Thierry Alonso

Denis Favier

Grégory Chagnon

Characterizing transformation phenomena and elastic moduli of austenite and oriented martensite of superelastic thin NiTi wire through isothermal Dynamic Mechanical Analysis

In this paper, superelastic behavior of Nickel Titanium thin wires is characterized using the method of dynamic mechanical analysis. Nominal dynamic storage modulus 𝐸′ is measured as function of nominal strain and stress during isothermal superelastic tensile tests at three testing temperatures above the reverse martensitic transformation finish temperature. The method brings new information on deformation mechanisms compared to the consideration of only tensile stress-strain curves. It is shown that determination of the elastic moduli E, especially at high strain, requires to calculate the true storage modulus 𝐸 𝑡 ′ . Using 𝐸 𝑡 ′ and not 𝐸′, elastic modulus of oriented martensite 𝐸 𝑀 is determined equal to 73 GPa of the same order than the elastic modulus of austenite 𝐸 𝐴 equal to 70 GPa. Two models are then proposed to simulate experimental storage moduli evolution during the tests. A first model explains the 𝐸′ evolution during stress plateau by the localization phenomenon ; it leads to express 𝐸 ′ as function of the nominal strain. A second model describes the evolution of 𝐸 𝑡 ′ after the stress plateau as function of true stress and test temperature. This model permits to determine the Clausius-Clapeyron coefficient of the forward transformation.

Introduction

Superelastic behavior of near equiatomic Nickel-Titanium based shape memory alloys (NiTi SMA) is due to the complex interaction between several deformation mechanisms including lattice elasticity, phase transformations between austenite (A), martensite (M) and R-phase (R), reorientations of R and/or M variants, twinning and detwinning phenomena and usual plasticity due to dislocations movements [START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF]. These phenomena are difficult to identify by considering only stress-strain curves of superelastic tensile tests. Using Dynamical Mechanical Analysis (DMA) to study shape memory alloys is a classical method to investigate phase transformation temperatures [START_REF] Besseghini | DMA characterization of a Ni50.5at%Ti shape memory alloys[END_REF][START_REF] Fan | Effects of frequency, composition, hydrogen and twin boundary density on the internal friction of Ti50Ni50-xCux shape memory alloys[END_REF][START_REF] Inamura | Stress Amplitude Dependence of Internal Friction in TiNbAl Shape Memory Alloy[END_REF][START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF][START_REF] Artiaga | DMTA study of a nickel-titanium wire[END_REF][START_REF] Suresh | Microstructure dependent elastic modulus variation in NiTi shape memory alloy[END_REF], damping properties [START_REF] Van Humbeeck | Damping capacity of thermoelastic martensite in shape memory alloys[END_REF][START_REF] Liu | On the damping behaviour of NiTi shape memory alloy[END_REF][START_REF] Chang | Internal friction of R-phase and B19' martensite in equiatomic TiNi shape memory alloy under isothermal conditions[END_REF][START_REF] Yoshida | Damping characteristics of Ti50Ni47Fe3 alloy[END_REF][START_REF] Chen | Damping capacity of TiNi-based shape memory alloys[END_REF] or effect of fatigue [START_REF] Roy | Mechanical characterization of NiTi SMA wires using a dynamic mechanical analyzer[END_REF]. Classically DMA tests are performed during a temperature sweep under a constant low stress; storage modulus and loss factor are measured as function of temperature. Consequently, phase transformation temperatures and damping properties are determined at low stress, what prevents from studying stress influence whereas this is of major importance.

In this paper, superelastic behavior of NiTi thin wires was investigated by using a nonconventional DMA which allows to measure the nominal storage modulus as function of nominal strain and stress during isothermal tensile tests at high level of stress. DMA tests were performed during cyclic superelastic tensile tests up to 10% of strain and monotonic tests up to failure at three temperatures above Af the reverse martensitic transformation finish temperature. The goal is to identify and study elasticity, phase transformations, localization and plasticity, by analyzing storage modulus evolution during isothermal superelastic tensile tests. A detailed analysis of the storage modulus evolution allows to estimate the elastic moduli of austenite and oriented martensite. From experimental results two models are proposed to simulate storage modulus evolution in cases where deformation is localized (during the stress plateau) or uniform (after the stress plateau), respectively. The last model and the results of the three isothermal tests are used to determine the Clausius-Clapeyron coefficient for the forward transformation.

Materials and experimental set-up

Materials

A commercial NiTi (Ti-50.9 at.% Ni) wire provided by Fort Wayne Metals (reference : FWM#1) with a diameter d=0.1mm was used. This is a straight annealed NiTi wire for medical applications. Fig. 1a shows the nominal stress-strain curve of a tensile test performed at room temperature. Fig. 1b shows the calorimetric curve obtained by Differential Scanning Calorimetry (DSC) with heating and cooling rates of 10°C min -1 . A-R transformation was detected during cooling with a peak at 9.2°C. During heating, reverse finish transformation temperature Af is determined equal to 26°C.

Experimental set-up and performed tests

All tests were performed with a commercial DMA Netzsch-Gabo Eplexor 500N device. Unlike a classical DMA device, this non-conventional tensile test combines a standard tensile test with a DMA device. Fig. 2a describes schematically the device which is composed of:

1. a static unit on the upper side 2. a dynamic unit on the lower side 3. a thermal chamber A static load is applied by means of the static unit. The static load can be driven in displacement, nominal strain, force, or nominal stress. A dynamic load is applied by the dynamic unit which is a shaker. The dynamic load can be driven in displacement, nominal strain, force, nominal stress. The thermal chamber permits to perform tests with a controlled temperature.

DMA and Static Sweep (DMA-SS) test -Nominal storage modulus E'

Fig. 2b describes test named DMA and static sweep (DMA-SS). It consists in applying a sinusoidal displacement ∆𝐿 𝑑𝑦𝑛 (𝑡) of amplitude ∆𝐿 𝑑𝑦𝑛 with a frequency f superimposed on a static length 𝐿 𝑠𝑡𝑎𝑡 . The length 𝐿 of the sample at any moment t is given by: 𝐿 = 𝐿 𝑠𝑡𝑎𝑡 + ∆𝐿 𝑑𝑦𝑛 (𝑡) with ∆𝐿 𝑑𝑦𝑛 (𝑡) = ∆𝐿 𝑑𝑦𝑛 sin(2𝜋𝑓𝑡) (Eq. 1)

During this test, force 𝐹 is measured; this force is composed by a static part 𝐹 𝑠𝑡𝑎𝑡 and a sinusoidal dynamic part 𝐹 𝑑𝑦𝑛 (𝑡) of amplitude 𝐹 𝑑𝑦𝑛 and is given by: 𝐹 = 𝐹 𝑠𝑡𝑎𝑡 + 𝐹 𝑑𝑦𝑛 (𝑡) with 𝐹 𝑑𝑦𝑛 (𝑡) = 𝐹 𝑑𝑦𝑛 𝑠𝑖𝑛(2𝜋𝑓(𝑡 -𝑡 𝛿 )) (Eq. 2)

𝑡 𝛿 being the phase difference between ∆𝐿 𝑑𝑦𝑛 (𝑡) and 𝐹 𝑑𝑦𝑛 (𝑡).

The nominal (or engineering) stress 𝜎 and nominal dynamic stress amplitude 𝜎 𝑑𝑦𝑛 are determined by: 𝜎 = 𝐹 𝑠𝑡𝑎𝑡 𝑆 0

and

𝜎 𝑑𝑦𝑛 = 𝐹 𝑑𝑦𝑛 𝑆 0 (Eq. 3)
with S0 being the initial cross sectional area of the sample

The nominal (or engineering) strain 𝜀 and nominal dynamic strain amplitude 𝜀 𝑑𝑦𝑛 are determined by:

𝜀 = 𝐿 𝑠𝑡𝑎𝑡 -𝐿 0 𝐿 0 and 𝜀 𝑑𝑦𝑛 = ∆𝐿 𝑑𝑦𝑛 𝐿 0 (Eq. 4)
with 𝐿 0 the initial length of the sample.

For each dynamic cycle, the nominal storage modulus 𝐸′ is given by the quotient between 𝜎 𝑑𝑦𝑛 the amplitude of the nominal dynamic stress and 𝜀 𝑑𝑦𝑛 the amplitude of the nominal dynamic strain :

𝐸 ′ = 𝜎 𝑑𝑦𝑛 𝜀 𝑑𝑦𝑛 (Eq. 5)

Performed tests

A first reference DMA-SS test was performed at T=60°C which is 34°C higher than Af . Two other tests were performed at two other temperatures above Af , i.e. at T=80°C and T =40°C respectively. The test at 60°C was used as the reference test because the test at 40°C can be influenced by the presence of R-phase (as shown by the DSC in Fig. 1.b). The test at 80°C implies higher stress and is more likely to be influenced by plasticity.

All samples were loaded up to 10% of nominal strain, unloaded then reloaded up to break. DMA-SS cycles were performed with 𝜀 𝑑𝑦𝑛 =0.1% and f =100Hz. These parameters have been chosen to reduce the noise during the measure of E' and to allow measurements of E' at low static strain value of the order of 0.2% (the static strain value has to be greater than the dynamic strain amplitude).

Definition of true storage modulus 𝑬 𝒕

′

The nominal storage modulus 𝐸 ′ is, as shown in Eq. 5, calculated from the nominal stress and strain defined in Eq. 3 and Eq. 4 respectively. The true storage modulus 𝐸 𝑡 ′ is calculated from the true strain 𝜀 𝑡 = ln ( 𝐿 𝐿 0

) and from the true stress 𝜎 𝑡 = 𝐹 𝑆 with S the current cross section area.

The true strain is given by:

𝜀 𝑡 = ln ( 𝐿 𝐿 0 ) = ln (1 + 𝜀) (Eq. 6)
By noting  the Poisson's ratio, the current cross section area is :

𝑆 = 𝑆 0 exp(-2 𝜀 𝑡 )= (1 + 𝜀) -2 (Eq. 7)
The true stress 𝜎 𝑡 is deduced from the nominal stress 𝜎 and strain 𝜀 by:

𝜎 𝑡 = 𝐹 𝑆 = 𝜎 exp(2 𝜀 𝑡 )= 𝜎 (1 + 𝜀) 2 (Eq. 8)
The true storage modulus 𝐸 𝑡 ′ is then given by:

𝐸 𝑡 ′ = 𝑑𝜎 𝑡 𝑑𝜀 𝑡 = 𝐸 ′ (1 + 𝜀) (1+2) + 2 𝜎 (1 + 𝜀) 2 (Eq. 9)
Perfect plastic or transformation deformation mechanisms occur without volume change. By assuming isotropy and incompressibility, the Poisson's ratio  is equal to 0.5. Then the true stress and true storage modulus are given by:

𝜎 𝑡 = 𝜎(1 + 𝜀) and 𝐸 𝑡 ′ = 𝐸 ′ (1 + 𝜀) 2 + 𝜎(1 + 𝜀) (Eq. 10)
In the following, when not specified, the true stress and true storage modulus are calculated Eq. 10, assuming no volume change.

In order to illustrate the differences between nominal and true values, Figure 3 shows the stress-strain curves (Fig. 3a) and the storage modulus-strain curves (Fig. 3b) during a DMA-SS test performed on a soft annealed Cu wire of diameter 0.5mm. The DMA-SS parameters (𝜀 𝑑𝑦𝑛 = 0.1% and f = 100Hz) were identical to those used for the studied NiTi wire. The nominal values are plotted in red, the true values with  = 0.5 using Eq.10 are plotted in black. The difference between nominal and true stresses (Fig. 3a) is well-known, especially in plasticity. The difference between nominal and true storage moduli (Fig. 3b) is less known. In the majority of publications this difference is not clarified because conventional DMA are performed during a temperature sweep under small stress and strain [START_REF] Besseghini | DMA characterization of a Ni50.5at%Ti shape memory alloys[END_REF][START_REF] Fan | Effects of frequency, composition, hydrogen and twin boundary density on the internal friction of Ti50Ni50-xCux shape memory alloys[END_REF][START_REF] Inamura | Stress Amplitude Dependence of Internal Friction in TiNbAl Shape Memory Alloy[END_REF][START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF][START_REF] Artiaga | DMTA study of a nickel-titanium wire[END_REF][START_REF] Suresh | Microstructure dependent elastic modulus variation in NiTi shape memory alloy[END_REF]. The results shown in Fig. 3b illustrate that for a nominal strain greater than 2%, the difference between nominal and true storage moduli is significant. For a strain equal to 10%, the true storage modulus 𝐸 𝑡 ′ is 25% higher than 𝐸 ′ the nominal storage modulus. The true value of the storage modulus using Eq. 9 and  = 0.33 is plotted in blue; the true modulus is still more than 20% higher than 𝐸 ′ the nominal storage modulus.

A second conclusion of the DMA-SS test performed on the Cu wire concerns the determination of the elastic modulus for an elastoplastic material. A first method consists in measuring the slope of the stress-strain curve in the elastic zone, i.e. for low values of strain and stress; in that case, it is not necessary to distinguish nominal and true values of strain and stress because the differences are very weak. The DMA-SS test performed in our study provides a second method. In fact, this method consists in performing small elastic unloadings during the tensile loading in the plastic zone ; the storage moduli are the slopes of the stress-strain unloading curves. As shown in Fig. 3b, the nominal storage modulus 𝐸 ′ decreases with increasing strain. On the contrary, the true storage modulus 𝐸 𝑡 ′ is almost independent of the plastic strain. Thus the estimation of the elastic modulus from the nominal storage modulus during a DMA-SS test would imply that the elastic modulus would depend on the plastic strain, which is not relevant. On the contrary, the estimation of the elastic modulus from the true storage modulus during the DMA-SS test on a elastoplastic material is not dependent on the plastic strain. This result is consistent with the definition of elastic modulus used in theoretical elastoplasticity [START_REF] Bruhns | 60 Years of Research in Plasticity The Contributions of Th. Lehmann and his Group[END_REF].

Thus, during a DMA-SS test, the elastic modulus has to be evaluated from the determination of the true storage modulus variation 𝐸 𝑡 ′ and not from that of the nominal storage modulus 𝐸 ′ ; this conclusion is important for large static strain, typically higher than 2%.

Experimental Results

Fig. 4, Fig. 5 and Fig. 6 show DMA-SS results at 60°C, 80°C and 40°C, successively. The measure of loss factor is too noisy and will not be used. Upper and lower curves of subfigures a) show nominal stress 𝜎-nominal strain 𝜀 and nominal storage modulus 𝐸 ′ -nominal strain 𝜀 curves, respectively for the specimen loaded up to 10% of strain and then unloaded. Subfigures b) show storage modulus 𝐸 ′ as function of nominal stress 𝜎 during this cycle. Singular points have been numbered, (0) to (4) during loading and (5) to (8) during unloading.

Similar curves are plotted using solid lines for the reloading up to break in the subfigures c) and d). Singular points (5') and (6') are associated to this test up to break.

Test at 60 °C

Cyclic stress 𝜎-strain 𝜀 curve (Fig. 4a) exhibits a classical superelastic behaviour. Loading and unloading plateaus occur for nominal stresses of 770 MPa and 530 MPa, respectively. During loading, 𝐸′ decreases monotonically between (0) and ( 1) with an initial value of 70 GPa. At point (1), matching with the start of the loading stress plateau, 𝐸′ reaches 52 GPa. Plateau nominal stress is globally constant, after a little drop at point [START_REF] Besseghini | DMA characterization of a Ni50.5at%Ti shape memory alloys[END_REF]. Between ( 1) and ( 3), 𝐸′ continues to decrease almost linearly to reach 35 GPa in [START_REF] Fan | Effects of frequency, composition, hydrogen and twin boundary density on the internal friction of Ti50Ni50-xCux shape memory alloys[END_REF]. At the end of the plateau in (3), 𝐸′ starts to increase monotonically to reach 47 GPa in (4).

During unloading, the reverse evolution is observed. 𝐸′ decreases between (4) and ( 5). The unloading plateau starts in [START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF] with a stress jump between ( 5) and ( 6), associated with a step increase of 𝐸′. During the stress plateau between ( 6) and ( 7), 𝐸′ increases almost linearly up to 56 GPa at point [START_REF] Suresh | Microstructure dependent elastic modulus variation in NiTi shape memory alloy[END_REF] that corresponds to the end of the plateau. During the last stage of the unloading between ( 7) and ( 8), 𝐸′ continues to increase to reach 70 GPa at the end of unloading in [START_REF] Van Humbeeck | Damping capacity of thermoelastic martensite in shape memory alloys[END_REF].

Figure 4b shows the evolution of 𝐸′ as function of stress during the previous cyclic test. During loading, 𝐸′ decreases monotonically before the stress plateau until point [START_REF] Otsuka | Physical metallurgy of Ti-Ni-based shape memory alloys[END_REF]. 𝐸′ is almost constant during the stress drop between ( 1) and ( 2). Then 𝐸′ decreases sharply during the stress plateau between ( 2) and (3). After the stress plateau, 𝐸′ increases between points (3) and (4).Reverse evolution is observed during unloading. Initial and final values of 𝐸′ at zero stress in (0) and ( 8) respectively, are identical and equal to 70 GPa as shown in Fig. 4b.

The test up to break (Fig. 4c and Fig. 4d), shows the same evolution than during the loading of the cyclic test. After the stress plateau, 𝐸′ increases ; the increase is slowing down with strain increasing. 𝐸′ reaches 53 GPa at point (6').

Test at 80°C

Figure 5 shows results for the test at T=80°C. Stress-strain curve (Fig. 5a) exhibits a classical superelastic behavior like at T=60°C. Loading and unloading stress plateaus occur for stresses of 900 MPa and 670 MPa respectively.

Storage modulus-strain curve follows similar evolution than at T=60°C. Initial and final values of 𝐸′ at zero stress are identical (Fig. 5b) and equal to 70 GPa which is identical to the value measured at 60°C. The storage modulus variations on one hand between (0) and ( 1) and on the other hand between ( 7) and ( 8) are both smaller than the same variations measured at 60°C.

The test up to break (Fig. 5c and Fig. 5d), shows the same evolution as at 60°C, and 𝐸′ increases up to point (6') to reach 50 GPa.

Test at 40°C

Figure 6 shows results for T=40°C. Similarly to the tests performed at 60°C and 80°C, the stress-strain curve (Fig. 6a) exhibits a classical superelastic behavior. Loading stress plateau occurs for a stress of 636 MPa and unloading plateau for a stress of 376 MPa.

Storage modulus-strain curve follows same evolution as T=60°C and T =80°C. Initial and final values of 𝐸′ at zero stress are identical (Fig. 6b), equal to 66 GPa, which is smaller than the value measured at 60°C and 80°C. The storage modulus variations on one hand between (0) and ( 1) and on the other hand between ( 7) and ( 8) are both larger than the same variations measured at 60°C and 80°C.

The test up to break (Fig. 5c and Fig. 5d), shows the same evolution at 60°C and 80°C, and 𝐸′ increases up to point (6') to reach 55 GPa.

Analysis of DMA results

Analysis of 𝑬′ evolution during loading

The stage before the stress plateau (i.e. between (0) and ( 1)) is usually considered to be associated with the elastic deformation austenite [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. Several authors [START_REF] Schlosser | Experimental Characterization of NiTi SMAs Thermomechanical Behaviour Using Temperature and Strain Full-Field Measurements[END_REF][START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF] have shown that strain is uniform along the wire during this stage. As shown in Figures 4,5,6, the storage modulus 𝐸 ′ decreases monotonically between (0) and ( 1). The monotonic decrease of 𝐸 ′ between (0) and (1) during a DMA and Static Sweep (DMA-SS) test on a NiTi wire proves that deformation mechanisms other than elasticity and plasticity are involved during this stage. It can be assumed that these additional deformation mechanisms are due to stress induced phase transformations, either A-R and/or A-M transformations [START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline NiTi thin tube under tension. Investigation via temperature and strain fields measurements[END_REF], even at low stress. At point (0) the wire is mainly in austenite with some amount of martensite and Rphase. The storage modulus is estimated by 𝐸 ′ = 𝜎 𝑑𝑦𝑛 𝜀 𝑑𝑦𝑛 . In this stage, the total dynamic strain 𝜀 𝑑𝑦𝑛 can be expressed as the sum of a lattice elastic dynamic strain (𝜀 𝑑𝑦𝑛 ) ). The monotonic decrease of the storage modulus 𝐸′ with increasing static strain (or static stress) during stage (0) and ( 1) is thus only due to the increase of transformation mechanisms.

During the stress plateau, between (2) and ( 3), the nominal stress is almost constant and the nominal storage modulus 𝐸′ decreases almost linearly with increasing strain. It was shown that deformation is localized along the wire in this stage [START_REF] Schlosser | Experimental Characterization of NiTi SMAs Thermomechanical Behaviour Using Temperature and Strain Full-Field Measurements[END_REF][START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF]. The number of regions and the number of fronts are increasing with increasing strain rate, as shown by Ng and Sun [START_REF] Ng | Stress-induced phase transformation and detwinning in NiTi polycrystalline shape memory alloy tubes[END_REF]. For low strain rate, the wire is divided into two regions (a) and (b), separated by a localization front as illustrated in Fig. 8. The fraction of martensite and the strain in region (a) are very high whereas they are very low in region (b). The front of localization moves from left to right with increasing wire strain. At point (3) localization is completed, the sample is fully in state (a) and uniform deformation replaces localized deformation. The quasi-linear evolution of the storage modulus with strain during the stress plateau can be explained by this deformation localization phenomenon and by a storage modulus difference between regions (a) and (b). A model will be proposed in subsection 6.1.

After the stress plateau, the deformation is usually associated with the elasticity of oriented martensite [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. The wire deformation is uniform, as shown in [START_REF] Schlosser | Experimental Characterization of NiTi SMAs Thermomechanical Behaviour Using Temperature and Strain Full-Field Measurements[END_REF][START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF]. Similarly to the stage between (0) and ( 1), the continuous increase of 𝐸′ during the stage between (3) and ( 4) shows that deformation is not only elastic and that phase transformations and/or variants reorientation occur.

The evolution up to rupture between (4) and (5') is presented in figures 4c and 4d at 60°C. 𝐸′ tends to be constant with increasing strain. This is explained by the decrease of the transformation and reorientation deformation mechanisms with increasing strain, being replaced by plasticity mechanisms between (5') and (6').

Analysis of 𝑬′ evolution during unloading

The stage between (4) and ( 5) is usually associated with the elastic deformation of oriented martensite [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. It was shown that the wire deformation is uniform during this stage [START_REF] Schlosser | Experimental Characterization of NiTi SMAs Thermomechanical Behaviour Using Temperature and Strain Full-Field Measurements[END_REF][START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF]. 𝐸′ continuous decrease shows that this stage is not only elastic and that reverse transformation occurs.

The stage during the lower stress plateau, between ( 6) and ( 7), is similar to the stage during the upper stress plateau, between (2) and ( 3), during loading. Nominal stress is nearly constant and 𝐸′ increases almost linearly with decreasing strain. Similarly than during the upper plateau, the 𝐸′ evolution between ( 6) and ( 7) is due to the localization phenomenon and to the existence of regions with different states along the wire.

The stage between [START_REF] Suresh | Microstructure dependent elastic modulus variation in NiTi shape memory alloy[END_REF] and [START_REF] Van Humbeeck | Damping capacity of thermoelastic martensite in shape memory alloys[END_REF] corresponds to the stage between (0) and (1) during the loading and a similar analysis can be done. The increase of 𝐸′ with decreasing strain shows that this stage is not purely elastic (Fig. 8), and that a reverse transformation occurs during this stage. Values at point [START_REF] Van Humbeeck | Damping capacity of thermoelastic martensite in shape memory alloys[END_REF] are the same as for the initial stage, i.e 70 GPa for 60°C and 80°C, and 65 GPa for 40°C (close to Af ).

Identification of deformation phenomena present in NiTi wire from stress and storage modulus 𝑬′ evolutions

In conclusion, analyzing both stress 𝜎(𝜀) and 𝐸′(𝜀) curves during a tensile test allows identifying the majority of the phenomena. 𝐸′ evolution measurement brings new information and clarifies the ambiguities we may have by analyzing only stress as function of strain. Table 1 shows the evolution of 𝜎(𝜀) and 𝐸′(𝜀) to identify elasticity, transformation and or reorientation, localization and plasticity during loading. To simplify interpretation, arrows are used to code evolution. →, ,  mean that the value is constant, increases and decreases with increasing strain successively.

For the NiTi wire studied, there is no stage with constant value of the storage modulus; it means that there is no deformation stage for which the deformation is purely elastic or purely plastic, even for tests performed in the temperature range of superelastic behavior. Phase transformations and/or variant reorientations occur at the beginning and the ending of the loading, separated by a deformation stage characterized by a deformation localization phenomenon.

How to determine elastic moduli of austenite and oriented martensite from DMA tests ?

Measurement methods

For a usual elastoplastic alloy such as the soft annealed Cu wire shown in Figure 3, a first method to determine the elastic modulus is based on the measurement of the slope of a tensile stress-strain curve. For an elastoplastic alloy, the elastic modulus is the initial slope. This first method is called "the slope method". A second method is to deduce this elastic modulus from the measurement of the storage modulus evolution during a DMA-SS test. It was shown in Fig. 3 that the elastic modulus for an elastoplastic alloy is equal to the true storage modulus independently of the strain/stress . This second method is called "the storage modulus method".

The behavior of a NiTi SMA is much more complicated than a usual elastoplastic alloy; it is due to complex interaction between several deformation mechanisms, including lattice elasticity, phases transformations and plasticity. In order to determine elastic moduli of austenite and oriented martensite, table 2 shows the values deduced from the slopes of the stress-strain curves during the tensile tests at 40°C, 60°C and 80°C. Table 3 presents measurements deduced from the storage moduli evolution during these same tests.

Table 2 presents values of slopes of the tensile stress-strain curves at different stages of the cyclic stress-strain curves plotted in Fig. 4.a, 5.a and 6.a. All the slopes are calculated with a stress range ∆𝜎 = 200 MPa. They are calculated from the nominal stress-strain curves (∆𝜎/∆𝜀) or from the true stress-strain curves (∆𝜎 𝑡 /∆𝜀 𝑡 ). The second column shows the slopes calculated in point (0). ∆𝜎/∆𝜀 and ∆𝜎 𝑡 /∆𝜀 𝑡 are identical in this deformation range. The third and fourth columns are the slopes calculated during loading after the stress plateau in (3). The fifth and sixth columns are the slopes calculated at the start of unloading in (4).

Elastic modulus of austenite 𝐸 𝐴

The first method to determine elastic modulus of austenite 𝐸 𝐴 consists in using the "slope method", which is the most popular method [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF]. From the second column of table 2, 𝐸 𝐴 is determined equal to 69 GPa at 60°C and 80°C and a little smaller 63 GPa at 40 °C. The DMA-SS tests performed in this paper allow using the "storage modulus method". By using the second column of table 3, 𝐸 𝐴 it is determined equal to 66 GPa at 40 °C and 70 GPa at higher temperatures. The values determined using the two methods are very close for the three test temperatures. The lower values at 40°C can be easily explained by the residual Rphase transformation which occurs between (0) and (1).

In conclusion, the elastic modulus of austenite 𝐸 𝐴 is estimated equal to 70 GPa using both "slope method" and "storage modulus method".

Elastic modulus of oriented martensite 𝐸 𝑀

Macroscopic elastic modulus of oriented martensite 𝐸 𝑀 is usually estimated from the slope of the nominal stress-strain curve after the stress plateau [START_REF] Liu | Apparent modulus of elasticity of near-equiatomic NiTi[END_REF][START_REF] Young | Phase volume fractions and strain measurements in an Ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF]. The slopes are taken either during loading after the stress plateau, which are the values in the column (3) of the table 2, or at the start of the unloading after the stress plateau which are the values in the column [START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF]. The values of the slopes during loading in column (3) are very low, of the order of 17 GPa, and almost independent of the temperature. Using column [START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF] gives higher values ranging from 38GPa at 40°C to 30GPa at 80°C. All the values in columns (3) and ( 5) are low compared to the elastic modulus of austenite. The low values of columns (3) are explained by the fact that the transformation is not completed after stress plateau as explained in paragraph 4.3. The low values of columns [START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF] mean that reverse transformation mechanisms occur upon unloading ; the decrease of the slope with increasing temperature is likely due to the increase of reverse transformation and/or reverse detwinning due to higher back stress, prior to the unloading stress plateau.

The slopes can be estimated from the true stress-strain curves rather than from the nominal stress-strain curves. The values in column (4) are increased compared to those of the column (3). Similarly, the values in column ( 6) are also increased compared to the column (5), ranging from 41GPa at 40°C to 34GPa at 80°C. The difference between values obtained from nominal or true stress-strain curves is easily explained by the fact that the reduction of current specimen section and the increase of current specimen length is taken into account when using true stress-strain curves. This precaution is scarcely taken.

Using either nominal or true stress-strain curves, the highest value of the estimation of martensite modulus is 41 GPa. As pointed out in [START_REF] Bucsek | Myths and Truths of Nitinol Mechanics: Elasticity and Tension-Compression Asymmetry[END_REF], the best measurement of martensite modulus which can be reached from stress-strain curve is obtained during unloading of stressinduced martensite. This value is of the same order that the elastic modulus proposed by the provider for wire in martensitic state [28]. However, it is still low compared to the elastic modulus of austenite.

The "storage modulus method" is an alternative way to estimate elastic modulus of oriented martensite. As shown in column (3) of table 3, at point (4) the nominal storage modulus decreases from 50 GPa at 40°C to 42 GPa at 80°C. These values are higher than the values determined by the slope method at the same point (38 GPa at 40°C to 30 GPa at 80°C) , as summarized in columns (5) of table 2. Indeed, the "storage modulus method" is based on the measurement of 𝐸 ′ = 𝜎 𝑑𝑦𝑛 𝜀 𝑑𝑦𝑛 as shown in eq [START_REF] Nespoli | Phase transition and mechanical damping properties: A DMTA study of NiTiCu shape memory alloys[END_REF]. 𝐸 ′ is measured with a dynamic strain of 0.1% which corresponds approximately to a dynamic stress 𝜎 𝑑𝑦𝑛 of the order of 50 GPa. The slope method uses a stress range ∆𝜎 = 200 MPa. As 𝜎 𝑑𝑦𝑛 is lower than ∆𝜎 , the moduli estimated at point (4) by the "storage modulus method" are higher than by the "slope method" due to activation of less inelastic deformation mechanisms by the first method. An other way to decrease the inelastic deformation mechanisms is to increase the strain till the rupture of the specimen. At the rupture, the nominal storage modulus is equal to 55 GPa at 40°C and of 50 GPa at 80°C. These values are higher than the values at point (4) to the activation of less inelastic deformation mechanisms.

The values deduced from the determination of the true storage modulus using Eq. 10 are given in columns ( 4) and ( 6) of table 3 for the point ( 4) and (6') of fig. 4,5 and 6 respectively. Comparison of columns ( 3) and ( 4) for the point (4) and columns ( 5) and ( 6) for the point (6') show that the differences between the nominal and the true storage moduli is important at this strain level. This effect was not taken into account in [START_REF] Sittner | Young's Modulus of Austenite and Martensite Phases in Superelastic NiTi Wires[END_REF] and leads to an underestimated value for the macroscopic elastic modulus of oriented martensite 𝐸 𝑀 .

In figure 9, the true storage modulus using Eq. 10 is plotted in full lines for stress higher than the plateau stresses as function of the true stress for the three tested temperatures, for the tests at rupture shown in Fig. 4.c, 5.c and 6.c. The best estimation for the macroscopic modulus of martensite is the asymptotic value of these three curves, which is equal to 73 GPa. Using Eq. 9 and a Poisson's ratio  = 0.33, the asymptotic value is equal to 70 GPa.

In conclusion, the elastic modulus of martensite can not be deduced from nominal or true stress-strain curves. It can be estimated using "storage modulus method" by considering the true storage modulus and not the nominal storage modulus. With this method, the macroscopic elastic modulus 𝐸 𝑀 of oriented martensite of the studied wire is estimated equal to 73 GPa.

Comparison with values of literature

Many articles proposed experimental values of elastic moduli of austenite and martensite [START_REF] Young | Phase volume fractions and strain measurements in an Ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF][START_REF] Rajagopalan | Elastic modulus of shapememory NiTi from in situ neutron diffraction during macroscopic loading, instrumented indentation, and extensometry[END_REF][START_REF] Qiu | On elastic moduli and elastic anisotropy in polycrystalline martensitic NiTi[END_REF][START_REF] Stebner | Young's modulus evolution and texture-based elastic-inelastic strain partitioning during large uniaxial deformations of monoclinic nickeltitanium[END_REF][START_REF] Sittner | Young's Modulus of Austenite and Martensite Phases in Superelastic NiTi Wires[END_REF]. In this paper, the elastic moduli have been determined at room temperature for a thin Ti-50.9 at.% Ni wire in austenitic state under stress-free condition. The elastic moduli of austenite and of stress-induced martensite were measured during a tensile test, at low and high stress respectively. The stress-strain deformation response include contributions from elastic, transformation, twinning and plastic deformation ; this lead to a strong coupling between activity of deformation mechanisms such as stress induced B2=> B19'=>B2T martensitic transformation into twinned austenite coupled with dislocation slip [25bis]. The main difficulty of the measure is to separate the elastic deformation mechanisms from all the other deformation mechanisms occurring during the deformation of the wire. In our experiments, the Dynamic Mechanical Analysis has been used .

The process of fabrication induces different textures, and elastic moduli are different between a rod [START_REF] Rajagopalan | Elastic modulus of shapememory NiTi from in situ neutron diffraction during macroscopic loading, instrumented indentation, and extensometry[END_REF][START_REF] Young | Phase volume fractions and strain measurements in an Ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF][START_REF] Qiu | On elastic moduli and elastic anisotropy in polycrystalline martensitic NiTi[END_REF][START_REF] Stebner | Young's modulus evolution and texture-based elastic-inelastic strain partitioning during large uniaxial deformations of monoclinic nickeltitanium[END_REF] and a thin wire [START_REF] Young | Phase volume fractions and strain measurements in an Ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF][START_REF] Sittner | Young's Modulus of Austenite and Martensite Phases in Superelastic NiTi Wires[END_REF]. Moreover, elastic moduli, especially for stress induced martensite, can be strongly anisotropic [START_REF] Sittner | Young's Modulus of Austenite and Martensite Phases in Superelastic NiTi Wires[END_REF][START_REF] Thomasova | Evolution of macroscopic elastic moduli of martensitic polycrystalline NiTi and NiTiCu shape memory alloys with pseudoplastic straining[END_REF]. As pointed out by Bucsek et al [START_REF] Bucsek | Myths and Truths of Nitinol Mechanics: Elasticity and Tension-Compression Asymmetry[END_REF], "the Young's modulus of polycrystalline NiTi alloys is not a fixed number and changes with both processing and in operando deformations".

The obtained values in the present study are compared to the values obtained by Young et al. [START_REF] Young | Phase volume fractions and strain measurements in an Ultrafine-grained NiTi shape-memory alloy during tensile loading[END_REF] which determined the elastic moduli of austenite and stress induced martensite during a tensile test of a similar wire. These last authors measured microscopic elastic moduli with in situ X-ray diffraction For the austenite B2, Young et al determine Young's moduli for (100),( 110),( 200) and (211) direction ranging between 51 GPa and 72 GPa, with an average value of 57 GPa. For the martensite B19' (001), they obtained a value of 69 GPa. The results obtained in the present work using Dynamic Mechanical Analysis are within range of values reported by Young et al. and confirm that for the studied wire the elastic modulus of oriented martensite is of the same order than of austenite and even a little higher.

From the modelling of DMA results to determination of Clausius-Clapeyron coefficients

Experimental results concerning the storage modulus measurement during superelastic tensile tests have been presented in section 4. In section 5, the true storage moduli evolutions 𝐸′ 𝑡 with strain have been used to determine the austenite elastic modulus 𝐸 𝐴 as the initial value of 𝐸′ 𝑡 at low strain-stress and the elastic modulus of oriented martensite, 𝐸 𝑀 , as the asymptotic value of 𝐸′ 𝑡 during the test at rupture for high values of strain/stress. This section is devoted to further analyze the evolution of storage modulus during all the tensile tests by proposing a first model to simulate the evolution of the nominal storage modulus during the stress plateau and a second model to simulate the evolution of the true storage modulus after the plateau. This second model permits to determine the Clausius-Clapeyron coefficient of the forward transformation.

During loading, three zones are identified in Fig. 4c and4d. Zones (I) and (III) correspond to uniform wire deformation stages, associated to the start and end of A/R-M uniform transformation, respectively. Zone (II) corresponds to the stress plateau during which the deformation is localized. In the following subsections, models are proposed to simulate the evolution of storage moduli for zones (II) and (III). The model proposed in the subsection 6.1 takes into account the deformation localization during the stress plateau. This model is based on mixture law ; and simulates the evolution of the nominal storage modulus 𝐸′ as function of the nominal strain during a tensile test.

The model proposed in the subsections 6.2 and 6.3 takes into account that the deformation is uniform during the stage after the stress plateau, and that the true nominal storage modulus 𝐸 𝑡 ′ is function of temperature and true stress. In subsection 6.2, the dependence of 𝐸 𝑡 ′ on true stress is modeled at 60°C. In subsection 6.3, it is shown that the dependence of 𝐸 𝑡 ′ on true stress at 40°C and 80°C can be deduced from the model established at 60°C by a Clausius-Clapeyron like relation; this allows to estimate the Clausius-Clapeyron coefficient of the forward transformation without using the plateau stress, which is now well-known not to be the onset of the transformation but that of the localization phenomenon [START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline NiTi thin tube under tension. Investigation via temperature and strain fields measurements[END_REF].

Modelling of nominal storage modulus 𝑬′ evolution during the stress plateau

Figure 8 shows a scheme of the deformation of the tested wire in the zone II. At point (0), the initial sample length is 𝐿 0 . The deformation is uniform between point (0) and (2).

During the stress plateau between ( 2) and ( 3), for a given static nominal strain 𝜀, the specimen length is 𝐿 0 (1 + 𝜀). The specimen is constituted of two regions (a) and (b) separated by a localization front. The lengths of regions are 𝐿 𝑎 (𝜀) and 𝐿 𝑏 (𝜀)respectively.

The nominal strains are constant and uniform in the regions (a) and (b), as shown in [START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF][START_REF] Favier | Homogeneous and heterogeneous deformation mechanisms in an austenitic polycrystalline NiTi thin tube under tension. Investigation via temperature and strain fields measurements[END_REF][START_REF] Jiang | Modeling of propagation of phase transformation fronts in NiTi under uniaxial[END_REF] ; they are noted 𝜀 𝑎 for the higher deformed region (a) and 𝜀 𝑏 for the region (b) which is mostly in austenitic phase. The nominal nominal stress 𝜎 𝑎 in region (a) is equal to 𝜎 𝑏 the nominal stress in region (b) and equal to the nominal stress 𝜎 applied to the wire. Delobelle et al. [START_REF] Delobelle | Study of electropulse heat treatment of cold worked NiTi wire: from uniform to localised tensile behaviour[END_REF] observed that for a small unloading that occurs when the deformation is localized, the front of localization does not move. This observation leads to assume that during a storage modulus measurement the localization front does not move, since the DMA consists of small cycles of loading/unloading. Thus during DMA tests around a static strain 𝜀, 𝐿 𝑎 (𝜀) and 𝐿 𝑏 (𝜀) are constant. During a DMA measurement, all regions are under the same dynamic stress 𝜎 𝑑𝑦𝑛 and the length variation ∆𝐿 𝑑𝑦𝑛 is the sum of ∆𝐿 𝑑𝑦𝑛(𝑎) and ∆𝐿 𝑑𝑦𝑛(𝑏) of regions (a) and (b). Thus : The initial and current lengths of the wire are respectively : 𝐿 0 = 𝐿 𝑎0 + 𝐿 𝑏0 and 𝐿 0 (1 + 𝜀) = 𝐿 𝑎 (𝜀) + 𝐿 𝑏 (𝜀) (Eq. 14)

∆𝐿 𝑑𝑦𝑛 = 𝜎
The combination of equations 12, 13 and 14 gives :

𝐸′(𝜀) = 𝐸 𝑎 ′ 𝐸 𝑏 ′ (𝜀 𝑎 -𝜀 𝑏 )
𝐸 𝑏 ′ (𝜀-𝜀 𝑏 )+𝐸 𝑎 ′ (𝜀 𝑎 -𝜀) (Eq. 15)

which permits to express the evolution of 𝐸 ′ as function of the given nominal strain 𝜀 of the nominal storage moduli 𝐸 𝑎 ′ and 𝐸 𝑏 ′ and of the nominal strains 𝜀 𝑎 and 𝜀 𝑏 of regions (a) and (b).

For a test performed at low strain rate, nominal plateau stress is constant, 𝜎 2 = 𝜎 3 so that :

𝐸 𝑎 ′ = 𝐸 3 ′ ; 𝐸 𝑏 ′ = 𝐸 2 ′
; 𝜀 𝑎 = 𝜀 3 ; 𝜀 𝑏 = 𝜀 2 and :

𝐸′(𝜀) = 𝐸 3 ′ 𝐸 2 ′ (𝜀 3 -𝜀 2 )
𝐸 2 ′ (𝜀-𝜀 2 )+𝐸 3 ′ (𝜀 3 -𝜀) (Eq. 16)

These 𝐸′ evolutions for zone (II) as predicted by Eq. 16 are plotted in dashed lines in Fig. 4c, Fig. 5c, Fig. 6c and fit well with the experimental curves.

Modelling of 𝐸 𝑡 ′ evolution during uniform deformation stage (III) at T=60°C

The experimental evolution of the true storage modulus 𝐸 𝑡 ′ during uniform deformation stage (III) at T=60°C is plotted in full lines in Fig. 9. During this stage, deformation mechanisms include elasticity, plasticity and martensitic transformation. When the deformation of an alloy is only due to the two first mechanisms, 𝐸 𝑡 ′ is constant as shown in Fig. 3 for a soft annealed Cu wire. Fig. 9 shows that for the studied NiTi wire, 𝐸 𝑡 ′ increases after the stress plateau to reach asymptotically the elastic modulus 𝐸 𝑀 of the oriented martensite. This increase is due to the decreasing transformation mechanism with increasing strain and stress. At the point (3), the beginning of zone (III), the true stress and true storage modulus at 60°C are 𝜎 𝑡3 and 𝐸 𝑡3 ′ respectively. It is proposed to fit the evolution of 𝐸 𝑡 ′ with the true stress 𝜎 𝑡 in the zone (III) at 60°C by a hyperbolic tangent function

𝐸 𝑡 ′ (𝜎 𝑡 , 𝑇 0 = 60°𝐶) = 𝐸 𝑡3 ′ + (𝐸 𝑀 -𝐸 𝑡3 ′ )𝑡𝑎𝑛ℎ [𝐾 3 𝜎 𝑡 -𝜎 𝑡3 𝐸 𝑀 -𝐸 𝑡3 ′ ]
(Eq. 17

)
𝐾 3 is a fitting parameter and represents the slope of the curve 𝐸 𝑡 ′ -true stress at point (3) in Fig. 9.

The equation 17 fitted with 𝐾 3 = 41 is plotted in dashed lines in Fig. 9 and fits very well with experimental true storage modulus evolution at 60°C.

Determination of Clausius Clapeyron coefficient for A-M transformation

Martensitic transformations are first order transformations which can be stress induced.

During a test performed at a reference temperature T0 , the transformation stress is noted 𝜎 𝑆𝐼𝑀 (𝑇 0 ). The transformation stress 𝜎 𝑆𝐼𝑀 (𝑇) at any other test temperature T above Af is deduced from 𝜎 𝑆𝐼𝑀 (𝑇 0 ) by using the Clausius-Clapeyron relation [START_REF] Ortin | Thermodynamics of thermoelastic martensitic transformations[END_REF] which writes : 𝜎 𝑆𝐼𝑀 (𝑇) = 𝜎 𝑆𝐼𝑀 (𝑇 0 ) + 𝐶 𝐴𝑀 (𝑇 -𝑇 0 ) (Eq. 18)

The Clausius-Clapeyron relation expresses that there is a linear relation between the transformation stress and the test temperature, 𝐶 𝐴𝑀 being the Clausius-Clapeyron coefficient for the A-M transformation.

For tensile tests of NiTi shape memory alloys, 𝜎 𝑆𝐼𝑀 is estimated with the value of the plateau stress in stress-strain curves, assuming that transformation starts only for stress equal to the plateau stress. In our experiments, values of forward plateau stresses were calculated for a strain of 4%. A linear fitting allowed to determine 𝐶 𝐴𝑀 = 6.6 ± 0.5 MPa K -1 . However, it is now well known that the stress plateaus are due to the localization phenomena and not only to the transformation [START_REF] Jiang | Modeling of propagation of phase transformation fronts in NiTi under uniaxial[END_REF]. Thus estimating the transformation stress as the plateau stress can lead to inaccurate measures of 𝐶 𝐴𝑀 .

Another method to evaluate 𝐶 𝐴𝑀 is proposed in this paper. It is based on the dependence on temperature T of the evolution of 𝐸 𝑡 ′ with the true stress 𝜎 𝑡 . Eq. ( 17) proposes an equation This relation expresses that the intensity of the transformation at the temperature T and for the true stress 𝜎 𝑡 is identical to that at the temperature T0 and for the true stress 𝜎 𝑡 -𝐶 𝐴𝑀 (𝑇 -𝑇 0 ).

The 𝐶 𝐴𝑀 Clausius Clapeyron coefficient in Eq. 19 was chosen to minimize the gap between model and experimental curves both at 40 °C and 80°C. Fig. 9 shows the results of the model (dashed line) with 𝐶 𝐴𝑀 = 7.9 ± 0.5 MPaK -1 . 𝐶 𝐴𝑀 determined by this method is close to 𝐶 𝐴𝑀 = 6.6 ± 0.5 MPaK -1 determined by using stress plateau.

The dependence on the temperature of the true storage evolution with true stress allows thus to determine Clausius-Clapeyron coefficient ; this method can be used in the absence of stress plateau, when the superelastic tensile tests are uniform.

Conclusion

The deformation mechanisms involved in the superelastic behavior of NiTi thin wires are still often explained by considering the tensile nominal stress-strain curve. In the usual presentation, four successive stages are distinguished during loading up to rupture. During the first stage, stress is considered as increasing linearly with strain, which is attributed to elastic deformation of austenite. The second stage is characterized by a stress plateau justified by the transformation from austenite to martensite. During the third stage, the stress is considered as increasing again linearly with strain which is explained by the elastic deformation of the oriented martensite. During the fourth stage, the slowdown of the stress increase is ascribed to martensite plasticity. The plateau stress during the second stage is considered as the transformation stress ; the Clausius Clapeyron coefficient is determined from the linear increase of this plateau stress with temperature.

In this paper, Dynamic Mechanical Analysis was used to measure storage moduli during superelastic tensile tests of NiTi thin wires as function of strain and stress at three temperatures above Af. The main conclusions are the following :

-during the first stage, nominal storage modulus 𝐸′ is neither constant. Its continuous decrease with increasing strain is due to the occurrence of non elastic deformation mechanisms due to phase transformations A-R, R-M, and A-M -during the third and fourth stages, continuous increase of the nominal storage modulus is due to deformation mechanisms other than elastic and plastic mechanisms, including phase transformation and twinning-detwinning phenomena, -the evolution of the nominal storage modulus 𝐸′ during the stress plateau is only due to the localization phenomenon as shown by a simple model proposed in this paper.

In this paper, it was shown that DMA results allows :

-to estimate the elastic modulus of austenite as the initial value at low stress of the nominal storage modulus during the first stage. Estimation is better for highest test temperature to reduce the effect of the R phase. With this method, elastic modulus of austenite EA is estimated equal to 70 GPa, independently of the test temperature. -to estimate the elastic modulus of oriented martensite from the evolution of the storage modulus during the fourth stage, just before the rupture. In this high stress and strain ranges, it is mandatory to calculate the true storage modulus 𝐸 𝑡 ′ from the true strain and stress. The elastic modulus of oriented martensite 𝐸 𝑀 is estimated as the asymptotic value of the true storage modulus 𝐸 𝑡 ′ during the last stage preceding the rupture. With this method, elastic modulus of oriented martensite EM is estimated equal to 73 GPa, slightly higher than the elastic modulus of the austenite.

-to propose a method to determine the Clausius Clapeyron coefficient 𝐶 𝐴𝑀 of the forward martensitic transformation without using the stress plateau stage characterized by localized deformation. This method is based on a model 𝐸 𝑡 ′ (𝜎 𝑡 , 𝑇 0 ) of the evolution of the true storage modulus 𝐸 𝑡 ′ as function of true stress 𝜎 𝑡 at a reference temperature T0. 𝐶 𝐴𝑀 is the coefficient which allows to extent this model to other temperature T under the form 𝐸 𝑡 ′ (𝜎 𝑡 , 𝑇) = 𝐸 𝑡 ′ (𝜎 𝑡 -𝐶 𝐴𝑀 (𝑇 -𝑇 0 ), 𝑇 0 ).

Acknowledgement

The authors wish to acknowledge the financial support of the ANR research program Guidage d'une Aiguille Médicale Instrumentée -Déformable (ANR12-TECS-0019).

  𝑒𝑙 and of a phase transformation dynamic strain (𝜀 𝑑𝑦𝑛 ) 𝑡𝑟𝑎𝑛𝑠𝑓 . Introducing the elastic modulus 𝐸 = 𝜎 𝑑𝑦𝑛 (𝜀 𝑑𝑦𝑛 ) 𝑒𝑙 , the storage modulus is expressed as 𝐸 ′ = 𝐸 (𝜀 𝑑𝑦𝑛 ) 𝑒𝑙 𝜀 𝑑𝑦𝑛 = 𝐸 (1 -(𝜀 𝑑𝑦𝑛 ) 𝑡𝑟𝑎𝑛𝑠𝑓 𝜀 𝑑𝑦𝑛

Table 3

 3 presents values deduced from the DMA-SS tests. The second column shows the values of the initial nominal storage modulus 𝐸′ at point (0); 𝐸′ is identical to the true storage modulus 𝐸′ in this deformation range. The third and fourth columns show the values of 𝐸′ and 𝐸′ 𝑡 in (4), respectively; as shown in Fig.4.b, 5.b, and 6.b, the values of the storage moduli in (4) at the end of the loadings are equal to those at the start of the unloadings. The fifth and sixth columns show the values of 𝐸′ and 𝐸′ 𝑡 at the rupture point in (6') of Fig. 4.d, 5.d and 6.d.

  𝜎 𝑑𝑦𝑛 𝐸 ′ 𝐿 0 = ∆𝐿 𝑑𝑦𝑛(𝑎) + ∆𝐿 𝑑𝑦𝑛(𝑏) 𝐿 𝑏0 , 𝐿 𝑎0 and 𝐿 𝑏0 being the undeformed lengths of regions (a) and (b) given by : nominal storage modulus 𝐸 𝑎 ′ and 𝐸 𝑏 ′ are uniform in region (a) and (b), the equation (11) leads to the following equation:

Table 1 :

 1 Identification of elasticity, plasticity, uniform and localized deformation phenomena from nominal stress-strain and storage modulus-strain evolutions during loading

	𝜎(𝜀)	𝐸′(𝜀)	Phenomena
		→	Elasticity
	→	→	Plasticity
			Starting transformation
			Ending transformation
	→	almost linear evolution	Localization

Table 2 :

 2 Values of the slopes determined from the nominal stress-strain and true stressstrain curves, respectively. The points (0), (3) and (4) are defined in figures 6, 4 and 5 for T= 40, 60°C and 80°C successively. The slopes are calculated with a stress range ∆𝜎 =200MPa.

				∆𝜎 ∆𝜀	and	∆𝜎 𝑡 ∆𝜀 𝑡
		Point (0)		Loading in (3)	Unloading in (4)
	Temperature	∆𝜎 ∆𝜀	=	∆𝜎 𝑡 ∆𝜀 𝑡		∆𝜎 ∆𝜀	∆𝜎 𝑡 ∆𝜀 𝑡	∆𝜎 ∆𝜀	∆𝜎 𝑡 ∆𝜀 𝑡
	40°C	63 GPa		17 GPa	19 GPa	38 GPa	41 GPa
	60°C	69 GPa		16 GPa	18 GPa	35 GPa	39 GPa
	80°C	69 GPa		18 GPa	21 GPa	30 GPa	34 GPa

Table 3 :

 3 Values of the nominal 𝐸 ′ and true 𝐸 t ′ storage moduli. The points (0), (4) and (6') are defined in figures 6, 4 and 5 for T= 40, 60°C and 80°C successively

	Temperature	Point (0) 𝐸 ′ = 𝐸 t ′	Loading and unloading in (4) 𝐸 ′ 𝐸 t ′	Loading in (6') 𝐸 ′ 𝐸 t ′
	40°C	66 GPa	50 GPa	62 GPa	55 GPa	73 GPa
	60°C	70 GPa	47 GPa	58 GPa	53 GPa	70 GPa
	80°C	71 GPa	42 GPa	52 GPa	50 GPa	70 GPa


Dr Hadjer Ouaddah

Dr Isabelle Périchaud

Prof Damien Olivier Barakel

Olivier Palais

Marisa Di Sabatino

Guillaume Reinhart

Gabrielle Regula

Nathalie Mangelinck-Noël
email: nathalie.mangelinck@im2np.fr

Dr Guillaume Palais

Dr Gabrielle Reinhart

Dr Nathalie Regula

Mangelinck-Noël

Prof Di Marisa

Sabatino

Role of impurities in silicon solidification and electrical properties studied by complementary in situ and ex situ methods

Keywords: Silicon, Photovoltaic, Growth defects, Impurities, Lifetime

All silicon (Si) ingot fabrication processes share challenges to control grain structure, defect and impurity contamination during the solidification step to improve the material properties. The final grain structure and inherent structural defects issued from the solidification step are responsible for the photovoltaic (PV) properties for a large part, all the more as they are often associated with impurity distribution. Impurities play a major role as they not only can modify the development of the grain structure formation but interact as well with structural defects creating regions of deleterious minority carrier lifetime recombination. Samples containing different levels of impurities and solidified with different processes were selected and analyzed as-grown or observed by X-ray imaging during re-solidification from as-grown seeds. The growth features and relative crystallographic orientation of neighbor grains were characterized. Moreover, minority carrier lifetime measurements were performed and correlated to the growth features. The complementarity of the different techniques allows improving the understanding of phenomena at stake during the formation of grains and twins, the effect of impurities and their impact on photovoltaic properties.

The results show the significant influence of light and metallic impurities such as copper on the grain structure and on the electrical properties.

Introduction

The development of solar photovoltaic (PV) electricity requires significant advances in research and development to answer to the constantly renewed societal demand for higher conversion efficiency and lower costs. Silicon technology has clear advantages over other technologies: advantages such as the good availability of basic material (silica), the relatively high energy conversion efficiency and the good stability of the solar cells resulting in an expected lifetime of about 30 years. Moreover, silicon is a well-established technology still representing about 90% of the world photovoltaic market. [START_REF] Sabatino | Technology Roadmap: Solar Photovoltaic Energy[END_REF] Existing fabrication processes produce either mono-crystalline or multi-crystalline ingots.

Recently, alternative methods have been proposed to improve the quality of the Si ingots. [START_REF] Lan | [END_REF]3] However, for all Si ingot fabrication processes, the grain structure, defect and impurity contamination control during the solidification step remains a major challenge to improve the material properties.

The final grain structure and associated defects (grain boundaries and dislocations) issued from the solidification step are responsible for PV properties for a large part. [4] In this context, impurities play a major role as they not only modify grain nucleation and competition during solidification [5,6] but interact as well with structural defects. [7] The nucleation of new grains during growth can happen either on the crucible walls or on precipitates or impurities in the molten silicon. Moreover, in a material contaminated with impurities, segregation along grain boundaries and extended defects (dislocations [7][8][9][10] , twins) can be detrimental for the cell conversion efficiency. On the one hand, grain boundaries can have a positive effect by purifying the grain core from impurities by segregation and trapping. On the other hand, the impurities accumulated at the grain boundaries may be responsible for the increase of the defect recombination strength. [11] Within this context, the grain crystallographic orientation deriving directly from nucleation and grain competition during growth [12,13] is of particular importance. Indeed, grain boundaries can have different propensity to segregation, mechanical, and electrical characteristics according to their crystalline quality and to the presence of dangling bonds. [14,[START_REF] Ervik | Proceedings 26th European Photovoltaic Solar Energy Conference and Exhibition[END_REF] Impurities in general, and light impurities such as carbon (C), oxygen (O), and nitrogen (N), are inherent to the feedstock and to the directional solidification process used to fabricate silicon for PV applications. The furnace environment (heaters, graphite plate…) and crucible are often responsible for most of the light impurity contamination [START_REF] Kvande | [END_REF]17] and in some cases for metallic impurity contamination. [18][19][20][21] In addition, impurities can be found in the starting feedstock used for PV applications.

The control of the final crystalline grain structure, the decrease of the structural defect density, and the control of the impurity concentration and segregation during crystal growth are objectives that can only be attained with a thorough understanding of the crystallisation mechanisms [22][23][24] and of the complex interaction between growth defects and impurities. Within this context, the objective of the present work is to correlate the growth mechanisms and defects to the electrical properties in presence of different impurity concentrations.

Experimental

Sample origin

The samples selected in the present work were obtained by different solidification processes. The selection of these samples aimed at encompassing representative solidification features and different levels of contamination.

Three samples (F1-F3) were cut from a zone melting ingot. [25] This process although costly allows producing mono-crystalline samples with a very low level of contamination (Sample type F in Table 1). Sample Cz was cut from a Czockralski ingot. This process allows producing mono-crystalline samples with a low level of contamination except for oxygen coming from the used quartz crucible (Sample Cz in Table 1). Another sample (Sample P in Table 1) was obtained from an ingot processed with the conventional multi-crystalline Si method [26] and corresponds to a conventional solar grade mc-Si quality. Samples C1, C2, V and J in Table 1 were processed using the cold crucible continuous casting solidification technique. This technique allows the growth of multi-crystalline silicon ingots without contact with the crucible walls thus reducing stresses and contamination by the crucible.

Copper inductors are crossed by an alternating current and produce a magnetic field [27,28] , which prevents the silicon to be in contact with the walls. A continuous flow of water cools the copper inductors. This process was previously used for instance by the French company EMIX. The main additional advantage of this technique, apart from the low level of contamination due to the absence of crucible, is that it provides the possibility to carry out continuous casting. In summary, this technique allows to increase the productivity and to decrease the impurity concentration in the ingots mainly in carbon and oxygen. [29] However, the use of the Cu inductors is at the origin of contamination by Cu as can be seen in Table 1.

The concentrations of the main elements measured in the different samples used for the experiments are gathered in Table 1. It refers to the as grown samples following the processes described in the above paragraph. It is worth noticing that only the detectable impurities are provided in the following table implying that the other metallic impurity concentrations are below the detection limit. As a consequence, Cu contamination is the sole significant metallic impurity contamination for samples V, C1 and C2.

Table 1. Concentrations of the detectable impurities in the as grown samples. DL is the detection limit: 1 ppbw and 1 ppm for GDMS (Glow Discharge Mass Spectroscopy) [30] and IGA (Instrumental Gas Analysis), respectively.

In situ X-ray radiography during the solidification

Sample type F (F1 to F3) Cz [31] V J P C1 C2

Oxygen concentration (at.cm -3 ) < 10

GDMS

Some of the samples were characterised directly after the above described processes. Additionally, some parts of the ingots were used as seeds for solidification studies in the IM2NP device named GaTSBI (Growth at high Temperature observed by Synchrotron Beam Imaging), which allows in situ X-ray imaging during solidification. The samples were directionally solidified in the furnace that consists in two independent heating elements, constituted by two graphite heating resistors that can reach 1800°C under a vacuum of 10 -6 mbar. The temperature is adjusted above and below the melting point in the top and bottom heating resistors, respectively. This creates a temperature gradient and allows the control of the position of the solid-liquid interface. In the present work, solidification was carried out by cooling down the heating element. The solid seed-liquid interface was first positioned in the field of view for X-ray imaging in between the two heating elements and a cooling rate was applied on both heating zones to trigger the solidification. The sample was fixed and the solid-liquid interface moved in the laboratory reference frame.

The melting and solidification is characterized in situ inside the GaTSBI device with synchrotron radiation X-ray radiography imaging technique at the ESRF (European Synchrotron Radiation Facility) in Grenoble, France. During solidification, the sample is illuminated by a polychromatic Xray beam. Then, the transmitted polychromatic beam is monochromatized by a post-monochromator and recorded with a specific X-ray camera. In this mode, the image contrast is due to absorption differences. However, it is worth noticing that in the case of a pure material such as silicon, the density difference between the solid (2.33 g.𝑐𝑐𝑐𝑐 -3 ) and the liquid (2.56 g. 𝑐𝑐𝑐𝑐 -3 ) creates a small X-ray absorption difference. For this reason, image processing by image division is performed to reveal the solidification front. Radiography X-ray imaging brings information on morphology and growth kinetics of the solid-liquid interface. The experimental set-up, the method and the image processing details are described in more details elsewhere. [13,32] During the experiments presented here, the images were recorded with an exposure time of 1 s with the ESRF FReLoN CCD camera equipped with an optics providing a resolution of about 12 µm.

Electron BackScatter Diffraction (EBSD) electron microscopy characterization

To obtain quantitative and precise information about the crystallography of the grain structure, electron backscatter diffraction (EBSD) measurements are performed using a FEG-SEM JEOL JSM 7001F equipped with a HKL Nordlys camera using either a 15 µm or 7 µm lateral step size. At the end of the analysis, post-processing software is used to obtain several maps related to the crystallographic orientations. In the inverse pole maps, the color of each pixel corresponds to the crystallographic orientation of the {hkl} plane which is perpendicular to one of the axis of the reference frame. (colors red, blue, yellow and green in the figures), respectively. The character (symmetric or asymmetric) of the grain boundary and the orientation of the crystallographic planes of the adjacent grains are determined by the analysis of the stereographic projections of the appropriate crystallographic planes as done for example in our previous work. [23] Micro Wave Phase Shift (µw-PS) minority carrier lifetime measurements Minority carrier lifetime scan maps localize extended crystallographic defects that are electrically active. They can give an idea of the intensity of their recombination strength and allow comparing the relative recombination activity of different defects. In this work, minority carrier lifetime was determined by the microwave phase shift (μw-PS) technique. [33] In this method, microwaves are driven towards the bottom surface of the sample by means of a microwave guide. Carrier generation is provided by a fiber-coupled laser diode with a wavelength of 942 nm, which is sine modulated in the range 0.5 -70 kHz. The sinusoidal variation of the carrier density induces a sinusoidal variation of the reflectivity coefficient of the microwaves. The intensity of reflected microwaves are accurately measured by a lock-in amplifier. The phase shift between the modulated exciting light and the microwave can be mapped with a spatial resolution better than 50 μm, which leads to lifetime mapping with the same spatial resolution. This technique can be sensitive up to a contamination level as low as 10 9 at.cm -3 . [34] In order to determine the minority carrier lifetime from the measured phase shift, the sample thickness, the optical absorption coefficient at 942 nm, the minority carrier diffusion coefficient, the modulation frequency and the front and bottom surface recombination velocities have to be computed.

In this study, the optical absorption coefficient was taken equal to 535 cm -1 and a value of 30 cm 2 . s -1 was taken for the minority carrier diffusion coefficient. The modulation frequency was 6 kHz. A value of 6000 cm.s -1 was applied for the front and bottom surface recombination velocities.

This value corresponds to a cleaned raw silicon surface without passivation.

Results

Segregation and solid-liquid interface morphology

Figure 1 shows the radiography images obtained during the solidification of three samples: Cz, C1 and C2. The liquid and solid phases are in the upper and lower parts, respectively (see indications in Generally, when growing solar grade (e.g. sample P in Table 1) or higher quality samples (e.g.

samples F and Cz in Table 1) in the GaTSBI furnace, no destabilization of the solid-liquid interface is observed by X-ray radiography during solidification as can be seen in Figure 1.a. The interface is smooth apart from grain boundary grooves and {111} facets. [35] The interface is also globally smooth in the case of samples containing light impurities (mainly C and O) as for example in samples J and P. Inversely, when studying the growth dynamics with X-ray radiography of C1 and C2 samples (Figure 1.b and 1.c) containing copper (Table 1), it appears that dendritic growth is observed under the same solidification conditions, even for a Cu concentration as low as 3.9 10 15 at.cm -3 in sample C1.

Grain nucleation and final grain structure

Figure 2 shows the final grain structure of samples F1, P, V and C2 after their growth inside the GaTSBI furnace.

Despite of the different growth conditions and level of impurities, all solidified samples exhibit the same kind of final grain structure, namely centimetric grain size and straight twin grain boundaries as can be seen in Figure 2. The level of Cu contamination is increasing from Oppositely, there is a clear impact of C concentration on the grain structure. When C concentration increases, a larger amount of grains is observed at the end of growth. This effect is not only seen in the directional solidification experiments performed inside the GaTSBI furnace but as well in the industrial as-grown ingots with the industrial cold crucible continuous casting process.

Figure 3 shows the inverse pole figure in samples V (Figure 3.a) and J (Figure 3.b) both cut in asgrown ingots processed using the conventional industrial process conditions (Figure 3). It allows us characterizing the grain density as well as the different crystallographic orientations (grain orientation legend in Figure 3.c). The first qualitative visual assessment of Figure 3 indicates clearly that there is no preferential growth crystallographic orientation in both samples. Moreover, about twice more grains are present inside sample J compared to sample V for the same surface although the same growth process was used. The grains of sample V can extend up to several millimeters while more grains, having a smaller size of some hundreds of microns, are observed in sample J. A quantitative assessment of the different grain boundary types was carried out from the EBSD measurements, the corresponding percentages of Σ3, Σ9, Σ27a, Σ27b twin boundaries and of random angle grain boundaries (RAGB) are gathered in Table 2. Regarding the samples type V and J, two kinds of samples were characterized by EBSD: as-grown samples characterized directly after the industrial process and others obtained after solidification in the GaTSBI furnace with an applied temperature gradient of 30 K/cm and a cooling rate of -2 K/min. Additionally, three solidification experiments were performed in the GaTSBI furnace with an applied temperature gradient of 30 K/cm and a cooling rate of -1 K/min (F1 to F3 in Table 2) using zone melting seeds (Type F in Table 1).

Type F sample analysis gives consistent results. Σ3 twin boundaries are the most abundant in a large amount (92.53% in average) in all F-type samples. It is worth noting that this observation in pure seed samples was confirmed in several other in situ experiments. [13,23] The amount of Σ3 twin boundaries is decreasing for samples containing impurities in this order: V then J samples, which corresponds to an increasing level of C concentration (Table 1). Concomitantly, the amount of high order twin boundaries and of random angle grain boundaries increases. Moreover, it is worth noting that the values of the percentages obtained in as-grown and samples processed in the GaTSBI furnace are consistent and comparable with a standard deviation of 2.2% at maximum in samples V and J even though the processing conditions and ingot sizes were different.

Table 2. Percentage of the distribution of twin boundary types in F, V and J samples. RAGB corresponds to random angle grain boundaries. As-grown samples were processed using the cold crucible continuous casting. GaTSBI refers to sample processed using the GatSBI furnace with a temperature gradient of 30 K/cm and cooling rate of -2 K/min in both samples V and J.

3 Correlation with the electrical and photovoltaic properties

To correlate impurity concentrations and growth features with electrical properties, minority carrier lifetime maps were measured. with values ranging from 20 to 25 µs. Sample V map would appear all uniform if the same scale as the one for sample J was used. However, even in sample V map, some local lifetime variations exist and can only be highlighted with this scale. These local variations will be discussed in the following. Random angle grain boundaries (appearing in black in Figure 5.a) are clearly all active from the electrical point of view. Regions with a high number of small nucleating grains (e.g. A in Figure 5) are clearly active as well. The situation is different concerning Σ3 twin boundaries as some of them are active (C, D, E in Figure 5) and others are not active (B in Figure 5). in black in Figure 6.a, are active (e.g C in Figure 6). Moreover, as for sample P, some Σ3 grains boundaries are inactive whereas there is at least one unambiguously identified as an active one (A in Figure 6).

Discussion

The solid-liquid interface destabilization observed in Figure 1 is attributed to the presence of copper in samples C1, C2 (Table 1). As mentioned above, only the main impurities are shown in Table 1.

Thus, Cu was the only detectable metallic impurity in samples C1 and C2. The interface destabilization is due to solute accumulation above the solid-liquid interface during growth. Indeed, the morphological destabilization is a mechanism of interface structuration during solidification to increase its exchange surface for better evacuation of the latent heat or excess solute. The Si-Cu phase diagram that can be found for example in [36] indicates that the partition ratio between Si and Cu is lower than 1 and is equal to 4. 10 -4 [37] . The value of the partition ratio implies that Cu is rapidly accumulating above the solid-liquid interface as solidification progresses creating favorable conditions for morphological destabilization (dendrites in our case). Morphological destabilization has been widely studied in the literature; a review can be found here [START_REF] Coriell | Handbook of Crystal Growth[END_REF] . Details concerning these models are beyond the scope of the present study. Essentially, the destabilization due to constitutional undercooling is favored by a higher solute concentration, a lower temperature gradient and a higher growth rate. In directional solidification of alloys, an initially smooth growth front can become cellular or dendritic when the growth rate exceeds a threshold considering these parameters. A simple approach is given by the Tiller et al. [START_REF] Tiller | [END_REF] model and gives a critical growth rate for the destabilization of the solid-liquid interface:

𝑉𝑉 𝐶𝐶 = 𝑘𝑘𝐺𝐺 𝐿𝐿 𝐷𝐷 𝐿𝐿 (𝑘𝑘-1)𝑚𝑚𝐶𝐶 0 (1)
where k is the partition coefficient of the impurity, GL is the liquid temperature gradient, DL the diffusion coefficient of the impurity in the liquid, m the liquidus slope of the corresponding phase diagram and C0 the impurity nominal concentration. In the case of Cu impurity in Silicon, the partition coefficient is 4. 10 -4 [37] , the liquidus slope is estimated at 6.2 K.at% -1 from Olesinski et al. [40] and DL ≈ 10 -8 m 2 .s -1 . [41] It is worth noting that this approach can only give an order of magnitude of the critical velocity for the interface destabilization. The average solid-liquid interface growth rates are measured during the experiments on three points at the solid-liquid interface before the dendritic destabilization: 56 µm.s -1 (standard deviation = 24 µm.s -1 ) for C1 and17 µm.s -1 (standard deviation = 6 µm.s -1 ) for C2. Then, the local temperature gradient at the solid-liquid interface is estimated according to the method described in Stamelou et al. [42] : 11 K.cm -1 and 39 K.cm -1 in samples C1 and C2, respectively. At last, equation ( 1) gives a critical velocity of 97 µm.s -1 and 6 µm.s -1 for samples C1 and C2, respectively. Concerning C2 sample, the growth rate is clearly above the estimated critical velocity which implies a destabilization of the interface as observed in Figure 1. The measured growth rate before the destabilization is lower than the critical velocity in sample C1. However, the order of magnitude and the standard deviation of the interface growth rate shows that the destabilization is also possible in this sample although less likely. Indeed, the maximum measured solid-liquid interface growth rate in sample C1 is 112 µm.s -1 .

Finally, in our experiments, the accumulating Cu above the solid-liquid interface during growth is sufficient to destabilize the interface and reach dendritic growth conditions. The destabilization of the solid-liquid interface is more likely in C2 sample compared to C1 sample because of the higher Cu composition in sample C2 (Table 1). Furthermore, during growth, Si solid phase and liquid rich in Cu coexist until the eutectic temperature is reached at 802°C for which an eutectic Si-Cu3Si is first formed. It is worth noting that this temperature cannot be seen in the X-ray radiography field of view at the same time as the solid-liquid interface is observed.

In samples issued from Czochralski (Figure 1.a) or zone melting processes, the level of impurities is expected to prevent such destabilization as observed in our experiments. Moreover, in the case of the samples containing a higher concentration in light impurities, no destabilization was observed during in situ experiments. For C contamination, solute can indeed accumulate by the same mechanism during growth but conditions of constitutional undercooling above the solid-liquid interface during growth are not reached as shown in our previous work. [43] Indeed, due to the low solubility limits of C (XCS = 4.5 x 10 17 at/cm 3 in the case of C), precipitation of SiC can take place regularly during growth which depletes the liquid above the solid-liquid interface preventing any destabilization of the solid-liquid interface as the critical concentration for destabilization is never reached. The occurrence of precipitates is thus time-dependent because they can form after some solidification length due to segregation phenomenon when the solubility limit is reached. Moreover, precipitation is dependent on the processing parameters and on the convection in the liquid that can modify the local impurity concentration. [6,[44][45][46] From Figure 3, it can be concluded that the presence of light impurities, mostly C here, is at the origin of the formation of new grains during growth. Directional solidification experiments performed in the GaTSBI furnace on model small scale samples confirmed this observation.

In both type of processes, the new grains can nucleate on SiC or more complex precipitates that can be easily formed during solidification as soon as the solubility limit is reached as seen above. In fact, it was proven that the presence of SiC precipitates reduces the nucleation undercooling of the first Si crystals as measured by electromagnetic levitation by Beaudhuin et al. [6] Thus, the undercooling needed for a new grain nucleation (nucleation undercooling) is lowered on precipitates so that nucleation can take place for lower growth undercoolings in the presence of SiC precipitates at the level of the solid-liquid interface. Finally, there is a higher probability of Si grain nucleation in the presence of SiC which affects directly the grain structure by increasing the number of grains. This mechanism was invoked to explain the so called "grits" formation in the mc-Si industrial processes in the model from Mangelinck et al. [46] This model was latter revisited by Beaudhuin et al. [47] The impact of a high concentration of C inherent to the process has also been studied in Si ribbons grown on carbon sacrificial template. Again, the consequence of the presence of a high C concentration was the formation of SiC precipitates which affected the grain structure and defects. [5] Moreover, the solid-liquid interface observed in situ during solidification by X-ray radiography for sample J shows more grain boundary grooves as compared to the case of samples F or V which is directly related to the fact that more grains are present. Indeed, the presence of more grain boundary grooves at the solid-liquid interface reveals the presence of more grain boundaries as grain boundary grooves are formed at the encounter of a grain boundary with the interface. [43] Additionally, it is expected that impurities or precipitates accumulate in grain boundary grooves as proposed by Fujiwara et al. [48] Inside grain boundary grooves the undercooling is higher as measured in our previous work [41] so that there is a higher probability of grain nucleation especially if some impurities / precipitates favorable for nucleation are present, enhancing the effect of the presence of nucleation sites. As nucleation takes place on precipitates/impurities, the new grain does not have to be necessarily in twin relationship at the level of the {111} facets that can exist in the grain boundary groove. Finally, all these facts concur to the increase of the grain nucleation rate during growth of samples containing C. It is worth noting that more complex precipitate arrangements could also participate to the nucleation of new grains during growth although the main impurity is C in the described experiments.

Because of the frequent nucleation during growth, the mechanism of crystallographic orientation selection cannot take place as seen in Figure 3.

Table 2 confirms the mechanism of random nucleation on precipitates when C is present. Indeed, Table 2 shows that during solidification from a seed of high purity zone melted samples F, the formation of new grains is governed by Σ3 nucleation which is a process that does not require impurity presence due to the very low formation energy of Σ3 twin grains. [49] Higher order twin boundaries are scarce and are only due to grain competition and not to genuine nucleation as was earlier demonstrated by in situ X-ray imaging. [23] The situation is completely different in sample J containing impurities for which random nucleation can be expected on precipitates as seen above. As a consequence, Table 2 shows the direct relation between the increasing level of C impurity and higher order or even random angle grain boundaries. Moreover, it is worth noting that the results are coherent when comparing the asgrown samples and those grown inside the GaTSBI furnace which suggest that the impact of C on the grain structure is slightly dependent on the processing conditions and sample size. Cu does not seem to play a major role in this mechanism as can be seen in Figure 2 showing the grain structure of several samples and in Figure 3.a compared to Figure 3.b.

The lifetime difference observed in Figure 4 can be explained by the difference of grain structure between samples J and V (Figure 3). More grains and thus more grain boundaries are present in sample J as explained above. Grain boundaries in general are prone to segregation of impurities and thus to lower PV properties if there is no particular initial seeding in the process which is the case here. The fact that the center part of the grains is generally brighter corresponding to a higher lifetime is due to the gettering by the grain boundaries that tends to deplete the grain center from impurities.

Moreover, regions where many grains nucleated are very active (A in Figure 5). These regions are the location of high crystalline deformation due to grain competition and generally of emission of dislocations which can degrade the PV properties.

However, according to our results, a finer approach is needed to better understand the effect of grain boundaries on the PV properties. For all samples (Figure 5 and6), random angle grain and higher order twin boundaries are more active. This can be expected as this type of grain boundaries are of low crystallographic quality and thus contain more dangling bonds which favour impurity gettering. Moreover, dislocations are easily emitted from high order twin boundaries like Σ27 during growth competition. [START_REF] Ervik | Proceedings 26th European Photovoltaic Solar Energy Conference and Exhibition[END_REF]23,50,51] Inversely, coherent Σ3 twin boundaries are expected to be inactive because of their high crystallographic quality [49] as can be seen in the minority carrier lifetime map in Figure 5 (e.g. label B). However, in the same sample P (Figure 5), the lifetime map shows the presence of both active and inactive Σ3 twin boundaries. The common planes for both neighbor grains shows that the inactive grain boundary (B in Figure 5) is a perfect coherent Σ3 {111}1/{111}2. However, active Σ3 twin boundaries (C to E) correspond to incoherent Σ3 {112}1/{112}2 as verified with the EBSD measurements. This type of incoherent Σ3 {112}1/{112}2 boundaries have also been identified by Ervik et al. [START_REF] Ervik | Proceedings 26th European Photovoltaic Solar Energy Conference and Exhibition[END_REF] They are more prone to segregation of impurities and/or to emission of dislocations because of their lower perfection compared to Σ3 {111}1/{111}2 which could explain their activity.

The situation is different concerning the sample V (Figure 6). As mentioned above, not all Σ3 twin boundaries are active in this sample but at least one is strongly active (A in Figure 6). It was checked from the EBSD measurement that this particular Σ3 twin boundary is a perfect coherent Σ3 {111}1/{111}2. For this particular sample, on top of the GDMS results that indicate that there is no contamination with other metallic impurities like Fe and Cr, a method allowing to evidence concentration in Fe and Cr as low as 10 9 at.cm -3 was used. [52] The phase shift was measured, using µWPS (600 mA ; 1.2 V; 6kHz), for the particular sample V shown in Figure 6 before and after a heat treatment (210°C during 15 min) known to dissociate Fe-B and Cr-B pairs. The phase shift measured before and after the heat treatment is equivalent, thus it can be concluded that there is no Fe and Cr contamination. Thus, the activity is most likely due to the segregation of Cu at the level of the grain boundary. As mentioned by Buonassisi et al. [53] , elements with high diffusivity (e.g. Cu and Ni) are predominantly found in silicide nanoprecipitates at grain boundaries or other structural defects. After solidification, these elements that segregated along growth are able to diffuse to the most stable sites (e.g., grain boundaries), where nucleation barriers for silicide precipitation are lower. The same authors mention that although it is less probable to occur than at the level of random angle grain boundaries, nanoprecipitates can also be found at the level of Σ3 twin boundaries which can explain the activity observed in our experiments in at least one Σ3 twin boundary.

Conclusion

The impact of the impurities on the grain structure was studied. It can be concluded that in the presence of light impurities (C in this work), a higher frequency of nucleation is observed. The occurrence of nucleation is most likely linked to the presence of precipitates that decrease the grain nucleation undercooling threshold. SiC precipitates are formed whenever the solubility limit is reached. The C melt composition is built during solidification by the segregation phenomenon and it originates from the initial C content and the C contamination during the process. The resulting grain structure is constituted by a higher proportion of high order twin and of random angle grain boundaries compared to the case of low contaminated sample grain structure. Thus, the formation of SiC precipitates should be avoided as far as possible by a close control of the C melt composition above the solid-interface.

Moreover, the effect of Cu was studied. Cu segregates during solidification and induces a dendritic destabilization of the interface. However, no significant modification of the grain structure is observed in samples solidified with Cu contamination indicating that Cu does not seem to be significantly involved in grain nucleation. The lifetime maps obtained with the different types of samples demonstrate the influence of light and metallic impurities on the electrical activity of the defects and of the grains. In the presence of C, the minority carrier lifetime is generally lower in average and inhomogeneous. At a lower scale, particular activities are observed at grain boundaries depending on their crystallographic characteristics. Random angle grain boundaries and high order twin boundaries are generally active as they are well known to be prone for recombination and for gettering of impurities. Σ3 high crystalline quality twin boundaries have been found to be active in some particular cases. First, Σ3 can be active when they are incoherent Σ3 {112}1/{112}2 implying that the crystalline quality is somehow degraded being more prone to impurity segregation. More strikingly, a coherent Σ3 {111}1/{111}2 was found to be active in the presence of Cu. This observation was related to the presence of Cu nanoprecipitates that can be found at the level of grain boundaries. Thus, for very different reasons, impurities are found to modify significantly the grain growth formation and the electrical properties. The present study focused mainly on the grain boundaries. Further work will be conducted on the dislocation clusters formation during growth in the presence of impurities and on their influence on the electrical properties. Table 1. Concentrations of the detectable impurities in the as grown samples. DL is the detection limit: 1 ppbw and 1 ppm for GDMS (Glow Discharge Mass Spectroscopy) and IGA (Instrumental Gas Analysis), respectively.

Sample type F (F1 to F3) Cz [31] V J P C1 C2

Oxygen concentration (at.cm -3 ) < 10

Figure 1 .

 1 Figure 1.a). The image processing consists in division of the current image by a preceding one, hence

Figure 1 .

 1 Figure 1. Radiographs obtained on three kinds of samples at different instants during the solidification from seed, using the same cooling rate R = -4 K.min -1 applied at t0 with an applied temperature gradient Gapp = 20 K.cm -1 a) Cz, b) C1 and c) C2 samples.

Figure 2 .Figure 2 .

 22 Figure 2. Images of the final grain structure of samples grown by directional solidification in the

Figure 3 .

 3 Figure 3. Inverse pole figure (IPF) along the growth direction (left column) and transverse direction

Figure 4 .a and 4 .

 44 b show the lifetime maps obtained in the case of asgrown samples extracted directly from the industrial ingots of type V and J, respectively. It is important to note the scale difference between the two maps. For sample J (Figure4.b), lifetime scale ranges from 0 to 22 µs, which means that the lifetime in the few white regions observed corresponds to 22 µs at maximum. Lifetime in sample V (Figure4.a) is in average higher and more homogeneous;

Figure 4 .In Figure 5 ,

 45 Figure 4. µw-PS minority carrier lifetime maps of samples a) V and b) J extracted from the industrial

Figure 5 .

 5 Figure 5. Sample P: a) Coincidence Site Lattice (CSL) map. Red, blue, yellow, and black lines

Figure 6 .

 6 Figure 6. Sample V: a) Coincidence Site Lattice (CSL) map. Red, blue, yellow, and black lines

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .

 123456 Figure 1. Radiographs obtained on three kinds of samples at different instants during the solidification from seed, using the same cooling rate R = -4 K.min -1 applied at t0 with an applied temperature gradient Gapp = 20 K.cm -1 a) Cz, b) C1 and c) C2 samples.

Table 2 .

 2 Percentage of the distribution of twin boundary types in F, V and J samples. RAGB corresponds to random angles grain boundaries. As-grown samples were processed using the cold crucible continuous casting. GaTSBI refers to sample processed using the GatSBI furnace with a temperature gradient of 30 K/cm and cooling rate of -2 K/min in both samples V and J.

		15	5-10 10 17	5 10 15	8.6 10 15	3.7 10 17	3.7 10 16	4.3 10 16
				SIMS	SIMS	IGA	SIMS	SIMS
						3.3 10 17
						FTIR
	Carbon	< 10 15	5-10 10 15	2.5 10 16	3.6 10 17	< DL	3.15 10 17	4.2 10 17
	concentration			SIMS	SIMS	IGA	SIMS	SIMS
	(at.cm -3 )					1.3 10 17
						FTIR
	Copper	-	-	6.2 10 15	< DL	< DL	3.9 10 15	2.2 10 17
	concentration			GDMS	GDMS	GDMS	GDMS	GDMS
	(at.cm -3 )

Acknowledgements

This work was funded by the ANR Project CrySaLID (N° ANR-14-CE05-0046-01). The ESRF (European Synchrotron Radiation Facility) BM05 team is greatly acknowledged for support during the X-ray imaging experiments. Laurent Barrallier and Fabrice Guittonneau (Arts et Métiers, Aix-en-Provence) are greatly acknowledged for the EBSD measurements.