysis

The model employed for the data analysis has been previously described [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF]; only a brief summary is reported here. A scheme of a substrate/oxide/coating/electrolyte system, and its equivalent circuit, are presented in Fig. 1. The impedance is assumed to consist of three components connected in series:

i. the uncompensated electrolyte resistance;

ii. the impedance of the coating, discussed below;

iii. the impedance of the oxide covering the Al 2024 alloy; this impedance is accounted for by a constant phase element (CPE) and is no longer discussed in the following.

The coating is assumed to be penetrated by the electrolyte which fills its pores. These pores are represented in Fig. 1 as being straight, although they may be tortuous and have variable lateral dimension. All pores have a mouth at the coating/electrolyte interface, but only a few (henceforth called "through pores") are as deep as the coating and reach the coating/oxide interface, providing a low-resistivity path, with resistance R pore , between the electrolyte and the oxide. The other pores (henceforth called "short pores") are less deep than the coating thickness , and thus different planes parallel to the interfaces, located within the coating at different positions x, cross different numbers of pores (see sketch in Fig. 2), and different elemental layers

Mise en forme : Puces et numéros of the coating, d thick, have different local electrolyte volume fractions (). Such a representation of the electrolyte-penetrated coating is in agreement with the one proposed by Mišković-Stanković et al. [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF][START_REF] Mišković-Stanković | The sorption characteristics of epoxy coatings electrodeposited on steel during exposure to different corrosive agents[END_REF] on the basis of optical microscopy coupled with image analysis.

The electrolyte volume fraction affects the local resistivity and permittivity of the coating, which are expressed through effective medium theory formulas as

11 1 wc () () [ 1 () ]           (1)
and

wc () () [ 1 () ]         (2)
where the subscripts w and c refer to the electrolyte and the coating material, respectively. Equations ( 1) and

(2) take into account that, in each elemental layer parallel to the interfaces, each position is occupied by either the coating material or the electrolyte, and so these two media are in parallel to each other. In this approximation it is unimportant whether () is the result of the presence of a large number of smaller pores or a small number of larger pores. However, () does not take into account the water present within the coating as individual molecules interacting with polymeric network and not forming aggregates big enough to allow dissolution of salts and generate a conductive medium. Such polymer-bounded water has been reported to be a very significant fraction of the total water uptaken by epoxy coatings [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF] and to cause strong plasticization of an epoxy/polyamidoamine coating [START_REF] Le Pen | Characterisation of water-based coatings by electrochemical impedance spectroscopy[END_REF]. Therefore, the water uptake determined through the analysis of experimental data by comparison with the model proposed in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF] is likely to be underestimated. In a following section (see 4.1.3), it will be discussed how a lower resistivity of the electrolyte within the pores, as compared to that of the external solution, may be a further cause for underestimating the water uptake.

Under assumption that each elemental layer may be represented by a parallel combination of a resistance d and a capacitance   d, the impedance of the electrolyte-penetrated coating (Z c ) is calculated as

1 c 1 0 0 1 () d () j () Z           (3)
The contribution of pores that extend to the oxide surface is included separately from the development of equation ( 3). The overall impedance Z() of the coating is thereby given by the pore resistance R pore in parallel with Z c , i.e.

1 11 cp o r e () () ZZ R       (4)
To account for the progressive variation in the number of pores along the coating thickness, the hypothesis is made that () changes from (0) = 0 at  = 0 to () at  = 1 according to a power-law, i.e.,

) 1 )( ( 1 ) ( ) (              (5)
The coating part corresponding to the impedance c Z contains only "short pores", meaning that no electrolyte reaches the interface between the coating and the oxide and, as a consequence,   00   .

In reference [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], calculated impedance plots were presented with () and R pore as parameters. It was shown that, according to a more general theory reported in [START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 1. Theory[END_REF], Z() corresponded to a CPE behavior, with a CPE exponent  linked to the power-law exponent  by the equation

   1   (6)
It was shown in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF] that fitting the above described model to experimental data allowed determination of , (), and R pore , by assuming known or reasonable values for  w ,  c ,  w and  c . The  profiles were calculated by entering the fitted  and () values into equation ( 5); hence,  profiles corresponding to different immersion times were obtained through equation ( 1). These profiles, presented in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], neglected the through pores, and, therefore, at all immersion times, (0) was identical to  c . In the present paper, the electrolyte content in the through pores is calculated as pore pore w pore film film pore film

VA VAR A     (7)
where V denotes a volume, and A is a surface area (with R pore normalized to unit surface area and A film = 1 cm 2 ). This allows estimation of the fraction of the uptaken electrolyte that is contained in the through pores and calculation of  profiles, taking into account all pores. Owing to the assumed cylindrical geometry of the through pores, R pore is not a function of position .

The above development retains the meaning of (0) as the resistance of the dry coating,  c . As may be seen in the subsequent discussion of experimental results, the power-law model does not account adequately for the high-frequency response. The origin of this discrepancy is, as yet, unknown. A tentative explanation might invoke capacitive interactions between adjacent pores. The regression analysis was therefore performed for frequencies up to an upper limit which, for different systems, varied from 0.5 to 5 kHz (see Figures 3 and6).

4.

Results and Discussion

The results are presented in terms of the influence of assumed values for parameters  w ,  c ,  w and  c on the fitted parameters, the relative amounts of electrolyte within through and short pores, and the influence of coating formulation and electrolyte composition.

4.1.

Effect of  w ,  c ,  w and  c on fitted parameters Coated samples were prepared, and impedance measurements were performed as described in the experimental section. For each combination of coating/electrolytic solution/immersion time the experimental data were compared with the model shown in Fig. 1, using a non-commercial software developed at the LISE CNRS, Paris. This software allowed regression of the model consisting of a combination of passive circuit elements, like R pore , the electrolyte resistance and the CPE parameters relevant to the oxide film, and the analytical expression of Z c (), but did not provide confidence intervals for the best fitted parameters. In our previous work [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], it was explained that, owing to the quasi-capacitive blocking behavior observed at low frequency, the oxide film resistance could be assumed to be very large and therefore its inclusion in the equivalent circuit, in parallel with the CPE, was not necessary. Equally unnecessary were circuit elements accounting for the corrosion reactions, because no sign of corrosion was observed for the immersion times explored in the present investigation.

In the regression procedure, ,  and R pore were the only adjustable parameters for the coating; whereas,  w ,  c ,  w and  c were given fixed values ( w = 2.22 × 10 1  cm,  c = 2 × 10 11  cm,  w = 82 and  c = 8). In our previous paper [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF] the effect of the assumed values on the fitted parameters was not discussed. New data and a more detailed discussion are presented below.

Parameters  w and  c

The permittivities ( w and  c ) are known with rather good precision; furthermore, it has been shown that the  profiles are quite flat and therefore do not affect the impedance of the system in a significant way [START_REF] Musiani | Constant-Phase-Element Behavior Caused by Coupled Resistivity and Permittivity Distributions in Films[END_REF].

Parameter  c

The resistivity of the dry coating  c is less well-known. Thus, recognizing that it is important to assess the magnitude of its effect, the model was regressed to the same set of data by assuming in each run a different  c value, covering a rather wide range. Figure 3 shows, as an example, an experimental impedance plot, obtained

with sample E immersed in 0.5 M NaCl after a 48-hour immersion time, compared with 7 best fitted diagrams corresponding to  c values varying from 10 10  cm to 2 × 10 13  cm. In all cases, the agreement between experimental and calculated curves is good at all frequencies (with some differences at frequencies above 5 kHz where, as discussed above, the model is not fully satisfactory). Fig. 4 shows that the fitted parameters ,  and R pore do not depend on  c in a significant way, so long as  c > 5 × 10 10  cm. Many polymeric materials have a resistivity higher than this threshold value [START_REF] Blythe | Electrical Resistivity Measurements of Polymer Materials[END_REF], possibly by some orders of magnitude. For example, a resistivity of 1 × 10 15  cm has been reported for a dry nylon film [START_REF] Baird | Nature of DC conductivity in polyamides[END_REF]. All the results presented in the following sections were obtained by assuming  c = 2 × 10 11  cm, as in our previous study [7] and in other studies [START_REF] Hinderliter | Interpretation of EIS data from accelerated exposure of coated metals based on modeling of coating physical properties[END_REF].

Parameter  w

The most obvious assumption about  w , made by ourselves in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF] and in the present study and by other authors, e.g. [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF], is that it is identical to the resistivity of the electrolytic solution used in the impedance measurement. This implies that the composition of the electrolyte inside the pores should be the same as its bulk composition or, in other words that the water and the ions dissolved in it be incorporated into the coating without any selectivity. Results presented in the literature show that such an assumption might not be correct for all systems. For example, Castela and Simoes [START_REF] Castela | Assessment of water uptake in coil coatings by capacitance measurements[END_REF] mention that the diffusivity of ions is lower than that of water; Hu et al. [START_REF] Hu | Determination of water uptake and diffusion of Clion in epoxy primer on aluminum alloys in NaCl solution by electrochemical impedance spectroscopy[END_REF] report that the diffusivity of chloride ion in epoxy coatings is ca. 3 orders of magnitude lower than the diffusivity of water; and, according to Mišković-Stanković et al. [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF], the ratio between the diffusivities of water and chloride ion is slightly higher than 10. Thus, water uptake may precede ions uptake [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF][START_REF] Mišković-Stanković | The sorption characteristics of epoxy coatings electrodeposited on steel during exposure to different corrosive agents[END_REF] , and, so long as equilibrium is not yet reached, the solution in the pores is likely to be less concentrated than the outer solution. It is also quite possible that the composition of the electrolyte within the pores would be a function of immersion time. The chemical nature of the electrolyte, especially its anion, may have important effects on the coating behavior. Duarte et al. [START_REF] Duarte | Sensing polymer inhomogeneity in coated metals during the early stages of coating degradation[END_REF], using negative-feedback scanning electrochemical microscope, have observed various topographic changes in coatings due to the water uptake and the ingress of dissolved ions depending on the electrolyte composition.

It is noteworthy that the permselective uptake of a solution less concentrated than the bulk solution would have significant consequences on (i) the determination of the  values and (ii) the comparison of different electrolytes. Since  c is typically 8 to 10 orders of magnitude larger than  w , equation (1) reduces to

) ( ) ( 1 1         w (8)
already for very low  values. If an incorrectly low  w is assumed, the water uptake is underestimated by the same factor, because larger amounts of less concentrated electrolytic solution are required to achieve a fixed  value. The permselectivity effects may be more or less marked for different salts and, thus, may affect the estimated water uptake to a different extent. Mišković-Stanković et al. [START_REF] Mišković-Stanković | The sorption characteristics of epoxy coatings electrodeposited on steel during exposure to different corrosive agents[END_REF] reported that the water diffusivity in epoxy coatings did not depend on nature of the anion (chloride, sulfate, naphtol-3,6-disulphonate). Since these anions have themselves different diffusivities, different electrolyte concentrations should be found in the pores of coatings immersed in electrolytes of the same molarity, but with different anions. In turn, different electrolyte concentrations would cause different resistivity profiles and different impedance responses.

Electrolyte in short and through pores

The volume fraction of the electrolyte within the through pores, henceforth denoted pore , was determined for different coating/electrolyte combinations and various immersion times using equation ( 7) and the fitted R pore values. Table 1 reports, as an example, data relevant to coating E in 0.5 M NaCl. The same table reports also the fitted parameters  , and R pore and the volume fraction of the electrolyte in the short pores, averaged along the coating thickness , henceforth denoted  av . It must be emphasized that, if  w is underestimated, for the reasons discussed in the previous section, both  pore and  av will be underestimated by the same factor, and their ratio will be correctly assessed. Inspection of Table 1 shows that, for all immersion times,  av > pore , and therefore most of the electrolyte is contained in pores that do not reach the metal. However, the  pore / av ratio increases as the immersion time increases. This suggests that, for this system, as the immersion time is prolonged and the water uptake progresses, (i) the number and/or diameter of the pores increases and (ii) the pore depth increases, such that a number of short pores become through pores at longer immersion time. All these conclusions agree with those drawn by Mišković-Stanković et al. [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF] from the analysis of results of liquid sorption and thermogravimetric experiments.

It is worth pointing out that the calculation was made for cylindrical pores, perpendicular to the interfaces, which allows the volume ratio in equation ( 7) to be replaced by the surface area ratio. Clearly, real pores must be tortuous. Considering their tortuosity would affect  pore (because a higher tortuosity would correspond to a higher apparent  and therefore a higher  pore ), but not  av (because Eq. ( 1) and ( 2) hold for any pore geometry). Unless tortuosity values are much higher than a few units, the conclusion that  av > pore still holds.

Once  pore is known, resistivity profiles taking into account the electrolyte in all pores may be calculated according to equation [START_REF] Beaunier | Etude électrochimique et par microscopie électronique a balayage du fer recouvert de peinture[END_REF], in which  is replaced by ' defined as

) 1 )( ( 1 ) ( ) ( '                pore (9)
Fig. 5 compares these new  profiles (empty symbols) with those reported in Fig. 10 of reference [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF] (full symbols), which were calculated by directly using the fitted  and  values and neglecting the electrolyte in the through pores. Clearly, taking  pore into account affects only marginally, thus the lower resistivity limit  is essentially identical in this new calculation and in the previous one. Instead, for each immersion time, the resistivity at  = 0 calculated according to equations ( 1) and ( 9) and based on ' is markedly lower than that calculated according to equations ( 1) and ( 5) and based on .

Additional differences caused by using 'instead of are: (i) a more extended region where  is independent of , next to the coating/oxide interface, and (ii) a progressive decrease of (0) with increasing immersion time The presence of inner and outer layers in organic films was experimentally shown by Kittel et al. [START_REF] Kittel | New methods for the study of organic coatings by EIS: New insights into attached and free films[END_REF] who highlighted the heterogeneity of the properties of coatings along their thickness.

Inspection of Table 1 shows that the total water uptake (averaged along the coating thickness), given by  av +  pore , remains below 10 -4 , i.e. below 0.01 %, even after 168 h of immersion. Such a value is significantly lower than values often reported in the literature for various coatings materials (typically between 0.1 and 10%, see e.g. [START_REF] Mišković-Stanković | Electrolyte penetration through epoxy coatings electrodeposited on steel[END_REF][START_REF] Mišković-Stanković | The sorption characteristics of epoxy coatings electrodeposited on steel during exposure to different corrosive agents[END_REF][START_REF] Le Pen | Characterisation of water-based coatings by electrochemical impedance spectroscopy[END_REF][START_REF] Duarte | Sensing polymer inhomogeneity in coated metals during the early stages of coating degradation[END_REF][START_REF] Kittel | New methods for the study of organic coatings by EIS: New insights into attached and free films[END_REF][START_REF] Park | Monitoring of water uptake in organic coatings under cyclic wet-dry condition[END_REF][22]).

Direct and reliable gravimetric data for our samples are not available. In previous sections we discussed reasons why the water uptake data in table 1 may be significantly underestimated. However, it must be stressed that, although the results of the fitting procedure are reported in part in terms of water uptake, e.g.

, what really matters is the conductivity, given by equation ( 1), and, with very good approximation by equation [START_REF] Hirschorn | Constant-Phase-Element Behavior Caused by Resistivity Distributions in Films: 1. Theory[END_REF]. In the development of the power-law model (referred to electrolyte-penetrated coatings), was assumed to be distributed along the thickness according to a power law, and the corresponding resistivity and permittivity distributions were calculated accordingly. It must be realized that  is just an intermediate in the calculation. The model might be directly described in terms of  and , both assumed to undergo a power-law variation, over a large and a small (essentially negligible) range, respectively.

Regression would then directly provide  and  values, allowing the calculation of  profiles identical to those shown in the present paper. In other words, the determined  profiles, e.g. those in Fig. 5, are reliable, whereas the  may be or not reliable, depending on permselectivity effects and on the presence/absence of significant amounts of polymer-bounded water.

Analyses of the water uptake based on the evolution of the coating capacitance with immersion time often lead to its overestimation, by a small factor, as compared with gravimetric methods. Since the permittivity of water does not change dramatically with the concentration of ions dissolved in it, estimates based on capacitance are intrinsically less sensitive to selective uptake of water/ions than our approach relying on resistivity variations.

Effect of coating formulation and electrolyte composition

The impedance of coatings E and O was measured during immersion in 0.5 M Na 2 SO 4 , during 168 h, and analyzed using the model described in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], with the same approach as described above for coating E exposed to 0.5 M NaCl. Coating O, less protective than coating E, was not studied in NaCl solution because signs of corrosion appeared rather soon and, to account for the corrosion process, the proposed model would need to be completed by including either additional circuit elements or an appropriate analytical expression.

The agreement between experiments and model was good for all immersion times, although, as discussed before, divergences appeared at frequencies higher than 1 kHz (which were therefore not taken into account in the fitting), as shown in Fig. 6. Tables 2 and3 summarize the fitting results, which are compared to those for the coating E/NaCl system in Fig. 7. Clearly, the different coating/solution combinations show significantly different time dependencies of both R pore and . For coating E, upon increasing the immersion time, R pore decreases in NaCl solution, but increases in Na 2 SO 4 solution. For coating O in Na 2 SO 4 solution, R pore undergoes only small variations, initially decreasing and then slightly increasing. For coating E, upon increasing the immersion time,  increases rapidly in NaCl solution and remains roughly constant at a much lower value in Na 2 SO 4 solution. For coating O in Na 2 SO 4 solution,  is initially much higher than for coating E and then increases slowly when immersion time increases.

These results suggest that coating E is intrinsically less defective than coating O, hence its lower water uptake and somewhat higher R pore value at short immersion time. Coating O undergoes then only minor changes, i.e. it does not uptake much additional water, nor significant amounts of sulfate ions diffuse from the external solution to the solution in the pores. Coating E shows diverging behavior, depending on the anion in the test solution. Supposing that the amount of water uptaken by the coating is not dramatically different in the two cases [START_REF] Mišković-Stanković | The sorption characteristics of epoxy coatings electrodeposited on steel during exposure to different corrosive agents[END_REF], the R pore decrease and  decrease might actually be the result of the progressive increase in the chloride concentration in the pores (with a resulting decrease in resistivity). Such an effect would not be observed in sulfate solution, due to a much slower diffusion of this anion. The increase in R pore values with exposure time, observed in Na 2 SO 4 solution, is not a common phenomenon, but increasing impedances have been observed in other studies [START_REF] Le Pen | Characterisation of water-based coatings by electrochemical impedance spectroscopy[END_REF][START_REF] Macedo | Contribution to a better understanding of different behaviour patterns observed with organic coatings evaluated by electrochemical impedance spectroscopy[END_REF]. To explain the experimental observations, either a further waterinduced cross-linking of the coating material or, less probably, a pore shrinking as a consequence of coating swelling may be hypothesized. Independent evidence allowing to assess which phenomenon predominates is not available.

Conclusions

The "power law model", whose application to anti-corrosion coatings had been proposed in [START_REF] Amand | Constant-Phase-Element Behavior Caused by Inhomogeneous Water Uptake in Anti-Corrosion Coatings[END_REF], has been revisited and employed to analyze new experimental data.

It has been shown that an inaccurate knowledge of the resistivity of the dry coating material  c is unlikely to affect the quality of the agreement between model and experimental data, nor the values of the regressed parameters, at least as long as  c is large. Instead, an erroneous assumption about the resistivity of the electrolyte ( w ) within the coating pores has a major impact on the estimated water uptake . If, as it has been often reported, the diffusion of ions into the coating is slower that water diffusion, then  w and  will be respectively overestimated and underestimated, possibly by a large factor.

The effect of "through pores" and "short pores" has been taken simultaneously into account to calculate resistivity profiles.

Finally, it was shown that the evolution of the coating properties with immersion time depends both on the coating material, e.g. on its microstructure, and on the electrolyte, e.g. the ability of its anions to diffuse into the coating.

1 Figure 1 . 2 Figure 2 .Figure 3 .

 11223 Figure 1. Schematic representation of a substrate/oxide/coating/electrolyte system. R e is the uncompensated electrolyte resistance; R pore accounts for the through pores joining the coating/electrolyte and coating/oxide interface; Z c is the impedance of the electrolyte-penetrated coating described by Eq. (3); the CPE represents the oxide layer covering the Al 2024 alloy.

4

 4

Figure 4 .

 4 Figure 4. Effect of the assumed  c value on the best fitted parameters (a) ,  (b) and R pore (c).

Figure 5 .

 5 Figure 5. Comparison of the resistivity profiles calculated for the Coating E/NaCl solution system, calculated on the basis of total water uptake (empty symbols) or water uptake in short pores only (full symbols) Immersion times are indicated on the figure.

Figure 6 .Figure 7 .

 67 Figure 6. Comparison of the experimental impedance plots obtained with sample E exposed to 0.5 M Na 2 SO 4 , after a 48-hour immersion time, with the best fitted curve obtained by assuming  w = 22.2  cm,  c = 2 × 10 11  cm,  w = 82 and  c = 8.

Table 1 .

 1 Dependence of the fitted parameters  and R pore and of the calculated quantities  av and  pore on the immersion time in 0.5 M NaCl for coating E.

	Immersion	/	 /	R pore /	 av /	 pore /
	time in 0.5	dimensionless	dimensionless	cm 2	dimensionless	dimensionless
	M NaCl /
	h
	24	1.5 × 10 -5	3.5	1.0 × 10 6	3.2 × 10 -6	4.3 × 10 -8
	48	2.2 × 10 -5	3.4	3.1 × 10 5	5.1 × 10 -6	1.4 × 10 -7
	72	3.2 × 10 -5	3.1	2.1 × 10 5	7.8 × 10 -6	2.1 × 10 -7
	168	1.2 × 10 -4	3.0	5.1 × 10 4	2.9 × 10 -5	8.6 × 10 -7

Table 2 .

 2 Dependence of the fitted parameters  and R pore and of the calculated quantities  av and  pore on the immersion time in 0.5 M Na 2 SO 4 for coating E.

	Immersion	/	 /	R pore /	 av /	 pore /
	time in 0.5	dimensionless	dimensionless	cm 2	dimensionless	dimensionless
	M Na 2 SO 4
	/ h
	6	2.3 × 10 -6	3.6	8.5 × 10 5	5.3 × 10 -7	3.9 × 10 -8
	24	3.9 × 10 -6	4.0	1.9 × 10 6	8.0 × 10 -7	1.8 × 10 -8
	48	1.6 × 10 -6	3.9	2.5 × 10 6	3.5 × 10 -7	1.3 × 10 -8
	72	3.8 × 10 -6	4.2	2.9 × 10 6	7.9 × 10 -7	1.1 × 10 -8
	168	2.2 × 10 -6	4.2	2.9 × 10 6	4.4 × 10 -7	1.1 × 10 -8

Table 3 .

 3 Dependence of the fitted parameters  and R pore and of the calculated quantities  av and  pore on the immersion time in 0.5 M Na 2 SO 4 for coating O.

	Immersion	/	 /	R pore /	 av /	 pore /
	time in 0.5	dimensionless	dimensionless	cm 2	dimensionless	dimensionless
	M Na 2 SO 4
	/ h
	6	3.7 × 10 -5	3.9	6.7 × 10 5	7.7 × 10 -6	4.9 × 10 -8
	24	2.8 × 10 -5	3.4	6.7 × 10 5	6.5 × 10 -6	5.0 × 10 -8
	48	4.3 × 10 -5	3.4	5.4 × 10 5	9.8 × 10 -6	6.2 × 10 -8

Acknowledgments

The experimental data were obtained in the framework of the SMILE project (Surface Mono Innovative Layer for Environment), with the financial support of the DGCIS (Direction Générale de la Compétitivité, de l'Industrie et des Services). N. Pébère thanks the partners of the project: Mapaero (Pamiers), Rescoll (Bordeaux), Airbus (Toulouse), Dassault (Paris), EADS IW (Suresnes), l'Electrolyse (Bordeaux), and the laboratories LCPO and US2B (Université de Bordeaux).