Mélanie Vital
email: ingagr.melanie.vital@gmail.com

Damien Daval
email: ddaval@unistra.fr

Gilles Morvan

Daniel E Martinez

Michael J Heap

Barite growth rates as a function of crystallographic orientation, temperature, and solution saturation state

Keywords: barite, anisotropic growth, geothermal reservoir, scaling, growth rate law Ba 2+

Barite growth kinetics was investigated as a function of crystallographic orientation for temperatures between 10 and 70 °C, and initial saturation indices (SI) of 1.1 and 2.1. The growth rates were estimated for (001), (210), and (101) faces using vertical scanning interferometry. Overall, face-specific barite growth rates (𝑟 ( ) ) can be successfully described by the following rate law:

where 𝐴 ( ) and 𝐸 ( ) represent the face-specific Arrhenius pre-exponential factor and activation energy, respectively, R is the gas constant, and T refers to the absolute temperature. In addition, because of the modest growth anisotropy of the various investigated faces, the following isotropic rate law can be used to satisfactorily account for the measured rate data:

with average values of A = exp(13.59) nm.h -1 and Ea = 35.0 ± 2.5 kJ.mol -1 . Over the range of conditions investigated in the present study, our results suggest that barite growth kinetics is surface-controlled, while possibly verifying the principle of detailed balancing and micro-reversibility. These results imply that previous modeling exercises of steady-state barite growth based on isotropic rate laws may remain valid, at least over the range of conditions investigated in the present study.

Introduction

Several industrial, environmental, and geological concerns require a detailed understanding of barite (BaSO4) growth kinetics. From an industrial standpoint, barite scale formation spontaneously occurs within geothermal wells [START_REF] Bozau | Hydrogeochemical modelling of corrosion effects and barite scaling in deep geothermal wells of the North German Basin using PHREEQC and PHAST[END_REF] and inside fractures within geothermal reservoirs [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] (Fig. 1), which disturb both the continuous running of geothermal power plants and the efficiency of fluid circulation due to the reduction in reservoir permeability [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] . The development of new reagents that could efficiently prevent or delay the barite scale formation crucially relies on our understanding of the mechanisms of barite precipitation [START_REF] Ruiz-Agudo | The effect of a copolymer inhibitor on baryte precipitation[END_REF][START_REF] Ruiz-Agudo | Mechanistic Principles of Barite Formation: From Nanoparticles to Micron-Sized Crystals[END_REF] .

Figure 1. Photograph of a section of 78 mm-diameter, cylindrical sandstone core (from the Buntsandstein unit) taken from the Soultz-sous-Forêts (France) EPS-1 exploration borehole, a borehole fully cored from a depth of 930 m to a depth of 2227 m in 1990-1991 (see [START_REF] Gérard | The Soultz-sous-Forêts project[END_REF][START_REF] Baria | European HDR research programme at Soultz-sous-Forêts (France) 1987-1996[END_REF][START_REF] Gérard | The deep EGS (enhanced geothermal system) project at Soultz-sous-Forêts (Alsace, France)[END_REF] for information on the Soultz-sous-Forêts drilling project). The sandstone core, red/green in color, was taken from a depth of 1374 m and contains a 1-2 cm-wide fracture (in the center of the core). The fracture is partially filled with barite crystals (white in color) that have precipitated from the circulating geothermal fluids.

From an environmental standpoint, the formation of barite is known to be frequently associated with the accumulation of elevated concentrations of radionuclides such as 226 Ra, which has stimulated several studies dedicated to quantify the uptake of radionuclides during barite precipitation (e.g., [START_REF] Curti | Radium uptake during barite recrystallization at 23±2°C as a function of solution composition: An experimental 133Ba and 226Ra tracer study[END_REF][START_REF] Rosenberg | Nucleation and growth kinetics of RaxBa1-xSO4 solid solution in NaCl aqueous solutions[END_REF][START_REF] Klinkenberg | Uptake of Ra during the recrystallization of barite: a microscopic and time of flight-secondary ion mass spectrometry study[END_REF][START_REF] Brandt | Replacement of barite by a (Ba, Ra) SO4 solid solution at close-to-equilibrium conditions: A combined experimental and theoretical study[END_REF] ).

Finally, from a geological standpoint, barite formation provides insight into past seawater chemistry (e.g. [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF][START_REF] Gonzalez-Muñoz | Precipitation of barite by marine bacteria: a possible mechanism for marine barite formation[END_REF] ), indirectly raising the question of the mechanisms of barite precipitation from environments that contained only trace amounts of sulfate. This issue has puzzled the geochemical community for decades [START_REF] Mcgrail | Field Validation of Supercritical CO2 Reactivity with Basalts[END_REF] , and further emphasizes the need for a more accurate picture of the mechanisms of barite nucleation and growth [START_REF] Ruiz-Agudo | Mechanistic Principles of Barite Formation: From Nanoparticles to Micron-Sized Crystals[END_REF] .

In spite of the indisputable interest associated with barite formation, our knowledge regarding barite growth kinetics remains incomplete. With the noticeable exception of a few recent studies focused on barite growth on the specific (001) face over a wide range of temperatures (from 30 °C to 108 °C) [START_REF] Kuwahara | In Situ AFM Study of Crystal Growth on a Barite (001) Surface in BaSO4 Solutions at 30°[END_REF][START_REF] Bracco | Hydrothermal atomic force microscopy observations of barite step growth rates as a function of the aqueous barium-to-sulfate ratio[END_REF][START_REF] Weber | Unraveling the Effects of Strontium Incorporation on Barite Growth-In Situ and Ex Situ Observations Using Multiscale Chemical Imaging[END_REF] , most existing studies were limited to the low temperature domain [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF][START_REF] Christy | The kinetics of barite dissolution and precipitation in water and sodium chloride brines at 44-85°C[END_REF] and were derived from experiments conducted on powders. While the resulting isotropic growth rate laws derived from powder experiments have proved useful for modeling exercises (e.g., [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] ), it is noteworthy that they remain essentially empirical, which may cast doubt on their predictive ability. In particular, a recent study by Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] has shown that barite growth is an anisotropic process. Importantly, isotropic kinetic rate laws were demonstrated to be unreliable and of low relevance to model anisotropic processes such as mineral dissolution/precipitation (e.g., [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF][START_REF] Daval | Linking nm-scale measurements of the anisotropy of silicate surface reactivity to macroscopic dissolution rate laws: New insights based on diopside[END_REF][START_REF] Godinho | Simulation of surface dynamics during dissolution as a function of the surface orientation: Implications for non-constant dissolution rates[END_REF][START_REF] Pollet-Villard | Does crystallographic anisotropy prevent the conventional treatment of aqueous mineral reactivity? A case study based on K-feldspar dissolution kinetics[END_REF] ), unless the same ratelimiting step controls the dissolution/growth kinetics of all faces. Among the main consequences of this anisotropic reactivity are (i) the non-uniqueness of the relation between dissolution/growth rate and the saturation state of the solution [START_REF] Pollet-Villard | Does crystallographic anisotropy prevent the conventional treatment of aqueous mineral reactivity? A case study based on K-feldspar dissolution kinetics[END_REF] , (ii) the existence of a transient regime where the crystal habit continuously evolves, resulting in an equal gradual evolution of the apparent dissolution/growth rate "constant" [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF][START_REF] Godinho | Simulation of surface dynamics during dissolution as a function of the surface orientation: Implications for non-constant dissolution rates[END_REF][START_REF] Pollet-Villard | Does crystallographic anisotropy prevent the conventional treatment of aqueous mineral reactivity? A case study based on K-feldspar dissolution kinetics[END_REF][START_REF] Briese | The effect of crystal size variation on the rate of dissolution -A kinetic Monte Carlo study[END_REF][START_REF] Robin | Experimental study and numerical simulation of the dissolution anisotropy of tricalcium silicate[END_REF] , and (iii) the existence of distinct equilibrium morphologies for a given crystal, which depend on the face-specific dissolution/growth rates of the considered crystal and therefore on reaction conditions such as temperature, fluid saturation state, and the potential catalytic / poisoning effects of specific (bio)molecules [START_REF] Montes-Hernandez | Synthesis of a Se-0/calcite composite using hydrothermal carbonation of Ca(OH)(2) coupled to a complex selenocystine fragmentation[END_REF][START_REF] Benzerara | Looking for traces of life in minerals[END_REF][START_REF] Guisbiers | Use of chemical functionalities to control stability of nanoparticles[END_REF] . Therefore, in addition to the expected improvement of reactive transport simulations, knowledge of the variables that influence the face-specific growth rates of minerals can also help decipher the factors that shape crystals, which might be useful to provide insights into the putative biological origin of minerals recovered from environments on Earth and on other planets ( 26and references therein).

To sum up, the study by Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] paved the way for a radically new description of barite growth kinetics, and provided the very first data in that respect. However, their study was limited to ambient temperature, whereas barite formation may occur over a wide range of environmental conditions, ranging from slightly above the freezing point in oceans to several tens to hundreds of degrees in geothermal reservoirs. To this end, the present study was principally aimed at extending this dataset to a wider range of temperatures and to eventually derive an activation energy for facespecific barite growth rate. We also provide rate data for an additional face that was not investigated in the study by Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] , at two different fluid saturation states. Taken together, we show that the activation energy is the same for all faces (within uncertainties), which is compatible with the assumption that all faces may have the same unique rate-limiting step. In addition, this activation energy is close to that determined by Zhen-Wu et al. [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF] for barite dissolution, which is compatible with their hypothesis of micro-reversibility of barite dissolution/precipitation process.

Materials and methods

Starting materials and preparation

Single crystals of barite (~50-100 mm in length / diameter) were collected in Puy de Chateix (France). The chemical composition of the crystals was analyzed following the protocol described by Bracco et al. [START_REF] Bracco | Hydrothermal atomic force microscopy observations of barite step growth rates as a function of the aqueous barium-to-sulfate ratio[END_REF] , revealing K, Sr and Al as the main impurities (0.14%, 0.10%, and 0.05% mol/mol Ba, respectively). The crystals were cleaved along (210) and (001) and cut along the (101) plane. The crystallographic orientations were first verified using Electron Backscatter Diffraction (EBSD) on a Tescan Vega 2 scanning electron microscope (SEM) and polished through a multi-step abrasive sequence with a final polishing step in a colloidal silica suspension. The oriented samples were divided into small pieces with a surface area (SA) between ~ 0.25 cm 2 and 0.50 cm². The faces not under study were protected with room-temperature vulcanizing (RTV) glue. For the faces under investigation, only a portion of their surface was protected with ~1 mm-diameter RTV glue spots to provide a reference surface (see similar protocol in [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF][START_REF] Wild | pH-dependent control of feldspar dissolution rate by altered surface layers[END_REF] ). The initial roughness of each surface was measured at different scales with a ZYGO® NewView 7300 vertical scanning interferometer (VSI). For each sample, the initial average arithmetic roughness (Ra), defined as the arithmetic average of the absolute values of the roughness measured by the VSI, ranged between 10 and 20 nm. These initial roughness parameters were measured on 270 × 360 µm 2 VSI images.

Growth experiments and analytical procedures

Barite samples were placed over polytetrafluoroethylene (PTFE) tripods and introduced into 120 mL perfluoroalkoxy alkanes (PFA) Savillex® reactors. The reactors were filled with a volume (V) of 100 mL of solution and continuously stirred with magnetic bars throughout the experiments, incubated at 10, 25, 40, and 70 °C for durations ranging from 10 minutes to 15 hours. Three selected crystallographic planes were studied: (210), (001), and (101).

All solutions were prepared immediately prior to each experiment by diluting 0.01 M BaCl2 and Na2SO4 stock solutions prepared from sodium sulfate and barium chloride powders (Acros Brand, purity ≥ 99%) to the desired concentrations. The in situ pH and ion speciation and activity were calculated using the CHESS software [START_REF] Van Der Lee | CHESS Tutorial and Cookbook[END_REF] implemented with the Debye-Hückel equation and Chess.tdb database (Lawrence Livermore National Laboratories EQ3/6 database, 8th version), and the saturation index (SI) with respect to barite was calculated from Eq. ( 1):

𝑆𝐼 = log 𝑎 𝑎 𝐾 (1)
where aBa and aSO4 refer to the activities of barium and sulfate ions in solution, respectively, and Ksp refers to the solubility product of barite. The CHESS software calculates the solubility product at the run temperature based on the interpolation between the Ksp values listed in Chess.tdb database, i.e., 10 - 10.49 , 10 -9.97 , 10 -9.61 and 10 -9.51 at 0 °C, 25 °C, 60 °C and 100 °C, respectively.

The following input solutions containing 1:1 ratio of SO4 Experiment durations ranged between 10 minutes and 15 hours, depending on the experimental conditions (SI and temperature). In order to calculate the evolution of the saturation index, solution sampling was carried out two to four times in each experiment. The aqueous samples were diluted 10 times and filtered using a 0.45-µm filter before analysis. Inductively coupled plasma atomic emission spectroscopy (ICP-AES -Thermo ICAP 6000) was used for the analyses of Ba 2+ and Na + , and ion chromatography (ICS-5000 Thermofisher Dionex) was used for the analyses of SO4 2-and Cl -. Because foreign cations such as Sr can impact barite growth rates [START_REF] Weber | Unraveling the Effects of Strontium Incorporation on Barite Growth-In Situ and Ex Situ Observations Using Multiscale Chemical Imaging[END_REF] , all input solutions were analyzed for major (Na, K, Ca, Mg, Fe, Si, Al) and minor (Sr, Ni, Mn, Ti, Co, Cr, Zn, Cu) cations prior to the immersion of the barite samples. The concentration of all of these elements was systematically below the detection limit (≤ 5 ppb), except for Cu, for which concentrations up to 20 ppb (8×10 -8 M) were occasionally detected. The pH was also checked at the beginning and at the end of each experiment. The pH of all solutions was 5.3 ± 0.1 and remained constant throughout the experiments.

At the end of each experiment, the sample was removed from the solution, rinsed with ultrapure water and sonicated for 10 min in ethanol to remove any possible secondary barite crystals nucleated from the solution and deposited on the surface, as classically done in literature to remove fine particles adhering on the surface of minerals [START_REF] Schott | Mechanism of pyroxene and amphibole weathering--I. Experimental studies of iron-free minerals[END_REF] . The RTV mask was then carefully removed and the sample topography was measured with VSI in stitching mode (magnification: ×5) to evaluate average changes in height between the unreacted reference surface and the reacted mineral surface. For each surface, 10 profiles across the interface were used to calculate the average thickness of the grown layer, which was used to calculate the face-specific growth rate. The uncertainties associated with these average values were estimated from the standard deviations calculated from the various profiles. For each oriented sample, growth rates were calculated according to Eq. ( 2):

𝑟 ( ) = ∆ℎ ∆𝑡 (2)
where 𝑟 ( ) (nm.h -1 ) is the growth rate of the (hkl) face, Δh is the average surface step resulting from barite growth (nm), and Δt is the duration of the experiment (s). Both SEM and VSI investigations confirmed that, over the course of the experiments, barite growth did not result from homogeneous nucleation in the solution, as revealed by the absence of euhedral secondary barite crystals on the surface of each investigated face, or the subsistence of polishing scratches in the grown layer of barite. The results show that the temperature variation seems to have no effect on the growth features of the faces under study. Pina et al. [START_REF] Pina | Molecular-scale mechanisms of crystal growth in barite[END_REF] described growth patterns which exhibit features specific to each investigated face (001) and (210), consistent with the anisotropy of the barite structure and the Periodic Bond Chain (PBC) theory. In the present study, the (001) and the (101) faces exhibit a growth morphology that resembles the circular sector shape described during the early stage of barite growth monitored using atomic force microscopy (AFM) by Sanchez-Pastor et al. [START_REF] Sánchez-Pastor | The effect of CO3 2on the growth of barite {0 0 1} and {2 1 0} surfaces: An AFM study[END_REF] . Conversely, the twodimensional islands formed in the experiments conducted at 25 and 70 ºC and a saturation index of 2.1 could not be easily related to the needle-shaped morphology described by Pina et al. [START_REF] Pina | Molecular-scale mechanisms of crystal growth in barite[END_REF] and Sanchez-Pastor et al. [START_REF] Sánchez-Pastor | The effect of CO3 2on the growth of barite {0 0 1} and {2 1 0} surfaces: An AFM study[END_REF] for the face (210). Possible explanations include the difference of resolution between AFM and SEM, as well as the difference in terms of reaction progress, since the study by Sanchez-Pastor et al. [START_REF] Sánchez-Pastor | The effect of CO3 2on the growth of barite {0 0 1} and {2 1 0} surfaces: An AFM study[END_REF] focused on the early stage of barite growth, whereas the present study reports on results obtained at a later stage of the reaction process. Therefore, part of the original islands documented in e.g. Sanchez-Pastor et al. [START_REF] Sánchez-Pastor | The effect of CO3 2on the growth of barite {0 0 1} and {2 1 0} surfaces: An AFM study[END_REF] might have been overgrown, and no longer identifiable by SEM at such late stages of observation.

Results and discussion

Face-specific growth patterns 160

Face-specific growth rate of barite as a function of solution saturation state

Time-resolved fluid analyses

The evolution of the fluid composition during all experiments is provided in Table 1.

The Ba 2+ to SO4 2-mole ratios in the aqueous solutions are consistent with the stoichiometric precipitation of barite. From the Ba 2+ , Na + , SO4 2-, and Cl -concentrations, the saturation index with respect to barite was determined at up to four different time steps in each experiment using the CHESS code.

The extent to which the evolution of the fluid composition resulted from barite growth of the oriented barite surface sample can be estimated based on the thickness of the grown layer measured by VSI following:

∆[𝑖] = ∆ℎ. 𝑆𝐴. 𝜌 𝑀 . 𝑉 (3)
where ∆[i] is the variation of Ba 2+ (or SO4 2-) concentration resulting from barite growth, ρBaSO4

and MBaSO4 are respectively the density (4.48 g.cm -3 ) and molar mass (233 g.mol -1 ) of barite.

Knowing that the surface area of each investigated sample never exceeded 0.5 cm², and using 150 nm as an upper bound for the thickness of the grown layer (consistent with our VSI measurements), the decrease in sulfate or barite concentration should never exceed 1.5×10 -6 M.

This value is about two orders of magnitude greater than the actual decrease measured in some of our experiments (see Table 1), suggesting that in addition to barite growth, homogeneous barite nucleation in the solution or heterogeneous nucleation on the walls of the reactor should have occurred as well. This result is consistent with the observations reported by Jindra et al. [START_REF] Jindra | Hydrothermal Atomic Force Microscopy Investigation of Barite Growth: Role of Spectator Ions in Elementary Step Edge Growth Kinetics and Hillock Morphology[END_REF] , who showed that solutions supersaturated in barite with SI ≥ 1 were unstable with respect to barite nucleation. This limitation illustrates that classical kinetics experiments based on the sole evolution of the fluid composition would be poorly informative to deconvolve the nucleation from the growth steps under such conditions, and further justifies the use of surface sensitivetechniques to probe growth kinetics, such as AFM [START_REF] Kuwahara | In Situ AFM Study of Crystal Growth on a Barite (001) Surface in BaSO4 Solutions at 30°[END_REF][START_REF] Bracco | Hydrothermal atomic force microscopy observations of barite step growth rates as a function of the aqueous barium-to-sulfate ratio[END_REF][START_REF] Weber | Unraveling the Effects of Strontium Incorporation on Barite Growth-In Situ and Ex Situ Observations Using Multiscale Chemical Imaging[END_REF][START_REF] Jindra | Hydrothermal Atomic Force Microscopy Investigation of Barite Growth: Role of Spectator Ions in Elementary Step Edge Growth Kinetics and Hillock Morphology[END_REF] or VSI [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] .

The rapid nucleation of barite complicated the conduction of experiments at a fixed value of SI, and the duration of the experiment had consequently to be adjusted to (i) yield an appreciable thickness of the grown layer of barite while (ii) limiting the intrinsic evolution of the solution composition resulting from barite nucleation. Importantly, for the experiments conducted under the less reactive conditions (i.e., T = 10 °C, SI = 1.1), in spite of several attempts, it has not been possible to run experiments for which the evolution of SI was modest while measuring an appreciable thickness of the grown layer at the same time. In other words, for these experiments, essentially barite nucleation occurred over short durations (with no observable grown layer), while over long durations (up to 15 hours), barite growth was observed, but with large variation of the fluid saturation index (between ± 18% and ± 34%), complicating the definition of the SI at which growth actually occurred. Therefore, the results

of experiments conducted at SI = 1.1 and T = 10 °C will have to be considered with caution (see Section 3.2.2 for a confirmation of this statement). In the end, the variation of SI was ± 6% for the 12 experiments run at an initial saturation index of 2.1, and ± 5% for the 9 experiments run at SI = 1.1 (excluding the 3 experiments run at 10 °C). The growth rates perpendicular to the three studied surfaces were calculated from the measured thickness of the layer grown whilst in solution. The results highlight an effect of the saturation index on growth rate. For example, in the experiments with a starting saturation index of 2.1, the growth rates are about 5 to 40 times faster (depending on the face and temperature) than in the solution with SI = 1.1 (Table 1).

Surface-controlled precipitation rates are commonly fitted to the following rate law (e.g. [START_REF] Lopez | The influence of temperature and seawater composition on calcite crystal growth mechanisms and kinetics: Implications for Mg incorporation in calcite lattice[END_REF] and references therein):

𝑟 = 𝑘. (10 -1) (4)
where k is the growth rate constant and n is a fitting coefficient generally referred to as the reaction order. Although Eq. ( 4) essentially remains empirical, the value of the reaction order is usually attributed to the rate-controlling reaction mechanism (a reaction order of 1 is typically attributed to a simple surface adsorption mechanism and a reaction order of 2 to spiral growth; see [START_REF] Shiraki | Kinetics of near-equilibrium calcite precipitation at 100°C: An evaluation of elementary reaction-based and affinity-based rate laws[END_REF] ). For each face, the value of the reaction order at each temperature (nT) can therefore be estimated following:

𝑛 = log 𝑟 -log 𝑟 log 10 -1 -log 10 -1 (5)
where 𝑟 refers to the growth rate measured at the corresponding saturation index 𝑆𝐼 (i.e., ~2.1 or ~1.1), for a given temperature T. Eq. ( 5) was used to estimate the value of n for each face and all investigated temperatures (Table 3), leading to average values of n = 1.39 ± 0.76, 1.26 ± 0.42, and 1.12 ± 0.19 for the (001), (210), and (101) faces, respectively. Of note, the large standard deviations associated with the value of n for the (001) and ( 210) faces are essentially due to the value of n derived for these faces at 10 °C (n = 2.50 and 1.88, respectively). The large uncertainties associated with the estimation of slow growth rates at low temperature and saturation index may partially contribute to this observation, as emphasized above (Section 3.2.1). In particular, the large variation of SI over the duration of these experiments, which result from the fact that barite precipitation occurred almost exclusively through homogeneous nucleation, considerably complicated the definition of the value of SI at which barite growth occurred. Therefore, the uncertainties associated with these experiments and the switch in barite precipitation mechanism at low temperature justifies the need to remove the rate data collected at 10 °C and SI = 1.1 for further processing of the dataset . If these rates are discarded from the dataset, the average values then become n = 1.02 ± 0.19; 1.05 ± 0.07 and 1.11 ± 0.13 for the (001), ( 210) and (101) faces, respectively. Therefore, with the noticeable exception of rate data obtained at 10 °C, our results show that a simple linear relation between r and (10 SI -1) can account for our observations for all faces, irrespective of the temperature (see Supporting Information for a plot displaying the general agreement over the whole dataset). Therefore, this study extends the similar conclusion previously reached by Zhen-Wu et al. [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF] from powder experiments to a broader range of temperatures and saturation indices, while suggesting that the reaction order (n) does not depend on the investigated face, at least for temperatures ranging from 25 °C to 70 °C. This result is also consistent with the second order reaction with respect to Ba concentration reported by Christy and

Putnis [START_REF] Christy | The kinetics of barite dissolution and precipitation in water and sodium chloride brines at 44-85°C[END_REF] , since a first order reaction with respect to barite saturation state is equivalent to a second order reaction with respect to either aqueous barium or aqueous sulfate concentration, as recalled by Zhen-Wu et al. [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF] .

Table 3. Estimation of the reaction order (n) of barite growth kinetics for all investigated faces and temperatures following Eq. ( 5). Note that the values in italic were not considered for the calculation of the mean value of n for each face. See text for details.

Face-specific growth rate of barite as a function of temperature

In experiments conducted with an input solution with SI = 1.1, growth rates range from 0.60 nm.h -1 to approximately 47 nm.h -1 , with the slowest rates at 10ºC and the fastest rates at 70 ºC (Table 1). For each face, the growth rate increases with temperature but generally, the (001) and (101) faces are the slowest growing faces (Table 1), in accordance with the results obtained by Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] . For face (001) reacted in a solution with SI = 1.1, an increase in temperature by 60 ºC increases the rate by almost a factor of 60 (Table 1). At a starting saturation index of 2.1, the growth rates are about one order of magnitude greater at 70 °C than at 10 °C for all faces.

The dependence of the mineral growth rate constant on temperature is usually described following the Arrhenius equation:

𝑘 = 𝐴. exp(-𝐸 𝑅𝑇 ⁄ ) (6)
where A represents the Arrhenius pre-exponential factor, Ea denotes the activation energy, R is the gas constant, and T refers to the absolute temperature. By combining Eq. ( 6) with Eq. ( 4) and assuming that n = 1 (see above), we can determine the value of A and Ea experimentally following the relation:

ln(𝑟 (10 -1) ⁄ )= ln 𝐴 -𝐸 𝑅𝑇 ⁄ (7)
This equation was applied to the data generated in this study using the Arrhenius plots shown in Fig. 3. The errors reported on the data points account both for the uncertainties on the growth rates (resulting from the variability of the thickness of the grown layer) and the variation of SI over the course of the experiment (resulting from the precipitation of barite). Excluding the rate data obtained at 10 °C and SI = 1.1 (see above), the activation energy was found to be virtually independent of the considered face, with values of 36.4 ± 4.0, 31.8 ± 3.3, and 35.8 ± 3.7 kJ.mol -1 for the growth on (001), (210), and (101) faces, respectively. The uncertainties on the activation energies were determined based on the errors associated with the data depicted in Fig. 3.

The activation energy determined for the growth on the (001) face is in excellent agreement with that determined by Higgins et al. [START_REF] Higgins | Kink Dynamics and Step Growth on Barium Sulfate (001): A Hydrothermal Scanning Probe Microscopy Study[END_REF] (37.3 ± 4.6 kJ.mol -1 ). More broadly, these values agree, within uncertainties, with those reported by Christy and Putnis 18 (22.0 ± 14.3 kJ.mol -1 ) for the growth of barite monitored on powder experiments over the temperature range 44-85 °C. This result further indicates that if a barite crystal form is developed based on the three faces investigated in the present study, the morphology of barite crystals will be negligibly affected by temperature variations, such that the morphology of barite crystals cannot be used as a criterion to determine the temperature at which they crystallized.

Regarding the pre-exponential factor, Fig. 3 reveals that it only slightly varies with the considered orientation, from exp(12.4) nm.h -1 for the (210) face to exp(14.1) nm.h -1 for the (001) face.

Following the reasonable assumption that the activation energy does not depend on the crystallographic orientation for the three investigated faces, this translates into a modest anisotropy of barite growth rate of a ~5-fold factor between the fastest and slowest growing faces, which is in reasonable agreement with our experimental data. The agreement between the multiple regressions described above and the experimental data can be assessed in Fig. 4, for growth rates varying over about two orders of magnitude. In Fig. 4a, the measured rates were compared with the corresponding values calculated following the overall facespecific growth rate law given by:

𝑟 ( ) = 𝐴 ( ) . exp -𝐸 ( ) 𝑅𝑇 ⁄ . (10 -1) (8a)
for which the values of A (hkl) and Ea (hkl) were derived from the plots depicted in Fig. 3. Of note, the agreement between calculated rates and measured rate data is negligibly affected if one considers an overall isotropic growth rate law for all faces following:

𝑟 ( ) = 𝐴. exp(-𝐸 𝑅𝑇 ⁄ ). (10 -1) (8b)
with average values of A = exp (13.59) nm.h -1 and Ea = 35.0 ± 2.5 kJ.mol -1 (Fig. 4b). In addition, both rate laws perfectly account for the rate data reported by Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF]

Comparison with previously published data and mechanistic insights

Overall, this study largely confirms both the measurements of face-specific barite growth rates and the reaction mechanisms reported in previous studies.

Regarding the anisotropy of barite growth, Godinho and Stack [START_REF] Godinho | Growth Kinetics and Morphology of Barite Crystals Derived from Face-Specific Growth Rates[END_REF] reported that the reactivity of the (210) face was about 1.5 times greater than that of the (001) face at ambient temperature and for SI conditions ranging from 1.1 to 2.1. These values are in excellent agreement with our measurements under the same conditions, which show that the reactivity of the (210) face is approximately between 1.7 and 1.9 times greater than that of the (001) face. At SI = 2.1, the absolute values of the growth rates of the two studies overlap as well, within uncertainties.

Regarding the reaction mechanisms, several lines of evidence suggest that barite growth may observe the principle of detailed balancing and micro-reversibility. First, our rate data are consistent with an isotropic activation energy with an average value of 35.0 ± 2.5 kJ.mol -1 , suggesting that the reaction is surface-controlled. Moreover, if the rate-limiting step of barite growth were to change with surface orientation, then the activation energies of barite growth rate would be face-specific. Therefore, and even though the activation energies estimated in the present study remain essentially apparent, the unicity of the activation energy is compatible with the suggestion that the same elementary step is ratelimiting for all faces. In addition, this value is consistent with the isotropic value reported by Christy and Putnis 18 (22.0 ± 14.3 kJ.mol -1 ), itself very similar to the activation energy of barite dissolution reported by Zhen-Wu et al. [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF] (25 ± 2 kJ.mol -1 ). Second, the growth rate was found to be a linear function of the saturation state of the solution (i.e., r ∝ 10 SI ) for all faces at temperatures ranging between 25 and 70 °C (this statement remains questionable at lower temperatures). This result is consistent with previously published studies [START_REF] Zhen-Wu | Experimental determination of barite dissolution and precipitation rates as a function of temperature and aqueous fluid composition[END_REF][START_REF] Christy | The kinetics of barite dissolution and precipitation in water and sodium chloride brines at 44-85°C[END_REF] and is typical of adsorption-controlled growth processes [START_REF] Shiraki | Kinetics of near-equilibrium calcite precipitation at 100°C: An evaluation of elementary reaction-based and affinity-based rate laws[END_REF] , for which the concept of micro-reversibility necessarily applies.

Implications for natural and anthropogenic systems

As highlighted above, a growing number of studies have emphasized the need to go beyond the classical development of isotropic rate laws to model the dissolution and/or the growth kinetics of minerals, partly because of their anisotropic reactivity. Notwithstanding, switching to more complex rate laws that are not implemented so far in classical reactive transport codes has to be justified through quantitative measurements of this anisotropy, as well as its dependence to fundamental rate-controlling parameters such as temperature or solution saturation state, in order to assess the extent to which isotropic models fail to reproduce quantitatively the actual mineral reactivity. In that respect, our results underline that (i) the anisotropic reactivity of barite growth remains modest, at least for the three studied faces and (ii) the growth activation energy is similar for all faces and possibly equivalent to the dissolution activation energy. These results have at least two practical implications for natural and anthropogenic systems:

(i) Regarding the reaction mechanisms, our study suggests that the principle of detailed balancing may be respected over the range of T and SI investigated in the present study (see section 3.4). Because the dissolution and precipitation rates of barite are relatively rapid, this result implies that at ambient temperature and close-to-equilibrium conditions, the isotope composition of barium may be quickly reset, consistent with the results of Curti et al. [START_REF] Curti | Radium uptake during barite recrystallization at 23±2°C as a function of solution composition: An experimental 133Ba and 226Ra tracer study[END_REF] . Because the isotopic and elemental compositions of barite are used as a proxy for past seawater chemistry ( 12 and references therein), our study contributes to the general warning regarding the preservation of isotopic signatures in minerals over geological timescales to trace paleoenvironmental conditions [START_REF] Critelli | Dissolution rate of antigorite from a whole-rock experimental study of serpentinite dissolution from 2< pH< 9 at 25 C: Implications for carbon mitigation via enhanced serpentinite weathering[END_REF][START_REF] Bernard | Burial-induced oxygen-isotope reequilibration of fossil foraminifera explains ocean paleotemperature paradoxes[END_REF][START_REF] Chanda | Low temperature stable mineral recrystallization of foraminiferal tests and implications for the fidelity of geochemical proxies[END_REF][START_REF] Oelkers | The rapid resetting of the Ca isotopic signatures of calcite at ambient temperature during its congruent dissolution, precipitation, and at equilibrium[END_REF] .

(ii) Regarding geothermal systems, a viable geothermal resource requires a reservoir with a high permeability, often provided by fractures (e.g., [START_REF] Kushnir | Assessing the role of fractures on the permeability of the Permo-Triassic sandstones at the Soultz-sous-Forêts (France) geothermal site[END_REF] ), to provide the flow rates required for efficient energy production. The hydrothermal brines circulating through these reservoirs typically contain a rich assortment of dissolved elements (e.g., [START_REF] Kushnir | Assessing the role of fractures on the permeability of the Permo-Triassic sandstones at the Soultz-sous-Forêts (France) geothermal site[END_REF][START_REF] Aquilina | Water-rock interaction processes in the Triassic sandstone and the granitic basement of the Rhine Graben: Geochemical investigation of a geothermal reservoir[END_REF][START_REF] Sanjuan | Main characteristics of the deep geothermal brine (5km) at Soultz-sous-Forêts (France) determined using geochemical and tracer test data[END_REF][START_REF] Sanjuan | Major geochemical characteristics of geothermal brines from the Upper Rhine Graben granitic basement with constraints on temperature and circulation[END_REF] ) that can precipitate in surface installations and within permeability-enhancing fractures within the reservoir as a result of, for example, changes in temperature (e.g., [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] ). Barite scaling can play havoc at geothermal surface installations, requiring timeconsuming and expensive mechanical removal (e.g., [START_REF] Christy | The kinetics of barite dissolution and precipitation in water and sodium chloride brines at 44-85°C[END_REF] ), and can dramatically reduce the permeability of fractures within the reservoir. For example, Griffiths et al. [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] highlighted that the Buntsandstein unit, a ~400 m-thick unit of sandstone that directly overlies the granitic reservoir at geothermal sites throughout the Upper Rhine Graben (e.g., [START_REF] Aichholzer | New stratigraphic interpretation of the Soultz-sous-Forêts 30-year-old geothermal wells calibrated on the recent one from Rittershoffen (Upper Rhine Graben, France)[END_REF][START_REF] Heap | Microstructural and petrophysical properties of the Permo-Triassic sandstones (Buntsandstein) from the Soultz-sous-Forêts geothermal site (France)[END_REF][START_REF] Heap | Rock mass strength and elastic modulus of the Buntsandstein: an important lithostratigraphic unit for geothermal exploitation in the Upper Rhine Graben[END_REF] ), contains abundant barite-filled fractures (e.g., Fig. 1).

Additionally, Griffiths et al. [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] estimated the time needed to seal a 2 mm-wide fracture with barite to be on the order of about a month. These calculations, however, were performed using data collected on powdered barite samples [START_REF] Christy | The kinetics of barite dissolution and precipitation in water and sodium chloride brines at 44-85°C[END_REF] . Our new face-specific data show that the activation energy is, within uncertainty, the same for all of the studied crystal faces. These new data provide additional confidence in the barite growth timescales provided in Griffiths et al. [START_REF] Griffiths | Geothermal implications for fracture-filling hydrothermal precipitation[END_REF] , which suggest that the permeability of the geothermal reservoir adjacent to the injection well could be, unless preventative measures are taken, greatly reduced over short timescales during production.

Summary and conclusions

Barite growth experiments combined with vertical scanning interferometry measurements of the surface topography have been conducted to study the effect of the saturation index (SI) and the temperature on the growth rates and surface features of the (001), (210), and (101) faces of barite. The results confirmed that barite growth is anisotropic with a rate that is promoted by elevated saturation indices and temperatures, while the growth morphologies observed by scanning electron microscopy did not significantly vary with temperature or SI. It was determined that barite growth rate observes a first order reaction with respect to barite saturation state for all faces at all temperatures, with the exception of rate data obtained at 10 °C. Therefore, our results show that a simple linear relation between 𝑟 and (10 SI -1) can account for barite growth. From the dependence of the rate of barite growth with temperature, the activation energy could be defined for each face using the Arrhenius equation, yielding an average value of 35.0 ± 2.5 kJ.mol -1 , with no significant difference between the three faces. This result indicates that the morphology of barite crystals will be negligibly affected by temperature variations for a crystal composed of the faces under study. Therefore, the morphology of barite crystals cannot be used as a criterion to determine the temperature at which they crystallized.

These data are also compatible with a reaction rate is surface-controlled, with the same rate-limiting step for all faces. The pre-exponential factor in the Arrhenius equation slightly varies with the considered orientation, from exp(12.4) nm.h -1 for the (210) face to exp(14.1) nm.h -1 for the (001) face, which represents a modest anisotropy of barite growth rate. Taken together, this study largely confirms both the measurements of face-specific barite growth rates and the reaction mechanisms reported in previous studies, and may contribute to improve modeling of barite growth rates in natural and anthropogenic systems.

Figure 2 .

 2 Figure 2. Barite growth features observed using scanning electron microscopy a) Face (001), SI = 2.1, T = 25 °C; b) Face (001), SI = 2.1, T = 70 °C; c) Face (210), SI = 2.1, T = 25 °C; d) Face (210), SI = 2.1, T = 70 °C; e) Face (101), SI = 2.1, T = 25 °C; f) Face (101), SI = 2.1, T = 70 °C.

Figure 3 .

 3 Figure 3. Determination of the face-specific activation energy (E a (hkl) ) and pre-exponential factor (A (hkl) ) of barite growth rate over the temperature range 10-70 °C for the (a) (001), (b) (210), and (c) (101) faces. Of note, the rate data obtained at 10 °C and SI = 1.1 for the faces (001) and (210) were dismissed from the regressions (see section 3.3 for details).

  at T = 22.2 °C and SI = 2.1 for the faces (001) and (210), while overestimating the rate values obtained at SI = 1.1. Possible explanations might reside in the difference in chemical composition of the crystals used to conduct the studies, since impurities such as strontium are known to decrease barite growth rates at low concentrations 17 .

Figure 4 .

 4 Figure 4. Comparison of barite growth rates calculated using (a) Eq. (8a) or (b) Eq. (8b) with those measured in the present study or reported in Godinho and Stack 19 .

Table 1 .

 1 Evolution of fluid composition (concentration of Ba 2+ , SO 4 2-, Cl -, and Na

	SI=2.1

+ in mol/L) measured by ICP-AES and ion chromatography for the growth experiments realized on the (001), (101), and (210) surfaces at 10, 25, 40, and 70 ºC. The fluid composition was measured at up to four time steps in each experiment. The saturation index of the experiments was determined from the fluid composition using the CHESS software.

Table 2 .

 2 Face-specific growth rates (nm.h -1 ) calculated from the thickness of the grown layer measured by VSI on the (001), (101), and (210) surfaces of crystals grown in two different solutions: (1) SI = 1.1 and (2) SI = 2.1. Experiments were performed at 10, 25, 40, and 70 °C. The numbers between parentheses indicate the uncertainties of the growth rates, estimated from the variability of the thickness of the grown layer averaged on 10 measurements (see Section 2.2).

	(1) SI = 1.1

Acknowledgements

This study was partly funded by LABEX grant ANR-11-LABX-0050_G-EAU-THERMIE-PROFONDE (this research therefore benefited from state funding managed by the Agence National de la Recherche (ANR) as part of the "Investissements d'avenir" program) and ANR grant CANTARE (ANR-15-CE06-0014-01). The authors are grateful to the National Scientific and Technical Research Council (CONICET) PIP2011-0392 which provided a post-doctoral grant. We also acknowledge the Mincyt-ECOS project for supporting international collaboration between the ICGyC/IIMyC in Argentina and the LHyGeS in France. Colin Fourtet and René Boutin (LHyGeS) are warmly thanked for their help with IC and ICP-AES measurements. Finally, we are grateful for the careful reviews and detailed suggestions made by two anonymous reviewers, which significantly improved an earlier version of the present paper.

Supporting information

Additional data processing and graph illustrating the linear relation between barite growth rates and the saturation state of the solution.

For Table of Contents Use Only

Manuscript Title

Barite growth rates as a function of crystallographic orientation, temperature, and solution saturation state

Author list

Mélanie Vital, Damien Daval, Gilles Morvan, Daniel E. Martinez, Michael J. Heap

TOC Graphic

Synopsis

Barite growth rate was monitored as function of crystallographic orientation, temperature, and solution saturation state by measuring the thickness of the grown layer using vertical scanning interferometry


Fairouze Guerza-Soualah

Abdelkader Hanna

Hiba Azzeddine
email: azzeddine@univ-msila.dz

Anne-Laure Helbert

Francois Brisset

Thierry Baudin

Djamel Bradai

The deformation and recrystallization behavior of an Mg-Dy alloy processed by plane strain compression

Keywords: Dynamic recrystallization, Static recrystallization, PSC, Mg-Dy alloy, Texture

The present study investigates the deformation and recrystallization behavior of an Mg-0.41Dy (wt. %) alloy processed by plane strain compression (PSC) at room temperature (RT), 200 and 400 °C under strain rate of 10 -4 s -1 . PSC at RT and 200 °C cause the introduction of a high amount of deformation features like extension, contraction and double twins.

Meanwhile, PSC at 400 °C is characterized by the occurrence of dynamic recrystallization.

The changing in the microstructures affects the mechanical response of the alloy. The deformation texture was characterized by the formation of basal <0001> and prismatic <101 ̅ 0> fiber. The texture intensity increases with increasing deformation temperature. The static recrystallization kinetics was investigated for all processed samples after subsequent annealing at 450 °C for a period time ranging from 30 min to 7 days. The evolution of grains size and microhardness depends strongly on the deformation temperatures. The recrystallization rate increases with increasing deformation temperature. The Avrami exponent of recrystallization n was in the range of 0.9-0.26 indicating that the recrystallization preferentially occurs along grain boundaries and within deformation features.

Introduction

Magnesium (Mg) based alloys have a great potential to be used in diverse industrial applications such as automobiles, biomedical, telecommunication and aerospace due to its low density, high specific strength, good damping capacity, corrosion resistance and recyclability [START_REF] Hirsch | Superior light metals by texture engineering: Optimized aluminum and magnesium alloys for automotive applications[END_REF][START_REF] You | Recent research and developments on wrought magnesium alloys[END_REF][START_REF] Atrens | Viewpoint -Understanding Mg corrosion in the body for biodegradable medical implants[END_REF]. Despite that, Mg-based alloys suffer from limited formability at room temperature because of an insufficient number of activated slip systems and the formation of strong basal texture [START_REF] Yoo | Slip, twinning, and fracture in hexagonal close-packed metals[END_REF][START_REF] Azzeddine | On the texture and grain growth in hot-deformed and annealed WE54 alloy[END_REF]. Thermo-mechanical processing and alloying elements have proven their efficiency to improve the formability and reduce the mechanical anisotropy of Mg-based alloy, especially by microstructural modifications. Hot deformation allows the activation of prismatic and pyramidal <c+a> slip systems and the occurrence of dynamic recrystallization (DRX) which improves the mechanical properties of Mg-based alloys by grain refinement [START_REF] Barnett | Influence of deformation conditions and texture on the high temperature flow stress of magnesium AZ31[END_REF][START_REF] Matsubara | Developing superplasticity in a magnesium alloy through a combination of extrusion and ECAP[END_REF][START_REF] Al-Samman | Dynamic recrystallization during high temperature deformation of magnesium[END_REF][START_REF] Abdessameud | Grain growth in AZ31 alloy after uniaxial compression[END_REF][START_REF] Sandlöbes | The Relation between Shear Banding, Microstructure and Mechanical Properties in Mg and Mg-Y Alloys[END_REF][START_REF] Al-Samman | Softening and dynamic recrystallization in magnesium single crystals during c-axis compression[END_REF][START_REF] Azzeddine | On some aspects of compressive properties and serrated flow in Mg-Y-Nd-Zr alloy[END_REF][START_REF] Mishra | Texture evolution during annealing of AZ31 Mg alloy rolled sheet and its effect on ductility[END_REF][START_REF] Elfiad | Investigation of texture, microstructure, and mechanical properties of a magnesium-lanthanum alloy after thermomechanical processing[END_REF]. Several mechanisms regrouped in two categories: discontinuous DRX (DDRX) and continuous DRX (CDRX) mechanisms have been proposed to explain the effect of DRX on the performance of Mg-based alloys. DDRX is characterized by nucleation and growth process derived by the stored energy during the deformation, while in CDRX mechanism the low grain boundaries (LAGB) are progressively transformed by absorbing dislocations to high grain boundaries (HAGB) and hence formed new grains [START_REF] Al-Samman | Softening and dynamic recrystallization in magnesium single crystals during c-axis compression[END_REF][START_REF] Ion | Dynamic recrystallisation and the development of microstructure during the high temperature deformation of magnesium[END_REF][START_REF] Liu | Effect of temperature on microstructure and texture evolution of Mg-Zn-Er alloy during hot compression[END_REF]. However, is often reported that the recrystallization texture is not drastically changed regardless of the DRX mechanism [START_REF] Al-Samman | Dynamic recrystallization during high temperature deformation of magnesium[END_REF][START_REF] Hanna | Evaluating the textural and mechanical properties of an Mg-Dy alloy processed by high-pressure torsion[END_REF][START_REF] Zeng | Role of deformation mechanisms and grain growth in microstructure evolution during recrystallization of Mg-Nd based alloys[END_REF]. Nowadays, rare earth (RE) alloying elements received special scientific attention due to its remarkable effect on the improving the mechanical anisotropy of Mg-based alloys through texture modifications during deformation and recrystallization as well [START_REF] Imandoust | Unraveling Recrystallization Mechanisms Governing Texture Development from Rare-Earth Element Additions to Magnesium[END_REF][START_REF] Jung | Role of RE in the deformation and recrystallization of Mg alloy and a new alloy design concept for Mg-RE alloys[END_REF]. Different mechanisms were proposed in the literatures to explain the role of RE elements on texture weakening such as activation of non-basal slip systems by changing of stacking fault energy of Mg [START_REF] Sandlöbes | On the role of non-basal deformation mechanisms for the ductility of Mg and Mg-Y alloys[END_REF], Particle-Stimulated Nucleation (PSN) mechanism due to the presence of second particles [START_REF] Al-Samman | Modification of texture and microstructure of magnesium alloy extrusions by particle-stimulated recrystallization[END_REF] and the enhancement of solute drag of grain boundaries and dislocations [START_REF] Jung | Role of RE in the deformation and recrystallization of Mg alloy and a new alloy design concept for Mg-RE alloys[END_REF][START_REF] Barrett | The effect of rare earth element segregation on grain boundary energy and mobility in magnesium and ensuing texture weakening[END_REF]. As a matter of fact, the texture modification or weakening of Mg-RE alloys depends strongly on the deformation conditions (processing type, temperature, strain and strain rate) and the choice of RE elements [START_REF] Imandoust | A review on the effect of rare-earth elements on texture evolution during processing of magnesium alloys[END_REF].

Up to now, the understanding of the impact of RE alloying elements on the evolution of mechanical properties during the deformation, recrystallization and grain growth processes is still not complete. Therefore, the effect of deformation temperature on the deformation and recrystallization behavior of binary Mg-0.41Dy (wt.%) alloy is investigated in the present study. First, the deformation microstructure and texture of the alloy are characterized after processing by plane strain compression (PSC) at RT, 200 and 400 °C under strain rate of 10 -4 s -1 . Second, the recrystallized kinetics is investigated after subsequent annealing at 450 °C for time ranging between 30 minutes to 7 days.

Experimental material and procedures

The Mg-0.41Dy alloy was kindly supplied in as-cast condition by the Institute für Metallkunde und Metallphysik (IMM-RWTH) in Aachen, Germany. Accordingly, the alloy was prepared by induction melting and casting under a protective gas atmosphere of Ar/CO2 and followed by annealing for 20 h at 420 °C.

Samples with dimensions of 14 mm (longitudinal direction, LD) x 10 mm (transverse direction, TD) x 6 mm (compression direction, CD) were machined from the as-cast bloc.

PSC tests were carried out at RT, 200 and 400 °C under strain rates of 10 -4 s -1 using a conventional screw-driven ZWICK testing machine. A final true strain ε = -1.2 corresponding to 70% of reduction of thickness was achieved for all the samples. After PSC tests, the samples were directly quenched in water.

Annealing treatments were performed in a radiation furnace at 450 °C for time periods ranging from 30 min to 7 days.

The microstructure of the samples was investigated in the LD-TD plane using optical microscopy after grinding with progressively finer SiC paper followed by electropolish with 5:3-part ethanol and phosphoric acid for 30min under 3V applied voltage and etching at room temperature in an acetic-nital solution (5% HNO3, 15% acetic acid, 20% distilled water and 60% ethanol) for 3 s. EBSD measurements were performed after mechanical and ionic polishing using a Gatan PECS II system at a high voltage of 5 kV for 15 min. The EBSD data acquisition and analysis were undertaken using the TSL Orientation Imaging Microscopy, OIM TM software.

The scanned areas were 94×58 μm 2 with a step size of 0.3 μm for the sample processed at RT, 120×107 μm 2 and 1080×870 μm 2 with a step size of 1 μm for the sample processed at 200 and 400 °C, respectively. The grain size data were obtained using a grain tolerance angle of 5° and the minimum grain size was chosen as 5 pixels. The confidence index (CI) which quantified the reliability of the indexed Kikuchi was taken higher than 0.05 to exclude poor quality pixels during the cleanup procedure [START_REF] Mikami | Effect of reference point selection on microscopic stress measurement using EBSD[END_REF].

Crystallographic texture was measured in the LD-TD plane using a Phillips X-ray texture diffractometer. A set of six pole figures {101 ̅ 0}, {0002}, {101 ̅ 1}, {101 ̅ 2}, {112 ̅ 0}, and {101 ̅ 3} was measured to calculate the orientation distribution functions (ODF) using MTex software [START_REF] Bachmann | Texture Analysis with MTEX -Free and Open Source Software Toolbox[END_REF].

The Vickers microhardness was measured in the LD-TD plane using a SHIMADZU type HMV-2 tester. An average microhardness value was obtained using at least five indentations under a load of 100 g (Hv0.1) with a dwell time of 10 s.

Results and discussions

Initial microstructure

Figure 1 shows the orientation imaging micrograph in inverse pole figure (IPF) and scanning electron microscopy (SEM) micrograph in the backscattering mode of the as-cast Mg-0.41Dy alloy, respectively. The as-cast microstructure exhibits large and coarse grains. In this case, the texture could be considered as random. Some casting twins are present as shown by arrows. Similar twins were observed in as-cast Mg-0.22Gd (wt.%) alloy [START_REF] Stanford | The effect of Gd on the recrystallisation, texture and deformation behaviour of magnesium-based alloys[END_REF]. The SEM micrograph (Figure 1b) revealed the presence of the second phase identified as Mg41Dy5 phase. It was expected that the Mg-0.41 alloy is a solid solution alloy since the binary phase diagram indicate that solubility of Dy is high in the Mg matrix (25.23%, wt. %) [START_REF] Rokhlin | Magnesium Alloys Containing Rare Earth Metals: Structure and Properties[END_REF]. The annealing at 420 °C for 20 h followed the as casting process failed to dissolve this metastable phase. The sample processed at RT exhibited higher flow. First, the stress shows a rapid increases due to the strain hardening to reach a maximum of 225 MPa at ε = -0.32 and decreases up ε = -0.6 to increases again. This secondary strain hardening (at ε ~ -0.6) is present in all processed samples (Figure 2) and it is usually seen in PSC processed Mg-based alloys and result from the result of the friction between the sample and channel-die [START_REF] Azzeddine | On some aspects of compressive properties and serrated flow in Mg-Y-Nd-Zr alloy[END_REF][START_REF] Elfiad | Investigation of texture, microstructure, and mechanical properties of a magnesium-lanthanum alloy after thermomechanical processing[END_REF][START_REF] Li | Texture and microstructure development during hot deformation of ME20 magnesium alloy: Experiments and simulations[END_REF][START_REF] Al-Samman | Effect of heavy metal impurities in secondary Mg alloys on the microstructure and mechanical properties during deformation[END_REF].

PSC stress flow

The first strain hardening is attributed to the accumulation of dislocations and twinning.

Actually, the twin grain boundaries act as barriers to dislocation motions and then led to a work hardening [START_REF] Barnett | Influence of grain size on the compressive deformation of wrought Mg-3Al-1Zn[END_REF][START_REF] Al-Samman | Uniaxial and Plane Strain Compression Behaviour of Magnesium Alloy AZ31: A Comparative Study[END_REF]. Besides, the formation of "hard" orientations in the microstructure caused by the twinning can also increases the flow stress [START_REF] Stanford | The effect of rare earth elements on the behaviour of magnesium-based alloys: Part 1-Hot deformation behaviour[END_REF]. The flow curve of sample processed at 200 °C shows a continuous increase with increasing strain. While, sample processed at 400 °C exhibits the lower strain hardening (15 MPa at ε = -0.02) and gradually decreases to reach a relative steady state indicating a softening effect due to the dynamic recovery or dynamic recrystallization (DRX).

Deformation microstructure

Figure 3 shows the evolution of the deformation microstructure of Mg-0.41Dy after PSC test at RT, 200 and 400 °C under strain rate of 10 -4 s -1 , respectively. As expected, the microstructure of sample processed at RT exhibits large scale of deformation heterogeneities.

Besides, the presence of high amount of twins shear bands could be also present as indicated by the highly distorted regions (dark regions as shown by arrow). The presence of shear bands is frequently observed in deformed Mg-based alloys and it appears in the form of dark streamlines inclined at 40° to the loading direction [START_REF] Liu | Effect of temperature on microstructure and texture evolution of Mg-Zn-Er alloy during hot compression[END_REF][START_REF] Barnett | Deformation microstructures and textures of some cold rolled Mg alloys[END_REF][START_REF] Mackenzie | The influences of alloying additions and processing parameters on the rolling microstructures and textures of magnesium alloys[END_REF][START_REF] Fatemi-Varzaneh | Shear banding phenomenon during severe plastic deformation of an AZ31 magnesium alloy[END_REF]. It was demonstrated that the shear bands were manifestly different in Mg-RE alloy compared to pure Mg and the AZ31 alloy [START_REF] Sandlöbes | The Relation between Shear Banding, Microstructure and Mechanical Properties in Mg and Mg-Y Alloys[END_REF][START_REF] Barnett | Deformation microstructures and textures of some cold rolled Mg alloys[END_REF]. The density of shear bands was higher in the deformed Mg-RE alloy and homogeneously distributed in the microstructure. The difference was attributed to the effect of RE element on the further activation of <c+a> pyramidal slip system and enhancement of contraction and double twins [START_REF] Sandlöbes | The Relation between Shear Banding, Microstructure and Mechanical Properties in Mg and Mg-Y Alloys[END_REF][START_REF] Barnett | Deformation microstructures and textures of some cold rolled Mg alloys[END_REF].

The twin fraction decreases with increasing deformation temperatures and less heterogeneity can be noticed in Figure 3. It is well known that the activation of twins is less significant during deformation at high temperature [START_REF] Tomé | The Relation between Texture, Twinning and Mechanical Properties in Hexagonal Aggregates[END_REF]. The twins in sample processed at 200 °C have steppes like morphology. Similar twins structure was already reported in hot-rolled AZ31 alloy [START_REF] Mackenzie | The influences of alloying additions and processing parameters on the rolling microstructures and textures of magnesium alloys[END_REF] and hot-compressed Mg-Zn-Er alloy [START_REF] Liu | Effect of temperature on microstructure and texture evolution of Mg-Zn-Er alloy during hot compression[END_REF]. The formation of such morphology depends on the strain level and the homogeneity of stress distribution inside the microstructure [START_REF] Liu | Effect of temperature on microstructure and texture evolution of Mg-Zn-Er alloy during hot compression[END_REF]. Figure 3c shows clearly the occurrence of DRX at high deformation temperature (400 °C). The grains boundaries in sample processed at 400 °C are serrated and bulged which is an indicative of a nucleation and formation of new DRX grains. The bulging at the grain boundaries is characteristic of the DDRX type. The bulging of grain boundaries is due to the interaction between moving lattice dislocations and the grain boundaries and it was found stimulated when the non-basal slip like pyramidal <c+a> is activated [START_REF] Barnett | Texture selection mechanisms in uniaxially extruded magnesium alloys[END_REF].

The SEM micrographs of all processed samples show the presence of second particles.

Unfortunately, their chemical composition identification was not possible because of their small size. However, it is obvious that their form and distribution differ significantly with increasing deformation temperature. At RT, the particles are small and well distributed in the microstructure. With increasing deformation temperature the particles are aligned along 45° toward LD and have plate form. The volume fraction of the particles increases from 0.2% at RT and 200 °C to 2.3% in sample processed at 400 °C. Previously, it was reported that HPT processing at RT of the same Mg-0.41Dy alloy causes the decomposition of the Mg24Dy5 second particles [START_REF] Hanna | An investigation of the thermal stability of an MgDy alloy after processing by high-pressure torsion[END_REF]. Moreover, annealing at 200 and 400 °C for 1 h of the HPT processed alloy result to more Dy element distribution [START_REF] Hanna | An investigation of the thermal stability of an MgDy alloy after processing by high-pressure torsion[END_REF].

Manifestly, the presences of twins have a great influence on the form and distribution of second phases in the processed sample. The high amount of twins in sample processed at RT prevents the second particles displacement. PSC at 200 °C results to less twins thus the second particles start to align 45° toward LD as shown in Figure 3d. In case of processing at 400 °C when no twinning was observed a further changing in the form and distribution of second particles is clearly observed (Figure 3g). In addition, increasing the volume fraction of second particles during processing at 400 °C could be an indication of dynamic precipitation.

In fact, it is often observed to occurrence of dynamic precipitation during hot processing of Mg-RE alloys [START_REF] Azzeddine | On the texture and grain growth in hot-deformed and annealed WE54 alloy[END_REF][START_REF] Xiao | Dynamic precipitation in a Mg-Gd-Y-Zr alloy during hot compression[END_REF][START_REF] Azzeddine | Texture and Microstructure of WE54 Alloy after Hot Rolling and Annealing[END_REF]. It was recently reported that the discontinuous precipitation is entirely hindered by the grain boundaries of extension twins during ageing at 150 °C of AZ80 (Mg-8Al-0.5Zn, wt. %) alloy [START_REF] Shi | Enhanced precipitation strengthening of extruded Mg-8 wt.%Al-0.5 wt.%Zn (AZ80) magnesium alloy by extension twinning[END_REF]. In addition, the high dislocation density inside the twins can promotes the occurrence of continuous precipitation within the twins [START_REF] Shi | Enhanced precipitation strengthening of extruded Mg-8 wt.%Al-0.5 wt.%Zn (AZ80) magnesium alloy by extension twinning[END_REF]. Consequently, the precipitates inside twins may block the dislocation motion and affect the mechanical properties of the alloy. In meantime, the presence of particles/precipitates can also affect the size and the total volume fraction of twins during deformation [START_REF] Stanford | Effect of particles on the formation of deformation twins in a magnesium-based alloy[END_REF][START_REF] Liu | On the interaction of precipitates and tensile twins in magnesium alloys[END_REF]. It was indicated that the effective critical resolved shear stress (CRSS) for twin growth increases linearly with precipitate volume fraction [START_REF] Liu | On the interaction of precipitates and tensile twins in magnesium alloys[END_REF].

Figure 4 shows some interesting features on the evolution of deformation microstructures using EBSD of Mg-0.41Dy alloy after PSC at RT, 200 and 400 °C under stain rate of 10 -4 s -1 , respectively.

The IPF map of Mg-0.41Dy alloy after PSC at RT and 200 °C (Figure 4a and4d) shows complex twins formation and distribution. The purple line correspond to sub-boundaries (2°-5°), the white line presents low grain boundaries (LAGB) with misorientation between 5-15° and the black line correspond to high grain boundaries (HAGB, >15°). The IPF map of the sample processed at RT presents some black zones resulting from the highly deformed areas in the sample. The identified twins in both samples are highlighted in Figures 4b and4e as: extension twin 86°<112 ̅ 0> 8° (in green), contraction twin 56°<112 ̅ 0> 8° (in blue), double twins 38°<112 ̅ 0> 8° (in red) and 22°<112 ̅ 0> 8° (in yellow).

Practically, all twins mode are present in both processed samples, except for 22°<112 ̅ 0> seems less activated in sample processed at 200 °C. The activation of the different twin modes in the same sample could be explained by the random texture of the as-cast Mg-0.41Dy alloy.

It can be observed that extension twin 86°<112 ̅ 0> and double twin 38°<112 ̅ 0> are dominated in both microstructures. As known in deformed Mg-based alloys, the extension twin is the most twin present due to its low CRSS compared to the other twin modes [START_REF] Knezevic | Deformation twinning in AZ31: Influence on strain hardening and texture evolution[END_REF]. In contrast, the presence of double twin is attributed to the rapid transformation of contraction twin to double twin during deformation processing [START_REF] Niknejad | The role of double twinning on transgranular fracture in magnesium AZ61 in a localized stress field[END_REF].

It is recognized by looking inside the twins for both processed samples that contain high fraction of sub-boundaries (purple line). It is expected that these LAGBs will be progressively transformed to HAGBs and therefore should form the new structure of fine recrystallized grains as known by continuous recrystallization [START_REF] Al-Samman | Softening and dynamic recrystallization in magnesium single crystals during c-axis compression[END_REF][START_REF] Sitdikov | Dynamic Recrystallization Based on Twinning in Coarse-Grained Mg[END_REF][START_REF] Wang | Effect of initial texture on dynamic recrystallization of AZ31 Mg alloy during hot rolling[END_REF]. It was shown that the recrystallized grains can develop different orientations from those of the host twins and parent grains [START_REF] Molodov | Mechanisms of exceptional ductility of magnesium single crystal during deformation at room temperature: Multiple twinning and dynamic recrystallization[END_REF][START_REF] Kim | Orientations of dynamically recrystallized grains nucleated at double twins in Mg-4Zn-1Sn alloy[END_REF].

One of the benefits of the activation of twin is the creation of new orientation. Figures 4c and4d show the {0002} pole figure indicating the orientation between grains and twins in samples processed at RT and 200 °C, respectively. The presence of extensive twinning causes the split of the basal pole approximately 15 ° towards LD. Similar splitting was already reported in cold-rolled AZ31 alloy and was explained by the occurrence of double twining [START_REF] Barnett | Deformation microstructures and textures of some cold rolled Mg alloys[END_REF]. Moreover, it was speculated that the deformation heterogeneities like shear band could contributed to this splitting [START_REF] Knezevic | Deformation twinning in AZ31: Influence on strain hardening and texture evolution[END_REF]. Generally, the splitting of basal pole reduces with increasing deformation temperature [START_REF] Knezevic | Deformation twinning in AZ31: Influence on strain hardening and texture evolution[END_REF]. Figure 4g shows the IPF map of the sample processed at 400 °C. No twins were detected and it is cleary seen that few grains start to form in the large deformed grains. The recrystallized grains can be reconnaized using the grains orientation spread (GOS) approach where GOS is defined as the mean standard deviation of all the orientations inside a grain [START_REF] Cho | Determination of a mean orientation in electron backscatter diffraction measurements[END_REF]. The GOS map shown in the Figure 4h indicates that a few grains are recrystallized (blue one) as grains having GOS values less than 2°. The remains small grains have GOS values between 2 and 4° which mean that they are dynamically recovered. The orientations of the deformed and recrystallized grains have the same basal texture as shown by the separated corresponding {0002} pole figures in Figure 4i. Several studies repotred similar finding where the dynamically recrystallised grains developped the same orientation like the deformed one [START_REF] Cottam | Dynamic recrystallization of Mg and Mg-Y alloys: Crystallographic texture development[END_REF][START_REF] Stanford | The effect of rare earth elements on the behaviour of magnesium-based alloys: Part 2recrystallisation and texture development[END_REF].

It is worthnoting that even at high deformation temeprature (400 °C) and low strain rate ( 10 -4 s -1 ) the present Mg-0.41Dy alloy shows low fraction of dynamic recrystallization in comparaison with conventional AZ31 [START_REF] Abdessameud | Grain growth in AZ31 alloy after uniaxial compression[END_REF][START_REF] Stanford | The effect of rare earth elements on the behaviour of magnesium-based alloys: Part 1-Hot deformation behaviour[END_REF] or even ternary Mg-RE alloy [START_REF] Azzeddine | On the texture and grain growth in hot-deformed and annealed WE54 alloy[END_REF]. It was observed that the dynamic recrystallization was inhibited in Mg-1.5Gd (wt. %) compared to AZ31 alloy during hot PSC and it was explain by the higher stored energy devellopted in Mg-1.5Gd alloy [START_REF] Stanford | The effect of rare earth elements on the behaviour of magnesium-based alloys: Part 1-Hot deformation behaviour[END_REF]. The retardation of DRX can be also result from the solute segregation to grain boundaries (pinning effect) [START_REF] Basu | The role of atomic scale segregation in designing highly ductile magnesium alloys[END_REF] and the change in the recovery behavior caused by the reduction in the stacking fault energy [START_REF] Sandlöbes | The relation between ductility and stacking fault energies in Mg and Mg-Y alloys[END_REF].

Deformation texture

Figures 5 presents the evolution of the texture of Mg-0.41Dy alloy after PSC at RT, 200 and 400 °C under a strain rate of 10 -4 s -1 in terms of recalculated {0002} and {101 ̅ 0} pole figure, respectively. As can be seen, PSC processing leads to the formation of two fibers: basal texture <0001> // CD (φ1 = 0-360°, Φ = 0°, φ2 = 0°) and weak <101 ̅ 0> // CD (φ1 = 0-360°, Φ = 90°, φ2 = 30°). Similar texture evolution was previuosly reported in hot-compressed WE54 (Mg-4.9Y-4.2Nd-0.56Zr, wt.%) alloy and binary cold proccesed Mg-1.33La (wt. %) alloy by PSC [START_REF] Azzeddine | On the texture and grain growth in hot-deformed and annealed WE54 alloy[END_REF][START_REF] Elfiad | Investigation of texture, microstructure, and mechanical properties of a magnesium-lanthanum alloy after thermomechanical processing[END_REF]. However, some difference can be noticed as function of deformation temepratures. First, the texture intensity increases significantly with increasing deformation temperature. Second, it can be shown that processing at RT lead to the development of only a weak basal <0001> // CD and a very weak prismatic <101 ̅ 0> // CD fiber. The weak texture can be attributed to the presence of high scale of deformation henetrogities resulting to the development of wide range of orientations [START_REF] Imandoust | A review on the effect of rare-earth elements on texture evolution during processing of magnesium alloys[END_REF][START_REF] Basu | Shear band-related recrystallization and grain growth in two rolled magnesium-rare earth alloys[END_REF]. In contrast, the prismatic <101 ̅ 0> // CD fiber despaired and a net spliting of basal pole towared LD can be notice in sample processed at 200 °C. The split of basal pole toward LD was attributed to the activation of twinnig and the pyramidal <c+ a> slip [START_REF] Mackenzie | The influences of alloying additions and processing parameters on the rolling microstructures and textures of magnesium alloys[END_REF]. Moreover, the recalculated {0002} pole figure shows the presence of two texture components <0001> // LD and <0001> // TD, respectively. The development of both texture components result from desorientation resultated from the activation of twins as shown in Figures 4e and4f. This finding is supported by the similar development of deformation texture of AZ31 alloy [START_REF] Song | Texture control by {10-12} twinning to improve the formability of Mg alloys: A review[END_REF]. The prismatic <101 ̅ 0> // CD fiber seems to appeared again after processing at 400 °C. However, Figure 6 shows cleary that the fibers are not continuous espacially for sample procesed at 200 and 400 °C where the high intensities of the fibers are more concentrated around specific orientations. For exemple, for sample processed at 400 °C the maximum intensity (~21 mrd) of basal <0001> fiber is located at (270°, 0°, 0°). While the maximum of <101 ̅ 0> // CD fiber is located at (50°, 90°, 30°), (115°, 90°, 30°), (230°, 90°, 30°) and (295, 90°, 30°), respectevely. Based on the evolution of deformation microstructure at 400 °C, the resulted texture is attributed to the DRX and the presence of the second particule. Howerver, it is known that DRX led to the weakenenig of deformation texture in Mg-RE alloy. As evidenced in Figure 4 h the DRX is strongly retained during deformation at 400 °C. It was suggested that the presence of second particle have more effect on texture weakening during hot compression rather than the concentration of RE element on the Mg matrix [START_REF] Wu | Effect of compression conditions on the microstructure and texture of a Mg-RE alloy[END_REF].

However, following the present finding, the dynamic precipitation during PSC and uniaxial compression at 300 °C of WE54 alloy show no effect on altering the deformation texture [START_REF] Azzeddine | On the texture and grain growth in hot-deformed and annealed WE54 alloy[END_REF].

Moreover, the presence of second phase is not necessary for weakening the deformation texture since several solid solution Mg-RE alloys exhibit weak texture [START_REF] Hantzsche | Effect of rare earth additions on microstructure and texture development of magnesium alloy sheets[END_REF][START_REF] Stanford | The origin of "rare earth" texture development in extruded Mg-based alloys and its effect on tensile ductility[END_REF].

Static recrystallization

Figure 7-9 illustrate the evolution of the microstructure of the processed Mg-0.41Dy at RT, 200 and 400 °C and subsequent annealed at 450 °C to 7 days, respectively. Interesting to note that sample processed at 400 °C still partially recrystallized after annealing for 30 min, while both processed samples at RT and 200 °C show a typical recrystallized microstructure with equiaxe grain distribution. In meantime, increasing annealing time causes a drastic coarsening of grain for sample processed at 400 °C as shown in Figure 9. The evolution of average grain size as function of annealing time for samples processed at RT and 200 °C is shown in Figure 10. The average grain size of sample processed at 400 °C could not be calculated because of lack of statistic since only few grains can be detected in the microstructures, as shown in Figure 9. As can be notice, the rate of increasing grain size increase with increasing deformation temperature. The average grain size of sample processed at RT increases slowly from 470.8 μm after annealing for 30 min to 712.1 μm after 7 days. While, processing at 200 °C lead to rapid increases from 492.9 μm after 30 min to 120.54.2 μm after 7 days of annealing.

It is expected that the stored energy in sample processed at lower deformation temperatures and higher strains will be higher and thereby increases the rate of recrystallization [START_REF] Humphreys | Chapter 7 -Recrystallization of Single-Phase Alloys[END_REF]. However, this is not the case in the present alloy. The rate of evolution of grains size is lower with decreasing the deformation temperature.

The presence of second particles is another factor could retard the recrystallization by pinning the migration of grain boundaries. However, it can be speculated that during the deformation at 400 °C the solute drag and Zener pinning would be negligible consequently the high rate of grain growth shown in Figure 9. The evolution of static recrystallization is rather faster at 400 °C indicating that DRX plays a significant role in this case.

Actually, the homogenous and fine grains distribution in sample processed at RT is attributed to the presence of high deformation mode such as shear band and twining as shown in Figure 2 compared to sample processed at 200 and 400 °C. It was evidenced that the recrystallized grains inside the twin do not grow beyond twin boundaries due to their low mobility [START_REF] Guan | Twin recrystallization mechanisms and exceptional contribution to texture evolution during annealing in a magnesium alloy[END_REF].

Static recrystallization kinetics

The recrystallization kinetics of deformed alloy can be investigated using the Johnson-Mehl-Avrami-Kolmogorov (JMAK) model [START_REF] Vandermeer | Microstructural modeling of recrystallization in deformed iron single crystals[END_REF]:

𝑋 = 1 -exp (-𝑘𝑡 𝑛 ) (1)
where X is the recrystallized fraction, k a pre-exponential constant and n is the Avrami exponent.

The recrystallized fraction can be estimated by mean grains size or Hv values following the equations:

𝑋 𝑑 = 𝑑 𝑡 -𝑑 0 𝑑 𝑓 -𝑑 0 ( 2
)
where dt is the mean grain size of the annealed sample at time t, d0 the mean grain size of the deformed alloy and df the mean grain size of the full recrystallized alloy (t = 7 days). It is to be noted that d0 was taken from samples annealed for 30 min.

𝑋 𝐻𝑣 = 𝐻𝑣 0 -𝐻𝑣 𝑡 𝐻𝑣 0 -𝐻𝑣 𝑓 (3)
where Hv0 is the microhardness of the deformed sample, Hvf the microhardness of the recrystallized sample and Hvt the microhardness of the annealed sample at time t.

The n avrami exponent can be calculated by the linear relationship from the equation (1):

𝑙𝑛𝑙𝑛 ( 1 1-𝑋 ) = 𝑙𝑛𝑘 + 𝑛𝑙𝑛𝑡 (4)
The plots of 𝑙𝑛𝑙𝑛 ( dimensional grain growth occurred due to migration of HAGB [START_REF] Fenghui | Recrystallization Kinetics of Fe-3%Si after Deformation at High Strain Rate and High Temperature[END_REF]. It was reported that the lower n value can be attributed to the non-random recrystallization sites present in the deformed alloy [START_REF] Humphreys | Chapter 7 -Recrystallization of Single-Phase Alloys[END_REF]. In this case, the recrystallization preferentially occurs along grain boundaries and within twins and shear bands. This is in good agreement with the evolution of deformation microstructure. The n values were reported to be as 0.68, 0.70 and 1.02 for hot-rolled Mg-Al-Zn-Mn-Ca alloy and annealed at 200, 300 and 450 °C, respectively [START_REF] Wang | Unveiling annealing texture formation and static recrystallization kinetics of hot-rolled Mg-Al-Zn-Mn-Ca alloy[END_REF]. Also, the n values were in range of 1.6-3.4 for cold rolled AZ31 alloy [START_REF] Su | Recrystallization and grain growth of deformed magnesium alloy[END_REF]. It is seems that n value increases with increasing annealing temperature and deformation strain [START_REF] Wang | Unveiling annealing texture formation and static recrystallization kinetics of hot-rolled Mg-Al-Zn-Mn-Ca alloy[END_REF][START_REF] Su | Recrystallization and grain growth of deformed magnesium alloy[END_REF]. In the present work the n value is sensitive to the deformation condition where it decreases with increasing deformation temperature.

Conclusions

 The binary Mg-0.41Dy alloy was processed by PSC at RT, 200 and 400 °C under strain rate of 10 -4 s -1 .

 The deformation and recrystallization microstructures and texture have a strong dependency on the deformation temperature.

 PSC at RT exhibits a large scale of heterogeneities composed of various mode of twining, i.e. extension, contraction and double twins. Less heterogeneity was observed during PSC at 200 °C and the twins have steppes structure morphology. While PSC at 400 °C led to the occurrence of dynamic recrystallization. However, the fraction of dynamically recrystallized grains was low compared to conventional Mg-based alloys due to the effect of the RE element and the presence of the second phase.

 The texture was characterized by the formation of basal {0001} and prismatic {101 ̅ 0} fibers. The texture intensity increases with increasing deformation temperature.

 The rate of evolution of grains size during annealing at 450 °C was lower with decreasing the deformation temperature.

 The Avrami exponent of recrystallization n was in the range of 0.9-0.26 indicating that the recrystallization preferentially occurs along grain boundaries and within deformation features such as twins.

Figure 1 :

 1 Figure 1: (a) IPF map and (b) SEM micrograph of as-cast Mg-0.41Dy alloy.

Figure 2

 2 Figure 2 illustrates the true stress-strain curves of Mg-0.41Dy during PSC tests at RT, 200 and 400 °C under the strain rate of 10 -4 s -1 . It can be clearly seen that the mechanical response of Mg-0.41Dy alloy is different as function of deformation temperature and strongly related to the evolution of the deformation microstructure during the PSC test.

Figure 2 :

 2 Figure 2: True stress-true strain curves of Mg-0.41Dy during PSC test at RT, 200 and 400 °C under strain rate of 10 -4 s -1 .

Figure 3 :

 3 Figure 3: Deformation microstructures obtained by OM and SEM of Mg-0.41Dy after PSC test at: (a, b) RT, (c, d) 200 and (e-g) 400 °C under strain rate of 10 -4 s -1 .

Figure 4 :

 4 Figure 4: (a) IPF map, (b) map showing the presence of different twins and (c) {0002} pole figure indicates the orientation between grains and twins for sample processed at RT. (d) IPF map, (e) map showing the presence of different twins and (f) {0002} pole figure indicates the orientation between grains and twins for sample processed at 200 °C. (g) IPF map, (h) GOS map and (i) the corresponding {0002} pole figure of deformed and recrystallized grains in sample processed at 400 °C.

Figure 5 :

 5 Figure 5: Recalculated (a) {0002} and (b){101 ̅ 0} pole figure of Mg-0.41Dy alloy after PSC at RT, 200 and 400 °C under strain rate of 10 -4 s -1 , respectively.

Figure 6 :

 6 Figure 6: Evolution of (a) {0002} and (b) {101 ̅ 0} fiber as function of 1 angle of Mg-0.41Dy processed at RT, 200 and 400 °C under strain rate of 10 -4 s -1 .

Figure 7 :

 7 Figure 7: Evolution of recrystallization microstructure of Mg-0.41Dy processed at RT and annealed at 450 °C to : (a) 30 min, (b) 10 h, (c) 24h and (d) 7 days.

Figure 8 :

 8 Figure 8: Evolution of recrystallization microstructure of Mg-0.41Dy processed at 200 °C and annealed at 450 °C to : (a) 30 min, (b) 24 h, (c) 3 days and (d) 7 days.

Figure 9 :

 9 Figure 9: Evolution of recrystallization microstructure of Mg-0.41Dy processed at 400 °C and annealed at 450 °C to : (a) 30 min, (b) 10 h, (c) 3 days and (d) 7 days.

Figure 10 :

 10 Figure 10: Evolution of grain size as function of annealing time of Mg-0.41Dy processed at RT and 200 °C, respectively.

Figure 11 :

 11 Figure 11: Evolution of microhardness as function of annealing time of Mg-0.41Dy processed at RT, 200 and 400 °C, respectively.

Figure 11

 11 Figure 11 shows the variation of microhardness as function of annealing time of Mg-0.41Dy processed at RT, 200 and 400 °C, respectively. It is obvious that the sample processed at RT has different tendency than those processed at 200 and 400 °C. First, the Hv values decrease slowly between annealing for 30 min and 3h followed by rapid decrease to reach a value of 38.861.2 Hv after 10 h of annealing and then a steady state is reached (38.70.5 Hv after 7 days of annealing). In contrast, the Hv values of samples processed at 200 and 400 °C show a continuous decrease with increasing annealing time indicate a continuous softening of the microstructure. As can be seen the Hv values depend strongly on the deformation temperature. It is decease with increasing deformation temperature.

)

  as function of lnt are shown in Figure 12. The n values for the annealed samples are shown in the graphs. The n value of sample processed at RT calculated from mean grain size was close to the unity (n = 0.91). n = 0.5 was found for sample processed at 200 °C. Both techniques, Xd or XHv give the same n value. For sample processed at 400 °C the n value decreases to n = 0.26. In the literature n = 1 indicate that one-

Figure 12 :)

 12 Figure 12: Evolution 𝑙𝑛𝑙𝑛 ( 1 1-𝑋 ) as function of lnt for the processed sample at RT, 200 and