Anne-Magali Seydoux-Guillaume

Bernard Bingen

Valérie Bosse

Emilie Janots

Antonin T Laurent

Desmond E Moser

Fernando Corfu

James R Darling

Steven M Reddy

Transmission Electron Microscope imaging sharpens geochronological interpretation of zircon and monazite

Keywords: nanoscale characterization, linking TEM and U-Th-Pb geochronology, resetting mechanism, zircon and monazite, nano-petrochronology

come

Introduction

The orthosilicate zircon (ZrSiO 4 ) and orthophosphate monazite (Light Rare Earth Element phosphate: LREEPO 4 ) are the most widely used minerals in U-Pb and U-Th-Pb (only for monazite) geochronology in a variety of geological contexts [START_REF] Parrish | U-Pb dating of monazite and its application to geological problems[END_REF][START_REF] Harrison | U-Th-. Pb dating of phosphate minerals[END_REF]Harley and Kelly, 2007;[START_REF] Corfu | A century of U-Pb geochronology: the long quest towards concordance[END_REF]. Ideally, the two decay chains, 238 U/ 206 Pb, that are straightforwardly interpreted as the age of the zircon or monazite crystallization.

However, discordant dates are frequent and indicate disturbance of the isotopic systems after initial crystallization. Yet, even in cases where the U-Pb data are entirely concordant, age disturbance cannot be ruled out (e.g. [START_REF] Ashwal | Slow cooling of deep crustal granulites and Pb-loss in zircon: Geochim[END_REF]). In a specific example from a monazite megacrystal from Arendal, Norway, high-resolution mineral characterization, in particular TEM, revealed partial replacement of primary monazite by secondary monazite due to late-stage alteration [START_REF] Seydoux-Guillaume | Contrasting response of ThSiO 4 and monazite to natural irradiation[END_REF]2012]. In the secondary altered domains, in-situ U-Pb dating leads to concordant dates but meaningless ages, generally older than the primary material, due to contrasting U, Th and Pb redistribution during alteration.

Sometimes, redistribution of the parent elements (U, Th), and radiogenic or non-radiogenic Pb takes place during the history of the mineral, leading to uncertain interpretation [START_REF] Harrison | U-Th-. Pb dating of phosphate minerals[END_REF][START_REF] Villa | Geochronology of metasomatic events[END_REF]. Two critical processes are responsible for U-Th-Pb mobility and thus resetting in minerals: chemical reactions and diffusion [START_REF] Cole | Rates and mechanisms of isotopic exchange[END_REF]. The relative efficiency and interplay of each of these processes depends mainly on temperature, time, grain-size or surface area, structural state of the lattice (defects induced by deformation and/or radiation damage) and presence of fluid or melt. The response of zircon and monazite to these parameters is briefly examined below.

Zircon is generally reactive (new domains with a size > 10's micrometer) in amphibolite-facies metamorphic conditions and beyond [Yakymchuk and Brown, 2014;[START_REF] Desormeau | Insights into (U)HP metamorphism of the Western Gneiss Region, Norway: A high-spatial resolution and high-precision zircon study[END_REF][START_REF] Marsh | Zircon U-Pb and trace element zoning characteristics in an anatectic granulite domain: Insights from LASS-ICP-MS depth profiling[END_REF]. While solid state diffusion can be excluded in well crystallized zircon [START_REF] Lee | Pb, U and Th diffusion in natural zircon[END_REF][START_REF] Cherniak | Diffusion in zircon[END_REF], zircon Achilles' heel is clearly the accumulation of radiation damage over time, critically affecting zircon physical and chemical properties. Typically, zircon contains U in the hundreds of ppm range, and consequently has received alpha-decay doses on the order of 10 18 -10 19 /g and is easily amorphized [START_REF] Ewing | Radiation-induced amorphisation[END_REF][START_REF] Ewing | Radiation damage in zircon[END_REF]. Self-irradiation enhances diffusion [e.g. [START_REF] Cherniak | Diffusion in zircon[END_REF], facilitates mineral-fluid interaction [e.g. [START_REF] Geisler | Low temperature hydrothermal alteration of natural metamict zircons from the Eastern Desert, Egypt[END_REF]2007;[START_REF] Nasdala | Retention of uranium in complexly altered zircon: an example from Bancroft, Ontario[END_REF], may lead to possible reduction of Pb 4+ to Pb 2+ [START_REF] Kramers | On the valency state of radiogenic lead in zircon and its consequences[END_REF][START_REF] Tanaka | Determination of the oxidation state of radiogenic Pb in natural zircon using X-ray absorption near-edge structure[END_REF][START_REF] Kogawa | Lead in zircon at the atomic scale[END_REF], and more generally can result in open system behavior even at low temperature (<200°C) and discordance [e.g. [START_REF] Geisler | Re-equilibration of zircon in aqueous fluids and melts[END_REF][START_REF] Xu | Reverse age zonation of zircon formed by metamictisation and hydrothermal fluid leaching[END_REF].

In contrast to zircon, the lattice of monazite is not affected by amorphization over time, in spite of alpha decay doses in the range of 10 19 -10 20 /g, generated by extremely high concentrations of Th (up to 25 wt% ThO 2 ) and to a lower extent of U (up to 1 wt% UO 2 ) [START_REF] Seydoux-Guillaume | Microstructure of 24-1928 Ma concordant monazites: implications for geochronology and nuclear waste deposits[END_REF]. Clear evidence of radiation damage is limited to isolated nm-sized domains within monazite crystal. Mottled diffraction contrast in TEM is a very characteristic feature of monazite, resulting from a mosaic structure of the crystal, i.e., a lattice comprising many smaller domains with slightly different orientations. [START_REF] Black | Pb isotopic composition, colour, and microstructure of monazites from a polymetamorphic rock in Antarctica[END_REF][START_REF] Meldrum | Radiation damage in zircon and monazite[END_REF]Seydoux-Guillaume et al., 2002b;2003;[START_REF] Wirth | Focused Ion Beam (FIB): a novel technology for advanced application of micro-and nanoanalysis in geosciences and applied mineralogy[END_REF]. Solid-state diffusion of U, Th and Pb is insignificant up to high temperature conditions [START_REF] Cherniak | Pb diffusion in monazite: a combined RBS/SIMS study[END_REF][START_REF] Gardés | Pb diffusion in monazite: An experimental study of interdiffusion[END_REF][START_REF] Gardés | Pb diffusion in monazite: New constraints from the experimental study of Pb 2+ =Ca 2+ interdiffusion[END_REF] and monazite can therefore yield precise ages without any isotopic disturbances after crystallization, at least under most crustal conditions. However, monazite is well known for its reactivity in the presence of fluids, even at low temperature [e.g. [START_REF] Poitrasson | Contrasted monazite hydrothermal alteration mechanisms and their geochemical implications for the U-Th-Pb geochronology and nuclear ceramics[END_REF]2000;[START_REF] Rasmussen | Dating low-grade metamorphic events by SHRIMP U-Pb analysis of monazite in shales[END_REF][START_REF] Bollinger | Evidence for Mio-Pliocene retrograde monazite in the Lesser Himalaya, far western Nepal[END_REF][START_REF] Janots | Evolution of the REE mineralogy in HP-LT metapelites of the Sebtide complex, Rif, Morocco: Monazite stability and geochronology[END_REF]2008;2011;[START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF][START_REF] Allaz | Microprobe analysis and dating of monazite from the Potsdam Formation, New York: A progressive record of chemical reaction and fluid interaction[END_REF][START_REF] Didier | Disturbance versus preservation of U-Th-Pb ages in monazite during fluidrock interaction: textural, chemical and isotopic in situ study in microgranites (Velay Dome, France)[END_REF]. Commonly monazite crystals show patchy compositional zoning. Each zone corresponds to a distinct crystallization event recording changes that took place during the history of the host rock. These changes can be metamorphic solid-state reactions [e.g. destabilization of feldspar, crystallization of garnet; e.g. [START_REF] Rubatto | Zircon and monazite response to prograde metamorphism in the Reynolds Range, central Australia[END_REF], reaction with a silicate melt, fluid-rock interaction during alteration, metamorphism or metasomatism [e.g. [START_REF] Heinrich | Monazite-xenotime miscibility gap thermometry. I. An empirical calibration[END_REF][START_REF] Finger | Three metamorphic monazite generations in a high-pressure rock from the Bohemian Massif and the potentially important role of apatite in stimulating polyphase monazite growth along a PT loop[END_REF][START_REF] Harlov | Fluid-mediated partial alteration in monazite: the role of coupled dissolution-reprecipitation in element redistribution and mass transfer[END_REF][START_REF] Janots | Constraints on fluid evolution during metamorphism from U-Th-Pb systematics in Alpine hydrothermal monazite[END_REF][START_REF] Didier | Disturbance versus preservation of U-Th-Pb ages in monazite during fluidrock interaction: textural, chemical and isotopic in situ study in microgranites (Velay Dome, France)[END_REF][START_REF] Gauthiez-Putallaz | Dating prograde fluid pulses during subduction by in situ U-Pb and oxygen isotope analysis[END_REF], or dynamic recrystallization in response to deformation event [START_REF] Erickson | Deformed monazite yields high-temperature tectonic ages[END_REF]2016]. This reactivity of monazite covers a large range of P-T-fluids conditions [START_REF] Spear | Apatite, Monazite, and Xenotime in metamorphic rocks[END_REF] ranging from low grade metamorphism [START_REF] Rasmussen | Monazite begets monazite: evidence for dissolution of detrital monazite and reprecipitation of syntectonic monazite during low-grade regional metamorphism[END_REF][START_REF] Janots | Evolution of the REE mineralogy in HP-LT metapelites of the Sebtide complex, Rif, Morocco: Monazite stability and geochronology[END_REF]2008;2011], granulite-facies metamorphism [START_REF] Kelly | Complexity in the behavior and recrystallization of monazite during high-T metamorphism and fluid infiltration[END_REF][START_REF] Kirkland | Discriminating prolonged, episodic or disturbed monazite age spectra: An example from the Kalak Nappe Complex, Arctic Norway[END_REF], to magmatic conditions [START_REF] Rubatto | Timescales of crustal melting in the Higher Himalayan Crystallines (Sikkim, Eastern Himalaya) inferred from trace elementconstrained monazite and zircon chronology[END_REF][START_REF] Dumond | Monazite as a monitor of melting, garnet growth, and feldspar recrystallization in continental lower crust[END_REF][START_REF] Wang | Timing of Partial Melting and Cooling across the Greater Himalayan Crystalline Complex (Nyalam, Central Himalaya): Insequence Thrusting and its Implications[END_REF]. In monazite, complex compositional zoning attests to successive crystallization events, either growth of new monazite and/or replacement of a previous one. These zones can be dated individually using microanalytical methods -Electron Microprobe analysis (EPMA), Secondary Ion Mass Spectrometry (SIMS), or Laser Ablation Inductively Coupled Plasma Mass Spectrometry (LA-ICP-MS) -and used to reconstruct the reaction history of the rock [e.g. [START_REF] Williams | Age mapping and dating of monazite on the electron microprobe: Deconvoluting multistage tectonic histories[END_REF]2007;2012;Janots et al., 2012;[START_REF] Kelly | Complexity in the behavior and recrystallization of monazite during high-T metamorphism and fluid infiltration[END_REF].

The steady development in micro-sampling methods (SIMS, LA-ICP-MS, EPMA), now routinely applied for in situ dating of zircon and monazite, requires clear evaluation of the balance between analytical precision, analytical volume and effective volume of closed system behavior of the isotopic systems [START_REF] Corfu | A century of U-Pb geochronology: the long quest towards concordance[END_REF][START_REF] Nemchin | High-spatial-Resolution Geochronology[END_REF][START_REF] Schoene | Precision and Accuracy in Geochronology[END_REF]. Any mobility of U, Th or Pb at a significant scale relative to the analytical volume or any overlap of the analytical volume between two volumes characterized by distinct isotopic closure leads to a disturbance, a modification of the initial crystallization age, or an analytically induced mixed age. In this context, [START_REF] Corfu | A century of U-Pb geochronology: the long quest towards concordance[END_REF], based on the results of [START_REF] Moser | Zircon U-Pb strain chronometry reveals deep impact triggered flow: Earth Planet[END_REF], showed the benefit applying several analytical methods, with contrasted precision and microsampling performances, on the same geochronological problem, to provide arguments for correct interpretation of the data. In that specific case, the more precise ID-TIMS data are discordant but plot on a discordia line, whereas the less precise SIMS data all overlap the Concordia curve within error. Nano-petrochronology, is the characterization of samples at different scales, including the nanometer scale [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF][START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF]2015;[START_REF] Didier | NanoSIMS mapping and LA ICP MS chemical and U-Th-Pb data in monazite from a xenolith enclosed in ndesite (Central Slovakia Volcanic Field)[END_REF][START_REF] Darling | Variable microstructural response of baddeleyite to shock metamorphism in young basaltic shergottite NWA 5298 and improved U-Pb dating of Solar system events[END_REF] or even atomic scale [START_REF] Valley | Hadean age for a post-magma-ocean zircon confirmed by atom-probe tomography[END_REF][START_REF] Valley | Nano-and micro-geochronology in Hadean and Archean zircons by atom-probe tomography and SIMS: New tools for old minerals[END_REF][START_REF] Peterman | Nanogeochronology of discordant zircon measured by atom probe microscopy of Pb-enriched dislocation loops[END_REF][START_REF] Piazolo | Deformation induced trace element redistribution in zircon revealed using atom probe tomography[END_REF] in order to link the mineral properties (nanostructure and nanoscale compositional heterogeneities) with geochronological data. Because Transmission Electron Microscopy (TEM) is the only technique able to combine nano-structural and nano-chemical information, it can provide evidences for the resetting mechanism of geochronological systems by identifying remnant of chemical reactions (e.g. secondary grains products and nano-inclusions; coupled dissolutionprecipitation interface; pore filled with fluid/mineral/melt) or structural defects enhancing diffusion and alteration processes (e.g. amorphisation, plastic deformation). Insights from nano-characterization thus help to validate or invalidate the interpretation of an isotopic ratio (a "date") into a geologically meaningful age. In the case of mineral replacement, it is common that the replaced domain contained tiny crystals (micrometric to nanometric) of distinct mineral phases [Figure 1 and review in [START_REF] Putnis | Mineral replacement reactions[END_REF]. For example, the presence of nanocrystals of thorite (ThSiO 4 ) in apatite (Figure 1A-B) or uraninite (UO 2 ) in zircon (Figure 1C) compromises U-Th-Pb or U-Th/He dating. In the presence of fluid, primary monazite can react to form a secondary, Th-U-depleted monazite, typically accompanied by numerous thorite/huttonite (ThSiO 4 ) and/or thorianite (ThO 2 ) inclusions [START_REF] Hetherington | Metasomatic thorite and uraninite inclusions in xenotime and monazite from granitic pegmatites, Hidra anorthosite massif, southwestern Norway: Mechanics and fluid chemistry[END_REF][START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF]Figure 1D]. In inclusion-rich altered domains, inaccurate ages, measured by EPMA or LA-ICP-MS, results from the presence of numerous nano-phases and nano-fractures filled with Th-U-silicates that are visible only under TEM [START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF]Figure 1D]. Information coming from such nanocharacterization may, however, help to improve the interpretation of dates. In this paper, we focus on zircon and monazite characterization with TEM and how it can sharpen interpretation of U-Th-Pb geochronological data. The paper is divided in three parts: the first part presents examples of nano-inclusions that interfere with the U-Th-Pb systematics and how we can address them; the second part is dedicated to experimental studies and how they simulate the resetting of the U-Th-Pb systems in monazite; and the third part focuses on a micro-to nano-chemical tracer that may help to improve the interpretation of geochronological data.

Sample preparation.

Because TEM requires very thin samples, less than 100 nm thick, sample preparation for TEM characterization is one critical and essential step to obtain reliable analyzes. The development of the Focused Ion Beam (FIB) technique for TEM preparation has considerably opened the possibilities for TEM characterization, especially for the geoscience community, for which imaging of tiny phases or inclusions, microstructures or interfaces in situ, in their thin section context, is often crucial. The FIB preparation allows to produce site-specific TEM foils, typically 10-15 µm large over 5-10 µm in depth and less than 0.1 µm thick slice, cut perpendicular to the surface of a (c. 30µm thick) petrographic thin section. The FIB is the sole technique that allows cutting TEM-ready foils with homogeneous sample thickness, in a section from a well-defined area such as mineral boundary, without preferential thinning or production of artifacts. Many examples and details of the technique are reviewed in [START_REF] Wirth | Focused Ion Beam (FIB): a novel technology for advanced application of micro-and nanoanalysis in geosciences and applied mineralogy[END_REF]2009]. Nowadays, most of the TEM studies are closely associated and dependent from FIB preparations.

Structural and chemical evidence for nano-inclusions affecting the U-Th-Pb

systematics in monazite and zircon.

Zircon

Heterogeneous distribution of radiogenic Pb in zircon at sub-micrometer scale was first identified by High-Angle Annular Dark Field Scanning Transmission Electron Microscope-(STEM-HAADF) imaging [START_REF] Utsunomiya | Nanoscale occurrence of Pb in an Archean zircon[END_REF]. The authors found nanoscale patches rich in Pb in two locations: inside the zircon lattice (concentrated at 3 at%) and inside amorphous domains from fission tracks. They suggested that beyond the first percolation point [START_REF] Salje | Is "metamictization" of zircon a phase transition?[END_REF][START_REF] Trachenko | Structural changes in zircon under αdecay irradiation[END_REF][START_REF] Geisler | Re-equilibration of zircon in aqueous fluids and melts[END_REF], when amorphous domains overlap and form an interconnected network, Pb will diffuse faster in zircon. These observations were confirmed a few years later by [START_REF] Utsunomiya | Radiation damage and alteration of zircon from a 3.3 Ga porphyritic granite from the Jack Hills, Western Autralia[END_REF] using the same technique. Even if the first percolation point will differ a little depending on the thermal history of the sample [START_REF] Geisler | Low temperature hydrothermal alteration of natural metamict zircons from the Eastern Desert, Egypt[END_REF], the dose is estimated to be of 2.2 x 10 18 /g [START_REF] Pidgeon | Zircon radiation damage ages[END_REF]. Beyond the first percolation point, the probability of enhanced Pb mobility in response to an external event, like for example metamorphism, is very high.

Lead mobility in polymetamorphic zircon of the Napier complex of Enderby Land, Antarctica, was documented earlier through SIMS micro-analysis by [START_REF] Black | Four zircon ages from one rock: the history of a 3930 Ma-old granulite from Mount Sones, Enderby Land, Antarctica[END_REF]. In the same sample, these authors observed reverse and normal discordance of the U-Pb isotopic system, attesting for Pb mobility with simultaneous Pb loss or Pb gain depending on the analytical spot. The nature and size of isotopic perturbation in high-grade polymetamorphic samples were recently addressed using SIMS imaging [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF][START_REF] Whitehouse | Behaviour of radiogenic Pb in zircon during ultrahigh-temperature metamorphism: An ion imaging and ion tomography case study from the Kerala Khondalite Belt, southern India[END_REF], and TEM [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF]. Thanks to HAADF-TEM and electron diffraction patterns, [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF] identified nanospheres of metallic Pb, 5 to 30 nm in diameter, randomly distributed in those zircons (Figure 2C-D). The nanospheres are commonly associated with an amorphous silica-rich phase and Ti-and Al-rich phases, interpreted to be melt inclusions formed during ultra-high temperature metamorphism. The patchy-distribution of Pb highlighted with SIMS imaging [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF]Figure 2B] causes the spreading of discordance and U-Pb ages (Figure 2A), and can be attributed to the presence of metallic Pb-nanospheres in zircon [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF]. Identification (type of phase, structure, size, morphology) of these nanospheres was feasible thanks to the combination of nano-structural and nano-chemical analyzes, only achievable with TEM. In contrast, the Atom Probe Tomography (APT) would not provide structural information, and would only reveal the presence of Pb-nanoclusters and their isotopic composition [e.g. [START_REF] Peterman | Nanogeochronology of discordant zircon measured by atom probe microscopy of Pb-enriched dislocation loops[END_REF][START_REF] Piazolo | Deformation induced trace element redistribution in zircon revealed using atom probe tomography[END_REF]. Their metallic nature can only be determined using nano-diffraction with TEM [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF]. In this example, TEM characterization helps to prove that apparent ages as old as 4.2 Ga do not correspond to relicts of ancient zircons (and thus have no geological meaning), but to Pb migration and coalescence into Pb metallic nanosphere during the recovery of crystallinity in radiation-damaged zircon under high-grade metamorphism [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF]. Such redistribution of radiogenic Pb at the nanoscale may compromise the precision and accuracy of U-Pb obtained by high-resolution in situ methods like SIMS, as shown by [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF] and [START_REF] Whitehouse | Behaviour of radiogenic Pb in zircon during ultrahigh-temperature metamorphism: An ion imaging and ion tomography case study from the Kerala Khondalite Belt, southern India[END_REF]. When larger analytical volumes are considered (e.g. by LA-ICP-MS or SIMS), the effect of the nanoscale heterogeneities may decrease. In [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF], the dates obtained with the highest spatial resolution (2 µm) generates meaningless age up to 4198 Ma [Figure 2B], whereas analyzes with typical SIMS spot size of c. 15µm yield broad distribution of ages between 3324 and 3605 Ma [Figure 3F in [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF]. This leads thus to the question of what should be the optimal spatial resolution for in situ analyses? Mobility of the parent element U in zircon has also been documented by [START_REF] Seydoux-Guillaume | Nanoscale evidence for uranium mobility in zircon and the discordance of U-Pb chronometers[END_REF] using STEM-HAADF and STEM-Energy Dispersive X-Ray Spectroscopy (EDS) analyses coupled with U-Pb isotopic analysis. In a pegmatite from Norway [START_REF] Duran | Fluid-mediated alteration of (Y,REE,U,Th)-(Nb,Ta,Ti) oxide minerals in granitic pegmatite from the Evje-Iveland district, southern Norway[END_REF], atypical zircon phenocrysts are affected by intense selfirradiation, associated with amorphization, microporosity, swelling, radial cracking, and uptake of non-formula cations like Ca. Fractures and pores represent the vectors for fluid mediated open system behavior. This alteration leads to increased discordance of the U-Pb system (Figure 3A-B). Again, STEM (HAADF and EDS) analyses demonstrate the presence of U-rich (uraninite) clusters (<70 nm) within the porosity and of high density walls (<50 nm), enriched in U or Ca or both, coating the pores (Figure 3C Both examples by [START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF] and [START_REF] Seydoux-Guillaume | Nanoscale evidence for uranium mobility in zircon and the discordance of U-Pb chronometers[END_REF] are notes of the promising role of nano-petrochronology for geochronological studies, especially when ages are discordant. They demonstrate how careful nano-scale characterization, associating the U-Pb analytical data with their structural context, may avoid spurious age interpretation. A meaningful age cannot be extracted from all zircon grains, because some are likely to be affected by radiation damage and/or chemical reactions, which will cause and enhance Pb and U mobility.

Monazite

Monazite is much less studied than zircon and, as far as we know, few studies attempted to link TEM characterization to geochronological interpretation. [START_REF] Black | Pb isotopic composition, colour, and microstructure of monazites from a polymetamorphic rock in Antarctica[END_REF] focused on monazite from polymetamorphic rocks of the Napier complex in Antarctica. The authors tried to link the color of monazite grains (brown, yellow and grey) with the nanostructure (TEM) and the discordance. Independently of the color, all monazite crystals present the same nanostructure, i.e. 100 Å crystalline domains misoriented from each other by no more than 2 x 10 -3 radians, leading to mottled diffraction contrasts on Bright Field or Dark Field (BF/DF)-TEM images. Such observation is typical for monazite crystals and the mottled contrast is attributed to radiation damage and was confirmed later by Seydoux-Guillaume et al. [2004] in a comparison between different monazite samples (from 24 Ma to 2 Ga). To address the problem of inconsistent electron-microprobe (EPMA) chemical ages in complex monazite from a polymetamorphic granulite (Andriamena unit, Madagascar, Seydoux-Guillaume et al. [2003] linked a TEM study with electron-microprobe U-Th-Pb chemical dating. The presence of 50nm Pb-rich inclusions is revealed in monazite domains where large variations in EPMA dates are detected (Figure 4). The nature of these inclusions is not completely understood, but it was proposed that they could be a Pb-Ca-Si-O phase like margarosanite (Ca 2 PbSi 3 O 9 ) or more probably represent material that crystallized in fluid inclusions (Figure 4B-C). The TEM study led to a new interpretation of EPMA dates for monazite previously considered to be the result of partial Pb loss from a 2.5 Ga monazite. The scatter of apparent ages ranging from 2.5 Ga to 790 Ma (Figure 4A) is reinterpreted to result from the replacement of a primary monazite by a secondary monazite during a metamorphic event at 790 Ma with random incorporation of various amount of Pb-rich inclusions in the analytical volume of the EPMA analysis in secondary monazite domains. The presence of Pbrich inclusions suggests that recrystallization of monazite does not result, in this case, in a total removal of previously accumulated radiogenic Pb, but partial preservation in Pb-rich clusters. This study therefore exemplifies that domains of unsupported Pb can exist in monazite. This result was recently confirmed by several studies [e.g. [START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF][START_REF] Didier | Disturbance versus preservation of U-Th-Pb ages in monazite during fluidrock interaction: textural, chemical and isotopic in situ study in microgranites (Velay Dome, France)[END_REF], that also suggest that Pb-rich nano-inclusions (or nanodomains) could contain a non-radiogenic Pb component.

Nanoscale constrains on the resetting mechanism of U-Th-Pb systems in experimentally altered monazite.

The first experimental study addressing the hydrothermal resetting of the U-Pb systems in monazite was performed in 1997 by [START_REF] Teufel | Partial resetting of the U-Pb isotope system in monazite through hydrothermal experiments: an SEM and U-Pb isotope study[END_REF] in pure water.

Partial resetting of the U-Pb systems was evaluated by multigrain ID-TIMS analysis of the experimental products. Partial resetting was only detected for a starting material made of fine monazite powder (<1-15µm) and could be shown to be the result of monazite dissolution and precipitation of neoformed monazite. The maximum Pb loss of 26% (ID-TIMS dating of the experimental product) was observed at the highest temperature (750°C). A major revelation of this study was that dissolution and precipitation is more efficient than volume diffusion [START_REF] Cherniak | Pb diffusion in monazite: a combined RBS/SIMS study[END_REF]Gardès et al., 2006 and2007] to reset the U-Pb clock in monazite.

Seydoux-Guillaume et al. [2002], confirmed the importance of the dissolution-precipitation mechanism by performing a series of experiments (800-1200°C) using different fluid compositions. Monograins (200-400µm) of the experimental products were analyzed by ID-TIMS, EPMA and SIMS show that 1) partial resetting was obtained only in the presence of Ca-rich fluid at 1000°C and involved dissolution followed by precipitation of Pb-free neoformed monazite; 2) experiments with Pb-bearing fluid produced sub-concordant monazite for which the 207 Pb/ 206 Pb apparent ages were older than the primary monazite used for the experiment, demonstrating that monazite may incorporate Pb from a fluid. This last result is consistent with TEM observations by Seydoux-Guillaume et al. [2003] that Pb-rich nano-domains observed in monazite samples from Madagascar may be incorporated from a fluid phase and may explain contamination by common Pb recorded in some natural monazites [e.g. [START_REF] Didier | Disturbance versus preservation of U-Th-Pb ages in monazite during fluidrock interaction: textural, chemical and isotopic in situ study in microgranites (Velay Dome, France)[END_REF][START_REF] Janots | Constraints on fluid evolution during metamorphism from U-Th-Pb systematics in Alpine hydrothermal monazite[END_REF][START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF].

Based on natural observations that metasomatic alteration of monazite is particularly developed in Na-and K-rich geological environments [Harlov et al., 2007;[START_REF] Hetherington | Metasomatic thorite and uraninite inclusions in xenotime and monazite from granitic pegmatites, Hidra anorthosite massif, southwestern Norway: Mechanics and fluid chemistry[END_REF], monazite alteration experiments were performed in the presence of Na-or Krich fluids, with different ligands, starting material and temperature [START_REF] Harlov | Partial high-grade alteration of monazite using alkali-bearing fluids: Experiment and nature[END_REF][START_REF] Hetherington | Experimental metasomatism of monazite and xenotime: mineral stability, REE mobility and fluid composition[END_REF][START_REF] Harlov | Fluid-mediated partial alteration in monazite: the role of coupled dissolution-reprecipitation in element redistribution and mass transfer[END_REF][START_REF] Williams | Resetting monazite ages during fluid-related alteration[END_REF][START_REF] Budzyn | Experimental determination of stability relations between monazite, fluorapatite, allanite, and REEepidote as a function of pressure, temperature, and fluid composition[END_REF]2015;Shazia et al., 2016]. All these experimental studies show that alkali fluids are the most efficient fluids to promote monazite replacement, even for temperature as low as 250°C [Budzyn et al., 2016]. Under these conditions, the monazite texture of experimentally replaced monazite is typical of those encountered in natural systems. It shows a secondary rim, that is normally porous, and with a composition that differs of the original monazite starting material. This replacement process is now believed to be preponderant in numerous natural environments (diagenetic, metamorphic or hydrothermal).

Using TEM characterization, [START_REF] Harlov | Fluid-mediated partial alteration in monazite: the role of coupled dissolution-reprecipitation in element redistribution and mass transfer[END_REF] could demonstrate that the replacement of experimental products is caused by an interfacially coupled dissolutionprecipitation mechanism [e.g. [START_REF] Putnis | The mechanism of reequilibration of solids in the presence of a fluid phase[END_REF][START_REF] Putnis | Mineral replacement reactions[END_REF]. Indeed, TEM observations reveal a sharp reaction interface between primary domains (starting material with radiation damage) and secondary domains free of radiation damage, with negligible evidence of diffusion even at the nanoscale. These experimental studies also indicate that the physicochemical conditions of the fluid play an important role on the U-Th-Pb resetting in secondary monazite. Remarkably, the U-Th-Pb system is not systematically reset and Pb is present in altered domain produced in presence of NaOH or KOH fluids [START_REF] Harlov | Fluid-mediated partial alteration in monazite: the role of coupled dissolution-reprecipitation in element redistribution and mass transfer[END_REF] or Na 2 SiO 5 bearing fluids below 350°C [START_REF] Budzyn | Monazite Th-Utotal Pb geochronology and P-T thermodynamic modelling in a revision of the HP-HT metamorphic record in granulites from Stary Gieraltow (NE Orlica-Snieznik Dome, SW Poland)[END_REF]. However, Pb was close or below the detection limit in monazite altered domain produced in presence of Na 2 SiO 5 bearing fluids at 450°C [START_REF] Williams | Resetting monazite ages during fluid-related alteration[END_REF] or in presence of K-and Na-rich melts at 800°C [Shazia et al., 2016]. As a consequence such altered chemical domains may be dated by in situ techniques [EPMA in [START_REF] Williams | Resetting monazite ages during fluid-related alteration[END_REF] and thus offer a great potential to date metasomatic events.

Recently, progress done on high spatial resolution in situ analytical techniques (e.g.

LA-ICP-MS), and on the development of site specific preparation (FIB-Focused Ion Beam)

for TEM analyses, allow [START_REF] Grand'homme | Partial resetting of the U-Th-Pb systems in experimentally altered monazite: Nanoscale evidence of incomplete replacement[END_REF] to formulate a new hypothesis to explain partial resetting observed in experimentally altered monazite samples (300-600°C) under alkali conditions. Experimentally altered monazite rims (Figure 5B; at 500°C), dated using in situ techniques (EPMA and LA-ICP-MS) systematically yield intermediate dates (Figure 5A; EPMA) between pristine monazite (555 Ma) and complete experimental resetting (0 Ma), with a mean date that decreases with increasing temperatures (down to 71 ± 27 Ma at 600°C). Nanometric characterizations using TEM of experimental products elucidated the origin of this partial resetting. At such nanoscale, the replacement is incomplete within the altered rim (Figure 5C): altered rim corresponds to a nano-mixture of Pb-free (EDS-TEM) secondary monazite (mnz2), with inherited monazite (mnz1). No Pb-bearing inclusions were observed in the altered domain. The partial resetting depends then on the nano-mixture of different proportion of mnz1 in the analyzed volume (Figure 5C). Even in the case of EPMA analyses, where the interaction volume is the smallest (< 35 µm 3 ; but with no way to have a 3D control during the analysis), a scattering of dates exists and lead to erroneous ages (Figure 5A; 163 ± 54 Ma at 500°C). Partial replacement in altered domains certainly applies to monazite naturally replaced in the presence of fluid. Indeed, the experimental results simulate well the microstructures of fluid-driven replacements (skeletal structures, microporosity, inclusion) and can account for the age scattering and discordance obtained in altered monazite domains [e.g., [START_REF] Bosse | Fluid-induced disturbance of the monazite Th-Pb chronometer: In situ dating and element mapping in pegmatites from the Rhodope (Greece, Bulgaria)[END_REF][START_REF] Kelly | Complexity in the behavior and recrystallization of monazite during high-T metamorphism and fluid infiltration[END_REF][START_REF] Seydoux-Guillaume | Low-temperature alteration of monazite: Fluid mediated coupled dissolution-precipitation, irradiation damage, and disturbance of the U-Pb and Th-Pb chronometers[END_REF][START_REF] Didier | Disturbance versus preservation of U-Th-Pb ages in monazite during fluidrock interaction: textural, chemical and isotopic in situ study in microgranites (Velay Dome, France)[END_REF]Grand'Homme et al., 2016a] but also in rock deformed by dislocation creep, where incomplete resetting was proposed but not explained [e.g. [START_REF] Wawrzenitz | Dating rock deformation with monazite: The impact of dissolution precipitation creep[END_REF][START_REF] Erickson | Deformed monazite yields high-temperature tectonic ages[END_REF]2016]. However, in the case of new monazite (over)growths involving spatially decoupled dissolution and precipitation [e.g. dissolution-precipitation creep in deformed rock; [START_REF] Wawrzenitz | Dating rock deformation with monazite: The impact of dissolution precipitation creep[END_REF][START_REF] Didier | Syn-deformation fluid-assisted growth of monazite during renewed high-grade metamorphism in metapelites of the Central Rhodope (Bulgaria, Greece)[END_REF][START_REF] Erickson | Deformed monazite yields high-temperature tectonic ages[END_REF]2016] and experimentally reproduced in Seydoux-Guillaume et al. [2002], such nano-scale incomplete replacement will not occur. It may be challenging to recognize possible disturbances linked with nanomixture, when large age data sets are determined with no reliable micro (to nano)-textural documentation on the dated grains (e.g., as for example in provenance studies).

Nano-petrochronology of monazite.

The structural flexibility of monazite permits substitution of more than 25 elements of the periodic table in its lattice [START_REF] Clavier | Crystal chemistry of the monazite structure[END_REF]. Monazite is monoclinic with space group P21/n and its structure is organized in chains, parallel to c-axis, with alternating PO 4 tetrahedrons and REEO 9 polyhedrons [START_REF] Ni | Crystal chemistry of the monazite and xenotime structures[END_REF][START_REF] Boatner | Synthesis, Structure, and Properties of Monazite, Pretulite, and Xenotime[END_REF][START_REF] Clavier | Crystal chemistry of the monazite structure[END_REF].

The nine-fold site preferentially incorporates LREE, but may accommodate a large variety of other (possibly larger) cations, especially Th, U, and Ca by two well-documented substitution types [START_REF] Förster | The chemical composition of REE-Y-Th-U rich accessory minerals in peraluminous granites of the Erzgebirge-Fichtelgebirge region, Germany, Part I: The monazite-(Ce)-brabantite solid solution series[END_REF][START_REF] Montel | X-ray diffraction study of brabantitemonazite solid solution[END_REF][START_REF] Clavier | Crystal chemistry of the monazite structure[END_REF]. The cheralite [CaTh(PO 4 ) 2 ; formerly called brabantite] exchange An 4+ + M 2+ = 2REE 3+ -with An = (Th, U) and M = (Ca, Ba, Sr, Pb) -involving the nine-fold site alone and the huttonite (ThSiO 4 ) exchange An 4+ + Si 4+ = 2REE 3+ + P 5+ that involve both sites. Furthermore, Y and HREE are incorporated through the xenotime (YPO 4 ) substitution; this property allows monazite to be used as a thermometer when associated with xenotime [START_REF] Gratz | Monazite-xenotime thermobarometry: experimental calibration of the miscibility gap in the binary system CePO 4 -YPO 4[END_REF][START_REF] Heinrich | Monazite-xenotime miscibility gap thermometry. I. An empirical calibration[END_REF]Seydoux-Guillaume et al., 2002b] and with garnet [START_REF] Pyle | Monazite-xenotimegarnet equilibrium in metapelites and a new monazite-garnet thermometer[END_REF]. More exotic cations may be incorporated into monazite lattice; these include S incorporated as anhydrite component [CaSO 4 ;[START_REF] Chakhmouradian | Niobian ilmenite, hydroxylapatite and sulfatian monazite: alternative hosts for incompatible elements in calcite kimberlite from Internatsional'naya, Yakutia[END_REF][START_REF] Prsek | Metamorphichydrothermal REE minerals in the Bacuch magnetite deposit, Western Carpathians, Slovakia: (S, Sr)-rich monazite-(Ce) and Nd-dominant hingganite[END_REF][START_REF] Krenn | Sulfur-rich monazite with high common Pb in ore-bearing schists from the Schellgaden mining district (Tauern Window, Eastern Alps)[END_REF], Sr [START_REF] Krenn | Sulfur-rich monazite with high common Pb in ore-bearing schists from the Schellgaden mining district (Tauern Window, Eastern Alps)[END_REF][START_REF] Holder | Monazite traceelement and isotopic signatures of (ultra) high-pressure metamorphism: examples from the Western Gneiss Region, Norway[END_REF][START_REF] Prsek | Metamorphichydrothermal REE minerals in the Bacuch magnetite deposit, Western Carpathians, Slovakia: (S, Sr)-rich monazite-(Ce) and Nd-dominant hingganite[END_REF], As incorporated as a gasparite component [CeAsO 4 ;[START_REF] Ondrejka | Arsenian monazite-(Ce) and xenotime-(Y), REE arsenates and carbonates from the Tisovec-Rejkovo rhyolite, Western Carpathians, Slovakia: Composition and substitutions in the (REE,Y)XO 4 system (X = P, As, Si, Nb, S)[END_REF][START_REF] Janots | Evolution of the REE mineralogy in HP-LT metapelites of the Sebtide complex, Rif, Morocco: Monazite stability and geochronology[END_REF]2008;2011;[START_REF] Gnos | Age of cleft monazites in the eastern Tauern Window: constraints on crystallization conditions of hydrothermal monazite[END_REF]V [McFarlane and[START_REF] Mcfarlane | Petrogenesis of the Kulyk Lake monaziteapatite-Fe (Ti)-oxide occurrence revealed using in-situ LA-(MC)-ICP-MS trace element mapping, U-Pb dating, and Sm-Nd isotope systematics on monazite[END_REF]. The structural incorporation of redoxsensitive elements in monazite opens the perspective of tracing the evolution of the oxidation state through time in a large number of rocks, thanks to the U-Pb and Th-Pb isotopic systems.

To show the importance of sulfur as a petrogenetic tracer and the contribution of TEM imaging to uncover it, we focus here on complexly patchy zoned monazite crystals (Figure 6A) from an osumilite-bearing granulite from Rogaland, Norway [START_REF] Laurent | Sulphate incorporation in monazite lattice and dating the cycle of sulphur in metamorphic belts[END_REF], in which in situ U-Pb dating (LA-ICP-MS) were performed (Figure 6B). The U-Pb dataset of monazite is concordant to sub-concordant (conc. > 95 %), with analyses plotting continuously along the concordia curve from ca. 1040 Ma down to 920 Ma (Figure 6B). When considering the chemical composition at both micro-and nano-scale (Figures 6E-F), more specific and useful information can be extracted to interpret the apparently continuous age spread. In the monazite crystals, three chemical domains are distinguished (Figure 6A, C-D): D1, a S-rich core (0.45-0.72 Wt. % SO 2 -Figure 6D, Th incorporated as cheralite component, i.e. Si-poor Figure 6C), D2, secondary S-bearing domains (SO 2 > 0.05 Wt. %, partly clouded with solid inclusions-Figure 6A), and D3, late S-free-Figure 6D, Y-rich, domains (0.8-2.5 Wt. % Y 2 O 3 , Th accommodated as huttonite component, i.e. Si-rich Figure 6C). Nano-characterization of the S-rich monazite domains (D1), using FIB preparation coupled with STEM-HAADF imaging, revealed the presence of black dots (5-10 nm in size) homogenously distributed all over the foil and representing negative density contrast in monazite (arrow in Figure 6E).

High resolution STEM-HAADF image (Figure 6F) demonstrates that these nanoclusters are made of anhydrite (CaSO 4 ) characterized by a coherent interface toward the host monazite. These data imply that S was incorporated in the monazite lattice as a clino-anhydrite component following the exchange Ca 2+ +S 6+ =REE 3+ +P 5+ , but then is partly exsolved in nanoclusters of CaSO 4 . Monazite with significant sulfate substitution is expected to crystallize from oxidized geological fluids or melts in which sulfur is present as S 6+ . The close combination of micro-chemical (EPMA) and nano-structural and nano-chemical (STEM) analyses with U-Pb dating allow to attribute precise ages to these three domains. The S-rich core D1 yields a concordia age of 1034 ± 6 Ma, S-bearing D2 gives a significantly younger age of 1005 ± 7 Ma and S-free D3 of 935 ± 7 Ma ( Figures C-D). From regional correlation [START_REF] Bingen | A four-phase model for the Sveconorwegian orogeny, SW Scandinavia[END_REF] and sulphide phase-stability diagrams [START_REF] Whitney | Fugacities of sulfurous gases in pyrrhotite-bearing silicic magmas[END_REF], the interpretation is that S-rich monazite cores (D1) crystallized at 1034 ± 6 Ma from an oxidized, sulphate (S 6+ )-dominated granitic melt during fluid-absent partial melting. Monazite may therefore be used to probe the presence of sulfur in anatectic melts from high-grade terrains at a specific point in time and date sulfur mobilization from reduced to oxidized state. This interpretation could be provided only because TEM analyses proves that sulfur substitutes as CaSO 4 in monazite lattice, and not as inclusions that may reflect and record other processes than primary crystallization [e.g. in figure 6A, sulfur inclusions in D2 reflect metasomatism event; monazite inclusions in Rasmussen et al., 2011 record metamorphic event).

Conclusions

This paper highlights how characterization at the nanoscale, with combination of nano-structural and nano-chemical information, using TEM, provides insights to understand the U-Th-Pb systematics of zircon and monazite and improve age geochronological interpretations. This nano-scale characterization, referred as nano-petrochronology, provides information not only on perturbation of the isotopic systems, but it also contributes to understand reaction mechanisms and distribution of petrogenetically important trace elements. This is well illustrated by the input of nano-chemical tracer (CaSO 4 ) in monazite or by the presence of Pb or U rich-nanophases in monazite and zircon crystals. This nano-petrochronology approach could possibly applied to other common U-Th-Pb chronometers (i.e. titanite, allanite).

The paper also raises the question of the optimal spatial resolution for in situ analyses, a critical issue coming from the improvement of the in situ analytical techniques (e.g. LA-ICP-MS, HR-SIMS, nano-SIMS, EPMA) and well-demonstrated by the studies of [START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF] and [START_REF] Valley | Hadean age for a post-magma-ocean zircon confirmed by atom-probe tomography[END_REF] for zircon, or Seydoux-Guillaume et al. [2003] and [2012] for monazite. We suggest that the use of high-spatially resolved techniques requires even more careful nano-characterization (with TEM) of the samples, to provide reliable geochronological interpretations of the data. Furthermore, considering the complementarity of the Atom Probe Tomography (APT)

technique, which provides isotopic information within nanoscale domains [e.g. [START_REF] Valley | Hadean age for a post-magma-ocean zircon confirmed by atom-probe tomography[END_REF][START_REF] Valley | Nano-and micro-geochronology in Hadean and Archean zircons by atom-probe tomography and SIMS: New tools for old minerals[END_REF][START_REF] Peterman | Nanogeochronology of discordant zircon measured by atom probe microscopy of Pb-enriched dislocation loops[END_REF][START_REF] Piazolo | Deformation induced trace element redistribution in zircon revealed using atom probe tomography[END_REF] in typical analyzed volumes of material of 50x50x200nm 3 , we assert that combination of APT with TEM, a highly complementary technique yielding both nano-structural and nano-chemical information, would considerably improve the interpretation of U-Th-Pb geochronological data; such correlative microscopy should certainly be developed in the future.

The need of experimental data has been demonstrated, in order to better constrain the behavior of parent (e.g. U, Th) and daughter (e.g. Pb) elements and the resetting mechanisms under a large range of conditions (P-T-fluids). Indeed, it has been demonstrated for monazite, that incomplete replacement at the nanoscale may lead to partial resetting of the U-Th-Pb systems and thus to meaningless ages. Such experimental studies coupling nanocharacterization with the behavior of isotopic systems (U-Pb but also Lu-Hf) are long overdue for zircon, and, in contrast to monazite, certainly highly dependent on its structural state due to its sensitivity to radiation damage [START_REF] Geisler | Re-equilibration of zircon in aqueous fluids and melts[END_REF]. Yakymchuk, C., and Brown, M. (2014). Behaviour of zircon and monazite during crustal melting: J. Geol. Soc. London, 171, 465-479.

Figure captions

  -D). It is only the close association of nano-characterization (TEM) to the geochronological data[START_REF] Seydoux-Guillaume | Nanoscale evidence for uranium mobility in zircon and the discordance of U-Pb chronometers[END_REF] Figure 3] that allows to interpret the spatial link between U, Ca and porosity as evidence for U uptake, and which demonstrates that discordance of the U-Pb system in zircon affected by radiation damage includes a component of U-mobility (U-gain -U-loss).

Figure 1 Transmission

 1 Figure 1

Figure 2 A

 2 Figure 2A-Tera-Wasserburg Concordia diagram and B-207 Pb/ 206 Pb ratio images of zircon grains from sample 16178-2 from Napier complex, East Antarctica, analyzed using Secondary Ion Mass Spectrometry (SIMS) [modified from[START_REF] Kusiak | Mobilization of radiogenic Pb in zircon revealed by ion imaging: Implications for early Earth geochronology[END_REF]. C-High Angle Annular Dark Field (HAADF) TEM image and D-High Resolution TEM image of Pb nanosphere in zircon crystal from the same sample [modified from[START_REF] Kusiak | Metallic lead nanospheres discovered in ancient zircons[END_REF].

Figure 3 A

 3 Figure 3

Figure 1 Figure 2 35

 12 Figure 1

Figure 3 Figure 4

 34 Figure 3

  Figure 5

  877

U/207 Pb, (and also the third one 232 Th/ 208 Pb in the case of monazite), yield concordant dates,

Acknowledgement

The authors thank L. Datas (UMS Castaing, Toulouse), C. Dominici (CP2M-Marseille) and B. Reig (LAAS, Toulouse) for technical assistance. The paper benefited from insightful discussions with many of our students, colleagues, and friends including Amélie Didier, Stéphanie Duchene, Charley Duran, Philippe Goncalves, Alexis Grand'homme, Damien Guillaume, Simon Harley, Wilhelm Heinrich, Jean-Marc Montel, Jean-Louis Paquette, Franck Poitrasson, Andrew and Christine Putnis, and Richard Wirth. The French RENATECH network allows to prepare most of the FIB samples. Collaborations were promoted thanks to PHC Aurora (Ministry of Foreign affairs of France and the Research Council of Norway).

Both the NEEDS (CNRS) French Research Group and the ANR-12-JS06-0001-01 (MONA) provided financial support. We thank Desmond Moser, Editor-in-Chief, S. Reddy, associate editor, A. Cavosie and an anonymous reviewer, whose comments significantly improved the paper.

association of UO 2 nanocrystals (bright points in C) within nanopores (black in C). Modified from Seydoux-Guillaume et al. [2015].


G Fraisse
email: gilles.fraisse@univ-savoie.fr

Y Bai

N Le Pierrès

T Letz

Comparative study of various optimization criteria for SDHWS and a suggestion for a new global evaluation

Keywords: Optimization, SDHWS, simulation, global evaluation, mixed criteria

This study compares various optimization criteria for a solar domestic hot water system (SDHWS). First of all, we present the various parameters used to evaluate a SDHWS. We consider the energetic, exergetic, environmental (CO 2 emissions) and financial (life cycle cost) analysis. Various optimization criteria of a standard solar hot water system are then proposed. The optimized solutions are compared with a standard hot water system. The most suitable criteria take into account both energetic (therefore environmental) and financial evaluations. The most powerful solutions tend to increase the collector area -increasing the solar fraction during the mid-season -and reduce the tank volume, thereby decreasing the thermal losses and financial cost. Some of the usual evaluation criteria for SDHWs cannot be used as optimization criteria because they do not consider the auxiliary heater, resulting in inaccurate indications of the system's performance. Therefore, it seemed important to propose a new evaluation method which integrates the life cycle savings, primary energy savings and CO 2 emission savings with regard to a referenced solution based on a radar diagram of these three fractions. This mode of representation is particularly useful when various auxiliary heaters are compared.

INTRODUCTION

In addition to being a clean energy, solar energy has the advantage of being an energy source that is available throughout the globe, with no losses attributable to transport. On the other hand, its highly variable input requires storage systems auxiliary heaters and specific control systems. Optimally combining solar and conventional systems for the most advantageous control, sizing, etc. is essential to obtaining high efficiency.

Although there is an international standard defining the solar energy vocabulary (ISO, 1999), the performance parameters used to evaluate SDHWSs are numerous and it is never indicated clearly which value is the most relevant to evaluate or optimize a solar heating system: collector productivity, the solar fraction, efficiency, the fraction savings considering energetic, exergetic, environmental or economic criteria? The energy and economic criteria are most often used to optimize solar heating systems.

The economic criterion is often taken as the life cycle cost (LCC): it gives the financial cost of a domestic hot water systems over the life cycle (approximately 20 years) All the annual costs and savings are converted into present worth at a given interest rate and discount rate [START_REF] Duffie | Solar Engineering of thermal processes[END_REF]. Various types of SDHWs were thus optimized: thermosiphon systems [START_REF] Lima | Optimization of tank and flat-plate collector of solar water heating system for single-family households to assure economic efficiency through the TRNSYS program[END_REF] (Salazar et al., Web), forced-circulation systems [START_REF] Kalogirou | Optimization of solar systems using artificial neural-networks and genetic algorithms[END_REF], direct-coupled photovoltaic and pump systems [START_REF] Cardinale | Economic optimization of low-flow solar domestic hot water plants[END_REF], and even a solar heating system combined with a wind power plant [START_REF] Bakos | Technoeconomic assessment of a hybrid solar/wind installation for electrical energy saving[END_REF].

The solar fraction (ISO, 1999) is widely used for both solar domestic hot water systems and combisystems.

Numerous recent studies used this parameter as a reference: [START_REF] Trillat-Berdal | Experimental study of a ground-coupled heat pump combined with thermal solar collectors[END_REF][START_REF] Badescu | Renewable energy for passive house heating: Model of the active solar heating system[END_REF][START_REF] Chow | Potential application of a centralized solar water-heating system for a high-rise residential building in Hong Kong[END_REF]Jordan and Furbo, 2005;[START_REF] Lund | Sizing and applicability considerations of solar combisystems[END_REF].

From an exergetic point of view, numerous studies have been conducted to optimize components (collector, storage tank) or the overall system [START_REF] Singh | Exergetic analysis of a solar thermal power system[END_REF][START_REF] Torres-Reyes | Optimal performance of an irreversible solar-assisted heat pump[END_REF][START_REF] Torres-Reyes | Optimal process of solar to thermal energy conversion and design of irreversible flat-plate solar collectors[END_REF][START_REF] Xiaowu | Exergy analysis of domestic-scale solar water heaters[END_REF][START_REF] Gunerhan | Exergetic modeling and performance evaluation of solar water heating systems for building applications[END_REF][START_REF] Fernandez-Seara | Experimental analysis of a domestic electric hot water storage tank. Part I: Static mode of operation[END_REF][START_REF] Zmeureanu | Energy and exergy performance of residential heating systems with separate mechanical ventilation[END_REF][START_REF] Hepbasli | Exergetic modeling and assessment of solar assisted domestic hot water tank integrated ground-source heat pump systems for residences[END_REF]. Such systems generally have low exergetical efficiency because of the energy quality degradation of the incident solar irradiation incident on the collector (stemming from a solar source with a temperature greater than 5800 K) towards a quantity of heat close to the ambient temperature. Another definition can be used by taking into account the Carnot coefficient at the collector's temperature instead of the sun's temperature [START_REF] Pierrès | Procédé solaire de production de froid basse température (-28°C) par sorption solidegaz[END_REF].

The objective of energy optimization is generally to minimize a function including energy consumption, solar production and a penalty when the water tap temperature is lower than the set-point (Prud'homme and Gillet, 2001): Mixed criteria are also used as criteria for optimization. For example, the ratio of the additional investment cost by the primary energy savings (€/kWh-ep) can be estimated and compared to a reference system. In the [START_REF] Bales | Generic system 11: Space heating store with DHW load side heat exchangers and auxiliary boiler[END_REF] study, the author compared the performance of a generic combisystem (space heating store with DHW load side heat exchanger and an external auxiliary), using several optimization criteria: the combisystem criterion and the fraction saving indicator, which includes the penalty function of the solar combisystem in the fractional energy savings. Up to ten parameters were optimized: e.g., the auxiliary heated tank volume, the thermostat setting for the auxiliary heated area, the position and size of the DHW heat exchangers, the insulation thickness and the collector area. Another study was based on the annuity of the overall investment (20 years of operation) over the annual solar heat delivered to the tank [START_REF] Krause | Investigations on optimizing large solar thermal systems[END_REF]. Compared with the initial system (the estimated solar heat cost of this system is 9.7 c€/kWh), the best configuration reduced the ratio (18%), because it increased solar energy use and reduced the initial investment.

One of the most widely used numerical tools to simulate solar heating systems is the TRNSYS software [START_REF] Klein | TRNSYS a transient system simulation program[END_REF]. As regards optimization, various software packages have been developed such as Optilib [START_REF] Gabriele | Optilib. An optimization program library: user manual[END_REF] and Genopt [START_REF] Wetter | GenOpt, Generic Optimization Program[END_REF]. Most optimization algorithms are adapted to the SDHWs: genetic algorithms, Hooke-Jeeves algorithms, etc. [START_REF] Krause | Investigations on optimizing large solar thermal systems[END_REF][START_REF] Wetter | Comparison of a generalized pattern search and genetic algorithm optimization method[END_REF].

This study provides the various quantities necessary to evaluate a system considering energetic, exergetic, environmental and economic criteria. We considered a conventional installation (without a solar heating system) and a standard solar installation, and compared the optimized solar installations based on energetic, environmental, economic and mixed criteria. The optimization was carried out by combining TRNSYS and GenOpt simulation program. The TrnOpt component of the TESS library (www. tess-inc.com) was the interface between both software packages. The optimized parameters were the collector area, the tank volume and the flow rate in the collector loop. Finally, a global approach to evaluation based on the radar diagram of three kinds of savings (primary energy, CO 2 emissions and financial cost) will be proposed.

The evaluation criteria

The energetic and thermal quantities for a conventional domestic hot water system (without a solar heating system) and a standard SDHW with an integrated auxiliary heater (electric or heat exchanger) will be described first. On the basis of these values, the evaluation criteria of SDHWs will be presented.

Description of a conventional installation

Qnet conv = Ql conv + Qd conv (kWh) ( 2 )
The final energy Qf conv , which is the energy as the consumer receives it, is obtained from Qnet conv with the transformation efficiency ηf-net conv . Also, the primary energy Qp conv is obtained from Qf conv with ηp-f conv .

Qf conv = Qnet conv / ηf-net con (kWh) ( 3 ) Qp conv = Qf conv / ηp-f conv (kWh) ( 4 )
The coefficient of performance of the conventional installation is:

COP conv = Qd conv / Qf conv ( 5 )
To take into account periods when the water tap temperature Thw is lower than the set-point temperature Tset 1 (typically 55°C), penalty energy Q 1 needed to heat the water from Thw to Tset 1 is considered. The temperature Thw can be less than Tset 1 because of the off-peak hour and peak hour rates. The peak hour period forbids electrical water heating. We note Qnet pen-conv the expression of Qnet conv integrating the penalties In addition, we added an equation that can penalize the installation when the water tap temperature Thw is lower than the set-point Tset 2 (Tset 2 ≤ Tset 1 ):

Q 1 and Q 2 .       - ⋅ ⋅ ⋅ ∆ = ) ( , 0 max 1 3 . 1 Thw Tset Cp m t Q (kWh) ( 6 )
      - ⋅ ⋅ ⋅ ∆ = X Thw Tset Cp m t Q ) ( , 0 max 2 3 . 2 (kWh) ( 7 )
The temperature Tset 2 corresponds to the temperature limit below which the discomfort is penalized. The penalty level is expressed with the parameter X, representing the degree of discomfort. A value X=4 means almost forbidding the periods when Thw ≤ Tset 2 , whereas a value X=2 is much less penalizing. In that case, we accept some periods of discomfort as long as they are rare and it reduces the installation size.

Similarly, as in the study on solar combisystems [START_REF] Weiss | Solar heating systems for houses, a design handbook for solar combisystems[END_REF], we defined the heat demand penalty Qd pen-conv , which integrates both quantities Q 1 and Q 2 . It compares several installations with the same loads and does not reach unacceptable solutions:

( ) ( ) [ ] { } X conv pen Thw Tset Thw Tset Cp m t Qd ) ( , 0 max ) ( , 0 max 2 1 3 . - + - ⋅ ⋅ ⋅ ∆ = - (kWh) ( 8 )
The final energy Qf pen-conv was obtained from Qnet pen-conv (which is the sum of Qnet conv and Qd pen-conv ) by integrating the efficiency ηf-net conv . Also, the primary energy Qp pen-conv was obtained from Qf pen-conv with ηp-

f conv . Qf pen-conv = Qnet pen-conv / ηf-net conv (kWh) ( 9 ) Qp pen-conv = Qf pen-conv / ηp-f conv (kWh) ( 10 )
2.2. Description of the solar installation with an integrated auxiliary heater [START_REF] Duffie | Solar Engineering of thermal processes[END_REF][START_REF] Peuser | Installations solaires thermiques[END_REF] The diagram of the solar installation is shown in Fig. 2. Since the outlet tank temperature can be greater than Tset 1 , mixing with cold water (Tcw) is necessary, not to exceed Tset 1 . The energy demand for the set-point

Tset 1 is Qnet aux . As previously, the heat demand penalty Qd pen (same definition as equation ( 8)) is added to Qnet aux . The tank's heat losses are Ql. The energy supplied to the cold water between the inlet and outlet of the tank is QL.

The final energy Qf aux (respectively Qf pen-aux ) is obtained from Qnet aux (respectively Qnet pen-aux which is the sum of Qnet aux and Qd pen ) by using the efficiency ηf-net aux . The additional electric consumption of the solar system (pump and controller) should be integrated into Qf aux . Also, the primary energy Qp aux (respectively Qp pen-aux ) is obtained from Qf aux (respectively Qf pen-aux ) with ηp-f aux .

The incident solar energy on the collector (area: Ac) is H (kWh/m².year). The energy transmitted by the solar collector to the heat transfer fluid is Qcol. The heat losses from the collector loop are Ql,pipe. The solar energy supplied by the collector loop to the tank is Qsol. The thermal balances are:

Qnet aux + Qsol = Ql + QL (kWh) ( 11 ) Qd = QL (kWh) ( 12 ) Qcol = Ql,pipe + Qsol (kWh) ( 13 )
We also defined the solar installation's coefficient of performance (COP):

COP = Qd / Qf aux ( 14 )

The various evaluation criteria

Energetic quantities

The main quantities used to evaluate the energetic performance of SDHWs are shown as:

-Solar collector productivity:

Prod = Qcol / Ac (kWh/m²) ( 15 )
-Collector efficiency:

ηcol = Qcol / (H.Ac) ( 16 )
-Solar fraction = "energy supplied by the solar plant of a system divided by the total system load" (ISO, 1999):

Fsol = Qsol / Qd ( 17 )
In the case of an integrated auxiliary heater, it is very difficult to determine the exact amount of energy supplied by the solar hot water system to the conventional part of the system. For this reason, we considered the energy supplied by the heat exchanger.

-Sometimes a different definition of the solar fraction Fsol is used (e.g. in the Polysun software http://www.solarenergy.ch and T-Sol http://www.solardesign.co.uk):

Fsol'= Qsol / (Qsol + Qnet aux ) ( 18 )
-Final energy savings:

Qf-sav = (Qf conv + Qf pen-conv ) -(Qf aux + Qf pen-aux + Qpar) (kWh) ( 19 )
with Qpar the parasitic energy for the collector loop (pump, the control unit, and possibly the boiler pump)

-Primary energy savings:

Qp-sav = (Qp conv + Qp pen-conv ) -(Qp aux + Qp pen-aux + Qpar * 2.58) (kWh) ( 20 )
(The term 2.58 is usually used in France to convert electrical energy into primary energy)

-The final energy fractional rate:

Fsav-f = Qf-sav / (Qf conv + Qf pen-conv ) ( 21 )
-The primary energy fractional rate:

Fsav-p = Qp-sav / (Qp conv + Qp pen-conv ) ( 22 )

Exergetic quantities

Exergetic efficiency not only takes into account the quantities of energy used, but also their quality. The exergetic efficiency ηex of the solar system with an integrated auxiliary heater is obtained with the equation:

( )

dt t Tabs t Ta Ac t G dt t S S t Ta t h h t m ex ux en ux cw Thw Tcw Thw ) ) ( ) ( 1 ( ) ( Qpar Qf Qf ] ) ( ) ( ) ( )[ ( a p a 3 - ⋅ ⋅ + + + - ⋅ - - = ∫ ∫ - & η ( 23 )
with Ta the ambient air temperature [K]; Tabs the absorber temperature [K]; h the water specific enthalpy [J/kg]; S the water entropy [J/(kg.K]; G the solar radiation [W/m²] The numerator of this equation represents the exergy gain of the water supplied to users [START_REF] Gunerhan | Exergetic modeling and performance evaluation of solar water heating systems for building applications[END_REF]. The denominator is the exergy entering the system: the electrical energy is pure exergy but the heat transferred to the solar panel is weighted by the Carnot factor [START_REF] Stitou | Development and comparison of advanced cascading cycles coupling a solid/gas thermochemical process and a liquid/gas absorption process[END_REF][START_REF] Sorin | Thermodynamic techniques for the conceptual design of thermochemical refrigerators using two salt materials[END_REF].

Environmental quantities (CO 2 emissions)

A solar installation can save energy that would have been supplied by a conventional heating system and that would have produced greenhouse gases. This quantity of greenhouse gases depends on the quantity of the conventional energy saved. By calculating the energy saved, we can calculate the reduction in greenhouse gases. In France, the quantity (kg) of CO 2 per kWh-PCI avoided for the DHWS is (ADEME, 2005): Natural gas 0.234 Fuel oil 0.300

Calor or butane gas (LPG) 0.234 Electricity 0.040

The electricity value is very low because of the huge proportion of nuclear energy used in France. We calculated the mass of non-emitted CO 2 (m CO2 ) and the fractional savings of CO 2 with regard to the conventional solution Fsav-CO 2 . [START_REF] Sae | Society of Automotive Engineers (SAE)[END_REF][START_REF] Fuller | Energy Price Indices and Discount Factors for Life-Cycle Cost Analysis[END_REF] The life cycle cost (LCC) method considers the costs and savings over the life cycle of the system, which is roughly 20 years for a SDHW. Optimizing the system means obtaining the lowest LCC or the highest life cycle savings (LCS). The annual costs and savings are converted into present worth values at a given interest rate and discount rate.

Economic quantities (life cycle cost)

The calculation of the solar heating system cost includes (see Annex 1):

-Initial investment: calculated according to the cost of the equipment and manpower associated with the installation of the system, and the tax reduction allowed (50% on the equipment in France, after having deducted the grants available)

-Annual maintenance and operation costs

The cost of a conventional installation consists in costs relating to the hot water tank, manpower, and annual operation and maintenance.

The annual savings (Sav) is the difference in maintenance and operation costs between the conventional and the solar installation. The additional investment cost between the solar and conventional installations are noted as AI.

By assuming equal yearly payments YP, the life cycle cost is calculated using the follow equation:

LCC = I + K * YP (€) ( 24 )
with I the investment [year 0]; YP the annual payments ( )

1 1 - - ⋅ = r r r K n ( 25 ) d i r + + = 1 1 ( 26 )
with i the interest rate (0.02 in France); d the discount rate (0.04 in France); n is given as 20 years (life cycle of the system)

The inflation rate corresponds to the increase in the cost of living. The discount rate makes it possible to return financial streams that are not directly comparable, because they occur at different dates, on the same basis. It can not only compare them but can also make arithmetic operations on them. The life cycle savings (LCS) over the life cycle can be calculated.

LCS = K * Sav -AI (€) ( 27 )
with AI the additional investment[€]; Sav the annual savings (€)

It is also possible to calculate the pay back time:

( ) r Ln r r Sav AI Ln N       - ⋅ + = 1 1 ( 28 )
Finally, we define the financial fractional savings compared with a conventional solution (without a solar heating system): The tank set-point temperature was Tset 1 = 55°C and the comfort penalty was calculated from the temperature

Tset 2 = 40°C (X=2).
The mass of non-emitted CO 2 was calculated by using the 0.089 kg value per kWh of electricity saved (this value was much lower than the European average).

The optimization criteria studied

The optimized parameters were the collector area, the tank volume and the collector loop flow rate. We limited the number of parameters to be optimized so as to simplify the comparisons. Various optimization criteria were used. The performance quantities were the following:

-Energetic criterion (to be minimized): Qf aux + Qf pen-aux + Qpar (kWh)

(optimizing in terms of Fsav-f)

-Exergetic criterion (to be maximized): ηex

(-)
-Financial criterion (to be minimized): LCC (€)

-Mixed criterion 1 (to be maximized):

Fsav-f * Fsav-LCC (-)
-Mixed criterion 2 (to be minimized): (Qnet aux + Qd pen + Ql) / Qsol (-)

-Mixed criterion 3 (to be minimized): AI / Qf-sav (€/kWh)

Mixed criterion 1 was designed to reflect the highest financial savings combined with the final energy savings to obtain a compromise between these two criteria. The second mixed criterion was the ratio of the energies to be minimized (tank losses, auxiliary heating and comfort penalties) divided by the energy to be maximized (the solar energy supplied to the tank). Therefore, mixed criterion 2 will be noted Qmin / Qsol. The third criterion was used in the [START_REF] Bales | Generic system 11: Space heating store with DHW load side heat exchangers and auxiliary boiler[END_REF] study; it reduced the additional investment with regard to the energy saving.

The optimization was carried out using the Hooke-Jeeves method [START_REF] Wetter | Comparison of a generalized pattern search and genetic algorithm optimization method[END_REF]. We used the default values of the interface TrnOpt, which combines TRNSYS [START_REF] Klein | TRNSYS a transient system simulation program[END_REF] and GENOPT [START_REF] Wetter | GenOpt, Generic Optimization Program[END_REF] We did not consider the initial values of these three parameters identical to the solar standard installation (see Annex 2) so as not to be too close to an a priori high-performance installation. The tank was modeled with the TRNSYS TYPE 140 considering 20 temperature nodes. The tank exit temperature was mixed with the cold water temperature using a three-way valve, noted Thw. Annual simulations were carried out with a 12-min time step.

The optimization results

The objective of the optimization was not to look for a common solution because the results depended directly on the hypotheses. These vary widely considering the SDHWS design (we assumed, for example, a relative height of the auxiliary heater in tank of 0.5), financial costs (interest and discount rates, energy price, the grants, etc. (see Annex 1)). On the other hand, comparing the solutions obtained can indicate which optimization criteria are the most relevant.

Improvements obtained on each of the optimized criteria

The improvement on the optimized solar installations compared to the standard solar installation with each of the criteria is shown in Table 1. The improvement on each criteria ranges from 10% to 256%.

Figure 3 shows the variation of each of the optimized parameters and the performance values during the iterations in the case of the LCC optimization. It can be seen that LCC performance decreased during the iterations. The variations in LCC did not decrease continuously because of the optimization method retained.

The Hook-Jeeves method tests new parameters based on the preceding parameters by considering an increase or a decrease, which explains the discontinuous progression of the LCC. The optimization stops automatically depending on the parameter defining the maximum number of step reductions. The performance values do not change much after 22 iterations. The minimum LCC value was obtained at the 40th iteration; the parameter values were then 8.3 m², 313 l and 17.2 l/(h.m²). These sizing parameters do not necessarily correspond to the material made available by manufacturers. Ideally, one would vary each parameter during the optimization phase by taking into account the data for each of the components actually available.

Presentation of all the results

Table 2 shows most of the evaluation criteria for the standard solar installation, the conventional solution (last column) and all the solar solutions obtained after optimization. The values of the parameters are indicated at the top of the table.

Optimizing a single criterion is not necessarily judicious in that because there is a risk of obtaining lower performance for the non-optimized criteria. The energy savings (Fsav-f) is always greater than the financial savings (Fsav-LCC) considering the low price of electricity and the additional investment for the solar plant.

A substantial increase in the price of energy could balance both savings rates.

If only these two criteria are relevant, it is easy to classify the optimized solutions with Fig. 4. A solution will provide greater performance in terms of these two criteria the farther the point is from the origin of axes, i.e.

the higher the product Fsav-LCC by Fsav-Ef is (mixed criterion 1). Mixed criteria 1 and 2 offer the best performance, followed very closely by the LCC and energy criteria. The last two criteria are AI/Qf-sav and exergy. The exergy criterion favors energy performance, whereas the AI/Qf-sav mixed criterion seems to benefit financial performance. Therefore, the most helpful criteria are the four criteria: mixed criteria 1 and 2, energy and LCC. The performance thus obtained for this group is approximately Fsav-LCC = 14% and Fsav-Ef = 67%. The standard solar configuration seems much less attractive (Fsav-LCC= 6% and Fsav-Ef = 47%).

This demonstrates the potential of performance improvement based on combining an optimization tool with an energy system simulation tool.

It is important to note that energy and mixed criteria 2 do not directly take the cost of the installation into consideration. Consequently, one risks obtaining the collector's oversize area if a large variation is brought into play. Moreover, the collector's area obtained with these two criteria corresponds to the maximum value authorized in our study (10 m²). This value corresponds to double the standard sizing (approximately 5 m²)

and can be considered to be oversized. A larger variation range was not used so as not to penalize the calculation time. The LCC and mixed criteria 1 integrate both installation cost and energy consumption. It will be necessary to consider the LCC if we wish to encourage reducing the installation pay-back time, and mixed criterion 1 to obtain an energy and financial compromise. Even if the progression in energy prices is an unknown over the life cycle of the installation, it seems necessary to take cost into account. Knowing that the price of energy can only increase, the pay-back time of the installation will actually be lower.

Other mixed optimization criteria were studied, but not presented in this study, for example mixed criterion 4, which consists in minimizing the cost and the final energy with regard to a reference solution (solar installation or not) in the following way:

-Mixed criterion 4 :

(LCC / LCC ref )² + ( (Qf aux + Qf pen-aux + Qpar) / Qf ref )² ( 32 )
-Solar Fraction: Fsol'

These criteria are very successful because they provide results that are very close to mixed criterion 1. The optimization criterion Fsol' presents the problem of not integrating the financial cost. The solution obtained therefore risks being oversized (moreover, the maximum collector area was obtained in this case).

Nevertheless one should not lose sight of the technological feasibility of the optimized solution. For example, over-sizing the solar collector requires a specific device that can prevent the risks of deterioration caused by over-heating (with an additional cost that should be taken into account).

The disadvantage of Fig. 4 is that it does not integrate the environmental issue (primary energy and greenhouse gas emissions). The following will illustrate how to define a global evaluation.

-

Energy issue

The high solar fraction Fsol is obtained in the summer period (roughly 150%), providing the best solutions because the areas are high (9.9 m² for mixed solution 1), providing an incentive to over-size the collector area so as to increase solar gains during the mid-season. Nevertheless, monitoring the system for over-heating during the summer period will remain necessary (technological systems are used with combisystems).

For all the solar installations, the parasite energy used by the pump and controller is not low since it accounts for approximately 5.7% of the final energy from the auxiliary heater Qf aux when LCC is optimized (flow rate, 17.2 l/h.m²) and 7.7% for the exergetic optimization (flow rate, 65 l/h.m²). It is therefore normal that the energy optimization tends to reduce the fluid flow rate compared to a standard solar installation. A direct coupling of a photovoltaic module with the pump would save parasite energy.

The coefficient of performance (COP) of the best solution is approximately 2.5, 1.4 in the standard solar installation and only 0.7 in the conventional solution.

High collector efficiency is not necessarily synonymous with optimal performance. The efficiency in the standard configuration is 43% versus 28% in the optimized solution with mixed criterion 1. This result is not surprising because by decreasing the thickness of the tank insulation, we can expect to increase the collector efficiency.

-

Exergy issue

The solution obtained with the exergy optimization shows very high efficiency (ηex=97%). The exergy criterion made the collector work with a high flow rate and a large tank volume so as to reduce the collector temperature. The quality of the input exergy is therefore very low. In addition, the exergy criterion tends to minimize the use of the auxiliary heater, which means a large collector area and high solar energy Qsol supplied to the tank. The maximum values authorized by the three optimized parameters (flow rate, area and volume) were determined by the optimization. On the other hand, the tank's thermal losses and the initial investment are penalized and the pay-back time is the highest. Furthermore, the large tank volume leads to high Ql losses but limits the over-heating in spite of the large collector area.

If we consider the mixed criteria, the most advantageous solutions do not have the highest exergy efficiency (ηex = 33% in mixed criterion 1), making it difficult to classify the solutions solely on the exergetic criterion.

It would be necessary to mix the exergetic and financial criteria.

We have also considered a modified exergy parameter taking into account not the exergy of the heat entering the solar panel as in equation ( 23), but rather the exergy of the incident solar radiation, following the Petela expression [START_REF] Petela | Exergy of undiluted thermal radiation[END_REF]. This criterion presented low values between 17% (using equation ( 23) for optimization) and 25% (using the Petela expression for optimization), because solar radiation has a high exergy content. The optimization also leads to a large solar collector (9.75 m 2 ), a large storage tank (587 l) and operation at a high flow rate (65 l/hm 2 ).

-Environmental impact

To reduce greenhouse gas emissions, optimized solutions with mixed criterion 1 can save 309 kg of CO2 per year compared to the conventional solution, corresponding to 94 kg more (309-215) compared to the standard solar heating installation. Concerning the fossil and fissile resources, the annual primary energy savings is 10,548 kWh-ep with solar energy according to mixed criterion 1, and 7386 kWh-ep in the standard solar installation.

-

Cost

The pay-back time of the basic solar installation is 15.2 years, and 12.9 years for the solution obtained with the optimization using mixed criterion 1. The annual savings (auxiliary heater, pump and controller) thus obtained is 288 € (taxes included); the annual cost of electricity consumption in the conventional installation being estimated at 553 €. The payoff period is relatively high given the current cost of electricity in France. This is not the case in all European countries such as Denmark, where the price of electricity is twice as high.

Even if energy prices continue to increase, thus reducing the payoff period, it is crucial to attempt to reduce the investment cost of a SDHWS installation using, for example, polymer materials.

-

Comfort

All the solar solutions (standard or optimized) present a more advantageous distribution of hot water temperature Thw in terms of comfort than the conventional solution (see the temperature distribution when

Thw is below 50°C in Fig. 5). The Qd pen penalty integrated into the various optimization criteria makes it possible to optimize the solutions with a high comfort level (Thw is close to the set-point 55°C).

The conventional solution is penalized by thinner tank insulation. Moreover, the solar energy supplied to the tank increases its temperature during the day, tapping being defined in an unpredictable way according to the method described in [START_REF] Shah | A solar combisystem based on a heat storage with three internal heat exchangers[END_REF]. It can be seen that solar energy provides both energy savings and improved comfort.

Relations between the energetic criteria

We estimated three important energetic criteria (Fsol, Fsav-f and COP). Improving these three criteria requires an increase in collector area (which reduces the productivity of the collector and increases the investment cost). The criterion, that seems a priori the most valuable is fractional savings Fsav-f compared to an equivalent conventional installation. The relation between this criterion and the two others needs to be calculated for better comparison.

It is easy to determine the relation between Fsav-f and COP from the equations. Assuming a water tap temperature of 55°C (Qf pen equals zero):

Fsav-f = 1-(Qd / COP + Qpar) / Qf conv ( 33 )
The function Fsav-f = f(COP) increases as long as Qd and Qpar remain fairly constant and Qf conv is fixed. It is thus possible to replace the Fsav-f optimization criterion with the COP criterion. The relation between Fsav-f and COP is easily defined in the following conditions:

when Fsav-f = 0:

COP = Qd / (Qf conv -Qpar) ( 34 ) -when COP increases, Fsav-f becomes 1-Qpar / Qf conv ( 35 )
Fig. 6 shows how this function evolves.

The solar fraction Fsol gives important information on the installation sizing: how much of the energy supplied by the system is solar? Nevertheless, the solar fraction Fsol is not representative of the installation's global energy performance because the solar energy supplied to the tank can be useless (high solar production during summer) or lost (high tank losses because of large volume or insufficient insulation). The solar fraction

Fsol determined in summer contributes additional information on the sizing of the installation for high solar radiation.

Qualitative considerations allow one to study the function Fsav-f = f(Fsol):

-When Fsol tends towards 0, the solar energy supplied to the tank becomes very low. We thus approach the conventional installation and we can consider that Fsav-f tends towards 0. It is actually a bit more complicated because the standard solar and conventional installations are very different (for example, there is more tank insulation in the first case).

-It is relatively easy to obtain Fsol=100% by over-sizing the collector area and the tank volume (we could even reduce its insulation) so as to supply a large amount of solar energy and thus obtain Qsol = Qd. On the other hand, obtaining Fsav-f =100% is unrealistic because of inadequate solar energy incident at certain times during the winter.

-The various solutions studied are presented in Fig. 7. The maximum for Fsav-f is slightly lower than 70%, which corresponds approximately to Fsol = 90%.

Since the function is not monotonic, we cannot use Fsol as a criterion for optimization.

Comparison of the various optimization methods

We also used the Hooke-Jeeves method and the Nelder-Mead-O'Neill and Coordinate Search methods to compare LCC optimization (the same initial values are used). We used the Hooke-Jeeves method with initial penalizing values (2 m², 0.2 m 3 and 65 l/hm²) for the LCC as well. The LCC values obtained at the end of the optimization are very close for all the methods.

Global evaluation approach

The most advantageous optimization criteria have been shown through the financial and energy savings. Here we study a global evaluation of an installation based on a radar diagram of savings with regard to a reference installation compared to the conventional installation. The radar diagram indicates three fractional savings that we believe to be essential in savings relative to the LCC (Fsav-LCC), primary energy (Fsav-p) and greenhouse gas emissions (Fsav-CO 2 ). The first evaluation criterion is important for the installation owner and the last two are important from an environmental point of view (limitation of greenhouse gas emissions and conservation of fossil or fissile resources). For readability, the best performance corresponds to the outside triangle (and conversely).

Fig. 8 shows that the standard solution is the least beneficial. The lines relative to the various solutions are parallel between Fsav-p and Fsav-CO 2 because these two criteria are equivalent in terms of the electricity choice for the auxiliary heater (it would obviously be interesting to compare installations with various auxiliary energies).

Conclusion

We have compared various optimization criteria for a SDHWS with an electric auxiliary heater. It is important to consider criteria integrating both energy and financial aspects. The LCC optimization criteria, mixed 1 and mixed 4, are the most indicative. The solutions obtained after optimization show that it is better to oversize the collector area than the tank volume (the thermal losses and financial cost are high).

Even if the results obtained are closely related to conditions in France through the parameters selected, a number of general conclusions can be drawn. For example, the evaluation criteria used for the solar installations are often not representative of actual performance because they are often based on the solar energy supplied to the tank (solar fraction and productivity) without considering the auxiliary heater. In addition, the current environmental problems require that the greenhouse gas emissions and the consummate primary energy also be considered. This is the reason why we developed a new representation of global evaluation in the form of radar diagram based on fractional savings Fsav-p, Fsav-CO 2 and Fsav-LCC.

The highest-performance solutions obtained should not be considered ideal because they are not necessarily realistic given, on the one hand, the sizing restrictions related to the materials available, and, on the other hand, the simplifying hypotheses advanced. For example, only three optimization parameters were considered and we did not take into account the technological precautions necessary when the collector is oversized.

Nonetheless, this article proposes a starting point for studies that are more complete (taking other countries into account, etc.) and more realistic.

Substantial work remains to be done to obtain a reliable tool for evaluation and optimization of the numerous SDHW configurations. It would be particularly necessary to:

-integrate the various design possibilities: direct coupled photovoltaic and pump, the energy choice for the auxiliary heater.

-modify the hypotheses so that countries other than France could be studied. Nevertheless, the price of a kWh of electricity is relatively low in most European countries (on average approximately 14 c€/kWh).

The payoff period therefore risks being relatively long in general. Even if a rise in price can be expected in the coming years, it is important to design new, less expensive SDHWS using, for example, polymer materials and simplifying the systems.

-evaluate more precisely the financial cost: manpower depending on the component sizing (collector area, etc.), the price of the devices to prevent over-heating during summer, the cost of the insulation (tank and pipes), etc. It would also be necessary to integrate various changes in the price of energy.

-add parameters to be optimized: height of the auxiliary heater in the tank, the thickness of insulating material, etc.

-integrate other optimization parameters related to the production of primary energy. For example, the availability of water in certain arid areas poses the problem of producing electricity based on a cycle vapor power plant.

Nomenclature

  the electrical consumption of the auxiliary heaters [W]; P sol the solar energy collected [W], P pump the power required by the pump to drive the fluid in the collector loop [W]; T s1 the temperature in the upper part of the storage tank [°C], T set the set-point temperature[°C], α is a trade-off factor.

Fig. 1 .

 1 Fig. 1. shows the diagram of a conventional installation with an integrated auxiliary heater. The net energy demand for the tank set-point temperature Tset 1 is Qnet conv . The tank thermal losses are Ql conv . The energy supplied to the cold water between the inlet and outlet of the tank is Qd conv . The tank energy balance is:

m

  the water tap mass flow rate [kg/s]; ∆t the time step [s]; Cp specific heat [J/kgK]

Fsav

  The characteristics of the conventional and solar installations are described in Annex 2. Only the electric auxiliary heater was studied. The energy demand for hot water was 3450 kWh per year (200 l at 55°C). The weather conditions corresponded to the city of Lyon. The cold water temperature varied between 9 and 19°C over the year. The price of electricity was 0.1074 €/kWh with the auxiliary heater functioning during the periods2h-6h, 12h-14h and 20h-24h. This price was applied to Qf conv , Qf pen-conv , Qf aux , Qpar and Qf pen-aux . The manpower cost of the solar installation was set at 1688 € (taxes included) for reasons of simplification. On the other hand, the cost of the equipment depended on the size of the collector and the tank (see Annex 1).The efficiency coefficients of the conventional and solar installations (electrical auxiliary heater) were: ηf-net conv = ηf-net aux = 0.95 ( 30 ) ηp-f conv = ηp-f aux =1 / 2.58 ( 31 )

  software. The optimization algorithm makes it possible to set the minimal and maximal values for each of the optimized parameters. The following values were considered: -Collector area: 2-10 m² (initial: 6 m²) -Tank volume: 0.2-0.6 m 3 (initial: 0.4 m 3 ) -Flow rate in the collector loop: 5-65 l/h.m² (initial: 35 l/h.m²)

Qpar-

  parasitic energy for the collector loop pump and the control unit (kWh/y)Qsolenergy delivered by the solar loop to the store over the year (kWh/y) / water mixture mass flow rate (kg/s) Tax reduction: 50 % on the equipment (grant have to be deduct) solar investment (taxes included)-Tax reduction = 50% * ( equipment -grants * equipment / (equipment + manpower ) )-equipment € = c1 + c2 * Sc + c3 * V + c4 * Lpipe c1 coefficients a 1 =4.26 W/m².K and a 2 =0.0047 W/m².

Fig. 1 .Fig. 4 .Fig. 5 .Fig. 6 .

 1456 Fig. 1. Conventional domestic hot water system

Table 1 .

 1 Improvement obtained for each optimized criterion

	Criterion	Performance function	Standard	Optimized	Improvement
	Energy	Fsav-f	47.0%	68.1%	45%
	Exergy	ηex	39.9%	97.4%	144%
	Cost	LCC	9653 €	8729 €	10%
	Mixed 1	Fsav-f * Fsav-LCC	2.7%	9.7%	256%
	Mixed 2	Qmin / Qsol	1.8	0.8	54%
	Mixed 3	AI / Qf-sav	0.94 € /kWh	0.82 €/kWh	13%

Table 2 .

 2 Evaluation criteria for the installations (standard solar, optimized solar and conventional systems)