Haytem Bazzaoui

Cécile Genevois

Emmanuel Véron

Michael J Pitcher

Mathieu Allix

Alessio Zandonà
email: alessio.zandona@cnrs-orleans.fr

Towards new zero-thermal-expansion materials: Li-free quartz solid solutions stuffed with transition metal cations

Keywords: stuffed quartz solid solutions, zero thermal expansion, aerodynamic levitation, transition metal cations, glass crystallization

Stuffed aluminosilicate quartz solid solutions (Qss) represent the main functional component of state-ofthe-art zero-thermal-expansion glass-ceramics. For the first time, we present the synthesis of Li-, Mg-and Zn-free Qss solely stuffed with transition metal cations (Fe 2+ , Ni 2+ , Co 2+ ); partial Li + co-doping enables also significant Mn 2+ incorporation. They were obtained by glass powder crystallization; Qss crystals exhibit compositionally tunable coefficients of thermal expansion (CTEs), from -2x10 -6 K -1 to ~10x10 - 6 K -1 in the range 30-300 °C. Co 2+ -bearing crystals exhibit the closest-to-zero CTE value (0.2x10 -6 K -1 ) and the most isotropic behavior in the whole Qss family, opening up new perspectives for the development of Li-free low-expansion materials. From a structural point of view, we identified the unit cell volume (and not the pseudo-hexagonality of the aluminosilicate framework) as the key structural parameter leading to low or negative CTEs in Qss, with a linear correlation extending to non-stuffed nonsilicate quartz-like phases.

1.Introduction

Low quartz (or α-quartz, SiO 2 ) represents the most common crystalline component of Earth's continental crust. Its structure (enantiomorphic space groups: P3 1 21 or P3 2 21) consists of interconnected helical chains of SiO 4 tetrahedra (Fig. S1 of the Supplementary Materials), defining continuous trigonal channels running along the c-axis [START_REF] Bragg | The structure of α and β quartz[END_REF]. This framework transforms reversibly upon heating above 573 °C, assuming hexagonal symmetry (high or β-quartz, enantiomorphic space groups: P6 4 22 or P6 2 22).

Notably, the same temperature marks a striking transition from positive to almost null thermal expansion, as reported by Le Châtelier at the end of the XIX century [START_REF] Chatelier | Sur la dilatation du quartz[END_REF].

Although its piezoelectric properties have been essential for metrology and telecommunications since the 1930s [START_REF] Ballato | Basic Material Quartz and Related Innovations[END_REF], the technological interest of quartz boomed by the 1960s, due to: (i) the discovery of the negative thermal expansion behavior [START_REF] Hummel | Thermal Expansion Properties of Some Synthetic Lithia Minerals[END_REF] of β-eucryptite (LiAlSiO 4 ); (ii) the identification of this phase as a stuffed derivative [START_REF] Buerger | The stuffed derivatives of the silica structures[END_REF] of the high quartz structure [START_REF] Winkler | Synthese und Kristallstruktur des Eukryptits[END_REF], in which Al 3+ isomorphously substitutes Si 4+ , enabling the incorporation of Li + at otherwise vacant channel sites; (iii) the serendipitous invention of glass-ceramics [START_REF] Stookey | Catalyzed Crystallization of Glass in Theory and Practice[END_REF], in which stuffed quartz solid solutions [START_REF] Petzoldt | Metastabile Mischkristalle mit Quarzstruktur mit Oxidsystem Li2O-MgO-ZnO-Al2O3-SiO2[END_REF][START_REF] Beall | Crystallization and Chemical Strengthening of Stuffed β-Quartz Glass-Ceramics[END_REF] (Qss) with generalized formula M x Al x Si 1-x O 2 (M = Li, Mg 0.5 and Zn 0.5 ) are easily synthesized by controlled glass crystallization [START_REF] Deubener | Updated definition of glass-ceramics[END_REF]. This combination rapidly led to the development of zero-thermal-expansion glass-ceramic materials, which are still today essential for applications spanning from transparent fire-viewing windows to telescope mirror substrates [START_REF]Low Thermal expansion Glass Ceramics[END_REF][START_REF] Beall | Design and Properties of Glass-Ceramics[END_REF]. Between 0 °C and 50 °C, commercially available materials based on Qss such as Zerodur ® can achieve a coefficient of thermal expansion (CTE) as low as (0 ± 0.007) x 10 -6 K -1 , with spatial variations that are less than 0.005 x 10 -6 K -1 over the whole volume of a telescope mirror blank with a diameter of 4 meters [START_REF] Hartmann | Glass ceramic ZERODUR®: Even closer to zero thermal expansion: a review, part 1[END_REF][START_REF] Hartmann | Glass ceramic ZERODUR®: Even closer to zero thermal expansion: a review, part 2[END_REF]. Indeed, the thermal expansion of Qss can be finely tuned, taking advantage of their variable Al/Si ratio and of the full solid solution [START_REF] Petzoldt | Metastabile Mischkristalle mit Quarzstruktur mit Oxidsystem Li2O-MgO-ZnO-Al2O3-SiO2[END_REF][START_REF] Beall | Crystallization and Chemical Strengthening of Stuffed β-Quartz Glass-Ceramics[END_REF][START_REF] Ray | Study of Ordering in High-Quartz Solid Solutions by Substitutions Affecting Superlattice Reflections[END_REF][START_REF] Ray | High-Quartz Solid Solution Phases from Thermally Crystallized Glasses of Compositions (Li2O,MgO).Al2O3.nSiO2[END_REF] existing between: (i) Mgbearing Qss, exhibiting trigonal symmetry and positive thermal expansion [START_REF] Schreyer | Metastable solid solutions with quartz-type structures on the join SiO2-MgAl2O4[END_REF][START_REF] Sternitzke | Crystal structure and thermal expansion of quartz-type aluminosilicates[END_REF][START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF][START_REF] Zandona | Mg-bearing quartz solid solutions as structural intermediates between low and high quartz[END_REF]; (ii) Li-and Znbearing Qss, which were reported to be fully hexagonal (at a sufficiently high level of chemical stuffing) and to contract upon heating [START_REF] Petzoldt | Metastabile Mischkristalle mit Quarzstruktur mit Oxidsystem Li2O-MgO-ZnO-Al2O3-SiO2[END_REF][START_REF] Beall | Design and Properties of Glass-Ceramics[END_REF][START_REF] Xu | Thermal expansion and structural transformations of stuffed derivatives of quartz along the LiAlSiO4-SiO2 join: a variable-temperature powder synchrotron XRD study[END_REF][START_REF] Zandona | Inversion of quartz solid solutions at cryogenic temperatures[END_REF][START_REF] Zandona | Spray-dried sol-gel glassceramic powders based on the tunable thermal expansion of quartz and keatite solid solutions[END_REF][START_REF] Müller | Hydrogen substitution in lithium-aluminosilicates[END_REF][START_REF] Li | The crystal structure of LiAlSi2O6 III (high-quartz solid solution)[END_REF][START_REF] Behruzi | Struktur und thermische ausdehnung des ungeordneten β-Eukryptits[END_REF].

Qss additionally allow for a wide range of substitutions at their tetrahedral framework sites, also in absence of channel stuffing: quartz-like BeF 2 , PON, GeO 2 , BPO 4 , AlPO 4 , FePO 4 , GaPO 4 , BAsO 4 , AlAsO [START_REF] Hummel | Thermal Expansion Properties of Some Synthetic Lithia Minerals[END_REF] and GaAsO 4 (referred to as "non-stuffed" in the following) were described in the past [START_REF] Shafer | Studies of Silica-Structure Phases: I, GaPO4, GaAsO4, and GaSbO4[END_REF][START_REF] Shafer | Studies of Silica Structure Phases II: Data on FePO4, FeAsO4, MnPO4, BPO4, AlVO4 and others[END_REF][START_REF] Schwarzenbach | Verfeinerung der Struktur der Tiefquarz-Modifikation von AlPO4[END_REF][START_REF] Kosten | Die III-V-Analoga des SiO2[END_REF][START_REF] Smith | The crystal structure of quartz-like GeO2[END_REF][START_REF] Wright | The preparation and structure of the α-and β-quartz polymorphs of beryllium fluoride[END_REF][START_REF] Haines | The use of composition and high pressure to extend the range of α-quartz isotypes[END_REF],

although only P 5+ incorporation has found application in glass-ceramics [START_REF] Beall | Crystallization and Chemical Strengthening of Stuffed β-Quartz Glass-Ceramics[END_REF][START_REF]Low Thermal expansion Glass Ceramics[END_REF][START_REF] Beall | Design and Properties of Glass-Ceramics[END_REF][START_REF] Petzoldt | Der Einbau von P2O5 in mestabile Mischkristalle mit Quarzstruktur des Grundsystems Li2O-MgO-ZnO-Al2O3-SiO2[END_REF]. Moreover, the available literature hints at an even wider range of compositional variability for Qss than systematically investigated so far. Although precise structural characterization is lacking, previous authors inferred the fortuitous formation of SiO 2 -rich Qss stuffed to a limited extent with Fe 2+ and Cu 2+ ions during the reaction between metal oxides and kaolinite [START_REF] Schreyer | Metastable solid solutions with quartz-type structures on the join SiO2-MgAl2O4[END_REF][START_REF] Segnit | Metastable Quartz-Type Structures Formed from Kaolinite by Solid State Reaction[END_REF][START_REF] Takeuchi | Formation of Quartz Solid Solution on Firing Kaolinite-CuO Mixtures and Its Structure[END_REF][START_REF] Takeuchi | Mechanistic Study of Solid-State Reaction between Kaolinite and Ferrous Oxide at High Temperatures[END_REF]; concurrently, minor accommodation (at doping levels) of Mn 2+ , Co 2+ and Ni 2+ has been inferred spectroscopically in lithium aluminosilicate Qss crystals formed in glass-ceramics [START_REF] Beall | Crystallization and Chemical Strengthening of Stuffed β-Quartz Glass-Ceramics[END_REF][START_REF] Kang | Structural states of Co(II) in β-eucryptite-based glass-ceramics nucleated with ZrO2[END_REF][START_REF] Dymshits | Structural states of Ni(II) in glasses and glass-ceramic materials of the lithium-aluminium-silicate system[END_REF]. We similarly assumed partial incorporation of Ti 4+ in the tetrahedral sites of Li-bearing Qss (Al/Si ratio equal to 1) after devitrification [START_REF] Zandonà | Glass formation and devitrification behavior of alkali (Li, Na) aluminosilicate melts containing TiO2[END_REF]. Finally, Qss with formulas HAlSi 2 O 6 , Na 0.5 H 0.5 AlSi 2 O 6 and Al 0.27 Al 0.82 Si 2.18 O 6 were obtained by ion exchange and leaching [START_REF] Müller | Hydrogen substitution in lithium-aluminosilicates[END_REF][START_REF] Müller | Synthesis and structure of β-quartz type Na0.5H0.5AlSi2O6 as compared to LiAlSi2O6[END_REF].

In this study, we demonstrate the first deliberate synthesis of aluminosilicate Qss solely stuffed with Co 2+ , Ni 2+ and Fe 2+ ; we additionally explored simultaneous stuffing with two transition metals (i.e.

Co 2+ and Ni 2+ ) and exploited Li + co-doping to introduce also Mn 2+ in the Qss structure. We obtained such crystalline phases through controlled crystallization of aluminosilicate glasses containing high amounts of transition metal oxides, produced by aerodynamic levitation coupled to laser heating (ADL). The crystallographic characterization of these phases and the elucidation of their thermal expansion behavior enabled us to develop an unprecedentedly complete description of structure-property relationships in this class of materials. As for Co-bearing Qss in particular, their CTE was found to be the closest to zero and the most isotropic in the system, opening up alternative scenarios for the sustainable production of Li-free low-expansion materials in a global economic system that features increasing pressure on lithium resources for battery production [START_REF] Xu | Future material demand for automotive lithium-based batteries[END_REF].

Experimental

Glass synthesis and crystallization

The main target stoichiometry was Laboratory-grade precursors were used for the syntheses: SiO 2 (99.9%, Chempur), Al 2 O 3 (99.999%, Strem Chemicals), Li 2 CO 3 (99%, Aldrich), MgO (99.5%, Strem Chemicals), ZnO (99.999%, Strem Chemicals), Co 3 O 4 (99.9985%, Alfa Aesar), NiO (99.9%, Aldrich), MnO 2 (99.9%, Alfa Aesar), FeO (99.9%, Aldrich). The raw powder materials were dispersed in ethanol, mixed thoroughly in an agate mortar and pressed into pellets of 1 g. Small chunks (20-40 mg) were then melted using the ADL system already detailed elsewhere [START_REF] Zandonà | Glass-forming ability and ZrO2 saturation limits in the magnesium aluminosilicate system[END_REF], consisting of two CO 2 lasers (maximum power: 250W) heating the sample from above and below while it levitates on an Ar gas flow. The laser power was increased until full melting was achieved, kept for 5 seconds for homogenization and then cut off to quench the melt into glass beads. The maximum achieved temperatures were approximately 1800-2000 °C as from pyrometer reading; no evident signs of volatilization (e.g. visible smoke or material deposition on the nozzle walls) were detected using this procedure.

MO/M 2 O · Al 2 O 3 · 2.
Crystallization heat treatments were applied to powders obtained after grinding together several melt-quenched beads in an agate mortar. For most of the samples, annealing was performed in air in a laboratory furnace at temperatures carefully selected (summary in Tab. 1) by trial and error to maximize the amount and unit cell volume of Qss (i.e. their degree of structural stuffing) while minimizing the appearance of secondary phases, also based on previous works on Li-and Mg-bearing Qss [START_REF] Zandona | Mg-bearing quartz solid solutions as structural intermediates between low and high quartz[END_REF][START_REF] Zandona | Inversion of quartz solid solutions at cryogenic temperatures[END_REF]. FeQss proved more sensitive to the redox conditions and could therefore only be transformed into Qss when the heat treatments took place in vacuum: we particularly used an Anton Paar's HTK1200N heating chamber attached to a diffractometer to monitor the process, quickly cooling the sample after the formation of Qss to avoid its decomposition.

We also unsuccessfully explored the possibility of crystallizing Qss stuffed with other transition metal cations (Cr and Cu): in the case of chromium, the strong liquid immiscibility in the system [START_REF] Degterov | Critical evaluation and optimization of the thermodynamic properties and phase diagrams of the CrO-Cr2O3-SiO2 and CrO-Cr2O3-SiO2-Al2O3 systems[END_REF] prevented the synthesis of a homogeneous aluminosilicate glass even with the addition of some Li 2 O; as for copper, our synthesis conditions (even trying with an O 2 atmosphere) appeared to stabilize Cu + rather than Cu 2+ and, after adjusting the stoichiometry, we were only able to obtain keatite solid solutions (another aluminosilicate stuffed derivative of SiO 2 [START_REF] Keat | A New Crystalline Silica[END_REF][START_REF] Li | The crystal structure of LiAlSi2O6-II ("β spodumene")[END_REF]) as described by previous authors [START_REF] Kamiya | Structural changes of glass-ceramics of the Cu2O-Al2O3-SiO2 system on heating in air[END_REF].

Powder diffraction

Laboratory X-ray powder diffraction (PXRD) measurements were performed in Bragg-Brentano geometry on a D8 Advance Bruker diffractometer (Cu Kα radiation) equipped with a LynxEye XE line detector. The samples were either dispersed on low-background flat Si holders using an ethanol mull, or compacted in front-loading Si holders, according to the available amount of powder. To estimate the amorphous fraction in sample CoQss, a weighed-in CaF 2 fully crystalline standard was used. Samples ZnQss, CoQss, NiQss and FeQss were additionally characterized by synchrotron powder diffraction (SPD) using the mail-in service of the 11BM beamline at the Advanced Photon Source at Argonne National Laboratory (USA). The powders were loaded into 0.8 mm Kapton capillaries and measured in the range 0-50° 2θ (calibrated wavelength: 0.458162 Å) with increments of 0.001° 2θ. CoQss was characterized by neutron powder diffraction (NPD) on the instrument MEREDIT at the Nuclear Physics Institute of the Czech Academy of Sciences (λ = 1.46 Å, range 4-144° 2θ, 0.08° increments, total measuring time ~27 h). For this specific measurement, approximately 4 g of powder were obtained by crushing numerous amorphous beads synthesized by ADL, subsequently crystallizing them at 850 °C for 45 min.

The thermal expansion of Qss phases contained in the samples was determined by variable temperature PXRD (VT-PXRD) using an Anton Paar HTK1200N furnace mounted on a D8 Advance Bruker laboratory diffractometer, acquiring a diffractogram every 50 °C between 30 and 600 °C. Silicon powder was added to the samples as an internal standard, to correct at each temperature for the height displacement due to the dilatation of the Al 2 O 3 sample holder; pure quartz (99.9% SiO 2 , Chempur) was measured in the same way for comparison. Lattice parameters were extracted from PXRD data by Le Bail refinements, performed with the software HighScore Plus (Panalytical); full structural refinements of SPD and NPD data were carried out using TOPAS Academic (v.6) [START_REF] Coelho | TOPAS and TOPAS-Academic: an optimization program integrating computer algebra and crystallographic objects written in C++[END_REF].

Transmission electron microscopy (TEM)

Electron energy loss spectroscopy (EELS) measurements were performed on Qss crystallites from samples CoQss, NiQss, and MnLiQss. A JEOL ARM200F cold FEG TEM operating at 80 kV was used due to the very limited stability of the samples under the electron beam. Small amounts of powder were ground and suspended in ethanol in an agate mortar; a small droplet of each suspension was then deposited onto a copper grid layered by a holey amorphous carbon film. After quick exploration in TEM mode, the samples were characterized in scanning mode (STEM). Energy-dispersive X-ray spectroscopy (EDX) was used to confirm the composition of the areas to be measured, which invariably matched the nominal one within uncertainty. Subsequently, we acquired EELS spectra (dispersion: 0.4 eV/channel) simultaneously at low (zero-loss-peak) and high (the target absorption edge) energy loss, integrating data over raster maps of several nanometers to minimize the effects of beam damage. The thickness-towavelength ratio during EELS spectra was kept in the range 0.4-0.6 to minimize plural scattering events.

During the processing, EELS spectra were realigned according to the respective zero-loss peak and deconvoluted to correct plural scattering events using the software DigitalMicrograph GMS 3 (Gatan). They were then qualitatively compared to the spectra obtained from available laboratory-grade single-valence simple oxides measured under the same conditions, namely LiCoO 2 , NiO, MnO and MnO 2 .

Results

Table 1. Overview of the synthesized samples, summarizing their nominal compositions, applied heat treatments, lattice parameters and PXRD-derived CTE values obtained for Qss (space group P3 2 21) and the presence of minor secondary phases (identified from PXRD data shown in Fig. 1 and SPD data in Fig. 4). Abbreviations: n.m. for "not measured", mul for mullite, sp for aluminate spinel, NiO for nickel (II) oxide, Ni for metallic nickel, mag for magnetite, un for unidentified secondary phase(s), rg for a substantial fraction of residual glass. Based on the negligible volume nucleation observed for Li-, Mg-and Zn-bearing Qss in glasses and melts devoid of nucleating agents [START_REF] Petzoldt | Metastabile Mischkristalle mit Quarzstruktur mit Oxidsystem Li2O-MgO-ZnO-Al2O3-SiO2[END_REF][START_REF] Zandona | Crystallization and SiAl ordering in cordierite glass-ceramics[END_REF][START_REF] Zandona | Chemical gradients at the surface of TiO2doped cordierite glass-ceramics[END_REF][START_REF] Hsu | Comparison of the Effects of Titania and Tantalum Oxide Nucleating Agents on the Crystallization of Li2O • Al2O3• 6SiO2 Glasses[END_REF], finely ground glass powders (Fig. S2) were subjected to very short heat treatments to induce instantaneous surface nucleation and growth of Qss phases, minimizing diffusion and phase separation tendency in the initially homogeneous glasses and avoiding thermal decomposition of the formed phases. As confirmed by PXRD (Fig. 1), this approach allowed Qss to be obtained as the main crystalline component in all samples listed in Table 1, despite the pronounced metastability of these phases. The applied chemical substitutions induced substantial changes in the relative intensities of their diffraction peaks, evident for example from the (010) and (110) reflections, respectively at ~19.5° and ~34° 2θ: while well-visible in LiQss and MgQss, these reflections almost completely disappeared in the samples stuffed with heavier transition metal cations. The chosen synthesis route led to a considerable amount of residual vitreous phase, as evidenced by some of the PXRD measurements shown in Figure 1: an estimation of the amorphous fraction in sample CoQss yielded 64 [START_REF] Winkler | Synthese und Kristallstruktur des Eukryptits[END_REF] wt% using a weighed-in CaF 2 standard, for instance. The most frequent secondary phases (between 2 and 10 wt% of the crystalline fraction, depending on the sample) included mullite and aluminate spinel, as expectable from the starting compositions and from previous investigations of Mg-bearing Qss [START_REF] Zandona | Mg-bearing quartz solid solutions as structural intermediates between low and high quartz[END_REF].

Sample

Although a-and/or c-doubling have been reported in the literature for Li-and Mg-bearing Qss and assigned to Al/Si ordering in the tetrahedra and non-random distribution of stuffing cations in the channels [START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Xu | Phase transitions induced by solid solution in stuffed derivatives of quartz: A powder synchrotron XRD study of the LiAlSiO4-SiO2 join[END_REF], low-temperature glass crystallization was shown to yield virtually fully disordered crystals [START_REF] Xu | Structural mechanisms underlying near-zero thermal expansion in β-eucryptite: A combined synchrotron x-ray and neutron Rietveld analysis[END_REF]. While PXRD measurements performed on LiQss and MgQss did not allow unambiguous identification of such structural features, SPD and NPD measurements (presented below) clearly excluded the occurrence of superstructure peaks in ZnQss, CoQss, NiQss and FeQss. We therefore based our further analysis uniquely on the substructure common to all Qss (space group P3 2 21), also to facilitate their comparison. We performed EELS measurements on Qss crystals contained in samples CoQss, NiQss and MnLiQss to investigate the oxidation state of the transition metal ions in the phases, although their instability under the electron beam and their relatively low metal contents led to spectra with poor signalto-noise ratios compared to the standards LiCoO 2 , NiO, MnO and MnO 2 (Fig. 2). The qualitative comparison with spectra collected from reference materials strengthened the assumption of a prevalently reduced state (Co 2+ and Ni 2+ ) in CoQss and NiQss, corresponding to a shift of the L 3 /L 2 edge of the transition metal to lower energies as compared to higher oxidation states [START_REF] Abraham | Microscopy and Spectroscopy of Lithium Nickel Oxide-Based Particles Used in High Power Lithium-Ion Cells[END_REF][START_REF] Tan | Oxidation state and chemical shift investigation in transition metal oxides by EELS[END_REF][START_REF] Schmid | Oxidation states of Mn and Fe in various compound oxide systems[END_REF]. As for the spectrum obtained from MnLiQss, the average manganese oxidation state was clearly higher than M 2+ , possibly accounting for the need to co-dope the material with Li + to induce the formation of Qss. The oxygen Kedge spectra (Fig. S3) were dominated by the interactions with Si 4+ and Al 3+ in the structure: they were therefore very similar in all three samples and markedly different from that of the reference transition metal oxides. Rietveld refinements were performed on SPD data acquired from ZnQss, CoQss, NiQss, and FeQss. Preliminary attempts using non-stuffed quartz models (i.e. trigonal or hexagonal frameworks with empty channel sites) expectedly resulted in evident peak intensity misfits; the substitution of the M 2+ cations (M = Zn, Co, Ni, Fe) for Li + in the hexagonal structure of Li-bearing Qss similarly led to poor fits (Tabs. S5-S16 of the Supplementary Materials). Consequently, we adopted a refinement strategy comparing the visual quality and goodness of fit (GOF) for trigonal frameworks in which M 2+ cations occupied: (i) solely the distorted 4-coordinated channel sites located at (0,0,0) as in Li-bearing Qss [START_REF] Li | The crystal structure of LiAlSi2O6 III (high-quartz solid solution)[END_REF], labelled as T ch in the following; (ii) only the distorted 6-coordinated channel sites located at (0,0,½) as in Mg-bearing Qss [START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Zandona | Mg-bearing quartz solid solutions as structural intermediates between low and high quartz[END_REF], labelled as O ch ; (iii) both T ch and O ch simultaneously and independently. The latter solution invariably yielded far better looking refinements (Fig. 4 and Tabs. S1-S4), showing the lowest GOF values and the best-fitting total site occupation factors (sof) as compared to the nominal composition of the samples (Fig. 5, theoretical M 2+ sof = 0.22). Although the GOF improvement was only marginal in ZnQss and FeQss as compared to structural models comprising only one M 2+ position (respectively T ch and O ch ), we refined all data using this same approach to favor the comparison across the four samples. Notice that mixed channel site occupation was also reported in the only previous structural refinement of Zn-bearing Qss [START_REF] Behruzi | Struktur und thermische ausdehnung des ungeordneten β-Eukryptits[END_REF]. We chose to fix all B iso thermal displacement parameters to the value of 1 Å 2 , which was in the range of those previously determined for all atoms by coupled SPD and NPD in a Qss of composition LiAlSiO 4 [START_REF] Xu | Structural mechanisms underlying near-zero thermal expansion in β-eucryptite: A combined synchrotron x-ray and neutron Rietveld analysis[END_REF]. In fact, our B iso refinement for the M 2+ cations invariably exhibited a strong correlation with their sof (observed from the correlation matrix); we estimated a variation by as much as ~15% and ~7% of the sof determined at T ch and O ch respectively (Fig. S4) when B iso values varied in the range 1-2 Å 2 . This uncertainty originated from the lack of information at high diffraction angle, likely produced by an inherent disorder of channel sites which also exhibited the geometrical irregularity expected from literature [START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Li | The crystal structure of LiAlSi2O6 III (high-quartz solid solution)[END_REF][START_REF] Xu | Phase transitions induced by solid solution in stuffed derivatives of quartz: A powder synchrotron XRD study of the LiAlSiO4-SiO2 join[END_REF]: O ch particularly represented a strongly distorted octahedron, with two short and two long M-O distances of ~1.9 Å and ~2.8 Å respectively. Bond valence sums (BVS) for the two differently coordinated sites (Tabs. S17-S20) mirrored this disorder, with values ranging between 1.312(1) and 1.825 [START_REF] Bragg | The structure of α and β quartz[END_REF]. Nevertheless, we detected an increasing preference of the M 2+ cations for the T ch site along the sequence FeQss -NiQss -CoQss-ZnQss; moreover, M-O distances at T ch increasingly converged towards a single value (Tab. S21), exhibiting therefore higher regularity as in fully hexagonal Li-bearing Qss [START_REF] Li | The crystal structure of LiAlSi2O6 III (high-quartz solid solution)[END_REF]. This tendency showed a linear correlation with the unit cell volume of the crystals (Fig. 6-a). CoQss was additionally characterized by NPD to achieve better spatial resolution on the oxygen positions. Due to the higher amount of powder required for reasonable counting times during such measurement, a bigger batch (~4 g) was synthesized, obtaining in this case a phase-pure sample. We first performed a combined lab PXRD/NPD Rietveld refinement fixing all atomic thermal parameters to the value of 1 Å 2 (Tab. S22), again due to the strong correlation between sof and thermal displacement parameters of Co 2+ . Due to the very similar neutron scattering factors of Al 3+ and Si 4+ (3.449(5) fm and 4.149(1) fm, respectively), we were unable to reliably refine the Al/Si ratio in the tetrahedral framework sites, hence we constrained their occupations according to the refined Co 2+ sof, to maintain charge balance in the structure. This approach yielded a good fit for lab PXRD, but intensity misfits were clearly observed for NPD data (R wp = 3.99% GOF= 2.37); the obtained structural model agreed well with that extracted from SPD, yielding similar Co-O distances and pseudo-hexagonality (δ = 8.4(3)°). In the quest of improving the quality of neutron data fit, we additionally refined the B iso parameter of the only nonequivalent oxygen atom, which increased to 2.93(7) Å 2 . Such a high value suggests a strong positional disorder of Co 2+ cations within the c-channels, inducing local displacements of the surrounding oxygen atoms. This structural model yielded a much better overall NPD data fit (Fig. S5 and Tab. S23) and lower reliability factors (R wp = 3.50% GOF= 2.08) without repercussions on the PXRD fit; this structure resulted interestingly far more hexagonal (δ = 4(1)°), revealing the sensitivity of the δ parameter to the applied refinement strategy.

VT-PXRD measurements were performed between room temperature and 600 °C (lattice parameters in Tabs. S24 and S25) to study the thermal expansion behavior of the synthesized Qss (Fig. 7)

and to compute their CTE in the range 30-300 °C (Tab. 1). All Qss exhibited a lower CTE than pure SiO 2 quartz (12.5 x 10 -6 K -1 ) but distributed over a wide range of values: while the unit cell of ZnQss decidedly contracted upon heating (-2.2 x 10 -6 K -1 ), LiQss and CoQss closely approached zero thermal expansion (-0.7 x 10 -6 K -1 and 0.2 x 10 -6 K -1 , respectively). All other Qss expanded during the measurements, with Co 0.5 Ni 0.5 Qss exhibiting the lowest positive CTE among them (1.9 x 10 -6 K -1 ). In all cases, expansion along the a-direction was at least comparable or substantially higher than along the c-axis (Fig. S6): as manifested by the relative changes in c/a ratios (Fig. 7-b), LiQss showed the least isotropic behavior, whereas CoQss, NiQss and FeQss retained the initial values throughout the measurements.

Discussion

Glass-ceramic powders containing Qss crystals stuffed with transition metal cations were synthesized for the first time within this work, relying on surface crystallization of glass powders and preferring short heat treatments to avoid the formation of more stable competing phases at these compositions, such as indialite/cordierite, spinel, non-stuffed quartz and cristobalite (see for instance Fig. S7 and previous literature [START_REF] Haefeker | Structural investigations of the two polymorphs of synthetic Fe-cordierite and Raman spectroscopy of hexagonal Fecordierite[END_REF][START_REF] Knorr | Structural and magnetic properties of Co2Al4Si5O18 and Mn2Al4Si5O18 cordierite[END_REF][START_REF] Serrano | Crystal structure and microstructure of synthetic hexagonal magnesium-cobalt cordierite solid solutions (Mg2-2xCo2xAl4Si5O18)[END_REF][START_REF] Sugimura | The Metastable Zn-Indialite Crystallized out from the Glasses in the Ternary System ZnO-Al2O3-SiO2[END_REF][START_REF] Terada | Effect of Ni substitution on the microwave dielectric properties of cordierite[END_REF][START_REF] Wandschneider | The System Magnesium Cordierite-Cobalt Cordierite[END_REF][START_REF] Torres | Phase evolution by thermal treatment of equimolar cobalt-magnesium cordierite glass powders[END_REF]). Qss is commonly obtained as the first metastable devitrification product in a variety of aluminosilicate glasses containing Li, Mg and/or Zn [START_REF] Petzoldt | Metastabile Mischkristalle mit Quarzstruktur mit Oxidsystem Li2O-MgO-ZnO-Al2O3-SiO2[END_REF][START_REF] Beall | Crystallization and Chemical Strengthening of Stuffed β-Quartz Glass-Ceramics[END_REF], possibly due to close structural analogies with these amorphous precursors, in which Si 4+ and Al 3+ build up a network of interconnected tetrahedra with M + and M 2+ cations acting as charge compensators.

Aerodynamic levitation coupled to laser heating proved to be a very suitable method to synthesize the starting glasses, due to the minimization of heterogeneous nucleation in contactless conditions and the comparatively fast achievable cooling rates [START_REF] Zandonà | Glass-forming ability and ZrO2 saturation limits in the magnesium aluminosilicate system[END_REF], of the order of 300 K s -1 (average rate over the first 1000 K). The choice of Ar as a levitation gas and the high melting temperatures (1800-2000 °C) were moreover crucial to retain transition metal ions predominantly in a reduced M 2+ valence state, although EELS measurements could not unequivocally confirm the full stabilization of M 2+ cations (particularly in sample MnLiQss). As known from literature, multivalent cations generally exhibit multiple coexisting oxidation states in glasses [START_REF] Johnston | Oxidation-Reduction Equilibria in Molten Na2O.2SiO2 Glass[END_REF], which might have contributed to the difficult retrieval of high fractions of Qss in some of the samples (e.g. CoQss, FeQss and MnLiQss) and to the formation of some secondary phases (e.g. metallic Ni in NiQss). Nonetheless, only FeQss required heat treatments in vacuum (an oxidizing atmosphere led to the formation of Fe 3 O 4 , Fe 2 O 3 and mullite), while all other samples seemed less sensitive and could be crystallized in air. Even during high-temperature decomposition (Fig. S7), CoQss yielded non-stuffed quartz and CoAl 2 O 4 , hinting at a remarkable stability of the main initial oxidation state.

We achieved the best-fitting description of the average structure of Qss stuffed with transition metals using the less constrained trigonal model of low quartz, in agreement with recent results [START_REF] Sternitzke | Crystal structure and thermal expansion of quartz-type aluminosilicates[END_REF][START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF][START_REF] Zandona | Mg-bearing quartz solid solutions as structural intermediates between low and high quartz[END_REF] that refuted the previously assumed [START_REF] Schreyer | Metastable solid solutions with quartz-type structures on the join SiO2-MgAl2O4[END_REF][START_REF] Schulz | The average structure of Mg[Al2Si3O10], a stuffed derivative of the high-quartz structure[END_REF] full hexagonality (i.e. the high-quartz-like character) of Mgstuffed Qss. Furthermore, we allowed for Zn 2+ , Co 2+ , Ni 2+ and Fe 2+ to freely distribute over the two distinct channel positions reported for Li + and Mg 2+ ionsrespectively a distorted 4-coordinated site T ch and a distorted 6-coordinated site O ch (Fig. 5-b). Indeed, occupation of both tetrahedral and octahedral sites was previously reported in Zn-bearing Qss [START_REF] Behruzi | Struktur und thermische ausdehnung des ungeordneten β-Eukryptits[END_REF] and assigned spectroscopically to Ni 2+ and Co 2+ ions incorporated at impurity levels in Li-bearing Qss [START_REF] Kang | Structural states of Co(II) in β-eucryptite-based glass-ceramics nucleated with ZrO2[END_REF][START_REF] Dymshits | Structural states of Ni(II) in glasses and glass-ceramic materials of the lithium-aluminium-silicate system[END_REF]. Mixed occupancy was identified in all Qss despite the inevitable uncertainties originating from the inherent disorder of channel sites, manifested by BVS calculations yielding lower values than 2 (Tabs. S4, S8, S12, S16) and by the high oxygen B iso values obtained from the refined NPD data. Still, a preference for T ch was invariably coupled to a bigger unit cell, even at constant Al/Si ratio in the tetrahedral sites (i.e. for all samples synthesized within this work, see Fig. 6), possibly hinting at higher size constraints for the occupation of these specific positions.

The combined channel disorder, crystalline site distortion and defective local charge balancing (of which our best-fitting structural models clearly provide only an average description) may partially account for the pronounced metastability of Qss, as previously speculated by other authors [START_REF] Li | Transformation mechanism between high-quartz and keatite phases of LiAlSi2O6 compositon[END_REF].

The synthesis of Qss stuffed with Co 2+ within this work specifically adds a new member to the restricted family of zero thermal expansion materials. The extremely low CTE (0.2 x 10 -6 K -1 ), negligible thermal expansion anisotropy (the c/a ratio changed only by 0.05% in the range 30-600 °C, compared to the 0.5% of LiQss) and simultaneously good stability against thermal decomposition (up to 900 °C at least, see Fig. S7) qualify this phase as a valid candidate for the future development of temperatureinvariant or thermal-shock-resistant materials, potentially both in poly-and monocrystalline form. More generally, our results isolate the key structural factors determining the thermal expansion behavior of quartz-like phases, enhancing the fundamental understanding of these technologically relevant materials.

As shown in Fig. 8, unit cell volumes exhibit a good linear correlation to the CTEs of quartz-like phases:

this trend notably extends far beyond the results of this work, encompassing also several non-stuffed members of this family and without breaking down even in the transition from high-to low-quartz-like structures. On the contrary, a correlation between c/a ratios (mirroring cell distortions) and CTEs is only able to cluster the existing quartz-like phases into non-stuffed frameworks and stuffed Qss, adding little information to what is obtainable from Fig. 3. Structural pseudo-hexagonality (i.e. the intertetrahedral tilt angle δ, Fig. S8) appears similarly inconclusive in rendering the thermal behavior of the crystals, although our coupled PXRD/NPD Rietveld refinements manifested a strong dependence of this latter parameter from the chosen refinement strategy, which could possibly affect the comparison between different literature sources.

Other authors [START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF][START_REF] Xu | Structural mechanisms underlying near-zero thermal expansion in β-eucryptite: A combined synchrotron x-ray and neutron Rietveld analysis[END_REF] previously connected the thermal expansion of Qss (LiAlSiO 4 and Mg 0.5 AlSiO 4 ) to three interdependent structural mechanisms: (i) a subtle deformation of Si/Al tetrahedra;

(ii) positional disordering of channel cations with increasing temperature, so far identified only in superstructured Qss but possibly crucial in the crystals synthesized within this work, exhibiting simultaneous stuffing at T ch and O ch sites; (iii) tetrahedral tilting, which also represents the main process for the transformation from low to high quartz [START_REF] Grimm | On the mechanism of the α-β phase transformation of quartz[END_REF]. This latter mechanism appeared substantially hindered by channel stuffing (e.g. in Mg 0.5 AlSiO 4 as compared to low quartz) [START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF], while it accounts for the slightly negative thermal expansion of hexagonal high quartz in the form of low-frequency, highamplitude rigid unit modes [START_REF] Tucker | A detailed structural characterization of quartz on heating through the α-β phase transition[END_REF]. It is therefore not surprising that highly Co-stuffed Qss exhibit the closest-to-zero CTE among all known Qss, given their almost full hexagonality at room temperature (i.e.

an only limited range for tetrahedral tilting) and the most random distribution of channel stuffing cations between T ch and O ch .

To sum up, thermal expansion appears to be minimized in this system by any chemical substitution that is able to enlarge the room-temperature quartz unit cell, either by increasing the volume of the available framework sites (e.g. introducing Al 3+ for Si 4+ ) or by occupying the otherwise vacant channel sites, thereby leading to a gradual distortion towards the hexagonal symmetry. Chemical stuffing of Qss is therefore a particularly efficient compositional strategy to synthesize crystals with low or negative CTE since it combines both mechanisms, i.e. tetrahedral expansion and framework gaping. In this context, full hexagonal symmetry does not appear a required condition for the synthesis of zero or negative thermal expansion quartz-like crystals. [START_REF] Xu | Synthesis and structure of a stuffed derivative of α-quartz, Mg0.5AlSiO4[END_REF][START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF][START_REF] Xu | Thermal expansion and structural transformations of stuffed derivatives of quartz along the LiAlSiO4-SiO2 join: a variable-temperature powder synchrotron XRD study[END_REF][START_REF] Müller | Hydrogen substitution in lithium-aluminosilicates[END_REF][START_REF] Schwarzenbach | Verfeinerung der Struktur der Tiefquarz-Modifikation von AlPO4[END_REF][START_REF] Kosten | Die III-V-Analoga des SiO2[END_REF][START_REF] Carpenter | Calibration of excess thermodynamic properties and elastic constant variations associated with the alpha <-->beta phase transition in quartz[END_REF][START_REF] Philippot | Neutron and X-Ray Structure Refinements between 15 and 1073 K of Piezoelectric Gallium Arsenate, GaAsO4: Temperature and Pressure Behavior Compared with Other α-Quartz Materials[END_REF][START_REF] Goiffon | Etude comparée à diverses températures (173, 293 et 373°K) des structures de type quartz α des phases MIIIXVO4 (MIII = Al, Ga et XV = P, As)[END_REF][START_REF] Haines | A Neutron Diffraction Study of the Thermal Stability of the α-Quartz-Type Structure in Germanium Dioxide[END_REF][START_REF] Muraoka | The temperature dependence of the crystal structure of berlinite, a quartztype form of AlPO4[END_REF]; c) unit cell volume and CTE of quartz-like phases, with data from various literature sources [START_REF] Xu | Structural behavior of a stuffed derivative of α-quartz, Mg0.5AlSiO4, at high temperature: an in situ synchrotron XRD study[END_REF][START_REF] Xu | Thermal expansion and structural transformations of stuffed derivatives of quartz along the LiAlSiO4-SiO2 join: a variable-temperature powder synchrotron XRD study[END_REF][START_REF] Kosten | Die III-V-Analoga des SiO2[END_REF][START_REF] Philippot | Neutron and X-Ray Structure Refinements between 15 and 1073 K of Piezoelectric Gallium Arsenate, GaAsO4: Temperature and Pressure Behavior Compared with Other α-Quartz Materials[END_REF][START_REF] Grimm | On the mechanism of the α-β phase transformation of quartz[END_REF][START_REF] Muraoka | The temperature dependence of the crystal structure of berlinite, a quartztype form of AlPO4[END_REF][START_REF] Philippot | A General Survey of Quartz and Quartz-like Materials: Packing Distortions, Temperature, and Pressure Effects[END_REF]. See also Fig. S8 of the supplementary materials. Error bars are smaller than the depicted symbols.

Conclusion

In addition to the well-known Li-, Mg-and/or Zn-stuffed quartz solid solutions, we report in this work the first synthesis of quartz-like aluminosilicate phases exclusively and heavily doped with transition metal cations (Fe 2+ , Co 2+ , Ni 2+ , Mn 2+ ). The thermal expansion of these phases can be tuned finely and continuously from substantially negative (-2 x 10 -6 K -1 ) to strongly positive (~10 x 10 -6 K -1 ) values, relying on a smart compositional design within this enlarged chemical domain. Structural examination revealed that an overall expansion of the unit cell through framework substitutions and/or chemical stuffing is crucial for lowering the CTE of quartz-like phases, while framework distortion and pseudo-hexagonality appear less decisive. Most notably, the incredibly low thermal expansion coefficient (0.2 x 10 -6 K -1 in the range 30 -300 °C) of the newly synthesized Co-bearing quartz solid solutions is expected to stimulate the development of potentially Li-free zero-thermal-expansion (glass-)ceramics, reducing the technological dependence on critical raw materials such as lithium ores, whose market prices and availability have been impacted by the ever-growing battery industry.

  5SiO 2 (M 0.44 Al 0.44 Si 0.56 O 2 , with M = Li, Mg 0.5 , Zn 0.5 , Co 0.5 , Ni 0.5 , Mn 0.5 or Fe 0.5 ) corresponding to the mineral cordierite (Mg 2 Al 4 Si 5 O 18 ) and therefore slightly SiO 2 -richer than the Li-bearing endmember β-eucryptite (Li 0.5 Al 0.5 Si 0.5 O 2 ). These samples are recalled as MQss in the following (e.g. CoQss for the Co-bearing sample). To explore the mutual solid solution between different stuffing cations, also a sample corresponding to a 1:1 mixture of CoQss and NiQss was produced (Co 0.5 Ni 0.5 Qss). In the case of MnLiQss, a molar substitution of 25% MnO 2 by Li 2 O in the starting batch was applied, since the Li-free sample invariably yielded a complex crystalline mixture, in which Qss (if present at all) only represented a minor phase. To explore the solid solution extent towards pure SiO 2 , the composition CoO · Al 2 O 3 · 4.6 SiO 2 (corresponding to 70 mol% SiO 2 ) was also tested (sample named CoQssSi). All nominal compositions are associated to the respective label in

Figure 1 .

 1 Figure 1. Excerpts of PXRD patterns (y-axis in natural logarithmic scale) collected from quartz solid solutions (Qss) synthesized within this work. Gray-shaded areas highlight peaks pertaining to Qss phases (note the systematic shifts produced by stuffing with different cations), which were the main crystalline component in all samples; otherwise labelled peaks arose from secondary phases: m for mullite, s for aluminate spinel, NiO for nickel (II) oxide, * for unassigned peaks. K β marks the Cu-K β line diffracting as the main peak of Qss.

Figure 2 .

 2 Figure 2. EELS measurements collected at the L 2 , 3 -edge of the respective stuffing transition metal cation in samples: a) CoQss, b) NiQss, c) MnLiQss. The spectra have been renormalized and shifted vertically to facilitate the comparison with the available reference materials (LiCoO 2 , NiO, MnO, MnO 2 ).

Figure 3 .

 3 Figure 3. a) All relevant crystalline phases with a stuffed or non-stuffed quartz-like framework reported in the literature, plotted according to their a and c lattice parameters from the most reliable sources (see related text for all citations). b) Magnification of the area shaded in green, comprising all known stuffed derivatives of the quartz structure; the samples synthesized within this work are depicted as stars. Error bars are smaller than the depicted symbols.

Figure 4 .

 4 Figure 4. Rietveld refinements of SPD data (y-axis in natural logarithmic scale) collected from: a) ZnQss (R wp = 9.61, GOF = 1.94), b) CoQss (R wp = 6.08, GOF = 1.54), c) NiQss (R wp = 7.52, GOF = 1.76) and d) FeQss (R wp = 6.16%, GOF = 1.33). The shown results correspond to the best fits, obtained with structures comprising stuffing cations occupying both T ch and O ch positions simultaneously and independently. Measured intensities are shown in red, calculated intensities in black, residuals in blue. The peaks designated with * could not be indexed in CoQss (absent in lab PXRD and NPD); † marks the peaks of a weighed-in Si standard in FeQss.

Figure 5 .

 5 Figure 5. a) Best-fitting Qss structure obtained for the SPD data of sample CoQss, exhibiting a trigonal framework of tetrahedra (Al 3+ and Si 4+ statistically distributed) with Co 2+ occupying both T ch and O ch , i.e. 4-and 6-coordinated channel sites (blue wedges represent the partial occupation at each site); b) details of the same structure, visualizing the 4-and 6-coordinated polyhedra occupied by Co 2+ ions in the c-

Figure 6 .

 6 Figure 6. Correlations between values obtained from Rietveld refinements of SPD data: a) unit cell volume and fraction of the M 2+ occupying the 4-coordinated channel site labelled as T ch in Qss; b) unit cell volumes of Qss and respective pseudo-hexagonality angle δ. Data for Li-and Mg-bearing Qss (sof, δ angles) were retrieved or assumed from literature [19,53].

Figure 7 .

 7 Figure 7. Results of VT-PXRD measurements performed up to 600 °C to determine the thermal expansion of Qss phases contained in the studied samples: a) relative average length change obtained from the unit cell volume of the crystals, referring to the radius of a sphere with the same volume; d) relative variation of the c/a ratio, providing the legend for both panels; error bars are smaller than the depicted symbols.

Figure 8 .

 8 Figure 8. Correlations between: a) c/a lattice parameter ratio and CTE of quartz-like phases, with data from various literature sources [19,20,22,25,30,31,58,60,61,64,77]; c) unit cell volume and CTE of quartz-like phases, with data from various literature sources [20,22,31,60,65,77,78]. See also Fig. S8 of the supplementary materials. Error bars are smaller than the depicted symbols.

Table

  .

Acknowledgement

Alessio Zandonà wishes to acknowledge the Deutsche Forschungsgemeinschaft (DFG) for funding his research through the Walter Benjamin Program, project n. 448961237, ZA 1188/1-1. Use of the Advanced Photon Source at Argonne National Laboratory was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. This project has benefited from the facilities of the Platform MACLE-CVL co-funded by the European Union and Centre-Val de Loire Region (FEDER). The authors are grateful to Dr. Charles Hervoches and Dr. Premsyl Beran for performing the neutron diffraction measurement: it was carried out at the CANAM infrastructure of the Nuclear Physics Institute ( ež, Czech Republic) and infrastructure of reactors LVR-15 and LR-0, supported through the Ministry of Education, Youth and Sports project LM2015056 and LM2018120, respectively.


Elyes Ben Hamida
email: elyes.ben-hamida@insa-lyon.fr

Guillaume Chelius
email: guillaume.chelius@inria.fr

A Line-Based Data Dissemination protocol for Wireless Sensor Networks with Mobile Sink

In wireless sensor networks, data dissemination is generally performed from the sensor nodes towards a static sink. In this paper, we address the particular case where the sink is mobile, according to an unpredictable mobility pattern. First, we study existing approaches. As an alternative, we present the Line-Based Data Dissemination (LBDD) protocol. Next, we analytically evaluate the communication cost of this protocol and we compare it to other approaches. Finally, realistic simulations are performed and results show that LBDD outperforms previous approaches and presents the best tradeoff among the evaluated protocols.

I. INTRODUCTION

A Wireless Sensor Network (WSN) is a multi-hop wireless network composed of sensor nodes. A sensor node is generally a constrained device with relatively small memory, restricted computation capability, short range wireless transmitter-receiver, and limited built-in battery. Furthermore, sensor networks usually operate on an N-to-1 communication paradigm, where sensors collect environmental data and forward them towards a static base station or sink. This procedure is called data dissemination. In many situations, a static sink may be unfeasible because of deployment or security constraints. Sink mobility may also improve the lifetime of a WSN by avoiding excessive transmission overhead at nodes that are close to the location that would be occupied by a static sink [START_REF] Wang | Using mobile relays to prolong the lifetime of wireless sensor networks[END_REF]. The sink mobility assumption may be useful for many applications such as target tracking, emergency preparedness, and habitat monitoring. In such a context, the difficulty for sensor nodes is to efficiently track the sink and report to it. As flat architectures and flooding-based protocols do not scale, overlaying a virtual infrastructure over the physical network has often been investigated as an interesting strategy for efficient data dissemination. This virtual infrastructure acts as a rendez-vous area for queries and data reports.

Several protocols have been proposed to implement a scalable and energy-efficient data dissemination architecture for WSNs. GHT (Geographic Hash Table) [START_REF] Ratnasamy | Ght: A geographic hash table for data-centric storage in sensornets[END_REF] has introduced the concept of data-centric storage (DCS). GHT hashes a sensed-data-type into geographic coordinates and stores the corresponding data at the sensor node (a.k.a. home-node) the closest to these coordinates. Two-Tier Data Dissemination (TTDD) [START_REF] Luo | TTDD: Two-tier data dissemination in large-scale wireless sensor networks[END_REF] provides a scalable and efficient data delivery to multiple mobile sinks. Each source pro-actively builds a grid structure by dividing the sensor field into cells with dissemination nodes located at the crossing points of the grid. Queries and data are then transmitted along the grid. Railroad [START_REF] Shin | Railroad: virtual infrastructure for data dissemination in wireless sensor networks[END_REF] builds and exploits a virtual infrastructure, called a rail. This rail is placed in the middle area of the sensor field so that each node can easily access it. When a source detects a new event, the data remains locally stored and the corresponding meta-data is sent to the rail. This infrastructure is then used by the mobile sink to retrieve meta-data, with the queries traveling around the rail. The Column-Row Location Service (XYLS) [START_REF] Das | Performance comparison of scalable location services for geographic ad hoc routing[END_REF] is a proactive location service which can be used to disseminate data reports. In XYLS, source nodes disseminate and replicate the data reports in the north and south directions from their current location, such that sink queries can intersect it subsequently.

In this paper, we present the Line-Based Data Dissemination (LBDD) protocol and compare it to existing approaches for data dissemination in WSNs with mobile sinks. LBDD relies on the concept of a rendez-vous region to decouple the operation of data dissemination from the one of data collection. We show that LBDD outperforms other approaches in event-driven and query-based scenarios. In general, we also show how the choice of an efficient dissemination structure may impact on dissemination protocol performances.

The remainder of this paper is organized as follows. Section II describes the LBDD protocol. In Section III, the analytical communication costs of LBDD and existing approaches are evaluated and compared. Simulation results are analyzed in Section IV. Section V finally concludes this article.

II. THE LINE-BASED DATA DISSEMINATION PROTOCOL

The Line-Based Data Dissemination (LBDD) protocol defines a vertical virtual line of width w which divides the sensor field into two parts, as shown in Fig. 1. This line is also divided into groups of size g. These two parameters are used to address the hotspot problem and the scalability issue. The line is placed in the center of the sensor field so that each node can easily access it. Nodes within the boundaries of this wide line are called inline-nodes, while the other nodes are referred to as ordinary nodes. This line acts as a rendez-vous region for data storage and lookup.

As in [START_REF] Ratnasamy | Ght: A geographic hash table for data-centric storage in sensornets[END_REF]- [START_REF] Shin | Railroad: virtual infrastructure for data dissemination in wireless sensor networks[END_REF], the LBDD design assumes that each node knows its geographic location as well as the network ge-ographic boundaries. This is not an utopian assumption as in most sensor applications (e.g., target tracking, military surveillance, etc.) the data are strongly correlated to the geographic location. A node position can be obtained through the use of GPS or some virtual coordinate system. Eligibility of nodes as inline-nodes is then easily performed based on this geographic information. Thus, the overhead for building the virtual infrastructure is avoided. Finally, we suppose that there are multiple sinks moving randomly in the sensor field. The operation of LBDD is composed of two main steps: (i) Dissemination: when an ordinary sensor node generates some new data, it forwards the data to the nearest inlinenode (see Fig. 1(a)); and (ii) Collection: in order to retrieve a specific data, a sink sends a query towards the line in a perpendicular fashion. The first inline-node which receives this query propagates it in both directions along the line until it reaches the inline-node storing the data (see Fig. 1(b)). The data is then sent directly to the sink (see Fig. 1(c)). To avoid the transfer of duplicated data, we suppose that a sink receives a response to its query only if the inline-node owns a new data.

To ease the data lookup process, two data-storage schemes are possible: the data can be either stored in all nodes of a group or just in their group-leader. The first scheme needs a fine-tuning of w and g to prevent an increase of the congestion under high traffic load conditions, while the second one requires a periodic group-leader election and a replication mechanism. Finally, the sink mobility is supported using the progressive-footprint-chaining strategy [START_REF] Shen | Sensor Information Networking Architecture and Applications[END_REF].

Several optimizations are possible to increase the efficiency of LBDD. First, in scenarios where the number of queries is higher than the number of data reports, replicating the data on the entire virtual line could decrease the cost of datalookup. In the rest of this paper we will refer to LBDD R as the basic LBDD protocol with data replication. Second, in target-tracking scenarios, a mobile target can evolve and remain in a given area. In that case, once the sink receives a data report from a given inline-node, the next sink query's can be sent directly to this inline-node instead of covering the entire virtual infrastructure. This allows to decrease the overhead and to minimize the energy consumption inside the virtual infrastructure.

III. PERFORMANCE ANALYSIS

This section provides an average-case communication cost analysis. This communication cost represents the total amount of messages generated in the network during the data dissemination and lookup process. We first present the models and assumptions, then we evaluate the average distances and the communication cost of LBDD and we compare them to GHT [START_REF] Ratnasamy | Ght: A geographic hash table for data-centric storage in sensornets[END_REF], RailRoad [START_REF] Shin | Railroad: virtual infrastructure for data dissemination in wireless sensor networks[END_REF], TTDD [START_REF] Luo | TTDD: Two-tier data dissemination in large-scale wireless sensor networks[END_REF] and XYLS [START_REF] Das | Performance comparison of scalable location services for geographic ad hoc routing[END_REF].

A. Models and assumptions

We consider a network with a large number of nodes being uniformly and independently distributed over a 1 × 1 area. We define H(l) as the number of hops on a path between two arbitrary nodes x and y such that |x, y| = l. According to [START_REF] De | Bounds on hop distance in greedy routing approach in wireless ad hoc networks[END_REF], given a geographic routing protocol, we have H(l) = ζ l r with r the communication range and ζ ≥ 1 a scaling factor which depends on the density of nodes. For numerical applications, we will assume that:

ζ = 1.
The communication cost of data dissemination and data collection is proportional to the distance between the different communicating entities. Then, to compare LBDD to other protocols, it is important to evaluate the distances between the protocol entities. In the rest of the paper we define the following metrics: (i) D src,rdv is the distance between the source-node and the rendez-vous area; (ii) D sink,rdv is the distance between the sink and the rendez-vous area; and (iii) D rdv,sink is the distance between the rendez-vous area and the sink. We assume that D sink,rdv and D rdv,sink may be different as a query and the data transfer do not necessarily use the same path. We consider four types of messages: event notification, query, data, and control messages, whose sizes are p e , p q , p d , and p c , respectively. We consider m sinks moving randomly in the sensor field as well as n sources. Each sink generates a number of queries equal to q and each source generates a number of events equal to ē. Thus, the total number of queries and events can be written as mq and nē. Finally, we disregard the LBDD line's width for the theoretical evaluation.

B. Average distances evaluation

LBDD:

In the average-case, a source node is located at a distance D src,rdv = 0.25 from the virtual line for data dissemination. To compute the average distance from the sink to the rendez-vous point (i.e., the inline-node which stores the requested data), we evaluate the average distance between the sink and the virtual line, which is equal to 0.25, plus the average distance covered by the query inside the line during the data lookup process.

Let Y 1 and Y 2 the Y-coordinates of two randomly chosen points in the unit square. We consider the following random variable:

D = |Y 1 -Y 2 | with a PDF f D (d) = 2(1 -d) for 0 < d < 1
and 0 otherwise. The average distance covered by a query inside the virtual line is equal to

E[D] + 1-E[D] 2 .
The average distance between the sink and the inline-node is finally equal to: D sink,rdv = 0.25 + 1 3 + 1 3 ≈ 0.9166. The average distance from the inline-node to the sink for the data transfer is equal to the average distance between two randomly chosen points in a unit square and the central line, respectively. This distance can be expressed as follows:

D = |X 1 -0.5| 2 + |Y 1 -Y 2 | 2 ,
with X 1 , Y 1 and Y 2 three independent and uniform random variables. The random variable D is defined by the following PDF [START_REF] Hamida | Analytical evaluation of virtual infrastructures for data dissemination in wireless sensor networks with mobile sink[END_REF]:

f D (d) =      2d(π -2d) 0 < d ≤ √ 0.25 2d(arccos(1 -1 2d 2 ) -1) √ 0.25 < d ≤ 1 2d(arccos(1 -1 2d 2 ) -arccos( 2 d 2 -1) +2 √ d 2 -1 -1) 1 < d ≤ √ 1.25 0 otherwise
The average distance is evaluated numerically using a Riemann sum and is computed as D rdv,sink ≈ 0.45.

Comparison with other protocols:

The average distances of the other protocols are computed in a similar way. For further details please refer to [START_REF] Hamida | Analytical evaluation of virtual infrastructures for data dissemination in wireless sensor networks with mobile sink[END_REF]. Results are summarized on Tab. I. From these results we can make three observations. First, we notice that a random rendez-vous node (i.e., GHT) presents a lower distance for data lookup (D sink,rdv ) than the other infrastructures (except replication-based solutions). As the virtual infrastructure is limited to a single node, there is no need to search for the requested data and the cost of data lookup is then avoided. However, this characteristic may induce a hotspot problem and causes congestion (i.e., lost of packets) as all queries and data reports are concentrated on a single node.

Protocols

D src,rdv D sink,rdv D rdv,sink P stretch GHT [START_REF] Ratnasamy | Ght: A geographic hash table for data-centric storage in sensornets[END_REF] 0.52 0.52 0.52 1.5 TTDD [START_REF] Luo | TTDD: Two-tier data dissemination in large-scale wireless sensor networks[END_REF] 0.0 0.88 0.88 1.69 Railroad [START_REF] Shin | Railroad: virtual infrastructure for data dissemination in wireless sensor networks[END_REF] 0 Second, the use of a large virtual infrastructure like LBDD, TTDD or RailRoad decreases the cost of dissemination (D src,rdv ) compared to the random rendez-vous area. Indeed, as the size of the rendez-vous area increases, source nodes get usually closer to this infrastructure. In addition, using a large infrastructure allows to distribute the communication load through the nodes belonging to the rendez-vous area. However, the use of large infrastructures for data dissemination induces a higher cost for the lookup (D sink,rdv ) as the data need to be searched. Third, replication-based solutions, like LBDD R and XYLS, allow to minimize the cost of data lookup and collection (D rdv,sink ). However, the replication induces an increase in the dissemination cost, while minimizing the average path-stretch.

From this analysis a tradeoff emerges between the virtual infrastructure size and the data dissemination/lookup path distances. Large virtual infrastructures are more suitable to applications inducing a large number of data reports compared to the number of queries (e.g., periodic sensing or event driven applications), while small infrastructures surpass the latter in scenarios with a large number of queries compared to the number of data reports (e.g., query based applications). This fact is further analyzed in the following Section.

C. Communication cost

The communication cost represents the total amount of messages generated in the network during the data dissemination and lookup processes. It is defined as: C protocol = C DD + C DL + C DT , where C DD , C DL , C DT are the costs of data dissemination, data lookup, and data transfer, respectively. The subscript "protocol" refers here to one of the solutions listed in Table I. To better investigate the tradeoff analyzed in previous section, we consider two application-scenarios for evaluating the communication costs. In scenario 1, we consider an event-driven application with a fixed average number of query per sink (q = 50) while varying the average number of data reports per source. In scenario 2, we consider a query-based application with a fixed average number of data report per source (ē = 50) while varying the average number of queries per sink. For both scenarios we consider 10, 000 sensor nodes deployed on a square sensor field of size 1 × 1. The sensor coverage area radius is r = 0.1 and we suppose that the size of a TTDD cell, α, is 0.3.

According to Tab. I, the total average communication cost of LBDD is defined as: C LBDD = nē p d H(0.25) + mq p q H(0.91) + nē p d H(0.45), where nē p d H(0.25) is the cost of the data dissemination, mq p q H(0.91) is the cost of the data lookup, and nē p d H(0.45) is the cost of data transfer. The communication costs of GHT, XYLS, Railroad, and TTDD are computed in a similar way (for further details, refer to [START_REF] Hamida | Analytical evaluation of virtual infrastructures for data dissemination in wireless sensor networks with mobile sink[END_REF]). Scenario 1. The average communication costs for scenario 1 are shown in Fig. 2. We notice that TTDD presents a rather high communication cost stemming from its need to build grids and its routing strategy along the grid. Moreover, we observe that the replication mechanism of LBDD R and XYLS engenders a high communication cost, as the number of data reports increases. On the other side, protocols implementing a large virtual infrastructure like RailRoad and LBDD are more suitable to scenarios with a high number of data reports. However, we notice that for a high number of data reports per source node (≥ 200 data reports per source), RailRoad performs slightly better than the others protocols. As RailRoad implements a large infrastructure (i.e., a square of width l = 0.7), sensor-nodes are closer to the rendez-vous area and the cost of data dissemination is reduced. Scenario 2. The average communication costs for scenario 2 are shown in Fig. 3. Note that TTDD presents a rather high communication cost and is not represented on this figure. Rail-Road and XYLS also present a high communication cost as the number of queries increase. This cost is due to the increase of the data lookup cost. In addition, we notice that for q ≤ 170, the average communication cost of LBDD is lower than the ones of the other approaches. Then, for q ∈ [170 . . . 340], GHT presents a lower average communication cost. And finally for a high number of query per sink (q ≥ 340), the replication mechanism of LBDD R allows to decrease the communication cost compared to the other schemes. The reason is that the more the number of queries increases, the more the cost of data lookup becomes crucial. Replication of data reports on the virtual infrastructure allows in that case to minimize the total cost. For example, to find the threshold from which the replication mechanism of LBDD becomes beneficial, we have to satisfy this condition :

C LBDD C LBDD R > 0 ⇔ q > 1.21 n p d m p q ē (1)
This relation between the number of queries and data reports (Eq. 1) can be easily used by LBDD or other approaches to decide when a replication mechanism has to be used or not. In that case, each group-leader (i.e., one of the inline-nodes) evaluates for each data type the number of received queries' and data reports during a given period of time, and depending on Eq. 1 can adapt its behavior on-the-fly to cope with the application needs. Such an adaptive protocol could decrease the overhead and congestion inside the virtual infrastructure.

IV. REALISTIC SIMULATIONS

For the purpose of performance evaluation under more realistic network factors (e.g., collisions, interferences, radio propagation, etc.) we have carried out simulations. The objective is to investigate tradeoffs involving congestion as well as energy consumption. A comparative study is performed between GHT, XYLS and LBDD with a varying number of source-sink pairs. In the simulations, we used the basic version of LBDD without the replication mechanism.

A. Simulation environment

In order to guarantee a fair comparison between LBDD, XYLS, and GHT, we have implemented the three protocols in the WSNet 1 simulator. We consider two scenarios : (i) scenario 1: with a sink datarate of 1 query per second and a source data-rate of 2 datapacket per second, and (ii) scenario 2: with a sink data-rate of 2 query per second and a source data-rate of 1 data-packet per second. Source nodes are randomly deployed into the sensor field and a Linear mobility model is used for the sinks with an average speed of 10m/s. Given that GHT was not designed to support sink mobility, we also use the progressive footprint chaining. In addition, each GHT source-sink pair chooses a different randomly placed home-node at each run. This homenode is then used by all sensor nodes as a rendez-vous area for queries and data reports.The LBDD virtual infrastructure's parameters, g and w, are set to 50m and 50m, respectively. All the simulation results are averaged over 10 different runs.

We use four metrics for the performance evaluation of the three protocols: (i) Energy consumption: defined as the global energy consumption during a simulation; (ii) Data delivery ratio: defined as the ratio between the total number of data packets received by the sink and the total number of data generated by its corresponding source; and (iii) Delay: defined as the total time elapsed between the data generation by a source and its reception by a sink.

B. Simulation results

Fig. 4(c-f) shows the average energy consumption which impacts strongly the network lifetime. For the three protocols, we notice that the energy cost is linear in the number of pairs as the source-sinks pairs are independent. We observe that XYLS presents a rather high communication cost in both scenarios. This is a direct consequence of the data replication along the vertical line. Next, we notice that LBDD presents a lower energy consumption than GHT on scenario 1 (see Fig. 4(c)). Given that all source-sink pairs in GHT have the same home-node, the average path length between the sink and the source via this home-node is higher than the average distance in LBDD. This result confirms the analysis made for the average case and presented in section III. However, on scenario 2 GHT performs better than LBDD (see Fig. 4(f)).

As in this scenario the number of queries is twice the number of data reports, the cost of data lookup inside the virtual infrastructure increases the total communication cost of LBDD compared to GHT. Fig. 4 (a-d) shows the average data delivery ratio. We notice that the success rate slightly decreases as the number of source-sink pairs increases because of congestion in the network. We notice on both scenarios that GHT presents a lower delivery ratio. Given that all data reports and queries converge towards the same single home-node, the congestion as well as the collisions increase. On the other side, LBDD and XYLS presents a higher delivery ratio stemming from the use of a virtual infrastructure which allows to better distribute the load among the nodes inside the rendez-vous area. However, in scenario 2, we observe that the delivery ratio of LBDD decreases slightly compared to scenario 1. But the delivery ratios of LBDD and XYLS remain close in both scenarios. Fig. 4 (b-e) shows the average delay. We notice on both scenarios that LBDD presents a lower delay compared to the other protocols.

V. CONCLUSION

In this paper we analyzed and compared various approaches for data dissemination in wireless sensor networks with mobile sinks. We also proposed a new virtual infrastructure called LBDD allowing an efficient data dissemination. LBDD relies on the concept of a rendezvous region to decouple the operation of data dissemination from the one of data collection. Through analytical analysis and realistic simulations, we showed that LBDD outperforms other approaches both in event-driven and query-based scenarios, and that it presents the best tradeoff among the evaluated protocols.

In the future, we plan to better investigate the problem of data persistence against node failure and malicious nodes inside the virtual infrastructure. We also plan to investigate a near optimal structure for the data dissemination and lookup problem.

Fig. 1 .

 1 Fig. 1. The LBDD virtual infrastructure.

Fig. 2 .

 2 Fig. 2. Average communication cost for scenario 1 (q = 50, m = 5 sinks, n = 10 sources, pe = pc = pq, p d = 2 × pq).

Fig. 3 .

 3 Fig. 3. Average communication cost for scenario 2 (ē = 50, m = 5 sinks, n = 10 sources, pe = pc = pq, p d = 2 × pq).

Fig. 4 .

 4 Fig. 4. Simulation Results.

TABLE I AVERAGE

 I DISTANCES EVALUATION FOR EXISTING APPROACHES.

  Table II presents the simulation parameters.

	Parameters	Values
	MAC protocol	IEEE 802.11 DCF
	Radio Propagation	BPSK, TX≈ -20dBm, sensitivity = -70dBm Free-space with a pathloss β = 3
	Interference model	SINR/Shotnoise model (Full interference model)
	Routing protocol	greedy geographic routing
	Sensor field size Node density	500 × 500m 2 1400 nodes)
	Scenario 1	1 query per 1s and 1 data report per 0.5s
	Scenario 2	1 query per 0.5s and 1 data report per 1s

1

  The WSNet simulator is available at: http://wsnet.gforge.inria.fr/

		1						16			LBDD			60000	LBDD
	Data delivery ratio	0.75 0.8 0.85 0.9 0.95			GHT LBDD		Average delay (s)	2 4 6 8 10 12 14			GHT XYLS		Energy consumption	10000 20000 30000 40000 50000	GHT XYLS
		0.7			XYLS			0						0
		2	4	6	8	10	12	2	4	6	8	10	12		2	4	6	8	10	12
				Source-sink pairs					Source-sink pairs						Source-sink pairs
		(a) Scenario 1 : Delivery ratio		(b) Scenario 1 : Average delay		(c) Scenario 1 : Energy consumption
		1						0.8			LBDD			40000	LBDD
	Data delivery ratio	0.75 0.8 0.85 0.9 0.95			GHT LBDD		Average delay (s)	0.1 0.2 0.3 0.4 0.5 0.6 0.7			GHT XYLS		Energy consumption	10000 20000 30000	GHT XYLS
		0.7			XYLS			0						0
		2	4	6	8	10	12	2	4	6	8	10	12		2	4	6	8	10	12
				Source-sink pairs					Source-sink pairs						Source-sink pairs
		(d) Scenario 2 : Delivery ratio		(e) Scenario 2 : Average delay		(f) Scenario 2 : Energy consumption