Clément Mailhe

Fouzia Achchaq

Marie Duquesne

Clément Mailhé

Applications of an infrared thermography method for solid-liquid equilibria modeling of organic binary systems

Keywords: Solid-liquid equilibrium, Phase diagram, Polymorphism, Infrared thermography, Organic materials

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

Introduction

Phase diagrams are very useful tools in material synthesis. Their assessment allows a deep understanding of the behaviors of materials and phase change dynamics. Their establishment through standard methods of calorimetry allows the gathering of experimental data required as inputs in equilibrium modeling [START_REF] Campbell | Phase Diagrams: Understanding the Basics[END_REF][START_REF]Applications of Phase Diagrams in Metallurgy and Ceramics: Proceedings of a Workshop Held at the National Bureau of Standards[END_REF][START_REF] Dubost | Industrial applications and determination of equilibrium phase diagrams for light alloys. Progress and prospects[END_REF][START_REF] Yang | Application of phase diagram calculations to development of new ultra-high temperature structural materials[END_REF]. Amongst them we can distinguish dynamic methods such as Differential Scanning Calorimetry (DSC) or X-Ray powder Diffraction (XRD) and static methods such as Scanning Electron Microscopy (SEM). The wide range of available methods arises from the variety of information they can provide. Their complementarity is a reason why a combination of methods is often found in studies aiming to assess the phase equilibria of systems of materials [START_REF] Chelouche | Experimental and modeling studies of binary organic eutectic systems to be used as stabilizers for nitrate esters-based energetic materials[END_REF][START_REF] Chelouche | Solid-Liquid Phase Equilibria, Molecular Interaction and Microstructural Studies on (N-(2-ethanol)-p-nitroaniline + N-(2-acetoxyethyl)p-nitroaniline) Binary Mixtures[END_REF][START_REF] Chelouche | Solid + liquid equilibria and molecular structure studies of binary mixtures for nitrate ester's stabilizers: Measurement and modeling[END_REF]. The use of multiple methods is actually required to fully and reliably establish a phase diagram. In [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF] for instance, authors use several methods including some of the aforementioned ones to establish the phase diagrams of binary systems of fatty acids. Due to their large availability [START_REF]OECD-FAO Agricultural Outlook 2019-2028[END_REF], their relatively low-cost [START_REF] Midgley | The market outlook for fatty acids[END_REF], their renewable nature and their wide range of applications (food industry, pharmaceutics, bio-fuel, thermal energy storage …) [START_REF] Maximo | Trends and demands in the solid-liquid equilibrium of lipidic mixtures[END_REF], the study of fatty acids and bio-sourced compounds in general is an important research topic. A large scale study of such systems can be very time-consuming if an accurate and reliable phase diagram is expected. In fact, phase diagrams obtained with standard methods usually include only 10 to 20 experimental datasets as individual experiments are to be performed for each sample. To accelerate the screening process of materials and allow focusing calorimetry measurements on systems or compositions of interest, an innovative method based on infrared thermography (IRT) was developed [START_REF] Del Barrio | Infrared thermography method for fast estimation of phase diagrams[END_REF]. This method has recently provided the estimation of complex phase diagrams of fatty acids binary systems in a series of very short-time experiments [START_REF] Mailhé | Phase Diagrams of Fatty Acids as Biosourced Phase Change Materials for Thermal Energy Storage[END_REF][START_REF] Mailhé | Improved infrared thermography method for fast estimation of complex phase diagrams[END_REF][START_REF] Duquesne | Biosourced organic materials for latent heat storage: An economic and eco-friendly alternative[END_REF][START_REF] Mailhé | A fast and low-cost dynamic calorimetric method for phase diagram estimation of binary systems[END_REF][START_REF] Mailhé | Performance analysis of the infrared thermography method for complex phase diagrams estimation[END_REF]. In comparison with the phase diagrams depicted in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF], presenting eutectic, peritectic, metatectic transitions and solid solubility domains, the phase diagrams obtained using IRT show some inconsistent patterns including unsuspected transitions. The latter are partly confirmed by DSC measurements. It is assumed that additional occurring transitions might be of polymorphic nature as fatty acids are prone to such behavior as reported in [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF][START_REF] Gbabode | Fatty acids polymorphism and solid-state miscibility: Pentadecanoic acid-hexadecanoic acid binary system[END_REF]. The effect of such transitions on the liquidus line depiction can be accounted for in SLE modeling as reported in [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF][START_REF] Maximo | Enhancing the description of SSLE data for binary and ternary fatty mixtures[END_REF][START_REF] Lohmann | Solid-Liquid Equilibria for Seven Binary Systems[END_REF][START_REF] Maximo | On the solid-liquid equilibrium of binary mixtures of fatty alcohols and fatty acids[END_REF] but it is usually left aside either because of insufficient data or because of an assumed negligible influence. The form of the model presented in literature suggests that the IRT data regarding polymorphic transitions temperatures could be used as inputs, leading to the model adjustments based on polymorphic transitions enthalpies. The significant amount of gathered data as well as the potential detection of polymorphic transitions indicates that IRT could be a tool of great interest for SLE modeling.

In that frame, the present study aims at assessing the potential applicability of IRT for SLE modeling based on two main criteria. On the one hand, the accuracy of SLE models, relying on both predictive and experimental-based approaches, is evaluated by comparison with the IRT data obtained for three binary systems of fatty acids. On the other hand, the polymorphic transitions temperatures measured with IRT are used as inputs in a SLE model using polymorphic enthalpies as fitting parameters. Both measured temperatures and adjusted enthalpies are then compared with the literature values and with the DSC measurements to confirm the applicability of IRT for polymorphic transition detection and its ability to provide consistent enthalpy values.

Materials & Methods

Samples

Three systems of fatty acids are studied in this work: Capric acid + Lauric acid, Lauric acid + Palmitic acid and Palmitic acid + Stearic acid. The phase diagrams of those systems are thoroughly studied in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF]. They are characterized by complex phase transitions, making the study of their solid-liquid equilibria a challenging task. Several works aiming at characterizing the SLE of fatty acids systems can be mentioned [START_REF] Zhang | Thermophysical Properties of Some Fatty Acids/Surfactants as Phase Change Slurries for Thermal Energy Storage[END_REF][START_REF] Kahwaji | Fatty acids and related phase change materials for reliable thermal energy storage at moderate temperatures[END_REF][START_REF] Zhang | Thermal studies on the solid-liquid phase transition in binary systems of fatty acids[END_REF][START_REF] Cedeño | Measurements of temperature and melting heat of some pure fatty acids and their binary and ternary mixtures by differential scanning calorimetry[END_REF][START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF][START_REF] Schaake | Thermodynamic properties of the normal alkanoic acids II. Molar heat capacities of seven even-numbered normal alkanoic acids[END_REF]. Although, they all hold the same purpose, considerably different results are presented. In fact, it appears that experimental conditions play a major role in the determination of their phase diagrams. As stated by Moreno et al. [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF], fatty acids are prone to polymorphism which is heavily influenced by crystallization kinetics and that can drastically alter the SLE of studied systems. Consequently, for the comparison with the literature data hereafter, only results obtained with similar operating conditions are considered. The pure compound properties from the remaining literature are still accounted for and are compared with the properties obtained in this work in Table 1.

Experimental data

The use of the IRT method for the study of phase change processes in organic materials has been increasingly studied in the past years [START_REF] Godin | Microscopic infrared thermography for fast estimation of the thermal properties of thin films[END_REF][START_REF] Duquesne | Experimental analysis of heterogeneous nucleation in undercooled melts by infrared thermography[END_REF][START_REF] Godin | Analysis of crystal growth kinetics in undercooled melts by infrared thermography[END_REF] as a way to efficiently contribute to the growing research efforts in this field [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF][START_REF] Maximo | Trends and demands in the solid-liquid equilibrium of lipidic mixtures[END_REF][START_REF] Zhang | Thermophysical Properties of Some Fatty Acids/Surfactants as Phase Change Slurries for Thermal Energy Storage[END_REF][START_REF] Kahwaji | Fatty acids and related phase change materials for reliable thermal energy storage at moderate temperatures[END_REF][START_REF] Zhang | Thermal studies on the solid-liquid phase transition in binary systems of fatty acids[END_REF][START_REF] Cedeño | Measurements of temperature and melting heat of some pure fatty acids and their binary and ternary mixtures by differential scanning calorimetry[END_REF][START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF][START_REF] Schaake | Thermodynamic properties of the normal alkanoic acids II. Molar heat capacities of seven even-numbered normal alkanoic acids[END_REF][START_REF] Duquesne | Crystal growth kinetics of sugar alcohols as phase change materials for thermal energy storage[END_REF][START_REF] Godin | Bubble agitation as a new lowintrusive method to crystallize glass-forming materials[END_REF]. Its ability to rapidly estimate phase diagrams of systems of organic materials including fatty acids has recently been proven in several works [START_REF] Del Barrio | Infrared thermography method for fast estimation of phase diagrams[END_REF][START_REF] Mailhé | Phase Diagrams of Fatty Acids as Biosourced Phase Change Materials for Thermal Energy Storage[END_REF][START_REF] Mailhé | Improved infrared thermography method for fast estimation of complex phase diagrams[END_REF][START_REF] Duquesne | Biosourced organic materials for latent heat storage: An economic and eco-friendly alternative[END_REF][START_REF] Mailhé | A fast and low-cost dynamic calorimetric method for phase diagram estimation of binary systems[END_REF][START_REF] Mailhé | Performance analysis of the infrared thermography method for complex phase diagrams estimation[END_REF]. In this work, the presented IRT results are validated based on phase transitions data extracted from literature and from DSC measurements. We then rely only on the experimental data obtained using IRT to assess the accuracy of the SLE model and to fit model parameters.

IRT

The estimated phase diagrams and liquidus lines are obtained using IRT according to the protocol thoroughly described in [START_REF] Del Barrio | Infrared thermography method for fast estimation of phase diagrams[END_REF]. Briefly, the method consists in monitoring the photonic flux emitted by multiple samples with different molar compositions and submitted to a controlled heating ramp. A phase transition in a sample is associated to heat exchanges, structural and morphological changes resulting in an abrupt variation in the measured photonic flux. The signal variations are simultaneously identified for all the samples throughout the heating experiment and allow consequently obtaining the estimated phase diagram of each studied binary system. For each considered I-J binary system, 101 compositions are studied ranging from 0 % I + 100 J % to 100 % I + 0 % J by 1 % increments approximately. The 1% mentioned is actually given as an indication to have enough data to correctly depict the liquidus line. The obtained phase diagrams then consist of 101 datasets.

For each composition, mixture batches of approximately 50 mg are prepared in aluminum weighing pans. The appropriate amount of the compound I is weighed in powder form with a Mettler Toledo weighing scale with a 0.03 mg accuracy. The content of the pan is melted at a temperature 5 °C above the highest melting point of the two species and is then weighed again after it has completely recrystallized. The compound J is added to the pan so that the total mass of two compounds allows obtaining the objective molar fraction. It is then melted and weighed again after solidification to obtain the actual molar percentage. Furthermore, with the LA+PA binary system for example, amongst the 101 compositions prepared, the maximum deviation between the objective molar percentage and the one obtained according to our weighing is around 0.045% (with an average deviation around 0.01%). The combined expanded uncertainty for the molar fraction is evaluated between 0.59 and 0.73% depending on purity which indicates that all prepared samples are at least within this range from the objective composition. From those negligible deviations, we can assume that each dataset obtained with the IRT method is given for the relevant composition.

DSC measurements

IRT measurements are followed with DSC experiments at selected compositions to validate the estimated phase diagrams. As getting 1 dataset at a time using this standard method is more timeconsuming than getting 101 datasets with IRT, the validity of the phase diagram is generalized from local DSC measurements made at a few selected compositions. Two DSC heating ramps are programmed per sample: one at 1 °C/min matching the rate of the experiments using IRT and one at 0.3 °C/min to get more information on transitions that could be missed with a higher rate. The determination of the transition temperatures is performed according to the guidelines given in [START_REF] Boettinger | Chapter 5 -DTA and heat-flux DSC measurements of alloy melting and freezing[END_REF]. The interpolated onset temperature is considered as the transition temperature if a flat baseline can be obtained with the DSC curves. Otherwise peak temperatures are considered. In case the peak temperature must be considered, the results obtained with both the 1 °C/min and 0.3 °C/min rates are compared to make sure their difference is at least within the uncertainty range of the calorimeter. By doing so, we ensure that transition temperatures are consistent with a zero scanning rate experiment.

The experiments are performed with a DSC 131 provided by SETARAM and the data treatment is performed using the SETSOFT 2000 software. The DSC is calibrated using Gallium (Purity: 99.9999 %), Indium (Purity: 99.995 %), Tin (Purity: 99.999 %) and Lead (Purity: 99.999 %) to validate the results precision in the 30-330 °C temperature range. Fatty acids samples of approximately 10 mg, and coming from the same batch used for the IRT experiment, are weighed using a Mettler Toledo scale with a 0.03 mg accuracy and are placed in open alumina crucibles.

SLE modeling

The thermodynamic model used for the modeling of the SLE of binary systems is given by Equation 1. This particular model, based on the fact that chemical potential of each substance in the solution is identical for the solid and liquid phases at equilibrium, has already been used and described in [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF][START_REF] Maximo | Enhancing the description of SSLE data for binary and ternary fatty mixtures[END_REF][START_REF] Lohmann | Solid-Liquid Equilibria for Seven Binary Systems[END_REF][START_REF] Maximo | On the solid-liquid equilibrium of binary mixtures of fatty alcohols and fatty acids[END_REF].

ln x i γ , z i γ , = ∆H , R 1 T , - 1 T + ∆C p,i R T , T -ln T , T -1 + ∆H , R 1 T , - 1 T + ∆C p,i R T , T - n tr=1 ln T , T -1 (1)
with i identifying the species, x i is the molar fraction of i in the liquid phase, z i is the molar fraction of i in the solid phase, γ L,i is the activity coefficient of i in the liquid phase, γ S,i is the activity coefficient of i in the solid phase, R is the universal gas constant, T m,i is the melting temperature of i, T is the solid-liquid equilibrium temperature, ∆H m,i is the melting enthalpy of i, ∆C p,i is the difference between the liquid and solid heat capacities of i, T tr,i is a solid-solid transition temperature, ∆H tr,i is the transition enthalpy associated to the latter transition.

This model is however more commonly found in the form given by Equation 2. This simplification comes from two main assumptions. First, it is assumed that the influence of the specific heat is negligible. Although, it was verified beforehand in our case that the specific heat difference is negligible compared to the latent heat of melting of fatty acids, it is a common practice especially for the SLE modeling of organic compounds. The second simplification considers the unity of the solid activity % & (% & = z i γ ',( ). It arises from the choice to consider both components in the mixture as immiscible in the solid phase. It implies that solid solubility domains do not exist or that their range is limited.

ln)x i γ , *= ∆H , R 1 T , - 1 T + ∆H , R 1 T , - 1 T n tr=1 (2)
In addition, the previous formulation implies that the equilibrium occurs below a transition temperature.

If we are in-between the melting temperature and the transition temperature, the right-hand part of Equation 2should not be considered. In that case, we are left with Equation 3 which is the most commonly found in literature for SLE modeling [START_REF] Maximo | On the solid-liquid equilibrium of binary mixtures of fatty alcohols and fatty acids[END_REF][START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF][START_REF] Gmehling | Solid-Liquid Equilibria Using UNIFAC[END_REF]. A thorough explanation of the modeling process in the presence of polymorphic transitions is given in [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF].

ln)x i γ , *= ∆H , R 1 T , - 1 T (3)
Moreover, binary systems of fatty acids differing by 2 or 4 carbon atoms are described in literature as having a complex phase diagram, specifically those including a peritectic transition. The methods to account for the effect of this reaction on the liquidus line have been studied [START_REF] Barbosa | On the description of the liquidus line of systems presenting peritectic reactions[END_REF][START_REF] Rocha | An approach to calculate solid-liquid phase equilibrium for binary mixtures[END_REF][START_REF] Slaughter | Calculation of solid-liquid equilibrium and crystallization paths for melt crystallization processes[END_REF][START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF] and consider the peritectic compound as the outcome of a stoichiometric chemical reaction between both I and J components in the mixture as represented in relation 4.

ν I I+ν J J ↔ ν P P (4)

with ν 3 , ν 4 and ν 5 respectively the stoichiometric coefficients for the pure components I, J and the peritectic compound P.

From this consideration come the models depicted by Equations 5 and 6 which respectively, consider and exclude the effect of the polymorphic transitions. More details regarding their determination can be found in [START_REF] Barbosa | On the description of the liquidus line of systems presenting peritectic reactions[END_REF][START_REF] Rocha | An approach to calculate solid-liquid phase equilibrium for binary mixtures[END_REF][START_REF] Slaughter | Calculation of solid-liquid equilibrium and crystallization paths for melt crystallization processes[END_REF].

ln)x i γ , * i = ν i 6 ∆H , R 1 T , - 1 T + 6 ∆H , R 1 T , - 1 T 7 n tr=1 7 - ∆ r G 0 RT i ( 5
)
With ∆ ; r G 0 the Gibbs free molar energy of the peritectic reaction.

ln)x i γ , * i = ν i 6 ∆H , R 1 T , - 1 T 7 - ∆ r G 0 RT i (6)
Although the model would be physically correct for a system in which a new compound is formed through the peritectic reaction, this transition actually often corresponds to a new crystalline structure from both compounds in a stoichiometric manner and not to the reaction itself. In this case, the value of the Gibbs energy of reaction is not representative of the actual reaction but just introduces an adjustable parameter for the model which allows taking into account indirectly the peritectic transition in the depiction of the liquidus line.

Solving the models described previously consists in a simple minimization procedure assuming that pure compounds properties are known and that liquid activity coefficient and Gibbs energy of reaction if needed are also known. If at least the melting enthalpies and temperatures of pure compounds are pre-required for SLE modeling, the liquid activity coefficients, Gibbs energy of reaction and polymorphs properties are often unknown. Different cases then arise for their determination.

Two approaches are possible for the evaluation of the liquid activity coefficient. Either experimental data is used to fit the value or a predictive method is employed. The first approach is widely used for SLE modeling and many methods exist such as the Margules-3-suffix, NRTL, Wilson or UNIQUAC [START_REF] Slaughter | Calculation of solid-liquid equilibrium and crystallization paths for melt crystallization processes[END_REF]. Mostly used in combination with the model presented in Equation 3, they aim at determining coefficients allowing the computation of the liquid activity coefficient as a function of the molar fraction, the temperature and the known pure compound properties. In this study, the Margules-3-suffix method is used to assess its efficiency when fitted to the large amount of data provided by IRT. This method is combined to the model described in Equations 3 and 6. This method provides a pair of coefficient % 12 and % 21 computed through Equation 7. The Margules-3-suffix method was chosen as it is widely used for the SLE modeling of fatty acids and shows good performances [START_REF] Maximo | Enhancing the description of SSLE data for binary and ternary fatty mixtures[END_REF][START_REF] Maximo | On the solid-liquid equilibrium of binary mixtures of fatty alcohols and fatty acids[END_REF][START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF].

= RT ln γ ,3 = (% >? + 3% ?> )(1 -x I ) ? -4% ?> (1 -x I ) A
RT ln γ ,4 = (% >? -3% ?> )x I ? + 4% ?> x I A [START_REF] Chelouche | Solid + liquid equilibria and molecular structure studies of binary mixtures for nitrate ester's stabilizers: Measurement and modeling[END_REF] with % 12 and % 21 the Margules coefficients, x I the molar fraction of the compound I and γ L,I and γ L,J the liquid activity coefficient of I and J respectively.

The second method used in this work is the UNIFAC-Dortmund method which is based on group contribution and is extensively discussed in [START_REF] Gmehling | A modified UNIFAC model. 2. Present parameter matrix and results for different thermodynamic properties[END_REF]. It allows predicting the values of liquid activity coefficients from the knowledge of pure compound properties and composition. This approach is particularly useful for the simple binary systems as it is able to provide a liquidus line without the need for experimental data. In our case, the presence of the peritectic transition implies fitting at least the Gibbs energy of reaction with the experimental data. However the liquidus lines depicted by either Equation 2 or 3 could theoretically be computed predictively assuming that polymorphic transitions are known beforehand.

As extensively discussed in [START_REF] Rocha | An approach to calculate solid-liquid phase equilibrium for binary mixtures[END_REF] the solving process implies the consideration of regions defining which equation to use for the liquidus line depiction. In fact, the determination of the Gibbs energy of reaction requires to be fitted to experimental data. This adjustment implies, as mentioned, the identification of the area in the liquidus affected by the peritectic reaction. In addition and as discussed previously, fatty acids are characterized by numerous polymorphs [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF][START_REF] Gbabode | Fatty acids polymorphism and solid-state miscibility: Pentadecanoic acid-hexadecanoic acid binary system[END_REF]. Including polymorphic transitions in the model implies the addition of new regions. In fact, as stated in [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF] and as explained previously, one should use either Equation 2or Equation 3 depending on the temperature as the pure compounds properties change according to which polymorph is present. The detail of this process is given in [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF]. Those polymorphic transitions are challenging to characterize even with standard methods which is why they tend to be neglected in SLE modeling. However, if the IRT method manages to provide polymorphic transition temperatures, then it would be possible to use polymorphic transition enthalpies as fitting parameters. An estimation could be provided for this thermal property assuming that the model assumptions are valid (negligible influence of the specific heat and immiscibility in the solid phase). As explained by Maximo et al. [START_REF] Maximo | The Crystal-T algorithm: a new approach to calculate the SLE of lipidic mixtures presenting solid solutions[END_REF], the form of the model depicted by Equation 1-6 is intrinsically right if we are to use the polymorphic transition enthalpies as adjusting parameters. Consequently, and as explained, adjusted values can be considered as coherent approximations if and only if model assumptions are valid. Therefore, to avoid the confusion between the actual polymorphic transition enthalpies and the adjusted parameters, ∆H will be referred as ∆H CDEC in the results hereafter.

Results

For the 3 studied systems, the estimated phase diagram using IRT is plotted together with the DSC measurements performed for the selected compositions. Both IRT and DSC data are available as Supplementary material.

The liquidus line composed of IRT data points is compared to three SLE models. One uses the Margules-3-suffix method for liquid activity coefficient computation. Its form is based on Equations 3 and 6. The two remaining models use the UNIFAC-Dortmund method for liquid activity coefficient computation. One is based on Equations 3 and 6 and the other on Equations 2 and 5. The latter, taking into account the polymorphic behavior of fatty acids, uses the transition temperatures provided by the IRT method as input. In this case, as the liquid activity coefficient calculation is predictive, the adjusting parameters are the enthalpies of polymorphic transitions. The transition temperatures obtained with IRT are compared with the known values from literature. The computed enthalpy values are also compared to the typical ones for each fatty acid.

The accuracy of the models is made based on the comparison of their absolute average deviation (AAD) calculated using Equation 8.

AAD =

HT I,3JK -T I,CDEC H T I,3JK × n k × 100 [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF] with T I,IRT the liquidus line temperature obtained using IRT for the sample k, T I,calc the liquidus line temperature for the sample k computed using the SLE model and n the number of samples studied using the IRT method.

Before presenting the results, one should be aware that pure fatty acids are known to present polymorphic transitions very close to the melting temperature as seen in Figure 1 showing the DSC thermograms of 4 pure fatty acids. Excluding the stearic acid, the double peaks are also observed in literature [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF][START_REF] Maximo | On the solid-liquid equilibrium of binary mixtures of fatty alcohols and fatty acids[END_REF] and have nothing to do with the impurities as the same results have been obtained for different samples of the same materials. Besides, the impurity ratio of the purchased compounds is very low (~1%). The melting enthalpy measured using our DSC is most likely the sum of the melting enthalpy and of the polymorphic transition enthalpy occurring slightly before the melting point.

Although our DSC measurements do not corroborate the occurrence of this polymorphic transition for the stearic acid, literature does [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF]. Consequently for all studied compounds a polymorphic transition is considered slightly below the melting point to account for its effect. From the DSC thermograms, it is also expected that the computed polymorphic transition enthalpies be rather low in comparison with the melting one. As can be seen in Figure 2A, both models using the Margules-3-suffix method and the one accounting for polymorphism provide a good fit with the IRT data. The purely predictive model based on the UNIFAC-Dortmund method appears here to progressively deviate from the experimental liquidus line. If we observe Figure 2B, we notice that the estimated phase diagram using the IRT method is sensibly different from the one depicted in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF]. Two transitions above the peritectic transition (in the 0%CA-25%CA and 0%CA-40%CA ranges respectively) are detected on the side rich in LA, that are neither fully confirmed by DSC measurements or literature. On the side rich in CA, one additional transition (in the 80%CA-100%CA range) above the eutectic transition appears in comparison with literature. On the exception of those transitions, a fairly good agreement between experimental results from literature, DSC measurements and the ones obtained using the IRT method is obtained. The polymorphism of CA and LA and of fatty acids in general has not been widely studied. In fact, no polymorphism has yet been observed for CA but on the other hand, [44] introduces a polymorphic transition for the LA occurring around 308 K, which is consistent with the one detected using the IRT method.

The values of the different parameters for each used model as well as their relative accuracies are listed in Table 2. with Ttr the considered polymorphic transitions temperatures, whose value is taken as the average temperature of the IRT measurements, ∆H CDEC the fitted polymorphic transition enthalpy, ∆rG 0 the adjusted value of the Gibbs energy of reaction for the peritectic transition, % >? and % ?> the Margules adjusting coefficients.

The average absolute deviation obtained using the Margules-3-suffix method and for the one using the UNIFAC-Dortmund method and accounting for polymorphism is very close with even a slight advantage for the latter. Regarding the values of the enthalpy of polymorphic transitions, a comparison is only possible for documented transitions with measured enthalpies. The values obtained for the polymorphic transitions occurring close to the melting temperatures are rather low with 2.46 kJ/mol for the CA and 1.09 kJ/mol for the LA, as expected from Figure 1. A polymorphic transition is detected using the IRT method around 308.32 K on the side of the phase diagram rich in LA. This transition is also documented in [44] but the value of its enthalpy is not indicated. A later study in [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF], gives enthalpies of polymorphic transitions for other fatty acids with the same type of polymorphs. The values are 6.4 and 7.6 kJ/mol for the MA and PA, respectively. The value computed with our model is 7.48 kJ/mol which can be considered as a reasonable estimation. In [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF][START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF][START_REF] Costa | The solid-liquid phase diagrams of binary mixtures of even saturated fatty acids differing by six carbon atoms[END_REF], the phase diagrams of fatty acids binary systems presented by the authors are characterized by metastable and solid solubility domains whose actual occurrence is arguable. The presence of these domains was in fact assumed based on the detection by DSC of horizontal transitions which are consistent with our measurements whether obtained by IRT or DSC. However, the gathered IRT data go against their assumptions and the existence itself of these domains is questionable. Consequently, undocumented transitions are considered as polymorphic in this work as a first approach, in order to include the effect of transitions such as the one detected at 296.33 K and 301.97 K on the liquidus line. An in-depth study of their nature via other techniques (XRD for instance) is however expected. The SLE modeling of this binary system is also made in [START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF] using the Margules-3suffix method and based on 15 data points obtained by DSC. The average absolute deviation measured by the authors is 0.12 % and the Margules parameters are evaluated at -1.60 kJ/mol for % >? and -1.30 kJ/mol for % ?> . As for the Gibbs energy of reaction, it is estimated at -1.20 kJ/mol. We notice that the model fit based on 101 data points obtained using the IRT method is as accurate as the one obtained for 15 DSC measurements, which is highly encouraging for the potential use of the IRT method for SLE modeling. Although the model parameters are comparable and in the same range, they need to be nuanced as they highly depend on the pure compound properties considered and on their purity.

Lauric acid -Palmitic acid

Figures 3A and 3B respectively show the liquidus line depiction of the SLE models in comparison with the IRT data and the LA+PA phase diagram estimated using the IRT method, DSC measurements and phase transitions data extracted from [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF]. Regarding the respective accuracies of the models in comparison with IRT data, the same conclusions as for the previous systems can be drawn. Indeed, from Figure 3A and Table 3, we see that both fitted models are relatively accurate in comparison with the predictive one. The latter presents significant deviations and cannot even depict the effect of the peritectic reaction on the liquidus line. We also notice that, even for experimentally fitted methods, the accuracy is lower than in the previous case. This loss in accuracy could be associated to the slight scattering of IRT data points in the 30%LA-50%LA range. It however does not affect the overall phase diagram as the amount of data compensates for this local deviation.

As can be seen in Figure 3B, the phase diagram obtained using the IRT method is relatively consistent with results from DSC measurements or extracted from literature [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF]. Although an horizontal transition occurring around 317 K between 10%LA-90%PA and 50%MA-50%PA could not be detected with certainty using the IRT method, another one around 312 K is detected and is confirmed by our DSC measurement (available in the Supplementary material). The latter can be associated to a polymorphic transition of PA similarly to the previous case. As for the LA, one polymorphic transition is considered with the same pattern as described in the previous system. The values of the different parameters for each used model as well as their relative accuracies are listed in Table 3. with Ttr the considered polymorphic transitions temperatures, whose value is taken as the average temperature of the IRT measurements, ∆H CDEC the fitted polymorphic transition enthalpy, ∆rG 0 the adjusted value of the Gibbs energy of reaction for the peritectic transition, % >? and % ?> the Margules adjusting coefficients.

The polymorphic transition detected above the eutectic line in the 85%LA-100%LA range is located at 310.77 K. This temperature is not found elsewhere in literature and as stated previously, it may suggest that the nature of this transition is not polymorphic. In that case, the value of the computed enthalpy can only be considered as an adjusting parameter and does not bear any relevant physical meaning. A polymorphic transition around 321.38 K is also detected for the PA in the 0%LA-50%LA range using the IRT method. Literature reports the occurrence of a transition at 318.8 K in [START_REF] Gbabode | Fatty acids polymorphism and solid-state miscibility: Pentadecanoic acid-hexadecanoic acid binary system[END_REF] while [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF] report several ones at 316.7 K, 317.5 K and 324.7 K. Despite our transition temperature is encompassed by the literature ones, we cannot conclude with certainty on the polymorphic nature of this transition based solely on literature. Our DSC measurements however corroborate the transition detected using the IRT method. DSC data even appear to depict one around 317.5 K which is consistent with the polymorphic transition reported in literature but could unfortunately not be detected for enough samples to be considered. The calculated enthalpy of polymorphic transition appears however to have been overestimated. As for the enthalpies of the transitions close to the melting point, the 5.77 kJ/mol computed for the PA can be considered low in comparison with the 54 kJ/mol measured for the melting enthalpy but 6.45 kJ/mol appears to be slightly high. If the nature of the transition considered at 310.77 K is not polymorphic it may imply the presence of a solid solubility domain as suggested in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids: differing by four carbon atoms[END_REF]. If it is the case, the assumptions of the model should be revisited or another way of accounting for this transition should be investigated. The Margules-3-suffix method is also used in [START_REF] Costa | Solid-Liquid Equilibrium of Binary Fatty Acid Mixtures[END_REF] to model the liquidus line of this binary system based on 12 DSC measurements. The average absolute deviation measured by the authors is 0.13 % and the Margules parameters are evaluated at -5.0 kJ/mol for % >? and -1.8 kJ/mol for % ?> . As for the Gibbs energy of reaction, it is estimated at -2.1 kJ/mol. If the Margules parameters are rather close, we notice that the accuracy obtained based on the IRT results is not as accurate as in literature. However, it should be mentioned that assessing the accuracy of the modeled liquidus line based on only 12 experimental points is hazardous when depicting complex phase diagrams such as this one as transitions and trend changes can easily be missed. We nevertheless notice that the modeled liquidus line obtained in this work successfully depicts the experimental one as can be seen in Figure 3A.

Palmitic acid -Stearic acid

Figures 4A and 4B respectively show the liquidus line depiction of the SLE models in comparison with the IRT data and the PA+SA phase diagram estimated using the IRT method, DSC measurements and phase transitions data extracted from [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF]. We can see from Figure 4A that both fitted models provide an estimated liquidus line almost not dissociable from the experimental one obtained using the IRT method as corroborated by the average absolute deviation computed in Table 5. The predicted liquidus line here fits rather well the right part of the experimental liquidus line but still present deviations in the left part. The phase diagram obtained using the IRT method as seen in Figure 4B is mostly consistent with DSC measurements and the one depicted in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF]. In this case, the transition occurring around 332 K in the 10%PA-40%PA range and the IRT Data DSC [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF] one occurring around 330 K in the 75%PA-90%PA range are both detected in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF] and the first is also verified in our DSC measurement (available in the Supplementary material). Another transition on both sides is detected consistently with the IRT method but could not be obtained with certainty using DSC.

The four mentioned transitions are assumed in the model to be of polymorphic nature. with Ttr the considered polymorphic transitions temperatures, whose value is taken as the average temperature of the IRT measurements, ∆H CDEC the fitted polymorphic transition enthalpy, ∆rG 0 the adjusted value of the Gibbs energy of reaction for the peritectic transition, % >? and % ?> the Margules adjusting coefficients.

Three polymorphic transitions are therefore considered in the model for the PA, one at 334.26 K, one at 332.17 K and one at 329.9 K. In [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF] and [START_REF] Stenhagen | The phase transitions in normal chain carboxylic acids with from twelve up to and including twenty-nine carbon atoms between 30° and the melting point[END_REF], a transition is reported to occur respectively around 331 K and 332.1 K which is consistent with the second one. Regarding the third one, we can once again not conclude on its polymorphic nature. In fact, we can see from Figure 4B that the IRT method also allows identifying a solid solubility domain. The presence of this domain suggests that this transition may not be polymorphic and its influence on the liquidus line depiction should be treated differently. As for the adjusted values of enthalpies of polymorphic transitions, they are not coherent with what should be expected. An explanation would be that as the predictive model is already rather accurate, either the influence of polymorphic transitions is negligible or, the melting point of the PA is misevaluated. In fact, from Figure 1 we see that the onset transition temperature is close to the melting point obtained using the IRT method whereas it actually represents the beginning of the polymorphic transition and not the melting point. It is likely then that the melting point is underestimated in this particular case. Regarding the SA, three polymorphic transitions are also considered, at 341.68 K, at 336.72 K and at 332.89 K. The occurrence of the first one is not supported by literature but the value of the adjusted enthalpy is coherent with our assumptions as 6.11 kJ/mol can be considered low in comparison with 60.02 kJ/mol. The second one is consistent with the one recorded in [START_REF] Stenhagen | The phase transitions in normal chain carboxylic acids with from twelve up to and including twenty-nine carbon atoms between 30° and the melting point[END_REF] at 337.1 K. It is also stated that its enthalpy of polymorphic transition is supposed to be low. This is also the case in our model with a value computed at 2.52 kJ/mol. As for the last one, its polymorphic nature is uncertain but it could eventually be related to the one present in [START_REF] Moreno | Polymorphism of even saturated carboxylic acids from n-decanoic to n-eicosanoic acid[END_REF] and occurring around 331.6 K. However the adjusted value is incoherent with the expected ones, suggesting that either this transition is not polymorphic as assumed in [START_REF] Costa | The solidliquid phase diagrams of binary mixtures of consecutive, even saturated fatty acids[END_REF] or that the enthalpy of a previous transition has been misevaluated.

Discussion

The IRT method is an emerging method primarily aiming at the fast estimation of phase diagrams. For numerous binary systems of organic materials, it has provided approached phase diagrams, consistent with literature data and measurements made with standard calorimetric methods. Its main asset is its capability to provide the phase transitions data of one system for a large amount of samples in a single and short-time experiment. Due to its fundamentally different functioning principle, as detailed in [START_REF] Del Barrio | Infrared thermography method for fast estimation of phase diagrams[END_REF], the IRT method shows a capacity to possibly identify polymorphic transitions. Indeed, the detected transitions have been compared with the polymorphic transitions found in literature for all studied systems. This relatively good agreement between the values shows that the IRT method brings valuable information for thermodynamic modeling as Equation 2 is commonly simplified to Equation 3 due to the lacking data regarding polymorphism. However, the adjusted enthalpies of polymorphic transitions are not always coherent with those collected from literature. Although the obtained values allow a good depiction of the experimental liquidus line, it would be ideal to be able to use the model as a way of estimating those thermodynamic properties. The model accounting for the polymorphic transitions for the CA+LA and the PA+SA binary systems is particularly efficient due to the very accurate polymorphic transitions detection. Besides, the computed enthalpies are consistent and in the range of literature values, especially for LA and SA. We can see from Figures 2B and4B that the phase diagrams obtained using the IRT method is relatively simple and refined in comparison with the one from Figure 3B. Because the left part of the phase diagram is less crowded with transitions, it is easier to accurately identify transitions which may have led to a better depiction of the modeled liquidus line. On another note, the authors show in [START_REF] Sato | Structure, Stability and Crystal Growth of Polymorphs and Polytypes of Long-Chain Aliphatic Compounds[END_REF] different DSC thermograms that could be associated to polymorphic transitions. It states that depending on the thermal history of the sample and the monotropic or eniantropic nature of the system, polymorphs may or may not be noticeable and their representation on a thermogram can have many aspects. It highlights the difficulties associated with the identification of polymorphic transitions and the many ways in which they can affect SLE modeling. In addition, we can mention that several recent works [START_REF] Dumas | New theoretical determination of latent heats from DSC curves[END_REF][START_REF] Gibout | Challenges of the Usual Graphical Methods Used to Characterize Phase Change Materials by Differential Scanning Calorimetry[END_REF] aim at improving DSC measurements for phase change materials, emphasizing the challenge of phase transition characterization even with established standard methods.

From the large amount of data obtained with the IRT method, we see that the depicted line is coherent and consistent with the DSC measurements. Despite a few local deviations, we have measured small average absolute deviations which suggest that the IRT method is reliable for SLE modeling by providing adequate experimental data and large datasets including, as we have seen, insightful information regarding polymorphic transitions. However, for systems presenting areas dense in transitions such as the LA+PA system, an improvement of the resolution of the IRT method may be needed for better performances and to eventually be able to estimate thermal properties from SLE modeling.

Conclusion

The applicability of the IRT method as experimental data input for SLE modeling has been evaluated upon assessing the performances of empirical and theoretical SLE models for the liquidus line depiction of 3 binary systems of fatty acids (CA+LA, LA+PA and PA+SA). The IRT method is used to build approached phase diagrams consisting of a large amount of datasets in a single experiment. The quantity of obtained data is an advantageous feature to supply databases and is particularly interesting in SLE modeling as it allows a better depiction of the equilibrium while making sure that every single region of the liquidus line is accounted for. The detected transitions using the IRT method and not reported in known phase diagrams have been compared to the polymorphic ones characterized in literature. For all presented systems, we have been able to relate some detected transitions to polymorphic transitions reported in literature. For the CA+LA and PA+SA systems, the computed enthalpies for those transitions were in the range of the documented polymorphic ones. As for the remaining transitions, further works are needed to fully identify their nature as neither literature nor our results allow any definitive conclusion. Although the IRT method has shown to be able to detect a solid solubility domain which is an interesting accomplishment for the estimation of complex phase diagrams, these results suggest that the model assumptions may not be fully valid. Nevertheless, whether the detected transitions have been successfully related to the polymorphism phenomenon or not, the model taking it into account has been able to provide a liquidus line with a good accuracy. The IRT method should be applied to systems presenting simpler phase diagrams and whose polymorphic behavior has been thoroughly studied in order to definitely conclude on the benefits of using the IRT method in SLE modeling. Our results are indeed encouraging as they suggest the IRT method could be used for other applications than providing a full complex phase diagram in a single and fast experiment. Indeed, it could also allow for the assessment of model assumptions beforehand and it may enable the incorporation of polymorphic transition temperatures in the model in order to estimate the enthalpies of polymorphic transitions.

Fig. 1 .

 1 Fig. 1. DSC thermograms of the 4 pure studied fatty acids.

Fig. 2 .

 2 Fig. 2. (A) Liquidus line depiction of SLE models in comparison with IRT data; (B) CA+LA phase diagram estimated using the IRT method and DSC measurements.

Fig. 3 .

 3 Fig. 3. (A) Liquidus line depiction of SLE models in comparison with IRT data; (B) LA+PA phase diagram estimated using the IRT method and DSC measurements.

Fig. 4 .

 4 Fig. 4. (A) Liquidus line depiction of SLE models in comparison with IRT data; (B) PA+SA phase diagram estimated using the IRT method and DSC measurements.

Table 1 .

 1 Characteristics and thermal properties of the used chemicals at pressure p = 0.1 MPa. a

	Capric acid	Lauric acid	Palmitic Acid	Stearic Acid

a The melting enthalpy ∆H and melting temperature T are obtained from DSC measurements at pressure p = 0.1 MPa following the protocol described in section 2.2.2. Standard uncertainties are u(T ) = 0.5 K, u(∆H ) = 5 % and u(p) = 10 kPa. The molar mass and purity are indicated as provided by the supplier.

The melting enthalpies and temperatures obtained in this work are within range of literature values. It is therefore assumed that that the pure compound properties considered for SLE modeling in this work is coherent and comparable to the SLE data extracted from literature.

Table 2 .

 2 Parameters and accuracies of the models for the CA+LA binary system at pressure p=0.1 MPa. b

		UNIFAC-Dortmund + polymorphism		Margules-3-suffix	UNIFAC-Dortmund
			AAD (%)			AAD (%)	AAD (%)
			0.12			0.13		0.33
		∆rG 0 (kJ/mol)			∆rG 0 (kJ/mol)	∆rG 0 (kJ/mol)
			-1.22			-1.50		-0.52
		CA: Ttr (K)		LA: Ttr (K)		% >? (kJ/mol)	% ?> (kJ/mol)
	303.72	296.33	314.12	308.32	301.97	-1.85	-0.43
	CA: ∆H CDEC (kJ/mol)	LA: ∆H CDEC (kJ/mol)
	2.46	3.91	1.09	7.48	6.60

b Standard uncertainties u are u(Ttr) = 1.54 K and u(p) = 10 kPa.

Table 3 .

 3 Parameters and accuracies of the models for the LA+PA binary system at pressure p=0.1 MPa. c

		UNIFAC-Dortmund + polymorphism	Margules-3-suffix	UNIFAC-Dortmund
		AAD (%)		AAD (%)	AAD (%)
		0.27			0.26	0.54
		∆rG 0 (kJ/mol)		∆rG 0 (kJ/mol)	∆rG 0 (kJ/mol)
		-1.20		-2.19	-
		LA: Ttr (K)		PA: Ttr (K)	% >? (kJ/mol)	% ?> (kJ/mol)
	314.97 310.77	333.8	321.38	-3.44	-1.75
	LA: ∆H CDEC (kJ/mol)	PA: ∆H CDEC (kJ/mol)
	6.45	1.78	5.77	11.09

c Standard uncertainties u are u(Ttr) = 1.54 K and u(p) = 10 kPa.

Table 4 .

 4 Parameters and accuracies of the models for the PA+SA binary system at pressure p=0.1 MPa. d

		UNIFAC-Dortmund + polymorphism		Margules-3-suffix	UNIFAC-Dortmund
			AAD (%)			AAD (%)	AAD (%)
				0.08			0.08	0.23
			∆rG 0 (kJ/mol)			∆rG 0 (kJ/mol)	∆rG 0 (kJ/mol)
			-1.18			-1.19	-0.61
		PA: Ttr (K)			SA: Ttr (K)		% >? (kJ/mol) % ?> (kJ/mol)
	334.26 332.17	329.9	341.68 336.72 332.89	-0.94	1.00
	PA: ∆H CDEC (kJ/mol)	SA: ∆H CDEC (kJ/mol)
	~ 0	-6.54	-1.87	6.11	2.52	-0.79
	d Standard uncertainties u are u(Ttr) = 1.54 K and u(p) = 10 kPa.

Acknowledgements

This work is carried out in the frame of SUDOKET project and is co-funded by the Interreg Sudoe Programme through the European Regional Development Fund (ERDF). The authors acknowledge them as well as the financial support of Region Nouvelle Aquitaine for subsidizing BioMCP project (Project-2017-1R10209-13023). We also would like to thank CNRS for promoting the I2M Bordeaux -CICe exchanges in the framework of the PICS PHASE-IR project.