Mélissa Ritouet
email: melissa.ritouet@univ-lorraine.fr

Patrice Berthod
email: patrice.berthod@centraliens-lille.org

Effect of NbC Addition on the High Temperature Oxidation Resistance of Co-and Ni-base Chromium-rich Alloys

Keywords: Cobalt-based alloys, Nickel-based alloys, Niobium carbides, High temperature oxidation, Mass gain rates, Scale spallation, Oxide characteristics

Two {M-25Cr}-based alloys, with M=Co or Ni, containing 0.5C and 3.9Nb (wt.%) to favor the crystallization of NbC carbides during solidification were elaborated by casting and subjected to oxidation in air at 1000, 1100 and 1200°C for 46 hours. The results were studied by comparison with recent similar ones obtained with Nb-free ternary alloys issued from the same elaboration protocol and tested in oxidation in the same conditions. In both cases interdendritic script-like shaped NbC carbides were obtained as expected, however not as single carbides in the nickel alloy which also contain chromium carbides. Preliminary exposure to 1200°C having shown a morphological evolution of the NbC, as-cast and preliminarily aged versions of both alloys were tested in oxidation. It appeared first that, whatever the aged state, the cobalt alloy behaved much worse than the nickel one, that the presence of Nb deteriorated the behavior in both cases but at a much lower scale and that aging alloys, which results in NbC fragmentation, was also detrimental for the high temperature oxidation behavior.

INTRODUCTION

Increasing service temperature to enhance both performance and efficiency creates a demand of new alloys superior to the best present superalloys, the single crystalline γ/γ' nickel-based ones [START_REF] Donachie | Superalloys: A Technical Guide[END_REF]. Candidates are cobalt-based particularly rich in heavy elements [START_REF] Gorr | [END_REF] or containing γ' precipitates [3], refractory High Entropy Alloys (HEA) [4], for instance. Another possible way consists in revisiting rather old metallurgical principles as cobalt-based and nickel-based carbide-strengthened chromia-forming by replacing the usual chromium or tantalum carbides by new hardening particles [5]. This was recently done with conventionally cast chromium-rich cobalt-based [6], iron-based [7] and nickel-based [8] alloys strengthened by eutectic HfC carbides. These new model alloys displayed interesting mechanical properties at temperatures as high as 1100°C [9] and even 1200°C [10]. Beside TaC and HfC, other MC carbides may be considered, involving metals not so expensive as tantalum and hafnium in particular. Many of them may crystallize with a mechanically favorable script-like morphology, as demonstrated with recently investigated polycrystalline cobalt-based [11] and nickel-based [12] alloys, with consequently potential creep resistance at very high temperature (e.g. 1200°C). NbC is one of these MC carbides crystallizing with a script-like shape forming an interdendritic eutectic compound in which it is imbricated with the periphery of the solidification dendrites. Niobium is already wellknown in the superalloys' world since it is used as a γ'-former element beside Al and it can be even employed as base element in some refractory metallic alloys [13,14]. In contrast, although the presence of NbC carbides was previously reported several times in literature, it seems that increasing the high temperature creep resistance of superalloys by exploiting NbC carbides as single strengthening particles was not envisaged. This is the reason why one undertook the exploration of the microstructures and high temperature properties of cast cobaltbased and nickel-based polycrystalline model alloys, containing Nb and C in quantities allowing obtaining sufficiently high density of NbC carbides (0.5C and 3.9Nb, in wt.%), as well as high chromium content (25wt.%) of the same level as many chromia-forming superalloys [START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF]. In parallel to the ongoing mechanical characterization at high temperature, the high temperature oxidation behavior of these alloys was studied.

These are the results of these investigations specifically concerning hot oxidation which are presented in this work, thus for two conventionally cast alloys, a Co-25Cr-0.5C-3.9Nb one and a Ni-25Cr-0.5C-3.9Nb one. In order to enlarge the characterization, by examining the behavior of a new alloy and the one of a already used one, the investigations concerned two distinct versions of these alloys: a as-cast one and a preliminarily aged one.

EXPERIMENTAL

The compositions of the elaborated alloys, named CoNb and NiNb, are given in Table 1. These alloys contain 25% of chromium (to favor a chromia-forming behavior) and the same molar quantity of C and Nb to promote the formation of MC type carbides. The alloys were elaborated from solids parts of pure elements (purity >99.9%) under argon. These ones were synthesized in a high frequency induction furnace (CELES, France), with melting and solidification achieved in water-cooled copper crucible. The obtained ingots weighed about 40g. These ingots were cut in two equivalent parts using a Delta Abrasimet saw (Buehler). The first part was studied in its as-cast state and the second part had undergone a heat treatment in order to simulate an aging in service. This dwell at high temperature was carried out in a tubular furnace, at 1200°C during 100h. The ingots were placed in the oven at the room temperature and the heating rate was of 20°C/min. After 100h at high temperature, the alloys were extracted from the furnace and they were cooled in room air. The aged versions of the NiNb and CoNb alloy were named NiNbR and CoNbR respectively. All the half-ingots were cut with Secotom 50 microsaw (Struers) to obtain the samples for the oxidation tests: their dimensions were 6 × 6 × 2 mm 3 . Then, these samples were polished with SiC papers from #400 to #1200 by rounding the edges to avoid local over oxidation and too high growth stresses in the oxide scales. A Setaram TG92 thermobalance (Setaram) was used for the thermogravimetry tests which were performed at 1000°C, 1100°C and 1200°C during 46h. The samples were suspended in chamber through which synthetic dry air (80% of N 2 and 20% of O 2 ) circulated (flow rate of 1.5L/hour). After the oxidation tests, the specimens were cut in two halves. The halves were embedded in CY230 resin mix with a HY956 hardener (Escil). The embedded samples were polished with SiC papers from #400 to #4000 then with a synthetic velor drape enriched with a solution containing alumina particles of 0.02 µm in diameter to have mirror-like state. And to finish, they were ultrasonically cleaned with ethanol. SEM images in back-scattered electrons (BSE) mode, EDS and XRD analyses were carried out on these samples in order to observe and to define their microstructures, the sub-surfaces state and the oxides formed. A JSM-6010LA (Jeol) electronic microscope equipped with an EDS analyzer was used. The images and the analyses were made with an acceleration voltage of 20kV. The X-ray diffractions were realized with a X'Pert Pro (Philips) diffractometer using a Bragg-Brentano (θ-2θ) assembly. The X-rays source was a copper anti-cathode (Kα1, λ=1.54056 Ǻ). The indexation of the obtained diffractograms was performed with a Diffrac AT software using the EVA module working with the PDF2 database.

RESULTS

The microstructures of both alloys in their as-cast and preliminarily aged states

The microstructures of the two alloys before and after the heat treatment are showed in Figure 1. NbC and chromium carbides have precipitated during solidification in the NiNb alloy, while in the CoNb alloy, only NbC carbides are present. These carbides are located in the interdendritic spaces and have a script-like shape. After heat treatment, niobium and chromium carbides have adopted a more circular shape; they get fragmented in order to reduce their matrix/carbide interfacial energy, a classical phenomenon. The obtained microstructures can be compared to the ones of as-cast ternary alloys earlier studied: Ni-25Cr-0.5 [9] and Co-25Cr-0.5C [START_REF] Conrath | [END_REF] (in wt.%), which can play the role of Nb-free references. In these ternary alloys, only chromium carbides were present. The chromium carbide population in the NiNb alloy is less important than in the corresponding reference alloy. In the CoNb alloy there are some chromium carbides in the reference alloy (but a small quantity) whereas no chromium carbide is present in CoNb alloy. In both Nb-containing alloys Nb appeared as being a stronger carbide-former element than chromium, but more in a cobalt base than in a nickel base.

Mass gain kinetics

Thermogravimetry tests were performed at 1000°C, 1100°C and 1200°C during 46h. The mass gain curves versus time plotted from these thermogravimetry results are showed in Figure 2. At all temperatures and for both states (as-cast or aged "R"), the mass gains are lower for the Ni-based alloy than for the Co-based one. Contrarily to the Ni-based alloys, at lower temperatures, the mass gains are not parabolic for the Co-based alloys. Previous works indicated that the mass gains after 50h for the Co-based reference alloy are approximately 5×10 -3 g/cm² at 1100°C [17] and 1.1×10 -2 g/cm² at 1200°C [START_REF] Conrath | [END_REF] and at 1100°C the mass gain curve is not parabolic. Therefore, the total mass gain for the Nb-containing cobalt alloy is greater than for the Nb-free one. Concerning the nickel alloys, according to an earlier work [START_REF] Donachie | Superalloys: A Technical Guide[END_REF], at 1100°C, the obtained mass gain after 46h of oxidation for the Ni-25-0.5C ternary alloy is near 1.3×10 -3 g/cm². For the NiNb alloy (in this work), the mass gain after 46h is 2.6×10 -3 g/cm², which is twice the one of the reference alloy. About the aging, it appears that it has an influence on mass gain for the CoNb alloy, more precisely it is harmful. In contrast, the aging doesn't influence the mass gain for the Ni-base alloy, except at 1000°C, temperature at which the mass gain is more important after the aging.

Since the corrosion products observed over the oxidized sample are never only chromia as this will be said farer in the text, and often a complex mix of several oxides, it was not attempted to value neither mass loss by chromia volatilization nor the kinetic constant associated to this phenomenon despite this one can be expected considering the high level of at least two of the three test temperatures. The mass gain curves were thus only exploited to determine the kinetic constant kp according to the classical equation ( 1):

(d(∆m))/dt=k p /∆m (1) where ∆m in the mass gain per surface unit area. The kp values are displayed in Table 2. They confirm the observations made above by comparing the obtained mass gain curves.

For the Ni-base alloy and for its two states (as-cast or preliminarily aged), the mass gain curves obey a parabolic law, at all temperatures. By looking to both curves and kp values, one finds again that the prior aging influences the mass gain kinetic, especially at 1000°C.

For the Co-base alloy, before and after the aging, at the lowest temperatures, the mass gain curves are not parabolic. At the 1200°C (and 1100°C too for the first aged alloy), the mass gain curves seem parabolic for the CoNb and the CoNbR alloys, this allowing determining values for kp. The obviously fast mass gains suggest that the alloys have not a chromino-forming behavior. This suggestion will be subsequently verified by the study of the formed oxides.

If compared to the data in the bibliography on ternary alloys being considered as Nb-free versions of the alloys studied in this work (see Table 3), the values obtained here for Nb-containing alloys are a little higher for the Nibased alloy. For exemple the kp value for Ni-25Cr-0.5C at 1100°C [9] is 5.5 ×10 -12 g².cm -4 .s -1 . For the Co-base alloy, there is a factor of 10 between the kp values obtained for CoNb and the corresponding Nb-free alloy [START_REF] Conrath | [END_REF]17]. The observations are the same than above with the total mass gain data.

Oxide spallation during cooling

With the results of the thermogravimetric tests, it is possible to know the tendency to the spallation of the oxide layer during the cooling by plotting the mass gain versus the temperature. In advance, it is necessary to deduce the Archimedean buoyant force because this one is function of the temperature. These results for both alloys at all test temperatures are represented in Figure 3. Two parameters can be deduced from this representation: a temperature of spallation start and the lost mass. These data are presented in Table 4. The annealing seems to have no influence on the spallation behavior. At 1000°C, the oxide layers on Co-based alloy put up with the spallation (temperatures of spallation's are about 310°C and the lost masses are the double of the mass gains after 46h of oxidation) whereas the oxide layers on the Ni-based alloy do not spall off at this temperature. At the highest temperatures (1100°C and 1200°C), it is the contrary. There is no spallation for the CoNb and CoNbR alloys but the Ni-based alloy puts up with spallation of oxide layers with as data: spallation's start temperatures of 745°C and 865°C and masses lost during the cooling of about 7×10 -3 g/cm² and 1.7×10 -2 g/cm² at 1100°C and 1200°C respectively. The temperature of start's spallation seems to increase with the temperature of test. In the cases where the oxide layer spalled off, the lost mass is practically the double of mass gain during the oxidation during 46h, therefore the spallation is too important in all cases. During the thermogravimetric tests carried out by Berthod et al. of the Co-25Cr0.5C alloy at 1100°C [17] and 1200°C [START_REF] Conrath | [END_REF], a spallation of oxide layers was noted during the cooling but it is not the case in the present study with the Nb-containing alloy. The addition of Nb seems to improve the adhesion of oxide layer at 1100°C and 1200°C.

To summarize, the exploitation of the cooling part of the mass variation files allowed finding that: -the thickest the chromia scale (due to high stage temperature, or scale also containing other oxides but chromia still being the most present one), the sooner the spallation start during cooling (cases of the Ni-based alloys), and no spallation for thin scales (NiNb and NiNbR, 1000°C) -the highest stage temperature of the Ni-based alloys, the sooner the spallation start spallation a little sooner for the aged Ni-based alloys than for the as-cast ones spallation is very late for the cobalt alloys (rather thick oxide scale) for a cooling after a 1000°C stage, while no spallation occurs after a stage at higher temperature (too thick -and then resistant enough -scales to spall off during cooling).

These observations well meet earlier ones done for more numerous alloys and reported in [18]. In addition, it appeared here that the presence of Nb seemed to be favorable for a better resistance of external oxides against spallation at cooling.

Analysis of the external oxides

To determine the formed oxides during the oxidation, EDS spot analyses and XRD analyses were carried out on the oxides formed on the oxidized samples. The SEM/BSE images of oxide layers are showed, for oxidation temperature of 1000°C, in Figure 4, for 1100°C in Figure 5 and for 1200°C in Figure 6. For the Ni-based alloy, the NbCrO 4 is formed at the matrix/oxide interface, except at 1000°C for as-cast state. Above, a chromia layer is present. And at lower temperature, the outer layer of Ni and Cr spinel oxide is formed. There is too NiO which is present in addition at 1000°C for this alloy after the aging. The formed oxides in the case of the Co-base alloy are very different. For all temperatures and all states, an inner layer of chromia is detected. Then, there is a complex mixture of several oxides. It is a mixture of CoCr 2 O 4 , Nb oxide and sometimes CoO and chromia. And to finish, for a few cases, the important outer layer of CoO oxide is formed. It should be noted that, in some cases, the oxide layer has peeled, as indicated above, therefore other oxides were maybe formed but they could not be detected and visualized in these cases. As the mass gain, the aging has only a little influence on formed oxides.

The Nb-free Ni-based alloy [9] has a chromia-forming behavior at 1100°C since only the Cr 2 O 3 layer is formed hence the mass gain is low. This chromia layer protected the alloy from oxidation by acting as diffusion barrier to O and Ni, which avoided the formation of NiO and NiCr 2 O 4 . This is not the case for the Nb-containing alloy where the formation of NbCrO 4 had obstructed the quick formation of chromia. For the Nb-free Co-based alloy oxidized at 1100°C [17] and 1200°C [START_REF] Conrath | [END_REF], the appearance of oxide layer is alike the Nb-containing alloy. An outer scale of CoO is present. Below, at 1100°C for the ternary alloy, only a chromia layer is formed while for CoNb and CoNbR alloys, CoCr 2 O 4 and Nb oxide in addition of Cr 2 O 3 also are detected. And at 1200°C, below the CoO oxide layer, for the reference alloy, a Co and Cr spinel oxide scale is formed, then there is an inner mixture layer of both oxides (Cr 2 O 3 + CoCr 2 O 4 ). And for Nb-containing alloy, it is almost similar with only the mix of the both oxides with in addition of Nb oxide.

Thicknesses of the oxide scales

The SEM images of oxide layers were used to measure their thicknesses in three zones for each alloy. The averages and standard deviations of these measures are represented in Figure 7. The points which were surrounded mean that the oxide layers had spalled off during the cooling and therefore the measured thicknesses are maybe lower than the real ones. It can be observed that, as the previous results, the heat treatment has only a little influence on this parameter and its influence is random. For the Ni-based alloy, the oxide scales have spalled off at higher temperatures, thus no conclusion is possible. For the CoNb and CoNbR alloys, the oxide layers are thick especially at higher temperature, it is due at the formation of CoO and spinel oxide which are porous oxides with a lot of defaults (gaps). In regarding the SEM images of oxide layers of ternary alloy in literature, the thicknesses of the oxide scales are about 20 µm and about 320 µm for the oxidized sample at 1100°C [17] and 1200°C [START_REF] Conrath | [END_REF] respectively but the oxide layers have spalled for this ternary alloy. Despite the spallation, the oxide layer at 1200°C is thick because CoO and spinel oxide are also formed.

Characterization of the subsurface affected by oxidation

EDS spot analyses were performed at sub-surfaces of samples to establish concentration profiles as seen in Figure 8. Obviously, not only a Cr-depleted zone is developed but also a NbC-free zone and a Cr 23 C 6 -free zone. The depths of these zones are measured and reported on the graphs which are showed in Figure 9. For the Ni-based alloy, the NbC-free zones are thinner than the chromium carbide-free zones. Then, two remarks can be made. Firstly, the oxidation is more important for the Co-based alloy than for the Ni-based one but all depths are higher for this last one. Secondly, as the above results, the depths of Cr-depleted zones and carbides-free zones obtained before and after the aging are a little different. These measured results before the heat treatment are little lower than these obtained ones after the aging whereas the oxidation of the aged samples was slightly higher than of as-cast samples.

The behavior of Nb in the subsurface during oxidation appears to be curious. Indeed, in contrast with chromium here (well-known phenomenon in the case of Cr), or tantalum in earlier studies (for example [5]), which both decrease in content when approaching the oxide/alloy interface, the Nb content obviously increases when coming close to the extreme surface of the alloy. This was previously noticed for a Co(bal.)-8Ni-30Cr-0.4C-3.3Nb (wt.%) alloy oxidized for 20 hours at 1300°C [5], as well as for the 625 Ni-based alloy after much longer oxidation at temperatures ranging from 700 and 1000°C [19,20].

DISCUSSION

Niobium introduction in well-known cobalt-chromium and nickel-chromium bases successfully led to the wished microstructure: dendritic matrix with significant interdendritic script-like NbC carbides mixed with the dendrites periphery. This is particularly true for the cobalt alloy for which all carbon was involved in NbC carbides, with furthermore keeping of this state during exposure at 1200°C for a rather long time (100 hours). This is less true for the nickel alloy in which chromium carbides co-exist with NbC. However the latter carbides are script-like shaped and it does not disappear during long exposure at elevated temperature. Unfortunately, in both cases the NbC carbide network becomes rather rapidly fragmented. Beside the detrimental effect that one can anticipate for the bulk mechanical properties at high temperature, this morphological evolution of the NbC carbides may influence the high temperature oxidation behavior of the alloys.

The thermogravimetry tests carried out here at three levels of high temperature first showed that the behavior of both alloys was seemingly deteriorated by comparison with Nb-free alloys of same compositions concerning Co or Ni, Cr and C earlier studied in the same conditions. Indeed, concerning the cobalt-based alloys, the Nb-free ternary versions of which were already not very oxidation-resistant, one noticed faster oxidation and very limited parabolic part of the mass gain kinetic. This led shortly to catastrophic oxidation, not really limited by the presence of external oxides the complexity and lack of quality of which were particularly great. This led to much thicker oxide scales (fast oxidation) but to less deep subsurface zones impoverished in chromium and in carbides than the nickel alloy. Indeed, when the cobalt alloy suffered general oxidation (oxidation of not only Cr but also the other metallic elements) instead selective oxidation of mainly chromium, the oxidation of the nickel alloy principally concerned chromium even if nickel oxide or spinel oxides involving Nb and Cr also formed. One finds again the same hierarchy as between the ternary Nb-free cobalt-based and the Nb-free nickel-based alloys. The presence of niobium induced an acceleration of oxidation in both cases, more sensible for the cobalt alloy than for the nickel one, but the morphology of NbC also influenced the global oxidation behavior, in a detrimental way. Thus, preliminarily aging the alloys for fragmenting the NbC interdendritic carbides to limit the outward Nb diffusion and reduce the detrimental effect of this element on the oxidation front is not a good idea: not only the high temperature mechanical properties may be lowered but the oxidation resistance too. Concerning the oxide spallation behavior the effect of niobium is rather limited, the main factor remaining the thickness of the external oxide scale. Indeed thin scales (i.e. when chromia is the main oxide present and the temperature low, 1000°C) better resist spallation at cooling while spallation appears when the oxide is thicker (chromia formed during 46h at higher temperatures, multi-constituted oxides). Inversely, when the oxides scales are thick enough to constitute veritable oxide shells spallation during cooling does not occur.

Concerning the chemical and microstructure changes due to high temperature oxidation in these Nb-containing alloys, the behavior of Nb was not expected. Although the selectively oxidized elements tends to present a local content decreasing when coming close to the oxidation front, an increase was noticed here instead. This phenomenon was reported in some recent works [19,20] dealing with the high temperature oxidation of a commercial Nb-containing nickel-based superalloy, in which it was explained as be caused by uphill diffusion resulting from negative niobium activity gradient in the chromium depletion zone appearing in the subsurface. Similar effect of the local decrease in chromium was found for carbon in the same alloy.

Despite that the alloys studied here were rather different from the 625 alloy and that the range of temperatures of interest was higher here, the same niobium uphill diffusion was observed. Concerning carbon, it seems that it has diffused towards the oxidation front to be oxidized and to leave the alloys as gaseous specie. This was demonstrated in earlier works [21,22] for similar (but Nb-free) alloys for temperatures higher than 1000°C in atmospheric air. Back diffusion of carbon was reported only for temperatures equal to or lower than 1000°C, in the same articles [21,22].. This oxidation-induced decrease in carbon activity in the subsurface is known as being responsible of the development of a carbide-free zone from the scale/alloy interface.

CONCLUSION

As this can be expected the addition of niobium to form MC carbides of a new nature tends to be detrimental for the behavior of cast chromium-rich cobalt-based and nickel-based alloys in oxidation at high temperature, as this was verified in the present work in the [1000, 1200°C]-range. However the effect is not as strong as one may think, the main effect remaining the one of the base element (cobalt or nickel), even if it must be taken into account. The main problem with the NbC carbides is its fragmentation phenomenon and its deleterious effect on the mechanical properties, which shows the importance of metallurgical studies to try stabilizing more their geometry at elevated temperature for long times (e.g. by finding chemical adjustments to reduce the matrix-NbC interfacial energy). Concerning hot oxidation, increase in chromium content or subsurface Cr-enrichment by pack-cementation may be efficient solutions. To finish one can highlight that the presence of Nb with rather high content and with the form of carbides led to complex corrosion products interesting to study for describing mechanisms. Further work may be envisaged by carrying out oxidation over different durations or XRD-followed oxidation tests, for instance.

FIGURE 1 .

 1 FIGURE 1. SEM images in BSE mode of the alloys: a) NiNb, b) CoNb, c) NiNbR and d) CoNbR.

FIGURE 2 .

 2 FIGURE 2. Mass gain versus time plotted of NiNb, NiNbR, CoNb and CoNbR alloys at 1000°C (a), 1100°C (b) and 1200°C (c) during 46h.

FIGURE 3 .FIGURE 5 .FIGURE 6 .

 356 FIGURE 3. Mass gain curves versus temperature of thermogravimetric tests for NiNb, NiNbR, CoNb and CoNbR alloys at 1000°C, 1100°C and 1200°C during 46h.

FIGURE 7 .

 7 FIGURE 7. Thicknesses of oxide layers of both alloys in as-cast and aged states after the oxidation at 1000°C, 1100°C and 1200°C.

FIGURE 8 .

 8 FIGURE 8. SEM/BSE image and concentration profiles of Cr, Ni and Ni at the sub-surface of oxidized NiNbR alloy at 1200°C.

FIGURE 9 .

 9 FIGURE 9. Depths of Cr-depleted and carbides-free zones created by oxidation at sub-surface of samples.

TABLE 1 .

 1 Compositions of CoNb and NiNb elaborated alloys.

	wt.%	Co	Ni	Cr	C	Nb
	CoNb	Bal.	/	25	0.5	3.9
	NiNb	/	Bal.	25	0.5	3.9

TABLE 2 .

 2 Values of the linear constant k l , of the parabolic constant k p and of the constant chromia volatilization constant k v , the two later ones being derived from the m × dm/dt = f(m) plots.

		1000°C	1100°C	1200°C
	Alloys	k p ×10 -12	k p ×10 -12	k p ×10 -12
		g².cm -4. s -1	g².cm -4. s -1	g².cm -4. s -1
	NiNb	2.4	18	90
	NiNbR	7.2	20	105
	CoNb	Non-parabolic curve	Non-parabolic curve	6960
	CoNbR	Non-parabolic curve	4050	7930

TABLE 3 .

 3 Comparison of kinetic constants obtained for Nb-containing alloys in this study and those obtained for the ternary alloys in recent works.

		1100°C	1200°C
	Alloys	k p ×10 12	k p ×10 12
		g².cm -4. s -1	g².cm -4. s -1
	NiNb	18	90
	Ni-25Cr-0.5C [9]	5.5	/
	CoNb	Non-parabolic curve	6960
	Co-25Cr-0.5C [17, 16]	15	59

TABLE 4 .

 4 Spallation's start temperatures of oxide layers and lost masses during the cooling of thermogravimetric samples of all alloys at all test temperatures.

		1000°C	1100°C	1200°C
		Spallation's	Lost mass	Spallation's	Lost mass	Spallation's	Lost mass
	Alloys	start	during the	start	during the	start	during the
		temperature	cooling	temperature	cooling	temperature	cooling
		°C	g/cm²	°C	g/cm²	°C	g/cm²
	NiNb	No spallation	≈ 740	6.8×10 -3	≈ 860	1.7×10 -2
	NiNbR	No spallation	≈ 750	7.4×10 -3	≈ 870	1.7×10 -2
	CoNb	≈ 320	1.0×10 -2	No spallation	No spallation
	CoNbR	≈ 300	9.5×10 -3	No spallation	No spallation


Patrice Berthod
email: patrice.berthod@centraliens-lille.org

Lydia Toubal
email: lydia.toubal8@etu.univ-lorraine.fr

Effect of the Ni/Co ratio on the high temperature oxidation behavior of {Ni&Co}-based {25Cr-0.4C-1.6Ti, wt.%}containing cast alloys

Keywords: Nickel-based alloys, Cobalt-based alloys, Chromium carbides, Titanium carbides, High temperature oxidation

Six alloys, based on Ni and/or Co with Co/(Ni+Co) ratio varying from 0 to 1, and containing 25wt.%Cr, 0.4wt.%C and 1.6wt.%Ti, were synthesized by conventional foundry way. Their as-cast microstructures were controlled by electron microscopy. Parts of these alloys, properly prepared on surface, were then exposed to atmospheric air during 20 hours at 1400K. The obtained surface states and sub-surfaces were characterized by optical observations and cross-sectional examinations by electron microscopy. The nickel-richest alloys contained more chromium carbides than titanium carbides while this was the opposite for the cobalt-richest alloys. This confirm the results of a recent study. It was furthermore found here that the alloys with many chromium carbides and few TiC were much better in oxidation at high temperature than the ones with many TiC and few chromium carbides. By considering the apparent involvement of titanium in the oxidation process it seems that the major influent parameter is directly the base element, independently on the nature of the obtained primary carbides.

INTRODUCTION

Despite the supremacy of the γ/γ' single crystals the polycrystalline superalloys issued from classical foundry process keep great interest by allowing obtaining geometrically complex pieces for high temperature applications [START_REF] Donachie | Superalloys: A Technical Guide[END_REF]. To be strong enough when used under stresses at elevated temperatures these alloys must be reinforced at the matrix level (e.g. by heavy atoms in solid solution or particles precipitation) as well as at the grain boundary level (e.g. by primary carbides). The most commonly used carbides for grain boundary strengthening are chromium carbides (containing, or not, other metallic elements) and the MC-type carbides. When the working temperature is higher than 1000°C, MC carbides must be preferred for their general better thermodynamic stability. Among the later ones, tantalum carbides are the most popular ones. TaC carbides are rather dense carbides (13.96g/cm 3 [START_REF] Shaffer | Plenum Press Handbooks of High Temperature Materials. Part 1: Materials Index[END_REF]), compared to TiC (4.85g/cm 3 [START_REF] Shaffer | Plenum Press Handbooks of High Temperature Materials. Part 1: Materials Index[END_REF]). With, in addition, a lower cost of Ti by comparison to Ta, using TiC instead TaC for the reinforcement of cast polycrystalline superalloys, appears to be envisaged.

Ti is a common element in superalloys [START_REF] Donachie | Superalloys: A Technical Guide[END_REF], in which it may participate to the γ' composition [START_REF] Donachie | Superalloys: A Technical Guide[END_REF] or complex MC carbides in association with other MC-forming metals (e.g. Haynes 282 superalloy [START_REF] Matysiak | [END_REF]). It is also often present in alloys employed for medical applications [e.g. cobalt-titanium alloys [4,5]. It was recently attempted to promote the formation of TiC carbides in chromia-forming quaternary cast alloys, based either on nickel [6] or on cobalt [7], by adding rather high quantities of Ti and of C with Ti/C atomic ratio equal to 1. TiC carbides were successfully obtained in the {Co-25wt.%Cr}-based alloys [7] in which they were almost the single carbides present. In contrast, the {Ni-25wt.%Cr}-based alloys were almost free of TiC with the presence of chromium carbides only in their microstructures [6]. By substituting more or less nickel by cobalt in {M-25Cr-0.4C-1.6Ti}-type alloys (M=Ni and/or Co) one thought that it was maybe possible to obtain a greater fraction of the carbides being TiC. This was attempted [START_REF]Report of trainee period, 1 st year of Master of Chemistry[END_REF] and, indeed it was observed that adding Co leads to an increasing part of TiC in the microstructure when the Co content increases. However, one can guess that the high temperature oxidation behavior of the Corichest alloys of the series may be deteriorated. A study of the effect of an increasing part of cobalt in the alloys on their behavior in air at 1400K was then undertaken.

EXPERIMENTAL

The chemical compositions of the six alloys considered in this study are displayed in Table 1. All alloys contain 25wt.%Cr for their resistance against high temperature oxidation and corrosion, 0.4wt.%C to promote the formation of carbides in quantity high enough to significantly increase the mechanical resistance of the alloys at high temperature, and 1.6wt.%Ti to have as many titanium atoms as carbon atoms. After having subtracted to 100% the sum of the weight contents in Cr, Ti and C, one have shared in six proportions the rest between Ni and Co.

For each alloy, 40 grams of mix of pure metals (Alfa Aesar, purity higher than 99.9wt.%) were prepared and put in the copper crucible of the High Frequency induction furnace (CELES, France, 50kW). A silica tube was placed around the crucible to isolate the metallic parts from laboratory air. Pumping was performed out until reaching 5×10 - 2 mbars in the fusion chamber and 800mbars of pure argon was injected inside. These two operations were repeated three times. Heating and melting were realized in 300mbar of Ar, with about 2 minutes of heating until obtaining the molten state, 3 minutes of isothermal stage to ensure total chemical homogenization and about ten minutes for solidification and cooling down to ambient temperature.

The obtained ingots presented an ovoid shape. They were divided in four equal quarters, by sawing them with a metallographic cutter (Delta Abrasimet of Buehler). One of the four parts was divided in two parts and metallographically prepared to characterize the as-cast microstructure. Another one was kept for the oxidation test. This one was ground all around with 1200-SiC paper with smoothing of the edges. Thereafter it was washed and dried. The six prepared oxidation samples were placed in six alumina nacelles with only three contact points with these later ones. These nacelles were placed in a muffle furnace (HFL 16/17 model, Nabertherm, 12kW, 1700°C max).

After heating at 20K/min, the samples were maintained at 1400K (1127°C) during 20 hours, in air. After the isothermal oxidation stage the samples slowly cooled in the furnace off, until reaching about 650°C. They were then exited from the furnace (air-cooling). The oxidized samples were photographed, then prepared for the crosssectional observations. The start of this preparation was to cathodically deposit gold all around to give the necessary electrical conductivity to the outer side of the oxides formed. Each gold-coated oxidized sample was then immersed in a Watt's bath heated at 50°C. It was then cathodically polarized to reduce the Ni 2+ ions of the solution in metallic nickel. After about two hours at 0.016A/cm 2 , a sufficiently thick nickel shell was obtained all around the sample to ensure good protection of the external oxide scales during the subsequent cutting. The Ni-covered samples were cut in two parts which were thereafter embedded in cold resin (ESCIL), as the samples for the as-cast microstructure examinations. After about two hours in a 40°C stove, the resin was totally rigidified. The samples were then extracted from the mold. All the embedded samples, as-cast or oxidized, were ground with SiC papers from 240 to 1200-grit. After ultrasonic cleaning in water the samples were polished with 1µm hard particles sprayed on textile disk.

The mirror-like metallic top of the embedded samples were examined using a JEOL JSM-6010LA Scanning Electron Microscope (SEM), in Back Scattered Mode (BSE), with a 20kV acceleration voltage. The chemical compositions of the as-cast alloys were controlled by ×250 full frame measurements on three randomly chosen areas. Spot EDS analysis was also carried out in the matrix (center of dendrites) in three locations too, as well as on the seen particles. The same measurements were done again in the centers of the oxidized samples.

The oxides formed, externally or internally, were observed in BSE mode, and identified by spot EDS analysis. For each oxidized sample, a concentration profile was carried out from the scale/alloy interface, perpendicularly to this one, towards the not oxidation-affected part of the alloy.

RESULTS

Chemical compositions and as-cast microstructures of the six obtained alloys

The chemical compositions really obtained are displayed in Table 2. One can see that all contents are well respected. One can only notice that the chromium ones tend to be slightly too high for many alloy. The carbon content, which cannot be measured with the single available technique (SEM/EDS) because of its low atomic mass and its too low content in the alloy, can be expected as being well respected. Indeed, the applied protocol is exactly the same as many times used for similar alloys. In addition it is periodically controlled by spark spectrometry on samples of the same category, and each time validated. By looking to the density of the primary carbides obtained in the microstructures (Figure 1 and Figure 2), which is typical of many 0.4wt.%-containing alloys of the same family, the good respect of the targeted carbon content is confirmed.

The microstructures of the six alloys in their as-cast states are illustrated by SEM/BSE micrographs taken with two magnifications (general view and detailed view). All alloys are unsurprisingly dendritic, with three possible types of carbides. As suggested by the more or less high local concentrations, either in carbon and in chromium, or in carbon and titanium, the as-cast Co-poorest (i.e. Ni-richest) alloys contain almost exclusively chromium carbides (Figure 1). According to the spot analyses carried out on the coarsest carbides which were found, the atomic Cr/C ratio is close to 2 for the Ni0/5Co, Ni1/5Co (and seemingly Ni2/5Co too) alloys. This suggests that these chromium carbides are of the Cr 7 C 3 stoichiometry, which is confirmed by the usual acicular shape of the carbide part of the interdendritic eutectic compound. In the Ni2/5Co alloy rare fine Cr 23 C 6 are seemingly also present. They can be seen thanks to their paler tint in BSE mode, but they were too rare and fine to be identified as previously by qualitative EDS results. However, particles with a similar tint are a little more numerous in the Ni3/5Co, Ni4/5Co and Ni5/5Co alloys (Figure 2). Nevertheless they are really not the main carbide type present since TiC carbides are present in great quantity, especially in the two Co-richest alloys. In the Ni5/5Co alloy the interdendritic carbide network is almost exclusively made of TiC carbides, which are obviously of a eutectic nature. Indeed they are script-like carbides mixed with the periphery of the matrix dendrites. So, when the Co/Ni ratio rises from 0 to 1, the acicularshaped eutectic Cr 7 C 3 are progressively replaced by the Chinese script eutectic TiC carbides, with simultaneously appearance of more Cr 23 C 6 carbides.

Results of oxidation

A quarter of each ingot, with preliminarily surface preparation, was subjected to a 20 hours exposure at 1400K in atmospheric air. Macrographs of the oxidized samples are presented in Figure 3 for the Co-poorest alloys and in Figure 4 for the Co-richest ones. All ingot quarter has obviously kept its oxide scale all around. However, one can see that spallation occurred, but this was rather limited. The metal is visible here and there, but the cumulative surface fraction of denuded alloy varies only between 95.8% to more than 99.5%, as this was measured by image analysis (tool of the Adobe Photoshop CS software). Furthermore, one can say that the surface fraction of denuded alloy tends to be a little greater for the Co-poorest alloys than for the Co-richest ones.

When observed on cross-sections, the three Co-poorest alloys are covered almost exclusively by Cr 2 O 3 (Figure 5). This chromia scale is very compact for the Ni0/5Co and Ni1/5Co alloys but it becomes a little porous for the Ni2/5Co alloy. This obvious chromia-forming behavior demonstrated by the three Co-poorest alloys is lost, at least locally, for the three Co-richest alloys (Figure 6). Indeed, beside some areas covered by chromia, one finds here and there (Co,Ni)O or CoO oxides juxtaposed or superposed with chromia, with additionally presence of different spinel varieties (CoCr 2 O 4 , TiCr 2 O 4 ). In all cases TiO 2 is also present: as an outer thin scale above chromia for the Copoorest alloys, and as internal TiO 2 in the subsurface for all alloys. For the Co-richest alloys, oxidation becomes catastrophic in some location, with a penetration inwards the alloys. Generalized oxidation was obviously near to occur.

It was attempted to measure the thickness of the oxide scales on the cross-sections. Table 3 (first results column) displays the values of the averages and standard deviations. The oxide (chromia) is the lowest for the Co-poorest alloys, while the average values of thickness is the greatest for the Co-richest ones. One can note the great values of standard deviation resulting from the existence of many points of catastrophic oxidation for the later alloys.

Carbides, chromium carbides as well as titanium carbides, disappeared from the scale/alloy interface. This phenomenon, classical for alloys containing primary carbides, is here more extended for the Co-richest alloys, due to the obvious faster oxidation (Table 3, second results column). Nevertheless, because of the points of penetrating oxidation, the local carbide-free depths can be lower, and taking them into account leads to lower average values and great values of standard deviation.

The carbide-free depths more or less correspond to the chromium concentration profiles (Figure 7 for the full scale profiles of the major elements Cr, Ni and Co of three of the six alloys), more precisely to the part with the greatest gradient (Table 4, second results column). The inner chromium depletion, with a Cr gradient a little lower, is included in the part of the alloy in which carbides are still here. Obviously, the disappearance of the interdendritic carbides induced little but existent difficulty for Cr diffusion towards the oxidation front. In this field, by noticing that the chromium gradients seem to be a little higher for the Co-richest alloys than for the Co-poorest ones, the diffusion of chromium may be less easy for the formers than for the latters. One can additionally note that the minimum Cr content in extreme surface was about the same (between 10 and 14 wt.%Cr) for all alloys. However the minimal chromium content seems to be maybe a little higher for the Co-poorest alloys than for the Co-richest alloys (Table 4, first results column).

Concerning titanium, depletions exist too for all alloys but the profile is often perturbed by the presence of internal titanium oxides or of TiC carbides. Ti seems having itself diffused towards the oxidation front and this explains the presence of titanium oxides and of spinels involving Ti as constituents of the external oxide scales.

DISCUSSION

As this was recently found concerning the dependence, on the nature of the base element, of the type of the primary carbides issued from solidification [START_REF]Report of trainee period, 1 st year of Master of Chemistry[END_REF], the Ni0/5Co (Ni-based in fact) alloy contains almost exclusively chromium carbides while the Ni5/5Co (Co-based in fact) alloy contains here too almost exclusively TiC carbides (thanks to the atomic equivalence between Ti and C). These chromium carbides are Cr 7 C 3 , as is to say the high temperature type of chromium carbides usually found in {20 to 30wt.%Cr}-containing alloys in presence of about 0.5wt.%C. After solidification the cooling was probably too fast to allow the transformation of the high temperature Cr 7 C 3 into the low temperature Cr 23 C 6 . These primary chromium carbides obviously solidified at the end of solidification, by forming a eutectic compound in the interdendritic spaces. This compound is composed of acicular carbides mixed with the periphery of the dendrites. The titanium carbides existing in the as-cast microstructure of the cobalt-based alloy (Ni5/5Co) are surely also eutectic carbides. They are themselves mixed with the periphery of the dendrites of matrix. Their morphology is different: Chinese script. A look to the bulk of the samples after their oxidation tests shows that the chromium carbides significantly evolved, more in term of morphology than in term of surface or volume fraction: they became rounder and more compact. This results in the loss of their matriximbricated character and probably of the reinforcement they may bring to the alloy at high temperature. In contrast, the TiC carbides kept their script-like morphology, even after 20 hours spent at the rather high temperature of 1400K (1127°C). Their presence and morphological stability let think to a good persistence of the strengthening effect at high temperature that they may bring to an alloy which is additionally already better than a nickel alloy in this field. If the Ni0/5Co cobalt-based alloy owns advantages to be mechanically resistant at high temperature, it is, in contrast, the worst in term of oxidation resistance: mainly cobalt oxides, spinels of different types, only a little chromia, numerous points of fast oxidation penetration, many voids in subsurface... The Ni0/5Co nickel-based alloys is the best in this field, with a protective chromia scale obviously of high quality (totally covering the sample, adherent to the metallic substrate, without too numerous and great porosities).

Between these two extreme alloys there are four other ones, with varying Co/(Ni+Co) ratios. The transition from chromium carbides to titanium carbides appears as being progressive, with transitorily low temperature Cr 23 C 6 chromium carbides co-existing with TiC. Consequently the resistance of the carbides population against morphological evolution at high temperature increases, with favorable consequences for the sustainability of the high temperature strength. In the same time, the initially very good high temperature oxidation degenerates to become almost catastrophic. The transition from the degenerating good chromia-forming behavior to the start and generalization of the catastrophic oxidation, for the conditions in this work (atmospheric air, 20 hours, 1127°C), seems to occur between the Ni2/5Co alloy and the Ni3/5Co one, as is to say when the alloy series finishes to be nickel-based to start to be cobalt-based.

The participation of titanium in the general oxidation behavior was of several types. As earlier reported [START_REF] Al-Hatab | [END_REF], this very reactive element was here internally oxidized in the subsurface of all alloys. This internal oxidation of titanium took place from its solid solution position in the case of the Co-poorest nickel-based alloys, as well as from the primary TiC carbides in the case of the Co-richest alloys. The formed titanium oxides are TiO 2 or spinel TiCr 2 O 4 , as suggested by the EDS spot analyses. In many cases an outer TiO 2 was also found above the chromia scale (mainly for the nickel-based of this study). The limited spallation affected it first, with as result only isolated parts of TiO 2 along the unaffected adherent chromia scale. This layout is what is commonly reported for chromia-forming alloys containing titanium [10][11][12][13][14]. But titanium oxides were never really found mixed with the other oxides in the cases of the parts of samples catastrophically oxidized. So Ti, added initially to form strengthening carbides, took part to the oxidation process. By taking into account the rather small quantity of titanium oxides formed, for the chromiaforming nickel-based alloys as well as for the not well oxidation resistant cobalt-based ones, one may think that the presence of this element, sometimes claimed as being deleterious for the high temperature oxidation behavior [15], did not really influence the behaviors of the six alloys. The explanations of the differences of behavior of the well oxidation resistant Ni0/5Co to Ni/2/5Co alloys and the badly resistant Ni3/5Co to Ni5/5Co alloys is certainly more to be found in the intrinsic characteristics of the matrixes [START_REF] Young | High Temperature Oxidation and Corrosion of Metals[END_REF]. In the ones rich in nickel, chromium easily diffuses towards the oxidation front. It is furthermore helped by the presence of chromium carbides which may be considered as chromium reservoirs available in the interdendritic spaces. These reservoirs Cr deliver particularly efficiently since the interdendritic spaces act as fast diffusion paths for Cr. On the contrary, Cr diffusion is much slower in the cobalt-rich alloys: interdendritic spaces occupied by TiC rather than by chromium carbides, and more difficult diffusion through the matrixes rich in cobalt. One can be easily convinced by considering the evolution of the Cr concentration gradients and the chromium contents in extreme surface at the end of experiment (Table 4). But this can be also additionally illustrated numerically by giving the following values of chromium diffusion coefficients available in literature for Co-Cr and Ni-Cr model alloys (both at 1100°C): 2 × 10 -11 cm²/s for Cr diffusion in Co-14Cr [START_REF] Weeton | Diffusion of chromium in alpha cobalt-chromium solid solutions[END_REF] and 26 × 10 -11 cm²/s for Cr diffusion in Ni-25Cr [START_REF] Berthod | [END_REF].

CONCLUSION

Thus, it was here found again that the progressive substitution of nickel by cobalt favored the formation of the morphologically stable script-like TiC carbides at the expense of the less stable chromium carbides. Unfortunately this had also obviously as consequence to harm the alloy's behavior in oxidation at high temperature. It appeared that it was mainly the mean used for promoting the TiC presence, i.e. an increasing cobalt part, than the result (TiC rather than chromium carbides) which is responsible of this decrease in high temperature resistance. Fortunately it is still possible to improve the oxidation resistance, for example by enriching in chromium the TiC-containing alloys (rich in cobalt), either at the level of the whole composition (25 → 30wt.%Cr, or more) or locally (surface and subsurface enrichment in Cr, by pack cementation for example. This remains to be tested in future work.

3 FIGURE 1 .

 31 FIGURE 1. As-cast microstructure of the three Co-poorest alloys (SEM/BSE; top: ×250; bottom: ×1000)

FIGURE 2 . 9 Ni0FIGURE 3 . 10 Ni4FIGURE 4 .FIGURE 5 .

 2931045 FIGURE 2. As-cast microstructure of the three Co-richest alloys (SEM/BSE; top: ×250; bottom: ×1000)

FIGURE 6

 6 FIGURE 6 Cross-sectional examination of the oxide scales and subsurfaces of the three Co-richest alloys (SEM/BSE; top: ×250; bottom: ×1000).

TABLE 1 .

 1 Targeted compositions for the six alloys of the study (weight contents).

	Alloys	Ni	Co	Cr	Ti	C
	"Ni0/5Co"	Bal.	0	25	1.6	0.4
	"Ni1/5Co"	Bal.	14.6	25	1.6	0.4
	"Ni2/5Co"	Bal.	29.2	25	1.6	0.4
	"Ni3/5Co"	29.2	Bal.	25	1.6	0.4
	"Ni4/5Co"	14.6	Bal.	25	1.6	0.4
	"Ni5/5Co"	0	Bal.	25	1.6	0.4

TABLE 2 .

 2 Obtained chemical compositions (in wt.%); average and standard deviation of three full-frame SEM/EDS measurements carried out at the ×250 magnification)

	Alloys	Ni	Co	Cr	Ti	C
	"Ni0/5Co"	Bal.	/	25.4 ±0.2	1.6 ±0.1	"0.4"*
	"Ni1/5Co"	Bal.	14.5 ±0.1	25.6 ±0.2	1.6 ±0.1	"0.4"*
	"Ni2/5Co"	Bal.	29.1 ±0.3	25.3 ±0.2	1.5 ±0.1	"0.4"*
	"Ni3/5Co"	29.4 ±0.1	Bal.	25.3 ±0.3	1.6 ±0.1	"0.4"*
	"Ni4/5Co"	14.8 ±0.1	Bal.	25.6 ±0.3	1.7 ±0.1	"0.4"*
	"Ni5/5Co"	/	Bal.	25.7 ±0.1	1.6 ±0.1	"0.4"*

*: carbon contents not controlled by EDS but assumed to be well respected according to the practical knowledge for the elaboration protocol for many alloys before, and confirmed by the obtained density of carbides

TABLE 3 .

 3 Obtained chemical compositions (in wt.%); average and standard deviation of three full-frame SEM/EDS measurements carried out at the ×250 magnification)

	Alloys	Oxide scale thickness (µm)	Carbides-free zone depth (µm)
	"Ni0/5Co"	18.1 ±0.5	26.1 ±1.0
	"Ni1/5Co"	21.1 ±1.0	37.8 ±3.5
	"Ni2/5Co"	16.1 ±2.6	28.9 ±3.5
	"Ni3/5Co"	3-10µm *	45.6 ±4.2
	"Ni4/5Co"	23.9 ±16.8	35.6 ±9.5
	"Ni5/5Co"	34.4 ±21.5	27.8 ±4.8

*: probably minimized because one suspects that an external part of the scale was lost by spallation

TABLE 4 .

 4 Average and standard deviation for the chromium content in alloy close to the external scale/alloy interface (three measurements); assessment of the chromium gradients from the concentration profiles

	Alloys	Chromium content in	Cr concentration gradients from the
		extreme surface (wt.%)	scale/alloy interface (wt.%/µm)
	"Ni0/5Co"	13.6 ±0.6	0.15 over ≈ 65µm
	"Ni1/5Co"	13.4 ±0.4	0.30 over ≈ 35µm, and 0.08 deeper
	"Ni2/5Co"	12.1 ±0.9	0.33 over ≈ 40µm, and 0.12 deeper**
	"Ni3/5Co"	12.5 ±0.3	0.23 over ≈ 45µm, and 0.11 deeper
	"Ni4/5Co"	10.3 ±4.1*	0.23 over ≈ 50µm
	"Ni5/5Co"	11.0 ±1.6	0.25 over ≈ 50µm

*: great variations (e.g. between 5.7 and 13.5wt.%Cr); **: illustration in Figure

7


Md Rasedujjaman

Guillaume Maire

Hugues Giovannini

Patrick C Chaumet

Kamal Belkebir

Anne Sentenac

Philippe Robert

3D optical microscopy for quantifying T lymphocyte activation

Acknowledgment

Abstract

The tomographic diffractive microscope (TDM) can be implemented in either transmission configuration or reflection configuration. TDM in reflection configuration has higher Fourier spatial frequency data along the optical-axis of the microscope in comparison to the transmission configuration and also reflective samples can be imaged. We have recently exploited the specific features of such a configuration. This optical tomographic microscope coupled to sophisticated inversion schemes could be a good candidate for detecting the immunological synapse of T lymphocyte activation. Presently, no technique permits to perform a fast detection of T lymphocyte activation at an early stage which is very promising in medical diagnosis applications. In doing so we have first considered polystyrene bead (comparable to the size of T-cell) in water medium and detected the interface. This same experiment could be used for detecting the immunological synapse.

Keywords: Reflection tomography, T lymphocyte, Immunological synapse

Experimental setup : reflection diffraction tomography

The signal as reflected by the sample E s and the referecne wave E ref interfere and the camera record the hologram.

The signal E s (both phase and intensity) is then seperated using 2D Fourier transform from the hologram in k-space.

I intf (r )= | E s (r ) | 2 + | E ref (r ) | 2 + E s E ref * + E s * E ref (a) (d) (b) (c)

Immunological synapse

To reconstruct a 3-D RI tomograms, multiple 2-D holograms of a cell are measured at various angles of illuminations using an interferometric microscope in transmission.

Fig. 1 .

 1 Fig. 1. Block diagram of the TDM in reflection configuration : M, rotating mirror; BE, beam expander; D, diaphragm; L1 ,tube lens; L2,...,5 lenses; BS1,...,3 , beamsplitters.

Fig. 5 .

 5 Fig. 5. Antigen presenting cell (APC) sitting on top of glass substrate: plane wave illumination at various angle

Fig. 4

 4 Fig. 4. 3-D Refractive Index (RI) tomograms of a T lymphocyte cell from mice peripheral blood

Fig. 2 .

 2 Fig. 2. (a) SEM image of a resin star, (b) Standard dark field image, (c) Tomographic reconstruction (d) Axial resolution (improved)


Vincent Roggerone

Marc Rébillat

Étienne Corteel

Parallel Hammerstein Models Identification using Sine Sweeps and the Welch Method

Keywords: Nonlinear system identification, Input and excitation design, Nonlinear system identification, Input and excitation design, Nonlinear system identification, Input and excitation design, Nonlinear system identification, Input and excitation design

Linearity is a common assumption for many real life systems. But in many cases, the nonlinear behavior of systems cannot be ignored and has to be modeled and estimated. Among the various classes of nonlinear models present in the literature, Parallel Hammertein Models (PHM) are interesting as they are at the same time easy to understand as well as to estimate when using exponential sine sweeps (ESS) based methods. However, the classical EESbased estimation procedure for PHM relies on a very specific input signal (ESS), which limits its use in practice. A method is proposed here based on the Welch method that allows for PHM estimation with arbitrary sine sweeps (ASS) which are a much broader class of input signals than ESS. Results show that for various ASS, the proposed method provides results that are in excellent agreement with the ones obtained with the classical ESS method.

INTRODUCTION

Systems are generally assumed to behave linearly and in a noise-free environment. This is in practice not perfectly the case. First, nonlinear dynamic behaviors are very common in real life systems. Second, the presence of noise is a natural phenomenon that is unavoidable for all experimental measurements. In order to perform reliable model estimation of such systems, one should thus keep in mind these two issues and care about them. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the system under study will be lost.

The first problem addressed here is related to the estimation of nonlinear models of real life systems [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF][START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF]. As nonlinear behaviors are complex and various and as it is not intended to build a model for each case, it is chosen here to rely on Parallel Hammerstein Models as they can be interpreted easily (see Fig. 1). Such models belong to the class of "Sandwich models" [START_REF] Chen | Modeling and identification of parallel nonlinear systems: Structural classification and parameter estimation methods[END_REF]. Even if the model presented in Fig. 1 involves only monomial nonlinearities, it has been shown to possess a good degree of generality [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]]. Classical estimation methods for this class of system rely on a leastsquare formulation of the problem but are computationally costly and prone to conditioning issues [START_REF] Gallman | Iterative method for identification of nonlinear-systems using a Uryson model[END_REF]. Hopefully, thanks to exponential sine sweeps (ESS), nonparametric versions of such models can be very easily and rapidly estimated [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]. However, one drawback of these ESS based methods is that they rely on a specific class of input signals (ESS) that limits their use in practice.

Fig. 1. Representation of parallel Hammerstein models

The second problem addressed here is related to the rejection of uncertainties caused by the presence of noise. One way to address this issue relies on a careful design of the excitation signal: special types of periodic excitations [START_REF] Pintelon | System Identification: A Frequency Domain Approach[END_REF], non-stationary excitation with an underlying periodic structure [START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF], or repeated ESS [START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF] for example. Another way to remove noise consists, independently of the input signal, to segment the output signal in several slices on which output noise is assumed to be independently distributed, as done in the classical Welch's Method [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF]. This kind of approach is very attractive in the context of nonlinear system identification as it puts no constraints on the input signal.

The aim of this paper is thus to provide a methodology that allows for the estimation of Parallel Hammerstein Models based on the Welch method and on arbitrary sine sweeps (ASS). The paper is organized as follows: the classical ESS and Welch Methods are first briefly recalled in Section 2 and Section 3. Then the proposed method is 1. INTRODUCTION Systems are generally assumed to behave linearly and in a noise-free environment. This is in practice not perfectly the case. First, nonlinear dynamic behaviors are very common in real life systems. Second, the presence of noise is a natural phenomenon that is unavoidable for all experimental measurements. In order to perform reliable model estimation of such systems, one should thus keep in mind these two issues and care about them. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the system under study will be lost.

The first problem addressed here is related to the estimation of nonlinear models of real life systems [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF][START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF]. As nonlinear behaviors are complex and various and as it is not intended to build a model for each case, it is chosen here to rely on Parallel Hammerstein Models as they can be interpreted easily (see Fig. 1). Such models belong to the class of "Sandwich models" [START_REF] Chen | Modeling and identification of parallel nonlinear systems: Structural classification and parameter estimation methods[END_REF]. Even if the model presented in Fig. 1 involves only monomial nonlinearities, it has been shown to possess a good degree of generality [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]]. Classical estimation methods for this class of system rely on a leastsquare formulation of the problem but are computationally costly and prone to conditioning issues [START_REF] Gallman | Iterative method for identification of nonlinear-systems using a Uryson model[END_REF]. Hopefully, thanks to exponential sine sweeps (ESS), nonparametric versions of such models can be very easily and rapidly estimated [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]. However, one drawback of these ESS based methods is that they rely on a specific class of input signals (ESS) that limits their use in practice.

Fig. 1. Representation of parallel Hammerstein models

The second problem addressed here is related to the rejection of uncertainties caused by the presence of noise. One way to address this issue relies on a careful design of the excitation signal: special types of periodic excitations (Pintelon and Schoukens, 2001), non-stationary excitation with an underlying periodic structure [START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF], or repeated ESS [START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF] for example. Another way to remove noise consists, independently of the input signal, to segment the output signal in several slices on which output noise is assumed to be independently distributed, as done in the classical Welch's Method [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF]. This kind of approach is very attractive in the context of nonlinear system identification as it puts no constraints on the input signal.

The aim of this paper is thus to provide a methodology that allows for the estimation of Parallel Hammerstein Models based on the Welch method and on arbitrary sine sweeps (ASS). The paper is organized as follows: the classical ESS and Welch Methods are first briefly recalled in Section 2 and Section 3. Then the proposed method is 1. INTRODUCTION Systems are generally assumed to behave linearly and in a noise-free environment. This is in practice not perfectly the case. First, nonlinear dynamic behaviors are very common in real life systems. Second, the presence of noise is a natural phenomenon that is unavoidable for all experimental measurements. In order to perform reliable model estimation of such systems, one should thus keep in mind these two issues and care about them. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the system under study will be lost.

The first problem addressed here is related to the estimation of nonlinear models of real life systems [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF][START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF]. As nonlinear behaviors are complex and various and as it is not intended to build a model for each case, it is chosen here to rely on Parallel Hammerstein Models as they can be interpreted easily (see Fig. 1). Such models belong to the class of "Sandwich models" [START_REF] Chen | Modeling and identification of parallel nonlinear systems: Structural classification and parameter estimation methods[END_REF]. Even if the model presented in Fig. 1 involves only monomial nonlinearities, it has been shown to possess a good degree of generality [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]]. Classical estimation methods for this class of system rely on a leastsquare formulation of the problem but are computationally costly and prone to conditioning issues [START_REF] Gallman | Iterative method for identification of nonlinear-systems using a Uryson model[END_REF]. Hopefully, thanks to exponential sine sweeps (ESS), nonparametric versions of such models can be very easily and rapidly estimated [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]. However, one drawback of these ESS based methods is that they rely on a specific class of input signals (ESS) that limits their use in practice. The second problem addressed here is related to the rejection of uncertainties caused by the presence of noise. One way to address this issue relies on a careful design of the excitation signal: special types of periodic excitations (Pintelon and Schoukens, 2001), non-stationary excitation with an underlying periodic structure [START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF], or repeated ESS [START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF] for example. Another way to remove noise consists, independently of the input signal, to segment the output signal in several slices on which output noise is assumed to be independently distributed, as done in the classical Welch's Method [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF]. This kind of approach is very attractive in the context of nonlinear system identification as it puts no constraints on the input signal.

The aim of this paper is thus to provide a methodology that allows for the estimation of Parallel Hammerstein Models based on the Welch method and on arbitrary sine sweeps (ASS). The paper is organized as follows: the classical ESS and Welch Methods are first briefly recalled in Section 2 and Section 3. Then the proposed method is 1. INTRODUCTION Systems are generally assumed to behave linearly and in a noise-free environment. This is in practice not perfectly the case. First, nonlinear dynamic behaviors are very common in real life systems. Second, the presence of noise is a natural phenomenon that is unavoidable for all experimental measurements. In order to perform reliable model estimation of such systems, one should thus keep in mind these two issues and care about them. Indeed, all the noise that is not correctly removed from the measurements could be misinterpreted as nonlinearities, thus polluting measurements. And if nonlinearities are not accurately estimated, they will end up within the noise signal and information about the system under study will be lost.

The first problem addressed here is related to the estimation of nonlinear models of real life systems [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF][START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF]. As nonlinear behaviors are complex and various and as it is not intended to build a model for each case, it is chosen here to rely on Parallel Hammerstein Models as they can be interpreted easily (see Fig. 1). Such models belong to the class of "Sandwich models" [START_REF] Chen | Modeling and identification of parallel nonlinear systems: Structural classification and parameter estimation methods[END_REF]. Even if the model presented in Fig. 1 involves only monomial nonlinearities, it has been shown to possess a good degree of generality [START_REF] Pearson | Discrete-Time Dynamic Models[END_REF]]. Classical estimation methods for this class of system rely on a leastsquare formulation of the problem but are computationally costly and prone to conditioning issues [START_REF] Gallman | Iterative method for identification of nonlinear-systems using a Uryson model[END_REF]. Hopefully, thanks to exponential sine sweeps (ESS), nonparametric versions of such models can be very easily and rapidly estimated [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]. However, one drawback of these ESS based methods is that they rely on a specific class of input signals (ESS) that limits their use in practice.

Fig. 1. Representation of parallel Hammerstein models

The second problem addressed here is related to the rejection of uncertainties caused by the presence of noise. One way to address this issue relies on a careful design of the excitation signal: special types of periodic excitations (Pintelon and Schoukens, 2001), non-stationary excitation with an underlying periodic structure [START_REF] Zhang | Fast detection of system nonlinearity using nonstationary signals[END_REF], or repeated ESS [START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF] for example. Another way to remove noise consists, independently of the input signal, to segment the output signal in several slices on which output noise is assumed to be independently distributed, as done in the classical Welch's Method [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF]. This kind of approach is very attractive in the context of nonlinear system identification as it puts no constraints on the input signal.

The aim of this paper is thus to provide a methodology that allows for the estimation of Parallel Hammerstein Models based on the Welch method and on arbitrary sine sweeps (ASS). The paper is organized as follows: the classical ESS and Welch Methods are first briefly recalled in Section 2 and Section 3. Then the proposed method is detailed in Section 4. Results comparing the classical ESS method with the proposed method for various ASS are provided in Section 5 before concluding in Section 6.

Note 1 : In this article, * will stand for the convolution product. The convolution product between a function a and a function b will be denoted as a(t) * b(t) instead of a * b(t) in order to simplify some notations.

Note 2 : f will refer to the continuous frequency whereas k will refer to its discrete numeric version. Both will be used depending on the context.

EXPONENTIAL SINE SWEEP METHOD

A Parallel Hammerstein Model (PHM) is a relatively simple model of a weakly non-linear system (see Fig. 1). The input-output relation of such a model can be written as follows:

y(t) = +∞ n=1 h n (t) * x n (t) (1)
where x(t) is the input of the system, y(t) its output and h n (t) the kernels of the Hammerstein model.

Let's consider an arbitrary sine sweep (ASS) x(t) as an input of a PHM truncated to the order N . Such a signal is defined as x(t) = sin [φ(t)] with φ(t) a strictly increasing function of time. By using trigonometry formula, one can then write [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]:

y(t) = N n=1 g n (t) * sin [nφ(t)] (2) With g n (t) = n m=1
A mn h m (t) a linear combination of the kernels h m (t) with m ≤ n. This clearly highlights the harmonics that commonly appear when measuring the frequency response of a nonlinear system.

The identification of a PHM consists in finding the set of kernels h n (t), by means of the known input x(t) and the measured output y(t). Let's consider the input x(t) as an Exponential Sine Sweep (ESS) of duration T starting from a frequency f 1 and ending with a frequency f 2 . Its instantaneous frequency can then be written as follows:

2πf (t) = φ (t) = 2πf 1 exp t T ln (f 2 /f 1 ) (3)
The integration of this formula then gives:

φ(t) = 2π f 1 T ln (f 2 /f 1 ) exp t T ln (f 2 /f 1 ) + C
with C a constant value. To be able to identify the kernels h n (t), the following relationship is desired and can be obtained by different means depending on the C value [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF]:

sin [kφ(t)] = sin [φ(t + ∆ t k )] (4)
Then, from Eq. ( 2) the following relationship is obtained:

y(t) = N n=1 g n (t) * sin [φ(t + ∆ tn )]
The function h(t) can then be computed as follows:

h(t) = y(t) * x -1 (t) = +∞ m=1 g n (t -∆ t k ) (5) with x -1 (t) defined such that: x(t) * x -1 (t) = δ 0 (t)
, where δ 0 is the Dirac distribution. This deconvolution operation is crucial for the ESS method and can be achieved by different means [START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF].

Another way based on the Welch to perform this step is proposed in this article.

Note from Eq. ( 5) that the linear combination g n (t) of high order Hammerstein kernels h m (t), m < n of the system are in advance in time of a given value ∆ tn thanks to Eq. ( 4). This justifies the usefulness of this property. Then, the different terms g n (t) can be easily isolated by means of a simple temporal windowing of h(t) and the kernel h n (t) can be found by inverting the linear combination defined by the matrix A = (A mn ) 1≤m,n≤N . Also note that according to the classical ESS method implementation, the length T cut of the temporal window used to extract h(t) has to be smaller than T ln((N + 1)/N )/ ln(f 2 /f 1 ) where N is the maximal order being considered. Consequently, the power of the noise polluting the estimation is relative to this length T cut . This temporal window is usually build with a Hanning window to avoid side effects.

WELCH METHOD

The Welch method was first introduced by [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF] and provides an estimation of the cross power spectral density of two signals through an unbiased estimator that belongs to the modified periodogram class.

Cross power spectral density estimation

Let's assume the signals x [n] and y[n] to be realizations of length N T of an ergodic and stationary process. By definition, the Cross Power Spectral Density (CPSD) of these signals is:

Γ xy (k) = DFT E x[n]y[n -m] (k)
where E denotes the expected value, a the conjugate product of a and DFT the Discrete Fourier Transform operation.

This CPSD can be estimated a certain bias by means of the classical periodogram S xy (k), defined as follows:

S xy (k) = 1 N T f s DFT [x](k)DFT [y](k)
Taking advantage of ergodicity and stationarity, another estimator of Γ xy (k) based on a temporal average process can also be defined. The idea is to split the signals x and y into P overlapping segments of length M containing each D overlapping sample using a window w, and to estimate the CPSD of each segment Γp xy (k) through the periodogram before averaging it. An unbiased CPSD estimator can thus be defined as [START_REF] Welch | The use of fft for the estimation of power spectra: A method based on time averaging over short, modified periodograms[END_REF][START_REF] Akcay | Estimation of cross-power and autopower spectral densities in frequency domain by subspace methods[END_REF]: The main advantage of this approach is that when considering deterministic signals polluted by ergodic noise, the error committed will be relative to the noise power considered as a length M instead of length N T . However, some spectral precision will necessarily be lost by using smaller windows than the single whole signal.

Γxy (k) = α P P p=1 Γp xy (k) = α P P p=1 S xpyp (k) (6)

Linear system identification using CPSD

The Welch method has been largely used to estimate the frequency response of a linear system such as:

y(t) = h(t) * x(t) + b(t)
time domain

Y (f ) = H(f )X(f ) + B(f ) frequency domain
where y(t) is the output, x(t) the input and b(t) the noise, and A(f ) = FT {a(t)}(f ) where FT denotes the Fourier transform. Note that b(t) is one realization of a ergodic and stationary process with zero mean and then that y(t) has the same statistical properties than this realization.

The simplest way to numerically estimate H is:

Ĥ(f ) ≈ Y (f )/X(f )
and the error committed by doing so is proportional to the power of the noise which last the whole signal length.

The idea is now to use the CPSD to define the transfer function and then to use the Welch method in order to reduce errors caused by noise. Using the fact that noise is uncorrelated with the input signal and the deterministic nature of H(f ), one obtains:

H(f ) = E Y (f )X(f ) X(f )X(f ) -E B Ω (f )X(f ) X(f )X(f ) = Γ xy (f ) Γ xx (f )
Then the error commited is here again proportional to the power of the noise as if it only lasts M samples.

WELCH SINE SWEEP OR CPSD METHOD

General idea of the method

Let's assume a PHM as defined by Eq. ( 1) and an input signal x(t) defined to be an arbitrary sine sweep (ASS) as done in Eq. ( 2). That way x(t) = sin [φ(t)] with φ a strictly increasing function of time. Then, at each time a single frequency is played by the input x(t), involving an output y(t) made up of a discrete set of harmonic frequencies. Indeed y(t) is a linear combination of convolution products between harmonics and g n (t) functions as described in Eq. ( 2). This is illustrated in Fig. 2 which shows for a given frequency of the input the frequencies contained in the output of the PHM when considering an ASS as input.

The idea behind the Welch Sine Sweep method (also called CPSD method) is to be able to estimate the kernels of the PHM with the same precision than the ESS method but by releasing all the constraints on the sweep itself.

As stated in Sec. 2, the ESS method is based on an ESS which leads, when provided as the input of a PHM and after a deconvolution, to time localized linear combination g n (t) of kernels k n (t), see Eq. ( 5). A temporal windowing operation is then performed to extract the impulses responses g n (t). By making use of the properties of the Welch Method, we directly estimate the impulse responses g n (t) by performing a temporal windowing operation on slices of input and output signals. The deconvolution operation is avoided and all constraints on the sweep are thus released.

However, in order to still be able to estimate the nonlinear model, this temporal windowing cannot be done arbitrarily. It must be done by taking into account the fact that the input signal is an ASS and the fact that the system under study is a PHM. A way to do so is to focus on the different harmonics one by one and to try to get information regarding them, and exclusively them, within a considered time window. The challenge is thus to be able to design temporal windows that ensure that the frequency content of the harmonic under study is contained within a frequency range that is not polluted by the other harmonics. If the harmonic of interest is the n th harmonic, and if the selected windows starts with a frequency f start , then the time at which this temporal window should stop is the time at which the the input frequency is such that f stop = (n + 1)/n × f start . Thus, the higher the harmonic order is, the smaller the largest window allowed is. This temporal window is called in the following the "optimal" window as it is the longest possible, but any shorter window is acceptable. Note that there is no constraints on the start frequency f start and thus that the windows can start at any arbitrary time, allowing to use overlapped windows. This process is illustrated in Fig. 2 that shows two examples of optimal windowing choice (without overlapping) when focusing on the linear part (blue dashed lines) and when focusing on the third harmonic (green dashed lines). It is also shown how this "optimal temporal windowing" choice derived

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017

when focusing on the third harmonic can be applied to the linear part in green dot-dashed line and thus become suboptimal with respect to the linear part. Bottom: application of Eq. ( 7) with regularization (see Sec. 4.3). Time windows are built from Eq. ( 9) for n = 1 with a security factor 3/4 (see Sec. 4.2). See Sec. 5.2 for details regarding the simulated system.

Then, due to the linearization of the trigonometric functions given in Eq. ( 2), it is possible isolate from the selected windows the exact harmonic contribution G n (f ) = FT [g n (t)](f ) of the n th harmonic in the frequency domain. However, as the information we try to infer from these windows is related to the n th harmonic, the n th harmonic of the input signal has to be used in the estimation process. The function G n (f ) can thus be directly by using the Welch method described in Section 3 over the selected temporal windows and using the n th harmonic x n (t) = sin [nφ(t)] as input instead of the actual input x(t) = sin [φ(t)]. This is illustrated in Fig. 3 where the parts of G 1 (f ) estimated by each window are plotted. Formally, using notations of Eq. ( 6), what has been done to estimate G n (f ) is:

Ĝn (f ) = Γxny (f ) Γxnxn (f ) = P p=1 Γp xny (f ) P p=1 Γp xnxn (f ) (7)
From the estimated Ĝn (f ), it is then possible to recover the kernels H n (f ) of the PHM by inverting the linear combination defined by the matrix A as in Sec. 2.

Note that contrary to the classical ESS implementations [START_REF] Novak | Nonlinear system identification using exponential sweptsine signal[END_REF][START_REF] Novak | Synchronized swept-sine: Theory, application, and implementation[END_REF][START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF], this method does not need an ESS satisfying any fundamental phase property as in Eq. ( 4). The only condition that needs to be satisfied here is related to the function φ(t) that needs to be known explicitely in order to determine the frequency bandwidth of each segment and thus the windows lengths that suit the Welch method.

Design of temporal windows for the CPSD estimator

As stated previously, a temporal window series relative to an harmonic of interest is said to be acceptable if : for all the segment of this temporal window series, the spectral content relative to the harmonic of interest is not overlapping with the spectral content of other harmonics (see Fig. 2). Then, it is possible to build the maximum length of each temporal window by building rectangles around each harmonic straight lines. Mathematically, it is demonstrated by recurrence that when focusing on the harmonic n, the p th temporal window of the series of temporal windows starting at frequency f n 0 and with no overlap should stop at a time such that the frequency of the input signal corresponds at this time to:

f n p = f n 0 n + 1 n p ( 8
)
This grid is defined in the frequency domain. It is then necessary to go back to the temporal domain by making use of the instantaneous frequency derived from the phase function φ(t). If we consider the phase given in Eq. ( 3), this gives the following series of times:

t n p = T ln(f 2 /f 1 ) ln(f n 0 /f 1 ) + p ln n + 1 n (9)
Note that this interestingly leads to regularly spaces times for this particular case of an ESS. Additionally, the step is the same as the maximum length of temporal window in the ESS method, see section 5.2. This will lead to an equivalent noise rejection for the two method.

If we consider now the phase of a linear sweep, i.e. a sweep with the instantaneous frequency such that :

f (t) = f 1 + f 2 -f 1 T t (10)
then the associated series of time is given by :

t n p = ( n+1 n ) p-1 f 2 -f 1 f 1 T (11)
and the series of "optimal windows" is defined by

T n p = t n p + t n p-1 = 1 n n + 1 n p-1 f 1 T f 2 -f 1 (12)
To remain simple, we keep a grid regularly spaced in time for each order. Therefore, the step of the temporal grid associated to the linear sweep is defined by the smallest time window, which is T n 1 in this case.

Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017

Regularization procedure

The computation of the inverse filter x -1 (t) (see Sec. 2) is crucial to reject unwanted harmonics. Because the CPSD estimation is based on the temporal windowing of the signals, the regularization procedure proposed by [START_REF] Rébillat | Identification of cascade of hammerstein models for the description of nonlinearities in vibrating devices[END_REF][START_REF] Rébillat | Repeated exponential sine sweeps for the autonomous estimation of nonlinearities and bootstrap assessment of uncertainties[END_REF] applies naturally here. Thus, the following regularization procedure is performed in order to reject efficiently unwanted harmonics:

Γxnxn (f ) = Γxnxn (f ) if nf 1 < f < nf 2 Γxnxn (f ) + x × max f [Γ xnxn (f )] else.
with x an arbitrary factor. However, this will only affect the exterior part of the frequencies played by the input signal. This will not reject the higher harmonics which are within the interval of the frequency played (see Fig. 3). Therefore, an additional regularization operation is performed for each segment to reduce the energy of the partial part Γp xny (f ) of the quantity Γxny (f ) outside of the frequency range relative to this segment (see Eq. ( 7)):

Γp xny (f ) = Γp xny (f ) if f n p-1 < f < f n p Γp xny (f )/ y else.
with y an arbitrary factor. Both x and y are taken equal to 10 10 . This effect is illustrated on Fig. 3. In each figures, 3 zones can be distinguished: the estimated part, the rejected harmonics, and the noise floor. The top figure is the straight application of the method whereas the bottom figure includes the regularization procedure proposed here. As can be seen from this figure, the proposed regularization really helps in rejecting unwanted harmonics components.

COMPARISON ON A SIMULATED SYSTEM

Implementation details

Demonstration of the efficiency of the CPSD method is done with two kind of ASS : an exponential sine sweep to compare results with the classical ESS method; and a linear sine sweep (LSS) to illustrate the potential of the method. All the ASS are 5 seconds long and starting at frequency 200 Hz and rising up to 10 kHz with a sampling frequency of 96 kHz. For the CPSD method, the classical value of 50% of overlapping segments pondered with Hanning windows has been retained.

For the ESS, the series of time used in CPSD method is defined according to Eq. ( 9). The step has been multiplied by a security factor equal to 3/4 which reduces the length of the time windows. This is done to avoid the spectral content from the next upper harmonic to be too close to the spectral content of interest. Because of the overlapping of 50%, the final series of time is given by:

t p = 1 2 × 3 4 × p × T ln[(N + 1)/(N )] ln(f 2 /f 1 ) (13)
Note that the p th segment is composed by the points [t p , t p+2 ] because of the 50 % overlapping.

In the same way, the series of times relative to the LSS for the CPSD method is (see Sec. 4.2) :

t n p = 1 2 × 3 4 1 n f 1 T f 2 -f 1 (14)
Note that this time, we used a different grid for each order in order n to keep a good precision in the low frequency range.

In the ESS method, after deconvolution of the output signal, each harmonic impulse response is localized in advance from the linear part and is extracted by a simple temporal windowing of length T cut (see Sec. 2). As stated in Sec. 4.1 and 3.2, this parameter has an equivalent role than the step of the CPSD series of time. Then the value T cut = 3/4T ln((N + 1)/N )/ ln(f 2 /f 1 ) ≈ 0.21 s is kept. Note that for the linear case, the equivalent value of T cut extracted from the CPSD method is

T cut = 3/4T f 1 /(n(f 2 -f 1 )) ≈ 0.076 s (n = 1) or 0.019 s (n = 4).
Because the windows are smaller for the LSS, the noise will have a lower influence on the estimated kernels in comparison with the ESS.

Simulated system

We simulated a PHM system of order 4. The linear part is modeled by a Chebyshev filter of Type II of order 4 with a cutoff frequency of 2 kHz. The harmonic parts are modeled with ARMA filters of order 2 defined by the coefficients given in Tab. 1. We also add a Gaussian white noise to the output signal y(t). Numerous SNR , PHM order N and different length of sweep have been tested showing identical results. We kept the representative value of N = 4, SN R = 40 dB and a duration of 5 s to present results. However a small discussion about computation time as a function of the sweep length and PHM order is given later.

Results

The PHM Kernels estimated through the classical ESS method and the proposed Welch Sine Sweep Method for the two kind of sweeps versus the theoretical ones are presented in the frequency domain in Fig. 4. From this figure, it can be seen that as expected both methods provides estimated Kernels that are in good agreement with the theoretical ones. The PHM Kernels estimation through a LSS is only possible with the CPSD method. As expected, the noise is more rejected with this sweep than with the ESS as the windows used are smaller. Note that because the LSS spends more time in high frequency than in low frequency as compared to the ESS, the estimation obtained with the LSS is better in high frequencies than in low frequencies. Finally, note that the ESS method only requires a single deconvolution whereas the CPSD method need as much FFT as the number of temporal windows being used. Then the longer is the signal, the higher is the computational cost of the CPSD method. Also, because the largest acceptable length of window depends on the Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 maximum order N of the PHM, the more complex is the system, the higher is the computational cost of the CPSD method. Some computation times are given in Tab. 2.

CONCLUSION AND PERSPECTIVES

In this article, a new method to estimate a a Parallel Hammerstein Models based on the Welch estimator of the cross power spectral density is proposed and validated. This method is compared to the classical ESS method on a simulated system with noise. It is shown that the proposed method releases all constraints on the input sweep in comparison with the classic ESS method at the cost of a longer computational time. Results provided by both methods are found to be almost identical. Because constraints on sine sweep are released and it is harmonic based, this new method is very attractive as it could be used with contextual sweeps (related to the power spectral density of the environmental noise for example) or be the basis of a nonlinear model estimation with input harmonic signals such as music, which corresponds to a strong demand of professionals in loudspeaker measurement.

Fig. 1 .

 1 Fig. 1. Representation of parallel Hammerstein models

  Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 with x p (k) = w[k]x[k +pD] and y p (k) = w[k]y[k +pD] and α a normalization factor such that: α = M/ M m=1 |w(m)| 2 .

Fig. 2 .

 2 Fig. 2. Input sweep frequency versus output signal frequencies for a 4 th order PHM and windowing choices examples (no overlap): linear part [optimal] (blue dashed lines), third harmonic [optimal] (green dashed lines), linear part [suboptimal] (green dot-dashed line).

Fig. 3 .

 3 Fig.3. Estimation of G 1 (f ) over unity time windows without overlap. Top: strict application of Eq. (7). Bottom: application of Eq. (7) with regularization (see Sec. 4.3). Time windows are built from Eq. (9) for n = 1 with a security factor 3/4 (see Sec. 4.2). See Sec. 5.2 for details regarding the simulated system.

Fig. 4 .

 4 Fig. 4. PHM Kernels estimated through the classical exponential sine sweep method (ESS) and the proposed Welch Sine Sweep Method (CPSD) for two kind of sweeps versus theoretical ones in the frequency domain (SN R = 40 dB).

Table 1 .

 1 Coefficients of the ARMA filters used to model high order kernels

	Order	a 0	a 1	a 2	b 0	b 2	b 2
	2	1.0000	-1.8996	0.9025	0.0500	-0.0951	0.0470
	3	1.0000	-1.9075	0.9409	0.0250	-0.0474	0.0226
	4	1.0000	-1.8471	0.8649	0.0087	-0.0115	0.0056

Table 2 .

 2 Comparison of computation times between CPSD (ESS) and ESS method.