domain: chim spi phys

text:
Camille Salsi

Daniel Monceau

Clara Desgranges

Thomas Gheno

Effect of Compressive Creep and Hydrostatic Pressure on Diffusion in NiCr and NiSi Systems

Keywords: Interdiffusion, creep, NiCr, NiSi, hydrostatic pressure, hot isostatic pressing test, stress, deformation

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

High temperature materials are optimized for their mechanical, chemical or physical properties mainly through their chemical composition and microstructure. However, the composition can locally evolve in service due to diffusion. In addition, these materials are often subjected to significant thermomechanical stress that may affect their diffusion properties. It is therefore desirable to evaluate the effect of stress on diffusion if one is to anticipate the behavior of high temperature materials in service conditions.

Several aspects of the effect of stress on diffusion should be distinguished. For example, a stress gradient can act as a driving force for atomic transport, as in the case of rafting of γ/γ' superalloys [1]. Otherwise, stress can directly affect the diffusion coefficient, which is the focus of the present work. The effect of stress and deformation on diffusivity has long been the subject of investigation: among other sources, a review of early theoretical developments and experimental results can be found in Ref. [START_REF] Girifalco | The Theory of Diffusion in Strained Systems[END_REF], a collection of papers on the subject can be found in Ref. [START_REF] Beke | Defect Diffus[END_REF], while a more recent review can be found in Ref. [START_REF] Mehrer | Defect Diffus[END_REF]. This subject has attracted considerable attention in the field of materials for electronics, while less effort has been spent on structure materials such as Ni-base alloys. For example, Nakamura et al. [5], studied the effect of high hydrostatic pressure on Ni-Al diffusion couples. They observe a decrease of the interdiffusion coefficient with the pressure in compression. However, their experiments were done under hydrostatic pressures of 3-5 GPa. These stress values are very high and do not necessarily represent the stress magnitude that materials may be exposed to in most of Ni-based alloys applications. Fewer studies address more complex stress states. The effects of different types of elastic loading (hydrostatic, uniaxial and shear stress) on mobility in pure metals was studied by atomistic calculations, e.g., by molecular dynamics in bcc-Fe [6], or by DFT in fcc metals [7]. References [6] and [7] also predict a decrease in mobility under compressive stress, in accordance with linear elasticity theory, with significant amplitude for stress levels in the GPa range. To our knowledge, in Ni-base systems, very few experiments have demonstrated the effect of creep on tracer diffusion [8], and none have addressed the effect of creep on interdiffusion.

This paper examines the effect of viscoplasticity on interdiffusion in the Ni-rich, fcc domain of the binary NiCr system. This was studied using uniaxial compression creep tests conducted on diffusion couples between 885 and 1000 • C, under 0 to 25 MPa. The NiCr system was chosen as it forms the most elementary basis of high temperature Ni-base alloys. Single-phase compositions were used to facilitate the quantitative evaluation of interdiffusion coefficients. The latter were determined with a numerical inverse method that takes into account the strain caused by creep. The absence of precipitation or solid solution strengthening results in low yield stress in these model alloys at the temperatures of interest. Consequently, relative low stress levels were applied in the creep experiments, to avoid excessive strain. In order to distinguish potential contributions of viscoplasticity and elasticity in the creep experiments, Ni-Cr couples of the same composition were subjected to hot isostatic pressing (HIP) experiments. The absence of plastic strain allowed using larger stress levels, up to 326 MPa.

From the standpoint of diffusion properties, the NiCr system is particular in that the intrinsic diffusion coefficients of Cr and Ni have a weak composition dependence, which results in the interdiffusion coefficient having a weak composition dependence as well [9]. Hot isostatic pressing experiments were also conducted on diffusion couples in the NiSi system. Since the diffusivities have a much larger composition dependence, this allows to examine possible element-specific effects of hydrostatic stress, which would affect the composition dependence of the interdiffusion coefficient.

Methods

Materials and experimental method

Diffusion couples were prepared from single-phase alloys with nominal compositions (at. %): Ni, Ni-10Si, Ni-20Cr and Ni-30Cr. Ingots were produced by vacuum induction melting (VIM), homogenized at 1250 • C during 24 h under vacuum, and then cut into 6 x 6 x 3 mm specimens. In order to consider only the diffusion in volume, and not the diffusion at the grain boundaries, the time and temperature of the homogenization treatment have been chosen to obtain alloys with millimetric grain size. To form a diffusion couple, one face of each alloy to be assembled was polished with SiC paper down to a P4000 grade. Particles of Gd 2 Zr 2 O 7 of average diameter 1 µm were deposited on one alloy, to serve as inert markers. The alloys were then placed between two plates of Ni-based superalloy (UDIMET 720) held together with molybdenum screws and nuts (Figure 1), and the assembly was heat treated 1 h at 1050 • C under vacuum. The alloys to be assembled have a much higher coefficient of thermal expansion than the molybdenum screws. The stress resulting from this difference, about 50 MPa in the conditions of interest, causes the alloys to bond. In addition, the conditions of the assembly treatment were chosen so that the diffusion distance (due to the assembly) is negligible compared to the diffusion distance of the test. Concentration profiles in Ni-Cr diffusion couples were measured after assembly to verify the latter condition. As shown on Figure 2, the diffusion distance is about 10 µm, whereas after diffusion in the conditions of interest it is between 100 and 200 µm.

After bonding, the couples were removed from the assembling device and subjected to diffusion heat treatments.

Uniaxial compression creep tests were carried out on Ni-Cr diffusion couples at constant temperature (885 -1000 • C), controlled by a type S thermocouple, in air under constant compressive stress. To select stress values, a compression test was carried out on Ni-30Cr alloy at 1000 • C and the yield strength was measured. The latter was about 37 MPa, thus the applied stress values were in the range of 0 to 25 MPa. The sample was oriented so that the stress direction is perpendicular to the interface. The heating rate was about 5°C/min and the load was applied and removed at high temperature. During creep test, the strain was measured by the extensometer. Figure 3 illustrates the strain as a function of time. It shows a plastic strain of about 5 % for 12 MPa and between 18 and 29 % for 25 MPa. The secondary creep stage is not exceeded. Furthermore, given the high temperature (T > 0.6 T f ) and the low stress level (σ < yield strength), deformation most likely occurs by the diffusion of vacancies (diffusion creep). Hot pressing tests were carried out on Ni-Cr and Ni-Si diffusion couples during 5 h under pressure (50-326 MPa) at 1200 • C with heating and cooling rates of 12 • C/min. The chamber was pressurized during the temperature rise. Table I summarizes creep and HIP test conditions.

After the diffusion treatment, cross sections were analyzed by scanning electron microscopy (SEM) in a FEG-SEM Zeiss Merlin. Concentration profiles were measured by energy dispersive X-ray (EDX) using a quantification procedure with pure metals as standards (SAMx acquisition software). These concentration profiles were then used to determine interdiffusion coefficient by a numerical inverse method, presented in the following part.

Repeatability tests were carried out to determine the uncertainty on the diffusion coefficient. For that, the same experiment was done three times and for each sample two concentration profiles were measured by SEM. This method provides a measurement random error about 9.3 % on the interdiffusion coefficient.

Determination of interdiffusion coefficient in a binary system

The estimation of interdiffusion coefficient can be performed either using an analytical direct method or a numerical inverse method. For a binary system, the main analytical methods are the Boltzmann-Matano and Sauer-Freise methods [10]. The Boltzmann-Matano method gives as accurate results as the Sauer-Freise method but it requires determining the Matano plane, which can be challenging, especially if experimental concentration profiles are noisy. In contrast, the Sauer-Freise method does not rely on the Matano plane. However, both methods result in inaccurate values of diffusion coefficients at the terminal concentrations. On the other hand, these problems are not encountered with numerical inverse methods. By imposing the composition dependence of the diffusion coefficient, these methods also reduce artifacts associated with noise in the concentration profiles. Numerical inverse methods consist in fitting parameters which define the composition dependence of the interdiffusion coefficient by minimizing the difference between experimental and simulated concentration profiles [11]. The simulated profiles are obtained by solving the diffusion equation, which corresponds to the combination of the Fick's law (1) and the continuity equation ( 2) [START_REF] Philibert | Diffusion et transport de matière dans les solides, les éditions de physique[END_REF].

J A = -DAB ∂c A ∂z (1)
∂c A ∂t = - ∂J A ∂z (2)
In Eqs. ( 1) and (2), J A and c A refer to the flux in the laboratory frame of reference (see Appendix) and the concentration of species A, respectively, and DAB to the composition-dependent interdiffusion coefficient for a binary system AB. In this work, the composition dependence of the interdiffusion coefficient is described by a polynomial function, where a i are the fitting parameters:

DAB = n i=0 a i * c i A (3)
The diffusion equations are solved by the finite difference method, implemented in a Python script. The optimization of the parameters describing the composition dependence of the interdiffusion coefficient is performed with the least squares function (Levenberg-Marquardt algorithm) from the Scipy library [13] and using the Numpy library [START_REF]NumPy v1.22 Manual[END_REF]. The minimization consists in determining the difference between the experimental and a simulated concentration profile by calculating a residual. Figure 4 explains schematically the detailed process of the optimization.

To estimate the order of the polynomial of the Eq. ( 3) best adapted to the experimental results, several tests were performed. Figure 5 illustrates the evolution of the residual on the concentration profile with the polynomial order n (up to 5) for the composition dependence of the diffusion coefficient. For NiCr system the residual stays nearly constant for all orders, while for NiSi the residual is constant only for an order greater than or equal to 1. Therefore, we chose a polynomial of order 0 (constant interdiffusion coefficient) in the NiCr system, and a polynomial 3) for NiCr and NiSi systems of order 1 (linear variation of the interdiffusion coefficient with the composition) in the NiSi system. These observations are consistent with the stress-free diffusivity data available in these systems. Calculations made with data from Du and Schuster [START_REF] Schuster | [END_REF]16] indicate that, at the temperatures of interest, DNiCr is nearly constant in the composition range investigated, while DNiSi varies significantly with Si composition. The composition range investigated (0-10 at. % Si) is narrow, and in this range the variations of DNiSi can be described as linear.

Since diffusion may be affected by the applied stress and the deformation, this method for measuring interdiffusion coefficients has to be adapted. The flux of species A in a binary alloy AB subjected to composition and stress gradients is written as [START_REF] Philibert | Diffusion et transport de matière dans les solides, les éditions de physique[END_REF]:

J A = -DAB (P ) ∂c A ∂z - DAB (P ) k B T dU dz (4)
with DAB (P ) the pressure-dependent interdiffusion diffusion coefficient, k B the Boltzmann constant, T the temperature and U the elastic interaction energy in the stress field. In the first term of Eq. ( 4), the diffusion coefficient will vary because of the hydrostatic contribution of stress, while the second term represents the flux induced by the stress gradient. In an uniaxial compression creep test (if contact friction between the traverses and the specimen is neglected), there is no stress gradient and only the first term of Eq. ( 4) remains, i.e., the usual flux expression, Eq. ( 1), can be considered, with a pressure-dependent interdiffusion coefficient. This also applies to the HIP tests, which generate no stress gradient.

In the case of the creep experiments, the dimensions of the specimens change over time, which affects the measured concentration profiles. As shown in the Appendix, with two simplifying assumptions, the 3D problem of creep-interdiffusion reduces to a 1D problem, with all deformation-related effects gathered into an apparent interdiffusion coefficient. The assumptions are the following: (i) creep can be described as a convective form of transport, and (ii) the diffusivity and mechanical properties are uniform throughout the couple.

The time evolution of the composition can then be computed by solving the diffusion equation in a frame of reference fixed to the sample that strains with time. The hypothesis of uniform mechanical properties implies that the strain in the load direction (z axis) is equally distributed over the diffusion couple. Strain is interpolated from the experimental data (Figure 3) and the strain value calculated at each time step is converted into a variation of the space step, as shown in Figure 6. Taking into account the strain is important as a strain of about 30 % can lead to an apparent variation of the interdiffusion coefficient of about 20 % if the strain is not taken into account.

Results

Cross sections of the diffusion couples were observed by SEM and show that no pore were formed due to the Kirkendall effect for Ni-Cr and Ni-Si systems. Concentration profiles were measured and the origin of the distance axis (z = 0) was placed at the location of the inert markers, i.e., at the Kirkendall plane. The position of the Matano plane was determined from the concentration profiles, according to the classical method [17], which then provided the Kirkendall shift. For the Ni-Cr diffusion couples, regardless of the temperature the Kirkendall shift was found to be less than 1 µm. In the Ni-Si couples, the measured values were slightly larger but also smaller than 1 µm.

On Figure 7, concentration profiles measured in the NiCr couple after compression creep tests with different stress levels are superposed for comparison at each temperature. We can see that the concentration profiles are almost identical for all stress values at each temperature. The profiles were used to evaluate the interdiffusion coefficients using the numerical inverse method described previously.

Figure 8 shows the evolution of the interdiffusion coefficient with temperature for all stress values. Without stress, the Ni-Cr interdiffusion coefficient is in agreement with literature data. Using diffusion data from Jönsson [18] and thermodynamic data from Lee [19], the interdiffusion coefficient is calculated to be fairly constant between 20 and 30 at.% Cr, with an average value of 7.5 x 10 -16 m 2 /s at 1000 • C. In comparison, the estimated diffusion coefficient at 1000 • C in this work is 1.5 x 10 -15 m 2 /s. Table II compares the values of the constant Ni-Cr interdiffusion coefficient determined in this work and that determined using data of Jönsson and Lee for all temperatures.

On Figure 8, a small variation of the interdiffusion coefficient with stress is observed at 1000 • C: an increase of the stress from atmospheric pressure to 25 MPa results in a decrease in the interdiffusion coefficient of about 76 %. However, the data at 940 and 885 • C do not follow any consistent trend. At 885 • C, the point at 12 MPa is far from the others, probably due to an experimental error. Overall, the small variations observed here may not be distinguished from experimental dispersion with confidence: no significant effect of stress is found in the conditions of these experiments.

Figure 9 shows the concentration profiles and the evolution of the interdiffusion coefficient with pressure in NiCr after HIP tests at 1200 • C. A slight decrease in the interdiffusion coefficient with pressure is observed.

For the NiSi system, Figure 10 presents the experimental concentration profiles for each tests and the interdiffusion coefficients, which depend linearly on the composition. The concentration profiles are very close to each other, but there are visible differences on the diffusion coefficients. As for the previous tests the decrease observed with pressure is small: for an increasing pressure ratio around 6.5 (between 50 and 326 MPa), a decrease of up to 66 % on the interdiffusion coefficient is ob-

Temperature (°C)

Ni-Cr interdiffusion coefficient (m 2 /s) using data from Jönsson [18] and Lee [19] this work served, that is greater than for the variation observed for the creep test at 1000 • C. Finally, one can notice that the pressure effect varies with the composition: there is almost no effect of pressure in pure nickel and a significant effect can be observed in Ni-10Si.

Discussion

In a binary AB system, the interdiffusion coefficient is related to the tracer diffusion coefficients via the Darken relation:

DAB = (c A D * B + c B D * A )φ (5)
where D * i is the tracer diffusion coefficient of species i, and φ is the thermodynamic factor, defined by

φ = c A RT ∂µ A ∂x A (6)
In Eq. ( 6), µ A is the chemical potential of A, R is the ideal gas constant and T the temperature. The thermodynamic factor is not expected to vary significantly with the pressure. The discussion therefore focuses on the tracer diffusion coefficients. In a cubic crystal, the latter may be written in the general form

D * i = gf a 2 ν 0 exp - ∆G RT (7)
where g and f are geometry and correlation factors, respectively, a is the lattice parameter, ν 0 is the jump attempt frequency, and ∆G is the Gibbs free energy of diffusion. All terms in Eq. ( 7) may be affected by stress and deformation, to extents that depend on the diffusion mechanism and the loading conditions [START_REF] Girifalco | The Theory of Diffusion in Strained Systems[END_REF]. The discussion first addresses the case of elastic deformation, before turning to plastic deformation.

The effect of hydrostatic pressure on diffusivity is mainly due to the pressure dependence of ∆G [START_REF] Mehrer | Defect Diffus[END_REF], ∆G = ∆U + P ∆V -T ∆S (8) where ∆U is the activation energy, ∆V the activation volume and ∆S the entropy of activation. Diffusion in metals occurs by a vacancy mechanism; ∆G may then be split into vacancy formation and migration energies, ∆G = G F + G M , where the pressure dependence of each term follows Eq. ( 8). It follows that the tracer diffusivity can be written

D * i = gf a 2 c eq V (T, P ) Γ (T, P ) (9)
where the equilibrium vacancy concentration is

c eq V (T, P ) = exp - U F + P V F -T S F RT (10)
and Γ(T, P ) is defined by

Γ (T, P ) = ν 0 exp - U M + P V M -T S M RT (11)
Experimental data have shown that diffusion in fcc metals occurs by a monovacancy mechanism at the temperatures of interest, with activation volumes that range between 0.7Ω and 0.9Ω (Ω is the molar volume) ( [START_REF] Mehrer | Defect Diffus[END_REF][START_REF] Philibert | Diffusion et transport de matière dans les solides, les éditions de physique[END_REF]). These values are consistent with recent DFT results for diffusion in pure fcc Ni [7]: V F = 0.61Ω, V M = 0.15Ω, which yield ∆V = 0.76Ω. Using these values, it is possible to evaluate the ratio of two tracer diffusion coefficients at two different pressures. When the pressure increases from 50 to 326 MPa, we expect D* to be divided by 1.2 at 1200 • C. The present experimental results are consistent with this expected variation: in the HIP tests, a ratio about 1.5 is obtained between the interdiffusion coefficients measured at 326 and 50 MPa.

Non-hydrostatic loading conditions (uniaxial or pure shear for instance) introduce a loss of crystal symmetry, and the terms of Eq. ( 9) become orientationdependent. This has been studied by analytical methods [20,21] and first-principle calculations [6,7]. However, the phenomena described previously are caused exclusively by elastic deformation and the most significant feature of diffusion under uniaxial loading, as in the present work, is related to the occurrence of viscoplastic deformation. The latter may be due to vacancy diffusion or to dislocation glide or climb, which corresponds respectively to diffusion creep or dislocation creep. As mentioned in Section 2.1, in the present work viscoplastic deformation is due to diffusion creep. Since the two creep mechanisms might have a different impact on diffusion, it is relevant to study their effects separately. In the following section, we will discuss the effect of dislocation creep observed in the literature and compare it with the results of this work, in order to deduce the impact of each mechanism on diffusion. No direct comparison of the present results can be made with experimental data from the literature, as no other study on the effect of diffusion creep on interdiffusion could be found.

According to the early review by Girifalco and Grimes [START_REF] Girifalco | The Theory of Diffusion in Strained Systems[END_REF], in the conditions typical of dynamic strain, the main contribution of deformation to diffusivity is due to the vacancy concentration, which is believed to increase and depart significantly from its equilibrium value as a result of dislocation climb. Note that the vacancy concentration, and therefore the tracer diffusivity, are expected to increase whether the applied load is in tension or compression. At steady-state, the diffusivity variation is expected to vary linearly with the strain rate [2]:

D * s D * u = 1 + K 1 ε K 2 c u V , (12)
where the indices s and u refer to strained and unstrained quantities, respectively, ε is the strain rate, K 1 reflects the vacancy production rate due to the plastic strain, and K 2 reflects the strength of other vacancy sources/sinks (dislocation jogs, grain boundaries, precipitates, pores). The linear dependence represented by Eq. ( 12) was observed experimentally: in Ref. [22], the self-diffusivity of bcc Fe was measured using the tracer method during uniaxial compression, and was found to increase linearly with the strain rate. The variation was described by

D * s D * u = 1 + 50 ε (13)
with ε in h -1 . The strain rates measured in the present creep experiments were in the range 7 x 10 -5 -3.70 x 10 -3 h -1 , which would not produce significant diffusivity variations if the same relationship was obeyed. Nevertheless, the slight decrease of the interdiffusion coefficient measured in the creep tests at 1000 • C (Figure 8) is at variance with the above formalism [START_REF] Girifalco | The Theory of Diffusion in Strained Systems[END_REF], since the latter anticipates an increase of the diffusion coefficient under dynamic strain, even in compression. Therefore, the decrease observed at 1000 • C in this study is not due to the effect of viscoplastic deformation introduced by Buffington and Cohen [22]. Furthermore, if we refer to the first-principles results of Refs [6,7], given the stress levels involved in the present creep experiments (12 and 25 MPa), neither the hydrostatic nor the deviatoric stress is expected to significantly affect the diffusivity. Factors other than vacancy concentration and stress must in fact have played a role in the measured diffusivity. Literature data regarding the role of viscoplastic deformation on diffusivity in Ni-base alloys are very limited. Nguejio [8] studied the tracer diffusion of Cr in Ni at 500 • C by tensile creep tests, with stress levels between 50 and 75 MPa and strain rates in the order of 1.4 x 10 -3 -1.19 x 10 -2 h -1 . They measured an increase of the diffusivity by a factor of 10, which they attributed to (i) the excess of vacancies which are created during creep, (ii) a fast diffusion along dislocations. If we refer to their data, the ratio of the diffusion coefficients with and without stress does not depend linearly on the creep strain rate as described by Buffington and Cohen [22], Eq. (13). Furthermore, the variation of the diffusion coefficient with stress observed by Nguejio et al. [8] is larger than the effect observed in the present experiments. Once again, the experiments of Ref. [8] involve dislocation creep, while the present experiments involve diffusion creep. These results would then suggest that dislocation creep has a larger effect on diffusion than diffusion creep. However, it would be useful to compare the two creep mechanisms for the same material to verify this observation.

Finally, it is useful to put the present results in perspective by considering the typical dispersion observed in diffusion data measured in different laboratories. Campbell and Rukhin studied different estimations of Ni diffusion coefficients available in the literature and determined an associated uncertainty [23]. They have shown that at high temperature (977-1394 • C), once the literature data have been filtered to retain only the most reliable datasets, one can still observe a difference of a factor of ten between diffusion coefficients at one given temperature. This is much greater than the variation due to the effect of external stress observed in the present study.

Conclusion

This work illustrates the influence of moderate compressive uniaxial load and of hydrostatic pressure on the interdiffusion coefficient. In the NiCr and NiSi systems, a small decrease of the diffusion coefficient with compressive hydrostatic stress at 1200 • C is observed. However, the variation is very low, even at 326 MPa. This result is consistent with previous studies, which show that the effect of elastic stress becomes significant at much higher pressures. This work also highlights the lack of any visible influence of the viscoplastic deformation during a uniaxial compression creep test on the interdiffusion coefficient in the NiCr system.

To conclude, our results indicate that a very high stress value is needed to see a significant effect on diffusion in Ni-base alloys at high temperature. In this work, the diffusion coefficient is at most multiplied by 1.7, which is smaller than the experimental dispersion typically observed between different laboratories in high temperature diffusion studies. Therefore, the selection of the diffusion data for a simulation is more important than the consideration of the effect of stress on diffusivity. For example, for structural Ni-based alloys used for turbine blades subjected locally to stress values up to 300 MPa, the effect of a uniform uniaxial or hydrostatic stress on the diffusion coefficient values is likely to be not significant and does not necessarily have to be taken into account in lifetime modelling.

Figure 1 :

 1 Figure 1: Scheme of the assembling device for the diffusion couples.

Figure 2 :

 2 Figure 2: Concentration profile after diffusion couple Ni-20Cr / Ni-30Cr assembly at 1050 • C during 1 hour, where x Cr corresponds to the atomic fraction of chromium.

Figure 3 :

 3 Figure 3: Strain vs. time for compression creep tests carried out on Ni-20Cr / Ni-30Cr diffusion couples at (a) 885 • C, (b) 940 • C and (c) 1000 • C in air during 670, 216 and 77 hours respectively.

Figure 4 :

 4 Figure 4: Diagram explaining the running of the python code for determining interdiffusion coefficients by a numerical inverse method. The Levenberg-Marquardt algorithm (from the function "least squares" of the Scipy library [13]) is used for the minimization between experimental and simulated profiles.

Figure 5 :

 5 Figure 5: Residual on concentration profiles as a function of polynomial order of Eq. (3) for NiCr and NiSi systems

Figure 6 :

 6 Figure 6: Pseudo-code of the diffusion solver, in which the red highlighted line represents the expression to add to take into account the deformation with time ε n during creep tests. Indices n and i refer to the time and space, respectively.

Figure 7 :

 7 Figure 7: Concentration profiles of Ni-20Cr / Ni-30Cr diffusion couples heat treated at (a) 885 • C, (b) 940 • C and (c) 1000 • C during 670, 216 and 77 hours, respectively, with 0, 12 and 25 MPa (compression creep tests)

Figure 8 :

 8 Figure 8: Evolution of the Ni-Cr interdiffusion coefficient with creep strain rate at 885, 904 and 1000 • C. Error bars represents the measurement random error determined from the repeatability tests.

Table 2 :

 2 Comparison of the constant Ni-Cr interdiffusion coefficient determined in this work and using data from the literature.

Figure 9 :

 9 Figure 9: (a) Concentration profiles and (b) evolution of the Ni-Cr interdiffusion coefficient with pressure for hot isostatic pressing tests on Ni-Cr diffusion couples at 1200 • C during 5 hours. Error bars represents the measurement random error determined from the repeatability tests.

Figure 10 :

 10 Figure 10: (a) Normalized concentration profiles and (b) evolution of Ni-Si interdiffusion coefficient with Si composition at several pressures after hot isostatic pressing tests on Ni-Si diffusion couples at 1200 • C during 5 hours. The colored background areas represent the measurement random error determined from the repeatability tests.

Figure A1 :

 A1 Figure A1: 2D illustration of interdiffusion and convection fluxes in an AB diffusion couple subject to a uniaxial creep test.

Table 1 :

 1 Conditions for creep and HIP tests on Ni-Cr and Ni-Si diffusion couples. Composition is given in at. %.

	Mechanical test	Diffusion couple	Temperature ( • C)	Time (h)	Stress (MPa)	Atmosphere
			885	670
	Creep	Ni-20Cr / Ni-30Cr	940	216	0, 12, 25	Laboratory air
			1000	77
	HIP	Ni-20Cr / Ni-30Cr Ni / Ni-10Si	1200	5	50, 100, 326	Argon

Acknowledgements

The authors would like to thank C. Lopes (ONERA) for producing the ingots, D. Mézières (ONERA), C. Liard (ONERA) for conducting the creep and compression, respectively, as well as C. Rio (ONERA), N. Horezan (ONERA) and D. Boivin (ONERA) for help with SEM analysis. The HIP experiments were conducted at the CARAMEL platform of the Laboratoire de Sciences des Procédés et des Matériaux, Université Sorbonne Paris Nord, by A. Hocini, who is gratefully acknowledged. We would also like to thank members of the COnCOrD GdR (CNRS, GdR 2081), as this work benefited from fruitful discussions at meetings of the GdR. This work was funded in part by the Agence Nationale de la Recherche (France), grant number ANR-19-CE08-0004.

Conflict of interest

The authors declare that they have no conflict of interest.

Appendix Consideration of the viscoplastic deformation in the determination of interdiffusion coefficients

We consider a diffusion couple made of two AB alloys of different compositions, with the initial interface in the (x, y) plane. The couple is subject to a uniaxial compression creep test, with the load applied along direction z. The problem geometry is represented in 2D in Figure A1, where direction y is omitted for clarity.

Creep causes the couple to shrink in the z direction, and to expand in the x and y directions. We assume that the mass transport associated with creep occurs via the bulk motion of the material, i.e., it is a convective form of transport. This implies that in the absence of an initial composition gradient, creep alone would not produce any composition change. This hypothesis neglects fluxes of A and B generated by the vacancy flux when A and B have different mobilities. We note c A and C A the concentration of species A in the laboratory reference frame and in the material reference frame, respectively. The laboratory reference frame is attached to one corner of the sample at the initial state and does not change over time. On the contrary, the material reference frame is attached to one corner of the sample, which moves with time (in the laboratory reference frame), and strains along with the sample. The material derivative, noted ĊA , is defined as:

where v is the material velocity field. In Eq. (A1) and in the following, bold letters indicate vector quantities.

In the laboratory frame of reference, the continuity equation for A reads:

where J A is the 3D flux of A, which contains two contributions, the interdiffusion flux J A and the convection flux due to creep Ĵ A :

Substituting Eq. (A3) in Eq. (A2) yields:

Combining Eqs. (A1) and (A6) yields:

In continuum mechanics, the divergence of the convective velocity field is defined as the relative volume variation rate:

where V is the volume of a representative element of material, typically of the order of a µm 3 . We make the hypothesis that during creep, the total volume of material is conserved, i.e., we neglect the elastic deformation. Further, we assume that the diffusivity and mechanical properties are uniform throughout the couple.

In particular, the convection associated with creep is unaffected by microstructural features such as grain boundaries. It follows that the volume is conserved at the scale of the representative element. This yields div v = 0 by virtue of Eq. (A8). Equation (A7) then reads: ĊA = -div J A (A9) Equation (A9) is a continuity equation in the material reference frame. It expresses the fact that in the case of a constant-volume transformation, the mass balance defined in the deforming material is directly obtained from the divergence of the interdiffusion flux. Finally, with the hypothesis of uniform material properties, the initial symmetry of the problem is conserved throughout the transformation. It follows that the x and y coordinates of J A are 0. Equation (A9) then reduces to a 1D equation:

Numerically, this equation is solved by the finite difference method, where the deformation is taken into account by updating the grid size ∆z at each time step, as illustrated in Figure 6.


P.-H Merrer

A Bouchier
email: bouchier@laas.fr

H Brahimi

O Llopis

G Cibiel

High-Q Optical Resonators for Stabilization of High Spectral Purity Microwave Oscillators

Different high-Q optical resonators are presented in this communication. Our aim is to use them to develop compact high performances microwave oscillators. The optical quality factor of a quartz disk used as a whispering gallery modes resonator has been measured to be higher than 10 9 . We also investigate the potentialities of high-Q fibre ring resonators. The performances of an optoelectronic oscillator at 10 GHz based on such a passive fibre ring resonator are presented here.

I.

INTRODUCTION

In order to design compact and high performances microwave oscillators, new resonators are investigated. But the dimensions of microwave resonators are intrinsically large, especially for cm-long wavelengths, and their quality factors also decrease when the frequency increases. Actually, the whispering gallery modes sapphire resonators present the best performances around 10 GHz, with an unloaded quality factor close to 10 5 .

An interesting method to reduce resonators dimensions is to carry the microwave on another wave, like acoustic or optical waves for example. In this study, we have chosen to use an optical carrier, here a 1.5 µm-laser. Thus microwave resonators become optical ones, with lower dimensions due to shorter wavelengths. It is thus possible to design devices that are more compact than the actual microwave ones. In this case, the optical resonator creates an optical frequency comb with microwave spacing. To each mode of this optical comb corresponds an optical Q factor (Q opt ) and an equivalent RF Q factor (Q RF ). Q RF is the product of Q opt and the RF to optical frequencies ratio [START_REF] Yao | Progress in the optoelectronic oscillator -a ten year anniversary review[END_REF].

opt RF opt RF f f Q Q = (1)
Due to this relation, in opposition to the case of microwave resonators, the RF Q factor increases with the RF frequency. Nevertheless the high frequency ratio between microwave (around 10 to 20 GHz) and optics (about 194 THz) constrains the optical quality factor to be higher than 10 9 at 1.55 µm to guaranty a RF quality factor at 20 GHz higher than 10 5 . In this case, these optical resonators can allow better performances than microwave components.

In the microwave photonics domain, optical resonators are an alternative method to the classical optical delay lines, which are easy to use and actually well known. These lines are few-km-long optical fibres, and their RF quality factors can reach a value of 10 6 at 10 GHz. They have been used to stabilize oscillators [START_REF] Yao | Progress in the optoelectronic oscillator -a ten year anniversary review[END_REF] but are bulky devices and there thermal stabilization is difficult.

An interesting alternative to these optical delay lines is proposed by the mini optical resonators [START_REF] Gorodetsky | Ultimate Q of optical microsphere resonator[END_REF][START_REF] Armani | Ultrahigh-Q toroid microcavity on a chip[END_REF]. These devices are spheres, torus or disks, in which the optical wave circulates on the whispering gallery modes. These structures have to be transparent at 1.55 µm, and are constituted from silica, quartz, calcium (CaF 2 ) or magnesium fluoride (MgF 2 ) and so on… As presented here, these resonators have really high optical quality factors (from 10 8 to 10 10 at 1.55 µm), but their use is delicate. A last approach is to use the ability of fibered optics, which has been developed for telecommunications around 1.55 µm, and we present here the principle and use of fibre ring resonators [START_REF] Merrer | Laser stabilization on a fiber ring resonator and application to RF filtering[END_REF]. We also present an optoelectronic oscillator (OEO) based on one of these passive fibre ring resonators.

II. STUDY OF DIFFERENT MICROWAVE PHOTONICS RESONATORS

A. The Whispering Gallery Modes Resonators

We have studied two types of whispering gallery modes (WGM) resonators presenting a low volume: a silica sphere with a diameter of 3.3 mm and a quartz disk, with a diameter of 7.7 mm. The whispering gallery modes are excited with the laser at 1.55 µm, which carries the microwave. The laser used for the characterization of these resonators is an amplified erbium-doped fibre laser from Koheras. It presents a really low linewidth of around 1 kHz for a delay of 120 µs. It is tunable on a frequency range of 120 GHz thanks to the variation of its temperature, with a precise tunability on a range of 2 GHz with a piezoelectric component. This pigtailed laser is launched in the WGM resonators thanks to the evanescent waves coming from a tapered or a half-tapered fibre. This fibre presents a waist of around 2 µm on a length of a few millimetres (Fig. 1), and is fabricated by Laseo (France) [START_REF] Merrer | Microwave filtering using high Q optical resonators[END_REF]. Due to the tunability of the laser, we have measured the optical quality factor of the silica sphere and the quartz disk with a method of scan. This consists in a scan of the frequency of the laser to explore the different optical modes of the frequency comb due to the resonator. We can thus measure the distance between two modes (the free spectral range or FSR), and the full-width at half-maximum of a mode. The linewidth of the laser is really low and is smaller than the width of a resonant mode. We can thus have a good resolution of the main resonant modes.

For the fused silica sphere, we have determined an optical quality factor of about 2 10 8 at 1.55 µm with a strong decrease to 10 6 after two years. This drop of the optical quality factor is due to the degradation of the resonator surface quality, probably caused by chemical reaction with the atmospheric water. We are thus mainly dealing with the second WGM resonator in the follow of this communication. In the case of the quartz disk, with a diameter of 7.7 mm, we have measured a free spectral range of 8 GHz and a width of the main resonant modes of 47 kHz (Fig. 1). With these measurements, we can determine an optical quality factor for this resonator of 4 10 9 at 1.55 µm (~194 THz). This really high optical quality factor let us expect RF quality factors around 4 10 5 at 20 GHz. Nevertheless these compact resonators present some disadvantages. Firstly, the coupling of the laser in the resonator with tapered fibres and the excitation and selection of whispering gallery modes is difficult and can be unstable. Secondly, these optical resonators are highly transverse multimode and numerous gallery modes can be excited in the resonator. This is clearly visible with a scan of the frequency of the single-frequency low linewidth laser. The stabilization of the laser frequency on a frequency resonance is thus really difficult. We are studying alternative coupling techniques to improve the coupling efficiency and stability, like the use of prisms [START_REF] Ilchenko | Miniature oscillators based on optical whispering gallery mode resonators[END_REF] or angle-polished fibres instead of the tapered fibres. Thus, although we still study these compact WGM resonators, which are really promising in terms of high optical quality factors, investigations on different types of resonator are also interesting.

B. The Fibre Ring Resonators

The optical fibres are particularly adapted for utilization at telecommunication wavelengths. We can thus use transverse single-mode fibres at 1.55 µm like the classical SMF28 fibre. A resonator can be built with such fibres, leading to a transverse single frequency comb. On another side, laser coupling in the resonator become really easy if we use pigtailed fibre lasers. In this case, the coupling is stable, and the system can be integrated in a compact device. We have thus studied a resonator based on the fibre ring described in [START_REF] Merrer | Laser stabilization on a fiber ring resonator and application to RF filtering[END_REF].This ring is fabricated with two single-mode fibres (Corning SMF28) linked with two low-loss fibered couplers (typically with a ratio of 99/1 %) as described on Fig. 2. The connection between fibres inside the ring is made by splicing the fibres. The total length of the ring is from 1 to 20 m, leading to a free spectral range comprised between 10 and 200 MHz. The simulation of such a fibre ring is explained in [START_REF] Merrer | Laser stabilization on a fiber ring resonator and application to RF filtering[END_REF]. The optical couplers are described with their coupling coefficients and losses. The model takes also into account the losses of the fibre splices (about 0.02 dB). The fibered couplers used for the rings are experimentally characterized. Optical quality factors from 2 10 8 for a ring length of 1 m to 5 10 9 for a 20m-long ring can be expected according to the simulation. We have thus studied experimentally a ring with a total length of 20 m.

The scan method used for the study of the whispering gallery modes resonators allowed us to check a free spectral range for this ring of 10 MHz. We have also observed that these resonators are really sensitive to some external influences due to their high quality factor. Thermal effects are the most difficult to control. They are mainly due to temperature variations of fibre environment and also to the laser launching inside the fibre ring. This effectively modifies the local temperature in the fibre, especially in the coupler, and leads to local changes of index and length. For the couplers, a change in temperature can also lead to a modification of coupling characteristics and to the appearance of losses. Globally, these thermal variations shift the resonance frequency of these resonators. It is thus necessary to stabilize the laser on a resonance frequency of the ring, as shown on the experimental setup of the Fig. 3.

The laser stabilization is obtained thanks to a Pound-Drever-Hall (PDH) loop [START_REF] Pound | Electronic frequency stabilization of microwave oscillators[END_REF][START_REF] Drever | Laser phase and frequency stabilization using an optical-resonator[END_REF]. The phase modulation is realized with a lithium niobate modulator. The error signal from this loop is sent on a proportional differential integrator. The correction is applied on the piezoelectric component of the Koheras laser. The laser frequency is thus precisely tuned. When the laser is stabilized on the resonator, it is possible to modulate it with microwaves. In this aim, we use a Mach-Zehnder Modulator (MZM). Lateral modes are thus explored by scanning the modulator frequency, thanks to a microwave network analyzer (Fig. 3). The resonator spectrum is then stable and we can measure a full-width at half-maximum of 65 kHz (Fig. 4). The optical quality factor determined with this method is comparable to the simulation, with a value of 3 10 9 for a 20mlong fibre ring resonator composed with 99/1 % fibered couplers. In the microwave range, this performance leads to an equivalent RF quality factor of 3 10 5 at 20 GHz, which is largely higher than what can be obtained directly with microwave resonators. Nevertheless some difficulties have been observed when the oscillator has been built with this resonator: an important phase noise component has been measured. A new fibre ring resonator, which solves this problem, has thus been developed (patent pending). But it presents a lower optical quality factor, around 2 10 8 at 1.55 µm. Despite its low quality factor, a first oscillator has been assembled with this fibre ring resonator.

III. DEMONSTRATION OF AN OPTOELECTRONIC OSCILLATOR BASED ON AN OPTICAL RESONATOR

From the setup on the Fig. 3, we have realized an optoelectronic oscillator (OEO) with a frequency of 10 GHz. The principle of the experimental setup is depicted on Fig. 5.

The output signal from the photodiode is amplified. It contains the different frequency beats between the resonant modes of the fibre ring resonator. In order to build an oscillator around 10 GHz, we have used a dielectric resonator that filters a frequency range of about 3 MHz around 10 GHz in order to select only one mode from the frequency comb of the optical resonator (Fig. 6). Thus, the RF oscillation starts on this mode only (Fig. 7). It is stable and no other spurious modes are observed.

The spectrum of this oscillator and its phase noise spectrum have been measured thanks to an Agilent E5052 signal analyzer. The results are presented on Fig. 8. The phase noise spectrum presents different slopes. Between 10 Hz and 1 kHz, the phase noise is dominated by the 1/f frequency noise, which is characterized by a slope of -30 dB/dec. The spurious peaks that are visible between 0.1 and 1 kHz can be due to thermal or mechanical fluctuations of the fibre ring resonator. The white frequency noise is characterized by a slope of -20 dB/dec from 1 to 400 kHz and is related to the signal to noise ratio in the RF oscillating loop. The phase noise floor appears after 400 kHz, and has a value of about -120 dBc/Hz. The final decrease of the noise is linked to the filtering by the dielectric resonator.

The frequency f c at the frontier between white frequency noise and the phase noise floor is related to the loaded microwave quality factor (Q RF ) by the relation ( 2)

RF RF c Q f f . 2 = (2)
with f RF the microwave frequency [START_REF] Leeson | A simple model of feedback oscillator noise spectrum[END_REF]. With a corner frequency of 0.4 MHz, the loaded microwave quality factor is 1.2 10 4 at 10 GHz, which is in good agreement with an optical quality factor of 2 10 8 at 1.55 µm (eq. 1).

In spite of its good quality factor, this oscillator presents a relatively high phase noise level. This phenomenon is mainly due to the transmission losses in the optical device. These losses are added to the low coupling rate between the resonant ring and the transmission output of the resonator (Fig. 2), and contribute to decrease the signal to noise ratio at the output of the resonator, and thus to increase the phase noise of the oscillator. A second factor is linked to the different 1/f phase noise sources in the optical components that have not been optimized for this first demonstration. We are actually working to improve these parameters in our future oscillators.

IV. CONCLUSION

Optical resonators allow thus to achieve really high quality factor in the microwave range, while they guarantee low dimensions. Whispering gallery modes resonators are potentially interesting for really compact oscillators, if solutions for an efficient assembling can be found to propose integrated (or semi-integrated) coupling for these devices. Fibered resonators like fibre ring resonators are easier to integrate and use. The optoelectronic oscillator presented in this communication is based on such a resonator and is, up to our knowledge, the first one of this type to be published. Its performances in terms of phase noise are still modest and close to the ones of a dielectric resonator oscillator (DRO). Nevertheless, its potentialities in terms of microwave quality factors are really promising and the current development of a new optimized resonator will allow us to highly improve the performances of this oscillator.

Figure 1 .

 1 photodiode transmission

Figure 2 .

 2 Figure 2. Scheme of the fibre ring resonator and resonant modes in transmission and absorption.

Figure 3 .

 3 Figure 3. Setup for the test of microwave filtering by the optical resonator.

Figure 4 .

 4 Figure 4. Resonance of the stabilized 20m-long fibre ring and free spectral range in inset.

Figure 5 .

 5 Figure 5. Principle of the optoelectronic oscillator at 10 GHz.

Figure 6 .

 6 Figure 6. Comparison around 10 GHz of the transfert function of the dielectric resonator and the free spectral range of the 20m-long fibre ring.

Figure 7 .

 7 Figure 7. Resonance around 10 GHz of the fibre ring resonator after filtering with the dielectric resonator.

Figure 8 .

 8 Figure 8. Phase noise measurement and spectrum (in inset) for the optoelectronic oscillator at 10 GHz.