Bruno Lanson
email: bruno.lanson@univ-grenoble-alpes.fr

Pierre Mignon

Mélusine Velde

Andreas Bauer

Martine Lanson

Nathaniel Findling

Carlos Perez Del Valle

Determination of layer charge density in expandable phyllosilicates with alkylammonium ions: A combined experimental and theoretical assessment of the method

Layer charge deficit is a key crystal-chemical parameter for the classification and nomenclature of 2:1 phyllosilicates that also controls some of their fundamental properties such as cation exchange capacity, expandability, water content, or rheology. The clay community has thus devoted significant efforts to determine this parameter either from the smectite/vermiculite chemical composition (the structural formula method) or experimentally using the alkylammonium method, that theoretically allows the determination of the mean layer charge density and of its distribution.

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The validity of the method appears however limited to smectite-group minerals [layer charge density ranging from ~0.5 to ≤1.2 per O20(OH)4] most likely owing to the coexistence of both stable "layered" and paraffin-like interlayer configurations for higher values of layer charge. In addition, this consistency challenges the ability of the method to quantitatively describe distributions of layer charge in expandable phyllosilicates even when using the complete series of alkylammonium cations.

Introduction

Classification and nomenclature of 2:1 phyllosilicates rely to a large extent on their layer charge deficit [START_REF] Guggenheim | Summary of recommendations of nomenclature committees relevant to clay mineralogy: report of the Association Internationale pour l'Etude des Argiles (AIPEA) Nomenclature Committee for[END_REF]. This key crystal-chemical parameter allows indeed differentiating talc-pyrophyllite, smectite, vermiculite, interlayer-deficient mica, true mica, and brittle mica groups of 2:1 phyllosilicates [START_REF] Guggenheim | Summary of recommendations of nomenclature committees relevant to clay mineralogy: report of the Association Internationale pour l'Etude des Argiles (AIPEA) Nomenclature Committee for[END_REF]. Furthermore, layer charge controls fundamental properties of the phyllosilicates such as cation exchange capacity [START_REF] Jackson | Soil chemical analysis -Advanced course[END_REF]Dohrmann, 2006 and references therein), expandability [START_REF] Laird | Influence of layer charge on swelling of smectites[END_REF], or water content [START_REF] Sato | Effects of layer charge, charge location, and energy change on expansion properties of dioctahedral smectites[END_REF][START_REF] Ferrage | Investigation of dioctahedral smectite hydration properties by modeling of X-ray diffraction profiles: Influence of layer charge and charge location[END_REF]. Similarly, layer charge density controls to a large extent clay swelling through the hydration of interlayer cations [START_REF] Ferrage | Hydration properties and interlayer organization of water and ions in synthetic Na-smectite with tetrahedral layer charge. Part 2. Towards a precise coupling between molecular simulations and diffraction data[END_REF][START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF][START_REF] Vinci | Influence of layer charge on hydration properties of synthetic octahedrally-charged Na-saturated trioctahedral swelling phyllosilicates[END_REF], and is thus an essential parameter for the design of engineered clay barriers, including their self-healing ability [START_REF] Gates | Bentonite clay keeps pollutants at bay[END_REF], and rheological properties of clay suspensions [START_REF] Christidis | Influence of layer charge and charge distribution of smectites on the flow behaviour and swelling of bentonites[END_REF]Paineau et al., 2011aPaineau et al., , 2011b)). Abundance and nature of expandable 2:1 layer silicates (smectite and vermiculite groups) also have, together with organic matter, a key influence on soil fertility.

The pivotal influence of layer charge density on properties of expandable phyllosilicates has induced significant efforts from the clay community to determine theoretically this crystal chemical parameter from the smectite/vermiculite chemical composition (the so-called structural formula method, SFM - [START_REF] Newman | The chemical constitution of clays[END_REF][START_REF] Cicel | Structural formulae of layer silicates[END_REF][START_REF] Laird | Evaluation of structural formulae and alkylammonium methods of determining layer charge[END_REF] or experimentally.

In this context, the so-called alkylammonium method (AAM) initially developed by Lagaly, Weiss, and co-workers (Lagaly andWeiss, 1969, 1971;Lagaly, 1994) appears especially well suited, allowing the determination of the mean layer charge density and of its distribution. The method relies on the experimental determination of unit-cell d001 parameter for phyllosilicates intercalated with alkylammonium cations of increasing chain lengths (Lagaly, 1994). As the original method is time consuming for routine analysis, simplified methods have been proposed [START_REF] Olis | The rapid estimation of the layer charges of 2:1 expanding clays from a single alkylammonium ion expansion[END_REF][START_REF] Dohrmann | Beyond the methylene blue test: Introduction of the Cu(II)-triethylene tetramine method for smectite estimation in bentonite[END_REF]. These methods commonly use a limited number of alkyl chain lengths to determine the mean layer charge density. Considerable efforts have also been devoted to reconcile the results obtained from SFM and AAM [START_REF] Laird | Evaluation of structural formulae and alkylammonium methods of determining layer charge[END_REF][START_REF] Kaufhold | Comparison of methods for the determination of the layer charge density (LCD) of montmorillonites[END_REF][START_REF] Kaufhold | Layer charge density of smectites -Closing the gap between the structural formula method and the alkyl ammonium method[END_REF].

In parallel to the experimental AAM, molecular dynamics (MD) has been used to provide insights into the molecular-level interactions responsible for the experimentally observed changes in layer-to-layer distances as a function of alkyl chain length [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF][START_REF] Liu | Interlayer structure and dynamics of alkylammoniumintercalated smectites with and without water: A molecular dynamics study[END_REF]. In these studies, computational results were assessed against experimental literature data obtained on natural smectites whose layer charges scatter essentially over the whole range allowed for smectite [START_REF] Lagaly | Anordnung und Orientierung kationischer Tenside auf Silicatoberflächen[END_REF][START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF][START_REF] Vaia | Interlayer structure and molecular environment of alkylammonium layered silicates[END_REF]. Comparison of MD simulations computed properties with data supported the robustness of the modelling approach and its ability to describe the organization of large organic molecules in smectite interlayers although some specific aspects remained unclear. The first issue is related to the underestimation of layer charge densities derived from the strict application of the AAM compared to those resulting from the SFM, this AAM known bias challenging the consistency between computed and experimental results. A second issue is related to smectite interlayer structure in transition zones between the well-defined configurations described by Lagaly and coworkers (Lagaly andWeiss, 1969, 1971;Lagaly, 1994). In the method description (Lagaly, 1994), layer-to-layer distances intermediate between those of the well-defined configurations correspond to their coexistence/interstratification whereas they correspond to a unique "stable" configuration in the simulations [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF][START_REF] Liu | Interlayer structure and dynamics of alkylammoniumintercalated smectites with and without water: A molecular dynamics study[END_REF].

The present study aims at investigating these two specific aspects. For this purpose the AAM was used on four synthetic samples whose layer charge densities range from medium-charge smectite to medium-charge vermiculite [0.8 to 1.4 per O20(OH)4]. The use of synthetic samples aimed at minimizing the heterogeneity of layer charge density for a given sample and at improving crystallinity to minimize experimental bias related to peak broadening [START_REF] Stanjek | The determination of layer charge by curve-fitting of Lorentz-and Polarization-corrected X-ray diagrams[END_REF][START_REF] Stanjek | Improved evaluation of layer charge of n-alkylammoniumtreated fine soil clays by Lorentz-and polarization-correction and curve-fitting[END_REF][START_REF] Mermut | Problems associated with layer charge characterization of phyllosilicates[END_REF]. Simulations were performed with both density functional theory (DFT) to determine interlayer configurations most symmetrical and most stable at 0 K and MD approaches that allow calculation of more realistic configurations after thermalization at 300 K.

Materials and Methods

Smectite synthesis

A set of expandable phyllosilicates with a common structural formula inter [Nax] oct [Mg6] tet [Si8.0-xAlx]O20(OH)4 and a layer charge (x) varying from 0.8 to 1.4 per O20(OH)4 were synthesized hydrothermally from gel precursors in an externally heated Morey-type pressure vessel with an internal silver tubing [START_REF] Hamilton | The preparation of silicate compositions by a gelling method[END_REF][START_REF] Robert | Fluorine in micas: Crystal-chemical control of the OH-F distribution between trioctahedral and dioctahedral sites[END_REF][START_REF] Bergaoui | Al-pillared saponites. Part 3.-Effect of parent clay layer charge on the intercalation-pillaring mechanism and structural properties[END_REF], as described previously [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF]. A temperature of 400 °C, a water pressure of 1 kbar, and a duration of 4 weeks were used for synthesis. After synthesis, all samples were Nasaturated at room temperature by contact with 1 mol•L -1 aqueous solutions of NaCl. Excess NaCl was removed by rinsing the solid by immersion for 24 h in deionized water (Siemens UltraClear, 18.2 MΩ.cm) and separation of the solid fraction by centrifugation. Resulting monomineralic synthetic saponites are hereafter referred to as Sap-OH-x. Previous studies of samples prepared using the same experimental protocols for gel preparation and synthesis supported, from nuclear magnetic resonance and Ar adsorption data, an homogeneous distribution of Al substitutions in saponites, whatever the layer charge [START_REF] Sanz | Influence of structural factors on 29 Si and 27 Al NMR chemical shifts of phyllosilicates 2:1[END_REF][START_REF] Michot | Assessment of surface energetic heterogeneity of synthetic Na-saponites. The role of layer charge[END_REF][START_REF] Sanz | Influence of charge location on 29 Si NMR chemical shift of 2:1 phyllosilicates[END_REF].

Alkylammonium method

Preparation of alkyl chains

Homoionic saturation of saponites with alkylammonium cations having different carbon chain lengths required the preparation of solutions with these cations from alkylamines of proper chain length (Table S1). Preparation of alkylammonium cation solutions was described by Lagaly (1994).

In brief, the required amount of alkylamine (Table S2) was weighted and stirred and dissolved in 100mL of ethanol in a 1000 mL beaker. Next, 400 mL of ultrapure water were added to the solution while maintaining the stirring, commonly leading to the formation of a precipitate. Suspension was then titrated with hydrochloric acid to pH 7.0. Concentrations and approximate volumes of hydrochloric acid solutions required for the titration are reported in Table S2. For short chain lengths, heat production during titration of the solutions required controlling temperature with a water bath. For chain lengths ranging 14-18 carbons, the precipitate did not disappear when acid was added; ethanol (approximate volumes listed in Table S2) was then added to the stirred suspension to allow for a complete dissolution of the precipitate. If necessary, precipitate dissolution was enhanced by heating the suspension to ~50 °C using a water bath. When clear and pH neutral solutions were obtained, ultrapure water was added to a final volume of 1000 mL of appropriate alkylammonium cation concentrations (Table S2).

Smectite saturation with alkylammonium cations

Homoionic saturation of saponites with alkylammonium cations having different carbon chain lengths was performed in 10 mL Teflon tubes using ~100 mg of synthetic saponite. After addition of 2 mL of the appropriate alkylammonium solution (4 mL for chains longer than 13 C atoms), the suspension was shaken vigorously. Tubes were shaken until foam, that may contain clay particles, disappeared. Tubes were then heated to 65 °C in an oven for 24 hrs, tube caps remaining slightly open. The cation-exchanged smectites were then centrifuged and rinsed with ethanol (2 mL per rinsing step). Rinsing steps were repeated 8-10 times until the ethanol supernatant did not leave white traces when dried on a glass slide. Alkylammonium-exchanged smectites were then finally air dried at room temperature overnight.

XRD data collection and processing (alkylammonium method)

For all samples, oriented slides were prepared by drying at room temperature a clay suspension in ethanol on glass slides. XRD patterns were then recorded using a Bruker D8 diffractometer equipped with a MHG Messtechnik humidity controller coupled to an Anton Paar CHC+ chamber. Intensities were measured with a SolXE Si(Li) solid-state detector (Baltic Scientific Instruments) for 4 s per 0.04° 2θ step over the 2-50° 2θ Cu Kα angular range. Divergence slit, the two Soller slits, the antiscatter, and resolution slits were 0.3°, 2.3°, 0.3°, and 0.1°, respectively.

Samples were kept at 23 °C in the CHC+ chamber during the whole data collection. Samples were kept also under a constant flow of mixed dry/saturated air to maintain a 5% relative humidity (RH) to minimize hydration of smectite interlayers. RH was continuously monitored with a hygrometer (uncertainty of ∼2% RH) located close to the sample.

The alkylammonium method was performed assuming a systematic surface area of the unit cell of 49 Å 2 within the ab plane and no correction for particle size [Ac = 5.67 x Nc + 14; ' = 49 / (2 x Ac) in which Ac is the average planar area associated with each alkylammonium cation of carbon chain length Nc, ' being the layer charge corresponding to the critical carbon chain length causing the transition from h1 to h2 configurations - [START_REF] Lagaly | Characterization of clays by organic compounds[END_REF]]. The revised equation (" = 1.672 x ' -0.156) proposed by [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF] was preferred however to ease the comparison with layer charge density determined from the structural formula, that is in the present case with layer charge density determined from the stoichiometry of the synthetic products. In addition, the use of synthetic smectites with an enhanced homogeneity and extent of coherent layer stacking along the c* axis allowed minimizing reflection broadening and no correction of peak position was performed [START_REF] Stanjek | The determination of layer charge by curve-fitting of Lorentz-and Polarization-corrected X-ray diagrams[END_REF][START_REF] Stanjek | Improved evaluation of layer charge of n-alkylammoniumtreated fine soil clays by Lorentz-and polarization-correction and curve-fitting[END_REF][START_REF] Mermut | Problems associated with layer charge characterization of phyllosilicates[END_REF].

Density functional theory (DFT) simulations

Crystal models defined with general structural formula Mg6Si8.0-xAlxO20(OH)4.NH3(CH2)nCH3, x = 1.0 and 1.5, and n = 1, 2, etc were used for DFT simulations. Simulations were performed using a 2a × 1b × 1c super-cell for x =1.0, and a 2a × 2b × 1c super-cell for x = 1.5. Interlayer displacement was allowed to vary during the simulations. Within these super-cells, distributions of Al-substituted tetrahedra and of interlayer alkylammonium cations were generated to obey Löwenstein rule (no adjacent Al-substituted tetrahedra) and to be as regular as possible, consistent with the homogeneous distribution of Al substitutions in saponites reported in the literature [START_REF] Sanz | Influence of structural factors on 29 Si and 27 Al NMR chemical shifts of phyllosilicates 2:1[END_REF][START_REF] Michot | Assessment of surface energetic heterogeneity of synthetic Na-saponites. The role of layer charge[END_REF][START_REF] Sanz | Influence of charge location on 29 Si NMR chemical shift of 2:1 phyllosilicates[END_REF]. Periodic DFT calculations were performed with the Siesta 4.0b-485 code [START_REF] Soler | The SIESTA method forab initioorder-Nmaterials simulation[END_REF] using a generalized gradient approximation and a Perdew-Burke-Emzerhof correlation-exchange functional with a mesh cut-off of 500Ry and 6 and 9 k-points in the Monkhorst Pack grid [START_REF] Hernández-Haro | Compressibility of 2M1 muscovite-paragonite series minerals: A computational study to 6 GPa[END_REF]. Troullier-Martins norm-conserving pseudopotentials [START_REF] Troullier | Efficient pseudopotentials for plane-wave calculations. II. Operators for fast iterative diagonalization[END_REF] and numerical atomic orbital and double  plus polarization basis sets were also used for calculations. DFT methods are well suited to described systems containing cations as interlayer species as the present ones [START_REF] Hernández-Haro | Compressibility of 2M1 muscovite-paragonite series minerals: A computational study to 6 GPa[END_REF].

Molecular dynamics (MD) simulations

Classical MD simulations were performed for saponites having layer charges similar to those of the synthetic products: x = 0.8, 1.0, 1.2, and 1.4 per O20(OH)4. Initial atomic positions for both the clay layer and interlayer were derived by replicating the DFT optimized configurations in a 6a × 4b × 1c super-cell, and using the DFT-optimized layer displacement. For x = 0.8 and x = 1.0, initial positions were obtained from DFT configurations optimized for x = 1.0; and for x = 1.2 and x = 1.4, DFT configurations optimized for x = 1.5 were used. Owing to the replication of DFT optimized cells, substitutions are regularly distributed in the super-cells used for MD simulations. When necessary, layer charge was decreased by switching Al-substituted tetrahedral sites back to Si occupancy and removing the associated alkyl molecule (total number of substitutions in Table S3). Charge adjustments were performed to maintain the distribution of Al-substituted tetrahedra as regular as possible.

MD simulations were performed using ClayFF force field for the clay model [START_REF] Cygan | Molecular models of hydroxide, oxyhydroxide, and clay phases and the development of a general force field[END_REF] whereas alkylammonium parameters were obtained from DFT calculations at the B3LYP/6-31G* level performed with the Gaussian software. Alkylammonium molecules were optimized in the gas phase and RESP (restrained electrostatic potential) partial atomic charges, that allow reproducing the electrostatic potential of molecular boundaries, were used as molecular net charges in the forcefield simulations [START_REF] Woods | Restrained electrostatic potential atomic partial charges for condensedphase simulations of carbohydrates[END_REF]. The general Amber force field [START_REF] Wang | Development and testing of a general Amber force field[END_REF], which is extensively used for organic molecules modelling and reproduces well their structure and dynamics [START_REF] Hobza | Performance of empirical potentials (AMBER, CFF95, CVFF, CHARMM, OPLS, POLTEV), semiempirical quantum chemical methods (AM1, MNDO/M, PM3), and ab initio Hartree-Fock method for interaction of DNA bases: Comparison with nonempirical beyond Hartree-Fock results[END_REF][START_REF] Young | Intrusion of Counterions into the Spine of Hydration in the Minor Groove of B-DNA: Fractional Occupancy of Electronegative Pockets[END_REF][START_REF] Thyveetil | Computer simulation study of the structural stability and materials properties of DNA-intercalated layered double hydroxides[END_REF][START_REF] Swadling | Influence of surface chemistry and charge on mineral-RNA interactions[END_REF][START_REF] Szczerba | Ethylene glycol intercalation in smectites. Molecular dynamics simulation studies[END_REF][START_REF] Szczerba | Intercalation of ethylene glycol in smectites: Several molecular simulation models verified by X-ray diffraction data[END_REF][START_REF] Corbin | Adsorption of pharmaceuticals onto smectite clay minerals: A combined experimental and theoretical study[END_REF], was used to generate alkylammonium molecular parameters. Lorentz-Berthelot rules were applied for Lennard-Jones inter force fields parameters. Amber and ClayFF force fields are rather simple to combine as they both use harmonic potential for bond terms, although ClayFF does not use angle and dihedral terms. This approach has already been successfully used in various theoretical studies [START_REF] Thyveetil | Computer simulation study of the structural stability and materials properties of DNA-intercalated layered double hydroxides[END_REF][START_REF] Swadling | Clay minerals mediate folding and regioselective interactions of RNA: A large-scale atomistic simulation study[END_REF][START_REF] Swadling | Influence of surface chemistry and charge on mineral-RNA interactions[END_REF][START_REF] Mignon | Adsorption of nucleotides on clay surfaces: Effects of mineral composition, pH and solution salts[END_REF][START_REF] Corbin | Adsorption of pharmaceuticals onto smectite clay minerals: A combined experimental and theoretical study[END_REF]. Periodic boundary conditions were applied in all three spatial directions.

The large-scale atomistic/molecular massively parallel simulator (LAMMPS - [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] was used for all simulations, which were performed at 300 K and 1 atm. A 12 Å cutoff was used for electrostatic and Lennard-Jones interactions. Coulombic interactions were computed by Ewald summation and the particle-particle-particle-mesh method at a 0.0001 accuracy. The equilibration procedure consisted first in warming up the system in the NVT ensemble. The system was then allowed to relax along all directions in the NPT ensemble to model the interlayer space in real conditions and density. The system was then equilibrated through a two-step procedure increasing time steps from 0.1 to 0.5 fs. The initial equilibration run included a total of 1,200,000 time steps. A second NPT simulation was then run for 600,000 time steps during which the average d001 was determined to be compared with experimental results. A final NVT simulation (600,000 time steps) was run to compute structural data. Octahedral Mg atoms were kept fixed during all simulations by imposing forces to be null. This allowed minimizing translations within the layer plane and induced systematic location of Mg atoms at the origin of the unit cell [START_REF] Mignon | Adsorption of nucleotides on clay surfaces: Effects of mineral composition, pH and solution salts[END_REF][START_REF] Corbin | Adsorption of pharmaceuticals onto smectite clay minerals: A combined experimental and theoretical study[END_REF].

Results and discussion

X-ray diffraction

As expected for smectites hosting interlayer alkylammonium cations, positions of reflections present in the XRD patterns of Sap-OH-0.8 indicated an increase of the layer-to-layer distance (LLD) with increasing chain length (Figs. 12). For Sap-OH-0.8, two "plateaus" corresponding to LLDs of ~13.4 and ~17.6 Å were observed for n = 3-9, and n = 16-18, respectively. The two "plateaus" correspond to the LLDs of monolayers and bilayers of alkylammonium cations [defined as interlayer h1 and h2 configurations, respectively at 13.4 and 17.6 Å by Lagaly (1994)], for which alkyl chains are essentially parallel to the layer surface. Between these two plateaus, the apparent basal distance increased steadily as described by Lagaly (1994). When increasing the charge density from x=0.8 to x = 1.0, 1.2, and 1.4, h1 and h2 configurations were observed for decreasing chain lengths (Figs. 2, 1, S1-S3), consistent with the basics of the method (Lagaly, 1994). A third plateau at ~22 Å, typical for the pseudotrimolecular layers of alkylammonium cations (h3 interlayer configuration -Lagaly, 1994) was observed for high-charge samples and long alkylammonium chains (x = 1.2-1.4 and n = 16-18 -Figs. 2, S2-S3).

When plotting the full width at half-maximum intensity (FWHM) as a function of peak position, three local minima were visible (Fig. 3). Minimum FWHM values were observed for apparent LLDs corresponding to h1, h2, and h3 configurations. The significant increase in FWHM between these three minima results from the interstratification of different stable configurations within the same crystals [START_REF] Méring | L'interférence des rayons-X dans les systèmes à stratification désordonnée[END_REF][START_REF] Moore | X-ray Diffraction and the Identification and Analysis of Clay Minerals[END_REF][START_REF] Aristilde | Interstratification Patterns from the pH-Dependent Intercalation of a Tetracycline Antibiotic within Montmorillonite Layers[END_REF], consistent with the prediction of Lagaly (1994). Apparent LLDs intermediate between those of the stable configurations thus correspond to the coexistence within the same crystals (interstratification) of two stable configurations rather than to intermediate configurations as consistently observed for smectite hydration [START_REF] Ferrage | Investigation of smectite hydration properties by modeling experimental X-ray diffraction patterns: Part I. Montmorillonite hydration properties[END_REF][START_REF] Ferrage | Hydration properties and interlayer organization of water and ions in synthetic Na-smectite with tetrahedral layer charge. Part 1. Results from X-ray diffraction profile modeling[END_REF][START_REF] Ferrage | Investigation of the interlayer organization of water and ions in smectite from the combined use of diffraction experiments and molecular simulations: A review of methodology, applications, and perspectives[END_REF] and cation/anion exchange in clays and layer double hydroxides [START_REF] Glaeser | Isothermes d'hydratation des montmorillonites bi-ioniques (Ca, Na)[END_REF][START_REF] Méring | Sur le rôle de la valence des cations échangeables dans la montomorillonite[END_REF][START_REF] Feng | Selective anion-exchange properties of second-stage layered double hydroxide heterostructures[END_REF][START_REF] Möller | Facile Scalable Synthesis of Rectorites[END_REF][START_REF] Taviot-Guého | Intercalation chemistry in a LDH system: anion exchange process and staging phenomenon investigated by means of time-resolved, in situ X-ray diffraction[END_REF]Lanson, 2011 and references therein).

Layer charge densities deduced from the evolution of XRD peak position with alkylammonium chain length increased consistently with the stoichiometry of the synthetic products. When using the original equation of Lagaly (1994), experimentally determined charge densities were 0.52, 0.69, 0.86, and 0.88 for Sap-OH-x with x = 0.8, 1.0, 1.2, and 1.4, respectively. As shown previously by [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF], values deduced from the original alkylammonium method (Lagaly, 1994) are systematically and significantly shifted towards lower values compared to those deduced from smectite stoichiometry and may be corrected to minimize this discrepancy. When using the revised equation proposed by [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF], layer charge densities determined using the modified alkylammonium method became 0.72, 0.99, 1.28, and 1.31 for Sap-OH-x with x = 0.8, 1.0, 1.2, and 1.4, respectively. Corrected values were significantly closer to those deduced from smectite stoichiometry, consistent with the conclusions of [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF]. No significant difference was observed between Sap-OH-1.2 and Sap-OH-1.4, however, possibly indicating a limited sensitivity of the method for high layer charge densities.

DFT and MD simulations

In the present work, DFT simulations allowed determining interlayer configurations most stable at 0 K. These optimal configurations, highly symmetrical owing to the limited system size, were then used to derive initial inputs for MD simulations that allowed calculation of more realistic configurations after thermalization at 300 K. As a result, layer-to-layer distances computed from both DFT and MD simulations were essentially similar for a given layer charge and alkyl chain length (Fig. 4b,d), although thermal expansion was noticeable. Specifically, the two computational approaches used in the present study allowed describing the three stable configurations initially proposed by Lagaly (h1, h2, h3 -1994), consistent with previous computational studies [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF][START_REF] Liu | Interlayer structure and dynamics of alkylammoniumintercalated smectites with and without water: A molecular dynamics study[END_REF].

Interlayer structure of the different stable configurations

Figure 4 also shows that LLDs derived from both DFT and MD simulations were close to the values obtained experimentally for a given layer charge and alkyl chain length. Slightly larger LLDs were systematically obtained from MD simulations compared to DFT however. Unit-cell expansion most likely results from thermal motion which is implicit in MD simulations performed at 300 K, compared to the initial structures optimized at 0 K in DFT calculations. Computed LLDs were also consistent with those computed from classical MD simulations by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF], although systematically lower (by ~0.50-1.25 Å -Fig. 4a,c), possibly as the result of different charge location (tetrahedral substitutions in the present case) and of the induced contrasting undersaturation of surface O atoms. In addition, the starting model used in the present study for MD simulations was derived from DFT simulations optimized at 0 K and thus corresponding to minimal LLD. In their study [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] compared their results with LLDs from the literature that are systematically higher than those obtained experimentally in the present study. As a result, the apparent discrepancy between computed and experimental values was larger in the present study although calculated LLDs were lower, and closer to experimental LLDs obtained in the present work. This may be due to the improved crystallinity of the synthetic smectite used in the present study that allowed minimizing positional shifts of diffraction peaks related to size-broadening [START_REF] Stanjek | The determination of layer charge by curve-fitting of Lorentz-and Polarization-corrected X-ray diagrams[END_REF][START_REF] Stanjek | Improved evaluation of layer charge of n-alkylammoniumtreated fine soil clays by Lorentz-and polarization-correction and curve-fitting[END_REF][START_REF] Mermut | Problems associated with layer charge characterization of phyllosilicates[END_REF].

Alkylammonium cations located in saponite interlayers were distributed with the ammonium heads pointing towards layer surfaces to compensate efficiently for the layer charge deficit through H-bonding (Figs. 5a,6a,7a,S4a,S5a). In DFT-optimized structures, ammonium NH3 + heads were located essentially above/below the ditrigonal cavity and bound to the layer through N-H••O-H and N-H••O-(Si,Al) H-bonds (see inset in Fig. 5b). Location of R-NH3 + head groups above the ditrigonal cavity is consistent with results obtained by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] and allows for a maximum compression of the system at 0 K.

DFT-computed N-H••O-(Si,Al) bond lengths between H atoms of the ammonium head and

surface O atoms from (Si,Al)O4 tetrahedra scattered from 1.7-1.8 Å to 1.85-1.95 Å for Al and Si tetrahedra, respectively. These distances were consistent with the long N••Al distances (3.45-3.55 Å) obtained from thermalized MD simulations(Fig. S6) but significantly larger than the 1.5 Å reported by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] for O••H distances, despite the shorter LLDs obtained in the present study, consistent with XRD data. In the model used by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF], only octahedral substitutions were considered and electrostatic forces may be over-estimated. Indeed, ammonium head charges were adjusted arbitrarily to +0.7 e -charge per ammonium head for all alkyl chains whereas parameters DFT computed RESP charges obtained in the present study indicated lower positive charges. Despite the induced longer H-bonds observed in our models compared to those reported by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF], shorter LLDs were obtained for stable h1 and h2 domains, consistent with experimental LLDs (Fig. 4). In addition to the position above/below the ditrigonal cavity, MD simulations performed in the present study indicated that in thermalized systems ammonium heads may also be located above/below Al-substituted tetrahedra leading to shorter N••O-(Si,Al) distances (2.85-2.95 Å -Fig. S6), consistent with previous reports [START_REF] Chen | Molecular dynamics simulation of Alkylammoniumintercalated vermiculites[END_REF]. This alternative configuration allows for an improved local charge compensation of undersaturated surface O atoms from Al-substituted tetrahedra and for an improved electrostatic stabilization but results in slightly larger LLD values, as computed for thermalized systems. From the present MD simulations, this configuration appeared to be favored for saponites having the lowest charge (x = 0.8 -data not shown) owing to the presence of localized tetrahedral charges and to minimum steric constraints.

Increasing layer charge density results in a layer charge distribution that is more homogeneous and in the sole location of ammonium heads above/below ditrigonal cavities thus allowing a stronger steric compression in the interlayers of higher charge saponites. Although no specific simulations were performed, location of ammonium heads above/below the ditrigonal cavity is thus most likely favored by the presence of multiple Al-substituted tetrahedra among those defining this cavity. As a consequence, such a distribution of Al-substituted tetrahedra that could result, for example, from a non-homogeneous distribution of isomorphic Al-for-Si substitutions does not appear to challenge the validity of the alkylammonium method which relies primarily on steric constraints. Consistently, the alternative configuration of ammonium heads above/below Al-substituted tetrahedra was absent from the simulations performed by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] whatever the layer charge considered, most likely as the result of the octahedral substitutions present in their clay layer that induced a more diffuse undersaturation of surface oxygen, compared to saponites. Consistently, Chen et al.

(2017) reported only a minor contribution of ammonium heads located above/below Al-substituted tetrahedra from MD simulations performed for a model equivalent to Sap-OH-1.2, although a nonhomogeneous distribution of tetrahedral substitutions and initial random location of alkylammonium cations were assumed. The actual location of ammonium heads above Alsubstituted tetrahedral or ditrigonal cavities thus appears to result from the balance between local charge compensation and steric hindrance.

Alkyl chains forming the tails of the alkylammonium cations were essentially parallel to the basal surface of the 2:1 layers to optimize steric constraints. Consequently, h1 and h2 configurations essentially correspond to 1 and 2 planes of C atoms from the alkyl chains bound to adjacent layers as initially proposed by Lagaly (1994 -Figs. S4b, S5b). These interlayer configurations were also consistent with the ones reported by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] for stable h1 and h2 configurations computed for C10 and C22 chain lengths (NH3 head groups) and a CEC 91meq/100g (see Fig. 1a in [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF]. Finally, MD calculations suggested that the h3 configuration proposed by Lagaly (1994 -Fig. 7), included two major planes of C atoms similar to those observed for the h2 configuration. An intermediate and apparently less organized zone sandwiched in between these two planes hosted sections of the alkyl chains that were not accommodated in the two main planes, similar to the organization of interlayer H2O molecules in highly hydrated smectites [START_REF] Dazas | Interlayer structure model of tri-hydrated low-charge smectite by X-ray diffraction and Monte Carlo modeling in the Grand Canonical ensemble[END_REF].

Stability domains for the different stable configurations

As LLDs calculated from both DFT and classical MD simulations were close to experimentally determined ones, the number of carbon atoms in the alkyl chains corresponding to the experimental transition from one stability domain to another also corresponded nicely to simulation results (Fig. 4). For example, the stability domain of the h1 configuration (apparent LLD within the 12.8-13.7 Å range) extended experimentally from short chain lengths to C10, C7, C5, and C5 for x = 0.8, 1.0, 1.2, and 1.4, respectively. Similar LLD values (12.8-13.7 Å -Fig. 4) were computed for the h1 configuration from both DFT and classical MD simulations. The decrease of the maximum (critical) chain length allowing this h1 configuration with increasing layer charge density was consistent with the initial model [START_REF] Lagaly | Anordnung und Orientierung kationischer Tenside auf Silicatoberflächen[END_REF]Lagaly, 1994). Except for DFT calculations that were performed for x = 1.5, these chain lengths corresponded to the maximum alkyl chain length allowing the h1 interlayer configuration in the simulations performed. Similarly, the shortest alkyl chains leading experimentally to the h2 configuration comprised 16, 12, 9, and 9 carbon atoms for x = 0.8, 1.0, 1.2, and 1.4, respectively (Fig. 4). These chain lengths were consistent with those determined using computational approaches and leading to LLDs typical for the h2 configuration. For MD simulations, the LLD typical of the h2 configuration was slightly higher (18.0-18.7 Å -Fig. 4b) than the distance usually reported experimentally for the h2 configuration. Similar LLDs exceeding 18.0 Å were also reported by [START_REF] Heinz | Self-Assembly of Alkylammonium Chains on Montmorillonite: Effect of Chain Length, Head Group Structure, and Cation Exchange Capacity[END_REF] for the h2 configuration. Compared to the maximum chain length leading experimentally to a stable h2 configuration (18, 15, 11, and 11 C atoms for x = 0.8, 1.0, 1.2, and 1.4, respectively), the stability domain computed for the h2 configuration was extended slightly towards longer alkyl chain lengths (e.g., 16-17 C atoms for x = 1.0), possibly as the result of the larger LLD computed for this h2 configuration. The stability domain of the h2 configuration computed for lower charge density extended also towards shorter alkyl chain lengths however, as a LLD of 17.7 Å, consistent with this interlayer configuration was computed for x = 0.8 and C12 (Fig. 4a).

Interstratification of different stable configurations and heterogeneity of layer charge density

The extents of the stability domains corresponding to the stable configurations defined by Lagaly and coworkers [START_REF] Lagaly | Anordnung und Orientierung kationischer Tenside auf Silicatoberflächen[END_REF]Lagaly, 1994) and of the transition zones were essentially similar in both experimental and computational approaches (Fig. 4). LLD evolutions within the transition zones differed strongly in both cases, however. Experimentally determined LLD values increased steadily from those corresponding to one stability domain to the next one, consistent with the random interstratification of different stable interlayer configurations within the same crystals [START_REF] Méring | L'interférence des rayons-X dans les systèmes à stratification désordonnée[END_REF]Lanson, 2011 and references therein). This steady evolution is also consistent with the initial definition of the method. XRD data, and more especially the observed broadening of XRD maxima located between these LLDs (Fig. 3) supports such interstratification.

Lagaly and coworkers [START_REF] Lagaly | Anordnung und Orientierung kationischer Tenside auf Silicatoberflächen[END_REF]Lagaly, 1994) related the coexistence of different interlayer configurations within a given crystal to the heterogeneity of layer charge density. The use of synthetic smectites for the experimental section of the present study was expected to reduce significantly this heterogeneity but no specific decrease of the extent of these intermediate domains was observed compared to natural smectites (Lagaly andWeiss, 1970, 1971;[START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF]Lagaly, 1994).

Simulation results indicated also the existence of LLDs intermediate between those corresponding to stable configurations (Fig. 4). In contrast to the experimental steady evolution however, computed LLD values usually exhibited a major increase for a small increase of the alkyl chain length typical of the destabilization of one configuration. The rapid LLD evolution possibly results from the lower stability of configurations having LLD values intermediate between those of the stable h1, h2, and h3 configurations. Similarity of experimental and computed transition domains challenges however the original assumption of Lagaly concerning the origin of the interstratification of different interlayer configurations since DFT and MD simulations were systematically performed for homogeneous charge distributions within successive layers and within a given layer. In the present theoretical calculations, the distribution of Al-for-Si substitutions was periodic and thus maximized the homogeneity of layer charge distribution within the layer plane.

The mixture (interstratification) of Lagaly's stable configurations is thus likely energetically more favorable than an intermediate configuration to "average" the interlayer space required to compensate for a given layer charge density using a single alkyl chain length.

Implications for the layer charge determination method using alkylammonium cations

The overall consistency observed in the present study between the stability domains (in terms of alkyl chain length) calculated for h1 and h2 configurations using computational approaches and those determined from XRD data for smectites having well-defined layer-charge deficit confirms the validity of the method that relies mainly on the extent (in terms of alkyl chain length) of these stability domains. The ability of the alkylammonium method to provide layer charge densities consistent with those derived from smectite structural formula was strongly improved however by using the revised equation proposed by [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF]. In the present case, layer charge densities deduced from the alkylammonium method and this revised equation were 0.72, 0.99, 1.28, and 1.31 for Sap-OH-x with x = 0.8, 1.0, 1.2, and 1.4, respectively. Note that layer charge density values obtained using the alkylammonium method may provide a more realistic description of smectite properties, and in particular of their cation exchange capacity however [START_REF] Kaufhold | Comparison of methods for the determination of the layer charge density (LCD) of montmorillonites[END_REF][START_REF] Kaufhold | Layer charge density of smectites -Closing the gap between the structural formula method and the alkyl ammonium method[END_REF]. Adequate corrections to measured cation exchange capacity values may however allow improving the agreement with layer charge densities obtained using the structural formula method [START_REF] Środoń | Surface area and layer charge of smectite from CEC and EGME/H2O-retention measurements[END_REF].

The similar layer charge density values obtained for Sap_OH_1.2 and Sap_OH_1.4 challenge however the ability of the alkylammonium method to accurately determine the layer charge density of high-charge expandable phyllosilicates (vermiculites). For the two samples investigated, the evolution of the position of the first basal reflection as a function of alkyl chain length was typical for low-to-medium charge vermiculites (x ≤ 1.5 per O20(OH)4), with stepwise, or wave-like, changes of the apparent basal distance with alkyl chain length (Lagaly, 1994). The reduced extension of the domains corresponding to stable h1 and h2 configurations possibly limits the applicability of the routine alkylammonium method. The non-linearity of this evolution allows rejecting also a paraffinlike configuration of alkylammonium cations in these low-to-medium charge vermiculites (Lagaly, 1994). Consistently, the  angle determined from a linear fit to the data (pos. = 10.172 + 0.744 × n, R 2 = 0.9865) was unrealistically low (36.2°) to allow layer-charge determination using the various empirical curves proposed for high-charge vermiculites [START_REF] Lagaly | Determination of the layer charge in mica type layer silicates[END_REF][START_REF] Ghabru | Layer-charge and cation-exchange characteristics of vermiculite (Weathered biotite) isolated from a gray lusivol in northeastern saskatchewan[END_REF][START_REF] Mermut | Problems associated with layer charge characterization of phyllosilicates[END_REF]. For these low-to-medium charge vermiculites (1.2 ≤ x ≤ 1.5 per O20(OH)4), interstratification of interlayers exhibiting stable h1/h2 configurations and paraffin-like configuration of alkylammonium cations thus appears as a possible alternative hypothesis for the inefficiency of both methods, thus hampering the use of the alkylammonium method for such expandable phyllosilicates. The stability of paraffin-like configurations for a layer charge x = 1.5 (per O20(OH)4) and short alkyl chain lengths (4-5 carbons -Fig. 8) evidenced by DFT calculations supports this hypothesis. The reported modification of interlayer alkylammonium configuration from h1/h2 to paraffin-like with increasing layer charge density may be due to contrasting ordering of isomorphic substitutions in smectites and vermiculites (random and ordered distributions, respectively) or to the contrasting layer charge densities. The alkylammonium method does not hypothesize however on the ordering, or location, of isomorphic substitutions and [START_REF] Dazas | Influence of Tetrahedral Layer Charge on the Organization of Interlayer Water and Ions in Synthetic Na-Saturated Smectites[END_REF] did not evidence any direct relationship between ordering of isomorphic substitutions and that of interlayer cations.

Finally, both DFT and MD simulations showed that even for highly homogeneous layer charge densities, a significant range of carbon chain lengths allowed the presence of LLDs intermediate between those of stable configurations. These intermediate distances correspond experimentally to the interstratification of different stable configurations that were initially thought to be characteristic of layer charge heterogeneity [START_REF] Lagaly | Anordnung und Orientierung kationischer Tenside auf Silicatoberflächen[END_REF]Lagaly, 1994). Simulations

show that intermediate distances, and thus interstratification of different stable configurations, occur also for homogeneous layer charge distributions, thus challenging the ability of the alkylammonium method to describe quantitatively layer charge distributions. Similarity of peak position evolution as a function of alkylammonium chain length determined in the present study from synthetic smectites with those determined from natural smectite of similar layer charge supports this limitation. As a consequence, the benefice expected from the use of the complete series of alkylammonium cations is vanishing. The use of simplified methods using a reduced number of alkyl chain lengths [START_REF] Olis | The rapid estimation of the layer charges of 2:1 expanding clays from a single alkylammonium ion expansion[END_REF][START_REF] Mermut | Problems associated with layer charge characterization of phyllosilicates[END_REF] is thus to be preferred as their main drawback was supposed to be their inability to provide layer charge distribution [START_REF] Kaufhold | Comparison of methods for the determination of the layer charge density (LCD) of montmorillonites[END_REF].

Conclusion

Layer-to-layer distances computed from DFT and MD simulations are consistent with those determined experimentally on a series of synthetic saponites with layer charge ranging from medium-charge smectite to medium-charge vermiculite over the whole range of alkyl chain length considered in the present study. Despite minor differences, this consistency confirms the ability of computational methods to accurately predict the organization of large organic molecules in smectite interlayers and to unravel the molecular interactions ruling this organization. DFT simulations allow high-precision predictions of structure models but computational costs do no always allow probing a model with proper stoichiometry. By deriving DFT-optimized models of similar stoichiometry, MD simulations allow describing structure models for all possible layer compositions. MD simulations also implies thermalization of the system and allows probing alternative configurations allowed by thermal expansion of DFT-optimized models. For example, location of ammonium heads above/below Al-substituted tetrahedra is not observed in DFT simulations owing to the limited interlayer space, whereas the slight increase of LLD values upon thermalization allow these electrostatically-favored configurations. In addition, although LLD values intermediate between those corresponding to the stable h1, h2, etc configurations described by Lagaly (1994) may be computed, experimentally determined intermediate distances rather result from the interstratification of different stable configurations, as proposed initially.

The overall consistency of computed and experimental LLD values confirms further the ability of AAM to accurately determine a mean value of layer charge density consistent with smectite structural formula when using the revised equations proposed by [START_REF] Laird | Evaluation of the alkylammonium method of determining layer charge[END_REF]. The validity of AAM appears however limited to smectite-group minerals, that is to 2:1 phyllosilicates with layer charge density ranging from ~0.5 to ≤1.2 per O20(OH)4. For high-charge smectite to medium-charge vermiculites (1.2 ≤ x ≤ 1.5 per O20(OH)4), the coexistence of both stable "layered" and paraffin-like configurations in phyllosilicate interlayers most likely hampers a proper use of the method. Finally, the alkylammonium method is most likely unable to describe quantitatively layer charge distributions even when using the complete series of alkylammonium cations thus questioning the interest of the original method compared to simplified methods involving the use of a reduced number of alkyl chain lengths. configurations of Lagaly (1994) at ~13.4, 17.6, and 22 Å, respectively]. as the number n of carbon atoms in the alkyl chain). Data were collected at 5 %RH. Dashed lines indicate the layer-to-layer distances typical for 32 monolayers and bilayers of interlayer alkylammonium cations [h1 and h2 configurations of Lagaly (1994), respectively].

Figure captions

Fig. 1 .

 1 Fig. 1. X-ray diffraction patterns recorded on Sap-OH-08 as a function of the alkyl chain length of the interlayer alkylammonium cation (expressed as the number n of carbon atoms in the alkyl chain). Data were collected at 5 %RH. Dashed lines indicate the layer-to-layer distances typical for monolayers and bilayers of interlayer alkylammonium cations [h1 and h2 configurations of Lagaly (1994), respectively].

Fig. 2 .

 2 Fig. 2. Evolution of the position of the first experimental basal reflection as a function of the alkyl chain length. Greyed zones correspond to the layer-to-layer distances typical for monolayers, bilayers, and pseudotrimolecular layers of interlayer alkylammonium cations [h1, h2, and h3

Fig. 3 .

 3 Fig. 3. Evolution of the full width at half maximum intensity (FWHM) of the first low-angle reflection as a function of its position. Greyed zones as in Fig. 2.

Fig. 4 .

 4 Fig. 4. Comparison of the position of the layer-to-layer distance as a function of the alkyl chain length obtained experimentally (Fig. 2) with those derived from density functional theory (DFT) or molecular dynamics (MD) simulations. Patterns as in Fig. 2.

Fig. 5a

 5a Fig. 5a DFT optimized configuration of saponite interlayer hosting a monolayer of alkylammonium cations [interlayer h1 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 6 carbons. Left and right, views along a and b axes, respectively. Snapshots generated using Materials studio.

Fig. 5b

 5b Fig. 5b DFT optimized configuration of saponite interlayer hosting a monolayer of alkylammonium cations [interlayer h1 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 6 carbons. View along the c* axis. Inset: Closeup on N-H••O-(Si,Al) bonding. Snapshots generated using Materials studio.

Fig. 6a

 6a Fig. 6a DFT optimized configuration of saponite interlayer hosting a bilayer of alkylammonium cations [interlayer h2 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 14 carbons. From top to bottom, view along the a axis. Snapshots generated using Materials studio.

Fig. 6b

 6b Fig. 6b DFT optimized configuration of saponite interlayer hosting a bilayer of alkylammonium cations [interlayer h2 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the

Fig. 7

 7 Fig. 7 Representative MD snapshots of saponite interlayer hosting a pseudotrilayer of interlayer alkylammonium cations. Layer charge: x = 1.4, alkyl chain length of the interlayer alkylammonium cation: 12 carbons. View along the b axis.

Fig. 8

 8 Fig. 8 DFT optimized configuration of saponite interlayer hosting alkylammonium cations in a paraffin-like configuration. Layer charge: x = 1.5, alkyl chain length of the interlayer alkylammonium cations: 4 and 5 carbons (left and right, respectively). View along the a axis. Snapshots generated using Materials studio.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. X-ray diffraction patterns recorded on Sap-OH-08 as a function of the alkyl chain length of the interlayer alkylammonium cation (expressed as the number n of carbon atoms in the alkyl chain). Data were collected at 5 %RH. Dashed lines indicate the layer-to-layer distances typical for monolayers and bilayers of interlayer alkylammonium cations [h1 and h2 configurations of Lagaly (1994), respectively].

Fig. 5b

 5b Fig. 5b DFT optimized configuration of saponite interlayer hosting a monolayer of alkylammonium cations [interlayer h1 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 6 carbons. View along the c* axis. Inset: Closeup on N-H••O-(Si,Al) bonding. Colors as in Fig. 5a. Snapshots generated using Materials studio.

Fig. 6a

 6a Fig. 6a DFT optimized configuration of saponite interlayer hosting a bilayer of alkylammonium cations [interlayer h2 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 14 carbons. From top to bottom, view along the a axis. Colors as in Fig. 5a. Snapshots generated using Materials studio.

Fig. 6b Fig. 7

 6b7 Fig. 6b DFT optimized configuration of saponite interlayer hosting a bilayer of alkylammonium cations [interlayer h2 configuration -Lagaly (1994)]. Layer charge: x = 1.0, alkyl chain length of the interlayer alkylammonium cation: 14 carbons. View along the c* axis. Colors as in Fig. 5a. Snapshots generated using Materials studio.

Fig. 8

 8 Fig. 8 DFT optimized configuration of saponite interlayer hosting alkylammonium cations in a paraffin-like configuration. Layer charge: x = 1.5, alkyl chain length of the interlayer alkylammonium cations: 4 and 5 carbons (left and right, respectively). View along the a axis. Colors as in Fig. 5a. Snapshots generated using Materials studio.

  Fig. S2. X-ray diffraction patterns recorded on Sap-OH-1.2 as a function of the alkyl chain length of the interlayer alkylammonium cation (expressed

Fig. S4b

 S4b Fig.S4bDistribution of atomic density derived from MD calculations for alkylammonium saponite hosting a single plane of interlayer alkylammonium cations [h1 configuration fromLagaly (1994)] in projection along the c* axis. Layer charge: x = 0.8, alkyl chain length : 6 carbons. Solid red, black, and blue lines represent O atoms from the 2:1 layer surface and C and N atoms from the interlayer alkylammonium cation, respectively.

TABLE S3

 S3 Details of the atomic models used for molecular dynamics simulations. Number of tetrahedral substitutions (Nb sub) in the 6a × 4b super-cell used for the simulations, and resulting layer charge deficit.

		x=0.8	x=1.0	x=1.2	x=1.4
	Nb Sub	18	24	28	34
	charge	0.75	1.00	1.17	1.42

Acknowledgments

Part of the simulations presented in this article were performed using the Froggy platform of the GRICAD infrastructure (https://gricad.univ-grenoble-alpes.fr), which is supported by the Rhône-Alpes region (GRANT CPER07_13 CIRA) and the Equip@Meso project (reference ANR-10-EQPX-29-01) of the "Investissements d'Avenir" program supervised by the Agence Nationale pour la Recherche. ISTerre is part of Labex OSUG@2020 (ANR10-LABX56).

Supporting information

Supporting information includes 3 Tables and6 Figures (X-ray diffraction data and simulation results)

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this article.

Determination of layer charge density in expandable phyllosilicates with alkylammonium ions


Alexis Dos Santos
email: dossantos.alexis73@gmail.com

J Hosdez

N Limodin

A El Bartali

A Tandjaoui

J.-F Witz

C Niclaeys

P Quaegebeur

D Najjar

2D and 3D characterization of damage mechanisms in A319 alloy

Keywords: Al-Si alloys, Mechanical cyclic test, Damage mechanisms, Digital image correlation, Digital volume correlation

. Due to the very slow cooling rate with Lost Foam Casting (LFC) foundry process, the alloy microstructure is coarse with shrinkage defects [2 -7]. Previous studies show the key role of microshrinkage cavities in the initiation and the propagation of cracks: pores induce stress concentration in the close microstructure [8 -20]. Objective: the aim of this work is to determine alloy damage at the microstructure scale in a model material with a simplified defect during a cyclic mechanical test. Methods: High Isostatic Pressing treatment and specific drilled hole were made on the sample test to control the crack initiation. A new speckle pattern was used for digital image correlation. The mechanical test was interrupted to couple 2D observations on the two sample surfaces with optical cameras and 3D observations with X-ray microtomography. Results: the resulting von Mises strain fields acquired in situ at the surface or ex situ in bulk show similarities and allow to study the strain evolution during a cyclic test. The study reveals crack initiation around the artificial hole in the gauge length of the specimen while propagation is mainly driven by the surrounding microstructure. Measured strain also allowed to identify the areas of damage and even to anticipate the cracking path. Conclusions: due to the microstructural complexity of a cast aluminum-silicon-copper alloy, the proposed approach based on a simplified defect material and two simultaneous observation methods with digital and volume image correlation is a strong advantage for quantitatively characterizing damage mechanisms at the microstructure scale.

EXPERIMENTAL PROCEDURE

To analyse microcrack initiation and propagation at a high spatial resolution in a limited observation area, all the pores in a sample cut from a cylinder head inter-valve bridge were closed with Hot Isostatic Pressing (HIP). The absence of porosity was controlled by X-ray microtomography at a fine resolution (voxel = 2.2 µm).Flat specimen has a representative microstructure of the Al-7Si-3Cu cylinder head alloy from LFC with a coarse dendritic structure (Secondary Dendrite Arm Spacing, SDAS = 75 µm). This hypoeutectic microstructure is composed of aluminum-rich alpha dendrites, eutectic silicon, copper (Al2Cu) and iron-rich (α-AlFeMnSi and β-AlFeSi) constituents [START_REF] Tabibian | TMF-LCF life assessment of a Lost Foam Casting A319 aluminum alloy[END_REF]. The gauge length has a square cross section of 2.5 x 2.5 mm² after mirror polishing. A circular through hole is drilled in the gauge length with a thin and hard drill bit made of tungsten carbide. The drilling machine has a high speed and slow feed with regular swarf removal in order to avoid material build-up on the hole surface. The surface finish quality was controlled with microtomography (2.2 µm voxel) before the test beginning to ensure that there are no cracks related to the drilling procedure. The hole is tilted by 6° thanks to an inclined holder made with 3D printing, hence it is centred on one surface of the specimen while it is off-center on the opposite surface (Fig. 2 for the hole geometry). Various holder tests were carried out but over 6° the drill bit broke. The goal is to increase the stress on the off-center surface and obtain microcrack initiation in the narrowest zone between the sample free surface and the hole. The last step is to make a speckle pattern on both drilled surfaces of the gauge length. The final aim is to perform Digital Image Correlation (DIC) at a small scale, i.e. at eutectic Si or intermetallics size. Contrary to the traditional pattern with airbrush [START_REF] Li | Influence of Fe content on the damage mechanism in A319 aluminum alloy: Tensile tests and digital image correlation[END_REF] the paint is replaced by powder to obtain the speckle; the new protocol is:

-Spray black paint for the background.

-Spray speckle solution: 50 mg of a mixture of calcium carbonate and clay with 20 ml of ethanol.

-Drying time of 5 minutes.

The white speckle with the chalk mixture is finer than the paint speckle allowing to increase the pattern density hence contributing to a better measurement resolution for DIC. Indeed, the grey level variation inside the white spots is greater in chalk than in paint [START_REF] Santos | Étude des mécanismes d'endommagement de la microstructure d'un alliage d'aluminium-silicium pour application automobile à partir d'éprouvette modèle à défaut contrôlé[END_REF]. The size measurement of the white spots also reveals finer particles with the use of chalk (21µm on average) than with paint (33µm on average). For the displacement field measurement by Digital Volume Correlation (DVC), no sample preparation is required because the intermetallics, visible in the microtomography images, provide a natural speckle pattern.

The specimen is placed in the clamping jaws of a miniature tensile test rig Kammrath & Weiss (Fig. 1). In order to follow the two drilled surfaces during the test, two Ximea XiC cameras (12.4 Mpixels) with two Techspec Compact Tl telecentric lenses (magnification of x3) and two LED lights are used. A Python program controls the loading in displacement and the unloading to zero-load with a speed of 5 μm/s. The maximum displacement level increases in each cycle (see the top graph in Fig. 3). The test includes 31 cycles, i.e. 31 loadings and 31 unloadings, the last one corresponding to the sample final failure. The cameras take pictures on each load extremum during the cyclic test. The test is also interrupted at some unloading (eleven times in total) in order to scan the specimen in the microtomograph. This allows to complete the microstructure monitoring around the hole with 3D images acquired during elastic and plastic strain regimes. This strategy of combining surface and volume observation has already been used to determine the damage of specimens in a cyclic test and it proved to be complementary [START_REF] Vrgoč | Application of different imaging techniques for the characterization of damage in fiber reinforced polymer[END_REF].

Figure 1: Experimental test device and test specimen with speckle pattern on the two drilled surfaces

Table 1 compares the characteristics of the two methods used to follow damage in the specimen microstructure: in situ optical monitoring and ex situ X-ray microtomography. Optical surface monitoring has a higher measurement resolution than X-ray microtomography due to a better spatial resolution of the optical images and an improved speckle pattern. It also allows following a larger area of the gauge length (4.7 mm x 3.5 mm) with a quick recording time in contrast to the microtomography scan (2.7 mm x 2.7 mm x 2.7 mm). Another advantage is the possibility to link the eutectic silicon with damage mechanisms (observed post mortem with SEM) and strain localization measured in situ with DIC. Because of their close atomic number, silicon has an X-ray absorption coefficient close to aluminum. This leads to low contrast on the X-ray microtomography images between eutectic Si and Al from the alpha phase of primary aluminum (dendrites) and eutectic aluminum. Moreover, the low contrast difference between the copper and iron intermetallics also causes difficulty in post-processing constituents' identification. The optical microscope mapping of the two drilled surfaces before spraying the speckle pattern allows to overlay the finely resolved optical microstructure with the displacement field from DIC and to identify the behaviour of all constituents.

DIC and DVC measurements were performed with YaDICs (Yet another Digital Image Correlation software), a homemade programming platform [START_REF] Dahdah | Damage Investigation in A319 Aluminium Alloy by X-ray Tomography and Digital Volume Correlation during In Situ High-Temperature Fatigue Tests[END_REF] [13] [START_REF] Hosdez | Plastic zone evolution during fatigue crack growth: Digital image correlation coupled with finite elements method[END_REF]. Digital image and volume correlation consist in finding the transformation between an initial and a deformed image for measuring displacement fields. Here, the initial image is the picture acquired at the test beginning with a 0N load. The moving image corresponds to the different steps of the mechanical test either at maximum or minimum load (Fig. 3); the computed strains shown hereafter are then cumulative strains. In this study, the metric is based on the Sum of Squared Differences (SSD) completed by an optimisation strategy via the gradient algorithm. A bi-cubic interpolator applies this transformation to the deformed image at each iteration by evaluating the grey levels of this image. The resolution is based on a multi-scale strategy in pyramid form and global displacements are determined using finite element functions [START_REF] Besnard | Finite-Element" Displacement Fields Analysis from Digital Images: Application to Portevin -Le Châtelier Bands[END_REF]. Fields regularization is applied with a median filter of 7 pix/vox (more details in [START_REF] Seghir | YaDICs[END_REF]). A rigid body movement was applied on the specimen before the test in order to determine the uncertainty of the displacement fields (Tab.1) [START_REF] Dahdah | Damage Investigation in A319 Aluminium Alloy by X-ray Tomography and Digital Volume Correlation during In Situ High-Temperature Fatigue Tests[END_REF] [28] and to demonstrate the feasibility of DIC and DVC with the A319 alloy. The measurement resolution choice in DIC (10x10) and DVC (16x16x16) was determined by the convergence of the digital correlation calculation: a smaller element size causes a non-physical field discontinuity in the results. Results of DIC measurements were validated on a comparative test with a specimen equipped by a strain gauge [START_REF] Santos | Étude des mécanismes d'endommagement de la microstructure d'un alliage d'aluminium-silicium pour application automobile à partir d'éprouvette modèle à défaut contrôlé[END_REF].

RESULTS AND DISCUSSION

The designed hole allowed a preferential initiation of microcracks at the hole surface but the inclination was not sufficient to generate a single crack on the specimen surface (Fig. 2). Analysis of X-ray microtomography slices through the whole thickness reveals microcracks on the off-center surface but also in volume. Thus, in the plastic regime, multiple microcracks initiation occurs. Furthermore, fracture surface investigation with Scanning Electron Microscope (SEM) shows that the additive elements such as silicon, copper and iron impurity, which reflect the presence of eutectic Si, copper and iron based intermetallics respectively, are characterised by brittle fracture. Only the primary α-phase of aluminum shows ductility with some dimples but at a local scale. The brittleness of the Si and intermetallic compounds did not allow the asymmetrical defect to enforce the fracture initiation.

Figure 2: X-ray microtomography slice at the hole cross section (on the left) and slices perpendicular to the z axis at different depths (on the right) showing multiple microcracks initiation along the hole after the 28 th cycle

This overview of the cyclic test shows the necessity of 3D images to study the damage mechanisms. The first microcrack observed is located in the thickness of the specimen at a depth of 623 µm (Fig. 3). The limited cycles number between crack initiation (27th or 28th cycle) and fracture confirms the material brittleness. The microcrack starts at the edge of the hole in the Al-Si matrix. The path of the final crack in cycle 31 shows that propagation occurs preferentially in the iron-rich intermetallics. The superposition of the strain fields calculated by DVC with the corresponding tomography slices allows establishing several observations for the three last microtomography scans at unloaded state before failure. The residual von Mises strain values increase progressively with the applied load: from 1% at cycle 26, they reach locally nearly 4% before cycle 31. Locally high levels of strain can be reached early in the test. High correspond to the location of the microcrack whose propagation seems to be correlated to the areas of the highest local strain recorded in the previous cycle.

Figure 3: X-ray microtomography slices (depth of 623 µm) and von Mises strain calculated from DVC displacement fields measurement at different steps of the test

To compare DVC and DIC at the surface of the sample, the optical image was registered (rigid body motion and scaling) with the corresponding microtomography slice. As the acquisition method is not identical, the two images do not have the same dimensions and orientation. Thus, a Python program is used to detect the edges of the specimen and the hole. It is then possible to correct the angle of rotation, to modify the scale and to apply a rigid body movement to the optical image to superimpose perfectly the surface of the specimen on the 2D image with the same surface on the tomographic slice. Strains were calculated by applying the derivative of the displacement fields in a Python program. The resulting registered DVC and DIC von Mises strain fields at the unloaded state on the surface with the off-center hole (Fig. 4) confirm the stress concentration effect of the hole with residual plasticity that increases throughout the test. Both methods give similar fields with identical deformation levels. However, the higher resolution reached with the optical camera means that the local deformations are less smooth than with DVC. Hence, the strain heterogeneity is better studied in 2D images. Image correlation is a sensitive measure and noise may appear on the results. The 2D strain fields on some cycles may show measurement errors locally. Consequently, only DIC and DVC results in a restricted area showing similar tendency were considered to ensure the analysis is related to physical phenomenon and not to numerical noise. To take advantages of the DIC, another area with a microcrack was selected to observe the damage of an intermetallic close to the hole edge at the specimen surface. This zone contains eutectic silicon and intermetallics rich in copper and iron. Post mortem images from SEM (Fig. 5 (a)) show a microcrack between two intermetallics near the final crack initiation site at the hole periphery. The 3D monitoring indicates that this crack appears at the end of the test during the last loading before failure.

The von Mises strain fields calculated by DIC at different loads during the test are projected on the X-ray microtomography slice to follow the strains evolution at intermetallics (Fig. 5 (a)). Before the final crack on the cycle 30 (load of 986N), the most deformed area is located on a copper intermetallic. A high-level strain band (Fig. 5 (b)) is however visible between iron and copper intermetallics, i.e. at the position of the future microcrack.

A maximum strain higher than 4%, an increase of nearly 80% compared to the previous measurement, is then measured at the surface during the failure of the specimen (cycle 31); this large strain corresponds to crack opening (Fig. 5 (c)).

Figure 5: Focus on some intermetallics before the start of the cyclic test (cycle 0), before the final crack (cycle 30) and in post-mortem analysis (cycle 31) (a). Von Mises strains fields measured with DIC at the last cycles of the test and position of the maximum value (indicated with star symbols) (b). Evolution of the maximum von Mises strain during the cyclic test (c)

CONCLUSION

Al-Si-Cu alloy has a complex microstructure but the importance of rigid inclusions in the cracking phenomenon has been highlighted. Crack initiation occurs on the "artificial defect" and propagation depends on the surrounding microstructure. As a result, drilling a non-symmetrical hole in the structure did not create a single crack initiation zone. This study used two complementary measurement methods to calculate the deformation fields: DVC and DIC. 3D observations are required to have a monitoring along the hole due to the presence of multi-cracking during the mechanical test while 2D observations allow access to more resolved and continuous information in time thanks to in situ recording. Strain fields revealed the plasticity evolution in the microstructure during the mechanical test. It also allowed to identify the areas of damage and even to predict the cracking path at the scale of the microstructure. As indicated in the literature, the brittle microstructure constituents appear to have a strong influence on the propagation path, in particular eutectic silicon which has the lowest deformations before failure.

Figure 4 :

 4 Figure 4: von Mises Strain fields from DVC and DIC at the off-center hole surface after the 30 th cycle before the final failure of the sample (size in pixels × pixels)

Table 1 :

 1 Digital image and volume correlation settings

	Observation	Optical cameras (2D)	Microtomography (3D)
	Size	1.15 µm/pixel	2.22 µm/voxel
	Region Of Interest (ROI)	4112 x 3008 pixels	1200 x 1200 x 1216 voxels
	Recording time	< 1 second	30 minutes
	Resolution measurement	DIC: 10 x 10 pixels	DVC: 16 x 16 x 16 voxels
	Images acquisition	Before the test beginning and at each load and unload extremum	Before the test beginning and at the end of cycles 3, 7, 12, 15, 19, 21, 23, 26, 28, 30, 31
		Center	Off-center	U: 0.06 vox.
	Displacement uncertainty	U: 0.17 pix.	U: 0.12 pix.	V: 0.04 vox.
		V: 0.24 pix.	V: 0.18 pix.	Z: 0.05 vox.
		Center	Off-center	Exx = 0.029 %
	Strain uncertainty
		Exx = 0.002 %	Exx = 0.064 %	Eyy = 0.024 %
	(X, Y, Z axis from Fig.1)
		Eyy = 0.20 %	Eyy = 0.031%	Ezz = 0.015 %

ACKNOWLEDGEMENTS

The authors wish to thank Centrale Lille, Hauts de France region and ELSAT2020 for funding the PhD thesis of Alexis Dos Santos and some of the equipment. The authors would like to thank also ISIS4D (https://isis4d.univlille.fr/) X-Ray CT platform (Lille, France) for making the ex situ test possible.

Ethics declarations

Conflict of interest: the authors have no conflicts of interest to declare.