Pauline Glatz

Monique Comte

Lionel Montagne

Bertrand Doumert

Fabrice Cousin

Laurent Cormier
email: laurent.cormier@sorbonne-universite.fr

Structural evolution at short and

Introduction

The incorporation of specific oxides, known as nucleating agents, in the glass melt strongly influences nucleation and crystal growth. It is actually the best technical route to tailor glass-ceramics with desired properties (transparency, optical, mechanical, thermal . . . ) for industrial applications. [START_REF] Comte | From Glass to crystal -Nucleation, growth and phase separation, from research to applications[END_REF] The mechanism of how nucleating agent favors nucleation is still debated but major recent advances have clarified their role, particularly for TiO 2 and ZrO 2 in various aluminosilicate glasses. From heterogeneities or amorphous phase separation preexisting in the as-cast glasses, [START_REF] Dargaud | Mesoscopic scale description of nucleation processes in glasses[END_REF][START_REF] Dargaud | Multi-scale structuration of glasses: Observations of phase separation and nanoscale heterogeneities in glasses by Z-contrast scanning electron transmission microscopy[END_REF][START_REF] Kleebusch | The evidence of phase separation droplets in the crystallization process of a Li 2 O-Al 2 O 3 -SiO 2 glass with TiO 2 as nucleating agent -An X-ray diffraction and (S)TEM-study supported by EDX-analysis[END_REF] the agents precipitate heterogeneously upon heat treatments, leading to structural and compositional modifications of the glass that allow nucleation of a major phase. [START_REF] Kleebusch | The evidence of phase separation droplets in the crystallization process of a Li 2 O-Al 2 O 3 -SiO 2 glass with TiO 2 as nucleating agent -An X-ray diffraction and (S)TEM-study supported by EDX-analysis[END_REF][START_REF] James | Crystallisation of silicate and phosphate glasses[END_REF][START_REF] Höche | Temporal Evo-lution of Diffusion Barriers Surrounding ZrTiO 4 Nuclei in Lithia Aluminosilicate Glass-Ceramics[END_REF][START_REF] Patzig | Temporal evolution of crystallization in MgO-Al 2 O 3 -SiO 2 -ZrO 2 glass ceramics[END_REF][START_REF] Kleebusch | The formation of nanocrystalline ZrO 2 nuclei in a Li 2 O-Al 2 O 3 -SiO 2 glass -a combined XANES and TEM study[END_REF] Though TiO 2 and ZrO 2 are the most commonly used nucleating agents, P 2 O 5 has been well recognized to aid nucleation in various silicate glasses [START_REF] Maier | Mechanism of Oxide Nucleation in Lithium Aluminosilicate Glass-Ceramics[END_REF][START_REF] Ananthanarayanan | The effect of P 2 O 5 on the structure, sintering and sealing properties of barium calcium aluminum boro-silicate (BCABS) glasses[END_REF][START_REF] Goswami | MAS-NMR study of lithium zinc silicate glasses and glass-ceramics with various ZnO content[END_REF][START_REF] Headley | Crystallization of a Glass-Ceramic by Epitaxial Growth[END_REF][START_REF] Holland | Early stages of crystallisation of lithium disilicate glasses containing P 2 O 5 -An NMR study[END_REF] and is particularly effective when introduced to LiO 2 -SiO 2 glasses. [START_REF] James | Crystallisation of silicate and phosphate glasses[END_REF][START_REF] Höche | Temporal Evo-lution of Diffusion Barriers Surrounding ZrTiO 4 Nuclei in Lithia Aluminosilicate Glass-Ceramics[END_REF][START_REF] Patzig | Temporal evolution of crystallization in MgO-Al 2 O 3 -SiO 2 -ZrO 2 glass ceramics[END_REF][START_REF] Kleebusch | The formation of nanocrystalline ZrO 2 nuclei in a Li 2 O-Al 2 O 3 -SiO 2 glass -a combined XANES and TEM study[END_REF][START_REF] Maier | Mechanism of Oxide Nucleation in Lithium Aluminosilicate Glass-Ceramics[END_REF][START_REF] Ananthanarayanan | The effect of P 2 O 5 on the structure, sintering and sealing properties of barium calcium aluminum boro-silicate (BCABS) glasses[END_REF][START_REF] Goswami | MAS-NMR study of lithium zinc silicate glasses and glass-ceramics with various ZnO content[END_REF][START_REF] Headley | Crystallization of a Glass-Ceramic by Epitaxial Growth[END_REF][START_REF] Holland | Early stages of crystallisation of lithium disilicate glasses containing P 2 O 5 -An NMR study[END_REF] Deep insights on the role of P 2 O 5 have been mostly obtained using NMR techniques, showing that P 2 O 5 can precipitate as a phosphate phase, usually Li 3 PO 4 , but this phase is not considered as a precursor for the nucleation of Li 2 Si 2 O 5 . [START_REF] Holland | Early stages of crystallisation of lithium disilicate glasses containing P 2 O 5 -An NMR study[END_REF] Indeed, a first step in the nucleation mechanisms has been recently proposed with the formation of poorly crystallized Li 3 PO 4 prior any crystallization. [START_REF] Bischoff | Phase evolution in lithium disilicate glass-ceramics based on non-stoichiometric compositions of a multicomponent system: structural studies by 29 Si single and double resonance solid state NMR[END_REF][START_REF] Huang | Crystallization of a high-strength lithium disilicate glass-ceramic: An XRD and solid-state NMR investigation[END_REF][START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF] P 2 O 5 is also well documented to promote phase separation, [START_REF] Holland | Early stages of crystallisation of lithium disilicate glasses containing P 2 O 5 -An NMR study[END_REF][START_REF] James | Liquid-phase separation in glass-forming systems[END_REF][START_REF] Hill | Real-Time Nucleation and Crystallization Studies of a Fluorapatite Glass-Ceramics Using Small-Angle Neutron Scattering and Neutron Diffraction[END_REF] but there is yet no direct confirmation of a relationship between the disordered Li 3 PO 4 and any phase separation or any heterogeneities.

Though transmission electron microscopy could be used to investigate phase separation and the early stages of nucleation, amorphous phase separation is sometimes undetectable in these glasses and Li 3 PO 4 crystals are unstable under the electron beam and difficult to observe. [START_REF] Iqbal | Metastable phase formation in the early stage crystallisation of lithium disilicate glass[END_REF] An alternative technique is small-angle scattering, which is widely used to detect phase separation and follow nucleation and growth processes. [START_REF] Hill | Real-Time Nucleation and Crystallization Studies of a Fluorapatite Glass-Ceramics Using Small-Angle Neutron Scattering and Neutron Diffraction[END_REF][START_REF] Wright | Crystallization of cordierite glass ceramic. part 1. Small angle neutron scattering measurement and simulation[END_REF][START_REF] Lembke | Phase Formation Kinetics in a Glass Ceramic Studied by Small-Angle Scattering of X-rays and Neutrons and by Visible-Light Scattering[END_REF][START_REF] Bras | The development of monodispersed alumino-chromate spinel nanoparticles in doped cordierite glass, studied by in situ X-ray small and wide angle scattering, and chromium X-ray spectroscopy[END_REF][START_REF] Fernandez-Martin | Nucleation and Growth of Nanocrystals in Glass-Ceramics: An In Situ SANS Perspective[END_REF][START_REF] Sigaev | Native amorphous nanoheterogeneity in gallium germanosilicates as a tool for driving Ga2O3 nanocrystal formation in glass for optical devices[END_REF] In particular, due to the good contrast in neutron scattering lengths between P and other components in lithium silicate glasses, small-angle neutron scattering (SANS) is advantageous. [START_REF] Donald | Influence of transition metal oxide addi-tions on the crystallization kinetics, microstructures and thermal expansion character-istics of a lithium zinc silicate glass[END_REF] Advanced solid-state 31 P NMR techniques have been recently developed to characterize the medium range order in glasses, revealing nanometric-sized chemical and structural heterogeneities corresponding to clusters of PO 4 tetrahedra in bioactive glasses. [START_REF] Fayon | Evidence of Nanometric-Sized Phosphate Clusters in Bioactive Glasses As Revealed by Solid-State 31 P NMR[END_REF][START_REF] Mathew | Direct Probing of the Phosphate-Ion Distribution in Bioactive Silicate Glasses by Solid-State NMR: Evidence for Transitions be-tween Random/Clustered Scenarios[END_REF][START_REF] Stevensson | Assessing the Phosphate Distribution in Bioactive Phosphosilicate Glasses by 31P Solid-State NMR and Molecular Dynamics Simulations[END_REF] DQ-DRENAR (Double-Quantum based Dipolar Recoupling Effects Nuclear Alignment Reduction) has been used to probe the spatial distribution of orthophosphate units and has evidenced their nano-segregation in a lithium silicate glass and glass-ceramics. [START_REF] Ren | Applications of DQ-DRENAR for the structural analysis of phosphate glasses[END_REF] In this paper, we investigate whether heterogeneities related to the presence of P 2 O 5 are observable in glasses or whether changes appear with thermal treatments. To complement local changes previously observed, [START_REF] Bischoff | Phase evolution in lithium disilicate glass-ceramics based on non-stoichiometric compositions of a multicomponent system: structural studies by 29 Si single and double resonance solid state NMR[END_REF][START_REF] Huang | Crystallization of a high-strength lithium disilicate glass-ceramic: An XRD and solid-state NMR investigation[END_REF][START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF] transmission electron microscopy (TEM), Small-Angle Neutron Scattering (SANS) and correlation technique 31 P-31 P DQ-DRENAR were used to probe medium range distance (1 nm-2 nm) and mesoscale microstructure in a Li 2 O-Al 2 O 2 -SiO 2 (LAS) glass containing 1 mol% P 2 O 5 .

Experimental Section

Materials

Glasses in the Li 2 O-Al 2 O 3 -SiO 2 ternary system with 0 and 1 mol% P 2 O 5 were prepared as reported previously. [START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF] The P 2 O 5 content was chosen to avoid phase separation that we have observed for P 2 O 5 ≥ 3 mol%. Analytical grade reagents Li 2 CO 3 (Sigma Aldrich, 99%), Al 2 O 3 (Merck, 99.9%), SiO 2 (Alfa Aesar, 99.5%) and (NH 4 ) 2 HPO 4 (Sigma Aldrich, 98%) were mixed using an agate pestle and mortar, dried and melted in a platinum crucible at 1550°C and maintained 2 h at this temperature. Then the samples were quenched by dipping the bottom of the crucible into water. The glasses were ground and remelted to ensure a good homogeneity. Glasses are transparent and bubble free.

The glass compositions were examined by Electron Probe Micro-Analyser (EPMA, CAM-PARIS, Sorbonne Université, Paris, France) with a CAMECA SX-Five apparatus equipped The standards for calibration were orthoclase for silicium and aluminum, and apatite for phosphorus. The nominal and analyzed compositions (averaged over 20 measurements at different points) are reported in Table 1.

Solid-State NMR

29 Si and 31 P MAS-NMR (Magic Angle Spinning Nuclear Magnetic Resonance) spectra were acquired using a Bruker AVANCE I solid state spectrometer operating at a magnetic field of 9.4 T. A 7 mm probe and a spinning frequency of 5 kHz were used for 29 Si. A 4 mm probe and a spinning frequency of 12.5 kHz were used for 31 P. The Larmor frequencies were 79.2 and 161.9 MHz for 29 Si and 31 P, respectively. For 31 P, the pulse durations were 4.8 µs (π/2) and the recycle delay (rd) were optimized at 140 s to enable enough relaxation and get quantitative spectra. For 29 Si, the pulse duration was 2 µs (π/5), and the rd was 900 s, which is insufficient to enable full relaxation despite its large value; these non-quantitive conditions were used anyway in order to keep acceptable measurement time. The chemical shifts are measured relative to tetramethylsilane (TMS) for 29 Si and relative to 85% H 3 PO 4 solution for 31 P. Spectral decompositions were performed using the DMFIT program, [START_REF] Massiot | Modelling one-and two-dimensional solid-state NMR spectra[END_REF] following the procedure explained previously. [START_REF] Glatz | Quantitative determination of the phosphorus environment in lithium aluminosilicate glasses using solidstate NMR techniques[END_REF] In order to get insight on the spatial proximity of similar nuclei ( 31 P) and to obtain the dipolar coupling constants, a DQ-DRENAR (Double-Quantum, Dipolar Recoupling Effects Nuclear Alignment Reduction) sequence was used, following Ren et al. [START_REF] Ren | A new NMR technique to measure site-resolved magnetic dipole-dipole interactions in multispin-1/2 systems: Theory and validation on crystalline phosphates[END_REF] The DQ-DRENAR signal is obtained from two acquisitions. In the first acquisition, the spectrum (S') contains the homonuclear dipolar coupling thanks to the introduction of two C blocks of pulses (two sequences POST-C7 with seven pulses [START_REF] Hohwy | Broadband dipolar recoupling in the nuclear magnetic resonance of rotating solids: A compensated C7 pulse sequence[END_REF]). The signal is thus attenuated by the interaction.

In the second acquisition S 0 , there are no dipolar couplings due to the out of phase of the two C block pulses, which cancel the dipolar recoupling. These blocks are incremented so that the reinsertion time of the dipolar interaction increases.

The DQ-DRENAR signal is defined by the amplitude S 0 -S' that depends on the intensity of the coupling constant between the nuclei, as well as the time during which the coupling is introduced. This time is controlled during the experiment by the term N.T r , where N corresponds to the number of rotation cycles of the rotor and T r is the rotation period. It is then possible to plot S 0 -S S 0 as a function of N.T r to get information on the strength of the dipolar coupling and to derive the coupling constants by modeling the curve for S 0 -S S 0 <0.3, following the equation: [START_REF] Ren | A new NMR technique to measure site-resolved magnetic dipole-dipole interactions in multispin-1/2 systems: Theory and validation on crystalline phosphates[END_REF] S 0 -S S 0 = 0.86π 2 15

k d 2 jk N T r 2 (1)
with d jk the dipolar coupling constant between spins j and k. The closer the phosphorus sites are, the higher the phase shift is and, consequently, the S 0 -S S 0

values will be higher.

The DQ-DRENAR experiments were performed on a Bruker Avance I spectrometer at a magnetic field of 9.4 T with a 4 mm rotor and a speed of 12.5 kHz. A pre-saturation pulse is added to ensure that all acquisitions start with the same magnetization state. A pulse block lasts two rotor periods. The power of the radiofrequency field is optimized so as to have a frequency ω rf = 7 * ω rot .

2.4 Small-Angle Neutron Scattering (SANS)

The small-angle neutron scattering (SANS) experiments were carried out on the PAXY spectrometer at the Léon Brillouin Laboratory in CEA Saclay, France. [START_REF] Cousin | Small angle neutron scattering[END_REF] Measurements have been carried out in a q-range of 2.5.10 -2 nm -1 ≤ q ≤ 5nm -1 (q=4π sinθ/λ with 2θ the scattering angle and λ the neutron wavelength). To cover these q ranges, four different configurations were used (Table 2).

Table 2: Configurations used in this study: couples (D/λ) for the different q ranges for the two spectrometers sample-detector distance wavelength D λ very small q 6.7 m 1.5 nm small q 5 m 0.85 nm medium q 3.5 m 0.5 nm large q 1 m 0.5 nm

The samples were polished to the micron on both sides and reduced to a thickness of 1 mm in order to avoid multiple scatterings. Samples are placed in a sample holder between two windows of silica glass. The data were corrected from the background using Cd, from the contribution of empty beam and the contribution from quartz windows, from the detector efficiency using water as a reference sample and then normalized using the scattering crosssection of H 2 O for 1mm thickness.

The incoherent scattering (fluctuations in scattering lengths due to nuclear spins) can be neglected as it is q-independent and only the coherent scattering cross-section is considered.

The relevant parameter for estimating the probability of an element to scatter neutrons is the coherent scattering length density (SLD) by volume unit, ρ, defined as:

ρ = n i bi ν (2)
with b i the coherent scattering lengths of atoms i within the elementary scatterer and ν the volume of the elementary scatterer. In Table S1 Detailed information on SANS formalism can be found in various reviews. [START_REF] Cousin | Small angle neutron scattering[END_REF][START_REF] Glatter | Small angle x-ray scattering[END_REF][START_REF] Lelong | Nanoporous Materials: Advanced Techniques for Characterization, Modeling, and Processing[END_REF] In a binary system of n rigid and centrosymmetric objects that contains N elementary scatterers homogeneously distributed within a surrounding medium, the measured intensity can be written:

I(q)(cm -1 ) = nN 2 V (ρ obj -ρ medium ) 2 ν 2 obj P (q)S(q) (3)
with ρ obj et ρ medium the SLD for the objects and the surrounding medium, respectively. V is the sample volume and ν obj is the volume of an elementary scatterer of the object. P(q)

is the form factor of the object that contains information about interferences between the elementary scatterers of the object (intra-object correlations) and S(q) is the structure factor that contains information about interference between the centers of gravity of the objects (inter-object correlations).

With the hypothesis that objects are centrosymmetric and well dispersed (no inter-object interactions), S(Q) is equal to 1 in equation 3. The scattered intensity can be described by the Guinier and Porod laws: [START_REF] Glatter | Small angle x-ray scattering[END_REF]

I(q) = I(0)e -q 2 3 R 2 g , f or qR g < 1 (4)
I(q) = K 1 + K 2 q 4 , f or qR g > 1 (5)
R g is the radius of gyration. The particle size for spherical objects can be obtained from R g with the relationship:

R g = 3 5 R (6)
When objects are hard spheres of radius R, the form factor can be written:

P sphre (q) = 3sin(qR) -qRcos(qR) (qR) 3 2 (7)

Transmission Electron Microscopy

Scanning transmission electron microscopy (STEM) was performed in high angle annular dark field imaging mode (STEM-HAADF) using a 2100F microscope operating at 200 kV.

This microscope was equipped with a field emission gun and an ultra-high resolution (UHR) pole piece. Samples were ground in agate mortar under acetone and the powder in suspension was collected and deposited on a Cu grid. to Supplemental Materials (Figure S1). The scattering of the samples when q tends towards 0 is very weak (around 0.05 cm -1 ). The errors bars are thus very large at the small q values since the contribution from the empty beam has the same order of magnitude as those of samples in this q-range. The two SANS profiles are almost identical, with a very low and nearly flat scattering between 0.2 and 3 nm -1 . This indicates that the introduction of P 2 O 5 160 does not create additional scattering. Therefore, according to SANS patterns, the glasses resulting from the melt-quench route are homogeneous. [START_REF] Benedetti | SAXS study of the micro-inhomogeneity of industrial soda lime silica glass[END_REF] Figure 1: SANS curves of the glasses without P 2 O 5 (LAS0.2-P0) and with P 2 O 5 (LAS0.2-P1). To improve curve readability, error bars are not shown but can be found in Figure S1.

X-ray diffraction

Using TEM and STEM-HAADF, the LAS0.2-P1 sample appears as an homogeneous glass (Figure 2), consistent also with the absence of macroscopic opalescence. For this composition, the contrast is low because the constituting elements have similar atomic numbers. This may explain the difficulty to observe compositional fluctuations. A good contrast is indeed observed in studies with glasses involving zirconium taking advantage of its high atomic number. [START_REF] Dargaud | Mesoscopic scale description of nucleation processes in glasses[END_REF][START_REF] Dargaud | Multi-scale structuration of glasses: Observations of phase separation and nanoscale heterogeneities in glasses by Z-contrast scanning electron transmission microscopy[END_REF] A study on a glass 72.8SiO

2 -20.8Li 2 O -2.7Al 2 O 3 -2.8K 2 O -0.8P 2 O 5 -0.1Sb 2 O 3
(% mol) by Kim et al. [START_REF] Kim | Nucleation behavior and microstructure of Al 2 O 3 -poor LAS glass-ceramics[END_REF] is consistent with our observation that no phase separation is detected by TEM.

Evolution of local environments with heating treatments

Different heating treatments at 525°C, 550°C and 575°C during 1 hour were applied to the LAS0.2-P1 glass. All the samples remained transparent after heat treatments.

X-ray diffraction

X-ray diffractograms are presented in Figure 3a. No Bragg peaks are observed for the as-175 cast glass or the glasses heat-treated at 525 °C and 550°C. Weak broad peaks are observed at 575°C but the amorphous signal remains dominant. Despite the low intensity of the peaks and their large widths, a Rietveld refinement for the glass heat-treated at 575°C was carried out in order to estimate the amount of the crystallized phases. Results from the refinement are summarized in Table 3. The sample appears mostly amorphous (≈ 90%). As revealed by 180 the broad peak at ∼ 33.3°, (plane (220) at 33.92°, in the JCPDS 00-015-0760 card), the XRD pattern allows the detection of the γ-Li 3 PO 4 phase in very low amount (1 to 2 weight%).

XRD analysis (Figure S2) also indicate the presence of lithium metasilicate (LS), lithium disilicate (LS 2 ) and petalite (LiAlSi 4 O 10 ).

29

Si MAS-NMR 29 Si MAS-NMR spectra are shown in Figure 3b. Small changes in the MAS-NMR spectra are visible after the first heating treatments at 525°C and 550°C but they are not significant in terms of the signal/noise ratio for our spectra. At 575°C, resonances at -74.7 ppm and at -93.0 ppm are clearly visible and are assigned to the crystalline phases of lithium metasilicate (at -75 ppm in [START_REF] Mackenzie | Multinuclear solid-state NMR of inorganic materials[END_REF]) and lithium disilicate (at -92.6 ppm in [START_REF] De Jong | Mixed Alkali Systems: Structure and 29 Si MASNMR of Li 2 Si 2 O and K 2 Si 2 O 5[END_REF]). Their presence is confirmed by XRD (Figure 3a). The petalite phase detected by XRD is not visible on the 29 Si spectrum, probably because its relaxation time is very long, as observed for other crystalline silicate phases with large polymerization degree.

To compare with results obtained by XRD, the weight% proportions of each phase were obtained by simulation of the 29 Si NMR spectra (Table 4). Note that the recycling time of 900 seconds does not ensure complete relaxation of the phases and can lead to errors in quantities.

31 P MAS-NMR

The 31 P MAS-NMR spectra are presented in figure 4. The spectrum of the as-cast glass is composed of two peaks that are typically lying in the ranges assigned to orthophosphate and pyrophosphate units, i.e. Q 0 and Q 1 , respectively. Following the methodology developed in a previous paper, [START_REF] Glatz | Quantitative determination of the phosphorus environment in lithium aluminosilicate glasses using solidstate NMR techniques[END_REF] the Q 1 units can be decomposed into two contributions Q 1 (0Al) and Q 1 (1Al) corresponding to PO 3 - 4 units connected to zero or one Al atom, respectively. The decomposition and the parameters of the various contributions are reported in Table 5. The as-cast glass is obtained with a different batch compared to our previous study. [START_REF] Glatz | Quantitative determination of the phosphorus environment in lithium aluminosilicate glasses using solidstate NMR techniques[END_REF] This explains the small differences in the intensities of the contributions and the FWHM, which are close to fitting errors and may also indicate a slightly different thermal history between the two glass batches.

For the heat-treated samples, in order to fit the main peak at 9.3 ppm (Figure 4), an additional contribution is required, with a FWHM larger at 525°C than at 575°C (see Figure S3 for a typical fit). Due to its relatively large broadness, this resonance has been assigned to a disordered Li 3 PO 4 species (disordered-LP) that appears when a heating treatment is applied. [START_REF] Bischoff | Phase evolution in lithium disilicate glass-ceramics based on non-stoichiometric compositions of a multicomponent system: structural studies by 29 Si single and double resonance solid state NMR[END_REF][START_REF] Huang | Crystallization of a high-strength lithium disilicate glass-ceramic: An XRD and solid-state NMR investigation[END_REF][START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF][START_REF] Glatz | Quantitative determination of the phosphorus environment in lithium aluminosilicate glasses using solidstate NMR techniques[END_REF] This disordered-LP species is formed at the expense of the Q 0 (0Al) and Q 1 (0Al) units and occurs before the detection of any crystallization by XRD (Bragg peaks are detected only at 575°C in figure 4). At 550°C, 16% of the phosphorus atoms are in disordered-LP species. At 575°C, a majority of P atoms (52%) in disordered-LP is determined. At this temperature, the peak is narrow, which characterizes a more orderly state than in the case

225

of heating treatments at lower temperatures. The amount of P atoms in disordered-LP corresponds to 0.94 mol% of disordered Li 3 PO 4 or 1.9 weight%, which is consistent with XRD results that give 1.7 weight% (Table 3). 5.

Table 5: Isotropic chemical shifts (δ iso ), full width at half maximum (FWHM) and relative intensities (I) for the fitted contributions for the as-cast glass LAS0.2-P1 and for samples heat-treated during 1 hour at various temperatures and at 525°C for various times. The gaussian/lorentzian ratios are fixed to 0.80 for all glassy contributions. For the disordered-LP contribution, the ratios are fixed at 0.3 at 525°C and 550°C and at 0.1 at 575°C. The spinning bands are taken into account in the simulation. Using TEM and STEM-HAADF, the sample heat-treated at 550°C/1 h does not show any crystalline phase nor the development of phase separation (Figure S4). For this heat treatment, the P local environment is modified with the apparition of disordered-LP species (Figure 4) but no Bragg peaks are observed on the X-ray diffraction pattern (Figure 3a).

sample disordered-LP Q 0 (0Al) Q 1 (1Al) Q 1 (0Al) δ iso FWHM I δ iso FWHM I δ iso FWHM I δ iso FWHM I (ppm) (ppm) (%) (ppm) (ppm) (%) (ppm) (ppm) (%) (ppm) (ppm) (
Similarly, no modification has been seen by TEM for a glass 72.8SiO 2 -20.8Li 2 O -2.7Al 2 O 3 -2.8K 2 O -0.8P 2 O 5 -0.1Sb 2 O 3 (% mol) heat-treated at 460°C/9 hours. [START_REF] Kim | Nucleation behavior and microstructure of Al 2 O 3 -poor LAS glass-ceramics[END_REF] The first changes that have been observed using TEM were after a heat-treatment at 500°C/9 hours, revealing circular zones of 20 nm in diameter, associated with Bragg peaks in the corresponding X-ray diffraction pattern.

Evolution at mesoscopic scale with heating treatments

SANS

SANS curves obtained for the different thermal treatments are presented in Figure 5 for the LAS0.2-P1 sample heat treated at various temperatures during 1h. There is no change in scattering intensities for the sample heat treated at 525°C/1 h compared to the as-cast glass.

Though a small change in the local phosphorus environment has been observed by NMR (Figure 4), any rearrangement at medium range distance is too small to be observed by SANS. At 550°C, a significant change in scattered intensity appears, which is characteristic of a size of scattering objects or a quantity of scattering objects large enough to introduce a neutron contrast with a maximum at about 0.6 nm -1 . At 575°C, the scattered intensity increases significantly below 1 nm -1 , indicating that more scattering objects are present. This is due to the appearance of the crystalline phases detected by XRD (Figure 3a). The maximum intensity is progressively shifted to the low q values as the heating temperature treatment increases, which means that the size of scattering objects increases. In Figure S6, the SANS curve is presented for a heating treatment of 1 h at 650°C, for which the scattered intensity is even greater (the sample is more crystallized). In order to determine the size of the scattering objects observed after thermal treatments, a Guinier analysis (equation 4) was performed for the samples heat-treated at 550°C and 255 575°C (Figure 6). The scattered intensities were fitted in Guinier's linear domain for q < 1.25 nm -1 for the sample heat-treated at 550°C and q < 0.687 nm -1 for the sample heattreated at 575°C. The ln(I)=f(q 2 ) plots show the linear domains in Figure 6 with a slope that allows the determination of the radius of gyration. No assumption on the shape of the objects is postulated. SasView software has been used for all simulations, using a scale factor 260 to take into account the term ∆ρ 2 Φ obj V obj . A radius of gyration of 2.79±0.03 nm for the sample heat-treated at 575°C is obtained. This is larger than the radius of 1.56±0.06 nm obtained for the glass heat-treated at 550°C. This confirms a larger size of the scattering objects in the former sample, for which both XRD and 29 Si MAS-NMR reveal crystallization (Figure 3).

For the treatment at 550°C, with the hypothesis that in the early stages of crystallization the germs are spherical, it is possible to determine a sphere radius using the equation 6. A radius R=2.01±0.07 nm is determined, or a sphere diameter of 4.0±0.2 nm. It is also possible to fit the curve considering a form factor P(q) for a sphere (equation 7). The simulation is presented in Figure S7. The simulated radius is 1.89±0.06 nm, which is consistent with the Guinier's analysis. In the SANS analysis the scattering is ascribed to the LP phase since the contrast for the Li-silicate phases will not be very different to that of the glass matrix. This gives typical size for LP phase of 5 nm.

In order to determine the volume fraction of scattering objects, the contrast term ∆ρ 2 = ρ obj -ρ medium in equation 3 must be defined. For the surrounding medium, the SLD ρ medium is chosen equal to that of the glass. At 550°C, no crystalline phase is observed by XRD while a local rearrangement around P is observed using MAS-NMR, with the presence of disordered-LP species. It seems thus reasonable to consider the disordered-LP species as the scattering objects and the SLD can be approximated by considering that ρ obj equals that of γ-Li 3 PO 4 . The real density of disordered-LP entities is smaller than the density of γ-Li 3 PO 4 and the real SLD is thereby smaller. Consequently, the volume fraction values are overestimated. Details on the calculation of scattering length density for objects and the surrounding medium are presented in Table S3. A volume fraction of Φ obj =4.9±0.4%

is obtained. This small fraction explains the difficulty to observe such heterogeneities using TEM (Figure S4). The amorphous and crystalline Q 0 species have no P-O-P bonds. The measurement of dipolar coupling constants 31 P-31 P DQ-DRENAR is thereby reflecting the spatial distribution of these species. [START_REF] Ren | Applications of DQ-DRENAR for the structural analysis of phosphate glasses[END_REF] The data were measured on the peak located at ca. 9.3 ppm for the ascast glass and the sample treated at 550°C/1h, as presented in a previous publication. [START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF] Note that for the latter glass, the contribution at 10 ppm assigned to the disordered-LP 295 species and the glass contribution at 9.3 ppm, assigned to Q 0 (0Al) species, are not distinguished in the DQ-DRENAR measurements.

For the sample heat-treated at 900°C/1h, the main peak corresponds to the Li 3 PO 4 crystalline phase observed at 10.3 ppm. For comparison, we reported in Figure 7 the crystalline reference γ-Li 3 PO 4 with a chemical shift at 9.6 ppm. [START_REF] Glatz | Quantitative determination of the phosphorus environment in lithium aluminosilicate glasses using solidstate NMR techniques[END_REF] The curves in Figure 7 were fitted using equation (1) for S 0 -S S 0 <0.30. The fit is shown as dashed lines and the fitted parameters are reported in Table 6. The dipolar constants are proportional to the interactions of the 31 P-31 P pairs in the system. Their values are close to those reported by Ren and Eckert [START_REF] Ren | Applications of DQ-DRENAR for the structural analysis of phosphate glasses[END_REF] for glasses and crystalline references. They increase continuously when the temperature of the heat-treatment increases. The k d 2 jk value for the as-cast glass is much lower than that of γ-Li 3 PO 4 . Following Ren and Eckert, this difference results from the greater dispersion of phosphorus atoms within the glass. Indeed, the dipolar coupling constants are lower because of the small amount of P 2 O 5 in the glass and, if the P units are well dispersed, this induces very little proximity and therefore a small k d 2 jk value. At 550°C, there is an increase in the k d 2 jk value, which is intermediate between the as-cast glass and the treatment at 900°C. This evolution indicates that segregation of P occurs at this temperature.

Isothermal changes at 525°C

DQ DRENAR indicates that P atoms are close to each other (Figure 7) at 550°C and a change in the 31 P NMR spectrum is observable after 1 h heat-treatment (Figure 4) at 550°C

but modifications in the P environment appear also at 525°C . We therefore investigated the evolution with time for samples heat-treated at 525°C for 1 h (previously presented) and for 4 h. For this long heat-treatment, the sample is still XRD amorphous (figure 8a) and transparent. This remains uncertain given the small shift, but can be considered in view of the evolution of the 31 P MAS-NMR spectrum (see below) that clearly indicates a local reorganization of P atoms within the glass.

The 31 P MAS-NMR spectra of the heat treatments at 525°C are presented in Figure 9 325 and the parameters of the various contributions obtained by the simulation are presented in Table 5. 5.

There is an increase in the intensity of the disordered-LP contribution when the duration of the heat treatment is increased from 1 h to 4 h, with a decrease of the Q 1 (0Al) contribution.

For the treatment 525°C/4 h, 16% of the phosphorus atoms are in the form of disordered-LP 330 species (close to the value obtained for the heating treatment at 550°C/1 h in Table 5).

Figure 11 compares the SANS patterns for the as-cast glass and the samples heat-treated at 525°C for 1 h and 4 h. There is a significant change in the SANS intensity after 4 h compared to the as-cast glass and the glass heat-treated at 525°C/1 h. A maximum appears at about 0.7 nm -1 , indicating a mesoscopic change in the glass organization. A simulation using a sphere form factor model for the heat treatment 525°C/4 h (Figure S9) gives a sphere radius of 1.67±0.06 nm, i.e. a sphere diameter of 3.3±0.1 nm. This can be compared to the size previously obtained for the sample heat treated at 550°C/1 h (3.8±0.2 nm). To estimate the volume fraction of the objects, we used the SLD of the as-cast glass for the surrounding medium, and that of γ -Li 3 PO 4 for the objects (Table S3). A

Discussion

Structural modification prior nucleation

In the LAS0.2-P1 glass composition, phosphorus is mostly in orthophosphate species surrounded by lithium ions, but isolated from the aluminosilicate network. SANS curves do not show compositional fluctuations, confirming that the phosphate species are well dispersed within the glass, as revealed by 31 P-31 P DQ-DRENAR experiments. Finally, TEM observations are unable to detect heterogeneities. Therefore, we conclude that P atoms are introduced in depolymerized, dispersed sites within the glass structure with no phase separation or compositional heterogeneities at the mesoscopic scale.

Slightly above Tg (Tg=508±1°C for LAS0.2-P1 glass) [START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF], the first structural modifications upon heating treatments appear at 525°C/1 h with a reorganization of the P environment involving the formation of disordered-LP entities at the expense of the Q 0 (0Al) and Q 1 (0Al) units. However, no mesoscopic effects are discernible by SANS or XRD, indicating that this structural reorganization is mainly local.

The amount of disordered-LP entities increases at 550°C, which can help to interpret changes at medium range distances detected by DQ-DRENAR and SANS measurements.

The k d 2 jk value is higher for the sample heat-treated at 550°C/1 h compared to the as-cast glass and it increases further for the glass heat-treated at 900°C/1 h. The simulation for the latter sample gives a k d 2 jk value of 2.9x10 5 Hz 2 , which is close to that obtained for the γ-Li 3 PO 4 reference. This confirms the presence of Li 3 PO 4 crystals after the treatment at 900°C/1 h, as observed by XRD and 31 P 1D-NMR. [START_REF] Glatz | Different roles of phosphorus in the nucleation of lithium aluminosilicate glasses[END_REF] For the glass heat-treated at 550°C/1 h, the value of the dipolar coupling constant increases compared to the value for the as-cast glass, from 0.33x10 5 Hz 2 to 0.89x10 5 Hz 2 . This increase illustrates that orthophosphate entities are closer to each other at high temperature. However, the value remains lower than the crystalline Li 3 PO 4 one. This is consistent with the formation of disordered-LP species. This medium-range reorganization with a segregation of P species appears before any crystallization observable by XRD.

Ren and Eckert also studied the evolution of the k d 2 jk dipolar constant values for different thermal treatments on a MKA glass sample (MKA glass: 66.9SiO [START_REF] Ren | Applications of DQ-DRENAR for the structural analysis of phosphate glasses[END_REF], previously studied by Bischoff et al.. [START_REF] Bischoff | Phase evolution in lithium disilicate glass-ceramics based on non-stoichiometric compositions of a multicomponent system: structural studies by 29 Si single and double resonance solid state NMR[END_REF] They reported a value of 0.65x10 5 Hz 2 . However, the selected thermal treatments did not reveal a significant difference in dipolar constant values for orthophosphates entities before the observation of crystallization of lithium metasilicate by XRD.

2 -1.9Al 2 O 3 - 28Li 2 O-1.9K 2 O-1.3P 2 O 5 )
The evolution of the SANS signal can be explained by the segregation of P atoms within disordered-LP objects, for which sizes of about 4 nm in diameter are determined. An investigation using STEM-HAADF did not allowed the observation of a chemical contrast at 550°C/1 h (Figure 2b). In addition to the low chemical contrast in STEM-HAADF, this can be explained by the small volume fraction of objects (4.9% determined by SANS).

Changing the duration of the heat treatment induces similar changes than increasing the temperature of the heat treatment. A local rearrangement around phosphorus is observed in the 31 P MAS-NMR spectrum at 525°C/4 h, with 16% of the phosphorus atoms are in the form of poorly crystallized Li 3 PO 4 (Figure 9, Table 5). This structural ordering is associated with a contrast in neutron scattering, with object sizes of about 3.3 nm in diameter. Similar to the heat treatment at 550°C/1 h, these areas are likely due to the disordered-LP entities.

Schematic scenario for nucleation

The as-cast glass is homogeneous and no heterogeneity is observed. Lithium orthophosphate (and pyrophosphate) species are well dispersed within the aluminosilicate network. Upon heating treatment, the nanoscale segregation of phosphate entities precedes the appearance of the crystalline phases observed at higher temperatures. These domains may correspond to amorphous or poorly organized regions composed of lithium orthophosphate groups, which act as seeds of crystallization for Li 3 PO 4 crystals. [START_REF] Glatz | Influence de P 2 O 5 sur la structure et la nucleation de verres d'aluminosilicates d'alcalins[END_REF] Figure 11 illustrates the influence that the disordered-LP species can have on the volume crystallization of LAS0.2-P1 glass. The segregation of phosphate entities (stage 1) triggers the precipitation of Li 3 PO 4 and leads to low-polymerized, Li-rich domains. These regions could be favourable to the crystallization of lithium metasilicate and disilicate (stage 2). The Li-rich areas could thus promote low interfacial energy between nuclei and glass, as mentioned by James for stoechiometric compositions of lithium disilicate. [START_REF] James | Crystallisation of silicate and phosphate glasses[END_REF] Alternatively, the formation of disordered-LP allows the segregation of P atoms that are otherwise inhibiting crystallization when dispersed in the silicate network. It is not possible to assert at present whether the silicate phases appear in direct contact with Li 3 PO 4 or whether they appear simultaneously with Li 3 PO 4 or as a result of the formation of Li 3 PO 4 . On the other hand, these Li-enriched domains that promote the appearance of lithium metasilicate and disilicate phases entail an impoverishment in lithium and an enrichment in aluminum of the residual glass. This residual glass, more polymerized, can thereby favors the crystallization of secondary phases containing aluminum, first petalite (stage 3) and eventually solid solution of β-quartz (stage 4).

It has been shown that the ability of ZrO 2 to nucleate magnesium aluminosilicate phases can be related to the preferential arrangements between Zr-O x polyhedra, implying heterogeneities present in the as-cast glass. [START_REF] Dargaud | Mesoscopic scale description of nucleation processes in glasses[END_REF][START_REF] Dargaud | Multi-scale structuration of glasses: Observations of phase separation and nanoscale heterogeneities in glasses by Z-contrast scanning electron transmission microscopy[END_REF] Here in the composition LAS0.2-P1, P atoms show a preferential environment in the as-cast glass, but the phosphate entities appear homogeneously dispersed. A thermal treatment is required to compel the phosphate entities to segregate, confirming a heterogeneous mechanism for the first stages of nucleation. Table S3: Calculation of scattering length density for the scattering objects and the surrounding medium for the fit using a sphere form factor for the glass heat-treated at

  XRD patterns were collected using a X-PERT Pro MPD Panalytical (Almelo, Netherlands) diffractometer on powdered samples. The following configuration was used: Ge[111] monochromator selecting CuK α1 at λ=1.540593 Å for an angular range from 5°to 140°( θ steps of 0.008°and a counting time of 80/,s per step). A real time multiple strip (RTMS) detector X-Celerator operating in scanning mode with an active length of 0.518°w as used. Rietveld refinement was performed based on the presence of the following phases: Lithium Di-Silicate (Li 2 Si 2 O 5 ), JCPDS card N°04-009-8780; Petalite 1M (LiAlSi 4 O 10 ), JCPDS card N°04-009-9405; Lithium Meta-Silicate (Li 2 SiO 3 ), JCPDS card N°00-029-0829; Lithium Phosphate (Li 3 PO 4 ), JCPDS card N°00-015-0760; α-quartz (SiO 2 ), JCPDS card N°01-077-8621. The instrumental resolution file had been determined using the SRM 1976 α-Al 2 O 3 XRD standard from NIST. From the Rietveld refinement (Fullproff software code), estimation of phase amounts were extracted.

3 Results 3 . 1 5 Figure 1

 33151 Figure1shows the SANS curves recorded for the glasses with and without P 2 O 5 , LAS0.2-P1 and LAS0.2-P0, respectively. Error bars have not been added here but the reader can refer 155

Figure 2 :

 2 Figure 2: STEM-HAADF image of the LAS0.2-P1 glass.

Figure 3 :

 3 Figure 3: (a) X-ray diffractograms and (b) 29 Si MAS-NMR spectra for the as-cast glass LAS0.2-P1 and for samples heat-treated during 1 hour at various temperatures. LS 2 : Li 2 Si 2 O 5 , LS: Li 2 SiO 3 , P:LiAlSi 4 O 10 , a: α-quartz et LP: Li 3 PO 4 .

Figure 4 :

 4 Figure 4: 31 P MAS-NMR spectra for the as-cast glass LAS0.2-P1 and for samples heattreated during 1 hour at various temperatures. The parameters for the different fitted contributions are detailed in Table5.

Figure 5 :

 5 Figure 5: SANS curves for the as-cast glass LAS0.2-P1 and for samples heat-treated during 1 hour at various temperatures. To improve curve readability, error bars are not shown but can be found in Figure S5.

Figure 6 :

 6 Figure 6: (a) SANS curves (a) LAS0.2-P1 heat-treated at 550°C/1 h (b) LAS0.2-P1 heat-treated at 575°C/1 h and their fits using a Guinier analysis. In inserts, the ln(I)=f(q²) plots showing the linear domains.

3. 3 . 2

 32 31 P-31 P DQ-DRENAR31 P-31 P DQ-DRENAR experiments were carried out on the LAS0.2-P1 as-cast glass and samples heat-treated at 550°C/1 h and 900°C/1 h. The latter one has been studied because phosphorus is entirely in the crystallized form Li 3 PO 4 . 31 P-31 P DQ-DRENAR curves are presented in Figure7.

Figure 7 :

 7 Figure 7: DQ-DRENAR curves for the as-cast glass LAS0.2-P1 (black circles), for the samples heat-treated at 550°C/1 h (blue squares) at 900°C/1 h (green diamonds) and for the crystalline reference γ-Li 3 PO 4 (rose hexagons). N represents the cycle number for the measurement and T r the rotation period (B 0 =9.4 T, ν rot =12.5 kHz). For each spectra S' and S 0 , 32 accumulations have been obtained. In dashed lines, simulations at low NT r values for the different sites for S 0 -S S 0 <0.30.

Figure 8 :

 8 Figure 8: (a) X-ray diffractograms and (b) 29 Si MAS-NMR spectra of the as-cast glass LAS0.2-P1 and the samples heat-treated at 525°C for 1 h and 4 h.

Figure 8b shows the 29

 29 Figure 8b shows the 29 Si NMR spectra. No resonances attributed to crystalline phases are observed. The features are slightly displaced towards more negative chemical shifts with longer duration of heat treatment, which could imply a reorganization of the glassy network.

Figure 9 :

 9 Figure 9: 31 P MAS-NMR spectra for the as-cast glass LAS0.2-P1 and for samples heattreated at 525°C during 1 h and 4 h. The fitting parameters for the different contributions are detailed in Table5.

  335

Figure 10 :

 10 Figure 10: SANS curves for the as-cast glass LAS0.2-P1 and for samples heat-treated at 525°C during 1 h and 4 h. To improve curve readability, error bars are not shown but can be found in Figure S8.

Figure 11 :

 11 Figure 11: Schematic representation of the crystallization sequence of the glass LAS0.2-P1.

  and NMR were used to investigate structural changes in a P 2 O 5 -Li 2 O-Al 2 O 3 -SiO 2 glass, allowing local environments and mesoscale organization to be probed complementary. The glass containing 0.9% mol of P 2 O 5 is homogeneous on a mesoscopic scale. No heterogeneities or segregation regions appear at 525°C/1 h on this scale. On the contrary, the local P environment evolves slightly as observed on 31 P MAS-NMR spectra. For treatments at 525°C/4 h and 550°C/1 h, where no crystallization is still observed by XRD, the local P rearrangement occurs concomitantly with changes at the mesoscopic scale revealed by SANS and 31 P-31 P DQ-DRENAR. Modifications of the P local environment are related to changes at the mesoscopic scale: the 3-4 nm zones observed by SANS can be associated with the first disordered Li 3 PO 4 entities. This segregation develops with increasing time or increasing temperature. The evolution of P local environment and the segregation of phosphorus species precede the crystallization of the silicate and aluminosilicate phases, which are observed by XRD at higher temperature (575°C/1 h). P is thus playing an essential role in the initial stages of heat treatment by promoting nanoscale heterogeneities that allow a description of the crystallization mechanisms.

Figure S3 :

 S3 Figure S3: Typical fit of a 31 P MAS-NMR spectrum for the the LAS0.2-P1 glass after heat-treatment at 550°C during 1 hour

Figure S4 :

 S4 Figure S4: STEM-HAADF image of the LAS0.2-P1 sample after heat-treatment at 550°C during 1 hour.

Figure S5 :

 S5 Figure S5: SANS curves with error bars for the as-cast glass LAS0.2-P1 and for samples

550°C/ 1

 1 h. Na: Avogadro number [a] Calculated from the structure of γ-Li 3 PO 4 JCPDS 630 00-015-0760. [b] Measurement by pycnometry, using a gaz Accu Pyc 1340 pycnometer from Micromeritics.

Figure S8 :

 S8 Figure S8: SANS curves with error bars for the as-cast glass LAS0.2-P1 and for samples

Table 1 :

 1 Analyzed compositions (mol%) by EPMA and F-AES of the glasses investigated in this study. Nominal compositions are in parenthesis. Errors are typically less than 0.3% relative on SiO 2 , less than 0.7% relative on Al 2 O 3 , less than 3% relative on P 2 O 5 and 2% relative on Li 2 O. Dispersive X-ray Spectrometers (WDSs) for SiO 2 , Al 2 O 3 , P 2 O 5 and by Flame-Atomic Emission Spectroscopy (F-AES) with a Agilent AA280FS apparatus for Li 2 O content. EPMA instrument operates at 15 kV and 10 nA with a spot size of 5 µm.

	Glass	SiO 2	Al 2 O 3	Li 2 O	P 2 O 5
	LAS0.2-P0 75.7 (74.23) 4.6 (4.13) 19.7 (21.64) 0.02 (0.00)
	LAS0.2-P1 74.6 (73.47) 4.3 (4.08) 20.2 (21.42) 0.9 (1.02)
	with five Wavelength-

Table 3 :

 3 Weight% of phases (relative errors are 5% for amorphous and 8% for crystals) obtained by Rietveld refinement on LAS0.2-P1 heat-treated at 575°C/1h. Phases Amorphous Li 2 Si 2 O 5 LiAlSi 4 O 10 Li 2 SiO 3 Li 3 PO 4 α-quartz *

			(LS 2 )	(P)	(LS)	(LP)	(a)
	weight%	≈ 90	≈ 2	≈ 5	< 0.5 1 < LP < 2	< 1

* α-quartz is likely present due to a pollution resulting from the grinding.

Table 4 :

 4 Calculation of weight% of lithium metasilicate and disilicate phases for the sample heattreated at 575°C/1h. Details on the calculation can be found in TableS2. *obtained from the simulation of the29 Si MAS-NMR data.Li 2 SiO 3 Li 2 Si 2 O 5Both methods (NMR and XRD) show that the sample contains Li 2 SiO 3 and Li 2 Si 2 O 5 crystals in small quantities (Table3 and Table 4). However the weight proportions obtained by these two techniques differ. NMR results indicate 2.3 weight% for Li 2 SiO 3 while only 0.3 weight% is detected by XRD. Similarly for the Li 2 Si 2 O

	% Si within crystals*	2	3
	mol% crystalline phases	1.5	1.1
	weight% crystalline phases 2.3	2.9

5 phase, 2.9 weight% is determined by NMR and 1.8 weight% by XRD. However, quantification is subject to large errors since we are at the detection limits for these two techniques. Moreover, these two techniques can have different sensitivity since

29

Si NMR probes local environment and XRD probes coherently scattering domains.

Table 6 :

 6 Experimental determination of the dipolar coupling constants k d 2 jk (±10%) obtained from 31 P-31 P DQ-DRENAR experiments.

	Sample	k d 2 jk x10 5 Hz 2 ±10%
	LAS0.2-P1 as-cast glass	0.33
	LAS0.2-P1 550°C/1h	0.89
	LAS0.2-P1 900°C/1h	2.9
	γ-Li 3 PO 4	3.3

volume fraction Φ obj of 6.5±0.6% is obtained.

Acknowledgments

We would like to acknowledge the Characterization Science and Reliability group (CETC, Corning SAS). Thanks to Nicolas Menguy (IMPMC, Sorbonne Université) for TEM exper-

iments. The ANRT is acknowledged for PhD funding (P.G.) under the CIFRE Contract 2015/0529. The authors thank the Chevreul Institute (FR 2638) for its help in the development of this work. Chevreul Institute is supported by the "Ministère de l'Enseignement 435 Supérieur et de la Recherche", the "Région Hauts-de-France" and the "Fonds Européen de 29 Si NMR spectra for the glass heat treated at 575°C/1h.


Kamal Alaili

come

Simultaneous determination of thermal diffusivity and

thermal conductivity of a thin layer using double modulated thermal excitations

Simultaneous determination of thermal diffusivity and thermal conductivity of a thin layer using double modulated thermal excitations Kamal Alaili, 1, a) Jose Ordonez-Miranda, 1 and Younès Ezzahri 1

1 Institut Pprime, CNRS-U niversité de Poitiers, ISAE-ENSMA, Futuroscope Chasseneuil F-86962, France (Dated: 17 October 2019)

A theoretical model is developed to determine simultaneously and in different ways thermal diffusivity and thermal conductivity of thin layers. This is done by using the accurate expression of the temperature distribution derived from the parabolic heat equation when the front surface of the thin layer is excited by a periodic heat flux, while the rear surface is maintained at one of three different types of boundary conditions; modulated periodic heat flux, modulated temperature, or constant temperature. Our approach exploits the modulation frequencies at which the normalized front surface temperature reaches its first maximum and first minimum. It is shown that (i) these characteristic frequencies can be used to obtain the thermal diffusivity of the finite layer under three different types of boundary conditions. (ii) the ratio between the values of the maxima and minima of the temperature can be utilized to determine the thermal conductivity of the finite layer. These two thermal properties are sensitive to the nature of the boundary conditions as well as the modulation frequency of the heat excitation. This paper provides a theoretical basis for the determination of the thermal diffusivity and thermal conductivity of the finite layer using laser-based heating photothermal techniques.

I. INTRODUCTION

Frequency-domain and time-domain photothermal methods have been extensively utilized to determine the thermal properties of materials by comparing the theoretical results to the measured experimental data. [1][2][START_REF] Hess | Photoacoustic and Photothermal Phenomena: Proceedings of the 5th International Topical Meeting[END_REF][START_REF] Ordonez-Miranda | [END_REF] The principle of these spectroscopic methods is based on the measurement of the thermal profile at one surface of the sample, when the latter is illuminated by a laser beam of modulated intensity, which is optically absorbed and converted into heat at the surface of the sample. [START_REF] Mandelis | Diffusion-wave fields: mathematical methods and Green functions[END_REF] Two common photothermal techniques for detecting the signal are the photoacoustic spectrometry and infrared photothermal radiometry. [START_REF] Balderas-Lopez | [END_REF] Since the pioneering work of Rosencwaig and Gersho 7 in thermal wave phenomena, significant efforts have been devoted to analyze the amplitude and phase of the photothermal signal in a wide variety of physical configurations. [START_REF] Balderas-Lopez | [END_REF][START_REF] Mandelis | Semiconductors and electronic materials[END_REF][START_REF] Salazar | [END_REF] In general; the analysis of the photothermal signals may be complicated because they include the transfer function of the electronic instruments. [START_REF] Balderas-Lopez | [END_REF] In most applications, it is convenient to normalize the measured signal in order to eliminate the transfer function. The thermal properties are then obtained by fitting the experimental data to the appropriate theoretical models. A major application of photothermal experiments is related to measurement of the thermal diffusivity of materials. [10][11][12][13][14] Balderas-Lopeza and Mandelis 6 developed a self-normalization procedure in order to measure the thermal diffusivity of thin metallic layers using two photothermal methods: photoacoustic and radiometric detection. Based on this self-normalization technique, the authors showed that the a) corresponding author : Kamal.alaili@univ-poitiers.fr corresponding measurement procedures involve linear fits in the photothermally thin and/or thick limits. By using the photothermal radiometry technique, Depriester et al. 15 proposed a method, which does not involve the multiparameter-fit to simultaneously obtain the thermal diffusivity and effusivity of solid materials. This was done by utilizing the ratio of photothermal radiometry signals obtained with and without the substrate. They used two characteristic modulation frequencies at which the normalized amplitude reaches its maximum value and the normalized phase passes through zero to determine the thermal diffusivity of vitreous carbon and lead-itanatezirconate ceramic samples. More recently, Vales-Pinzon et al. 16 have determined the thermal properties of the glassy carbon sample using four characteristic modulation frequencies in two-layer systems based on the analysis of the amplitude and phase of the photothermal signal. In this latter work, the authors showed that these four characteristic frequencies are independent of the thermal properties of the layers, and their values can be used to determine the thermal diffusivity and the thermal effusivity of the illuminated layer. Even though their results are in good agreement with those reported in the literature, [START_REF] Balderas-Lopez | [END_REF]15,16 their approach is however limited to the two-layer configuration. Therefore, the simultaneous determination of thermal diffusivity and thermal conductivity of a thin film layer is very desirable. Besides, in the case of periodic boundary conditions, an interesting phenomenon comes to play; the occurrence and propagation of thermal waves. Extensive efforts have been devoted to analyze the propagation processes of these thermal waves with analytical and numerical methods. [17][18][19][20][21][22] In this regard, an interesting work was performed by Torii and Yang. 17 The authors, also used numerical methods to study the effect of laser heating on the propagation phenomenon of a thermal wave in a very thin film subjected to a symmetrical heat excitation on both sides. Marin 23 discussed the heat conduction in solids excited by a periodically time-dependent source, commonly encountered in photothermal experiments. Thermal wave transmission and reflection phenomena induced by a pulsed boundary heat flux in a two-layer slab with imperfect interface are studied numerically by Ramadan and Al-Nimr. 24 Ordonez-Miranda and Alvarado-Gil 18 investigated a one-dimensional thermal wave transport in a semi-infinite layer using the exact solution of dualphase-lag heat conduction model. Geiler et al. [START_REF] Geiler | Photoacoustic and Photothermal Phenomena III[END_REF] used double laser beams with different wave lengths in order to generate the differential frequency in the sample by photothermal modulation of the refractive index. In this context, Dietzel et al. [START_REF] Dietzel | [END_REF] developed the double modulated thermoreflectance microscopy of semi-conductor devices based on combined optical and electrical excitation. However, in these latter works, the problem of a semi-infinite layer has been studied extensively, and those with a finite layer do not offer analytical solutions. [START_REF] Mandelis | Diffusion-wave fields: mathematical methods and Green functions[END_REF]27 In this paper, we solve analytically the parabolic heat equation (PHE) in order to determine the thermal diffusivity and thermal conductivity for a thin film layer, in which both sides are excited with periodic sources.

The purpose of this study is to analytically solve the PHE for the temperature profile in a finite layer, when both sides are excited by a modulated heat source. It is shown from the theoretical analysis of the photothermal signal that the characteristic frequencies at which the temperature of the front surface reaches its first maximum and minimum can easily be determined. It is also shown that these characteristic frequencies can be used to obtain the thermal diffusivity of the finite layer. Moreover, the ratio between the values of the maxima and minima of the temperature can be utilized to extract the thermal conductivity of the finite layer. Our approach is applied to a finite layer when the front surface is excited by a periodic heat flux, while the rear surface is maintained at one of three different types of boundary conditions; modulated periodic heat flux, modulated temperature, or constant temperature (see also the supplementary material). The paper is organized as follows: Sec. II presents the formulation of the physical problem using the PHE, while its key features are presented and discussed in Sec. III. We finish our work with summary and concluding remarks in Sec. IV.

II. FORMULATION OF THE PROBLEM AND SOLUTIONS A. Governing equation and physical problem

Let us consider the configuration shown in Fig. 1, in which the layer system is confined between x = 0 and x = l. We assume that the layer has a uniform thickness and constant thermophysical properties. Initially, the layer is at temperature T (x, 0) = T 0 which is the ambient temperature. For time t > 0, the boundaries at x = 0 and x = l are excited with modulated heat sources at frequencies ω 1 and ω 2 , respectively. By assuming a one dimensional propagation through the layer, the thermal wave field at any position is given by:

∂T (x, t) ∂t -α ∂ 2 T (x, t) ∂x 2 = 0, (1)
where α = k/(ρc p ) is the thermal diffusivity, k is thermal conductivity, ρ is the density, and c p is the specific heat capacity of the material. For the sake of simplicity, we introduce the normalized position z, time τ , temperature θ, heat flux Q, and frequency Ω, respectively, defined as:

z = x/l, τ = t/(l 2 /α), θ = T /T 0 , Q = q/(kT 0 /l), Ω = ω/(α/l 2 ). (2)
After some simplifications, Eq. ( 1) can be written in a dimensionless form as:

∂θ(z, τ ) ∂τ - ∂ 2 θ(z, τ ) ∂z 2 = 0. (3)

B. Boundary and initial conditions

The boundary and initial conditions for the physical problem are:

∂θ(0, τ ) ∂z = f 1 (τ ) = - Q 1 2 [1 + cos(Ω 1 τ )], (4a)
θ(z, 0) = 1. ( 4b
)
where q 1 = F η (1 -R) q 0 , with F being a parameter determined by the optical, thermal and geometric properties of the finite layer, 28 η the efficiency at which the absorbed light is converted into heat, R the reflection coefficient of the surface at z = 0, and q 0 [Wm -2 ] the intensity of the laser beam. [29][30][START_REF] Almond | Photothermal science and techniques[END_REF] The surface z = 0 is illuminated by a modulated heat flux, which is the case when the opaque surface of a medium is uniformly excited by a laser beam of modulated intensity. [START_REF] Ordonez-Miranda | [END_REF] Furthermore, the opposite surface at z = 1 considered in this work will be studied for three separate cases:

• Case 1: A modulated temperature imposed on the right-hand side surface

θ(1, τ ) = f 2 (τ ) = 1 + mcos(Ω 2 τ ), ( 5
)
where m is a positive constant. The modulated temperature (Eq. ( 5)) may be fulfilled when the medium at z = 1 is in contact with a Peltier module producing an oscillating temperature. 18,[START_REF] Almond | Photothermal science and techniques[END_REF] In practice, the temperature oscillations of a surface can be set with a Peltier cell, 33 which typically operate with a resolution better than 0.1 K and modulation frequencies from 0.1 up to 100 Hz. 34 • Case 2: A modulated heat flux applied on the righthand side surface

∂θ(1, τ ) ∂z = f 2 (τ ) = Q 2 2 [1 + cos(Ω 2 τ )]. (6)
• Case 3: Two independent heat flux centered on two different frequencies Ω 1 and Ω 2 are applied at the surface z = 0. This type of excitation was applied by Grégoire et al. 35 to examine non-linear photothermal and photoacoustic phenomena. The opposite side at z = 1, is maintained at a constant temperature.

∂θ(0, τ ) ∂z = f 1 (τ ) = - Q 1 2 [1 + cos(Ω 1 τ )] - Q 2 2 [1 + cos(Ω 2 τ )], (7a) θ(1, τ ) = 1. (7b)
The above physical problem is described by a heat conduction equation with nonhomogeneous boundary conditions. In order to determine the temperature distribution within the layer, we make use of the solution structure theorem. It is worth mentioning, that this theorem is applicable only in the case of heat conduction problems with homogeneous boundary conditions. 19,36,[START_REF] Wang | Heat conduction: mathematical models and analytical solutions[END_REF] Hence, to use this theorem, the homogeneous problem should first be established.

C. Auxiliary homogeneous conditions

As mentioned previously, the above mathematical problem includes nonhomogeneous boundary conditions that need to be converted into a problem with homogeneous boundary conditions such that the solution structure theorem can be applied to determine the temeprature distribution. We assume that θ has the following form:

θ(z, τ ) = v(z, τ ) + w(z, τ ). ( 8
)
By using this transformation, the original problem can be expressed in a form which consists of a corresponding homogeneous problem with the function v(z, τ ), and an auxiliary function, w(z, τ ). Note that the auxiliary function w(z, τ ) depends on the type of boundary conditions.

For more details, the readers are referred to reference [37]. In the present work, we will focus only on case 1; the other cases are detailed in the supplementary material.

According to Eqs. (4a) and ( 5), w(z, τ ) is given by:

w(z, τ ) = f 1 (τ ) (z -1) + f 2 (τ ). (9)
By inserting Eq. ( 8) into Eqs. ( 3), (4a), (4b), and (5), we obtain the following system of equations:

                 ∂v(z, τ ) ∂τ - ∂ 2 v(z, τ ) ∂z 2 = H(z, τ ), ∂v(0, τ ) ∂z = 0, v(1, τ ) = 0, v(z, 0) = f (z), (10a) (10b) (10c) (10d)
where

H(z, τ ) = ∂ 2 w(z, τ ) ∂z 2 - ∂w(z, τ ) ∂τ = - Q 1 2 Ω 1 sin(Ω 1 τ )(z -1) + mΩ 2 sin(Ω 2 τ ), (11)
and

f (z) = θ(z, 0) -w(z, 0) = Q 1 (z -1) -m. (12)
Note that after the transformation procedure, the new function H(z, τ ) appearing in Eq. (10a) plays a role similar to that of an internal heat source.

D. Complete solution of the non-homogeneous problem

The system of Eqs. (10a)-(10d) is homogeneous, and the solution can now be determined by utilizing the solution structure theorem. According to the superposition principle, we can write Eqs. (10a)-(10d) as two subproblems relatives to v 1 (z, τ ) and v 2 (z, τ ), respectively.

• Sub-problem 1: homogeneous equation and boundary conditions.

                 ∂v 1 (z, τ ) ∂τ - ∂ 2 v 1 (z, τ ) ∂z 2 = 0, ∂v 1 (0, τ ) ∂z = 0, v 1 (1, τ ) = 0, v 1 (z, 0) = f (z). ( 13a
) (13b) (13c) (13d)
• Sub-problem 2: inhomogeneous equation and homogeneous initial and boundary conditions.

                 ∂v 2 (z, τ ) ∂τ - ∂ 2 v 2 (z, τ ) ∂z 2 = H(z, τ ), ∂v 2 (0, τ ) ∂z = 0, v 2 (1, τ ) = 0, v 2 (z, 0) = 0. ( 14a
) (14b) (14c) (14d)
By using the method of Fourier series, Eq. (13a) has the following solution:

v 1 (z, τ ) = +∞ n=0 v n (τ ) [cos(λ n z) + sin(λ n z)] . (15)
With the application of the first boundary condition (Eq. (13b)), Eq. ( 15) reduces to

v 1 (z, τ ) = +∞ n=0 v n (τ )cos(λ n z). (16)
By appling the second boundary condition (Eq. (13c)), the eigenvalues in Eq. ( 16) take the form:

λ n = (n + 1/2)π. ( 17
)
Substituting Eq. ( 16) into Eq. (13a) leads to

dv n (τ ) dτ + λ 2 n v n (τ ) = 0. ( 18
)
The solution of Eq. ( 18) is straightforward and is given by:

v n (τ ) = a n e -λ 2 n τ . (19)
Therefore, Eq. ( 16) can be expressed as

v 1 (z, τ ) = +∞ n=0 a n e -λ 2 n τ cos(λ n z). (20)
The coefficients a n can be calculated by appling the initial condition (see Eq. (13d)). After some mathematical manipulations, we obtain:

a n = 2 1 0 f (z)cos(λ n z)dz. (21)
Inserting Eq. ( 12) into Eq. ( 21), leads to

a n = - 2Q 1 (1/2 + n) 2 π 2 - 2m(-1) n (1/2 + n) π . ( 22
)
For the second sub-problem, v 2 (x, τ ), the solution can be obtained by using the solution structure theorem, [START_REF] Wang | Heat conduction: mathematical models and analytical solutions[END_REF] which takes the form:

v 2 (z, τ ) = τ 0 1 0 G (z, ξ, τ -τ ) H(ξ, τ )dξdτ , (23)
where G (z, ξ, τ -τ ) is the Green function, that is defined by:

G (z, ξ, τ -τ ) = +∞ n=0 2e -λ 2 n (τ-τ ) cos (λ n ξ) cos (λ n z) . (24)
By combining Eqs. ( 11), (23), and (24), we get the following solution for the second sub-problem

v 2 (z, τ ) = +∞ n=0 [A n ψ n,1 (τ ) + B n ψ n,2 (τ )] cos(λ n z), (25)
where

A n = Q 1 (1/2 + n) 2 π 2 1 + χ 2 n,1 , (26a)
B n = 2m (-1) n (1/2 + n) π 1 + χ 2 n,2 , (26b)
ψ n,j (τ ) = e -λ 2 n τ -cos (Ω j τ ) + χ n,j sin (Ω j τ ) (j = 1, 2), (26c)
χ n,j = λ 2 n Ω j . ( 26d
)
The complete temperature distribution inside the finite layer is thus the sum of Eqs. ( 9), (20), and (25), which leads to:

θ(z, τ ) = w(z, τ ) + v 1 (z, τ ) + v 2 (z, τ ). ( 27
)
For any of the modulated thermal excitations, the temperature distribution at any point of the medium can be written as follows 28 θ(z, τ

) = 1 + θ dc (z) + θ ac (z, τ ), (28)
where the unity corresponds to the ambient temperature, θ dc (z) and θ ac (z, τ ) are the stationary and periodic components of the temperature, respectively. These last two terms are associated to the first and second terms of the thermal source excitation, respectively. The combination of Eqs. ( 27) and ( 28) fully determines the expressions of θ dc (z) and θ ac (z, τ ). Equation ( 28) satisfies all the imposed boundary conditions along with the temperature differential equation and therefore it is the correct solution for the temperature profile, as established by the fundamental theorem of differential equations. [START_REF] Spiegel | Applied differential equations[END_REF] In addition, in the limit Ω 2 = 0 and m = 0, Eq. ( 27) reduces to previous results reported by Rodney and Diana. [START_REF] Hechavarría | AIP Conference Proceedings[END_REF] Taking into account that the transient heat conduction is driven by both the thermal diffusivity and thermal effusivity (thermal conductivity), [START_REF] Balderas-Lopez | [END_REF]16,18 their simultaneous determination is therefore desirable.

III. DETERMINATION OF THE THERMAL PROPERTIES

In this section, the explicit expressions for the thermal diffusivity and thermal conductivity are obtained using the complete temperature distribution inside the material (Eq. ( 27)). Given that this temperature distribution oscillates in time with respect to the boundaries conditions (Eqs. (4a) and ( 5)), we calculate the average temperature over a period Γ 2 (Γ 2 = 2π/Ω 2 ).

The average temperature over Γ 2 is given by:

θ(z) = 1 Γ 2 Γ2 0 θ(z, τ )dτ. ( 29
)
Even though most photothermal techniques use lockin amplifiers to measure the amplitude and phase of the temperature field averaged over many heating periods, [START_REF] Balderas-Lopez | [END_REF]15 the time variations of the physical temperature and its average value θ can also be visualized and recorded through an oscilloscope, for different modulation frequencies, as reported by S. Dilhaire et al. [START_REF] Dilhaire | [END_REF] After inserting Eq. ( 27) into Eq. ( 29), the following expression for θ(z) is obtained

θ(z) = 1 - Q 1 2 1 + Ω 2 2πΩ 1 sin 2πΩ 1 Ω 2 (z -1) + Ω 2 2π +∞ n=0 a n + A n + B n λ 2 n + 2A n Ω 1 sin πΩ 1 Ω 2 • λ 2 n Ω 1 sin πΩ 1 Ω 2 -cos πΩ 1 Ω 2 cos (λ n z) . ( 30
)
As in the case of the temperature, the average heat flux over Γ 2 at the illuminated surface is straightforward and is given by:

Q = Q 1 2 1 + Ω 2 2πΩ 1 sin 2πΩ 1 Ω 2 . ( 31
)
Note that both Eqs. ( 30) and ( 31) can be written as the sum of a steady state and modulated components. According to Eq. ( 30), the temperature field expression depends on the thermal diffusivity α and thermal conductivity k of the material, and therefore this expression can be used to determine these thermal properties. The determination of α was already done by using one frequency modulated excitation source applied at the surface z = 0. 15,16 In the present work, both thermal diffusivity and thermal conductivity are obtained based on the expression of the average temperature over a period Γ 2 (Eq. ( 30)). This is achieved by means of a meticulous graphical examination of the features of this average temperature at its first maximum and first minimum values. The normalized average temperature θ and heat flux Q at the illuminated surface z = 0 as functions of the normalized frequency Ω 1 are shown in Figs. 2(a)-2(b), respectively; for two values of Ω 2 . Note that the average temperature presents the typical behavior of traditional thermal wave phenomena for Ω 2 = 0.1. When the modulation frequency increases (Ω 1 1), the temperature oscillations manifest a strong attenuation. Furthermore, the penetration depth of the temperature oscillation (thermal wave) which depends on the modulation frequency of the heat source (µ 1 = 2α/ω 1 ) reduces as the excitation frequency increases. These results agree well with the the work reported by Torii and Yang 17 and Ordonez-Miranda et al. 28 . In the intermediate regime, the average This was expected, with respect to Eqs. (4a) and (5). For low frequencies ( Ω 1 1), the behavior of the temperature and heat flux takes a constant value and becomes independent of the modulation frequency with respect to the steady-state conditions. The difference between the first maximum and first minimum of the temperature is respectively, 3.43 K and 1.71 K for Ω 2 = 0.1 and Ω 2 = 4. Experimentally, this difference is easily measurable with (T 1,max -T 1,min ) ≥ 1 K. We obtain similar results for the Case 2 and Case 3. We provide more details in the supplementary material. According to complex the derivations in Sec. II, it is almost impossible to get analytically the expressions of the first maximum and minimum of the temperature as well as the frequencies at which they occur. The numerical solution of Eq. ( 30) makes it possible to calculate the frequencies at which the temperature reaches its first maximum and minimum, as shown in Fig. 3. In this figure, we plot the frequencies at which the temperature reaches its first maximum and minimum as functions of the normalized frequency Ω 2 . We note these frequencies Ω 1,max and Ω 1,min , respectively. These frequencies increase linearly as functions of Ω 2 which is consistent with the results shown in Figs. 2(a)-2(b). Furthermore, the relationship between Ω 1 and Ω 2 , where the normalized average temperature reaches its first maximum and first minimum values can be obtained through fitting. Given that for small values of Ω 2 (0.001 ≤ Ω 2 < 1), Ω 1,max and Ω 1,min varies linearly with Ω 2 (see Fig. 3 (a)). The results obtained through fitting lead to Ω 1,max ≈ (9/7)Ω 2 and Ω 1,min ≈ (3/4)Ω 2 . Therfore, the proposed method based on subtracting the minimum and maximum frequencies cannot be used to determine α. By contrast, in the high frequency regime (Ω 2 > 1), the first minimum and maximum of θ occur for a linear dependence between Ω 1 and Ω 2 (see Fig. 3 (b)) and we can write

1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 .
       Ω 1,min ≈ 9 11 Ω 2 - 1 17 , θ = θ 1,min , Ω 1,max ≈ 15 11 Ω 2 - 2 35 , θ = θ 1,max , (32a) (32b)
where Ω j = l 2 ω j /α. Given that the Eqs. (32a) and (32b) are derived for the modulated components of the applied heat flux and temperature in Eqs. (4a) and (5), they are valid for high frequency regime (Ω 2 > 1). Note that Eqs. (32a) and (32b) are independent of the particular values of m and Q 1 . The variation of these parameters changes the temperature θ 1,max and θ 1,min but not the modulation frequencies of their minima and maxima. One can obtain the thermal diffusivity of the sample by combining Eqs. (32a) and (32b), as follows:

α ≈ πl 2 (48.9f 1,max -81.5f 1,min ) , (33)
where ω j = 2πf j . Therefore, Eq. ( 33) will lead to obtaining the thermal diffusivity of the sample, by performing a simple frequency scan to determine the modulation frequencies where the average temperature at the illuminated surface z = 0 reaches its first minimum and maximum values. According to Eq. ( 33), the thermal diffusivity α is highly sensitive to the frequencies f 1,max and f 1,min at which the maximum and minimum values of the temperature field occur, respectively. In practice, the accurate determination of α will thus need to repeat the required experiments a number of times to measure representative values of these two frequencies together with their uncertainties. According to Eq. ( 30), the average temperature over a period Γ 2 at the illuminated surface z = 0 can be conveniently rewritten as follows: FIG. 3: Behavior of Ω 1,max Ω 1,min as functions of Ω 2 for Case 1. Calculations have been made for l = 10 -3 m, k = 100 Wm -1 K -1 , T 0 = 300 K, m = 1/15, and q 1 = 2.10 6 Wm -2 . Dashed lines correspond to the linear fit and solid one correspond to the numerical solution of Eq. (30).

θ(0) = 1 + Q 1 F 1 (Ω 1 , Ω 2 ) + F 2 (Ω 2 ), (34)
where

F 1 (Ω 1 , Ω 2 ) = 1 2 1 + Ω 2 2πΩ 1 sin 2πΩ 1 Ω 2 + Ω 2 2π +∞ n=0 a n + A n λ 2 n + 2A n Ω 1 sin πΩ 1 Ω 2 . λ 2 n Ω 1 sin πΩ 1 Ω 2 -cos πΩ 1 Ω 2 , (35)
1 . 0

1 . 1 1 . 2 1 . 3 1 1 0 1 0 0 Ω 2 = 0 . 1 Ω 2 = 1 Ω 2 = 4 T h e r m a l c o n d u c t i v i t y , k ( W m -1 K -1 ) ؑ 1 , m a x / ؑ 1 , m i n
FIG. 4: The computed behavior of the ratio θ 1,max /θ 1,min between the first maximum and minimum values of the temperature for Case 1 as a function of thermal conductivity k for three values of Ω 2 . Calculations have been made for l = 10 -3 m, T 0 = 300 K, m = 1/15, and q 1 = 2.10 6 Wm -2 . and

           F 2 (Ω 2 ) = Ω 2 2π +∞ n=0 1 λ 2 n B n - 2m(-1) n (1/2 + n) π , a n = - 2 (1/2 + n) 2 π 2 and A n = A n /Q 1 . (36a) (36b)
Given that the detected photothermal signal is proportional to the modulated temperature defined in Eq. ( 34), it is convenient to normalize the measured maximum signal by the minimum one. By doing so, we can write:

θ 1,max θ 1,min = 1 + Q 1 F 1 (Ω 1,max , Ω 2 ) + F 2 (Ω 2 ) 1 + Q 1 F 1 (Ω 1,min , Ω 2 ) + F 2 (Ω 2 ) , (37)
where Ω 1,min and Ω 1,max are related to Ω 2 by Eqs. (32a) and (32b), respectively. After simplifications, Eq. ( 37) yields

k = q 1 .l T 0 F 1 (Ω 1,max , Ω 2 ) -θ 1,max /θ 1,min F 1 (Ω 1,min , Ω 2 ) θ 1,max /θ 1,min -1 (1 + F 2 (Ω 2 )
) .

(38) Note that k depends on the ratio θ 1,max /θ 1,min and the absorbed heat flux q 1 , which can be considered nearly constant and independent of the modulation frequency, as is usually done in similar problems. 29,[START_REF] Almond | Photothermal science and techniques[END_REF] Equations ( 33) and (38) indicate that the simultaneous determination of both the thermal diffusivity and thermal conductivity of a layer only requires the measurement of two characteristic frequencies of the temperature signal. This fact represents one of the advantages of the proposed method with respect to other photothermal ones, such as thermoreflectance, which requires recording the amplitude and phase of the temperature field over wide intervals of modulation frequencies to retrieve the same properties. 41 The determination of Ω 1,max and Ω 1,min and their corresponding temperature magnitudes θ 1,max and θ 1,min allows therefore the extraction of the thermal conductivity k of the thin film layer trough application of Eq. (38). The precision of k will depend on the precision of these input parameters as well as the number of terms considered in functions F 1 and F 2 . We report in Fig. 4 the computed behavior of the thermal conductivity k as a function of the ratio between the first maximum and first minimum values of the temperature θ 1,max /θ 1,min . Note that k depends strongly on the ratio θ 1,max /θ 1,min as well as the normalized frequency Ω 2 . k decreases as the ratio θ 1,max /θ 1,min increases and when the latter approaches unity, k tends to infinity. We notice also that the ratio θ 1,max /θ 1,min takes higher values for low frequencies Ω 2 ; the determination of the thermal conductivity k becomes more precise.

Taking into account that the Biot number Bi = hl/k represents the fraction of the thermal resistance (l/k) of the material that opposes the heat convection characterized by the heat transfer coefficien h, the convective losses are expected to be negligible for good thermal conductors Bi 1 excited with relatively high modulation frequencies, as demonstrated by Martinez et al. 42 In addition, given that the proposed method involves temperature gradients of a few degrees only (see Fig. 2), the radiation effects can also be neglected during the realization of the required experiments, as established by the Stefan-Boltzmann law of heat radiation. Finally, it is worthwhile to mention that even though all our results have been obtained for a heating with a sinusoidal profile, the developed analytical methodology is also expected to be valid for other profiles, as is the case of the squarewave one that is commonly used in chopped experiments.

IV. CONCLUSION

A simple method for determining simultaneously the thermal diffusivity and thermal conductivity of a thin layer has been developed, based on the analytical solution of the one-dimensional parabolic heat equation. This is achieved by using the exact expression of the normalized temperature at the illuminated front surface of a thin layer when a periodic heat flux is applied at this surface, while the rear surface is maintained at one of three different types of boundary conditions; modulated heat flux, modulated temperature, or constant temperature. We have determined the frequencies at which the normalized surface temperature reaches its first maximum and first minimum. It has been demonstrated that these characteristic frequencies can be used to determine the thermal diffusivity of the finite layer. Furthermore, the combination of these frequencies and their corresponding temperature magnitudes, can easily lead to the determination of the thermal conductivity of the thin layer. Once the thermal diffusivity and thermal conductivity are known, one could extract the specific heat per unit volume. Therefore, application of the suggested method to a single photothermal experiment would allow determination of a set of three thermal properties of the thin layer.

SUPPLEMENTARY MATERIAL

For more details are provided in the supplementary material.

FIG. 1 :

 1 FIG. 1: Schematic diagram of the studied finite thickness layer of thermal conductivity k and thermal diffusivity α. The modulated thermal excitations are applied at x = 0 and x = l.

Ω 2 = 0 . 1 NFIG. 2 :

 212 FIG. 2: Normalized average temperature and heat flux at the illuminated surface z = 0 for Case 1 as functions of the normalized frequency Ω 1 for (a) Ω 2 = 0.1 and (b)Ω 2 = 4. The up down arrow corresponds to the difference between the first minimum and first maximum of the temperature. Calculations have been made for l = 10 -3 m, k = 100 Wm -1 K -1 , T 0 = 300 K, m = 1/15, and q 1 = 2.10 6 Wm -2 .

5  ؑ 1

 51 m a l i z e d f r e q u e n c i e s , Ω 1 , m a x , Ω 1 , m i n N o r m a l i z e d f r e q u e n c y , , m a x  ؑ 1 , m i n N o r m a l i z e d f r e q u e n c i e s , Ω 1 , m a x , Ω 1 , m i n N o r m a l i z e d f r e q u e n c y , Ω 2 E q u a t i o n P l o t W e i g h t I n t e r c e p t S l o p e R e s i d u a l S u m o f S q u a P e a r s o n ' s r R -S q u a r e ( C O D ) A d j . R -S q u a r e ( b )