Ivan Miro-Panades
email: ivan.miro-panades@cea.fr

Benoit Tain

Jean-Frédéric Christmann

David Coriat

Romain Lemaire

Clement Jany

Baudouin Martineau

Fabrice Chaix

Anthony Quelen

Emmanuel Pluchart

Jean-Philippe Noel

Reda Boumchedda

Adam Makosiej

Maxime Montoya

Simone Bacles-Min

David Briand

Jean-Marc Philippe

Alexandre Valentian

Frédéric Heitzmann

Edith Beigne

Fabien Clermidy

SamurAI: a 1.7MOPS-36GOPS Adaptive Versatile IoT Node with 15,000x Peak-to-Idle Power Reduction, 207ns Wake-up Time and 1.3TOPS/W ML Efficiency

published or not. The documents may come

Introduction

An event-driven IoT node is a way to reduce the power consumption of sporadic computing. SamurAI (Fig. 1) combines an event-driven WuC using asynchronous logic (low-energy, clock-less, and fast wake-up time) in the AR subsystem with an energy efficient synchronous RISC-V CPU including specialized accelerators in the OD sub-system to make a versatile IoT node. Depending on the application needs, one or both cores can be used as shown in Fig. 2.

Always-Responsive Sub-System

The WuC (Fig. 3), a clock-less 32b MCU [9] with 16b RISC ISA, is the master on the AR sub-system having 1.7MOPS at 0.45V and 1.6µW idle power. Program and data are stored on an asynchronous 8kB Two-Port SRAM (TP-SRAM) [10] with auto power-down capabilities down to 0.4V and 4.6µW idle power. Key component of this architecture, TP-SRAM is also connected to the OD sub-system through an AHB asynchronous interface to create a shared memory space between the two sub-systems. Thanks to asynchronous logic, the wake-up time from idle state to first instruction fetch takes 207ns, i.e. a third of an instruction cycle. The AR sub-system contains multiples wake-up sources: an internal timer, OD interrupts, GPIOs connected to sensors, and a Wake-up Radio (WuR). The WuR senses the radio channel with 10x less power than the main radio. Thanks to its mixerfirst topology (as in [7]) and the use of three distinct oscillators, the RF front-end enables operation in all the main IoT bands: 433MHz, 868MHz and 2.4GHz. The digital baseband (DBB) supports data-rates up to 100kbps. It decodes an 8b identifier to selectively wake-up the WuC and a 32b message payload for application specific purposes. At 50kbps data-rate, the WuR achieves -73dBm sensitivity with 4.1µW power consumption at 5% duty cycle and this drop to 40nW in idle mode.

On-Demand Sub-System The 4-stage pipeline RISC-V CPU is the master of the OD sub-system (Fig. 1). Its memory sub-system is composed of 64kB for program (TCPM), 128kB for data (TCDM) and external NVM FeRAM memories. 32kB of TCDM have retention capability with 1.03pA/bit leakage at 0.5V. An instruction cache and a data interface allow direct FeRAM operations. Both cores (in AR and OD) share the APB peripherals while synchronizing through interrupts and locks. The Crypto IP embeds AES, TRIVIUM and PRESENT stream-cipher accelerators to support various encryption formats. The adaptive voltage scaling (AVS) module manages 128 sensors (TFS) and a programmable replica path (TFR) to estimate and track the Fmax/Vmin according to the applicative needs. To offer ML inference capability in this event driven IoT node, we implement PNeuro [8] (Fig. 4), a SIMD programmable accelerator composed of 2 clusters of 32 8-bit PEs each and 264kB multi-banked SRAM. Designed to accelerate neural networks, it performs up to 64MACs/cycle.

Measurements Results

The 4.5mm² circuit (Fig. 8) has been fabricated in 28nm FDSOI technology and contains 6 switchable power domains. For fair comparison with the SoA, measurements are done without using body-bias. Fig. 6 depicts Fmax and energy per cycle of the OD sub-system with RISC-V running Dhrystone, showing 19pJ/cycle at 25MHz, 0.48V and up to 350MHz at 0.9V. Fig. 7 shows the performance of PNeuro block, reaching 1.3TOPS/W and 2.8GOPS at 0.48V and up to 36GOPS at 0.9V for 8b precision fully-connected layers. Fig. 5 reports power consumption for different mode: 96mW at full activity and 6.4µW at 0.45V. The 15,000x ratio between peak and idle power, highlights the adaptive and versatile performance of this architecture. Fig. 9 shows a scenario where SamurAI is used to classify a scene based on the presence of people signaled by a pyroelectric detector (PIR). To minimize power consumption, the WuC filters PIR activity based on previous scenes and powers up the OD part only when required: PNeuro classifies the images acquired by the CPU and shares results with the WuC. AES encrypted messages can be transmitted through an external low-power radio. The WuR is also used to receive user commands. The daily average power for this application is 105µW where 26% is consumed on SamurAI (Fig. 10). Using RISC-V instead of PNeuro would increase the total average power consumption by 2.3x. Fig. 11 compares the circuit to prior art and shows significant improvements in terms of versatility, performance, wake-up time and power reduction.

Fig. 9 :

 9 Fig. 9: Presence classification scenario using SamurAI with off-the-shelf components.

Fig. 10 :

 10 Fig. 10: Daily average power breakdown of presence classification scenario (70% PIR filtering), 105µW total power.

Fig. 3 :

 3 Fig. 3: Wake-up Controller and Radio architecture details.

Fig. 6 :

 6 Fig. 6: Fmax and energy per cycle of OD sub-system.

Fig. 2 :

 2 Fig. 2: SamurAI power modes.

Fig. 4 :

 4 Fig. 4: Two-cluster PNeuro accelerator with 64 PEs.

Fig. 1 :Fig. 7 :

 17 Fig. 1: SamurAI system architecture, with Always-Responsive and On-Demand sub-systems and associated power domains.

Fig. 5 :

 5 Fig. 5: Power consumption measurements and reduction w.r.t. power modes.

Always-Responsive (AR) TP-SRAM On-Demand (OD) Power mode

		Voltage (V)	WuC state	WuR State	Voltage (V)	State	Voltage (V)	RISC-V Freq (MHz)	Periph Freq (MHz)
	IDLE	0.45	Sl eep	OFF	0.48	Sl e ep	OFF	-	-
	WuC only	0.45	Run	OFF	0.48	Run	OFF	-	-
	WuC+WuR	0.45	Run	ON	0.48	Run	OFF	-	-
	WuC+Periph. (cpu sleep)	0.45	Run	ON/OFF	0.48	Run	0.48	Ga ted	10
	CPU running	0.45	Run/Sleep ON/OFF	0.48 -0.9	Run	0.48 -0.9	1 -350	1 -350

Acknowledgments

WAKeMeUP (ECSEL 783176), SERENE-IoT (Penta 16004), MACS (FUI20) projects, ST Microelectronics, and PULP project.

This work JSSC 2018 Lallement [1]

JSSC 2018 Yu Pu [2] JSSC 2017 S. Paul [


domain: chim sdu sde

text:
Brian Grégoire
email: brian.gregoire@univ-poitiers.fr

Baptiste Dazas

Fabien Hubert

Emmanuel Tertre

Eric Ferrage

Laurent Grasset

Sabine Petit

Orientation Measurements of Clay Minerals by Polarized Attenuated Total Reflection Infrared Spectroscopy

Keywords: Attenuated total reflection spectroscopy, clay minerals, orientation, linear dichroism, thin layer

The orientation and organization of molecular guests within the interlayer of clay minerals control the reactivity and performance of tailored organo-clay materials. Such a detailed investigation of hybrid structure on the molecular scale is usually provided by computational methods with limited experimental validation. In this study, polarized attenuated total reflection infrared spectroscopy was used to extract quantitative orientation measurements of montmorillonite particles. The validity of the evanescent electric field amplitude calculations necessary to derive the order parameter was critically evaluated to propose a methodology for determining the orientation of the normal to the clay layer relative to a reference axis, enabling comparison with the results obtained from X-ray scattering experiments and molecular dynamic simulations. Subsequently, the orientation of the interlayer water dipole and surface hydroxyls with respect to the normal of the clay layer was experimentally determined, showing good agreement with molecular simulations. This methodology may provide quantitative insights into the molecular-level description of interfacial processes between organic molecules and clay minerals.

I. Introduction

The increasing development of methods enabling fine tuning of the nanostructure of materials at the molecular level has strongly contributed to the boost in nanotechnology. Beyond the realm of graphene and its promising commercial potential, other 2D natural-based materials have attracted considerable attention from the scientific community [START_REF] Wang | Nanoscale Clay Minerals for Functional Ecomaterials: Fabrication, Applications, and Future Trends[END_REF]. Owing to their naturally occurring nanoscale structures and associated large surface areas, clay minerals are subjected to increasing interest for numerous applications. For instance, the interactions of clay minerals with colloidal particles or molecular species result in interesting bulk-scale physical and chemical properties. Clay-based nanotechnology has thus led to promising innovation in nanoscience-related fields, such as materials science, electronics, food, and healthcare, contributing to the improvement of human quality of life [START_REF] Carretero | Clay minerals and their beneficial effects upon human health. A review[END_REF][START_REF] Zhou | An overview on strategies towards clay-based designer catalysts for green and sustainable catalysis[END_REF][START_REF] Choi | Bioinspired Layered Nanoclays for Nutraceutical Delivery System[END_REF][START_REF] Ma | Visible light photocatalytic activity enhancement of Ag3PO4 dispersed on exfoliated bentonite for degradation of rhodamine B[END_REF][START_REF] Ogawa | Photofunctions of Intercalation Compounds[END_REF]. Their abundance in nature, convenient preparation and biocompatibility are in part responsible for the large quantity of studies devoted to applications of clay minerals.

Sodium montmorillonite is certainly one of the most often used clay minerals owing to its low layer charge density, which is responsible for its swelling behavior and exfoliation properties. This material consists of a 2:1 layer composed of two tetrahedral sheets and one octahedral sheet. Isomorphous substitution of aluminum by magnesium in the octahedral sheet produces a negative charge that is counterbalanced by hydrated Na + cations located in the interlayer space [START_REF] Brigatti | Chapter 2 -Structure and Mineralogy of Clay Minerals[END_REF]. The interlayer cations can be further exchanged by a variety of molecules or polymers, conferring the ability to finely tune the functionality of the composite [START_REF] Darder | Bio-Nanocomposites Based on Layered Double Hydroxides[END_REF][START_REF] Wicklein | Bio-organoclays based on phospholipids as immobilization hosts for biological species[END_REF][START_REF] Jordan | Organophilic Bentonites. I. Swelling in Organic Liquids[END_REF][START_REF] Yamagishi | Optical resolution of metal chelates by use of adsorption on a colloidal clay[END_REF]. The low charge density of the clay surface and its flat structure at the nanoscale level prevent aggregation of intercalated compounds and further enable molecular control of the molecule-molecule distance and orientation through a size matching effect [START_REF] Eguchi | Role of Hydrophobic Interaction in Controlling the Orientation of Dicationic Porphyrins on Solid Surfaces[END_REF]. This method opens the way for advanced reactivity, such as enantioselectivity and photonic properties. The performance of such materials is strongly dependent on the orientation of the guest molecules, which is sometimes difficult to assess. The molecular orientation is most often derived from the basal spacing obtained from X-ray diffraction (XRD) measurements in comparison with the size and shape of the guest molecules. In some cases, the interlayer domain is highly disordered, preventing local information from being directly obtained through XRD analyses [START_REF] Ferrage | Investigation of smectite hydration properties by modeling experimental X-ray diffraction patterns: Part I. Montmorillonite hydration properties[END_REF]. Further complications arise when more than one single layer is examined when the orientation function of the clay particles has to be equally determined [START_REF] Hubert | Investigating the anisotropic features of particle orientation in synthetic swelling clay porous 34 media[END_REF][START_REF] Ferrage | Influence of crystal structure defects on the small-angle neutron scattering/diffraction patterns of clay-rich porous media[END_REF]. The preferred orientation of clay particles is of crucial importance, as it controls the geometric properties of the whole porous medium [START_REF] Dabat | Mesoscale Anisotropy in Porous Media Made of Clay Minerals. A Numerical Study Constrained by Experimental Data[END_REF] and can vary from almost isotropic to perfectly oriented. The orientational distribution function of clay particles can be experimentally obtained through X-ray diffraction or X-ray microtomography measurements, but these techniques are insensitive to intercalated organic molecules. As a consequence, the interfacial interaction between organics and clay minerals remains inaccessible by these techniques. Vibrational spectroscopies are some of the few molecular-level probes that are inherently orientation sensitive and provide an alternative method for probing the organization of both clay particles and organic molecules in the same preparation [START_REF] Ikeda | Basics of Orientation Measurements in Infrared and Raman Spectroscopy[END_REF]. Moreover, Fourier transform infrared (FTIR) spectroscopy allows the study of dynamic processes on a 10 -14 -10 -12 s time scale, which is attainable for molecular dynamic simulations. By explicitly taking into account the particle orientation, the experimental determination of molecular orientation with respect to a clay mineral surface, achievable by FTIR, spectroscopy opens a new way to validate or constrain MD simulations at the molecular level.

The absorption of infrared light depends on the mutual orientation of the transition moment and the electric field components. The polarization of infrared light enables the selection of the orientation of the electric field and therefore offers additional control to measure the orientation of molecules and/or clay particles. The transition moments associated with minerals and organics are often well separated in frequency so that independent orientation measurements can be carried out on the same sample. While the literature is rich in studies on molecular orientation determination, little focus has been placed on clay mineral orientation by FTIR spectroscopy. In a recent paper, the general methodology to extract the anisotropic optical constants of Na-montmorillonite by attenuated total reflectance infrared spectroscopy (ATR-FTIR) was presented, and a strong out-of-plane transition moment associated with the Si-O stretching vibration was evidenced [START_REF] Grégoire | Optical theory based simulation of Attenuated Total Reflection infrared spectra of Montmorillonite[END_REF]. Therefore, such a Si-O stretching vibration band appears to be an adequate descriptor for the determination of the orientation of clay particles.

In the few studies devoted to this objective, the authors systematically used thin-film approximation equations for the calculation of the electric field amplitude to derive the order parameter measuring the anisotropy of the clay deposit [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Holmgren | A Polarized Fourier Transform Infrared Spectrometry Attenuated Total Reflection Study of Bentonite Settled onto Magnetite[END_REF]. To further assess the validity of this approach, the matrix formalism for calculating the electric field amplitude proposed by Hansen, considering the number of layers, the complex refractive index and the deposit thickness, was used and compared with the approximate equations [START_REF] Hansen | Electric Fields Produced by the Propagation of Plane Coherent Electromagnetic Radiation in a Stratified Medium[END_REF][START_REF] Hansen | Expanded formulas for attenuated total reflection and the derivation of absorption rules for single and multiple ATR spectrometer cells[END_REF]. This paper aims to investigate the ability to probe clay orientation by polarized ATR-FTIR spectroscopy. To enable comparison with the results obtained by MD simulations or other experimental techniques, a methodology for determining the orientation of the normal to the clay with respect to a reference axis is proposed. To illustrate the ability to probe the interfacial phenomenon, the orientation of structural hydroxyls and water dipole moments of Namontmorillonite is experimentally determined and successfully compared to the MD results.

II. Materials and Methods

II.1. Sample preparation

SWy-3 montmorillonite clay was obtained from the Source Clays Repository of The Clay Minerals Society. Clay with a size of <1 µm was collected by centrifugation and Na-saturated using three saturation cycles in a 1 M NaCl solution. Excess salt was removed by dialysis until the Cl -anion was no longer detected by the AgNO3 test.

II.2. Polarized ATR-FTIR spectroscopy

The ATR-FTIR spectra were recorded on a Nicolet IS50 equipped with a KBr beam splitter and a DLaTGS detector. Polarized radiation was achieved by a ZnSe wire grid polarizer set before the sample compartment. A VeeMax 3 accessory consisting of a thermally controlled horizontal 45° single-bounce ATR plate was installed, and the temperature was set at a constant value of 25 °C for all experiments. A diluted clay suspension was pipetted and deposited on the Ge IRE (20 mm diameter), and the film was allowed to dry under ambient conditions. Polarized reflectance spectra were acquired at a resolution of 4 cm -1 , and 100 scans were coadded.

For vibrational bands that are not well resolved, such as the Si-O stretching vibration at 1085 cm -1 , band fittings of the region 600-1300 cm -1 were applied with a combination of Lorentzian and Gaussian functions, as in Ras et al. [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF] When band fitting was not necessary, the dichroism ratio was calculated from the maximum intensity.

II.3. Molecular Simulations

The molecular structure of the montmorillonite sample was predicted using grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulations at selected hydration states corresponding to the homogeneous one-layer hydrate (1W) and the dehydrated state (0W). The system was composed of three clay sheets (with dimensions of 8-by-5 unit cells in the ab plane) and three interlayers. Isomorphic octahedral substitutions of Al 3+ by Mg 2+ were randomly of montmorillonite determined by experimental X-ray diffraction modeling [START_REF] Ferrage | Investigation of smectite hydration properties by modeling experimental X-ray diffraction patterns: Part I. Montmorillonite hydration properties[END_REF] of similar samples. Accordingly, the Na-montmorillonite sample shows the least hydration heterogeneity for one layer hydrated at a relative humidity (RH) of 0.4 and under vacuum (~0RH) conditions for the dehydrated state. Consequently, the basal spacing for the present simulations was set, in agreement with the former relative humidity, at 12.46 Å and 9.75 Å for the 1W and 0W states, respectively. For both GCMC equilibrium and MD simulations, the following sets of parameters were used. Interatomic interactions were modeled using the SPC/E water model and the CLAYFF model of silicate minerals [START_REF] Cygan | Molecular Models of Hydroxide, Oxyhydroxide, and Clay Phases and the Development of a General Force Field[END_REF]. The van der Waals "size" parameter of oxygen atoms belonging to the clay layer was increased by 7 % relative to the original CLAYFF model as recommended [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 2. Toward a Precise Coupling between Molecular Simulations and Diffraction Data[END_REF]. Interatomic interactions were calculated in real space up to a distance of 15 Å. Beyond this distance, long-range Coulomb interactions were resolved by Ewald summation. Ewald sum calculations were computed with a precision of 10 -5 .

The two systems were first equilibrated in the µH2OVT ensemble (fixed volume, temperature, and water chemical potential) in the GCMC simulation using a homemade program [START_REF] Ferrage | Hydration Properties and Interlayer Organization of Water and Ions in Synthetic Na-Smectite with Tetrahedral Layer Charge. Part 2. Toward a Precise Coupling between Molecular Simulations and Diffraction Data[END_REF][START_REF] Delville | Modeling the clay-water interface[END_REF][START_REF] Rinnert | Hydration of a Synthetic Clay with Tetrahedral Charges: A Multidisciplinary Experimental and Numerical Study[END_REF][START_REF] Porion | 1H and 7Li NMR Pulsed Gradient Spin Echo Measurements and Multiscale Modeling of the Water and Ionic Mobility within Aqueous Dispersions of Charged Anisotropic Nanoparticles[END_REF][START_REF] Porion | Structural and Dynamical Properties of the Water Molecules Confined in Dense Clay Sediments: a Study Combining 2H NMR Spectroscopy and Multiscale Numerical Modeling[END_REF][START_REF] Jiménez-Ruiz | Anisotropy on the Collective Dynamics of Water Confined in Swelling Clay Minerals[END_REF].

The one-layer hydrates were equilibrated using an RH of 0.4 at 298 K. The target relative humidity was converted to a water chemical potential using the ideal fluid relation H2O =  0 H2O + kBT ln(RH), where  0 H2O = kBT ln( 3 ),  is the de Broglie wavelength of H2O (0.238376 Å), kB is Boltzmann's constant, T is the absolute temperature, and  is the density of water vapor at 298 K (7.7087481 × 10 -7 molecules Å -3 ). Each GCMC simulation was carried out for 6 × 10 7 steps. At each step, one of the clay interlayers was selected randomly, and with an equal probability, an attempt was made to remove a H2O molecule, add a H2O molecule in a random configuration, or move a randomly chosen Na + ion or H2O molecule. Interlayer water quantities were found to be 4.95. This result is in agreement with previously reported water vapor desorption isotherm experiments on similar samples [START_REF] Bérend | Mechanism of Adsorption and Desorption of Water Vapor by Homoionic Montmorillonites: 2. The Li+ Na+, K+, Rb+ and Cs+-Exchanged Forms[END_REF]. For the dehydrated state, no water molecule addition was attempted. Thus, at each step, one of the clay interlayers was selected randomly, and an attempt was made to simply move a randomly chosen Na + ion.

Thermalized GCMC configurations were used as the starting points for the MD simulations.

Molecular dynamics simulations were performed with LAMMPS software [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. Each simulation cell was first equilibrated in the NVE ensemble (fixed number of atoms, volume and energy), weakly coupled to a Berendsen thermostat, for a total of 200000 steps at T = 298 K.

During these steps, the timestep size for subsequent molecular dynamics simulations was gradually increased from 0.001 to 0.5 femtoseconds to allow a smooth transition from GCMC simulations performed with rigid clay layers to MD simulations with flexible layers. Finally, each system was simulated for 1 nanosecond in the NVE ensemble and weakly coupled to a

Berendsen thermostat with a 0.5 fs timestep at T = 298 K. Polarized ATR-FTIR spectroscopy is commonly used to characterize dipole orientation or molecular order given that the symmetry of the system is known. Recently, the uniaxial a)

III. Order parameter and mean-squared electric field amplitude calculations

a)

symmetry of Na-SWy-3 clays was verified, enabling the calculation of the anisotropic optical constants [START_REF] Grégoire | Optical theory based simulation of Attenuated Total Reflection infrared spectra of Montmorillonite[END_REF]. For a beam propagating on the ATR crystal surface along the x direction, the evanescent field is directed normal to this surface in the z direction. Using p-polarized radiation, the evanescent electric field has components in the x and z directions, while for s-polarization, it has components only in the y direction. Therefore, the mean-squared amplitude of the electric field is separated into three orthogonal components:

< 𝐸 2 > = < 𝐸 𝑥 2 > + < 𝐸 𝑦 2 > + < 𝐸 𝑧 2 > (1)
By using a projection similar to that for the electric field vector, μ, the vector representing the dipole change associated with a vibration, can be written as:

𝜇 = 𝜇 𝑥 + 𝜇 𝑦 + 𝜇 𝑧 (2)
To calculate the absorption intensity, the individual dipoles must be summed:

𝐴 = ∑ |𝜇 𝑖 . 𝐸|² 𝑖 (3)
Making use of the polar angle as presented in Figure 1a, the absorbance can be written as:

𝐴 𝑥 = 𝑘 < (𝜇. 𝐸 𝑥 ) 2 > = 𝑘 < 𝐸 𝑥 2 > 𝑠𝑖𝑛 2 (𝜃)𝑐𝑜𝑠²(𝜙) (4a)
𝐴 𝑦 = 𝑘 < (𝜇. 𝐸 𝑦 ) 2 > = 𝑘 < 𝐸 𝑦 2 > 𝑠𝑖𝑛 2 (𝜃)𝑠𝑖𝑛²(𝜙) (4b)
𝐴 𝑧 = 𝑘 < (𝜇. 𝐸 𝑧 ) 2 > = 𝑘 < 𝐸 𝑧 2 > 𝑐𝑜𝑠²(𝜃) (4c)
For transition moments that are axially distributed around the z axis, as is the case for clays, all values of φ are equiprobable:

< 𝑐𝑜𝑠 2 (𝜙) > = < 𝑠𝑖𝑛²(𝜙) > = 1 2 (5)
The dichroic ratio is obtained by dividing the p-polarized absorbance by the s-polarized absorbance:

𝑅 𝐴𝑇𝑅 = 𝐴 𝑝 𝐴 𝑠 = 𝐴 𝑥 +𝐴 𝑧 𝐴 𝑦 = <𝐸 𝑥 2 >𝑠𝑖𝑛 2 (𝜃)+2<𝐸 𝑧 2 >𝑐𝑜𝑠²(𝜃) <𝐸 𝑦 2 >𝑠𝑖𝑛²(𝜃) (6)
For any orientations of the dipole defining an isotropic system, the isotropic dichroic ratio is written as:

𝑅 𝑖𝑠𝑜 = <𝐸 𝑥 2 >+<𝐸 𝑧 2 > <𝐸 𝑦 2 > (7)
The order parameter S characterizing the degree of preferential orientation is calculated on the basis of 2 nd -order Legendre polynomials as:

𝑆 = < 𝑃 2 cos(𝜃) > =< 3 2 𝑐𝑜𝑠 2 (𝜃) - 1 2 > (8)
By substituting the angular factor in equation 8, the order parameter S is then defined as:

𝑆 = <𝐸 𝑥 2 >-𝑅 𝐴𝑇𝑅 <𝐸 𝑦 2 > + <𝐸 𝑧 2 > <𝐸 𝑥 2 >-𝑅 𝐴𝑇𝑅 <𝐸 𝑦 2 > -2 <𝐸 𝑧 2 > (9)
The S value can reach values from -0.5 to 1. A value of -0.5 indicates that all transition moments are precisely and uniformly oriented in the x or y direction, while a value of 1 indicates transition moments along the z axis. Values of 0 are obtained for isotropic materials, where all dipole moments have random orientation around the z axis or are all oriented at the magic angle (θ = 54.7°).

The evaluation of the order parameter requires the calculation of the electric field <E²>. The general procedure for calculating the electric field amplitude at any point along a stratified system was developed by Hansen [START_REF] Hansen | Electric Fields Produced by the Propagation of Plane Coherent Electromagnetic Radiation in a Stratified Medium[END_REF]. Here, only the analytical expression will be given for a 3-phase system, as depicted in Figure 1b [START_REF] Axelsen | Orientational order determination by internal reflection infrared spectroscopy[END_REF][START_REF] Ohta | Matrix formalism for calculation of electric field intensity of light in stratified multilayered films[END_REF][START_REF] Ohta | Matrix formalism for calculation of the light beam intensity in stratified multilayered films, and its use in the analysis of emission spectra[END_REF]. Because these expressions explicitly take into account the thickness and absorption properties of the sample, they are called thicknessand absorption-dependent equations:

-

------------------------------------ < 𝐸 𝑦(𝑗=1) 2 (𝑧) > = {(1 + 𝑅 𝑠 ) + 2√𝑅 𝑠 cos (𝛿 𝑠 - 4𝜋𝑑𝜉 1 𝜆 )} (10a) < 𝐸 𝑦(𝑗=2) 2 (𝑧) > = |(1 + 𝑟 𝑠 ) cos ( 2𝜋𝑑𝜉 2 𝜆 ) + 𝑖 ( 𝜉 1 𝜉 2 ) (1 -𝑟 𝑠 )𝑠𝑖𝑛 ( 2𝜋𝑑𝜉 2 𝜆 )| 2 (11a) < 𝐸 𝑦(𝑗=3) 2 (𝑧) > = |𝑡 𝑠 | 2 exp ( -4𝜋 𝐼𝑚(𝜉 3 ) 𝜆 ) (𝑧 -𝑑) (12a) ------------------------------------- < 𝐸 𝑥(𝑗=1) 2 (𝑧) > = 𝑐𝑜𝑠 2 (𝛼) {(1 + 𝑅 𝑝 ) -2 √ 𝑅 𝑝 cos (𝛿 𝑝 - 4𝜋𝑑𝜉 1 𝜆 )} (10b) < 𝐸 𝑥(𝑗=2) 2 (𝑧) > = |cos(𝛼) (1 -𝑟 𝑝 ) cos ( 2𝜋𝑑𝜉 2 𝜆 ) + 𝑖 ( 𝜉 2 𝑛 1 𝑛 ̂2 2 ) (1 + 𝑟 𝑝 )𝑠𝑖𝑛 ( 2𝜋𝑑𝜉 2 𝜆 )| 2 (11b) < 𝐸 𝑥(𝑗=3) 2 (𝑧) > = | 𝜉 3 𝑛 ̂3 𝑡 𝑝 | 2 exp ( -4𝜋 𝐼𝑚(𝜉 3 ) 𝜆 ) (𝑧 -𝑑) (12b) ------------------------------------- < 𝐸 𝑧(𝑗=1) 2 (𝑧) > = 𝑠𝑖𝑛²(𝛼) {(1 + 𝑅 𝑝 ) + 2 √ 𝑅 𝑝 cos (𝛿 𝑝 - 4𝜋𝑑𝜉 1 𝜆 )} (10c) < 𝐸 𝑧(𝑗=2) 2 (𝑧) > = | 𝑛 1 2 sin(𝛼) 𝑛 ̂2 2 (1 -𝑟 𝑝 ) cos ( 2𝜋𝑑𝜉 2 𝜆 ) + 𝑖 ( 𝑛 1 sin(𝛼) 𝜉 2 ) cos (𝛼)(1 + 𝑟 𝑝 )𝑠𝑖𝑛 ( 2𝜋𝑑𝜉 2 𝜆 )| 2 (11c) < 𝐸 𝑧(𝑗=3) 2 (𝑧) > = | 𝑛 1 sin(𝛼) 𝑛 ̂3 𝑡 𝑝 | 2 exp ( -4𝜋 𝐼𝑚(𝜉 3 ) 𝜆 ) (𝑧 -𝑑) (12c)
where 𝜉 𝑗 = 𝑛 ̂𝑗cos (𝛼 ̂𝑗), j is the layer taking a value from 1 to 3 for a 3-phase system, 𝑛 ̂𝑗 is the isotropic complex optical constant (𝑛 ̂𝑗 = 𝑛 𝑗 + 𝑖𝜅 𝑗 ), α is the angle of incidence, Rs and Rp are the reflectance signals, and rs, rp, ts and tp correspond to the Fresnel coefficients calculated with the expressions given in [START_REF] Grégoire | Optical theory based simulation of Attenuated Total Reflection infrared spectra of Montmorillonite[END_REF].

Alternatively, Harrick proposed equations to calculate the amplitude of the electric field in the second phase of a 3-phase system when the film thickness is very large compared to the evanescent wave depth of penetration [START_REF] Harrick | Electric Field Strengths at Totally Reflecting Interfaces[END_REF][START_REF] Harrick | Effective Thickness of Bulk Materials and of Thin Films for Internal Reflection Spectroscopy[END_REF]. In this approximation, the film is neglected, and equations 11a-c are resolved at z = 0 for d = 0 and n2 = 1 (air). This corresponds to a 2-phase system, and therefore, these approximations are referred to as the 2-phase approximations. This gives the following analytical expressions (equation 13a-c): 2 )𝑠𝑖𝑛 2 (𝛼)-𝑛 21 2 ] (13b) 2 )𝑠𝑖𝑛 2 (𝛼)-𝑛 21 2 ]

< 𝐸 𝑦(𝑗=2) 2 (𝑧 = 0) > = 4 𝑐𝑜𝑠²(𝛼) 1-𝑛 21 2 (13a) < 𝐸 𝑥(𝑗=2) 2 (𝑧 = 0) > = 4 𝑐𝑜𝑠²(𝛼)(𝑠𝑖𝑛 2 (𝛼)-𝑛 21 2 ) (1-𝑛 21 2 )[(1+𝑛 21
< 𝐸 𝑧(𝑗=2) 2 (𝑧 = 0) > = 4 𝑐𝑜𝑠²(𝛼)𝑠𝑖𝑛 2 (𝛼) (1-𝑛 21 2 )[(1+𝑛 21
(13c)

where 𝑛 21 = 𝑛 2 /𝑛 1 , with 𝑛 1 and 𝑛 2 being the real parts of the refractive indexes of phases 1 and 2, respectively.

In the other extreme case, where the film is very small compared to the penetration depth of the evanescent wave, the electric field amplitude in the second phase can be calculated from equations 11a-c at z = 0 by setting d = 0 and n2 = nfilm. In this approximation, the film is optically considered but treated as non-absorbing (κ2 = 0), and the following analytical equations are referred to as the thin-film approximation: 2 )𝑠𝑖𝑛 2 (𝛼)-𝑛 31 2 ] (14b)

< 𝐸 𝑦(𝑗=2) 2 (𝑧 = 0) > = 4 𝑐𝑜𝑠²(𝛼) 1-𝑛 31 2 (14a) < 𝐸 𝑥(𝑗=2) 2 (𝑧 = 0) > = 4 𝑐𝑜𝑠²(𝛼)(𝑠𝑖𝑛 2 (𝛼)-𝑛 31 2 ) (1-𝑛 31 2 )[(1+𝑛 31
< 𝐸 𝑧(𝑗=2) 2 (𝑧 = 0) > = 4 𝑛 32 4 𝑐𝑜𝑠²(𝛼)𝑠𝑖𝑛 2 (𝛼) (1-𝑛 31 2 )[(1+𝑛 31 2 )𝑠𝑖𝑛 2 (𝛼)-𝑛 31 2 ] (14c)
It is therefore of interest to evaluate the various theoretical treatments given above by carefully examining the plausibility of the order parameter as obtained. More specifically, constraints having the ability to dismiss a given approach if the results are irrelevant must be added. This methodology offers advantages in understanding the origin of failures in theory. For example, if the 2-phase system adequately describes the molecular orientation, it suggests that the optical properties of the thin film are best ignored. Although questionable in terms of physics, it is certainly more reasonable than systematically employing thin-film expressions when thin films are deposited on an ATR crystal. This mismatch between theory and experiment originates from the inherent difficulty of optically describing a thin film layer. Modifications of the dielectric functions at the interface accounting for surface rugosity may substantially influence the electric field amplitude within the thin film. Using an effective dielectric constant may partially solve the problem, even if difficulties in experimentally determining this effective dielectric function of a very thin layer still subsist [START_REF] Easwarakhanthan | An ellipsometric procedure for the characterization of very thin surface films on absorbing substrates[END_REF]. Anderson and Hansen calculated the electric field amplitude of an idealized single film composed of multiple anisotropic layers and showed that optical constants may be chosen to reflect the physical behavior of the interface. This approach assimilates optical constants not as a bulk phase optical property but rather as arbitrary fitting parameters [START_REF] Anderson | Optical characterization of thin films*[END_REF]. In Figure 2a, the electric field amplitudes along the x, y and z axes were calculated using the thickness-and absorption-dependent equations for an isotropic 3-phase system (Ge IRE -Clay -Air; Figure 1b). The calculations were performed with an assumption of isotropic refractive indexes of 4 for Ge IRE, 1.4 for the clay film and 1 for air at 1085 cm -1 and an incidence angle of 45°. The three orthogonal components were computed at the interface between the ATR crystal and the clay film for different thicknesses of the clay film. The clay film was treated as a non-absorbing (k = 0) and absorbing (k = 0.7675) medium. Among the three orthogonal components of the electric field, <Ez²> exhibits the largest variation, with values ranging from 0.350 and 0.592 to 1.712 and 2.593 for a 2 µm thick film for absorbing and non-absorbing media, respectively (Figure 2a). In the same thickness range, <Ex²> and <Ey²> show only a slight variation, which is more pronounced for absorbing media, where an overall decrease in the amplitude is observed compared to that in a non-absorbing medium. The thin-film and 2-

a) b) a) b)
phase approximations correspond to the amplitudes at which the thicknesses are 0 and 2 µm, respectively, for the non-absorbing media. For the absorbing medium, these approximations give values closer to the exact calculations for very thin films than for thick films. Difficulties arise for intermediate film thicknesses of up to 1 µm, where the two approximate sets of equations do not allow a reliable estimation of the electric field components. Therefore, neglecting the imaginary part of the complex optical constants can induce significant variations in the computation of electric field amplitude, leading to an erroneous determination of order parameter S.

In Figure 2b, the evolution of the electric field amplitude along the thickness for a 100 nm thick film is shown. The electric field components are not constant within the film and decrease almost linearly. The isotropic ratio, R iso , increases when the film thickness increases, illustrating the inherent uncertainty when the calculations are performed only at the interface between the ATR crystal and the film. Therefore, the thinner the film is, the more accurate the determination of the molecular orientation. Nevertheless, a mean value of the three orthogonal components of the electric field within the film thickness is preferred to achieve more reliable results [START_REF] Picard | Quantitative orientation measurements in thin lipid films by attenuated total reflection infrared spectroscopy[END_REF].

IV.2. Optical constants and definition of the order parameters S for the

IV.3. Experimental determination of 𝑺 𝜹 𝑺𝒊𝑶 from a single layer deposit

In the limit case where a single layer of smectite is deposited on the ATR crystal (𝑆 𝛽 is assumed to be equal to 1), equation 15 is reduced to 𝑆 𝜃 𝑆𝑖𝑂 = 𝑆 𝛿 𝑆𝑖𝑂 , offering the possibility to determine the angle between the transition moment under investigation and the normal to the clay surface.

Such an experiment can be achieved with the Langmuir-Blodgett technique, as performed by

Ras et al. [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF] This film preparation technique generally leads to a higher degree of organization than the layer-by-layer assembly (LbL) because in the former case, the film is formed at the gas/liquid interface and offers additional control on the lateral distribution of clay mineral particles [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Ras | Ultrathin hybrid films of clay minerals[END_REF]. Although previous experiments were performed on Na-SWy-1, we assumed that the optical constants that were previously determined for Na-SWy-3 are valid [START_REF] Grégoire | Optical theory based simulation of Attenuated Total Reflection infrared spectra of Montmorillonite[END_REF]. The dichroic ratio measured by these authors on a unique layer of Na-SWy-1 was found to be 8.33 using a ZnSe IRE. Based on the extraction of the isotropic optical constants from Figure 3b at 1085 cm -1 (n2 = 0.467; κ2 = 0.768), the mean square electric field amplitudes were calculated using the different sets of equations presented in the theory section (Table 1a). For the three sets of equations, similar results were obtained for the calculations of <Ex²> and <Ey²>, whereas important variations were noted for <Ez²>. Given the very thin thickness of the sample, the difference observed between the thin-film approximation and the thickness-and absorptiondependent equations originates from the attenuation coefficient, which drastically reduced the <Ez²> value from 61.55 to 4.49. This last value is close to that calculated with the 2-phase approximation, where the film is not considered. The order parameter 𝑆 𝜃 𝑆𝑖𝑂 , which is strongly influenced by the value of <Ez²> according to equation 9, was also calculated, as shown in Table 1a, along with the < 𝜃 > value, assuming that < cos²(𝜃) > = 𝑐𝑜𝑠 2 (< 𝜃 >), which holds only for a very narrow distribution of θ. The value of <θ> calculated using the thin-film approximations is irrelevant since it would signify that the transition moment is directed within the plane of the clay layer. For the other equations used to compute the electric field amplitude, <θ> values of 30° and 35° were obtained. In that case, the tilt angles appear to be very large, which would suggest large deformations of the tetrahedral units, which is contradictory to the XRD analysis and molecular dynamics simulations [START_REF] Cygan | Molecular Models of Hydroxide, Oxyhydroxide, and Clay Phases and the Development of a General Force Field[END_REF][START_REF] Larentzos | An ab Initio and Classical Molecular Dynamics Investigation of the Structural and Vibrational Properties of Talc and Pyrophyllite[END_REF][START_REF] Cygan | Molecular models and simulations of layered materials[END_REF][START_REF] Brindley | Crystal Structures of Clay Minerals and Their Xray Identification[END_REF]. Therefore, using the as-determined optical indexes, none of the equations for the calculations of the mean square electric field amplitudes is reliable. This suggests that the electromagnetic theory fails to properly describe thin films of clay deposited on ATR crystals.

Table 1. Values of the three orthogonal mean square electric field amplitudes at 1085 cm -1 calculated using the formalisms presented in the theory section, the order parameter of the transition moment and the tilt angle. e RATR taken from refs [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Ras | Ultrathin hybrid films of clay minerals[END_REF]. Note that the authors in these references expressed the dichroic ratio as Ap/As and was converted to be consistent with our definition -see equation 6.

< 𝐸 𝑥 2 > < 𝐸 𝑦 2 > < 𝐸 𝑧 2 > 𝑆 𝜃 𝑆𝑖𝑂 < 𝜃 𝑆𝑖𝑂 > d a)
To overcome this difficulty, we analyzed the order parameter 𝑆 𝛿 𝑆𝑖𝑂 as a function of the isotropic value of the refractive index of the clay film (Figure 5). The extinction coefficient, κ(v), which represents the dynamic dipole moment and describes the absorption of the electromagnetic radiation, was set to 0.768 [START_REF] Fraser | Conformation in fibrous proteins and related synthetic polypeptides[END_REF][START_REF] Pauling | Introduction to Quantum Mechanics with Applications to Chemistry[END_REF]. Although the values of <Ex²> and <Ey²> seem insensitive to the refractive index, n2, drastic variation in <Ez²> is noted, which in turn affects the calculated 𝑆 𝛿 𝑆𝑖𝑂 value. Changes in 𝑆 𝛿 𝑆𝑖𝑂 are the most important for the n2 value in the range of 0-1 and correspond to less than 2% of the variation per 0.1 unit of refractive index above n2 = 1.4. The determination of the refractive index value is therefore of prime importance and strongly affects the determination of the particle orientations. For these out-of-plane Si-O stretching vibrations, values of n2 between 0.61 and 1.15 have been reported for Na-SWy layers that lead to strongly different order parameters [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Holmgren | A Polarized Fourier Transform Infrared Spectrometry Attenuated Total Reflection Study of Bentonite Settled onto Magnetite[END_REF][START_REF] Johnston | Polarized ATR-FTIR Study of Smectite in Aqueous Suspension[END_REF]. To obtain more insight into the maximum value of 𝑆 𝛿 𝑆𝑖𝑂 , MD simulations were performed on a montmorillonite layer. The angle between the Si-O apical bond and the normal to the clay layer was extracted and averaged over a simulation period of 125 fs. Subsequently, the distribution function 𝑓(𝛿) was extracted, and the 𝑆 𝛿 𝑆𝑖𝑂 (𝑀𝐷) value was calculated according to:

𝑆 𝛿 𝑆𝑖𝑂 =< 𝑃 2 𝑐𝑜𝑠(𝛿) > = < 3 2 𝑐𝑜𝑠 2 (𝛿) - 1 2 > (16)
The 𝑆 𝛿 𝑆𝑖𝑂 (𝑀𝐷) value obtained was 0.987, corresponding to a <δ> value of 5.33°. Reporting this value in Figure 5 enables determination of a refractive index value of 2.4. Since this value is extracted on the basis of structural considerations from molecular simulations, it has to be regarded as the maximum value relating the transition moment to the normal of the clays.

Submicron natural clay minerals may present structural faults and layer flexibility, leading to a more disordered structure that would in turn decrease the experimental 𝑆 𝛿 𝑆𝑖𝑂 value [START_REF] Suter | Large-Scale Molecular Dynamics Study of Montmorillonite Clay: Emergence of Undulatory Fluctuations and Determination of Material Properties[END_REF][START_REF] Thyveetil | Emergence of undulations and determination of materials properties in large-scale molecular dynamics simulation of layered double hydroxides[END_REF]. Due to the inherent difficulty in precisely determining this value and given the low intensity of the 1085 cm -1 ATR band, an arbitrary value of refractive index of 1.4 will be used, resulting in an 𝑆 𝛿 𝑆𝑖𝑂 value of 0.929 (Table 1b). This corresponds to the refractive index in a region where there is no absorption in Figure 3b and may indicate that the experimental value of 0.467 for n2 is decreased substantially by the strong anomalous dispersion associated with the very intense Si-O stretching vibration at 1039 cm -1 .

IV.4. Experimental determination of 𝑺 𝜷 for thin film deposits

For the 22 nm thick Na-SWy-3 deposit on the Ge IRE, an experimental dichroic ratio of 4.1 was measured. Using the newly determined value of the refractive index, the electric field amplitudes were calculated with the approximate and exact equations (Table 1c). Both thinfilm approximations and thickness-and absorption-dependent equations give similar 𝑆 𝜃 𝑆𝑖𝑂 values of 0.776 and 0.847, respectively, while a value of 0.396 was obtained using the 2-phase approximations. This suggests that the optical properties of the clay film have to be taken into account for reliable calculation of the electric field amplitude. Using equation 15 and the calculated value of 𝑆 𝛿 𝑆𝑖𝑂 (0.929), the order parameter representing the orientation of the normal to the clay film with respect to the z axis, 𝑆 𝛽 , is 0.912, corresponding to a <β> value of 14°. In a recent publication, measurements of the order parameters of a dozen-micrometer-thick Na-SWy-2 film led to an experimental Sβ value of approximately 0.65 [START_REF] Carrier | Effect of Water on Elastic and Creep Properties of Self-Standing Clay Films[END_REF]. For thinner sample preparation, as used for FTIR spectroscopy measurements, a higher 𝑆 𝛽 value would be expected, accounting for the lower number of layers investigated and their less disordered organization. Therefore, our results are consistent and fully support the determined value of equations, the variation in the dichroic ratio for Sβ below 0.6 is very limited. This suggests that precise orientation measurements of nearly isotropic clay particles may not be achievable using ATR spectroscopy. In this case, other transition moments should be considered. Figure 6b also suggests the need to determine the dichroic ratio with the best accuracy. The highest precision of the orientation measurement was achieved for Sβ greater than 0.6, where large variations in the dichroic ratio were correlated to small variations in the clay orientation. It can thus be concluded that the orientation of an anisotropic clay particle organization can readily be determined using ATR-FTIR spectroscopy according to the methodology proposed here.

Moreover, using the approximate equations instead of the most complete equations may underestimate the Sβ value. Future directions for this work could be the experimental determination of the orientation of all transition moments of clays with respect to the clay normal, but the validity of the assumptions for the electric field calculations still need to be proven.

IV.5. Determination of structural hydroxyl and interlayer water orientations with

respect to the clay layer surface.

The benefit of ATR-FTIR spectroscopy lies in its high sensitivity for probing interfacial phenomena. For a more accurate description of the organization of organic molecules on mineral surfaces, from the nanometer to the micrometer scales, the mineral organization must be taken into account. This is illustrated in this section, where the orientations of structural hydroxyls and water molecules are determined with respect to the clay mineral surface, which allows comparison with the MD results.

To improve the signal-to-noise ratio, spectra were calculated from the anisotropic optical constants determined on a thick deposit of Na-SWy-3 (>2 µm). Subsequently, s-and ppolarized ATR spectra were computed assuming a 1 nm thin film, where the information about the orientation of all transition moments corresponds to that of the bulk deposit. This procedure guarantees a high-quality spectrum with an improved signal-to-noise ratio. The normal to the clay orientation with respect to the z axis was determined from the Si-O stretching vibration dichroic ratio at 1085 cm -1 according to the method described in this paper. The clay mineral hydroxyl orientation was determined from their stretching vibrations at 3630 cm -1 . Due to the strong overlap between the vibrations of hydroxyl groups and those of water, the same thick deposit was dehydrated under an air-dried stream for 24 h, and the dehydrated-clay optical constants were extracted for the simulation of polarized ATR spectra. Using dehydrated or monohydrated samples was not observed to significantly affect the experimental dichroic ratio of the Si-O stretching bands, so the orientation of the particles remained equivalent for both hydration states, in agreement with the MD simulations. For water molecules, the dichroic ratio from the bending motion at 1637 cm -1 was calculated. For both transition moments, the uniaxial orientation of the transition moment was determined with respect to the z axis (𝑆 𝜃 𝑂𝐻 ). To discard the influence of particle orientations, an order parameter relative to the normal to the clay, 𝑆 𝛿 𝑂𝐻 , was obtained according to:

𝑆 𝛿 𝑂𝐻 𝑜𝑟 𝐻 2 𝑂 = 𝑆 𝜃 𝑂𝐻 𝑜𝑟 𝐻 2 𝑂 𝑆 𝛽 (17)
where 𝑆 𝛽 is determined using equation 15. The dichroic ratio of the SiO stretching vibrations extracted from the simulated s-and p-polarized spectra was found to be 2.77, corresponding to a 𝑆 𝛽 value of 0.847. This value is lower than that measured for a 22 nm thick sample (𝑆 𝛽 = 0.912), suggesting slightly less anisotropic particle organization for thicker deposits (Table 2a). Using the isotropic optical constants from Figure 3b at the wavenumbers of the two transition moments, i.e., 3630 cm -1 and 1638 cm -1 for ν MO-H and δ HOH , respectively, and the experimentally determined dichroic ratio, the order parameter 𝑆 𝜃 𝑂𝐻 was calculated according to

V. Concluding remarks and perspectives

The present study provides the general methodology for determining clay particle orientation using polarized ATR-FTIR spectroscopy and allows the derivation of three major conclusions.

First, Si-Oap transition moments are suitable for determining clay particle orientations of rather anisotropic organizations (S > 0.6). The appropriate optical constants associated to the Si-Oap vibration for reliable measurements of particle orientations were determined: the frequency dependence variation of the refractive index is best ignored (n2 set to 1.4), and the extinction coefficient was determined by pATR measurements to be κ2 = 0.768. Second, by comparison with the absorption-and thickness-dependent equations, the thin-film approximations often used in literature [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Ras | Orientation and conformation of octadecyl rhodamine B in hybrid Langmuir-Blodgett monolayers containing clay minerals[END_REF] were observed to slightly underestimate the order parameter, while the 2-phase approximation yielded unreliable results. Third, as a proof of concept in the present study, structural hydroxyl and water molecule orientations were experimentally determined on Na-SWy-3 smectite with respect to the normal of the clay using equation 15 with an 𝑆 𝛿 𝑆𝑖𝑂 value of 0.929 (which depends on the ATR IRE and the clay mineral studied) to extract 𝑆 𝛽 and equation 17 to determine 𝑆 𝛿 𝑂𝐻 𝑜𝑟 𝐻 2 𝑂 . This result was found to be in fairly good agreement with the results of molecular dynamics simulations.

The present work extends the quantitative treatment of clay minerals using pATR methods to include sample thicknesses larger than a single monolayer of particles [START_REF] Ras | Polarized Infrared Study of Hybrid Langmuir-Blodgett Monolayers Containing Clay Mineral Nanoparticles[END_REF][START_REF] Ras | Ultrathin hybrid films of clay minerals[END_REF][START_REF] Ras | Orientation and conformation of octadecyl rhodamine B in hybrid Langmuir-Blodgett monolayers containing clay minerals[END_REF]. This upper scale, which could be referred as to the aggregate scale, is of particular importance for enabling connection between the orientation of particles and the macroscopic properties of hybrid or composite materials. Taking advantage of a reference function describing the orientation of clay particles [START_REF] Dabat | A general orientation distribution function for clay-rich media[END_REF], the order parameter as obtained from pATR measurements using the presented methodology is thus sufficient to determine the full shape of the orientation distribution function. Such an information is a key parameter for the design of clay-based engineered materials with tunable properties [START_REF] Podsiadlo | Ultrastrong and Stiff Layered Polymer Nanocomposites[END_REF][START_REF] Aulin | High strength, flexible and transparent nanofibrillated cellulose-nanoclay biohybrid films with tunable oxygen and water vapor permeability[END_REF][START_REF] Pan | Transforming ground mica into high-performance biomimetic polymeric mica film[END_REF][START_REF] Wang | Highwater-content mouldable hydrogels by mixing clay and a dendritic molecular binder[END_REF]. In addition to the refined analysis of clay particles orientation by pATR measurements, the logical perspective of this work is to follow up this study with orientation determination of organic molecules adsorbed on or intercalated within clay particles. Because of the great sensitivity of FTIR spectroscopy to organic functional groups, such investigation should help to identify how organic molecules interact with clay interfaces. This is of particular importance given the prominent role played by organo-clays in various applications.

  75] oct [Al3.25Mg0.75] tet [Si8.0]O20(OH)4. The basal spacing was set according to the results

Figure 1 .

 1 Figure 1. a) Definition of the laboratory (xyz) coordinate system used in this paper to define

Figure 2 .

 2 Figure 2. Orthogonal mean squared electric field amplitude a) at the interface between Ge IRE

Figure 3 .

 3 Figure 3. a) Experimental s-and p-polarized spectra of a 22 nm thick Na-SWy-3 film on Ge

Figure 3a presentsFigure 4 .

 4 Figure3apresents the ATR spectra of the 22 nm thick film analyzed with s-and p-polarization.The most obvious change is the increase in the band located at 1085 cm -1 assigned to a

a

  Na-SWy-n1 = 2.41; n2 = 0.467; RATR = 8.33 e ; ν = 1085 cm -1 Thin Film a -κ2 = 0; d = -SWy -n1 = 2.41; n2 = 1.40; RATR = 8.33 e ; ν = 1085 cm -1 Thin Film -κ2 = 0; d = -SWy -n1 = 4; n2 = 1.40; RATR = 4.10; ν = 1085 cm -1 Thin Film -κ2 = 0; d = Assuming an infinite and non-absorbing thin film -see equation 14a-c b The film is not consideredsee equation 13a-c c The electric field amplitude is averaged over the film thickness -see equation 11a-c d Assuming an infinitely narrow orientational distribution < 𝐜𝐨𝐬²(𝛉) > = 𝐜𝐨𝐬²(< θ >)

Figure 5 .

 5 Figure 5. Evolution of the mean square electric field amplitude calculated with equation 11a-c

Figure 6 .

 6 Figure 6. a) Evolution of the dichroic ratio for different film thicknesses in the case of perfectly

Figure 6a presents the

  Figure 6apresents the variation in the dichroic ratio as a function of the sample thickness for

equation 9

 9 and converted to 𝑆 𝛿 𝑂𝐻 using equation 17. MD simulations were comparatively performed, and the angular distributions of the transition moments were extracted and are reported in

  -n1 = 4; n2 = 1.4; κ2 = 0.768; RATR = 2.77; ν = 1085 cm -1 ; d = 1 nm Thickness-Absorption dependent 0.847 x x x x b) νMO-H -n1 = 4; n2 = 1.388; κ2 = 0.033; RATR = 1.01; ν = 3630 cm -1 ; d = 1 nm Thickness-Absorption dependent 0.847 -0.356 -0.421 77° [-0.437;-0.367] a c) δOHO -n1 = 4; n2 = 1.330; κ2 = 0.048; RATR = 1.23; ν = 1638 cm -1 ; d = 1 nm Thickness-Absorption dependent 0.847 -0.045 -0.053 57° [-0.213;-0.140] a a Extremum values from MD simulation assuming rigid or flexible clay

Table 2 :

 2 Table 2b-c. The structural hydroxyls give an experimental 𝑆 𝛿 𝑂𝐻 value of -0.421, which is very close to that determined by MD simulations (-0.436), confirming that the hydroxyls are oriented nearly parallel to the layer, as expected for dioctahedral smectites. The good agreement between the experimental and expected 𝑆 𝛿 𝑂𝐻 values fully supports the methodology presented in this work for extracting 𝑆 𝛽 and therefore validates the proposed values of the optical constants, n2 and κ2, to determine clay mineral orientation. The dichroism ratio determined from the bending vibrations of the water molecules was 1.23, corresponding to an 𝑆 𝛿 𝐻 2 𝑂 of -0.053. This value indicates that the orientation of the water molecules is nearlyisotropic. The MD-extracted value is slightly lower than the experimental value, reflecting a higher proportion of water molecules with their dynamic transition moments perpendicular to the z-axis. This slight discrepancy may originate from the contribution of surface-adsorbed water molecules and/or 2W layers, as commonly observed at ambient humidity and not taken into account in the simulations.[START_REF] Ferrage | Investigation of smectite hydration properties by modeling experimental X-ray diffraction patterns: Part I. Montmorillonite hydration properties[END_REF][START_REF] Bérend | Mechanism of Adsorption and Desorption of Water Vapor by Homoionic Montmorillonites: 2. The Li+ Na+, K+, Rb+ and Cs+-Exchanged Forms[END_REF][START_REF] Cases | Mechanism of adsorption and desorption of water vapor by homoionic montmorillonite. 1. The sodium-exchanged form[END_REF]]. On the basis of the MD simulations, Szczerba et al. showed that 2W montmorillonite layers contain a higher proportion of bidentate water molecules whose transition moments are parallel to the z axis, contributing to an increase in the 𝑆 𝛿 𝐻 2 𝑂 value[START_REF] Szczerba | Structure and Dynamics of Water-Smectite Interfaces: Hydrogen Bonding and the Origin of the Sharp O-DW/O-HW Infrared Band From Molecular Simulations[END_REF]. Determination of the order parameters 𝑺 𝜷 , 𝑺 𝜽 𝝁 and 𝑺 𝜹 𝝁 using the thickness-and absorption-dependent equations and comparison with the MD results

	𝑆 𝛽	𝑆 𝜃 𝑂𝐻	𝑆 𝛿 𝑂𝐻	< 𝛿 𝑂𝐻 >	𝑆 𝛿 𝑂𝐻 or 𝑆 𝛿 𝐻 2 𝑂
			or	or
			𝑆 𝛿 𝐻 2 𝑂	< 𝛿 𝐻 2 𝑂 >

Acknowledgments

The CNRS interdisciplinary "défi Needs", through its "MiPor" program (Project TRANSREAC), is acknowledged for providing financial support for this study. The authors acknowledge financial support from the European Union (ERDF) and "Région Nouvelle Aquitaine".

where 𝑛 31 = 𝑛 3 /𝑛 1 and 𝑛 32 = 𝑛 3 /𝑛 2

IV. Results and Discussion

IV.1. Electric field propagation within a clay sample deposited on a Ge ATR crystal

The availability of rigorous mathematical treatments for determining electric field propagation through a sample enables a better understanding of the origin of ATR band intensity [START_REF] Hansen | Electric Fields Produced by the Propagation of Plane Coherent Electromagnetic Radiation in a Stratified Medium[END_REF]. The macroscopic treatment of the electric field considers interfaces between two media as a step function of their respective dielectric constants [START_REF] Bagchi | Perturbative approach to the calculation of the electric field near a metal surface[END_REF][START_REF] Sipe | Bulk-selvedge coupling theory for the optical properties of surfaces[END_REF][START_REF] Feibelman | Surface electromagnetic fields[END_REF]. Implicitly, the macroscopic physical properties of a layer are assumed to be adequate descriptors of their microscopic properties, irrespective of their dimensions or chemical/physical environments. As a consequence, electromagnetic field theory accurately describes the electric field amplitude within a few angstroms from the surface. Although the expressions are mathematically exact, they are not exact in the sense of physics for describing the electric field amplitude within a film whose thickness is in the nanometer range. For this reason, it is not surprising to note that for similar systems, some authors may use the thin-film set of equations, while others prefer the 2-phase expressions [START_REF] Axelsen | Orientational order determination by internal reflection infrared spectroscopy[END_REF][START_REF] Higashiyama | Infrared attenuated total reflection spectra of adsorbed layers at the interface between a germanium electrode and an aqueous solution of sodium laurate[END_REF][START_REF] Takeda | Studies of poly-γ-methyl-lglutamate monolayers by infrared ATR and transmission spectroscopy and electron microscopy[END_REF][START_REF] Jang | Molecular Orientation of Langmuir-Blodgett and Self-Assembled Monolayers of Stearate Species at a Fluorite Surface As Described by Linear Dichroism Theory[END_REF][START_REF] Buffeteau | Linear Dichroism in Infrared Spectroscopy[END_REF][START_REF] Picard | Quantitative orientation measurements in thin lipid films by attenuated total reflection infrared spectroscopy[END_REF][START_REF] Citra | Determination of molecular order in supported lipid membranes by internal reflection Fourier transform infrared spectroscopy[END_REF]. This illustrates well that profound differences exist in the way that authors use theory to interpret the molecular orientation of thin films. Interestingly, Citra and Axelsen, for example, demonstrate that contrary to expectations, the 2-phase approximation is more appropriate for describing a 25-50 Å thick layer of phospholipids [START_REF] Citra | Determination of molecular order in supported lipid membranes by internal reflection Fourier transform infrared spectroscopy[END_REF]. Considering the above arguments, it is surprising that very little attention has been paid to determining which set of expressions is more appropriate and accurate for given experimental conditions.