Sébastien Genty

Philippe Tingaut

Claire Tendero

Maëlenn Aufray
email: maelenn.aufray@ensiacet.fr

Effects of infrared radiation on the mechanical properties of an epoxy-amine adhesive using a Central Composite Design method

Keywords: F, poxy adhesive lnfrared curing Central composite design, Thru-point bending test Tensile strength Flexuralstrongth 1

RACT

Recenùy, the aerospace induscry has been facing many challenges , induding an increase in the production rates to meet the market needs. In the context of adhesives and liquid shim applications, this means the possibility of on-demand curing. In other words, adhesives must cure slowly at room temperature and this process must be accelerated at any cime to allow for the fastest polymerization possible. However, while on-<len1and curing is possible in several ways (ultraviolet radiation, induction, or microwave) , the route chosen in this srudy is infrared (IR) radiation. This is because this method allows curing ac low temperatures (i.e., acound so•c) and is universal, hence requiring no modification in the adhesive formulation.

Given chat the acceleration of polymerization using thermal (temperature) and nonthern1al (radiation-matter interaction) effects has been demonstrated in another study, it is now important to srudy the properties of such an adhesive after curing under 1R radiation.

In this srudy, we measured the following properties: adherence on aluminum 2024-1'3 via three-point bending, tensile strength and modulus, and flexural strength and modulus. We also srudied the patan1eters of the 1R lanlp, including the Jan1p-adhesive distance and the rate and temperature of polymerization. For this purpose, a composite design of experin1ents was used, which generally has two main advantages: screening and response surface methodology. On the one hand, screening allows determining the factors, among chose selected, chat have a signifi cant infl uence on the studied responses. At the same rime, it allows deteflllÎlling the interactions (synergistic effects) between the influencing pacan1eters. On the other hand, response surface methodology al lows quantifying the influence of the pacan1eters and detern1ining the optimal ones.

Polyepoxide adhesives are widely used in the aerospace industty for stt•uctural bonding and liquid shin1 applications [START_REF] Anes | Bonded joints of dissimilar adherends at very low temperatures -an adhesive selection approach[END_REF]. However, currently, these two-component materials cure at room temperature and in a rather slow fashion (i.e., 5-10 hof curing tin1e for an application time of Jess than 1 h). Therefore, to reduce the cm i ng time of these adhesives without modifying their application time, it is necessary to consider on-den1and cming methods, called COD [START_REF] Javadi | Cure-on-command technology: a review of the current state of the art[END_REF]. In recent yeai• s, much attention has been paid to the new and so-called unconventional poly merization methods. Many of these methods are based either on heat tt• ansfer, such as induction polymerization [START_REF] Yungwirth | Induction curing of a Phase-Toughened adhesive[END_REF][START_REF] Fink | Ferromagnetic nano-particulate and conductive mesh susceptors for induction-based repair of composites[END_REF][START_REF] Yarlagadda | Resistive susciptor design for uniform heating during induction bonding of composites[END_REF] and microwave [START_REF] Jacob | Review: thermal and non-thermal interaction of microwave radiation with materials[END_REF][START_REF] Galema | Microwave chemistry[END_REF][START_REF] Marand | Comparison of reaction mechanisms of epoxy resins undergoing thermal and microwave cure from in situ measurements of microwave dielectric properties and infrared spectroscopy[END_REF][START_REF] Cheok | Study on the microwave curing of adhesive joints using a temperature-controlled feedback system[END_REF][START_REF] Majetich | Applications of microwave-accelerated organic synthesis[END_REF], or on reaction activation by electt•omagnetic radiation, such as ultt• avi olet radiation [11-13).

den1and curing method because it eliminates the need for adding chemical compounds to the adhesive formulation, making this method versatile (i.e., applicable to products already on the market). The other advantage of IR radiation resides in the low inertia [START_REF] Belhamra | Technology and applications of infrared heating in the industrial area[END_REF] (i.e., rapid temperature variation) and low energy consumption required to pro duce IR radiation [START_REF] Kumar | Development of infrared radiation curing system for fiber reinforced polymer composites: an experimental investigation[END_REF]. In addition, IR radiation is effective in direct contact with adhesives and in areas where the adhesive is not directly exposed to the light source (shadow zone). Hence, depending on the needs and constt• aints, several IR sources whose line of sight is perpen dicular to the smface of the adhesive can be used: a far-IR source ( 400-10 cm-1 ), which heats the adhesive on the surface; a mid-IR source (4000-400 cm -1 ), which heats the adhesive at the core; and a near-IR source (12,500-4000 cm-1 ), which penett•ates the adhesive until reaching the substrate and causes more homogeneous heating. In this work, we chose infrar• ed (IR) radiation as our preferred on-In a previous study [START_REF] Genty | Fast polymerization at low temperature of an infrared radiation cured epoxy-amine adhesive[END_REF], the kinetics of an epoxy-antine ntixture

Under the usual conditions of use, the IR lamp is placed at approximately 20 cm from the adhesive to be cured and the temperature is increased at a rate of 5.5 • C⋅min -1 until the adhesive reaches a temperature plateau of 50 • C. Notably, the emitted flux in this mode has been previously measured and presented [START_REF] Genty | Fast polymerization at low temperature of an infrared radiation cured epoxy-amine adhesive[END_REF], and this way of curing is called IR-Φ Auto .

For sample preparation and the measurement of the adhesive properties, the operating conditions are provided in the corresponding sections.

Mechanical characterization

All three mechanical tests described below were performed using a mechanical testing machine (Instron 3369, Élancourt, France) equipped with a 500 N sensor with a sensitivity of 0.5% for the measured values.

Three-point bending test

A three-point bending test was performed according to the ISO 14679:1997 [START_REF] Roche | A three-point flexure test configuration for improved sensitivity to metal/adhesive interfacial phenomena[END_REF][START_REF] Roche | Measurement of the practical adhesion of paint coatings to metallic sheets by the pull-off and three-point flexure tests[END_REF][START_REF] Bouchet | The formation of epoxy/metal interphases: mechanisms and their role in practical adhesion[END_REF] standard. This test helps determine the ability of a material to adhere to a substrate. It involves bending a rectangular polymer block (25 mm in length and 5 mm in width) bonded to the surface of a substrate (40 mm in length and 10 mm in width). For this study, although the thickness of the substrate is not specified by the standard, it was chosen as 1 mm of aluminum alloy 2024-T3. Then, the three-point bending specimens are tested at room temperature at a speed of 0.5 ± 0.001 mm min -1 .

It should be noted, however, that to measure the adhesion of a model adhesive under an IR lamp, the preparation setting differs slightly from that indicated by the standard. Indeed, different curing process types (oven or IR radiation) can lead to different conversion rates and, therefore, different adhesive modulus values. As shown in Fig. 2, the load at break (F Max ) depends not only on the adherence (W Adh ) itself, but also on the stiffness of the substrates (W Substrate ). Thus, measuring the adherence value by studying F Max is only possible if the stiffness of the substrate and that of the rectangular block remain the same from one experiment to the other.

Therefore, it is important to ensure that the rectangular block is cured under the same conditions across all tests to obtain an identical conversion rate. Hence, a new preparation setting for the three-point bending specimen has been proposed. In this method, on a degreased, deoxidized, sol-gel-coated aluminum alloy [START_REF] Genty | Experimental and statistical study of three adherence tests for an epoxy-amine/aluminum alloy system: pull-Off, Single Lap Joint and Three-Point Bending tests[END_REF], the model adhesive is applied at a thickness of 0.3 mm using spacers on both sides of the substrate. The sample is then placed immediately under an IR lamp to initiate the curing process. Once polymerization has been realized, three-point bending rectangular blocks are manufactured in accordance with ISO 14679:1997 [START_REF] Roche | A three-point flexure test configuration for improved sensitivity to metal/adhesive interfacial phenomena[END_REF][START_REF] Roche | Measurement of the practical adhesion of paint coatings to metallic sheets by the pull-off and three-point flexure tests[END_REF][START_REF] Bouchet | The formation of epoxy/metal interphases: mechanisms and their role in practical adhesion[END_REF] and cured in an oven. Fig. 3 shows the method chosen for preparing the adherence samples after curing under IR radiation.

It is worth noting that after curing in the oven, the adhesives and rectangular blocks are not completely cured, since the T g value of the Fig. 1. Chemical structures of the DGEBA and TETA molecules.

under IR radiation have been studied. IR polymerization has been shown to produce different curing kinetics from those observed with a simple increase in temperature. This additional IR effect was termed the nonthermal effect. Furthermore, after it has been demonstrated that the activation energy of the epoxide-amine reaction decreases under IR radiation, the most probable hypothesis suggested was that the epoxide functions to absorb IR radiation, causing local agitation at the reactive group, followed by an increase in reactivity without a "macroscopic" increase in temperature.

In this study, we focus on the mechanical properties (adherence, tensile and flexural strength at break, and tensile and flexural modulus) of a polyepoxy adhesive cured under IR radiation. The aim of the experiments is to identify which parameters influence the mechanical properties and to optimize these parameters to enhance the adhesive itself and its cohesive properties.

Experimental

Model adhesive

The polymer matrix that we used was a stoichiometric mixture between an epoxy prepolymer and an aliphatic amine hardener. The epoxy prepolymer (n 0.03) was a bisphenol A diglycidyl ether (DGEBA, epoxy equivalent weight 171-175 g⋅eq -1 ; D.E.R.™ 332 from Dow Chemicals), and the hardener was triethylenetetramine (TETA, amine hydrogen equivalent weight 24 g⋅eq -1 ; D.E.H.™ 24 from Dow Chemicals). The two chemical structures of these molecules used to form the model adhesive are shown in Fig. 1.

The model adhesive used in this study was a stoichiometric mixture of DGEBA with 13.9 parts per hundred grams of resin (phr) of TETA. Given these proportions, the infinite onset glass transition temperature measured using differential scanning calorimetry (i.e., at a conversion degree of 1.0) was 138 • C.

Infrared curing

Fast curing was performed using an IR lamp supplied by Sunaero (Lyon, France). This method is already used industrially, in aeronautics, to accelerate the polymerization of sealants or for composite repairs [START_REF] Comoglio | Method of sealing a hollow structure, for example a fuel tank for an aircraft[END_REF]. This lamp was initially supplied to increase the kinetics of polymerization only via radiation energy transfer. It consists of a transportable control panel and an IR lamp emitter. The lamp is fixed on a tripod so that the line of sight of the IR radiation is perpendicular to the surface of the adhesive, and the distance between the lamp and the adhesive is adjustable. The induced temperatures, heat-up ramps, and heat times are set up in the control panel so that the emitted flux is automatically regulated by the temperature measurement of the cured sample with a K-type thermocouple. As previously outlined [START_REF] Genty | Fast polymerization at low temperature of an infrared radiation cured epoxy-amine adhesive[END_REF], IR emittance is mainly within the medium range (i.e., 400-4000 cm -1 ), and the radiative flux is provided in Appendix B. model adhesive cured under IR radiation is 85 • C while it is T g ∞ 138 • C when fully cured. Generally speaking, the aim in this study is to assess the influence of the curing parameters on the final mechanical properties, regardless of the degree of curing. Therefore, since the stiffness of the rectangular block always remains the same, using a constant temperature in each trial allows the results to be reproducible.

Tensile strength and modulus

Dumbbell-type specimens were molded by pouring epoxy-amine monomers into a silicon mold and then placed directly under an IR lamp, according to the various parameters considered in this study. To control the IR emission, a thermocouple was placed at the end of the sample, in an area that would not interfere with the mechanical test.

The test was performed in accordance with ISO 37:2017 at a speed of 0.5 ± 0.001 mm min -1 . Fig. 4 shows the dimensions of the manufactured samples.

Data on the tensile strength and elastic modulus were collected from these tests. Both the tensile strength (σ r , in MPa) and the elastic modulus (E, in MPa) were determined according to Equation (1) and Equation (2), respectively:

σ r F r l × e Equation 1 E ΔF l × e × Δε Equation 2
where F r is the load at break (N), l and e are, respectively, the initial width and thickness of the specimen (both in mm), and ΔF (N) is the difference in the force applied in each deformation range Δε (between 0.25% and 0.5%, as proposed in NF ISO 527-2).

Flexural strength and modulus

Beam-type specimens were molded by pouring epoxy-amine monomers into a silicon mold and then placed directly under an IR lamp, according to the various parameters considered in this study. To control the IR emission, a thermocouple was placed at the end of the sample, in an area that would not interfere with the mechanical test.

Both the load at break and flexural modulus values were determined using a three-point bending test on beam specimens, as outlined by ISO 178:2010. According to this standard, the beam dimensions for samples of thermosetting materials are a cross section of 3.2 mm by 12.7 mm and a length of 127 mm.

Both the tensile strength (σ f , in MPa) and the elastic modulus (E f , in MPa) were determined according to Equation (3) and Equation (4), respectively: Fig. 2. Illustration of the load-displacement curve for a three-point bending adherence test, highlighting the stiffness of the sample and substrate according to [START_REF] Sauvage | Using the 3-point bending method to study failure initiation in epoxide-aluminum joints[END_REF].

f 3.F f .L 2.b.d 2
Equation 3

E f L 3 .m 4.b.d 3 Equation 4
where F f is the load at break (N), L is the distance between supports (mm), b and d are, respectively, the width and thickness of the beamtype specimen (mm), and m is the slope of the deformation-load curve (i.e., m |ΔF| |Δf| , with f for deformation in mm). Thus, the values of σ f and E f can be calculated from the breaking force F f , obtained graphically on the bending curves. It is important to note that for this test, the deflection (and, consequently, the flexural modulus) is determined by the displacement of the crossbeam.

Design of experiments

By varying all the parameters for each experiment and conducting a statistical study, the design of experiments (DoE) allows many factors to be studied and correlated with a minimum number of tests. It also allows modeling the studied responses and suggesting an optimization for the experimental parameters from the established models [START_REF] Kumar | Optimization of infrared radiation cure process parameters for glass fiber reinforced polymer composites[END_REF]. A composite DoE was selected as it allows a second-order polynomial equation to model the effects of various factors. Besides, among all the designs that allow second-order polynomials, this design is the one with the least variance and no covariance. In other words, this design allows properly calculating the effect of each factor, even if the constant a0 is not perfectly calculated. All DoE data were processed using Design-Expert® software version 10, and all the influential factors were also directly determined on it. To determine the influence of IR polymerization parameters, the effects of three factors were studied: the distance between the IR emitter and the sample in centimeters (A), the induced temperature of the adhesive in • C (B), and the heating rate to reach the correct temperature in • C/min (C). All the factors and associated levels are presented in Table 1. For axial points, α levels are used, which in this DoE are set at ±1.32.

It should be noted that these three factors have been chosen for two specific reasons. First, these are the three parameters that the lamp user can control. Thus, from an industrial standpoint, it is possible to suggest an optimized lamp configuration to obtain the desired mechanical properties. However, above all, these parameters allow the effects of temperature and radiative flux to be uncorrelated; in other words, both the thermal and nonthermal effects should be investigated separately. Indeed, the heating rate and distance factor have an impact on the radiative flux emitted, initially during the first minutes (i.e., the low conversion rates) and then for the entire duration of the test. Finally, the temperature factor has an impact on both the thermal and nonthermal effects at the same time. Therefore, studying significant factors can help determine the role of thermal and nonthermal effects in the evolution of mechanical properties.

Thus, the thermal effect of IR radiation (only the increase in temperature) is explained with temperature (factor B) as a unique influential parameter. If the temperature factor is significant with AC, AB, or BC interactions, then the thermal and nonthermal (coupled) effects will be at the origin of the response variations. However, if the influencing factors are parameters A and C or their quadratic effects or respective interactions, then only the nonthermal effect will cause variations in the mechanical properties.

Table 2 provides the number of tests performed according to the DoE. For each experiment, four specimens were prepared simultaneously under the same conditions. This allowed obtaining an average value for the four samples. The DoE-01 to DoE-06 series correspond to different tests with the same factor values. This allows determining whether the answers studied yield reproducible results and are adapted to the chosen DoE methodology. In other words, it allows testing the reliability of the system. Thus, these six trials allow determining the level of uncertainty. However, it is important to note that all six of these points were not considered for the resolution of the DoE, with the exception of the first experiment, DoE-01. In addition, other series were performed several times at different dates to check the level of reproducibility. The results showed that the DoE-11 and DoE-12 series correspond to the same parameters, as well as the DoE-17 and DoE-18 series.

For each series of tests, five answers were studied: adherence (threepoint bending), tensile properties (tensile strength and modulus), and flexural properties (flexural strength and modulus), and for each answer, four samples were evaluated. For each result, an analysis of variance (ANOVA) was performed.

For this purpose, in the Design-Expert® software, the Fisher factor (F) is provided for each response analysis. This factor corresponds to the ratio between the sum of the squares of the model and the sum of the squares of the residues. Then, the p-value is defined by the probability of observing F if the null hypothesis H 0 is verified, and it was set at 5% (pvalue less than 0.05).

ANOVA was performed in two rounds with the Design-Expert® software. First, it was performed in a raw form with the available data. Thus, a p-value was calculated for each of the parameters. A p-value less than 0.05 (i.e., a confidence level of more than 95%) was considered to confirm the significance of factors. Thus, a new ANOVA (Round 2) was performed with only those parameters whose p-value was less than 5%. Table 3 shows the ANOVA (both Round 1 and Round 2) results for the three-point bending adherence tests as an example.

In this example, the resolution of the DoE in the Design-Expert® software shows that the significant terms are the C parameters (heating rate for a p-value less than 0.0001). Both of the A 2 parameter (the quadratic effect of the distance for a p-value of 0.0041) and the A and C parameters are correlated and significant (a p-value of 0.0071 for the distance/ramp correlation). Factor A (the distance for a p-value of 0.1203) should be included in the analysis because of the requirement of the model hierarchy as it has an interaction effect. Fig. 5 represents the normal probability of studentized residuals for adherence. Each point corresponds to a test conducted experimentally, and the real value of each point can be identified by its color (from blue to red for values between 106 and 327 N, respectively), the line representing the model. As can be seen, the points are close to the normal line, which means that the errors are distributed normally, and the model is appropriate. This means that using a model allows predicting the results for any parameter values contained in the test design. Thus, an optimization for the process parameters is suggested.

Results and discussion

Thermal versus infrared curing

In the first step, before the DoE was performed, IR polymerization (IR-Φ Auto ) was compared to pure thermal curing (50 • C). IR curing was achieved at a distance of 20 cm with an increase in temperature at a rate of 5.5 • C⋅min -1 and a maximum temperature of 50 • C in the stationary mode. This first step was performed to highlight the effects of IR radiation, regardless of the parameters modified in the later part of the article.

For these two methods of curing, four samples were manufactured for each of the mechanical properties tested: three-point bending adherence and flexural and tensile stress/strain properties. All the results are presented in Fig. 6.

Starting with the analysis of the adherence results (first series of columns), all samples tested up to rupture were found to exhibit an interfacial fracture (i.e., the fracture looked adhesive to the naked eye). The loads at break were, therefore, representative of the adherence (adhesive initiation of fracture) for both curing pathways. Apart from uncertainty, the load required for the initiation of rupture was the same for both polymerization pathways. In other words, the level of adherence following IR curing was the same as that following thermal curing. Therefore, from these first tests, it seems that IR radiation does not significantly modify the polymer-metal interaction. For the purpose of comparison, in the literature, microwave polymerization (which is also characterized by a nonthermal effect) was found to result in an increase in the adherence measured by a single lap joint test [START_REF] Cheok | Study on the microwave curing of adhesive joints using a temperature-controlled feedback system[END_REF]. This non difference in adherence after IR curing was unexpected, as it has previously been observed that the interphase formation (i.e., the reaction between an amine and an aluminum surface) decreases the adherence [START_REF] Bouchet | The formation of epoxy/metal interphases: mechanisms and their role in practical adhesion[END_REF]. Indeed, when an epoxy-amine mixture is cured on an aluminum alloy surface, a competition arises between polymerization and the reaction between the amine and the metal [START_REF] Aufray | Properties of the interphase epoxyamine/metal : influences from the nature of the amine and the metal[END_REF], and an interphase is created [START_REF] Bouchet | The formation of epoxy/metal interphases: mechanisms and their role in practical adhesion[END_REF][START_REF] Aufray | Properties of the interphase epoxyamine/metal : influences from the nature of the amine and the metal[END_REF][START_REF] Roche | Formation of epoxy-diamine/metal interphases[END_REF]. Hence, it is reasonable to believe that a modification of the kinetics, and therefore of the vitrification, leads to a modification of this interphase, as well as of the internal stresses and consequently the adherence. Now, the flexural strength at break (second series of columns) was found to increase by more than 20% with IR curing (146 ± 15 MPa compared to 120 ± 13 MPa). This phenomenon has already been observed in the case of microwave polymerization of Carbon Fiber Reinforced Polymer (CFRP) [START_REF] Xu | Effect of microwave curing process on the flexural strength and interlaminar shear strength of carbon fiber/bismaleimide composites[END_REF]. In the case of the composite studied, the flexural strength at break was found to evolve from 1552 ± 44 MPa under microwave to 1326 ± 29 MPa under pure thermal conditions. In that study, Xu showed that the increase observed in the strength at break is explained by the decrease in the quantity of air present in the material (air bubbles). Thus, in an autoclave (under vacuum), the flexural strength at break was found to increase to 1651 ± 32 MPa. In our study, the presence of air bubbles and the internal constraints (both can be linked) are two serious avenues explaining the increase in the flexural strength at break of the DGEBA-TETA mixture. The flexural modulus (third series of columns) of the DGEBA-TETA mixture cured under IR radiation was found to be the same as that under thermal curing. Table 3 ANOVA results for the three-point bending adherence tests.

Generally, IR radiation allows obtaining a material with a higher tensile strength (+60%), in comparison to pure thermal cured adhesive. However, it also exhibits a module that is the same as thermal curing, apart from uncertainty. The increase observed in the mechanical tensile properties has already been observed in the case of IR curing [START_REF] Kumar | Optimization of infrared radiation cure process parameters for glass fiber reinforced polymer composites[END_REF]. The explanation provided in the case of this study, which seems compatible with the IR lamp polymerization performed in this work, consists of more homogeneous IR curing resulting in a reduction in the residual stresses in the material. It is also worth noting that some similarities have already been observed when microwave curing was considered. Thus, it can be concluded that microwave radiation leads to a modification of the mechanical properties of adhesives by releasing the internal stresses [START_REF] Johnston | Microwave and thermal curing of an epoxy resin for microelectronic applications[END_REF].

The role of infrared lamp parameters (DoE)

During IR curing, the changes that occur in the mechanical properties may be a combination of thermal and nonthermal effects. To study the effects of IR radiation more accurately, the effects of temperature (thermal effects) should be uncorrelated to those of radiative flux (nonthermal effects). However, since this is not experimentally feasible, using a DoE with a Central Composite Design (CCD) is proposed. This allows the investigation of influencing factors while considering quadratic effects in a minimum number of testing. The results of the experimental tests for adherence, tensile and flexural strength at break, and tensile and flexural modulus are available in Appendix A. Each line corresponds to an average of four samples with the same batch number (i.e., performed at the same time), and the errors for each of the properties are defined by the standard deviation of all the specimens from DoE-1 to DoE-6 (i.e., 24 (6 × 4) specimens).

Adherence

Given the adherence results obtained in three-point bending (last column of the table), the resolution of the DoE in the Design-Expert® software shows that the significant terms, which are the heating rate (parameter C for a p-value less than 0.0001), the square of the distance parameter (the quadratic effect of parameter A for a p-value of 0.0041), and the distance and heating rate (a p-value of 0.0071 for the distance-ramp correlation), are correlated and significant. Notably, the distance (parameter A for a p-value of 0.1203) should be included in the analysis because of the requirement of the model hierarchy as it has an interaction effect. The final equation with the coded factors for predicting the adherence of the DGEBA-TETA mixture after IR curing is provided in Equation ( 5), and the same equation with the real factors (in cm for the distance and • C/min for the heating rate) is provided in Equation ( 6 It is worth noting that the distribution of studentized residuals is in line with the theoretical values from Equation ( 5) and Equation [START_REF] Jacob | Review: thermal and non-thermal interaction of microwave radiation with materials[END_REF]. Therefore, within the limits of the parameters used for the DoE, these equations can be used to predict the adherence of the DGEBA-TETA mixture on AA 2024-T3 after IR curing. Fig. 7 shows adherence as a function of the AC parameters for a temperature value set at 65 • C.

Indeed, the dependence of the adherence on the heating rate (C) and the combined effect of the heating rate and distance (parameter AC) are easily observable. The value of adherence increases to 275 N for the maximum heating rate and distance to just over 150 N for the minimum heating rate and maximum distance. It is worth noting that both the distance and heating rate have a significant influence on the energy received by the adhesive. Moreover, at a constant heating rate, the adherence values change a little as a function of the distance parameter alone (parameter A not significant).

The DoE shows that IR curing leads to a change in adherence (from 150 to 300 N, depending on the parameters used). This first response from the DoE, however, does not confirm the observations made in the simple comparison between thermal and IR curing (cf. Fig. 6). Recall that IR curing did not change the adherence level of the DGEBA-TETA mixture on AA 2024-T3 (240 and 255 N, respectively). Therefore, the resolution of the DoE allows determining whether IR radiation (i.e., nonthermal effect) modifies the level of adherence and, hence, the substrate-adhesive interaction. In addition, by further studying the significant factors, it was found that it is the nonthermal effect that causes a change in adherence with the heating rate (parameter C), distance (parameter A), and their interaction (parameter AC) as significant factors. Thus, high radiative flux at the beginning of curing (i.e., a high heating rate) leads to a significant increase in adherence.

Tensile strength

The tests performed according to the DoE show that parameters A and C are significant (distance and ramp for p-values of 0.0002 and 0.0482, respectively) and that the correlation of distance (parameter A) and temperature (parameter B) is also significant (with a p-value of 0.0013). The final equation with the coded factors for the prediction of According to parameters A, B, C, and AB of Equation ( 7) and Equation ( 8), the tensile strength values are well in line with the tensile strength measured in the DoE tests. This means that the determined model is reliable for predicting the tensile strength according to the distance, heating rate, and temperature parameters contained in the terminals as used in the DoE.

Besides, the tensile strength at break is expressed as a function of distance (parameter A) and temperature (parameter B) in Fig. 8, whose heating rate (parameter C) is set at 5.5 • C/min.

The correlation between the distance (parameter A) and temperature (parameter B) can be observed on this graph with a significant increase in the fracture strength when the temperature decreases at a short distance as well as a slight decrease in strength when the temperature decreases at a larger distance. Although parameter AB is significant, the temperature (factor B) is not, whereas the distance (parameter A) and heating rate (parameter C) are. This clearly indicates that the variation in the tensile strength is governed by the nonthermal effect of IR radiation.

Tensile modulus

Among the parameters of the DoE, only the correlation of temperature (parameter B) and heating rate (parameter C), that is, parameter BC, is significant on the tensile modulus response (with a p-value of 0.0211). The final equation with the coded factors for the prediction of the modulus of the DGEBA-TETA mixture after IR polymerization is provided in Equation ( 9), and the same equation with the real factors (in • C/min for the heating rate and • C for the temperature) is provided in Equation ( 10):

E r [GPa] 1.74 + 3.1110 3 B + 0.06C 0.093BC 0.10B 2 Equation 9 E r [GPa] 1.02 + 0.067 × Temperature + 0.186 × Heatingrate 2.4810 3 × Temperature × Heatingrate Equation 10
Although the BC factor is significant, the weight of this factor on the model is small. In other words, the influence of radiation (i.e., the nonthermal effect) on the flexural modulus is not evident, and if it exists, it is negligible and cannot be used to optimize the modulus values. Studying studentized residues revealed a very good match between the model and experimental data, proving that the suggested model is robust. Fig. 9 expresses the module in tension as a function of parameters B and C. the tensile strength of the DGEBA-TETA mixture after IR polymerization is provided in Equation [START_REF] Galema | Microwave chemistry[END_REF], and the same equation with the real factors (in cm for the distance, • C/min for the heating rate, and • C for the temperature) is provided in Equation ( 8):

It can be observed that parameters B and C separately are not significant and that it is their correlation that plays a role in the response studied.

Flexural strength

The resolution of the DoE shows that the significant factors for flexural failure strength are the temperature and distance. It also shows that the significant terms are the temperature (parameter B for a p-value of 0.0009), the quadratic effect of temperature (parameter B 2 for a pvalue of 0.0009), and the quadratic effect of the heating rate (parameter C 2 for a p-value of 0.0263). However, parameters A and C alone are not significant, but parameter AB is to a lesser extent (a p-value of 0.0993). The final equation with the coded factors for predicting the flexural strength of the DGEBA-TETA mixture after IR curing is provided in Equation [START_REF] Ellis | Chemistry and Technology of epoxy resins[END_REF], and the same equation with the real factors is provided in Equation ( 12 It is worth noting that the flexural strength prediction model as a function of on-demand curing parameters using IR radiation is reliable. Indeed, the studentized residues are in accordance with the theoretical values of the model represented by the red line. In Fig. 10, the flexural strength is expressed as a function of distance (parameter A) and temperature (parameter B), with parameter A being significant and parameter AB not very significant, for a fixed heating rate of 5.5 • C/min.

Thanks to the DoE and Design-Expert® software, it was demonstrated that none of the parameters of the DoE is significant for the results of the bending modulus. Here, the model is not realistic because the experimental dispersion is smaller than the uncertainty. Therefore, the lamp parameters do not allow, besides control, the induced temperature to optimize the value of the flexural modulus. In other words, the nonthermal effect does not lead to any change in the flexural modulus value.

Optimization of IR lamp parameters (DoE)

The objective here is to determine the experimental parameters of the IR lamp to optimize the properties of the adhesive cured by IR radiation. For this purpose, the equations that have previously been obtained are reused.

Verification of the previously established models

Before initiating the optimization tests, it is important to ensure that the results obtained under the conditions of the IR-Φ Auto reference test are consistent with those predicted by the models in the DoE. This validation is particularly important since the conditions of the reference test (IR-Φ Auto : 50 • C with a distance of 25 cm and a heating rate of 5.5 • C/ min) are not part of the cube of the DoE. Table 4 outlines all of these results.

The results showed that the correlation between the properties predicted by the model and those measured in practice is very good for the flexural strength at break and adherence (less than 5% difference). Moreover, the predictions of the tensile strength at break and modulus were found to be acceptable (less than 10% deviation). Therefore, the predicted results are quite accurate as the error is below 10%, and hence the model can be considered as valid for further calculations and optimization.

From the model equations, it is possible to fix some input data and classify the desired responses according to their importance. Several scenarios have been developed, and some information is presented to analyze the optimization results. First, the optimal parameters are presented on diagrams with the minimum and maximum values of the DoE. It should be noted that all optimizations are included in the experimental domain (i.e., the presence of points outside the cube is not possible). Thus, the optimum value (-•-for factors and -•-for answers) is located on a diagram whose minimum value of the DoE is on the left and the maximum value is on the right. Studying the position of the responses in the experimental domain allows calculating the desirability, which corresponds to the average of the positions of each of the responses.

First optimization

The first optimization determines the lamp parameters to obtain an adhesive with the highest mechanical properties. For this purpose, the selected input data are a distance between 20 and 40 cm, a temperature between 40 • C and 100 • C, and a heating rate between 2 • C/min and 10 • C/min, all with no optimum required. Fig. 11 presents the optimal parameters and associated responses. Recall that desirability represents Fig. 10. Influence of the parameters on the three-point bending test according to parameter AC.

Table 4

Results of the mechanical properties of the IR-cured adhesive (IR-ΦAuto) compared to the predictions provided by the model. the degree of satisfaction with the results of the model as a function of the maximum value that can be achieved (maximum value observed in the DoE). For this first optimization, the desirability is 75%, which means that, on average, the predicted mechanical properties are at 75% of their maximum value. Desirability represents the degree of satisfaction with the results of the model as a function of the maximum value that can be achieved. This index is equal to 0 if the value of the modeled response is outside the tolerance interval associated with it.

σ r [MPa] E [GPa] σ f [MPa] E f [GPa] F Max [N]
According to the models determined from the DoE data, the optimal lamp parameters are a distance between the sample and transmitter of 20 cm, a heating rate of 8.4 • C/min, and an induced temperature of 47 • C. Fig. 12 shows the influence of the temperature and distance parameter evolution on this optimization as a 2D plot.

In general, the most sensitive parameter in optimization is the temperature. Indeed, the results change a little if the distance is changed. In addition, temperature is regarded as a determining factor, and any small change in it leads to large variations in responses.

Second optimization

The purpose of this second optimization is to determine the optimal

Conclusion

In this study, we focused on the mechanical properties (adherence, tensile strength and modulus, and flexural strength and modulus) of a model adhesive after IR curing. The mechanical properties studied were determined for an IR-cured model adhesive (according to the conditions of the reference test in this study: IR-Φ Auto ) compared to pure thermal curing at the same temperature (50 • C). The results showed that the adherence properties and tensile and flexural moduli have not been modified by the IR radiation. In contrast, the tensile and flexural strength at break increased when IR curing was used, with an increase of 60% and 20%, respectively, for tensile and three-point bending loads. This difference can be explained by the assumption that IR allows more homogeneous curing and reduces the internal stresses of the model adhesive. In addition, we also studied the most significant factors influencing IR lamp curing. For this purpose, we used a central composite DoE with distance, induced temperature, and heating rate as factors. The results of the DoE are the five mechanical properties studied so far (adherence, tensile strength at break and modulus, and flexural strength at break and modulus). Generally, the DoE allowed us to determine the significant factors for each of the responses by proposing a mathematical model for predicting the values of the mechanical properties. The results further showed that the prediction of the mechanical properties for the IR-Φ Auto reference test is in line with the values obtained experimentally, proving that the suggested model is valid. In addition, determining the significant factors showed that the improvements in the tensile and flexural strength at break after IR curing can be explained by the nonthermal effect of IR radiation. Finally, an optimization of the experimental parameters (i.e., factors) was proposed according to several scenarios: either for a process-oriented optimization (the lowest possible temperature with the highest possible mechanical properties) or for a product-oriented optimization (the highest possible properties). This allowed submitting different parameters for using an IR lamp depending on the properties of the model adhesive required. This can be achieved thanks to the DoE software by weighting the importance of the different answers: the weight of the adherence property is 5, whereas the weight for the other properties is 1. Fig. 13 shows the results (factors and responses) of this optimization.

Appendix A. Results of adherence, tensile and flexural strength, tensile and flexural modulus

It should be noted that in this second optimization, the optimal parameters are different from those in the previous one. That is, the distance used is 40 cm with an induced adhesive temperature of around 60 • C and a heating rate of 9.5 • C/min. These parameters allow obtaining an excellent level of adhesion (very close to the highest possible value) and maintaining an average desirability level of 80%. It is also true that the induced temperature is high compared to the industrial constraints. Fig. 14 shows the influence of temperature and distance parameter evolution on this optimization as a 2D plot.

In this optimization, modifying the lamp parameters leads to a strong variation in the response. Therefore, it is possible to propose parameters that can result in maximized adhesion, but with less consistent mechanical properties.

Fig. 3 .

 3 Fig. 3. Method used for preparing three-point bending samples.

Fig. 4 .

 4 Fig. 4. Dumbbell samples used to determine the tensile strength and modulus (dimensions in mm).

  σ

Fig. 5 .

 5 Fig. 5. "Studentized" residuals plot for adherence data measured in threepoint bending.

Fig. 6 .

 6 Fig. 6. Some mechanical characteristics of the DGEBA-TETA mixture according to both methods of curing (IR-Φ Auto and thermal curing).

2 5 Adh.[N] 484 18 . 4 ×

 25184 ): Adh.[N] 197.7 9.7A + 34.4C + 22.6AC + 37.7A Equation Distance 13.4 × Heatingrate + 0.9 × Distance × Heatingrate + 0.2 × Distance 2 Equation 6

Fig. 7 .

 7 Fig. 7. Influence of the parameters on the measurement of the three-point bending according to parameter AC.

  σ r [MPa] 44.45 + 8.09A 5.63B + 2.00C + 8.56AB Equation 7

Fig. 8 .

 8 Fig. 8. Influence of the parameters on the measurement of the tensile strength at break according to parameter AB.

Fig. 9 .

 9 Fig. 9. Influence of the parameters on the measurement of the tensile modulus according to parameter BC.

  σ r [MPa] 8.54 0.18 × Distance 0.81 × Temperature + 0.799 × Heatingrate + 7.19.10 3 × Distance × Temperature Equation 8

2 14.76C 2

 22 ): σ y [MPa] 148.83 + 0.21A 19.92B 0.43C + 12.79AB 21.65B Equation 11 σ y [MPa] 76.26 5.52 × Dis tan ce + 8.62 × Temperature + 25.79 × Heatingrate + 0.08 × Dis tan ce × Temperature 0.096 × Temperature 2 2.36 × Heatingrate 2 Equation 12

Fig. 11 .

 11 Fig. 11. IR lamp parameters calculated using models from the DoE.

Fig. 12 .

 12 Fig. 12. Influence of temperature and distance on the results from the optimization.

Fig. 13 .

 13 Fig. 13. IR lamp parameters calculated using the models from the DoE.

Fig. 14 .

 14 Fig. 14. Influence of temperature and distance on the results from the optimization.

Appendix B .

 . Infrared lamp emitter spectrum and the measurement of the temperature of the sample when the set point is 50 • C parameters to propose an adhesive with the highest adhesion possible.

Table 1

 1 Factors and levels chosen for the DoE.

	Factors	Levels
		-α	-1	0	+1	+α
	(A) Distance [cm]	16.8	20.0	30.0	40.0	43.1
	(B) Temperature [ • C]	45.2	50.0	65.0	80.0	84.7
	(C) Heating rate [ • C/min]	2.2	3	5.5	8	8.8

Table 2

 2 List of tests of the DoE. The first row (colored in red and underlined) is the reference point that is outside the DoE, whereas the labels written in italics are iterations.

	Number	A	B	C	(A)	(B)	(C)
					Distance	Temperature	Heating rate
					[cm]	[ • C]	[ • C/min]
	IR-	N/	N/	N/	25	50	5.5
	Φ Auto	A	A	A
	DoE-01	0	0	0	30	65	5.5
	DoE-02	0	0	0	30	65	5.5
	DoE-03	0	0	0	30	65	5.5
	DoE-04	0	0	0	30	65	5.5
	DoE-05	0	0	0	30	65	5.5
	DoE-06	0	0	0	30	65	5.5
	DoE-07	+1	-1	+1	40	50	8
	DoE-08	-1	-1	+1	20	50	8
	DoE-09	-1	-1	-1	20	50	3
	DoE-10	+1	-1	-1	40	50	3
	DoE-11	0	-α	0	30	45.2	5.5
	DoE-12	0	-α	0	30	45.2	5.5
	DoE-13	+α	0	0	43.1	65	5.5
	DoE-14	-α	0	0	16.8	65	5.5
	DoE-15	0	0	-α	30	65	2.2
	DoE-16	0	0	+α	30	65	8.8
	DoE-17	0	+α	0	30	84.7	5.5
	DoE-18	0	+α	0	30	84.7	5.5
	DoE-19	-1	+1	-1	20	80	3
	DoE-20	-1	+1	+1	20	80	8
	DoE-21	+1	+1	+1	40	80	8
	DoE-22	+1	+1	-1	40	80


Giuseppe Greco
email: g.greco@vu.nl

Michael Moortgat
email: mjmoortgat@gmail.com

Valentin D Richard
email: valentin.richard@live.fr

Apostolos Tzimoulis
email: a.tzimoulis@vu.nl

Lambek-Grishin Calculus: focusing, display and full polarization

Focused sequent calculi [START_REF] Andreoli | Logic programming with focusing proofs in linear logic[END_REF][START_REF] Andreoli | Focussing and proof construction[END_REF][START_REF] Miller | An Overview of Linear Logic Programming[END_REF] make use of syntactic restrictions on the applicability of inference rules achieving three main goals: (i) the proof search space is considerably reduced without losing completeness, (ii) every cut-free proof comes in a special normal form, (iii) a criterion for defining identity of sequent calculi proofs. Being able to identify or tell apart two proofs has far-reaching consequences.

We introduce a novel focused display calculus fD.

LG and a fully polarized algebraic semantics (see the last paragraph of this abstract for more details on this) F P.LG for Lambek-Grishin logic [START_REF] Moortgat | Proof nets and the categorial flow of information[END_REF] by generalising the theory of multi-type calculi [START_REF] Frittella | Multi-type sequent calculi[END_REF] and their algebraic semantics, admitting not only heterogeneous operators [START_REF] Birkhoff | Heterogeneous algebras[END_REF], but also heterogeneous consequence relations (see [START_REF] Jung | Multi lingual sequent calculus and coherent spaces[END_REF]) now interpreted as weakening relations [START_REF] Kurz | Stone duality for relations[END_REF] (i.e. a natural generalisation of partial orders). The calculus fD.LG has strong focalization and it is sound and complete w.r.t. F P.LG. This completeness result is in a sense stronger than completeness with respect to standard polarized algebraic semantics, insofar we do not need to quotient over proofs with consecutive applications of shifts operator over the same formula (see the last paragraph of this abstract for more details on this). We also show a number of additional results. fD.LG is sound and complete w.r.t. LG-algebras: this amounts to a semantic proof of the completeness of focusing, given that the standard (display) sequent calculus for Lambek-Grishin logic is complete w.r.t. LG-algebras. fD.LG and the focused calculus fLG of Moortgat and Moot are equivalent with respect to proofs, indeed there is an effective translation from fLG-derivations to fD.LGderivations and vice versa: this provides the link with operational semantics, given that every fLG-derivation is in a Curry-Howard correspondence with a directional λµ µ-term.

We conjecture that this approach, here tailored for the signature of the Lambek-Grishin logic, can be extended to a large class of logics, namely all lattice expansions logics extended with analytic inductive axioms (see [START_REF] Greco | Unified correspondence as a prooftheoretic tool[END_REF]). We conjecture that if a calculus belongs to this class, then it enjoys cut-elimination, aiming at generalizing the cut-elimination meta-theorem in the tradition of display calculi (see [START_REF] Wansing | Sequent Systems for Modal Logics[END_REF]). Moreover, we conjecture that any displayable logic [START_REF] Greco | Unified correspondence as a prooftheoretic tool[END_REF] can be equivalently presented as an instance of this class.

In what follows we summarise the main features of this analysis in general terms, without special reference to Lambek-Grishin logic. In the case of focused sequent calculi, the distinction between positive versus negative formulas is the key ingredient for organising proofs in phases. The distinction is proof-theoretically relevant in that it reflects a fundamental distinction between logical introduction rules. We observe that this distinction is also semantically grounded, indeed the main connective of a positive formula (in the original language of the logic) is a left adjoint/residual and the main connective of a negative formula (in the original language of the logic) is a right adjoint/residual. Proofs in focalized normal form (see [START_REF] Moortgat | Proof nets and the categorial flow of information[END_REF]) are cut-free proofs organised in three phases: two focused phases (either positive or negative) and one non-focused phase (also called neutral phase). A focused positive (resp. negative) phase in a derivation is a proof-section where a formula is decomposed as much as possible only by means of non-invertible logical rules for positive (resp. negative) connectives. This formula and all its immediate subformulas in this proof-section are said 'in focus'. All the other rules are applied only in non-focused phases. So, each derivable sequent has at most one formula in focus. Moreover, the interaction between two focused phases is always mediated by a non-focused phase.

Shift operators -usually denoted as ↑ and ↓ ([7, 8, 3])-are often considered to polarize a focused sequent calculus, i.e. as a tool to control the interplay between positive and negative formulas and the interaction between phases. Shifts are adjoint unary operators that change the polarity of a formula, where ↑ goes from positive to negative, ↓ goes from negative to positive, and ↑ ↓. In this paper, we consider positive and negative formulas as formulas of different sorts. We also distinguish between positive (resp. negative) pure formulas and positive (resp. negative) shifted formulas, i.e. formulas under the scope of a shift operator. So, we end up considering four different sorts, each of which is interpreted in a different sub-algebra. Therefore, in this setting shifts are heterogeneous operators, where ↑ gets split into ↑ (from positive pure formulas into negative shifted formulas) and (from positive shifted formulas into negative pure formulas), ↓ gets split into ↓ (from negative pure formulas into positive shifted formulas) and (from negative shifted formulas into positive pure formulas), ↑ and ↓. Moreover, the composition of two shifts is still either a closure or an interior operator (by adjunction), but we do not assume that it is an identity. We call a presentation of a logic with the features described above full polarization.


Synchrotron x-ray diffraction was used to investigate the insertion of helium in the archetypal siliceous zeolite Mobile-Five (MFI). Due to the very high degree of helium penetration into the pores and the framework, the monoclinic form of MFI is stabilized over a large pressure interval and the compressibility is much lower than when the pores are empty or filled with any other guest molecules or atoms except hydrogen. The structure of MFI transformed from monoclinic P21/n to orthorhombic Pnma with a discontinuous volume increase of 3% at pressures close to 10 GPa due to the filling process. The material is highly strained at pressures above 42 GPa, but remains crystalline. The strain is released on decompression and highly crystalline material is recovered.

Introduction

Helium has been found to easily penetrate the frameworks of certain forms of silica at high pressure such as -cristobalite [START_REF] Sato | Anomalous Behavior of Cristobalite in Helium under High Pressure[END_REF][START_REF] Matsui | Crystal Structures and Stabilities of Cristobalite-Helium Phases at High Pressures[END_REF] and silica glass [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces Its Compressibility[END_REF][START_REF] Shen | Effect of Helium on Structure and Compression Behavior of SiO2 Glass[END_REF][START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF][START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF][START_REF] Weigel | Polarized Raman Spectroscopy of v-SiO2 under Rare-Gas Compression[END_REF] , resulting in swelling of the structure. Such behavior does not occur on compression in argon [START_REF] Weigel | Polarized Raman Spectroscopy of v-SiO2 under Rare-Gas Compression[END_REF] , but can occur with slow kinetics above a threshold pressure in the case of the intermediate-sized neon atom [START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF] . In the case of porous, pure silica zeolites, larger atoms and molecules readily enter the pores at high pressure [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF][START_REF] Coasne | Enhanced Mechanical Strength of Zeolites by Adsorption of Guest Molecules[END_REF][START_REF] Santoro | Silicon Carbonate Phase Formed from Carbon Dioxide and Silica under Pressure[END_REF][START_REF] Santoro | High-Pressure Synthesis of a Polyethylene/Zeolite Nano-Composite Material[END_REF][START_REF] Scelta | High Pressure Polymerization in a Confined Space: Conjugated Chain/Zeolite Nanocomposites[END_REF][START_REF] Vezzalini | High-Pressure-Induced Structural Changes, Amorphization and Molecule Penetration in MFI Microporous Materials: A Review[END_REF][START_REF] Santoro | High Pressure Synthesis of All-Transoid Polycarbonyl [-(C=O)-](N) in a Zeolite[END_REF][START_REF] Arletti | Pressure-Induced Penetration of Guest Molecules in High-Silica Zeolites: The Case of Mordenite[END_REF][START_REF] Richard | Pressure-Induced Insertion of Ammonia Borane in the Siliceous Zeolite, Silicalite-1F[END_REF][START_REF] Santoro | Synthesis of 1D Polymer/Zeolite Nanocomposites under High Pressure[END_REF][START_REF] Santoro | Intermolecular Interactions in Highly Disordered, Confined Dense N2[END_REF][START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF][START_REF] Marqueno | An Ultrahigh CO2-Loaded Silicalite-1 Zeolite: Structural Stability and Physical Properties at High Pressures and Temperatures[END_REF][START_REF] Santoro | Dense, Subnano Phase of Clustered O2[END_REF][START_REF] Santoro | Insertion and Confinement of H2O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF][START_REF] Xu | High-Pressure Insertion of Dense H2 into a Model Zeolite[END_REF][START_REF] Paliwoda | Anomalous Volume Changes in the Siliceous Zeolite Theta-1 TON Due to Hydrogen Insertion under High-Pressure, High-Temperature Conditions[END_REF] . In the case of the archetypal all-silica zeolite, silicalite-1 with the monoclinic Mobile-Five (MFI) structure and a three-dimensional pore system [START_REF] Olson | Crystal-Structure and Structure-Related Properties of ZSM-5[END_REF][START_REF] Hay | Orthorhombic Monoclinic Phase-Changes in ZSM-5 Zeolite Silicalite[END_REF] , argon or simple molecules such as water, nitrogen and carbon dioxide can completely fill the pores system, but do not enter the framework [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF][START_REF] Santoro | Intermolecular Interactions in Highly Disordered, Confined Dense N2[END_REF][START_REF] Santoro | Insertion and Confinement of H2O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF] . The exception is hydrogen [START_REF] Xu | High-Pressure Insertion of Dense H2 into a Model Zeolite[END_REF] , which has very recently been found to enter the secondary building units of the framework. The structure of MFI [START_REF] Olson | Crystal-Structure and Structure-Related Properties of ZSM-5[END_REF][START_REF] Hay | Orthorhombic Monoclinic Phase-Changes in ZSM-5 Zeolite Silicalite[END_REF] is formed by linear 5.3 Å x 5.6 Å channels along the b direction and 5.1 Å x 5.5 Å sinusoidal pores in the xz plane (Figure 1). High silica, MFI undergoes a ferroelastic phase transition [START_REF] Ardit | Monoclinic-Orthorhombic Phase Transition in ZSM-5 Zeolite: Spontaneous Strain Variation and Thermodynamic Properties[END_REF] from the monoclinic P21/n to the orthorhombic Pnma form at 349 K. A similar phase transition is also observed at high pressure in the vicinity of 1 GPa 8 . When siliceous MFI is compressed in penetrating medium, such as H2O or CO2, this transition can occur at even lower pressures [START_REF] Marqueno | An Ultrahigh CO2-Loaded Silicalite-1 Zeolite: Structural Stability and Physical Properties at High Pressures and Temperatures[END_REF][START_REF] Santoro | Insertion and Confinement of H2O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF] . Filling with guest species was also found to stabilize the orthorhombic phase with respect to pressure-induced amorphization at very high pressures [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF][START_REF] Santoro | Intermolecular Interactions in Highly Disordered, Confined Dense N2[END_REF][START_REF] Xu | High-Pressure Insertion of Dense H2 into a Model Zeolite[END_REF] . In H2-filled MFI [START_REF] Xu | High-Pressure Insertion of Dense H2 into a Model Zeolite[END_REF] , the crystalline phase is still observed at 60 GPa.

In the present work, we explore siliceous MFI with a focus on the effect of helium insertion into the monoclinic and orthorhombic phases, their stability fields as well as the volume changes and compressibility of this material.

Figure 1.

Structures of monoclinic P21/n11, Space Group 14 (below) and orthorhombic Pnma, S.G. 62 MFI (above). Comparing the structures, we note that the monoclinic form has =90.6207°, and some small differences are observed in the Si-O-Si bridging angles. The unit cell of the monoclinic form is described in the non-standard space group setting in order to conserve the basis similar to the orthorhombic phase.

Methods

Calcined siliceous MFI, silicalite-1, was prepared by the sol-gel route in a fluoride medium followed by crystallization under hydrothermal conditions and then calcination as described previously [START_REF] Santoro | Insertion and Confinement of H2O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF] . The sample was placed in the hole of a gasket in a membrane diamond anvil cell Pnma P2 1 /n along with a ruby sphere used as a pressure sensor [START_REF] Mao | Calibration of the Ruby Pressure Gauge to 800-Kbar under Quasi-Hydrostatic Conditions[END_REF] . Fluid helium was introduced at high pressure and the cell was closed at 0.13 GPa. Angle-dispersive x-ray powder diffraction measurements were performed at the extreme conditions beamline P02.2 [START_REF] Liermann | The Extreme Conditions Beamline P02.2 and the Extreme Conditions Science Infrastructure at Petra III[END_REF] of the PETRA-III synchrotron at DESY using monochromatic radiation with a wavelength of 0.4828 Å. The focused beam spot dimensions were 9 m  3 m. Detection was performed with a Perkin Elmer XRD1621 (2048 x 2048 pixels, 200 m x 200 m) detector. The 2-D diffraction images were integrated, and then 1-D diffraction profiles were obtained using the program Dioptas [START_REF] Prescher | Dioptas: A Program for Reduction of Two-Dimensional X-Ray Diffraction Data and Data Exploration[END_REF] .

Le Bail refinements to obtain precise lattice parameters and volumes were performed using the program Fullprof [START_REF] Rodriguez-Carvajal | Magnetic Structure Determination from Powder Diffraction Using the Program Fullprof[END_REF] . The pressure was measured using the ruby fluorescence method [START_REF] Mao | Calibration of the Ruby Pressure Gauge to 800-Kbar under Quasi-Hydrostatic Conditions[END_REF] .

Results and discussion

At ambient pressure, the sample adopted the known monoclinic structure for calcined MFI [START_REF] Hay | Orthorhombic Monoclinic Phase-Changes in ZSM-5 Zeolite Silicalite[END_REF] .

Upon loading with helium at 0.13 GPa, the structure remained monoclinic, however, in contrast, to previous work on empty MFI 8,[START_REF] Quartieri | Elastic Behavior of MFI-Type Zeolites: 3-Compressibility of Silicalite and Mutinaite[END_REF][START_REF] Haines | Topologically Ordered Amorphous Silica Obtained from the Collapsed Siliceous Zeolite, Silicalite-1-F: A Step toward "Perfect" Glasses[END_REF] and MFI filled with many atoms of molecules, such as H2O [START_REF] Santoro | Insertion and Confinement of H2O in Hydrophobic Siliceous Zeolites at High Pressure[END_REF] and CO2 [START_REF] Marqueno | An Ultrahigh CO2-Loaded Silicalite-1 Zeolite: Structural Stability and Physical Properties at High Pressures and Temperatures[END_REF] , transformation to the more symmetrical orthorhombic Pnma was not observed below 1 GPa in spite of pore filling. In fact, even up to 9.1 GPa, the unit cell remained monoclinic with a clear splitting of hkl and hk reflections (Figures 2 and3). The monoclinic angle  was not found to decrease with pressure indicating no obvious tendency towards the orthorhombic form (Table 1, Figure 4). The unit cell of the monoclinic form is described in the non-standard P21/n11, S.G. 14 space group setting to conserve the same reference as the Pnma orthorhombic phase. The relative unit cell volume in the case of He filling is much higher than for the orthorhombic phases of empty [START_REF] Haines | Topologically Ordered Amorphous Silica Obtained from the Collapsed Siliceous Zeolite, Silicalite-1-F: A Step toward "Perfect" Glasses[END_REF] or Ar or CO2-filled MFI [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF] , resulting in a monoclinic MFI swelled with He (Figure 4). Helium is different from the other guest species mentioned above as it has been clearly shown to penetrate the frameworks of various silica forms such as silica glass [START_REF] Sato | Helium Penetrates into Silica Glass and Reduces Its Compressibility[END_REF][START_REF] Shen | Effect of Helium on Structure and Compression Behavior of SiO2 Glass[END_REF][START_REF] Weigel | Vitreous Silica Distends in Helium Gas: Acoustic Versus Static Compressibilities[END_REF] and cristobalite [START_REF] Sato | Anomalous Behavior of Cristobalite in Helium under High Pressure[END_REF] and thus can fill various rings and cages in the framework in addition to the 5.5 Å diameter pores. The situation is similar for H2 and in the orthorhombic form of MFI at high pressure, hydrogen molecules occupy the pores, but also the cages in the framework built up of 5-and 6-membered rings of SiO4 tetrahedra [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] . In the case of H2; however, measurements were not performed on the monoclinic form at low pressure and the monoclinicorthorhombic phase transition was not investigated [START_REF] Thibaud | Saturation, of the Siliceous Zeolite TON with Neon at High Pressure[END_REF] . In another siliceous zeolite TON, insertion of H2 at high pressure and high temperature was found to induce an anomalous volume increase and a phase transition to a higher symmetry structure [START_REF] Paliwoda | Anomalous Volume Changes in the Siliceous Zeolite Theta-1 TON Due to Hydrogen Insertion under High-Pressure, High-Temperature Conditions[END_REF] .

The a lattice parameter decreases up to pressures of 7GPa, but between 7 and 9 GPa it is almost incompressible indicating the beginning of pre-transition phenomena (Figure 4). This is followed by major changes in the diffraction pattern at 10.9 GPa, which still in the fluid phase and more than 1 GPa below the solidification pressure of helium at 12.1 GPa,. At 10.9 GPa the peaks sharpen and shift to lower 2 angles, the hkl reflections splittings disappear (e.g. 13 and 313), and additionally hkl reflections merge with khl reflections. At this pressure, the unit cell parameters are a = b = 19.443(2) Å and c = 13.071(3) Å with accidentally the same values for a and b, even though from a structural point of view these directions are very different. This accidental equivalence is not new and it was previously observed for MFI in Ar and CO2 at high pressure [START_REF] Haines | Deactivation of Pressure-Induced Amorphization in Silicalite SiO2 by Insertion of Guest Species[END_REF] . A volume increase of close to 3% is observed at the phase transition due to insertion of the dense fluid helium into the structure (Figure 5). A continuous or discontinuous decrease in monoclinic angle and volume should be expected for this monoclinic to orthorhombic, ferroelastic, phase transition at high pressure. The present behavior is clearly different from the previous reported cases due to the different nature of penetrating agent -He, which enters the voids in the framework of the crystal structure. Here, we also report that the a cell parameter is significantly more compressible in the orthorhombic phase of He-filled MFI, which corresponds to the direction of the sinusoidal pores. The b parameter along the direction of the linear pores is particularly stiff. The peaks of MFI in helium remain very sharp up to 42 GPa due to the low degree of deviatoric stress of this quasihydrostatic pressure-transmitting medium (Figure 2). Above 42GPa, broadening occurs linked to strain in the structure of MFI. The lines sharpen on pressure release, but we observe some hysteresis in structural behavior. After decompression to 6.2 GPa, the volume of MFI is similar to that observed on compression, but the structure remains orthorhombic.

Cell parameter (Å) P (GPa)

Table 1. Unit cell parameters of monoclinic (space group P21/n11, S.G.14) and orthorhombic silicalite-1 (space group Pnma) in He as a function of pressure. The structural transition occurs between 9.1 and 10.9 GPa upon compression. In comparison to He-filled MFI, the diffraction lines are slightly broader in H2-filled MFI, but a rapid increase was never observed and the broadening is more gradual. The unit cell is less compressible than in the case of He-filled MFI. As the He atoms and the hydrogen molecules are similar in size, the amount of helium in the MFI structure could be expected to be similar to that found in the case of H2 from GCMC simulations [START_REF] Xu | High-Pressure Insertion of Dense H2 into a Model Zeolite[END_REF] at pressures where the unit cell volumes are similar as for example, between 12 and 16 GPa, for which around 135 H2 molecules were inserted per unit.

P(GPa) a(Å) b(Å) c(Å) (°) V(Å 3 ) 0.

Conclusions

Helium readily fills the structure of monoclinic MFI, stabilizing this phase at pressures of ~10 GPa, which is much higher than reported for any other atomic or molecular guest species.

Above this pressure, but still 1-2 GPa below the solidification pressure of helium, a transition to the orthorhombic form is observed with a volume increase of about 3% due to further filling with dense fluid helium that completely changes the mechanism of this transition, which is normally ferroelastic in the absence of pore filling. Indeed, changes in the local structure of inserted helium, mimicking solidification-related changes in pure helium, are likely to occur and to be the driving force for the observed transition in He filled MFI.

Figure 2 .

 2 Figure 2. X-ray diffraction patterns of MFI in He as a function of pressure (=0.4828 Å). The data

Figure 3 .

 3 Figure 3. Experimental (black), calculated (red) and difference (blue) profiles (=0.4828 Å) for the Le Bail fits for the P21/n structure of MFI/He at 9.1 GPa (below) and for the Pnma structure of MFI/He at 10.9 GPa (above). Vertical bars indicate the calculated positions of the Bragg reflections.

Figure 4 .

 4 Figure 4. Cell parameters of MFI in He as a function of pressure on compression (solid

Figure 5 .

 5 Figure 5. Relative volume (V/V0) of MFI in He as a function of pressure on compression (solid


Isabelle Trenque

Stéphane Mornet

Antoine Villesuzanne

Manuel Gaudon

Discussion on the structural anisotropy of würtzite-type compounds

Keywords: Würtzite, Structure distortion, Empirical models, Bond valence model

The würtzite-type ionic compounds exhibit flattened tetrahedral sites with a decentering of the central atom. The tetrahedral distortion and the relationship between the atom parameter (z) and the c/a ratio were explained on the basis of cell volume optimization respecting the bond valence model, and the role of anharmonic atomic vibrations was emphasized.

Introduction

AX ionic compounds (A and X for cation and anion respectively) can be stabilized into various structure types as würtzite, blende, rocksalt, etc. Many works in literature deal with the phase transitions that can occur for zinc oxide between würtzite, blende and/or rocksalt forms [1][2][3][4][5] . For instance, the influence of the doping on the stability of the different allotropic forms of zinc oxide was investigated by first-principles calculations [6] , or experimentally from X-ray powder diffraction using synchrotron [7][8][9] . The würtzite-type compounds can be described as a hexagonal close-packed structure (space group P63mc) where both A cation and X anion are in tetrahedral coordination. Usually, the atomic parameter z = 3/8 and the ratio c/a = (8/3) ½ , corresponding to the perfect hexagonal eutaxy, i.e. values for which the four bond lengths are equal (L 0 bond length), are not observed in crystals of the würtzite type. In most of the würtzitetype compounds, the tetrahedral sites are flattened along c axis and the central atom is not located at the geometrical center (Fig. 1). The central atom decentering over-compensates the tetrahedral cage flattening: the A-X L 2 bond parallel to c is longer than the three L 1 bonds.

Experimental

The solver of Microsoft Excel ® was used in order to minimize the cell volume, while varying in a semi-free way the structural parameters. The chemical bond lengths of the wurtzite-type structures can be varied but these bond lengths should, in the same time, respect the Brown and Altermatt [START_REF] Brown | The Chemical Bond in Inorganic Chemistry, The Bond Valence Model[END_REF][START_REF] Brown | [END_REF] law for imposed valence and r 0 bond parameter: see hereafter, Equation (1).

Density Functional Theory (DFT) and the generalized gradient approximation (GGA) for the exchange-correlation (xc) functional, as implemented in the VASP package [12][13][14][15][16] , were used. Projected augmented wave potentials were chosen to describe ionic core-valence interactions. Thresholds were set to 10 -5 eV and 10 -4 eV/Å for electronic and structural convergence, respectively, with a plane-wave cutoff energy of 400 eV and a set of 30 k-points for the irreducible wedge of the Brillouin zone.

Results and discussion

Cation decentering

The cation decentering is directly correlated to the divergence of L 2 and L 1 bonds lengths. In 1978, O'Keeffe and Hyde proposed to consider a direct correlation between the non-bonded interactions (X-X or A-A non-bonded distances) and the A-X bond lengths [17] . Their critical assumption was that non-bonded distances are all the more short (i.e. under compression) than the A-X bond lengths are lengthened (i.e. under tension). They hence obtained a linear equation linking the c/a ratio and the z parameter (Fig. 2), in good agreement with the (z, c/a) parameters of various würtzitetype compounds. This important work was the raw material for lot of posterior publications [18][19][20][21][22][23] . Nevertheless, this proposition is neither more nor less the reformulation in a mathematical way of the observation of a systematic overcompensation of the tetrahedral cages flattening by the central atom decentering. Moreover, the tension and compression strains attributed to non-bonded interactions remained only a theoretical proposition. From our knowledge, no further works were devoted to the subject so far.

The situation cannot be easily explained, even from first principles. Using DFT and the generalized gradient approximation, we performed a careful structural optimization for ZnO within the P6 3 mc space group. The z atomic parameter was optimized for a set of (c/a, a) values (Fig. 3a). The optimized values were a = 3.2944 Å, c/a = 1.607, z = 0.3803; thus, the chosen DFT/GGA scheme retrieves the correct trends in c-axis flattening and metal atom decentering (z), but clearly underestimates both. In principle, the agreement with experimental data can be improved by testing other xc functionals. Nevertheless, it is not straightforward to identify, from DFT alone, the driving forces for the structural features in würtzite-type compounds. Also, our calculations did not reveal any unstable phonon mode, whose freezing could explain the observed distortions. ). The values obtained by Albertsson et al. [24] for zincite analyzed at different temperatures are also plotted in the gray area.

In this paper, we propose a new explanation of this phenomenon. The z versus c/a was calculated in order to minimize the cell volume, with a constrain to respect for all the metal-oxygen distances: the Brown's bond valence model [START_REF] Brown | The Chemical Bond in Inorganic Chemistry, The Bond Valence Model[END_REF] . Actually, it was considered that, whatever the structural distortion, the A-X bond lengths of würtzite-type compounds shall first satisfy the valences. The relation between the bond lengths and valences is given by the wellknown Brown & Altermatt equation [START_REF] Brown | [END_REF] . For A V+ X V-compositions, with V being the valence of the anion or cation, the equation becomes:

where r 0 is an empirically-determined parameter, characteristic of the A-X bond. The valence is then a function of L 1 and L 2 bond lengths, which can be then expressed as functions of z, c/a and c parameters only:

Finally, it can be noted that the tetrahedral sites or würtzite cell volumes depend only on c and c/a parameters. Hence, for various c/a fixed values, the c and z parameters can be treated both as variables and were refined in order to minimize the cell volume (or equivalently, the tetrahedral site volume) and to satisfy the Brown & Altermatt equation. Indeed, the two imposed conditions allow the refinement of the two parameters. Correlation laws between c/a and z parameters can be obtained using the solver of Microsoft Excel ® or other common software. The law was thus calculated on both sides of the isotropic state (c/a = (8/3) ½ , z = 0.375) (Fig. 3b andc). Laws exhibiting a high proximity with the linear dependence proposed by O'Keeffe & Hyde were found, illustrating the overcompensation mechanism of the tetrahedral cages distortion versus the decentering.

fixed c/a and a values), the optimum value of the z atomic parameter was calculated (in brackets). Relation between z and c/a from cell volume minimization obtained for würtzite-type compounds with: b) Different cation-anion L 0 distances (but with same ion valences); c) Different ion valences (but with

same bond lengths). In both cases, the calculation was made to satisfy Equations ( 1), ( 2) and to minimize the cell volume.

First, the law was calculated for various cation and anion sizes, i.e. for various isotropic L 0 bond lengths, i.e. with different r 0 values, and a fixed valence, for instance, V = 2 (Fig. 3b). The choice for the starting r 0 values of 1.5 and 2 was made from the typical values found for würtzite compounds; as examples, r 0 for CdS, one of würtzite compound with the longest bond lengths, is 2.3, and for BeO, one of the würtzite compounds with the shortest bonds, it is 1.4 [START_REF] Brown | The Chemical Bond in Inorganic Chemistry, The Bond Valence Model[END_REF] . The correlation law slightly depends on the A-X bond lengths. The results show that the shorter the A-X bonds, the higher the overcompensation of the central atom decentering: the z atomic parameters vary more intensely for the shorter bonds.

Second, the law was calculated for various anion/cation valences but fixed L 0 bond length (for instance, 1.75 Å) (Fig. 3c).

There is clearly no influence of the valence on the correlation law between z and c/a ratio.

The "overcompensation mechanism", i.e. the arrangement of würtzite-type structures to exhibit the longest A-X bond parallel to the cell flattening axis is illustrated in Fig. 4. It may be noticed that for flattened cages (c/a < (8/3) ½ ), the L 2 bond parallel to c is longer than the other three L 1 bonds. Moreover, the L 2 bond is all the more lengthened, and the L 1 bonds all the more shortened, than the tetrahedral cages are flattened.

Tetrahedral flattening

In a second step, we tried to propose an explanation of the c/a distortion alone, i.e. the source of the tetrahedral flattening itself at the origin of the cation decentering.

Albertsson et al. showed that anharmonic thermal vibrations take place into zincite compounds [24] . In this way, it is easy to consider, as the 6 3 axis is parallel to c, that the c axis corresponds to the anisotropic axis of the atomic thermal vibrations. In a very simple model, we treated the zincite structure as the temporal superimposition of two crystals exhibiting a central atom decentering equal to +Δz and -Δz, respectively, i.e. in a frozen phonon approach. In other words, the zincite was treated as a new crystal which exhibits central atoms distributed equally on two sub-positions as represented in Fig. 1 (right).

Then, for crystals with various c/a ratios and their associated z atomic parameters as previously established, the corresponding reticular energy (except for a constant) was calculated: the crystal reticular energy was estimated from the electrostatic potential between the cations and anions in first neighboring, i.e. considering the first term equal to 1/L 2 + 3/L 1 sum in an ionic approach.

These reticular energies calculated for "different temperatures" are reported in Fig. 5. It can be shown that for a nonvibrating central atom, the perfect hexagonal eutaxy (c/a = (8/3) ½ and z = 0.375) is the more stable situation. However, increasing the central atom splitting to simulate the temperature increase, it can be seen that the most stable situation is shifted to more and more flattened tetrahedral cages. It can be noted that the four more distorted würtzite compounds: GaN, BeO, ZnO and AlN, in ascending order of distortion degree, correspond to the more ionic and lighter compounds. Moreover, the covalent würtzite compositions built on heavier elements as cadmium/zinc sulfides and selenides are isotropic. These different behaviors can be interpreted as the increase of the ions thermal vibration with the decrease of the bond covalency and the ion atomic number.

Conclusion

In conclusion, the würtzite-type structures show distorted hexagonal close-packed structure with a decentering of the cations from the tetrahedral sites geometrical center. To our knowledge, the driving forces for the departure of the c/a and z parameters from the isotropic situation were not well known. Furthermore, anharmonic thermal vibrations play an essential role in the tetrahedral cage distortion. Indeed, even if for fixed atoms, the most stable situation in würtzite-type systems (i.e. the lowest reticular energy first term) is the perfectly isotropic situation, the occurrence of atomic thermal vibrations along c axis induces a significant tetrahedral flattening.

Fig. 1 .

 1 Fig. 1. Representation of an isotropic tetrahedral site (left), a würtzite tetrahedral site (center), the positioning of splitted sub-positions of the central atom to simulate thermal vibration (right).

Fig. 2 .

 2 Fig. 2. The linear relation between the z atomic parameter and c/a ratio for würtzite-like compounds as proposed by O'Keeffe and Hyde (the lawwas drawn in regard of experimental values from literature[4]

Fig. 3 .

 3 Fig. 3. a) Valence energy vs. c/a ratio and a cell parameter, calculated from DFT/GGA, within the P6 3 mc space group. In each case (i.e.fixed c/a and a values), the optimum value of the z atomic parameter was calculated (in brackets). Relation between z and c/a from cell volume minimization obtained for würtzite-type compounds with: b) Different cation-anion L 0 distances (but with same ion valences); c) Different ion valences (but withsame bond lengths). In both cases, the calculation was made to satisfy Equations (1),(2) and to minimize the cell volume.

Fig. 4 .

 4 Fig. 4. Evolution of the L 1 and L 2 bond lengths with the cell flattening/extension for a structure with ion valence V = 2 and r 0 = 1.723 Å (i.e., parameters as in ZnO würtzite-type compound).

Fig. 5 .

 5 Fig. 5. Plot of the -(1/L 2 + 3/L 1 ) sum calculated for various c/a ratios and central atom splitting: a) Δz = 0, b) Δζ = 0.005, c) Δz = 0.01.

  First-principles approaches (DFT), at least in their standard use, allow only a qualitative prediction of trends in c/a and z = f(c/a) values. In this work, it was firstly shown that the z = f(c/a) law is directly linked to the minimization of the cell volume. The satisfaction of the bond valence law as imposed criterion and the minimization of the tetrahedral site volumes let refine c and z parameters for various c/a values. The z = f(c/a) curve was determined for würtzite-type materials with different cell dimensions and for ions with different valences. Ion size has a small impact, valences have no impact.

Acknowledgments :

The authors acknowledge Philippe Aurel and the Institute of Molecular Sciences (University of Bordeaux, UMR 5255 CNRS) for computational support.