Julien Moriceau
email: julien.moriceau@univ-rennes1.fr

Patrick Houizot

Maciej Lorenc

Tanguy Rouxel

Healing of cracks by green laser irradiation in a nanogold particles glass matrix composite

Keywords: Glass matrix composites, Spark Plasma Sintering, Indentation, Laser treatment, Gold nanoparticles

A transparent composite material consisting of a soda-lime-silicate glass matrix containing up to 0.5 % w gold nanoparticles (10 to 20 nm) was successfully obtained by coating glass grains and a subsequent sintering using a spark plasma sintering (SPS) furnace. Coating was achieved thanks to the reduction of gold(III) chloride trihydrate by sodium borohydride in an ethanol solution. The sintered composites were dense and transparent, but for the largest gold content where the sample exhibits a deep red color. When submitted to laser irradiation (532 nm, up to 100 W/cm²), the composite experiences a fast, localized and much more significant increase of its temperature in comparison with the pristine glass material. A temperature as high as 550 °C was reached, allowing the healing of surface crack.

Introduction

Crack exhibiting healing phenomena were scarcely reported in the literature [START_REF] Amendola | Advances in self-healing optical materials[END_REF]. In healing polymers the common approaches use the vascular network [START_REF] Lee | Advances in self-healing materials based on vascular networks with mechanical self-repair characteristics[END_REF][START_REF] White | Autonomic healing of polymer composites[END_REF] or reactive particles [START_REF] Jackson | Transparent self-healing polymers based on encapsulated plasticizers in a thermoplastic matrix[END_REF] which fill the cracks that propagate across them. Another method is to favor some relaxation process by triggering atoms mobility with an external source of energy [START_REF] Froimowicz | Towards the generation of self-healing materials by means of a reversible photo-induced approach[END_REF]. In oxide glasses the atoms mobility is much lower than in polymers and high temperatures are required to observe some relaxation phenomenon within a "reasonable" time (say less than an hour). Besides oxide glasses are mostly elaborated at high temperature, typically above 1200 °C, and the second phase particles that are introduced with the aim to promote healing must be chosen with much care to survive the synthesis cycle. For example, Coillot et al. [START_REF] Coillot | Influence of the active particles on the self-healing efficiency in glassy matrix[END_REF] dispersed vanadium boride particles in a barium aluminosilicate glass by SPS and they observed the healing of indentation cracks after the particles melted and reacted with the matrix due to the combination of an oxidizing environment and a high temperature. Singh [START_REF] Singh | Kinetics of self-repair in inorganic glasses: modeling and experimental verification[END_REF] promoted healing at indentation cracking sites by heat treating a window glass in the transition range, hence inducing viscous relaxation. However, the drawback of these two latter methods is the global heating of the material up to high temperatures which might be detrimental to the conservation of the geometry of the component and may thus not be suitable to industrial application. Besides, the dispersion of particles mostly results in the loss of transparency [START_REF] Amendola | Advances in self-healing optical materials[END_REF][START_REF] Coillot | Influence of the active particles on the self-healing efficiency in glassy matrix[END_REF].

In this article we suggest a new approach based on the incorporation of gold nanoparticles in a transparent glass and a localized laser heating in damaged areas.

Nanogold glass composites were already produced by glass craftsmen centuries ago in order to brighten red glasses. Historically the coloration of the glass was obtained by direct addition of gold in the glass melt and further heat treatments [START_REF] Ruivo | Gold nanoparticles in ancient and contemporary ruby glass[END_REF]. Nowadays similar glasses can be obtained by various methods such as sol-gel [START_REF] Magyari | Novel bioactive glass-AuNP composites for biomedical applications[END_REF][START_REF] Simon | Gold nanoparticles developed in sol-gel derived apatite-bioactive glass composites[END_REF][START_REF] Lusvardi | Bioactive glasses containing Au nanoparticles. Effect of calcination temperature on structure, morphology and surface properties[END_REF] or high energy irradiation of Au containing glass [START_REF] Ruivo | Gold nanoparticles in ancient and contemporary ruby glass[END_REF][START_REF] Nedyalkov | Optical properties modification of gold doped glass induced by nanosecond laser radiation and annealing[END_REF][START_REF] Eichelbaum | On the chemistry of gold in silicate glasses: studies on a nonthermally activated growth of gold nanoparticles[END_REF]. Such composites are of interest for biomedical [START_REF] Magyari | Novel bioactive glass-AuNP composites for biomedical applications[END_REF][START_REF] Simon | Gold nanoparticles developed in sol-gel derived apatite-bioactive glass composites[END_REF][START_REF] Aina | Gold-containing bioactive glasses: a solid-state synthesis to produce alternative biomaterials for bone implantations[END_REF] and optical [START_REF] Nedyalkov | Optical properties modification of gold doped glass induced by nanosecond laser radiation and annealing[END_REF] applications and for historical purposes as well [START_REF] Ruivo | Gold nanoparticles in ancient and contemporary ruby glass[END_REF][START_REF] Eichelbaum | On the chemistry of gold in silicate glasses: studies on a nonthermally activated growth of gold nanoparticles[END_REF].

Indeed gold nanoparticles (GNP) can enhance the bone regeneration via the surface plasmon resonance (SPR) or significantly affect the optical properties (color, refractive index). The gold SPR occurring at 532 nm is also used to induce heating by laser irradiation [START_REF] Keblinski | Limits of localized heating by electromagnetically excited nanoparticles[END_REF][START_REF] Roper | Microscale Heat Transfer Transduced by Surface Plasmon Resonant Gold Nanoparticles, The journal of physical chemistry[END_REF]. This technique allows for a localized heating by promoting a strong laser/material interaction [START_REF] Roper | Microscale Heat Transfer Transduced by Surface Plasmon Resonant Gold Nanoparticles, The journal of physical chemistry[END_REF][START_REF] Nedyalkov | Laser heating of gold nanoparticles: photothermal cancer cell therapy[END_REF][START_REF] Sassaroli | Numerical investigation of heating of a gold nanoparticle and the surrounding microenvironment by nanosecond laser pulses for nanomedicine applications[END_REF].

In the present work, nanoparticles of gold were coated at the surface of window glass particles and the coated particles were then sintered by spark plasma sintering (SPS). Vickers indentation was performed at the surface of the obtained material and the indentation site was further submitted to laser irradiation to promote crack healing.

Materials and methods

Window glass spherical particles were purchased from HD-Outillage with a diameter in the 40 to 70 µm range. The coating of Au on glass particles was achieved by reduction of gold(III) chloride trihydrate (HAuCl 4 .3H 2 O, Sigma-Aldrich, >99.9%) by sodium borohydride (NaBH 4 , Sigma-Aldrich, >98%) in ethanol following the protocol reported by Kaneti et al. [START_REF] Kaneti | Hydrothermal synthesis of ternary α-Fe 2 O 3 -ZnO-Au nanocomposites with high gas-sensing performance[END_REF]. Different amounts (0.01, 0.02, 0.05, 0.1, 0.2 and 0.5 % w ) of gold were deposited by tuning the amount of reactants. The gold content was calculated by assuming that the reaction of ionic gold with sodium borohydride was complete and that the obtained gold is entirely deposited on the glass particles. The relative error on the gold content is then estimated at around 5 %. A broad range of colored glasses were obtained (see Figure 1). After the drying of the powder some purple aggregates of gold were observed for samples containing more than 0.1 % w of gold. These aggregates were removed from the powder mixture. A presintering step of 1 g of coated glass particles was performed in a silica mold (diameter 10 mm) at 700 °C during 20 min to limit the carbon contamination that may further occur during the subsequent SPS step [START_REF] Bertrand | A comprehensive study of the carbon contamination in tellurite glasses and glassceramics sintered by spark plasma sintering (SPS)[END_REF]. After presintering, a pellet of 10 mm was obtained and introduced in a graphite die protected by a 220 µm thickness carbon paper (Papyex, Mersen). SPS sintering step was done using a KCE-FCT-HP D 10-SD system with a temperature plateau at 625 °C during 1.5 min under a pressure of 64 MPa. The heating rate was set at 100 °C/min. After the heating stage, the sample was cooled down to room temperature by switching the power off. 30 g of the uncoated glass particles were also melted for 2 h at 1500 °C to be used as a reference. After quenching the liquid in a brass mould an annealing heat treatment was done for 2 h at 570 °C and followed by a slow cooling down to room temperature.

e)

The density of the sintered samples was measured from 5 measurements by the Archimedes method in water. The mean error on the relative density was around 0.1 %. The transmittance of the composites (thickness 1.7 mm) was recorded by a PerkinElmer Lambda 900 spectrophotometer between 200 and 800 nm with a precision slot of 2 nm. Vickers indents were performed with a Matsuzawa VMT-7S apparatus with a load of 10 N applied for 15 sec. The laser irradiation was implemented by an unfocused (spot size: 2.25 mm ± 10 %) continuous irradiation of a Coherent Verdi series V G5 laser emitting at 532 nm. The temperature measurement was done by a FLIR camera positioned 15 cm and 30° on the right of the sample and focused on the surface. The relative error on the temperature measurement is around 10 %. Observation of crushed glass grains coated with 0.5 % w of GNPs was performed by means of a JEOL 2100 TEM under a voltage of 200 keV. The glass transition and softening temperatures of the initial glass were measured with a L75 PT Linseis vertical dilatometer at a rate of 10 °C/min with a precision of ±5 °C.

Results and Discussion

The sintering of the pristine glass grains leads to fairly transparent samples (see Figure 1). When Au is coated on the glass particles, the sample after sintering are colored from pink to purple with a slight decrease of the transparency as the amount of gold is increased even though the relative density is maintained above 99.5% in all cases. The optical transmittance spectra in the visible range are shown in Figure 2a. The average transmission of the gold free sintered glass is slightly inferior to the one of the samples containing 0.01 and 0.02 % w of gold. This can be due to trapped bubbles created by uneven beads packing before the presintering step. As the gold amount is increased, the transparency decreases because of the formation of gold agglomerates, which are clearly visible in the glasses containing between 0.1 and 0.5 % w of gold. An absorption peak is visible at around 532 nm in all spectra of gold containing samples.

It is due to the SPR of GNP. The SPR is very dependent of the size and geometry of the nanocrystal [START_REF] Link | Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles[END_REF][START_REF] Vincenzo | Surface plasmon resonance in gold nanoparticles: a review[END_REF][START_REF] Rodríguez-Oliveros | Gold nanostars as thermoplasmonic nanoparticles for optical heating[END_REF] and was reported from 511 nm [START_REF] Ruivo | Gold nanoparticles in ancient and contemporary ruby glass[END_REF] to 556 nm [START_REF] Eichelbaum | On the chemistry of gold in silicate glasses: studies on a nonthermally activated growth of gold nanoparticles[END_REF]. The observation of the GNP with the TEM shown the particles diameter was comprised in between 10 and 20 nm (Figure 1). Although the particles were too small to observe any diffraction pattern, the lattice parameter of the observed NPs was found to be consistent with the (111) plans of metallic gold (ICSD: 00-004-0784). In order to better compare the absorption of the different samples, spectra were normalized by the maximum transmission (usually around 800 nm) and the results are plotted in Figure 2b. An increase of the wavelength of the SRP with the GNP amount is noticed, from 522 to 540 nm as the GNP fraction is increased from 0.01 to 0.1 % w . This implies an increase in the average size of the nanoparticles with an increase of gold weight percent [START_REF] Zhang | Investigation of gold and silver nanoparticles on absorption heating and scattering imaging[END_REF]. Figure 2b shows that the absorption peak is more and more intense as the gold amount is increased. However the sample with 0.2 % w absorbs nearly as much light as the sample with 0.5 % w which is likely due to the fact, that above 0.2 % w , gold cannot coat more surface and thus form larger aggregates in the solution.

Iron impurity is common in window glass and is characterized by two absorption peaks at 380 and 450 nm due to Fe 3+ . Moreover Fe 2+ creates a large shoulder for wavelength above 450 nm and is responsible for the decrease of the transmittance in this large wavelength range in the quenched glass sample [START_REF] Meulebroeck | The potential of UV-VIS-NIR absorption spectroscopy in glass studies[END_REF]. This latter trend is not observed in the sintered sample as the transmittance continuously decreases with the decrease of the wavelength, likely because of carbon contamination from the SPS treatment as a 40 nm layer of carbon can impact the transmittance over the entire visible spectrum [START_REF] Baydoğan | Evaluation of optical properties of the amorphous carbon film on fused silica[END_REF]. The irradiation of the sample was done by progressively increasing the laser beam power step by step up to 100 W/cm² with a 3 min long plateau at each step (1.3, 2.5, 12.5, 25, 50, 75 and 100 W/cm²). The temperature at the end of each step was recorded (Figure 3a). The temperature of the quenched glass reached 110 °C at the end of the irradiation cycle, thanks to a partial absorption at 532 nm due to the presence of Fe 2+ . The pristine sintered glass showed a lower temperature increase compared to the quenched one due to the diffusion of light in the sample resulting from the residual porosity. Samples containing GNP were submitted to the same cycle but showed a larger temperature increase under irradiation due to the laser/GNP interaction, which is effective even for gold content as small as 0.01 % w , as evidenced by the temperature of 200 °C reached in the latter case at maximum illumination. The increase of the achieved temperature is nearly linear with the laser power (Figure 3a) and evolves logarithmically with the gold content (R² > 0.95, Figure 3b). The maximum temperature is 550 °C and is reached with the 0.5 % w gold content composite.

The glass transition and softening temperatures of the original glass are at 570 and 616 °C respectively, as determined by dilatometry. The gold content being smaller than 0.5 % w , the influence of gold on these characteristic temperatures is supposed to be negligible. As the maximum temperature reached by the composite with the largest gold content is close to the glass transition temperature, some healing due to viscous relaxation is expected. To investigate this possibility, a Vickers indent of 10 N was made on the surface of the sample containing 0.5 %w of GNP. The indented area was then submitted to laser irradiation for 10 min with a fluence set at 100 W/cm². A temperature of 550 °C was recorded at the indentation site during the whole experiment, which was found to induce an efficient healing of the indentation cracks (Figure 4). The indent cracks are found to shorten and to significantly open, with a concomitant blunting of the tip, after the healing treatment. Besides the indentation imprint is found to shrink as the glass temperature gets close to the transition temperature (570 °C as measured with a dilatometer at 10 °C/min).

Conclusions

We have presented a method to coat gold on window glass particles by reduction of ionic gold by sodium borohydride. This method is versatile and can be applied to different glass composition. The SPS sintering of the glass grains leads to dense translucent samples. The coating of gold nanoparticles was tailored by changing the reactant content in the solution in order to synthesized materials with up to 0.5 % w gold resulting in a progressive shift of the color from light pink to bright purple. The laser illumination of gold containing samples enabled the heating of the glass due to the SRP of the GNP. This was shown to result in a partial healing of the indentation crack and imprint of an indent after only 10 min of treatment.

These preliminary investigations are encouraging and open new perspectives for the curing of mechanical damage at the surface of glass and possibly in the volume as well. In the future, other noble metal and less refractory glasses (lower T g ) will be investigated with the aim to decrease the cost.

Figure 1 :

 1 Figure 1: a) Pristine quenched glass; b) Sintered glass; c) Sample containing 0.01 % w of gold; d) Sample containing 0.2 % w of gold; e) TEM image of a GNP coated on the glass.

Figure 2 :

 2 Figure 2: a) Transmittance in the visible range of the different composites and glass between 200 and 800 nm; b) Normalized transmittance in the visible range of the different composites and glass between 200 and 800 nm.

Figure 3 :

 3 Figure 3: a) Evolution of the temperature with the laser power for different gold content (in weight percent); b) Increase of the temperature as a function of the gold content for different laser powers.

Figure 4 :

 4 Figure 4: Indent before (left) and after 10 min at 550°C (right). The red and black dashed lines represent guides for the eyes. The white arrows represent the decrease in crack length and by consequence the healing.

Acknowledgments

The European Research Council is greatly acknowledged for the Advanced Grant 320506 (DAMREG) of the 7 th framework program "Ideas". We would like to thank J. Troles and L. Calvez (Institut National des Sciences Chimiques de Rennes) and E. Gallou (Institut de Physique de Rennes) for their technical support and providing us with IR Camera and SPS equipment. We further thanks V. Dorcet (THEMIS platform, Université de Rennes 1) for TEM observations.


Laurent Carette

Philippe Jacquet

Dominique Cotton

Vincent Vignal

Sébastien Faure

(TaC/Ta 2 C) bilayer formed on carburized and annealed tantalum; development of a numerical growth model

Keywords:

This paper describes a model developed to simulate the evolution of carbide layers (TaC/Ta 2 C) formed during carburizing and annealing treatments of tantalum. The model is based on a parabolic evolution of the layer thickness during the growth, and on the carbon conservation during the annealing. In order to validate this model a MATLAB® program was developed. It enables the prediction of the evolution of the tantalum carbide layers (nature and thickness) for a desired treatment defined by a temperature, a carburizing and an annealing duration.

Introduction

Carburizing is a thermochemical treatment which increases the carbon concentration of a metallic surface in order to improve its properties such as hardness. The process used here for carburizing is described in a recent patent [START_REF] Cotton | Process for treating a piece of tantalum or of a tantalum alloy[END_REF]. The carburizing gas is ethylene (C 2 H 4 ). It is introduced continuously inside the furnace with a controlled flow while a vacuum pump maintains a vacuum level of a few millibars. The carburizing temperature is set in a domain where the diffusion coefficients and the solubility of carbon in tantalum are known [START_REF] Brizes | Diffusion of carbon in the carbides of tantalum[END_REF].

The process contains two steps; firstly an enrichment with carbon and secondly a diffusion sequence. Enrichment steps have already been studied and simulated in the literature for similar systems (FeB/Fe 2 B, Fe 2-3 N/Fe 4 N) [START_REF] Keddam | A kinetic model for estimating the boron activation energies in the FeB and Fe2B layers during the gasboriding of Armco iron: Effect of boride incubation times[END_REF][START_REF] Keddam | Simulation of the growth kinetics of the (FeB/Fe2B) bilayer obtained on a borided stainless steel[END_REF][START_REF] Keddam | A diffusion model for simulation of bilayer growth (epsilon/gamma') of nitrided pure iron[END_REF][START_REF] Keddam | Computer simulation of nitrided layers growth for pure iron[END_REF] and for tantalum [START_REF] Krikorian | The formation of carbide surfaces on tantalum and tantalum-tungsten alloys[END_REF][START_REF] Brizes | Diffusion of carbon in the carbides of tantalum[END_REF][START_REF] Resnick | Determination of diffusivity of carbon in tantalum and columbium carbides by layer-growth measurements[END_REF]. However, the simulation of the annealing step for a bilayer system is not developed in these studies. The purpose of the model presented in this paper is to define the duration of each step in order to obtain the desired microstructure.

The model, developed with MATLAB® software, is composed of several steps. Carburizing consists of a carbon enrichment phase (Step 1) during which two carbide layers appear: TaC on the surface and Ta 2 C below. Then, annealing, which is divided into two carbon diffusion steps, leads firstly to the disappearance of the TaC layer and to the growth of the Ta 2 C one (Step 2). Secondly, annealing leads to the disappearance of the Ta 2 C layer (Step 3). After the treatment, the cooling to room temperature changes the carbon solubility in tantalum, leading to a Ta 2 C precipitate layer in carbon saturated tantalum.

Model assumptions

In this model, the following assumptions are made:

• The carbon flow on the surface is sufficient to saturate the surface and obtain TaC.

• The kinetics are driven by the growth coefficient of the layers and carbon diffusion coefficient.

• Layer evolutions are due to the carbon diffusion perpendicularly to the plane surface of the sample.

• Carbon diffusion coefficients are independent of the carbon con- centration; in each layer they follow an Arrhenius law.

• The temperature is supposed to be uniform and constant throughout the sample.

• Layer thicknesses are small compared to the sample: the carbon concentration of the heart of the sample is zero.

• When the carbon source of a layer disappears, the time needed to homogenize the carbon concentration in the layer is supposed to be zero.

• The volume variation is neglected.

• Porosity, roughness and material defects are not taken into account. For all following equations, the index i indicates to which zone the parameter refers: = i 1 for the TaC layer, = i 2 for the Ta 2 C layer and = i 3 for the carbon solid solution in tantalum matrix.

Carbon enrichment phase (Step 1)

Enrichment phases have already been studied in the literature for similar systems like FeB and FeB 2 for boriding [START_REF] Keddam | A kinetic model for estimating the boron activation energies in the FeB and Fe2B layers during the gasboriding of Armco iron: Effect of boride incubation times[END_REF][START_REF] Keddam | Simulation of the growth kinetics of the (FeB/Fe2B) bilayer obtained on a borided stainless steel[END_REF], Fe 2-3 N and Fe 4 N for nitriding [START_REF] Keddam | A diffusion model for simulation of bilayer growth (epsilon/gamma') of nitrided pure iron[END_REF][START_REF] Keddam | Computer simulation of nitrided layers growth for pure iron[END_REF] and also for carburizing TaC and Ta 2 C [START_REF] Krikorian | The formation of carbide surfaces on tantalum and tantalum-tungsten alloys[END_REF][START_REF] Brizes | Diffusion of carbon in the carbides of tantalum[END_REF][START_REF] Resnick | Determination of diffusivity of carbon in tantalum and columbium carbides by layer-growth measurements[END_REF]. During carburizing, the calculation of layer thicknesses depending on the layer growth coefficient and on the treatment duration is given by Eq. (1)

= w k t • i i c a r b (1)
With w i the layer thickness (m), t carb the carburizing duration (s) and k i the layer growth coefficient (m 2 •s -1 ). = i (1, 2). In each phase the carbon concentration is a function of the depth and responds to Eq. ( 2).

⎜ ⎟ = + ⎛ ⎝ ⎞ ⎠ c x t A B x D t ( , ) •erf 2 . i carb i i i i carb (2)
With c x t ( , ) i carb the carbon concentration (at.%) at the depth x i (m) for the carburizing duration t carb (s), D i the carbon diffusion coefficient (m 2 •s -1 ), and, A i and B i two constants (at.%). = i (1, 2, 3). If layer thicknesses and their limit concentrations are known (Table 1 according to Fig. 1), it is possible to calculate the carbon concentration as a function of depth. Integrating concentrations over the entire profile, the total amount of carbon diffused in the sample I tot is obtained. During the following diffusion phases (annealing), this global quantity of carbon will be conserved inside the sample.

Carbon diffusion phases (Steps 2 and 3)

Carbon profile in the tantalum matrix

During the annealing, as long as TaC and Ta 2 C layers are not consumed, the mobile interface Ta 2 C/Ta(C) can be considered as the carbon source which enriches the tantalum matrix Ta(C). The new carbon profile in the matrix responds to an expression similar to the Eq. ( 2), described in Eq. ( 3) with a treatment duration + t t carb an :

+ = + + c x t t A B x D t t ( , ) •erf( 2 .( ) )
carb an carb an

3 3 3 3 3 (3)
I' 3 is the integral of the concentration of carbon in solid solution in the tantalum matrix. Its calculation leads to two possible cases.

• If < I I ' tot 3
, at least one carbide layer remains and its thickness can be calculated with the model.

• If not, the two carbide layers are consumed.

Carbon profile in carbide layers if

< I I ' tot 3
The model assumes that as soon as enrichment stops, the carbon concentration in the TaC layer instantaneously homogenizes to its minimum value C 1/2 . The same is also assumed for the Ta 2 C layer: the carbon concentration takes the value C 2/3 after the disappearance of the upper layer. Carbon profiles in Fig. 2 step 2 and step 3 illustrate these hypotheses.

Cotton [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF] showed that as long as the TaC layer is not totally consumed, the depth + w' TaC Ta2C of the interface Ta 2 C/Ta(C) follows the same law as during carburizing. This leads to Eq. ( 4).

′ = ′ + ′ = + × + + w w w t t k k ( ) TaC Ta C TaC Ta C carb an TaC Ta C 2 2 2 (4)
For a given treatment duration, the program precisely determines the position of Ta 2 C/Ta(C) interface according to Fig. 2 step 2 and to the Eq. ( 4). The thickness of the TaC layer is initialized to that obtain after carburizing.

Then the program iteratively decreases the depth of TaC/Ta 2 C interface until the integral of carbon concentrations in the new profile is equal to the integral I tot after carburizing. At each iteration A 2 and B 2 constants of the Eq. ( 2) for = i 2 and = + t t t carb an are recalculated in order to adjust the carbon profile. If the integrals are not balanced before the TaC layer totally disappears, the profile changes to the configuration of Fig. 2 step 3. The program therefore gradually reduces the thickness of Ta 2 C in turn until balancing integrals.

Results and discussion

In the literature, the values of carbon diffusion coefficient in tantalum differ slightly from one to another [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF][START_REF] Son | Diffusion of carbon in tantalum[END_REF][START_REF] Ké | Internal friction in the interstitial solid solutions of C and O in tantalum[END_REF][START_REF] Wert | Measurements on the diffusion of interstitial atoms in BCC lattices[END_REF][START_REF] Powers | Diffusion of interstitial solutes in the Group V transition metals[END_REF]: this is probably due to the dependence of the coefficient with the material microstructure (grain size, hardening…) which is not described in these publications. The carbon diffusion coefficient in tantalum calculated by Cotton [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF] by inverse analysis of the depth of Ta 2 C precipitates after cooling; its value at 1600 °C is = -D 6.02•10 m /s c Ta 12 2 . However, at the same temperature, the coefficient calculated by Son et al. [START_REF] Son | Diffusion of carbon in tantalum[END_REF] is four times greater ( -23.37•10 m /s 12 2 ). For example, for an enrichment of 600 s, 228 min of annealing are necessary for the disappearance of TaC and Ta 2 C layers with Cotton's diffusion coefficient while only 80 min are necessary with Son's coefficient. This highlights the importance of choosing appropriate data. For the present work, the values of Cotton [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF] are chosen because they were calculated with the same carburizing process as that used for experiment. Moreover, the model uses a minimum carbon concentration threshold to calculate the profile of carbon in solid solution in tantalum. The value of this threshold is 0.0075 at.% because the sensitivity study shows that considering lower thresholds changes annealing time for layers disappearance by less than one percent.

In order to execute the model, the chosen values set for the Arrhenius laws leading to carbon diffusion and layer growth coefficients are referenced in Table 2. By integrating these values into the model and by setting a carburizing time, the program can return the evolution of the layer thicknesses and the surface phase during an

Table 1

Limit concentrations of carbon in each phase at 1600 °C according to references [START_REF] Lesser | Karbidphasen des Tantals[END_REF][START_REF] Gusev | Atomic and vacancy ordering in carbide ZETA-Ta4C3-x (0.28x0.40) and phase equilibria in the Ta-C system[END_REF][START_REF] Hörz | High-temperature solid solubility limit of carbon in niobium and tantalum[END_REF].

C S 1/ C 1/2 C 2/1 C 2/3 C 3/2
Values (at.%) 50 42.5 33.3 29.0 0.40 annealing. For example, for 10 min carburizing duration Fig. 3 shows the microstructure as a function of the depth and of the annealing time. This makes it possible to choose the time of treatment in order to have a desired surface microstructure. Fig. 3 shows that after 10 min of enrichment at 1600 °C a bilayer is obtained: 2 µm of TaC on the surface and 0.5 µm of Ta 2 C below. After 6 min of annealing, the surface TaC layer completely disappears in Fig. 2. Schematization of the model algorithm.

Table 2

Carbon diffusion coefficient and layer growth coefficients for each phase from 1400 °C to 2000 °C. However, in Fig. 3 during the diffusion step, at the moment of the total disappearance of the TaC layer, a discontinuity is observed for the thickness of Ta 2 C layer. This discontinuity is due to the hypothesis which states that the homogenization of the carbon concentration is instantaneous in the Ta 2 C layer when the upper layer totally disappears.

In Fig. 4 and Fig. 5, the model is compared to experimental results obtained by Cotton [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF] after 1 h of carburizing (Fig. 4a), and 2 h of carburizing (Fig. 5 and Fig. 4b). The total thickness of layers predicted with the model is in good agreement with the tests of Cotton [START_REF] Cotton | Etude de formation d'hémicarbure de tantale (Ta2C) par l'intermédiaire d'un procédé de cémentation sous pression réduite[END_REF]. However, for the experiment, the presence of undesired decarburization on the surface was observed during annealing. This can be correlated with the layers disappearance faster than according to the model. In Fig. 6, the depth of the Ta 2 C/Ta(C) interface is plot versus the treatment duration for samples which contain a TaC layer (only carburizing or carburizing and annealing). Despite the surface decarburization, Fig. 6 shows that the TaC permits the enrichment of the underlying layer, and the position of the Ta 2 C/Ta(C) interface follow the law described with Eq. ( 4). Finally, in spite of the decarburization, there is a good correlation between the theoretical model and the experience.

Conclusion

• The evolution of TaC and Ta 2 C layers during carburizing and an- nealing is simulated with a model which firstly considers the layers growth coefficients and then the constant quantity of absorbed carbon in tantalum during the annealing.

• This model permits to adjust treatment duration in order to obtain a specific microstructure on the surface.

• The model gives thicknesses of carbide layers in good agreements with experimental results.

• The carbon diffusion coefficient in tantalum (depending on grain size, microstructure…) is a key parameter of the model. This parameter significantly differs in the literature. For this reason, the carbon diffusion coefficient has to be estimated prior carburizing and annealing treatment to take into account the state of the raw material.

• This model can be used for similar diffusion models with bilayers like steel boriding or nitriding.

T 3 .

 3 Layer evolution modelIf the carburizing cycle of tantalum is well set, a microstructure with three different layers as presented in Fig.1is obtained.

Fig. 1 .

 1 Fig. 1. Carbon profile after a carburizing process: Step 1 corresponding to carbon enrichment.

  •10 6 [7] E (J•mol -1 ) 197.5•10 3 [12] 298.1•10 3 [7] 324.9•10 3 [7] Layer growth coefficients k o (µm 2 •h -1 ) - 63.25•10 3 [12] 201.6•10 3 [12] E (J•mol -1 ) -168.1•10 3 [12] 141.2•10 3 [12]

Fig. 3 .

 3 Fig. 3. Evolution of layer thicknesses during a carbon enrichment of 10 min followed by annealing at 1600 °C.

Fig. 4 .

 4 Fig. 4. Confrontation of the model (M) with the experiment (E): for carburizing of 1 h (a) and 2 h (b).

Fig. 5 .

 5 Fig. 5. Tantalum carburized 2 h at 1600 °C, SEM image (left); model prediction (right).

Fig. 6 .

 6 Fig. 6. Depth of Ta 2 C/Ta(C) interface for carburized and annealed samples containing a TaC layer.

Acknowledgments

The authors wish to thank CEA/DAM, ECAM Lyon and CNRS for the financial support of this research.

Appendix A. Supplementary material

Supplementary data to this article can be found online at https:// doi.org/10.1016/j.apsusc.2018.10.142.