Eliane Younes
email: eliane.younes@univ-nantes.fr

Volfango Bertola
email: volfango.bertola@liverpool.ac.uk

Cathy Castelain
email: cathy.castelain@univ-nantes.fr

Teodor Burghelea
email: teodor.burghelea@univ-nantes.fr

Microscopic flows of a simple yield

Keywords: Carbopol gel, yield stress, rheology, microchannel, DPIV, wall slip

Introduction

Yield stress fluids represent a broad class of materials that naturally exist in a solid state but, if sufficiently forced (e.g. subjected to a large enough stress) they behave as fluids i.e. they flow [START_REF] Coussot | Rheometry of pastes, suspensions and granular materials[END_REF]. Such fluids are encountered in the daily life in a variety of forms including pasty materials commonly related to the food industry (e.g. ketchup, mustard, molten chocolate), construction industry (cements), oil industry (drilling oils), cosmetics (toothpaste, hair gel, creams), pharmaceutical industry (drug delivery gel capsules, vertebral repair gels etc.).

Most of the yield stress fluids present a shear thinning behaviour beyond the yield point classically described by the Herschel-Bulkley model. It was initially proposed in Refs. [START_REF] Herschel | Konsistenzmessungen von Gummi-Benzollösungen[END_REF][START_REF] Herschel | Measurement of consistency as applied to rubbery benzene solutions[END_REF] and it takes the form below:

σ = σ y + k| γ| n if σ ≥ σ y γ = 0 if σ < σ y ( 1
)
Here σ is the applied stress, σ y is the yield stress, γ is the shear rate, k is the consistency of the material and n is the power law index. This model assumes that the solid-fluid transition occurs at a well defined value of the applied stress (the commonly referred to yield stress).

Carbopol gel has been frequently used as a "model" yield stress fluid to study the yielding behaviour of viscoplastic materials [START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF]. Carbopol polymer is a cross-linked polyacrylic acid particles (COO -H + ) with high molecular weight. In an anhydrous state, it is commercialized in the form of a white powder soluble in aqueous solvents. After addition of a neurtralizing agent such as sodium hydroxide (NaOH), a clear gel is obtained. A chemical reaction is created between the acid group of the Carbopol polymer and the hydroxide group that leads to the formation of the gel networks. Carbopol microgels exhibit an elasto-viscoplastic rheological behaviour. The gel networks are maintained for small applied stresses (solid regime). Once the applied stress exceed a certain threshold, the gel networks break completely and the material starts to flow [START_REF] Putz | The solid-fluid transition in a yield stress shear thinning physical gel[END_REF].

For over a decade, it has been considered that the Carbopol gel follows the Herschel-Bulkley correlation [START_REF] Curran | Properties of Carbopol Solutions as Models for Yield-Stress Fluids[END_REF][START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF]. However, it has been shown recently that the solid-fluid transition does not occur at a well defined value of the applied stress but gradually, i.e. within a finite range of stresses [START_REF] Putz | The solid-fluid transition in a yield stress shear thinning physical gel[END_REF][START_REF] Divoux | Transient Shear Banding in a Simple Yield Stress Fluid[END_REF]. In the current study, we choose the Carbopol gel as a working yield stress fluid because of its low thixotropy, chemical stability and optical transparency.

The flow analysis of a yield stress fluid has been well studied using rheological measurements. Torsional rheometers are used to characterise a yield stress fluid provided that wall slip is eliminated. The presence of slip on the walls of the rheometer geometry complicates the determination of the yield stress [START_REF] Kalyon | Apparent slip and viscoplasticity of concentrated suspensions[END_REF]. However, the wall slip phenomenon is ubiquitous during flow of pasty materials and may affect the solid-fluid transition.

The study of flow kinematics in a real industrial flow has received limited attention. Ouvrier and Windhab in Ref. [START_REF] Ouriev | Rheological study of concentrated suspensions in pressure-driven shear flow using a novel in-line ultrasound Doppler method[END_REF] developed a novel rheological measuring technique based on ultrasonic pulsed echo Doppler technique and pressure difference method in order to characterise the flow behaviour of a nontransparent and highly concentrated suspensions in a cylindrical pipe. The local shear stresses were calculated from the measured pressure drop along the pipe and the wall slip velocity were calculated by extrapolating the velocity profile at the wall.

An experimental investigation was carried out by Ma et al. in Ref. [START_REF] Ma | Wall slip and rheological behavior of petroleum-coke sludge slurries flowing in pipelines[END_REF]. They propose a model to determine the rheological properties and wall slip behaviour of a petroleum-coke sludge slurry flowing in pipelines by using the traditional Moony method and Tikhonov regularization.

A study of the yielding process in a capillary tube of a 0.2 wt% Carbopol gel has been performed by Pérez-González et al. in Ref. [START_REF] Pérez-González | Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall[END_REF]. By combining the Digital Particle Image Velocimetry (DPIV) and classical rheological measurements they found the solid-fluid transition occurs gradually which complicates the determination of the yield stress. To overcome the difficulty of reliably measuring the yield stress they propose an alternative method based on the analysis of the velocity profiles.

To this day, we do not have a clear picture of the wall phenomenon during a yield stress fluid flow. 2 Experimental methods

Preparation and rheological characterisation of the working fluid 2.1.1 Fluid preparation

The working fluid was an aqueous solution of 0.1 wt% Carbopol Ultrez 10. The solution batch has been prepared according to the following protocol. First, the right amount of anhydrous Carbopol has been dissolved in deionised water using a magnetic stirring device at a speed of 1000 rpm. The degree of mixing/dissolution has been assessed visually by monitoring the optical isotropy of the solution. Next, the pH of the solution has been gradually increased from 3.2 to 7 by gradual titration with a small amounts of a 10 wt% aqueous NaOH solution gradually pipetted while gently mixing the solution.

The rheological tests were performed after seeding the carbopol solution with the fluorescent tracers as detailed in Sec. 2.2. The average density of the Carbopol solution was ρ f ≈ 1100 kg.m -3 .

Rheological measurments

The rheological measurements have been performed using a controlled stress rotational rheometer (Mars III, Thermofischer Scientific) equipped with a nano-torque module. Tests have been performed using a parallel plate geometry with a diameter D = 35 mm and a gap d = 1 mm. To prevent the wall slip, glass paper with an average roughness of 500 µm has been glued on each plate. To account for the addition of the glass paper on the rotating plate of the device, the inertia has been recalibrated. The absence of any wall slip effect was verified by measuring flow curves in subsequent tests performed with several values of the gap and showing that all measurements perfectly overlap. To prevent the evaporation of the solvent during the rheological measurements a thin layer of commercial oil has been added to the free meniscus of the sample.

Rheological measurements have been performed according to the following protocol. First, the sample has been pre-sheared at a constant applied stress larger than the yield stress for 300 s and allowed to relax for another 300 s. Then, to assess the rheological behavior of the Carbopol gel in different deformation regimes, a commonly used rheological tests consisting of loading the material according to an increasing stress ramp was applied to a fluid sample. The duration of each step was t 0 = 5 s and the data averaging time per stress value was δt 0 = 2 s.

To test the reproducibility and quantitatively assess the instrumental error, each rheological measurement was been repeated three times with a fresh sample.

The rheological flow curve is illustrated in Fig. 1. Three distinct deformation regimes are observed. For low values of applied stress, the rate of material deformation γ remains constant. This corresponds to a solid-like behaviour of the material, region (S) in Fig. 1. For high values of applied stress (beyond the yield stress), the material becomes fully yielded, region (F) in Fig. 1, and can be modelled with the Herschel-Bulkley equation (Eq. 1), the red line in Fig. 1. For intermediate values of the applied stresses, the solid-fluid transition is not direct and the behaviour of the material corresponds neither to a solidlike regime nor to a viscous flow regime, region (S+F) in Fig. 1. According to the Herschel-Bulkley fitting, the yield stress of the working material is around σ y ≈ 32.5 Pa.

Micro-channel design, microscopic flow control and data acquisition protocol

To characterise the kinematics of a yield stress fluid flow for a wide range of flow rate in the presence of wall slip, micro-fluidic experiments have been performed in a plane acrylic micro-channel. The width of the micro-channel is W = 200 µm, its depth is H = 200 µm and its length is L = 4 cm. The microchannel was machined in an acrylic block with the dimensions 5 cm × 3 cm × 0.5 cm.

The micro-channel chip is mounted on an inverted epifluorescent microscope (Axio Observer A1, serial no3832002215. ), Fig. 2. The microscopic flows have been visualised through a 20X magnification objective (Zeiss EC Plan-NEOFLUAR 20x/0,5) with a numerical aperture NA = 0.35 and a long working distance W D = 70 mm.

The microscopic flows were generated using a high precision micro-syringe pump (KdScientific, model Legato 110). To insure a steady flow rate Q which was crucial for this study, we have used a 10 ml gas tight syringe (Hamilton, model 1010LT). The flows were illuminated with by a powerful halogen lamp coupled to the inverted microscope.

To visualise the flow, the working fluid was seeded with a minute amount of buoyantly neutral fluorescent tracers with a diameter of 0.92 µm (Fluoresbrite Multifluorescent from Polysciences).

For each flow rate, we have waited for 5 mins before starting the image acquisition in order to avoid the transient states and then a series of 1000 flow images was acquired with a digital camera.

Data analysis

The main tool used to systematically characterise the microscopic flow was an adaptive multi-grid Digital Particle Image Velocimetry (DPIV) (see Refs. [START_REF] Scarano | Advances in iterative multigrid PIV image processing[END_REF][START_REF] Raffel | Particle Image Velocimetry: A Practical Guide (Experimental Fluid Mechanics)[END_REF] for a detailed description of the method) entirely developed in house under Matlab (together with the "Image Processing Toolbox") which followed the steps briefly described below. A a background was calculated by averaging all the images of the first time series we have acquired. Second, the average background was subtracted from images. Finally, pairs of pre-processed images separated in time by the inter-frame t 1 are passed to a multi-pass DPIV algorithm using a sequence of squared interrogation windows with sizes [128, 64, 32, 16, 8]. As a post-processing step, each computed velocity field obtained from the DPIV algorithm was filtered using a median filter. The spatial resolution of the velocity fields was 4.5 µm (44 times smaller than the width of the channel) which suffices for a reliable calculation of the velocity gradients in the proximity of the channel walls.

To extract the wall slip velocity we have interpolated each transversal profile of the axial velocity by a smoothing spline function and computed its value at the level of the wall. The velocity gradient near the was calculated by calculating the slope of the last four points of the smoothing spline.

Results

Description of the flow regimes in the presence of wall slip

We focus in the following on the experimental characterisation of the microscopic flow profiles of a Carbopol gel in a steady channel flow for several flow rates.

Profiles of the time averaged flow velocity U av measured according to the method described in Sec. 2.3 are presented in Fig. 3. The standard deviation of the velocity profiles was smaller than 5% for different values of flow rates. Depending on the magnitude of the constant imposed flow rate Q, three distinct flow regimes are observed in Fig. 3. At low driving flow rates (Q < 2 µl/min), a full plug flow regime is observed in Fig. 3(a). The entire Carbopol gel is un-yielded and "slides" over a very thin layer of depleted solvent (water) located in the proximity of the channel walls. The fully plug velocity profiles are characteristic of a solid-like behaviour visible in Fig. 1

, region (S).

As the flow rate is gradually increased past this first flow regime, a second flow regime is observed.

The Carbopol gel is partially yielded in the proximity of the channel walls (where the velocity gradients are the largest) but a central un-yielded plug may still be observed around the centre-line of the microchannel, Fig. 3(b). The shear stresses are not large enough to yield the material along the entire width. This behaviour may be related to the solid-fluid transition of the material observed in Fig. 1, region (S+F).

As the flow rate is increased even further (Q > 10 µl/min), the Carbopol gel is fully yielded across the entire width of the channel. The further increase of flow rate translates into additional shear stresses and a shear thinning flow regime is observed in Fig. 3(c) (the local shear stresses exceed the yield stress everywhere across the channel width). The shear thinning velocity profiles are characteristic of a fluid-like regime visible in Fig. 1, region (F).

Furthermore, the velocity profiles clearly exhibit a non zero slip behaviour at the channel walls (a non zero slip velocity at y = 1 × 10 -4 m ) meaning that the gel slip.

Scaling behaviour at the wall within various flow regimes

Understanding the scaling behaviour of the relevant hydrodynamic quantities (shear stress, shear rate, slip velocity) in the proximity of the wall is of paramount importance to developing reliable microscopic models able to explain the confined flows of viscoplastic materials in the presence of wall slip and for the implementation of numerical simulations. Based on the characterisation of the flow fields presented in Sec. 3.1, we study in this section the scaling behaviour for each flow regime previously identified.

The dependence of the wall velocity gradients computed according to the procedure detailed in Sec. 2.3 on the flow rate Q is presented in Fig. 4(a). In the solid region (S), the material is fully unyielded and, within the instrumental accuracy of our technique, no velocity gradients could be reliably measured near the solid wall. As the flow rate Q is gradually increased, one can clearly notice the existence of two different scaling regimes. For the solid-fluid transition regime (S+F), the wall velocity gradient scales with the flow rate as dU dy | w ∝ Q 2 (see full line in Fig. 4(a)). A further increase of the flow rate into the fluid regime (F) leads to a different scaling of the wall velocity gradient, dU dy | w ∝ Q 3 (see dashed line in Fig. 4(a)).

The measurements of the wall velocity gradients dU dy | w allow one to compute the wall shear stresses τ w using the macro-rheological measurements presented in Fig. 1. The result of this evaluation is presented in Fig. 4(b). Two different scaling regimes of the wall shear rate with the wall shear stress can be clearly observed. The first scaling regime corresponds to a partial yielding of the gel in the vicinity of the wall, dU dy | w ∝ τ 7.99 w (see full line in Fig. 4(b)). Above a critical value of wall shear stress and corresponding to the fluid deformation regime a second scaling is observed, dU dy | w ∝ τ 4.8 w (see dashed line in Fig. 4(b)). Due to confinement, the microscopic flows can not be generally described by the Herschel-Bullkley model, see Ref. [START_REF] Liu | Effect of confinement on the rheology of a yield-stress fluid[END_REF].

The value of the critical wall shear stress defined as the intersection between the two power laws is close to the yield stress measured in the rheometric flow and fitted by the Hurschel-Bulkley equation, see Fig. Wall slip behaviour is usually described as the evolution of the wall slip velocity U s with the wall shear stress τ w . This dependence is presented in Fig. 5. Because in the solid regime the wall velocity gradients are too small to be reliably measured, we could not use the method described above to reliably compute the wall shear stresses. For higher flow rates (beyond the (S) regime) , again, two distinct scaling laws are observed. Corresponding to the solid-fluid coexistence regime (S+F), U s ∝ τ 3.55 w . Within the fully yielded regime (F), U s ∝ τ w . regimes are observed. At low driving flow rates, a full plug flow regime is observed. As the flow rate is gradually increased, the Carbopol gel is partially yielded in the proximity of the channel walls but a central un-yielded plug may still be observed around the centre-line of the micro-channel. As the flow rate is increased even further, the Carbopol gel is fully yielded across the entire width of the channel, see Fig. 3.

The accurate measurements of the velocity profiles allow one to obtain the wall velocity gradients and slip velocities via the numerical differentiation of a spline interpolant of the velocity profiles. We use the dependence between the stress and the rate of shear obtained via classical rotational rheometry and we relate the measured wall velocity gradients to the wall stresses in order to obtain experimentally the scaling law(s) for the wall slip phenomenon. Two scaling regimes are observed corresponding to partial and fully yielded flow. For partial yielded flow regime the slip velocity, U s scales with τ w as U s ∝ τ 3.55 w . A linear scaling law of the slip velocity with the wall shear stresses is found, see Fig. 5.

This study helped us to characterise the wall slip phenomenon of a Carbopol flow in a plane microchannel for different flow regimes. The next step is to propose a model that can describe accurately the wall slip behaviour of a yield stress fluid flow depending on its rheological properties, wall shear stress and slip velocity.

Figure 1 :

 1 Figure 1: Dependence of the shear rate γ on the on the applied stress σ . The full line represents the Herschel -Bulkley fit that gives σ y = 32.5 ± 2.5 Pa, K = 2.8 ± 0.215 Pa.s n , n = 0.6403 ± 0.0299. The symbols marking the highlighted regions denote the deformation regimes and are explained in the text: (S) -solid, (S + F) -solid-fluid coexistence, (F) -fluid.

  For the experiments where the flow rate did not exceed 10 µL/min, we have used a Prosilica GE camera with 16 bit quantisation (model GE680C from Allied Technologies). The maximal frame rate achievable with this camera is 200 f ps. During the experiments with flow rates Q ∈ [10, 23] µL/min, a Mikrotron camera has been used up to a frame rate of 500 f ps. The images were acquired at the mid-plane of the micro-channel and at the mid-length where the laminar flow is fully developed at this position as the Reynolds numbers are very small (Re ≈ 10 -9 -10 -3 ).

Figure 2 :

 2 Figure 2: Schematic view of the microfluidic experimental setup: MCC: micro-channel chip, I: microchannel inlet, O: micro-channel outlet, AT: adhesive tap, MO: microscope objective, DM: dichroic mirror, LP: light emitting lamp, EXF: excitation filter, EMF: emission filter, CCD: digital camera.

Figure 3 :

 3 Figure 3: Velocity profiles for various values of flow rate Q. Panel (a) corresponds to the solid regime for: circles-Q = 1 µl/min, squares-Q = 1.4 µl/min, left triangles-Q = 1.6 µl/min, right triangles Q = 1.8 µl/min, stars-Q = 2 µl/min. Panel (b) corresponds to the transition for: circles-Q = 3.5µl/min, squares-Q = 5.5 µl/min, left triangles-Q = 8 µl/min, right triangles Q = 10 µl/min. Panel (c) corresponds to the fluid regime for: circles-Q = 14 µl/min, squares-Q = 15 µl/min, left triangles-Q = 21 µl/min, right triangles Q = 23 µl/min.

1

 1

  .

Figure 4 :

 4 Figure 4: (a) Dependence of the wall velocity gradients on the applied flow rate Q. The full line (-) is a guide for the eye, dU dy | w ∝ Q 2 and the dashed line (--) is a guide for the eye, dU dy | w ∝ Q 3 . (b) Dependence of the wall velocity gradients on the wall shear stress computed using the rheological measurements and computed wall velocity gradients. The full line (-) is a guide for the eye, dU dy | w ∝ τ 7.99 w and the dashed line (--) is a guide for the eye, dU dy | w ∝ τ 4.8 w . The symbols marking the highlighted regions denote the deformation regimes and are explained in the text: (S) -solid, (S + F) -solid-fluid coexistence, (F)fluid.

Figure 5 :

 5 Figure 5: Dependence of the wall slip velocity U s on the wall shear stress τ w . The full (-) and dashed (--) lines are guides for the eye, Us ∝ τ 3.55 w , Us ∝ τ w respectively. The symbols marking the highlighted regions denote the deformation regimes and are explained in the text: (S + F) -solid-fluid coexistence, (F) -fluid.

  Therefore, we present in the current work a detailed experimental investigation of the flow behaviour of a 0.1 wt% Carbopol gel in an acrylic micro-channel in the presence of wall slip using DPIV technique and classical rheological measurements.

	The paper is organised as follows. The rheological characterisation of the working fluid is presented in
	Sec. 2.1. The micro-channel setup design as well as data acquisition are presented in Sec. 2.2. The
	data analysis is explained in Sec. 2.3. A detailed characterisation of the velocity profiles is presented in
	Sec. 3.1. By combining the micro-fluidic and macro-rheological measurements, we identify in Sec. 3.2
	different scaling laws of the wall slip phenomenon corresponding to different flow regimes. A detailed
	comparison of the experimentally found scaling laws with results from the literature is presented in Sec.
	3.3. The paper closes with a summary of the main conclusions and several perspectives, Sec. 4.

Comparison with previous results from the literature

A study of flow kinematics was carried out by Pérez-González et al. in Ref. [START_REF] Pérez-González | Rheo-PIV of a yield-stress fluid in a capillary with slip at the wall[END_REF]. The apparent shear rate γa and the wall shear stress τ w are calculated as γa = 32Q πD 3 , τ w = ∆P 4 L D respectively, where ∆P is the pressure drop between the capillary ends, L and D are respectively the length and the diameter of the borosilicate glass capillary and Q is the flow rate calculated from the integration of the velocity profiles. In their study, three distinct flow regimes are observed: solid-like, transition and fluid-like regimes. The apparent shear rate γa scales with the wall shear stress τ w as γa ∝ τ 2.44 w in the fluid regime. This scaling law differs from the one we observe, dU dy | w ∝ τ 4.8 w (the dashed line in Fig. 4(b)). This difference may be related to the way γa is calculated. This formula takes into account the assumption that the velocity profile is parabolic and the slippage does not exist at the capillary walls.

To describe the wall slip behaviour Kaylon proposed in Ref. [START_REF] Kalyon | Apparent slip and viscoplasticity of concentrated suspensions[END_REF] the following relationship between the slip velocity and the wall shear stress based on Herschel-Bulkley equation:

Here β relates the slip layer δ to the consistency of the binder fluid m b and to its power-law index n b ,

. The Carbopol gel is considered as jammed system of swollen gel micro-particles [START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF]. As the carbopol gel was prepared with an aqueous solvent, the "binder fluid" might be considered as Newtonian and thus K b = 10 -3 Pa.s and n b = 1.

In the fluid regime, U s scales linearly with τ w , U s ∝ τ w (the dashed line in Fig. 5). This assumption confirms the assumption that the "binder fluid" is Newtonian, n b = 1. This is consistent with the scaling law observed in previous study, Ref. [START_REF] Poumaere | Burghelea Unsteady laminar flows of a Carbopol gel in the presence of wall slip[END_REF] while differs from the ones predicted theoratically by Piau in Ref. [START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF] where U s ∝ τ 1/3 w for loosely packed system and U s ∝ τ 2 w for closely packed system. The difference from ref. [START_REF] Piau | Carbopol gels: Elastoviscoplastic and slippery glasses made of individual swollen sponges: Meso-and macroscopic properties, constitutive equations and scaling laws[END_REF] might be related to the fact that the lubricating fluid is considered as non-Newtonian fluid.

In our study, the factor β = 70 • 10 -6 m.Pa -1 .s -1 leads to a slip layer of thickness δ ≈ 0.07 µm. This value is of the same order of magnitude as the one found by Jiang et al. in Ref. [START_REF] Jiang | The rheological characterization of HPG gels: Measurement of slip velocities in capillary tubes[END_REF], δ ≈ 0.1 µm.

By direct observation of a rheometric microgel flow , Meeker et al. in Ref. [START_REF] Meeker | Slip and flow in pastes of soft particles: Direct observation and rheology[END_REF] observe that U s remains constant for 1 < σ σ y < 1.5 (σ is the applied stress) while in our experiments we can observe that the slip velocity increase linearly with the wall shear stresses, Us ∝ τ w .

Conclusions, outlook

An experimental investigation of "wall slip" phenomenon in a Carbopol gel flow is carried out in this study. In the first part, we present in Sec. 2.1.2 a systematic rheological caracterization of Carbopol gel. Rheological measurements has been performed using an accurate rotational rheometer. The flow curve obtained during an increasing stress ramp reveals three distinct deformation regimes. An elastic solid regime that corresponds to low values of applied stresses (below the yield point), a solid-fluid transition corresponds for intermediate stresses (around the yield stress) and a viscous regime observed for high values of applied stresses.

To characterise the "wall slip" phenomenon, a systematic measurements of flow fields of Carbopol gel in a plane acrylic made micro-channel have been performed, Sec. 2.2. Bye using a custom Digital Particle Image tool developed in the house, times series of velocity fields were measured within a wide range of driving flow rates. Depending on the magnitude of the constant imposed flow rate Q, three distinct flow


Aydar Rakhmatullin

Mathieu Allix

Ilya B Polovov

Dmitry Maltsev

Andrey V Chukin

Rinat Bakirov

Catherine Bessada

Combining solid state NMR, powder X-ray diffraction, and DFT calculations for CsSc3F10 structure determination

Keywords: solid-state NMR, X-ray, CASTEP, Coordination number

come   L'archive ouverte pluridisciplinaire

Introduction

In recent years, alkali metal fluoroscandates have attracted a lot of attention due to their potential applications in solid-state lasers, solar cell, fluorescent probes and many other areas [START_REF] Ai | Lanthanide-doped NaScF4 nanoprobes: crystal structure, optical spectroscopy and biodetection[END_REF][START_REF] Yu | Controlled synthesis of high quality scandium-based nanocrystals as promising recyclable catalysts for silylcyanation reaction[END_REF][START_REF] Xiong | One-step surfactant-free synthesis of KSc2F7 microcrystals: controllable phases, rich morphologies and multicolor down conversion luminescence properties[END_REF][START_REF] Zhang | The colour tuning of upconversion emission from green to red in NaScF4:Yb 3+ /Er 3+ nanocrystals by adjusting the reaction time[END_REF][START_REF] Tyagi | Syntheses and structures of Li3ScF6 and high pressure LiScF4, luminescence properties of LiScF4, a new phase in the system LiF-ScF3[END_REF]. The necessary improvement of these different crystalline materials for these specific applications requires precise knowledge of the local structure and an accurate description of their structure. There are many reports on lithium, potassium, and sodium fluoroscandates (for example: LiScF4 [START_REF] Tyagi | Syntheses and structures of Li3ScF6 and high pressure LiScF4, luminescence properties of LiScF4, a new phase in the system LiF-ScF3[END_REF], KSc2F7 [START_REF] Yu | Controlled synthesis of high quality scandium-based nanocrystals as promising recyclable catalysts for silylcyanation reaction[END_REF][START_REF] Xiong | One-step surfactant-free synthesis of KSc2F7 microcrystals: controllable phases, rich morphologies and multicolor down conversion luminescence properties[END_REF],

NaScF4 [START_REF] Ai | Lanthanide-doped NaScF4 nanoprobes: crystal structure, optical spectroscopy and biodetection[END_REF][START_REF] Yu | Controlled synthesis of high quality scandium-based nanocrystals as promising recyclable catalysts for silylcyanation reaction[END_REF], and others [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF]), but relatively little is known about rubidium-and cesium-containing fluoride compounds. We suppose that the main reasons for this lack of investigations are related to the high cost of ScF3 and the high corrosiveness and hygroscopicity of Rb and Cs fluorides. In the CsF-ScF3 system, only the structure of CsScF4 has been reported [START_REF] Krylov | Crystal structure and phase transitions of a layered perovskite-like CsScF4 crystal[END_REF].

Up to now, the crystal structure of CsSc3F10 remains unknown. Champarnaud-Mesjard and Frit [START_REF] Champarnaud-Mesjard | Les systemes MF-ScF3 (M = Li, Na, K, Rb[END_REF] reported the synthesis of this material and proposed an orthorhombic symmetry and with a structure that might be isostructural with RbIn3F10 [START_REF] Champarnaud-Mesjard | Structure cristalline du fluoroindate de rubidium RbIn3F10[END_REF], which crystallizes in the P2221 space group. In this model, indium atoms present two different kinds of coordination: In1 exhibits octahedral coordination whereas In2 and In3 are 7-coordinated with pentagonal bipyramid environments. All these fluorinated polyhedra sharing either edges or corners form a three dimensional framework, (In3F10) -, with tunnels parallel to the z axis where are located (8+2)-coordinated rubidium atoms. CsSc3F10 was reported to be stable up to 477 °C, temperature above which it decomposes into CsScF4, and ScF3.

Solid state NMR combined with theoretical frameworks, such as DFT calculations, becomes a more and more powerful tool for determining the structure of crystalline materials, complementing the usual diffraction techniques. This combined approach has found many applications for a wide range of materials [START_REF] Martineau | Structure determination of Ba5AlF13 by coupling electron, synchrotron and neutron powder diffraction, solid-state NMR and ab initio calculations[END_REF][START_REF] Martineau | Chapter One -NMR Crystallography[END_REF][START_REF] Rollet | Synthesis and Structure Resolution of RbLaF4[END_REF]. In this work, we use such an approach to determine and describe finely the structure of CsSc3F10 by coupling X-ray diffraction, 1D and 2D D-HMQC solid-state NMR techniques and first principles calculations. The unusually high cesium coordination number in a pure fluorine 18 (12+6) environment is discussed in detail.

Experimental methods

Synthesis

Scandium fluoride was obtained by precipitation from a solution of scandium chloride using hydrofluoric acid. Scandium chloride solution was previously prepared by reacting scandium oxide (99.9 %, Sigma-Aldrich) with hydrochloric acid (analytical purity). The precipitate was centrifuged and then air-dried at 130 °C. At the final stage, ScF3 was dried under vacuum at 300 °C. The residual water content in ScF3 samples was below 0.01 wt%. The mechanically homogenized powder mixture of CsF (99 %, Sigma-Aldrich) and ScF3, previously weighed in stoichiometric ratios to match the CsSc3F10 composition, was placed in a nickel crucible and heated from room temperature to 402 °C using a rate of 2 °C/min under argon atmosphere. The holding time was 100 hours and then the sample was cooled down to room temperature with an approximate rate of 2 °C/min.

NMR measurements

Solid state 19 F, 45 Sc, and 133 Cs NMR spectra were obtained using Bruker AVANCE III HD 750 (17.6 T) and 850 (20 T) NMR spectrometers. 19 F, 45 Sc, and 133 Cs chemical shifts were referenced to a CFCl3, 0.11 M ScCl3 solution in 0.05 M HCl, and 0.5 M CsCl solution, respectively. Room temperature MAS NMR spectra were acquired using 1.3 mm Bruker MAS probes. Data were collected at spinning frequencies ranging from 40 to 67 kHz. π/2 pulse widths were 0.73 μs and 2.85 μs for 19 F and 45 Sc, respectively. In order to improve the resolution, we have used PISSARRO [START_REF] Weingarth | Probing the quenching of rotary resonance by PISSARRO decoupling[END_REF] 45 Sc and 19 F decoupling.

The SR4 2 1 recoupling pulses [START_REF] Brinkmann | Proton-Selective 17 O-H Distance Measurements in Fast Magic-Angle-Spinning Solid-State NMR Spectroscopy for the Determination of Hydrogen Bond Lengths[END_REF] in the dipolar 45 Sc( 19 F) HMQC experiments [START_REF] Trebosc | Through-space R3-HETCOR experiments between spin-1/2 and half-integer quadrupolar nuclei in solid-state NMR[END_REF] were applied with nutation frequencies ν( 19 F) = 42.4 kHz (π/2 pulse of 5.9 μs). The optimum DQ transfer was obtained for τ = 0.2 ms, corresponding to 12 rotor periods (two SR4 2 1 super-cycles). The radiofrequency field on the 45 Sc channel was ν( 45 Sc) = 100 kHz, the spinning speed was 60 kHz and the spectral width in the direct dimension was set to 60 kHz. 1052 complex points were recorded with 512 transients, and in the indirect dimension, 90 t1 increments. Phase sensitive detection in the indirect dimension was obtained using the States method [START_REF] States | A two-dimensional nuclear overhauser experiment with pure absorption phase in four quadrants[END_REF]. The recycling delay was set to 0.5 s.

Powder X-ray diffraction

XRPD data were recorded on a Bragg Brentano D8 Advance Bruker diffractometer (CuKα radiation) equipped with a LynxEye XE detector over an angular range of 5 °<2<130 ° using a 0.012 ° step size.

Structural refinements of the CsSc3F10 material were realized using the Rietveld [START_REF] Rietveld | A profile refinement method for nuclear and magnetic structures[END_REF] method with the JANA2006 [18] software.

CASTEP calculations

First-principles calculations of NMR parameters were performed using the CASTEP NMR program [START_REF] Segall | First principles methods using CASTEP[END_REF] running in the Materials Studio 7.0 environment using the experimental geometries determined via diffraction methods, pseudopotentials calculated "on-the-fly", and the Perdew, Burke, and Ernzerhof (PBE) [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] functionals in the generalized gradient approximation (GGA). The gauge including projector augmented wave (GIPAW) [START_REF] Yates | Calculation of NMR chemical shifts for extended systems using ultrasoft pseudopotentials[END_REF][START_REF] Pickard | All-electron magnetic response with pseudopotentials: NMR chemical shifts[END_REF] method implemented in CASTEP provides accurate computed values of the periodic system properties and its use is well-established in the solidstate NMR literature [START_REF] Charpentier | The PAW/GIPAW approach for computing NMR parameters: A new dimension added to NMR study of solids[END_REF][START_REF] Bonhomme | First-Principles Calculation of NMR Parameters Using the Gauge Including Projector Augmented Wave Method: A Chemist's Point of View[END_REF]. The cut-off energies were 600 eV and k-point spacing of 0.04 Å -1 .

Calculations were carried out after a geometry optimization of all atom positions, keeping the experimental cell parameters and symmetry constraints. Optimized structure was obtained by minimizing the residual forces (|F|max below 10 meV/Å) for all atoms. To convert the calculated isotropic chemical shielding into isotropic chemical shift, the definition of calibration curve relating isotropic chemical shift and shielding is necessary. For 45 Sc and 19 F the relationship reported in our previous study [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF] are : δiso = -0.852×σiso+667.42 and δiso = -0.589×σiso +20.24, respectively.

3.

Results and Discussions

X-ray powder diffraction and NMR spectroscopy

First, X-ray powder diffraction (XRPD) was performed to determine the nature of the synthesized phase corresponding to the CsSc3F10 nominal composition. Auto-indexing analysis of the room temperature XRPD pattern was performed using Dicvol [START_REF] Boultif | Indexing of powder diffraction patterns for low-symmetry lattices by the successive dichotomy method[END_REF] and Treor [START_REF] Werner | Trial-and-error computer methods for the indexing of unknown powder patterns[END_REF] indexing softwares.

The results led to a similar solution with good reliability factors: an orthorhombic cell with a = 8.08

Å, b = 7.58 Å and c = 6.81 Å. A subsequent ICDD database search using close cell parameters (volume tolerance of 10 %) and chemical restrictions (presence of alkali metals and fluorine) resulted in no match with any reported compound with similar cell parameters. A small amount of a secondary phase was identified as ScF3 (ICSD 77071) [START_REF] Fedorov | Ionic conductivity and dielectric relaxation of scandium fluoride[END_REF]. After unsuccessful attempts to find isostructural phases in both ICSD and ICDD databases, solid-state NMR spectroscopy was applied to gain more information on this new crystalline phase.

The 19 F MAS NMR spectrum of CsSc3F10 contains five peaks (Fig. 1a). Accurate simulation of the spectrum, including all spinning sideband patterns, yielded integral intensity ratios of 1:2:4:2:

1
for the resonances at -17.1 ppm, -23.0 ppm, -31.0 ppm, -38.8 ppm, and -45.6 ppm, respectively (Table 1). The very small peak of maximum at -33.9 ppm [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF] and showing a Lorentzian-Gaussian line shape, and was assigned to the secondary phase ScF3. Additionally spectrum shows 3 very small peaks at -3.2 ppm, -12.5 ppm and -55 ppm. The integral intensities of these peaks were very small, i.e. 0.15 %, 0.33 %, and 0.24 % respectively. We suppose that this is some unknown cesium fluoroscandates CsxScyFz. The chemical shift for the bridging fluorine atoms is ranging from -23 to -75 ppm [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF],

therefore, it can be concluded that each fluorine atom in the CsSc3F10 compound is bonded with two scandium atoms. respectively [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF]. To the best of our knowledge, there are no NMR data available for the eight coordination scandium environment [START_REF] Tyagi | Syntheses and structures of Li3ScF6 and high pressure LiScF4, luminescence properties of LiScF4, a new phase in the system LiF-ScF3[END_REF]. Therefore, from scandium NMR results, it is difficult to define exactly the coordination number of scandium atoms. Despite the isotropic chemical shift of the atoms resonating at a low frequency is beyond the range of sevenfold coordinated scandium, we suppose that they are most probably seven coordinated. However, the coordination number eight for these atoms cannot be excluded. The resonance at a higher frequency can be assigned to either a 6-or 7-fold coordination.

The presence of the secondary phase ScF3 was also observed in 45 Sc NMR spectra. Due to the small deviation of local 45 Sc environment from perfect spherical symmetry in ScF3, we observed at νrot=60 kHz a set of small spinning sideband patterns in the 45 Sc spectra. Almost all integral intensity is enclosed in an isotropic peak. It should be noted, that this peak is narrower compared to that of CsSc3F10, it has the Lorentzian-Gaussian line shape and FWHM=300 Hz. In case of CsSc3F10, the integral intensity is distributed over the whole spectrum from -3000 ppm up to 3000 ppm. Additionally, the line shape of the signals of CsSc3F10 shows a quadrupolar broadening. High amplitude of 45 Sc NMR signal of ScF3 compare to that of CsSc3F10 makes the impression that the content of scandium fluoride impurity in CsSc3F10 was significant. In order to gain additional information about the chemical bonding between scandium and fluorine, we have performed 45 Sc-19 F D-HMQC MAS NMR experiment (Fig. 2). On the 2D map, one can The 133 Cs spectrum recorded at 20 T and 60 kHz exhibits only one resonance with a Gaussian line shape and a maximum at -1.2 ppm and FWHM = 200 Hz (Fig. 2S). This observation indicates the existence of only one type of symmetrical environment for cesium atoms, even though fast exchange in NMR time scale between few cesium sites cannot be excluded.

Determination of the crystal structure of CsSc3F10.

As stated above no isostructural model could be found in the ICDD database to match the cell (a = In order to determine precisely the structure of CsSc3F10, a Rietveld refinement of room temperature XRPD data was performed. The starting model was based on the RbSc3F10 structure using the CsSc3F10 cell parameters previously determined and substituting Cs for Rb. Both cationic and fluorine positions, as well as atomic displacement parameters, were refined. Good reliability factors were obtained (wRp= 6.00 %, Rp= 3.48 % and GOF=7.21). The fit of the XRPD Rietveld refinement is shown in Fig. 3.

Crystallographic data, atomic coordinates, atomic displacement parameters and Cs-F and Sc-F interatomic distances are summarized in Tables 345. The amount of the secondary phase ScF3 was estimated at about 6.6 wt% and 6.3 wt% from powder XRD and 19 F NMR, respectively.

Table 3. Crystallographic data from XRD Rietveld refinement.

Chemical formula CsSc3F10

Formula weight (g.mol - The Sc atoms in CsSc3F10 present two different kinds of coordination environments, as well as in RbSc3F10 [START_REF] Yin | New Solid-State Fluorides: Synthesis, Crystal Chemistry, and Optical Properties[END_REF]: 1/3 of the Sc atoms (Sc1) occupy slightly distorted octahedra and 2/3 (Sc2) occupy pentagonal bipyramids (Fig. 4 and5). 5). The average Cs-F distance is 3.45 Å. Therefore, cesium exhibits an unusual high coordination number of 18, which can also be noted as 12+6, as 6 Cs-F distances are above 3.5 Å.

Cesium usually show a variable coordination environment ranging from 6 to 12 [START_REF] Shannon | Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides[END_REF], but it can also form complexes with unusually high coordination numbers above twelve. However, such a coordination number is rare because of the limited space available around the alkali atom. Pollak et al.

studied Cs[H2NB2(C6F5)6] metallocomplexe, in which a central cesium atom is coordinated by 16

fluorine atoms [START_REF] Pollak | Cs[H2NB2(C6F5)6] Featuring an Unequivocal 16-Coordinate Cation[END_REF]. To the best of our knowledge, this was the highest cesium coordination number reported to date. In this metallocomplexe structure, eight Cs-F distances range between 3.099(3) and It should be noted that the rubidium cation has smaller ionic radius than cesium cation. Consequently, rubidium atom in RbSc3F10 is only 14 coordinated, whereas cesium is 18coordinated in CsSc3F10 [START_REF] Yin | New Solid-State Fluorides: Synthesis, Crystal Chemistry, and Optical Properties[END_REF].

CsSc3F10, as well as RbSc3F10, has a structure similar to that of Rbln3F10, but the cesium and rubidium compositions are characterized by a higher symmetry, Pmma space group, than Rbln3F10

CASTEP data

After the crystal structure determination by XRPD, CASTEP calculations were performed on the (atomic positions) optimized structure. The geometry optimization leads to only very small shifts in atomic coordinates (|Δx|, |Δy|, |Δz| ≤ 0.0103 or |atomic displacements| ≤ 0.0851 Å; Table 1S).

Therefore, the crystal structure determined from the XRPD data is close to the minimum energy structure. It should be noted that the obtained values of the atomic displacements are close to those found for the structure determined from powder XRD, i.e. up to 0.0131 Å for K5Sc3F14 [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF] and up to 0.125 Å for K2YF5 and β-KY2F7 [START_REF] Dabachi | Insight into the factors influencing NMR parameters in crystalline materials from the KF-YF3 binary system[END_REF]. First-principles calculated 19 F and 45 Sc NMR parameters are given in Tables 1 and2 the NMR lines were ranked by increasing order of experimental δiso values while the corresponding F sites were ranked by increasing order of calculated 19 F δiso values. This assignment of lines to F sites agrees well with the measured integral intensities (Table 1). We also note that this assignment is in a very good agreement with 45 Sc-19 F D-HMQC spectrum: Sc1 atom is bonded to F3 and F2, and Sc2 is surrounded by F1, F3-F5. A discrepancy of ca. 14 ppm between experimental and calculated chemical shift values is observed for F4. The linear regression employed here was defined in the previous work [START_REF] Rakhmatullin | Combined Approach for the Structural Characterization of Alkali Fluoroscandates: Solid-State NMR, Powder X-ray Diffraction, and Density Functional Theory Calculations[END_REF] using data for lithium, sodium, and potassium fluoroscandates; no data for cesium fluoroscandates were available. Since F4 is the fluorine atom closest to the cesium atom (Table 5), we may attribute the discrepancy to an imperfect description of the ultrasoft pseudopotentials (USPP) of cesium. Similar discrepancy was also observed in case of CsF [START_REF] Sadoc | NMR parameters in alkali, alkaline earth and rare earth fluorides from first principle calculations[END_REF]. The attribution of 45 Sc signals was performed from the integral intensities. The resonance with integral intensity of 33 % corresponds to Sc1 atom and the signal with 67 % to Sc2 atom. The computed results agree well with the experimental data. The calculated CQ for Sc2 atom is 7.4 MHz, and it is found slightly lower than the experimental one 8.1

MHz. There is difference between calculated and observed asymmetry parameter ηQ for Sc1 site; the calculated values are higher, 0.8 and 0.33, respectively. A possible reason for this ambiguity is that accurate calculations of ηQ value require a high precision on the determination of all three EFG tensor components Vxx, Vyy and Vzz. However, the agreement between all calculated and experimental NMR parameters is satisfying and enables reliable assignment of the observed experimental resonances.

Conclusions

CsSc3F10 was synthesized by a solid state reaction route. It was structurally characterized using a combination of high field solid-state MAS NMR spectroscopy, X-ray diffraction, and first-principles calculations. CsSc3F10 exhibits an orthorhombic structure that is built by stacking layers of Sc-centered polyhedra. The resulting frameworks contain tunnels that are filled by the cesium atoms. An unusual high cesium coordination number of 18 (12+6) was observed. It is the highest coordination for cesium reported to date. NMR parameters were determined using first principle DFT calculations. These results could be useful for further designing novel fluoroscandate's materials with industrial and scientific applications.

  observe spots corresponding to Sc-F bonds. It is seen that Sc1 type atom is connected with fluorine sites of only two types. The integral intensities of two 2D peaks of Sc1 type atoms are different; it means that Sc1 atom is more strongly bonded (with a shorter bond length) with the fluorine site resonating at -31 ppm or that out of six/seven fluorine sites surrounding Sc1 atom, more fluorine sites resonate at -31 ppm than at -45.6 ppm. Second type scandium atom (Sc2) is surrounded by four different fluorine sites. We note that the spot Sc2 with F at 23 ppm has the highest integral intensity.

8 .

 8 08 Å, b = 7.58 Å and c = 6.81 Å) determined by auto-indexation. However, an extensive bibliographic search allowed us to find a PhD dissertation mentioning the structure of RbSc3F10 [28]. The proposed model of RbSc3F10 (Pmma, a = 7.790 Å, b = 7.581 Å and c = 6.648 Å) contains five crystallographically nonequivalent bridging F sites with relative multiplicities of 2:4:8:4:2, two Sc sites with relative multiplicities of 1:2, and one Rb site. This description perfectly matches our NMR results, and thus indicates that CsSc3F10 could be isostructural to RbSc3F10.

  The structure is formed by the insertion of octahedral chains between parallel sheets of pentagonal bipyramids. The bipyramids share edges and vertices in the c direction and vertices only along the a direction. The hexagonal tunnels formed in the threedimensional framework are occupied by Cs atoms. It should be noted that Cs atoms are situated in the tunnel central axis along the a-axis. The coordination environments of the scandium and cesium atoms are shown in Figure 5. F3 fluorine atoms form the octahedron's base of Sc1 and two F2 atoms are located in the apex. The Sc1-F2/F3 distances are almost equivalent, 2.021(1) Å and 2.008(3) Å respectively, and they are very close to the value reported for ScF3 (2.01 Å) [27]. The octahedral environment around Sc1 exhibits F2-Sc1-F2 bond angles of 180 ° and small deviations from ideal octahedral angles are observed in F3-Sc1-F3 bond angles, which are 180 ° and 90 ° ±2.90 °. Sc2 is coordinated to seven fluorine atoms. The average Sc2-F distance in the pentagonal bipyramid is 2.07(2) Å. Sc2 occupies the center of a distorted pentagonal bipyramid with F1-Sc2-F1 angles of 178.58 °. The two distinct Sc-centered polyhedra are joined together through F3 bonding. The connections between the Sc-centered polyhedra between the different layers are indicated by bond angles of 178.07 ° and 175.16 °, for Sc1-F2-Sc1 and Sc2-F1-Sc2 respectively. Cesium is sandwiched by two slightly distorted hexagonal F planes with bond distances ranging from 2.95(2) Å to 3.42(2) Å (12 coordination). Four additional F1 atoms from the vertices of the pentagonal bipyramids and two additional F2 atoms from the vertices of the octahedra are located at longer distances of 3.76(2), 3.89(2) Å and 3.79(2), respectively (Fig. 4 and 5, Table

3.303( 3 )

 3 Å and eight more distances are above 3.5 Å, ranging between 3.569(2) and 3.799(1) Å, with a mean distance of 3.423 Å. Thus, all bond distances values, as well as, the average value are very close to that observed in CsSc3F10. In the isostructural RbSc3F10 structure, the angles between the Sccentered polyhedra of the different layers are of 157.60 ° and 161.75 °, for Sc1-F2-Sc1 and Sc2-F1-Sc2 respectively. The polyhedra of each layer are more inclined relatively to each other. This leads to a change in the distances between alkali metal and transition metal/fluorine, d(Rb-Sc2) = 4.283(6) Å and d(Rb-F1) = 4.056(5) Å, d(Rb-F2) = 3.804(3) Å, respectively, which are almost identical. Only two additional F1 atoms from the vertices of the pentagonal bipyramids are bonded to the Rb atom at a distance 3.393(5) Å.

(

  P2221). In all three compounds, the chains of pentagonal bipyramids are linked by the vertices constituting sheets parallel to the xoz plane. These sheets are interconnected by vertex-sharing octahedra. The three-dimensional network contains fluorinated tunnels parallel to the z axis, into the alkali atoms are inserted. The coordination environments of indium atoms in Rbln3F10 are more distorted, which leads to the presence of two kinds of sevenfold coordinated indium atoms with multiplicities of 1:1, and also to different coordination environments for rubidium. The coordination of each rubidium atom is 8+2: two F4 atoms are situated at distances d(Rb-F4)= 3.37 Å, than the other eight are located at distances from 2.77 Å to 3.13 Å. These eight fluorine atoms are situated approximately at the vertices of a dodecahedron.

  along with the experimental results. A complete unambiguous assignment of 19 F resonances to fluorine crystallographic sites cannot be performed, if only the analysis of integral intensities is taken into account: the signals at -23.0 ppm, -38.8 ppm on the one hand and at -17.1 ppm, -45.6 ppm one the other have the same integral intensities. Based on relative intensities, an only partial assignment of the 19 F resonances to the F sites can be performed: lines at -17.1 ppm and at -45.6 ppm to F2 or F5; lines at -23.0 ppm and at -38.8 ppm to F1 or F4; line at -37.1 ppm to F3. Thus, the complete assignment of 19 F resonances to fluorine crystallographic sites was based on DFT computations, where

Figure 1 . 19 F

 119 Figure 1. 19 F (a) and 45 Sc (b) MAS NMR experimental (blue lines) spectra recorded at 60 kHz and at 20 T are presented with their simulations (red lines). Inserts show (a) Isotropic resonances of the 19 F spectrum and (b) expanded 45 Sc MAS spectrum in the region of the central transition (right); and expanded satellite transition sideband revealing their complex line shape (left).

Figure 2 .

 2 Figure 2. 45 Sc -19 F D-HMQC NMR spectrum recorded at spinning frequency of 60 kHz and at 20 T. Dashed lines show a correlation between fluorine and scandium atoms.

Figure 3 .

 3 Figure 3. Experimental (red), calculated (black) and difference (blue) XRPD Rietveld refinement of CsSc3F10 (green marks). Insert shows expanded area between 20 and 56 °. Final reliability factors are wRp= 6.00 %, Rp= 3.48 % and GOF=7.21. Light green marks correspond to the secondary phase ScF3.

Figure 4 .

 4 Figure 4. Layer structure viewed along x-axis and y-axis.

Figure 5 .

 5 Figure 5. Coordination environments of rubidium and scandium atoms. Dashed lines show Cs-F bonds longer than 3.5 Å. Cs-F bonds distances in Å.

Fig. 1

 1 Fig.1

Fig. 2

 2 Fig.2

  Fig.3

Fig. 4

 4 Fig.4

Table 1 .

 1 Experimental (isotropic chemical shifts δiso, chemical shift anisotropies Δ, asymmetry parameters ηcs, integral intensities I) and calculated (isotropic magnetic shielding σiso and isotropic chemical shifts δcalciso)19 F NMR parameters.

	Fluorine site	δiso, ppm (±0.1 ppm)	Δ, ppm (±10 ppm)	ηcs (±0.1)	I, % (± 1%)	σiso, ppm	δcalciso, ppm
	F5	-17.1	-167	0.24	10	72.3	-22.4
	F4	-23.0	-53	0.03	20	96.9	-36.9
	F3	-31.0	-141	0.02	40	100.2	-38.8
	F1	-38.8	-181	0.26	20	107.2	-42.9
	F2	-45.6	-168	0.2	10	118.0	-49.3

45

Sc MAS NMR spectra recorded at 17.6 T and 20 T contains two typical quadrupolar lines with relative signal intensities of 1:2 (Table

2

and Fig.

1b, 1S

). The 45 Sc chemical shift values for six and seven coordinated scandium environments range from -20 to 14 ppm and from -53.5 to -40 ppm,

Table 2 .

 2 of the 45 Sc MAS at 17.6 T and 20 T and calculated45 Sc NMR parameters (isotropic magnetic shielding σiso, |CQcalc| quadrupolar constant, and ηQcalc asymmetry parameter).

		δiso, ppm	CQ, MHz	ηQ	I, %		δiso calc,
	Atom					σ, ppm		|CQcalc|, MHz ηQcalc
		(±0.2 ppm)	(±0.1MHz)	(±0.1)	(± 1%)		ppm
	Sc1	-32.2	7.4	0.8	33	831.8	-33.9	7.4	0.33
	Sc2	-67.3	8.1	0.8	67	861.6	-55.4	7.4	0.94

45

Sc isotropic chemical shifts (δiso), quadrupolar constants (CQ), asymmetry parameters (ηQ), and Integral intensities (I) obtained from the simulation

Table 4 .

 4 Atomic coordinates, occupancy (Occ) and equivalent isotropic displacement parameters (Ueq)

			1 )	457.8
	Temperature, °C		20
	Z			2
	Crystal system		Orthorhombic
	Space group		Pmma (n°51)
				a = 8.0837(1)
	Unit cell dimensions, Å	b = 7.5764(1)
				c = 6.8127(1)
	Cell volume (Å 3 )		417.25(1)
	Density, calculated (g.cm -3 )	3.644
	(Å 2 ×10 3 ).
	Atom Site	X	Y	Z	Occ	Ueq
	Cs1	2e	1/4	0	0.501(1)	1	0.034(5)
	Sc1	2d	0	1/2	1/2	1	0.031(1)
	Sc2	4g	0	0.2342(3)	0	1	0.018(1)
	F1	4k	1/4	0.2309(7)	0.012(3)	1	0.016(2)
	F2	2f	1/4	1/2	0.495(8)	1	0.028(3)
	F3	8l	0.007(3)	0.308(1)	0.7028(5)	1	0.034(2)
	F4	4i	-0.015(2)	0	0.830(1)	1	0.023(2)
	F5	2b	0	1/2	0	1	0.044(3)

Table 5 .

 5 Selected bond lengths (Å) from XRD Rietveld refinement.

	Sc1-F2 × 2 2.021(1) Sc2-F1 × 2 2.023(1)	Cs1-F3 × 4 3.35(2)	Cs1-F1 × 2 3.76(2)
	Sc1-F3 × 4 2.008(3) Sc2-F3 × 2 2.101(3)	Cs1-F3 × 4 3.42(2)	Cs1-F1 × 2 3.89(2)
	Sc2-F4 × 2 2.121(3)	Cs1-F4 × 2 2.95(2)	Cs1-F2 × 2 3.79(1)
	Sc2-F5 × 1 2.013(2)	Cs1-F4 × 2 3.10(2)

Acknowledgment

This work was supported by the Ministry of Education and Science of the Russian Federation №02.G25.31.0210 of 27.04.2016. Financial support from the TGIR-RMN-THC Fr3050 CNRS for conducting the research is gratefully acknowledged. For the calculations, we thank the "Centre de Calcul Scientifique en region Centre" (Orleans, France).