S Cazottes

G Y Wang

A Fnidiki

D Lemarchand

P O Renault

F Danoix

A transmission electron microscopy and X-Ray diffraction study of microstructural evolution in magnetoresistive Cu-Fe-Ni ribbons

Keywords: EFTEM, GMR, granular metals, X-ray diffraction supplied): Cu-Fe-Ni system Cu-Fe-Ni system, EFTEM, X-ray, granular solids, GMR

The evolution of the microstructure of a granular Cu 80 Fe 10 Ni 10 (at%) melt spun ribbon is studied by means of transmission electron microscopy (TEM), energy filtered transmission electron microscopy (EFTEM) and X-Ray Diffraction. This system is interesting since large giant magnetoresistance (GMR) values have been measured for this composition. We have shown the presence of two face-centered cubic phases, an (Fe,Ni)-rich phase and a Cu-rich phase. The lattice parameters of those two phases are rather close and no diffraction or elastic contrast acts to evidence the two phases in TEM bright field mode. With EFTEM imaging, we have shown the presence of a fine scale (Fe,Ni)-rich precipitation, inside the Cu-rich fcc matrix. The precipitates are 2-4 nm large in the as-spun state, and 4-6 nm large after an annealing treatment of 2 hours at 400°C. The lattice parameter of the Curich phase in the as-spun sample is 0.3608 nm, and that of the (Fe,Ni)-rich phase is 0.3610 nm. After a 24 hours treatment at 600°C, the mean diameter of the particle is 20 nm and the lattice parameter of the (Fe,Ni)-rich phase has decreased to 0.3600 nm while that of the Cu-rich phase has increased to 0.3613 nm, which is constistent with a segregation of Fe and Ni in the precipitates. The composition and volume fraction of the two phases measured for this annealed sample are in good agreement with the Thermocalc® predictions.

Introduction

Granular solids consisting of magnetic fine particles embedded in a non magnetic medium have been intensively studied in the last decade because of their interesting physical properties and their potential technological applications. The aim of this work is to characterize the evolution of the microstructure of a Cu 80 Fe 10 Ni 10 (at%) ribbon, as-spun and annealed at different temperatures. The results could later be used in other studies to explain the magnetic behavior and GMR properties of these ribbons.

Indeed, relatively high GMR values were measured on CuFeNi ribbons [1,5]. This GMR effect was attributed to the presence of fine scale magnetic precipitates. The Giant Magnetoresistance (GMR) effect in granular alloys is strongly dependant on the microstructural parameters, such as the precipitates size and size distribution, the composition of the matrix and the precipitates, the number density of the precipitates, the distance between each particle and the morphology of the particles. However, the correlation between those parameters and the GMR is not fully understood yet. In order to be able to correlate the GMR properties of an alloy with its microstructure, a fine quantification of its microstructural parameters needs to be done. We present a method for the characterization of the chemical nature, the volume fraction of the precipitates and the structure of the two phases, i.e. the crystallographic structure and the lattice parameter of the matrix and the precipitates.

Indeed, the nature of the precipitation is strongly dependant on the ribbon studied. In [1], the composition of the studied ribbon was Cu 80 Fe 10 Ni 10 ( at%) and the nature of the precipitates was deduced from magnetization curves and Mossbäuer spectrum. Those magnetic analysis revealed the presence of fine bcc -Fe precipitates with a radius slightly larger than 1 nm. In [2], some spinodally decomposed (Fe,Ni)-rich precipitates, were observed with High Resolution Transmission Electron Microscopy in a Cu 60 Fe 20 Ni 20 (wt %) ribbon. In reference [3], X-Ray measurements were performed on Cu 100-x Fe 20 Ni x (x = 0, 5, 20) ribbons which evidenced the presence of -Fe for Cu 80 Fe 20 and Cu 75 Fe 20 Ni 5 ribbons. All those different microstructures can be explained by the fact that the ribbons are in a non equilibrium state and that their microstructures are highly dependent on the elaboration technique and elaboration parameters. Thus, this bibliographical survey indicates that each type of ribbon should be characterized to get microstructural informations .

This paper presents the evolution of the microstructure of a Cu 80 Fe 10 Ni 10 (at%) ribbon, as-spun and annealed at different temperatures. The microstructure evolution is observed by conventional TEM, Energy Filtered Transmission Electron Microscope (EFTEM), Energy Dispersion X ray spectrometry (EDX) and X Ray Diffraction (XRD). Energy-filtered (EF) imaging in the transmission electron microscope (TEM) is a widely used technique for acquiring qualitative data at high spatial resolution [8][9][10]. This technique provides images with a chemical contrast. The filtered images are element maps.

The aim is to provide structural information for the studies of the magnetic behaviour and GMR properties of those ribbons. The results can be used in order to interpret the magnetization data obtained by [1][2][3][4][5].

Experimental procedure

Master alloys of Cu 80 Fe 10 Ni 10 (at %) have been prepared by arc melting of pure elements. Ribbons were obtained by conventional melt spinning and rapid solidification processing in an helium atmosphere using a steel wheel rotating at a surface speed of ~25 m/s. The ribbons were 5 mm wide, 60 µm thick and several meters long. The ribbon was rapidly solidified, the cooling rate is approximately 10 5 K/s, and its microstructure is in a metastable state.

As the main objective of the project is to correlate the magnetic properties of this ribbon with its microstructure, the annealing temperatures were chosen to maximize the giant magnetoresistance ratio. For this system, the best GMR values were observed for specimens annealed for 2 hours at temperatures around 400°C [11]. A specimen was thus heat treated 2 h at 400°C. Moreover, in order to reach a stable state for this system and assess the Thermocalc® predictions, another specimen was annealed at 600°C for 24h TEM specimens were milled directly from ribbons in a GATAN PIPS system at low energy ( from 5 to 3 keV Ar) and low incidence (from ± 6° to ± 3°). The sample was milled on both faces in order to study the center of the ribbon. TEM observations and energy filtered images are obtained/acquired on a TECNAI F20 operating at 200kV. The microscope is equipped with a Field Emission Gun (FEG), a Gatan Imaging Filter (GIF) for EFTEM imaging and a EDAX-EDX spectrometer. The electrons with a specific energy loss range are used to form an image showing chemical contrast. Each specific energy loss range is characteristic of one element. The filtered images are thus element maps. The GIF is also used simply to remove electrons that have suffered an energy loss of more than a few electron volts to produce a so-called zero-loss image [9]. The zero loss image is created with the remaining electrons, it has a better contrast than the bright field images, particularly in the case of thick specimens.

When an element is present in a much higher concentration in the particles than in a matrix, the EFTEM images reveal it as bright features on a darker background.

X Ray Diffraction analyses were performed on a commercial Bruker D5005 diffractometer. Complementary theta/2theta measurements were performed on a high-resolution Seifert MZ VI HR diffractometer with a tworeflection Bartels monochromator. We used the symmetric Ge 220 reflections and a 0.1 mm slit in the monochromator to obtain the Cu-K 1 radiation (0.15406 nm) of a sealed X-ray tube (linear focus).

Results and discussion

Thermodynamics

There is a miscibility gap in the Fe-Cu phase diagram which is still present in the ternary system Cu-Fe-Ni. According to the literature [5][6], and for the studied compositions, the fcc phase, present at high temperature, decomposes into two fcc phases 1 and 2 . The 1 phase is the Cu-rich phase and the 2 phase is the (Fe,Ni)-rich phase. A bcc Fe-rich phase can also appear for annealing at temperatures lower than 400°C. Thus, at 400°C, Thermocalc® predicts the presence of 1 , 2 and the bcc iron phase, and only 1 and 2 for an annealing temperature of 600°C. The Thermocalc® calculated composition (at %) and phase fraction of each phase are presented in table 1 and the corresponding calculated phase equilibria is presented in figure.1.

'[Insert table 1 about here]' '[Insert figure 1 about here]'

Transmission Electron Microscopy

Transmission Electron Microscopy observations of the as-spun sample indicates that the ribbon is composed of Cu-rich fcc grains, see figure 2. Their size is in the range of 0.5 -1µm. The diffraction pattern indicates that the structure of these grains is face centered cubic with the same lattice parameter as copper, i.e. a = 0.361 ± 0.005 nm. No bcc phase was observed on the diffraction pattern. With conventional TEM, it was not possible to identify phase segregation in the ribbons. That means that no diffraction or elastic contrast acts to evidence any second phase in bright field mode. Indeed, as the 1 and 2 phases have the same face centered cubic crystallographic structure and the lattice parameter of fcc Cu is 0.3610 nm, the one of Fe is 0.3646 nm and the one of fcc Ni is 0.3524 nm. Thus the unit cell parameters of these two phases are rather close, and it is not possible to evidence their presence with conventional bright-field TEM. EFTEM is an alternative solution to localise the iron and nickel in the ribbon, and thus the formation of 2 in the ribbon.

'[Insert figure 2 about here]'

As spun sample

EFTEM indicates the presence of fine scale (Fe,Ni)-rich precipitates. The zero loss image, and the elements maps are presented on figure 3. Some dark regions are present in the zero loss image, with an irregular shape and a size of 10-20 nm. Those particles give no additive signal on the copper, iron and nickel elemental map. It is most likely surface oxides or surface damage due to the PIPS milling. Whereas no precipitation is visible on the zero loss image, the (Fe,Ni)-rich precipitates appear bright in the iron and nickel maps and dark on the copper map, showing that the particles are enriched in iron, and that their copper concentration is lower than the one of the matrix. In the nickel map, the contrast is less clear, indicating that there is still some nickel in the copper matrix. This result is consistent with the complete miscibility of Ni in Cu. On the contrary, the precipitates are clearly identifiable on the copper and iron maps. The precipitates appear spherical, with a high number density and their size is in the range of 2-4 nm. It was not possible with this microscope to probe a smaller area without avoiding irradiation damages. Due to the small size of the precipitates the EFTEM signal is very weak and the noise/signal ratio is high. Thus the interface between the precipitates and the matrix is not clear and it is difficult to identify the precise size of the precipitates. A more precise quantification of the size distribution and number density of the precipitates can not be calculated from this technique. The composition measurements with EDX is also not possible, because of the scale of the precipitates.

'[Insert figure 3 about here]'

Sample annealed 2 hours at 400°C

After a 2 hour heat treatment at 400°C, no significant evolution was observed in the microstructure of the ribbon. The grains have a face centered cubic structure, with a size in the range of 0.5-1µm. The element maps (see figure 4) show the presence of the fine precipitates with a size of 2-6 nm . The measurements were made on a large area to avoid specimen damages. The increase in size is rather small, and in order to estimate it, a technique with higher spatial resolution should be used, such as Field Ion Microscopy for example.

The copper map is very bright and gives no information about the presence of the precipitates, but the contrasts in the iron and nickel maps are clear which allow us to conclude that those precipitates are enriched in nickel and iron. The precipitates are in the range size of 2-6 nm. In order to avoid irradiation damage, it is not possible to get smaller scale filtered images.

'[Insert figure 4 about here]'

Sample annealed 24 hours at 600°C

After annealing at 600°C for 24h, a drastic change in the precipitate morphology is observed. The precipitates are much bigger and more enriched in iron and nickel than for the as-spun sample whereas the grain size and structure of the matrix are unchanged. The energy filtered images are presented in figure 5, and they show precipitates with a size range of 15-25 nm, with a mean value of 20 nm. By analyzing the iron map, the volume fraction of precipitates is 19%. This calculation was performed using the measured diameter of the precipitates, calculating the total volume of the precipitates and then the volume fraction. The thickness of the specimen is unknown because it is not the same on the border of the hole and 200 nm away from it. Without the thickness of the sample, this calculation induces an overestimation of the volume fraction of the precipitates. As the particles are bigger than in the as spun state and in the sample annealed at 400°C for two hours, it was possible to check the composition of the matrix and particles with an EDX analysis. Several zones were probed , in the center of the precipitates for the 2 phase, and away from the precipitates , i.e. in the matrix, for the 1 phase, and the results are summarized in Table 2. The EDX precision is approximately 1% and the composition measured for the 2 phase is almost the same as the one predicted by Thermocalc®. For the 1 phase, there is more iron and nickel in the experimental data than in the Thermocalc® prediction. The difference is around 2% for both elements. This difference is maybe due to the fact that neither the precipitates and the matrix are homogeneous in composition in the ribbon. Thermocalc® gives an mean composition value, while only the center of the two phases were probed. If the composition near the interface is not the same as in the center of the phases, then the measured value are not exactly the same as the one given by Thermocalc®. Moreover, the volume fraction of the 2 phase is slightly higher than the one given by Thermocalc® which is maybe due to the calculation of this value from the EFTEM images. However, considering the experimental precision of the EDX measurements, the estimation of the size of the particles and the volume fraction of the two phases are in good agreement with the Thermocalc® simulation. After 24 hours annealing at 600°C, the ribbon has reached a stable thermodynamic state.

'[Insert figure 5 about here]' '[Insert Table 2 about here]'

X Ray Diffraction

With the aim of determining the lattice parameter of the two phases and their composition, the samples were studied by X Ray Diffraction (XRD), as shown in Figure 6. In order to avoid as much as possible the overlapping of the Bragg peaks, a first set of XRD measurements were performed at high 2 angles (from 86° up to 148°) in a two-circle Bruker D5005 diffractometer working with CuK radiations , i.e. using both K 1 and K 2 radiations. From those patterns, a single fcc phase is observed, and no bcc phases are present, in agreement with the diffraction pattern obtained from TEM observations. The precision does not allow confirming the presence of two phases with different lattice parameters. Indeed, all the peaks can be indexed within the fcc structure with a lattice parameter of 0.3607 nm, which is close to the lattice parameter of pure Copper (a = 0.3610 nm).

'[Insert figure6 about here]'

In order to analyze the data for the presence of the (Fe,Ni)-rich and Cu-rich phases, a second set of XRD measurements was performed in a 4 circle Seifert 3000 diffractometer in order to increase the precision, i.e. no k 2 radiation and lower background. Indeed, the incoming monochromatic radiation is CuK 1 ( =0.15406 nm) with a Ge (220) 2 reflections Bartels monochromator, and, the diffracted rays are analyzed through a 0.1 mm slit on a 220 mm radius circle. The figure 7 presents an example of extraction made on the 331 peaks for the as spun and annealed samples. The peaks are clearly not symmetric and the asymmetry is changing as a function of the annealing temperature. As EFTEM observations showed the presence of two phases, all the peaks were fitted as the sum of two peaks. The diffraction peaks could be well fitted using symmetric Pearson VII function; the background function is a first order polynomial. The area and position of each peak is an indication of respectively the volume fraction and the lattice parameter of each phase. Thus, a least square algorithm is used to perform a least-squares minimization of the objective function, with some constraints on the total area under each peak profile in order to take into account the volume fraction estimated from TEM analysis or Thermocalc®.

'[Insert figure7 about here]'

In the case of the sample annealed at 600°C for 24h, the first peak is high, with a small FWHM and the second one is smaller, with a larger FWHM. The FWHM is related with the crystallite size, the microstrain in the sample and the experimental set up [12]. After a 24 hours heat treatment we can expect the micro strain in the ribbon have been released, and the width of the peak is then an indication of the size of the precipitates. Theoretically, the bigger the FWHM is, the smaller the precipitates are. Thus, the major peak is expected to correspond to the Cu-rich phase, while the minor one, with the large FWHM, is expected to correspond to the (Fe,Ni)-rich phase. The same fitting procedure was used for the as spun sample and the sample annealed at 400°C for 2 hours. The results are presented in Table 3. The main result of this fitting procedure is the peak position, and then the lattice parameter of each phase.

'[Insert table 3 about here]'

The lattice parameter evolution is presented in figure 8. In the as-spun state, the lattice parameter for the Cu-rich phase is 0.3609 nm and the one for the (Fe,Ni)-rich phase is 0.3614 nm. After a 2 hours heat treatment at 400°C, almost no evolution of the lattice parameter is observed This is in agreement with the EFTEM observations, there was almost no evolution between the as spun state and the sample annealed at 400°C for 2h. A significant evolution is observed after a 600°C annealing for 24h. The symmetry of the peaks has changed, and the lattice parameter of the Cu rich increases from 0.3609 nm to 0.3610 nm while the lattice parameter of the (Fe,Ni) rich phase decreases from 0.3614 nm to 0.3600 nm. The theoretical lattice parameter of the copper is 0.3610 nm, the one of the gamma iron is 0.3640 nm, the one of the nickel is 0.3524 nm and the one of L10 ordered FeNi is 0.3556 nm [13]. Assuming the Vegard's law, it is possible to calculate the composition of a solid solution from its lattice parameter. However, this calculation was not possible on this system. Using the experimental composition obtained from the fit for the (Fe,Ni) rich phase, and the theoretical lattice parameters of Cu (0.3610 nm), Ni (0.3519 nm) and Fe (0.3640 nm[14]), the calculated lattice parameter is 0.3593 nm instead of 0.3614 obtained from the model. As the Fe phase is not a stable phase, several values are given in the literature for the lattice parameter. In [13], the lattice parameter given for Fe is 0.3454 nm, which gives with the Vegard's law 0.3492 nm for our composition. The Vegard's law is not applicable in for the sample annealed at 600°C for 24h, it is thus not possible to deduce the evolution of the composition with those measurements. Precise composition measures are in progress with Atom Probe. However, from the evolution of the lattice parameter a qualitative description is possible, the decrease in lattice parameter of the (Fe,Ni)-rich phase after a 24 hours annealing at 600°C can be correlated with an enrichment of this phase in iron and nickel while the increase of the Cu-rich phase lattice parameter can be correlated with an enrichment of Cu Those results are in agreement with the EFTEM observations. During annealing, there is a segregation of iron and nickel in the (Fe,Ni)-rich precipitates.

'[Insert figure 8 about here]' Further GMR and magnetic measurements are in progress. The magnetic and GMR properties will be explained using our structural results. It is known that, in order to be fitted, the magnetization data and Mössbauer spectrum need the microstructural parameters such as the size distribution, the chemical nature and the volume fraction of the two phases.

Moreover, some theoretical models (Pogorelov et al, PRB, 58,1998, 425 ; Ferrari et al, PRB , 56, 1997,6086) could fit their magnetoresistance data using some microstructural parameters such as the size distribution, the chemical nature and the concentration of the magnetic particles. Our experimental results could be inserted in those models for the correlation of the magnetoresistance behavior of the CuFeNi system and their microstructural parameters

Conclusion

We have evidenced the presence of (Fe,Ni)-rich precipitation inside a Cu-rich phase in a Cu 80 Fe 10 Ni 10 (at%) melt spun ribbon. Those two phases have a similar lattice parameter, and no contrast is visible on bright field images. A chemical analysis is necessary to evidence the presence of the precipitates. In the as spun state, the precipitates are 2-4 nm wide. The lattice parameter of the Cu-rich phase was measured by X-Ray diffraction as 0.3609 nm and the one of the (Fe,Ni)-rich phase as 0.3614 nm. There is no significant evolution of the microstructure for the sample annealed two hours at 400°C, whereas the morphology of the precipitates has drastically changed after a 24 hours heat treatment at 600°C. The precipitates diameter reaches a size of 20 nm with an experimental composition of Cu 11 Fe 53 Ni 36 in agreement with Thermocalc® predictions. The decrease in the lattice parameter, from 0.3614 nm to 0.3600 nm, confirms the segregation of the iron and nickel in the (Fe,Ni)-rich precipitates. Further GMR and magnetic measurements are in progress. The magnetic and GMR properties will be explained using our structural results. Moreover, some theoretical models could fit their magnetic properties and magnetoresistance data using some microstructural parameters such as the size distribution, the chemical nature and the concentration of the magnetic particles. Our experimental results could be inserted in those models for the correlation of the magnetoresistance behavior of the CuFeNi system and their microstructural parameters.

F

Table 3 :Figure 1 :

 31 Figure 1: Thermocalc® CuFeNi equilibrium phase diagram at 400°C. The FCC_A1 is the 1 phase, the FCC_A1#2 is the 2 phase and the BCC_2 is the bcc iron phase

Figure 2 :

 2 Figure 2: TEM bright field micrograph of the as-spun ribbon and diffraction pattern of a <100> zone axis oriented grain.

Figure 3 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 35678 Figure 3: Energy filtered images of the as spun ribbon showing the presence of (Fe, Ni)-rich precipitates with a size of 2-4 nm.

Table 1 : Composition (at%) and phase fraction (%phase) calculated from Thermocalc® software for Cu 80 Fe 10 Ni 10 at different annealing temperature

 1 [11] C.S.Martins, F.P. Missell, J. Appl. Phys. 87 4840 (2000). [12] R. W. Cheary, A. A. Coelho, J. P. Cline, J. Res. Natl. Inst. Stand. Technol. 109 1 (2004) [13] Y. Mishin, M.J. Mehl, D.A. Papaconstantopoulos, Acta Mat. 53 4029 (2005) [14] T. Massalski (Editor), Binary Alloy Phase Diagrams, second edition, (ASM International, 1990) [15] S.H. Ge, Z.Z. Zhang, Y.Y. Lu, C. X. Li, R. J. Gan, Thin Solid Films 311 33 (1997)

	At%		1 (Cu)-rich phase		2 (Fe,Ni)-rich phase	bcc Iron
			%phase	Cu	Ni	Fe %phase Cu	Ni	Fe	%phase Cu	Ni	Fe
	Thermocalc	400°C	80.94	97.48 2.50 0.02	17	6.51 46.50 46.99	2.06	0.03 3.46 96.51
	predictions 600°C	82.08	95.48 4.26 0.26 17.92	9.10 36.31 54.59
		o r	1-Cu rich phase		2 -(Fe,Ni)-rich phase
		Volume fraction P Thermocalc® prediction 82.08 e r Cu 95.48 4.26 0.26 Ni Fe e Experimental measurements 81 91 7 2	Volume fraction 17.92 19	Cu 9.10 36.31 54.59 Ni Fe 11 36 53
					R e v
							i e
							w
								O n l
										y

Table 2 : Comparison of Thermocalc® predictions at 600°C and experimental measured compositions of the (FeNi)-rich phase and the Cu rich phase for the sample annealed at 600°C for 24h.

 2

		1-Cu rich phase lattice parameter (nm)	FWHM 1	2 -(Fe,Ni)-rich phase lattice parameter (nm)	FWHM 2	A 1/(A 2+ A 1)
	As Spun	0.36096 ± 3 10 -5 0.65±0.01	0.3614 ± 3.10 -4	1.2 ± 0.1	0.06
	Annealed at 400°C for 2h	0.36082 ± 3.10 -5 0.62±0.01	0.3616 ±3.10 -4	1.2 ± 0.1	0.06
	Annealed at 600°C for 24h	0.36103 ± 3.10 -5 0.47±0.01	0.36038 ± 6.10 -5 1.1 ± 0.1	0.17

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Aknowledgements

M. Barrico (Dipartimento di Chimica IFM and INFM/INSTM, Università di Torino, Torino Italy

) is acknowledged for providing the ribbons.


Sébastien ; Gravier

Jean-Jacques J Blandin

Patricia Donnadieu

Interactions between high temperature deformation and crystallization in zirconium based bulk metallic glasses

Keywords: amorphous metals, crystallization, nanocrystals, viscosity user supplied): high temperature deformation Bulk metallic glass, high temperature deformation, crystallization, viscosity, nanocrystal

Introduction

In the last decade, bulk metallic glasses (BMG) have been extensively studied since it is now possible to produce them with relatively large dimensions. In particular, the mechanical properties at room temperature of BMG have focused interest since these materials generally exhibit high value of the fracture stress with a particularly large elastic domain [1]. BMG also exhibit a superplastic-like behavior in the supercooled liquid region (SLR) [2,3]. It has been shown that the homogeneous viscous flow behavior of BMGs in SLR can be Newtonian or non-Newtonian, depending on testing temperature and strain rate [4,5]. The Newtonian flow (i.e. flow stress linearly dependent on the applied strain rate or viscosity independent on strain rate) is typically obtained at high temperature and low strain rates. The transition from Newtonian to non-Newtonian flow behavior when the strain rate is increased or when the temperature is reduced, has been attributed to stress-induced formation of defects in the bulk glassy alloy [6], which limits the increase of the flow stress.

However, since their elaboration requires relatively rapid solidification from the melt, BMG structures are out of equilibrium with respect to crystalline state. While reheated into the SLR, crystallization and/or transformation into other metastable phases may occur, leading to changes of the mechanical behavior at ambient as well as high temperatures. The effect of crystallization on the fracture stress at room temperature depends mainly on the volume fraction and the size of the crystals. When the crystallite size is no more than few tens of nanometers, the fracture stress often increases slightly until a critical content in crystallites and then falls significantly [7][8][9]. At high temperature, the effect of nanocrystallization on deformation remains poorly analyzed. On the Vit1 alloy (Zr 41.2 Ti 13.8 Cu 12.5 Ni 10 Be 22.5 , at. %), Waniuk et al. [10] carried out bending tests in the SLR and detected after an incubation time, a huge increase of the viscosity. This hardening was attributed to several causes: the formation of nanocrystals in the viscous matrix but also a phase decomposition which may occur in this alloy before crystallization or a change in the composition of the residual glass as far as crystallization occurs. Similar reinforcement due to nanocrystallization were also observed on different BMGs bases [11,12].

The coupling between deformation and crystallization was also partially studied recently and some studies have shown that high temperature deformation can accelerate crystallization kinetics [13,14,[START_REF] Gun | Formatted: Font: Bold Formatted: Font: Bold Formatted: Font: Bold 11[END_REF]: for instance, after tensile tests on a Zr based BMG, nanocrystals were observed in the sample gauge length whereas the alloy was still amorphous in the non deformed region. Conversely, in the case of the formation of quasicrystals, it was observed that deformation in the non Newtonian regime could slow down the quasi-crystallization kinetics while the deformation in the Newtonian regime had no detectable influence [START_REF] Saida | [END_REF]. Quasicrystallization requiring an icosahedral local structure in the glass, if viscous flow strongly increases structural disorder, this may hinder quasi-crystallization.

The aim of this paper is thus to get experimental information about the influence of the crystallization on the high temperature deformation and more precisely to investigate the interactions between high temperature deformation and crystallization in the Vit1 BMG.

Experimental procedure

The investigated alloy was supplied by Howmet Corp. (USA). Differential scanning calorimetry (DSC) at a heating rate of 10 K/min. was used to determine the characteristic temperatures and to get information about the crystallization energies. From DSC measurements, a temperature of 628 K was found for the onset of the glass transition temperature T g on in agreement with previously published data [10]. High temperature deformation was studied by compression tests in air. For the compression experiments, cylindrical samples of 3 mm in diameter and 5 mm in height were machined from a plate of about 3 mm thickness. The samples were heated to the testing temperatures at a heating rate of 20 K/min and held at the testing temperature for about 300 s to homogenize temperature before starting compression. During testing, the temperature is stabilized to ± 1 K. A thin superficial oxidized layer appears during testing but to thin to affect the macroscopic behavior. The temperature range of interest was between 623 K and 683 K. In this range, the glass was expected to be fully relaxed when deformation started. Indeed, structural relaxation has been studied in detail for the Vit1 alloy [6,17,18]: at 643 K, a relaxation time of about 20 s has been reported whereas a relaxation time close to 300 s is expected only for temperatures close to 623 K. X-Ray Diffraction (XRD) with Cu K radiation was used to investigate the microstructure evolution after static and dynamic annealing at the given temperatures. Transmission electron microscopy (TEM) was also carried out, the observations being performed at 300kV in a JEOL 3010 microscope. An ion beam milling method was used to prepare the thin foils.

Mechanical behaviours in the SLR

In order to get information about the rheology of the glass, strain rate jump tests were carried out for temperatures between 623 K (i.e. T g on -5 K) and 683 K (i.e. T g on + 55 K) and using the experimental data from the strain rate increase tests, the apparent viscosity app was calculated and the evolution of app with the applied strain rate is shown in figure 1 for the investigated temperatures. Depending on the temperature, the strain rate interval was more or less extended in order to avoid crystallization during the tests. To check that, a strain rate applied during the first jumps was repeated at the end of each test and the resulting stress was compared to the value initially obtained. These comparisons confirmed that in the studied experimental conditions, no significant hardening occurred during the strain rate jump tests. For the investigated conditions, the apparent viscosity varies roughly from 10 9 Pa.s to 10 12 Pa.s. Since the glass is expected to be fully relaxed when the test starts, the Newtonian viscosity which is measured corresponds to an equilibrium viscosity. For the same glass deformed in compression, a Newtonian viscosity close to 4x10 10 Pa.s was reported at 643 K [6] which is in good agreement with the value of about 4x10 10 Pa.s measured in the present work at 646 K and for strain rates slightly lower than 5x10 -4 s -1 . Apparent viscosities equal approximately to 3x10 10 Pa.s for bending tests at 643 K in the first 20 minutes of testing were also reported for the same alloy [10].

As usually obtained during high temperature deformation of bulk metallic glasses, a Newtonian domain is detected for high temperature and/or low strain rates. At a given temperature, the viscosity tends to decrease for high strain rates, the mechanical behavior becoming then non Newtonian. In the studied experimental domain, the critical strain rates corresponding to the transition from Newtonian to non-Newtonian regimes at a given temperature are in good agreement with previously published results [6]. For the lowest temperatures (T 633 K), it becomes more difficult to obtain direct experimental measurements of the Newtonian viscosity. In such conditions, it is fruitful to use a master curve in order to get an estimation of the Newtonian viscosity as performed by Bletry et al. [19]. This extrapolation confirms that the Newtonian viscosity is very sensitive to temperature since under the assumption of an Arrhenius law N = 0 exp(Q app /kT), a value of an apparent activation energy Q app of about 4.5 eV can be measured, in agreement with previously reported values of Q app for Zr-based BMG deformed in the SLR [5,20,21]. From these data, two temperatures were selected: 646 K and 683 K. This choice was motivated by the ability to have information about the thermal stability of the Vit1 during the mechanical tests. Figure 2a displays the evolution of the flow stress with strain for the following rates (1.5x10 -4 s -1 , 2x10 -4 s -1 and 3x10 -4 s -1 ) at 646 K. For all the applied strain rates, at least two domains can be detected for all curves: a first region in which the stress is roughly constant followed by a region where the stress significantly increases. In the first region (up to a strain of about 0.2), a quasi-Newtonian behavior is measured, in agreement with the results coming from the strain rate jump tests (see figure 1). The situation is more unusual for larger strains. For a strain close to 0.5, similar flow stresses are obtained for the three investigated strain rates and for larger strains ( > 0.5), apparent negative strain rate sensitivity parameters are measured. For instance, at = 1.0, the flow stress is reduced from 130 MPa to 70 MPa when the strain rate is multiplied by a factor 2. Since hardening is attributed to crystallization which may occur at this temperature, this result confirms the idea that care must be taken in the choice of the conditions of forming of BMG in the SLR if partial crystallization cannot be avoided.

[Insert figure 2a here: Flow stress vs. strain at 646 K for various strain rates] [Insert figure 2b here: Apparent viscosity vs. time at 646 K for various strain rates] These data can be rationalized if curves ( ,t) (with the apparent viscosity) are drawn instead of ( , ) (see Figure 2b). For the three investigated strain rates, similar curves are observed. Up to an incubation time t inc of about 1800 s (t inc appears independent on the applied strain rate), viscosities close to 4x10 10 Pa.s are measured. For t > t inc (keeping in mind that the compression tests start after 300 s of temperature stabilization), the viscosities increase similarly whatever the strain rate is. The mechanical test at the highest strain rate (3x10 -4 s -1 ) had to be stopped after a total annealing time of 5000 s due to maximum load considerations. These results are in agreement with Waniuk et al. [10], who reported also apparent viscosities equal approximately to 3x10 10 Pa.s at 643 K in the first 20 minutes of testing followed by a huge increase of viscosity. This increase in viscosity can be measured by a reinforcement factor R defined as the ratio of the viscosity after a given time by the viscosity of the glass before crystallization.

Figure 2-c shows the effect on the R factor of heat treatments at the same temperature before compression when the glass is deformed at 1.5x10 -4 s -1 . it is worth noting that when the strain rate is fixed, plotting an evolution against stress or against viscosity is similar. Two more heat treatment time before compression were selected: 3600 s and 7200 s. In both cases, when the glass is deformed, the R curve superposes with the one corresponding to the alloy deformed continuously. This behavior supports the idea that deforming the sample does not modify significantly mechanical properties and consequently suggests that crystallization is probably not strongly affected by deformation.

[Insert figure 2c here: Reinforcement factor vs. time at 646 K and 1.5x10 -4 s -1 for various incubation times at 646 K: 300 s, 3600 s and 7200 s] Similar mechanical characterizations were carried out for compression tests at 683 K at a strain rate of 5x10 -4 s -1 . Figure 3 shows the effect on R of the duration of the heat treatment before compression. Compared to what was observed at 646 K, there is a relatively short steady state before the increase of the flow stress. Two hardening domains (before and after about 2200 s) can be distinguished where the slopes are different. Hardening can be very important since a value of R 300 can be measured between 300 s and 3000 s. The curves in figure 3 confirms that strain is not the parameter which controls the variation of the mechanical properties of the alloy since various annealing times before deformation (300 s, 900 s 1500 s and 2050 s) were performed without changing neither the reinforcement factor nor the change in the variation with time of this factor. In other words, the reinforcement factor after a given time appears to be controlled by the total time spent at 683 K and not by the strain level.

[Insert figure 3 here: Reinforcement factor vs. time at 683 K and 5x10 -4 s -1 for various incubation times at 683 K: 300 s, 900 s, 1500 s and 2050 s]

Discussion

Crystallization at 683 K and 646 K

4.1.1. Nature and size of the crystallites. As previously mentioned, important strain hardening has been already reported in the case of BMG and particularly for the Vit1 alloy [10]. Such stress hardening was mainly attributed to crystallization and phase separation effects. To get information about the crystallization kinetics at the investigated temperatures, DSC analyses at a heating rate of 10 K/min. were carried out on the as cast sample and on two samples heat treated at 646 K for 1800 s (corresponding to the time for which strain hardening starts when the glass is deformed at this temperature) and 7200 s respectively. The resulting DSC curves are shown in figure 4a. In the amorphous state, three exothermic peaks respectively at 710 K, 729 K and 777 K are observed after the SLR. These peaks are associated to three distinct crystallization events, knowing that the first one can also include phase separation [22,23]. After the holding time of 1800 s at 646 K, the first crystallization peak has been shifted to a lower temperature and has also been reduced whereas after 7200 s the first peak has totally disappeared. One can note that despite some slight variations in characteristic temperatures of the two other peaks with the holding time at 646 K, the associated crystallization energies remains roughly unchanged. Similar analyses were carried out after treatments performed at 683 K and the results are shown in figure 4b for two durations of treatments: 1200 s and 3600 s. After 1200 s, the first crystallization peak has disappeared while the crystallization energies for the two other ones remain roughly unchanged. Finally, after 3600 s, only the third peak is present. A longer heat treatment at this temperature has no major effect on the third crystallization peak. These observations confirm the idea that the increase of viscosity which has been detected during compression tests is due to the microstructural changes in the Vit1 during isothermal annealing at 646 K and 683K, including phase separation and nanocristallyzation.

[Insert figure 4a here: DSC curves at a heating rate of 10K/mn of the as cast sample and of samples annealed 1800 s and 7200 s at 646K] [Insert figure 4b here: DSC curves at a heating rate of 10K/mn of the as cast sample and of samples annealed 1200 s and 3600 s at 683 K]] Figure 5ab display bright field TEM observations after isothermal treatment for 600 s and 2700 s at 683 K. After 600 s, dispersed crystals can already be observed with an average size of 30 nm (the particle size distribution appearing relatively narrow) whereas after a holding time of 2700 s, the number of crystals increases but without significant change of the average size. Figure 5b suggests that even after a holding time of 2700 s at 683 K, the crystal volume fraction is high but the alloy is not fully crystallized. Figure 5c displays a dark field TEM observation of a sample annealed 7200 s at 646 K, suggesting the presence of nanocrystals with a mean size less than 10 nm in the amorphous matrix. This size is in agreement with measurements carried out by Löffler et al. [23], who concluded that the size was constant up to about 6 hours of maintain at this temperature. Figure 6 shows the variations of the XRD profiles with the isothermal holding time at 683 K. After 600 s, some small peaks can be detected suggesting that crystallization process has started, which is in agreement with the TEM observation (figure 5.a) and with the hardening observed during the compression tests carried out at this temperature. After 1200 s, crystallization is more obvious since three significant peaks can be identified. These peaks are attributed to an icosahedral phase [22]. For longer holding times (t 1800 s), additional peaks are detected, revealing in particular the presence of Be 2 Zr and Zr 2 Cu crystals [22,24,25]. One can note that the icosahedral phase is still present after 3600 s, as confirmed by the high resolution TEM observation coupled with its Fourier transform shown in figure 7ab, where the typical five field symmetry of I-structure can be observed.

Volume fraction of crystallites.

In order to achieve a quantitative analysis of the reinforcement induced by the presence of nanocrystallites in the studied metallic glass, it is necessary to measure as accurately as possible the volume fraction of nanocrystals. Usually, this fraction is calculated by differential scanning calorimetry analyses [7,26,27] according to:

0 0 ( ) / ( ) t v f H t dt H t dt = (1)
where H(t) is the released heat flux. This equation assumes that the enthalpy of crystallization does not change with the degree of crystallization and that only one kind of crystals dominates the crystallization process. It also assumes that the alloy is fully crystallized at the end of the transformation, those assumptions being often experimentally not satisfied.

Other techniques like for instance density or resistivity measurements can be used but they require to know associated characteristics of the crystallites, which are frequently unknown [28]. One fruitful way to obtain information about the volume fraction of crystallites in partially crystallized metallic glasses is based on the integrated intensity of the X-ray diffraction pattern, as shown by Wesseling et al. [29]. This technique of quantification was performed in the present investigation and despite the small size of the crystallites, the deduced values were validated by a comparison with the values derived from TEM dark field observations of the partially crystallized samples [30]. The corresponding values of the volume fraction f v of crystals are given in table 1 for the XRD patterns of figure 6 and for an annealing time of 7200 s at 646 K.

[Insert table 1 here: Variation of crystal volume fraction calculated thanks to the XRD patterns with maintains time at 683 K and for 7200s at 646 K (errors are due to the area calculation in the XRD patterns)]

Effect of deformation on crystallization?

Before analyzing the various causes of the viscosity increase during deformation, one must wonder about the effect of deformation on crystallization since, as already mentioned, such effects have been reported for various BMGs. deformation carried out at 5x10 -4 s -1 and the other one in static conditions (i.e. no deformation). The two patterns are very similar with the same main peaks, suggesting no detectable effect of deformation on the nature of the crystallites. Moreover, the calculated volume fraction of crystallites corresponding to these XRD patterns show no significant differences since in both cases, a value of f v 0.45 is measured. This value is in agreement with the DSC scans (at 10 K/min) shown in figure 9. These scans correspond to four samples heat treated at 683 K for two durations (1800 s and 2500 s), in both static (i.e. no deformation) and dynamic (deformed at a strain rate of 5x10 -4 s -1 ) conditions. For the two durations, the curves associated with static and dynamic conditions appear quite similar: two peaks are systematically detected. The first peak corresponds to the remaining of the second crystallization event in the amorphous sample and the second one is due to the third crystallization peak in the amorphous sample. Some differences in the peak temperatures can be nevertheless detected for the samples annealed 1800 s. One can also note a slight difference between static and dynamic conditions in the remaining crystallization energy of the first remaining peak. For the two studied durations, the remaining energy of the non deformed sample is a little bit smaller suggesting that strain slightly reduces the crystallization kinetics. In the present study, the loss of the thermal stability of the glass is associated with the production of quasi-crystals. Since quasi-crystallization is expected to be promoted by the presence of an icosahedral short range order (ISRO), if viscous flow increases structural disorder, it may hinder this quasi-crystallization. Such effect has been previously reported in the case of Zr 65 Al 7.5 Ni 10 Cu 12.5 Pd 5 BMG deformed by compression in the SLR [START_REF] Saida | [END_REF]. These analyses support the idea that deformation in the investigated experimental domain has only a marginal effect on crystallization in the studied alloy. It also confirms the results ( figure 2c and3) that stress (or viscosity) level after a given time appears to be controlled by the total spent time at the temperature of deformation and not by the strain level.

[Insert figure 8 here: XRD spectra of two samples deformed and undeformed during 3600 s at 683 K] [Insert figure 9 here: DSC scans of samples deformed at 5x10 -4 s -1 and undeformed at 683K during respectively 1800 s and 2500 s]

Reinforcement effects due to partial crystallization

Various mechanical models can be used to predict the reinforcement effect resulting from the increase of particle volume fraction in a viscous medium. However, these models generally require hypotheses related to the characteristics of both the particles and the matrix. Concerning particles, rounded shapes and rigidity are frequently assumed. In the present study, TEM observations of partially crystallized samples before and after large strains ( 1.5) at high temperature did not reveal any significant change in the morphology of the crystallites [30]. This suggests consequently that the crystallites can be considered as rigid particles for the studied conditions of deformation. Moreover, as previously mentioned, quite spherical crystallites are also observed by TEM. Concerning the matrix, it is generally assumed that its characteristics do not vary when the volume fraction of particles increases. In the present case, since the change in particle volume fraction results from crystallization processes, it is of prime importance to get information about a possible change of the composition of the residual amorphous phase. Indeed, such a change can contribute to harden the alloy. For instance, if T g increases because of a change in the composition, it will induce a decrease of the ratio T/T g at the given temperature of deformation. A change of the glass transition temperature can be associated to a progressive solute enrichment of the residual amorphous phase during crystallization. Some authors have suggested that beryllium could be rejected from the growing crystals in Vit1 [25,31], since beryllium enrichment tends to increase T g in ZrTiCuNiBe BMG [32]. In the present investigation, as illustrated by the DSC curves (figures 4a and 4b), only very slight increases of T g were measured after maintains at 646 K or 683 K, typically less than 3 or 4 K. One must however indicate that for long maintains at 683 K (typically longer than 1800 s, measuring T g was no longer possible. If the measured apparent activation energy Q app of the Newtonian viscosity for the amorphous alloy is taken into account, such a change of T g would result in a reinforcement coefficient of about 1.3. It means that this matrix effect remains limited for the studied metallic glass in the selected conditions of deformation.

These comments suggest that simple mechanical models can be reasonably used to predict the reinforcement effect resulting from partial crystallization of the studied BMG. The simplest one is probably the Einstein relationship for which the reinforcement factor R associated to a composite with a crystal volume fraction f v is given by: ( )

0 1 2.5 V R f = = + (2)
with the flow stress of the composite and 0 the flow stress corresponding to the as-provided glass. This relationship has been frequently used in the case of partially crystallized BMGs [10,14]. However, relation (2) leads to very limited reinforcement effect compared to those measured experimentally in this work. For instance, a reinforcement coefficient less than 1.5 is predicted by relation ( 2) when f v 0.15. Such a volume fraction of crystallites is roughly measured after 7200 s of maintain at 646 K (either in static conditions or during continuous straining) and results in a reinforcement factor close to 7. Similar conclusions can be also drawn for other values of f v . Therefore the Einstein's relationship is probably not adapted to account for the reinforcement due to crystallization in the studied metallic glass, even for relatively small values of f v (typically f v < 0.2).

A more general (but less simple) relationship to describe the mechanical behavior of a suspension of particles in a viscous matrix, is given by [33,34], noting that for small crystal volume fraction, this equation is equal to the Einstein's relationship:

max 5 . 2 max 1 V V eff V R = (3)
with eff V the effective volume fraction of particles and max V the maximum packing fraction.

max V depends on the size and shape distribution of the particles. In the case of randomly dispersed spherical particles with the same diameter, max V corresponds to the random close packing (RCP) value, namely max V 0.64. Since in the present investigation, the crystallites are nearly spherical with a narrow size distribution, this assumption is expected to be satisfied. In the present work, as a first approximation, eff V is supposed to be equal to f v . Under this assumption, figure 10 compares the variation with the crystallite volume fraction of both the experimental reinforcement factor (for testing carried out at 683 K) and the factor predicted by relation (3). Except for f v = 0.07 for which the two values are in relatively good agreement, the experimental reinforcement appears much stronger than the predicted one. For f v = 0.32, the measured reinforcement factor is for instance nearly 90 times greater that what is predicted. One must noted that the simplicity of the model is not the cause for this large discrepancy since, as demonstrated by Wolff et al. [26] in a Mg based BMG, more sophisticated analyses lead to similar discrepancy. Values are given for various annealing times: 600 s, 1200 s, 1800 s and 2700 s] One must keep in mind that relations (2) and ( 3) are valid only for matrix displaying a Newtonian rheology (i.e. constant viscosity with various strain rates). One can wonder whether such a rheology still holds as far as the glass crystallizes. In order to get information about the rheology of the partially crystallized alloy, strain rate jump tests were carried out on pre-treated alloys. However, at 683 K, the continuous crystallization does not allow performing strain rate jumps after various annealing times with a constant microstructure. Consequently, only tests at 646 K were carried out. The variation of the apparent viscosity with the strain rate for an alloy pre-treated during 7200 s at 646 K is displayed by figure 11 and compared to the behaviour of the glass in as-provided conditions. The viscosities corresponding to the continuous deformation experiment at this temperature are also shown on figure 11 and suggest again that deformation does not affect significantly crystallization in the present studied conditions. The comparison between the as-provided and the pre-annealed alloy clearly shows that, whereas the behaviour is Newtonian for the as cast sample and for applied strain rates lower than 3x10 -4 s -1 , the viscosity is no longer constant with strain rate when the glass is pre-annealed for 7200 s at 646 K. This change in rheology means that the reinforcement factor which is experimentally measured is probably lower than what would be measured if a Newtonian regime could be preserved despite the crystallization process. In other words, the underestimation of the reinforcement factors predicted by relation (3) compared to the experimental ones is probably even greater than what is displayed by figure 10.

[Insert figure 11 here: Relation between viscosity and strain rate at 646K in compression for the as cast sample and for the samples annealed 7200s at 646K with and without deformation] Such underestimation of the values predicted by relation (3) compared to the experimental reinforcement factors can be related to the hypothesis eff V = f v . For instance, it is known that geometrical percolation of particles can occur for particle volume fraction close to 0.16 in the case of spherical particles with the same diameter and randomly distributed in a given matrix. However, such a geometrical percolation does not systematically induce a "mechanical" percolation. Indeed, in order to get a "mechanical" percolation, interfacial bonds between the particles are needed and the effect on the mechanical behavior of the composites will be directly related to the strength of the interfaces. In the present investigation, if a geometrical percolation cannot be excluded (despite the fact that crystal clustering is not observed by TEM), it is not expected to result in a significant "mechanical" percolation because crystallites appear spherical and without well established grain-(or phase-) boundaries between them. Moreover, if a skeleton of crystallites is assumed, it is expected to induce apparent strain softening in the first strain steps resulting from its breakdown. Such behaviors were not observed for the partially crystallized samples in the studied experimental conditions as shown in figure 2c and3. eff V may also differ from f v if some volumes of matrix are triggered in particle agglomerates or if the matrix in the direct vicinity of the particles displays specific mechanical properties. Such concepts have been already pointed out in the case of particles agglomerates considering that the matrix which is triggered in such agglomerates cannot contribute efficiently to the macroscopic strain [34]. Changes in mechanical behavior of the glass located at the vicinity of the particles have been already reported to explain the unusual reinforcement observed for polymers reinforced by nanoparticles [35,36]. For such nanocomposites, it has been suggested that the macromolecular chains which are anchored on the particle surface could have a reduced mobility compared to the bulk matrix [START_REF] De Gennes | La matière ultradivisée' in 'L'ordre du chaos[END_REF], an effect which can be huge in the case of nanometric particles. If a similar hypothesis is considered in the present work, the experimental reinforcements can be described by relation (3) as illustrated by figure 10, which displays also the predicted reinforcement under the assumption of a thickness of non-deformable matrix around each nanocrystal equal to 4 nm. This curve supports the idea that such effect could account for the reinforcement observed in the partially crystallized BMG even if the origin of such an effect in metallic glasses remains an open question.

Conclusion

The interactions between deformation and crystallization were investigated in a bulk metallic glass (Vit1). In the studied conditions, it is shown that the effect of crystallization on the flow stress at high temperature depends mainly on the time spent at the given temperature without significant strain effect. The crystallization was studied by different techniques and the volume fraction of crystals was measured using the XRD patterns for various heat treatments. According to a cross analysis of data, we conclude that the deformation does not change significantly the volume fraction of crystal, the crystal composition neither the crystal size.

The crystallization (in static or dynamic annealing) leads to a strong reinforcement effect which cannot be explained by a dispersion of rigid particles in an amorphous matrix. Additional reinforcement possible origins were discussed: change in residual glass, contribution of the crystals to deformation, connection between particles and nanometer crystallites size effect were proposed. It finally seems that this effect is mainly due to the nanometric size of the crystals and could be taken into account by introducing a non deformable amorphous layer around each crystallite.

F

Figure 4a

DSC curves at a heating rate of 10K/mn for the as cast sample, and for samples annealed 1800 s and 7200 s at 646K.

[ 6 [

 6 Insert figure 5a here: Bright field TEM observations of a sample heat treated at 683 K for 600 s] Insert figure 5b here: Bright field TEM observations of a sample heat treated at 683 K for 2700 s] [Insert figure 5c here: Dark Field TEM observation of a sample annealed 7200 s at 646 K]

[

  Insert figure 6 here: XRD spectra after various maintains at 683K] [Insert figure 7a here: High resolution TEM observation of a crystallite after a holding time of 3600 s at 683 K] [Insert figure 7b here: Associated Fast Fourier Transform showing an icosahedral structure of the crystallite after a holding time of 3600 s at 683 K]

Figure 8 displays

 8 XRD patterns of samples heat treated for 3600 s at 683 K, one under

[

  Insert figure 10 here: Curves represent the reinforcement factor versus the crystallized volume fraction according to the Krieger's model with or without modified matrix around the crystals. A modified matrix layer of 4 nm is assumed in the left case. The filled squares represent the lower bound of the reinforcement factor in Newtonian regime during the compression tests at 683K time of 300 s.

13 FIGURESFigure 1 :Figure 2a :Figure 3 :Figure 4a :

 1312a34a FIGURESFigure 1: Viscosity vs. strain rate in compression for the amorphous alloy at various temperatures Figure 2a: Flow stress vs. strain at 646 K for various strain rates Figure 2b: Apparent viscosity vs. time at 646 K for various strain rates Figure 2c: Reinforcement factor vs. time curves at 646 K and 1.5x10 -4 s -1 for various incubation times at 646 K: 300 s, 3600 s and 7200 s Figure 3: Reinforcement factor vs. time curves at 683 K and 5x10 -4 s -1 for various incubation times at 683 K: 300 s, 900 s, 1500 s and 2050 s Figure 4a: DSC curves at a heating rate of 10K/mn of the as cast sample, and of samples annealed 1800 s and 7200 s at 646K.Figure 4b: DSC curves at a heating rate of 10K/mn of the as cast sample, and of samples annealed 1200 s and 3600 s at 683 K.

Figure 4b :

 4b Figure 4a: DSC curves at a heating rate of 10K/mn of the as cast sample, and of samples annealed 1800 s and 7200 s at 646K. Figure4b: DSC curves at a heating rate of 10K/mn of the as cast sample, and of samples annealed 1200 s and 3600 s at 683 K.

Figure 5a :Figure 6 :

 5a6 Figure 5a: Bright field TEM observations of a sample heat treated at 683 K for 600 s Figure 5b: Bright field TEM observations of a sample heat treated at 683 K for 2700 s Figure 5c: Dark Field TEM observation of a sample annealed 7200 s at 646 K Figure 6: XRD spectra after various holding times at 683K.

Figure 7a :Figure 8 :Figure 9 :Figure 10 :

 7a8910 Figure 7a: High resolution TEM observation of a crystallite after a holding time of 3600 s at 683 K Figure 7b: Associated Fast Fourier Transform showing an icosahedral structure of the crystallite after a holding time of 3600 s at 683 K Figure 8: XRD spectra of two samples deformed and undeformed during 3600 s at 683 K Figure 9: DSC scans of samples deformed at a strain rate of 5x10 -4 s -1 and undeformed at 683K during respectively 1800 s and 2500 s Figure 10: Curves representing the reinforcement factor versus the crystallized volume fraction according to the Krieger's model with or without modified matrix around the crystals. A modified matrix layer of 4 nm is assumed in the left case. The filled squares represent the lower bound of the reinforcement factor in Newtonian regime during the compression tests at 683K with an incubation time of 300s. Values are given for various annealing times: 600s, 1200s, 1800s and 2700s.

Figure 11 :

 11 Figure 11:Relation between viscosity and strain rate at 646K in compression for the as cast sample and for the samples annealed 7200s at 646K with and without deformation.

Figure 1 1 Figure 2

 112 Figure 1Viscosity vs. strain rate in compression for the amorphous alloy at various temperatures

  Figure 4bDSC curves at a heating rate of 10K/mn of the as cast sample, and of samples annealed 1200 s and 3600 s at 683 K.

Figure 6 26 Figure 7b :Figure 8 Figure 9 Figure 10 Figure 11

 6267b891011 Figure 5a: Bright field TEM observations of a sample heat treated at 683 K for 600 s

TABLE Temperature

 Temperature Variation of crystal volume fraction calculated thanks to the XRD patterns with maintains time at 683 K and for 7200s at 646 K (errors are due to the area calculation in the XRD patterns).

	(K)		683			646
	Annealing time (s)	600	1200	1800	2700	7200
	f v	0.07 ±0.03 0.17 ±0.03 0.27 ±0.03 0.32 ±0.03	0.16 ± 0.04
	Table I

http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

Acknowledgements

The authors acknowledge Dr. A. Mussi for his help in some thin foils preparation and TEM observations.


J Woirgard

V Audurier

C Tromas

Elastic stress field beneath an arbitrary axisymmetric punch

Keywords: hardness, mechanical testing, nanoindentation user supplied): hardness, mechanical testing, nanoindentation

The nanoindentation test is commonly used for the local determination of mechanical properties (hardness, Elastic modulus, …) and also to study the initial stages of plasticity (dislocation nucleation, dislocation interaction mechanisms) at nanometer scale. In that case, the determination of the elastic stress field beneath the indenter is of primary interest. The purpose of this paper is to give an analytical expression for the elastic stress and strain fields beneath an axisymetric punch. Most solutions, in the literature, are given for simple indenter shapes, such as flat, conical or spherical indenters.

Introduction

Over the last two decades, the nanoindentation test has become a common technique for the determination of local mechanical properties such as hardness and elastic modulus [START_REF] Fischer-Cripps | Nanoindentation, 2nd edR[END_REF][START_REF] Oliver | [END_REF][3][4]. The principle consists in applying a low load with an indenter on the surface while the progressive penetration depth is monitored.

The main experimental parameter for hardness and elastic modulus determination is the contact area between the indenter and the surface under load. Several models have been proposed to determine this crucial parameter through the analysis of the unloading curve shape. One of the first attempts has been given by Doerner and Nix [6] who considered that the contact area was not modified during an elastic unloading, as for a flat punch. In a more realistic approach, , Oliver and Pharr [START_REF] Oliver | [END_REF] proposed to determine the initial value of the contact area by taking into account its variation during elastic unloading. For this purpose, the principle of the equivalent indenter, assumed to be parabolic was used. This principle, initially introduced by Solomon [START_REF] Solomon | Elasticité Linéaire[END_REF], suggests that the elastic unloading of an indenter on a plastically deformed surface is equivalent to that of a "modified" indenter giving the same displacement when pressed against a plane. However, the shape of the equivalent indenter which depends on the plastic behaviour of the material, is more complex than a paraboloid and should be determined for each unloading. One method consists in describing the equivalent indenter shape by a polynomial [4,[START_REF] Schwarzer | [END_REF][9] in cylindrical coordinates ( ,z):

= n n n C ) ( z (1)
In this case, for a rigid indenter the load P and the displacement h are functions of the contact radius a [10][11] :

( ) n n n a C 2 1 2 n 1 2 n a h + + = (2) ( ) n n n r a C 2 1 2 n 1 2 n 1 n n E 2 a P + + + = (3)
where is the gamma function and E r is the reduced modulus defined by The corresponding load-displacement relationship being:

( ) ( ) n 1 1 n 1 r h 1 2 n 2 1 2 n 1 n n C E 2 h P + + + + = (9)
The relation between the unloading curve exponent m and the indenter shape exponent n is thus given by:

1 n m 1 = ( 10
)
This method is commonly used to estimate the true penetration depth and thus the contact area under load A c . The hardness H and the reduced elastic modulus E r are then easily determined through the equations

c P H A = ( 11
) c r A S 2E = (12)
Beside the determination of elasto-plastic characteristics of materials, nanoindentation is also of fundamental interest for studying elementary deformation mechanisms and incipient plasticity, involving small material volumes and few dislocations [12][13][14][15][16][17][18][19][20][21]. The precise knowledge of the stress field generated in the material is required to understand the individual behaviour of these defects [20]. The surface stress has been previously calculated in the case of an elastic contact between a flat surface and an axisymetric indenter [START_REF] Sneddon | Fourier Transforms[END_REF][START_REF] Johnson | Contact Mechanics[END_REF], or a more complex indenter described by a power function with a non integer exponent n [4,5,[START_REF] Fabrikant | [END_REF]. This latter case is of great interest since it can be applied to any unloading curve, even for plastically deformed material, through the concept of the equivalent indenter [START_REF] Solomon | Elasticité Linéaire[END_REF].

In the following, analytical solutions will be proposed for the calculation of the complete stress field in the case of arbitrary shaped axysimetric indenters.

Theoretical results

Close form expressions of the stress field, either on the surface or in the volume, beneath simple indenters, including flat punches, cones or spheres have been previously published [START_REF] Hamilton | Proc. Instn. Mech. Engrs[END_REF][START_REF] Sackfield | [END_REF][27][28][29][30], but very few results are available for more complex profiles, except a recent work of Schwarzer [31][32] proposing a partial solution in the case of axisymetric punches described by polynomials.

However this theory, named the "extended Hertzian theory", is limited to even power of r which cannot be applied to real equivalent indenters with non integer exponent. For more complex shapes, the only results, to our best knowledge, have been obtained by finite element methods (FEM) [33][34].

In this work, Sneddon's [START_REF] Sneddon | Fourier Transforms[END_REF]35] results, will be used to determine the complete stress fields in the quite general case of power laws with exponents integer or not. Sneddon [36] showed that the normal stress and displacements can be derived from the f(t) function, solution of the dual integral equation:

( )

n n 0 0 z r ca h ) r ( f dt ) t r ( J ) t ( f a ) 2 ( r u = = µ µ + = for 1 r (13) ( ) 0 dt ) t r ( J ) t ( f t ) ( 2 r 0 0 z = µ + = for 1 r (14) With a r = (15)
And and µ the Lame's coefficient. To find a f(t) function satisfying the preceding equations, Sneddon proposed to introduce the function g(u) obeying:

= 1 0 du ) t u cos( ) u ( g ) t ( f (16)
With this form of function g(u) we find that: (17) and integrating by parts:

= 1 0 0 0 0 0 du ) ut cos( ) u ( g t dt ) t r ( J dt ) t r ( J ) t ( f t
= 1 0 0 0 0 0 0 0 du ) ut ( sin u ) u ( g dt ) t r ( J dt ) t sin( ) t r ( J ) 1 ( g dt ) t r ( J ) t ( f t ) 1 r ( > (18)
Interchanging the order of integration in the double integral and knowing that:

0 dt ) t u sin( ) t r ( J 0 0 = ( 19
)
we see that equation ( 14) is satisfied and we must have g(1)=0 since:

0 z = for r=1
Equation ( 13) can be written: If we interchange the order of integration and make use of: [START_REF] Sneddon | Fourier Transforms[END_REF] we see that g(u) is a solution of the integral equation:

n n 1 0 0 0 r ca h du ) t u cos( ) u ( g dt ) t r ( J a ) 2 ( = µ µ + (20)
2 2 0 0 u r 1 dt ) t u cos( ) t r ( J = ) u r ( > (21) 0 dt ) t u cos( ) t r ( J 0 0 = ) u r 0 ( < <
n n r 0 2 2 r Ca h u r du ) u ( g a ) 2 ( = µ µ + (23)
We can try for g(u):

n u B A ) u ( g + = (24)
and equation ( 20) may be written :

n r 0 2 2 r ) 1 2 n ( 2 ) 2 1 n ( B 2 A u r du ) u ( g + + + = (25)
We finally get: (28) and:

h a ) 2 ( 2 A µ + µ = (26) n Ca ) 2 1 n ( ) 1 2 n ( 2 a ) 2 ( B + + µ + µ = (27) Since g(1)=0 we have: n Ca ) 2 1 n ( ) 1 2 n ( h + + =
) u 1 ( Ca ) 2 1 n ( ) 1 2 n ( a ) 2 (
+ + µ + µ = 1 0 n n du ) tu cos( ) u 1 ( a C ) 2 1 n ( ) 1 2 n ( a ) 2 ( 2 ) t ( f (30)
Let k be : (31) We see that f(t) is the real part of the complex quantity:

n a C ) 2 1 n ( ) 1 2 n ( a ) 2 ( 2 k + + µ + µ =
= 1 0 iut n * du e ) u 1 ( k ) t ( f (32)
To compute the stress field we have to evaluate the following complex integrals:

dt ) rt ( J ) t ( f t e I 0 * 0 zt * = (33) dt ) rt ( J ) t ( f t e K 1 * 0 zt * = (34)
and the components of the stresses are the real parts of the complex stresses:

) z I z z I )( ( k 2 2 * 2 * * z + µ + = (35) 2 * 2 * r z K z ) ( k 2 µ + = (36) ) z K z K ( k r 2 z I k 2 * * * * µ + µ + µ + = (37) ) z I z z I 2 )( ( k 2 2 * 2 * * * r + µ + µ + µ + = + (38)
These components of the stress are easily derived from the I * and K * integrals and their derivatives relative to z: Finally the integrals and their derivatives take the form:

2 2 2 2 * 0 ) z i ( r z i z r z ln i I + + + + + + = (49) 2 2 2 2 * 0 ) z i ( r i z r i z I + + + = (50) 2 3 2 2 2 3 2 2 2 * 0 2 ) z r ( z i ] ) z i ( r [ 1 z i z I + + + = (51) r ) z i ( r i z r i 1 K 2 2 2 2 * 0 + + + + = (52) F o r P e e r R e v i e w O n l y 2 3 2 2 2 3 2 2 * 0 ) z r ( r z i ] ) z i ( r [ r 1 z i z K + + + = (53) 2 3 2 2 2 3 2 2 2 * 0 2 ) z r ( z r i ] ) z i ( r [ ) z i ( r i z K + + + + = (54)
Let us note, with F 1 (a,b,c,d,z 1 ,z 2 ) the Appell hypergeometric function:

) z i r 1 , z i r 1 , n 2 , 2 1 , 2 1 , n 1 ( F F 1 2 1 , 1 n + + + + = + ( 55
)
) z i r 1 , z i r 1 , n 3 , 2 1 , 2 1 , n 2 ( F F 1 2 1 , 2 n + + + + = + ( 56
)
) z i r 1 , z i r 1 , n 2 , 2 3 , 2 3 , n 1 ( F F 1 2 3 , 1 n + + + + = + ( 57
) ) z i r 1 , z i r 1 , n 3 , 2 3 , 2 3 , n 2 ( F F 1 2 3 , 2 n + + + + = + (58) ) z i r 1 , z i r 1 , n 2 , 2 5 , 2 5 , n 1 ( F
+ + + + = + (61)
The real part of the F 1 function can be easily calculated with mathematical packages. Then, we can write: The formulae are undetermined when n is an integer, but in that case we can use a recurrence relationship:

2 2 2 1 , 1 n * n z r ) n 1 ( F I + + = + (62) ) 2 n F 1 n F z ( ) z r ( 1 z I 2 3 , 2 n 2 3 , 1 n 2 3 2 2 * n + + + + = + + (63) ] F r 3 F ) z r ( 2 [ ) z r ( ) 1 n ( 1 z I 2 5 , 1 n 2 2 3 , 1 n 2 2
) F 2 n i F 1 n z ( z r r 1 r ) 1 n ( 1 K 2 1 , 2 n 2 1 , 1 n 2 2 * n + + + + + + + = (65) 2 3 , 1 n 2 3 2 2 * n F ) z r ( ) 1 n ( r z K + + + = (66) ) F 2 n i F 1 n z ( ) z r (
* 1 n * 2 n 2 2 2 2 * n I z i n 1 n 2 I ) z r ( n 1 n n ) z i ( r I + + + + + = (68) ] I z n I ) z r ( i n ) z i ( r i 1 [ r ) 1 n ( 1 K * n * 1 n 2 2 2 2 * n + + + + + + = (69) with: 2 2 2 2 * 0 ) z i ( r z i z r z ln i I + + + + + + = (70) 2 2 2 2 2 2 2 2 * 1 ) z i ( r z i z r z ln z ) z i ( r z r I + + + + + + + + + = (71) r ) z i ( r i z r i 1 K 2 2 2 2 * 0 + + + + = (72)
Again, following equations (35)(36)(37)(38), the stress components can be derived from the real parts and their first and second derivatives relative to z of the expressions (43) and (44).

When the exponent n is an integer, the I and K integrals, real parts of the I * and K * complex ones, can be calculated knowing that:

b i a ) z i (
+ + + + = (75) 2 ) 1 z r ( z 4 ) 1 z r b 2 2 2 2 2 2 + + + = (76)
For n=0 (flat punch) equations ( 68) and ( 69) fail but we have [START_REF] Sneddon | Fourier Transforms[END_REF]:

t t sin ) t ( f = (77)
and the I and K integrals are the imaginary parts of:

2 2 0 0 t ) i z ( * ) z i ( r 1 dt ) rt ( J e I + + = = + (78) r ) z i ( ) z i ( r dt ) rt ( J e K 2 2 1 0 t ) i z ( * + + + = = + (79)
For example, the normal stress is the imaginary part of:

) ] ) z i ( r [ ) z i ( z ) z i ( r 1 )( ( k 2 2 3 2 2 2 2 * z + + + + + + µ + = (80)
When n is different from zero we can use the (68) to (69) equations.

For example, for n=1 (cone):

] z r z ) z i ( r z i ln ) z i ( r z z r z [ ) ( k 2 2 2 2 2 2 2 2 * z + + + + + + + + + + µ + = (81) For n=2 (paraboloid) : ] ) z i ( r z 2 ) z i ( r i 2 [ ) ( k 2 2 2 2 * z + + + + µ + = (82) For n=3: (83)
] z r z ) z i ( r z i ln 2 ) z 2 r ( 3 ) z i ( r z 3 ) z i ( r 2 ) z i ( 3 z r 2 z 3 [ ) ( k 2 2 2 2 2 2 2 2 2 2 2 * z + + + + + + + + + + + + + + µ + =
Values of the I and K integrals, for n=1 to 5, and listed in appendix.

Using equations (61) to (64), the normal stresses on the surface (z=0) and below the surface (z=a/2), are plotted in Figures 1 and2 for non integer values of n: 1.5, 2.5 and 3.5.

Conclusion

In this paper, analytical expressions have been proposed for the determination of the complete stress field generated by an arbitrary axisymetric indenter described by an exponent n, integer or not. This calculus is not restricted to elastic indentation, but can also be applied through the concept of the equivalent indenter to elasto-plastic contact. The elastic stress field can thus be readily obtained once the exponent m of the elastic unloading curve has been determined. One of the main interests of this study is to give the ability to determine the forces acting on individual dislocations involved in the imprint formation. As an example, this result can be applied in discrete dislocation dynamic (DDD) study of the indentation test. This result, combined with recent insights in the description of the complete microstructure generated during a nanoindentation test, represent a significant step towards the comprehension of the early plastic deformation mechanisms in materials. Normal stress on the surface, z=0, for indenters described by power laws with exponents n: 1.5, 2.5 and 3.5

  the Young's modulus and the Poisson's ratio of the sample respectively. It has been shown elsewhere[4], that with the stiffness: stiffness relation can be fitted by a polynomial: of the C n coefficients and the n exponents.Unloading curves can often be fitted by simple power law the residual penetration depth. In this case, the shape function of the equivalent indenter reduces to a single term with a non integer exponent n

  Figures captions

Figure2:

  Figure2:

Page 13 of 22 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

	Page
	Appendix
	n=0:
	I	* 0	=	i		ln	i	+	z z + +	r	2 r	2 + z i ( + + 2	z	)	2
	K	* 0	=	r 1				i	F o r F o r F o r F o r r i ( r i r z r 2 2 2 + + +	+		z	)	2
	n=1:																				P e P e P e P e
	I	* 1	=	r	2	+	z	2					r	2	+	(	i		+	z	)	2	+	z	e r e r e r e r 2 2 r z r z i ln + ( i z + + 2 + + +	z	)	2
	K	* 1	=	r 1 2	+	r z 2			r	2	+	z	2			+	i	r z + 2		r	2	+	i (	+	R e v R e v R e v R e v 2 z r z i ln r 2 1 ) z + + +	2 r	2 + z i ( + + 2	z	)	2
																																													i e i e i e i e
																																													w w w w
	n=2: * 2 2 3 I =	i	z		r	2	+	z	2			(	2 1	+	3	2 i	z	)		r	2	+		i (	+	z	)	2	(	i	2 2 r	i	z	2	)	ln	i	+	O n l O n l O n l O l n 2 2 2 2 z r ) z i ( r + + + + + z z
																																													y y y y
	K	* 2	=	(				r r i 3 2	+	2 r z 3 i	)		r	2	+	z		2		+	(	r i 3	+	3 i 2	r	r z 3	+	2 r z 3 i	)	r	2	+	i (	+	z	)	2	i	r	z	ln	i	+	z z + +	r	2 r	2 + z i ( + + 2	z	)	2

17 of 22 http://mc.manuscriptcentral.com/pm-pml Philosophical Magazine & Philosophical Magazine Letters

	n=3:
	I	* 3		=	(	2	3 r	2		6 z 11	2	)	r	2	+	z	2		(	3 1	+	2	3 r	2		6 z 11	2	+	5	6 i	z	)	r	2	+	(	i	+	z	)	2	+	(	3	2 2 r	z	z	3	)	ln	i	+	z z + +	r	2 r	2 + z i ( + + 2	z	)	2
	K	* 3	=		(	8 r 13	z	4 z	r	)	r	2	+		z	2	+	(	4 i	r	+	3	8 i	r	r z 4		8 r 13	z		4 z i	2 r	+	4 z	3 r	)		r	2	+	(	i	+	z	)	2
	8 r n=4: 3 (	3			3	2 r 2 r z r z + + + + ( + F 2 i ln ) 2 z r o	z i	2 +	z	)	2
																		P
	2 z z 12 i 25 2 i z + 2 + 3 ln ) 4 i ) ) z r i 2 z r i 2 3 24 r i 55 4 8 ( r i 3 2 z r i 3 + 3 r 5 z 15 r i 8 15 r i 4 r + + i = 3 5 2 3 4 ( 4 * ( + I ( + * 30 z r i 83 15 r i 8 ( K + =	e 2 3 4 1 ( z + + e 2 8 r r 12 i 7 + 2 2 2 2 z r z ) z i ( r z + + + + + + R r z + e 24 r i 55 v 2 e 2 12 z 13 2 2 2 2 z r z 2 i ) z i ( r z ln r + + + + + + w 25 12 z i 2 3 2 4 3 2 2 ) z i ( r ) r 5 z i r 5 z 30 z r i 83 r 5 z i 30 z r 29 + + + + i 2 2 4 z r ) r 5 z i +	)	r	2	+	(	i	+	z	)	2
	n=5:																																																O n l
																																																								y


Sylvie Lartigue-Korinek

Serge Hagege
email: serge.hagege@icmpe.cnrs.fr

Christian Kisielowski
email: cfkisielowski@lbl.gov

Anna Serra
email: a.serra@upc.edu

Sylvie Lartigue
email: sylvie.lartigue@icmpe.cnrs.fr

Korinek Corresponding

Disconnection arrays in a rhombohedral twin in -alumina

Keywords: alumina, dislocations, grain boundary structure, transmission electron microscopy user supplied): alumina, dislocations, grain boundary structure grain boundary structure, dislocations, alumina, transmission electron microscopy 1-Introduction

come

Disconnection arrays in a rhombohedral twin in

α-alumina

The interfacial defects in the rhombohedral twin are discontinuities of the perfect twin structure that have both dislocation and step-like character. Such defects are known as disconnections [START_REF] Pond | Dislocations in Solids[END_REF]. Their dislocation character is characterized by their Burgers vector, which is related to the crystal symmetries that are broken when the bicrystal is created. In rhombohedral crystals with non-rational values of c/a, pure steps with no dislocation character are not possible due to the absence of periodicity perpendicular to the interface.

In previous work, the characteristics of interfacial dislocations were analyzed on the basis of near coincidence descriptions for the rhombohedral symmetry [8,[START_REF] Lartigue-Korinek | Intergranular and Interphase Boundaries in Materials[END_REF]. In the present paper we report an exact characterization of line defects linked to rhombohedral twins using the topological theory of interfacial defects based on geometrical considerations [START_REF] Pond | Dislocations in Solids[END_REF]. The disconnections analyzed present a wide core whose structure is visualized by atomic resolution imaging using the technique of exit wave reconstruction from a focal series of high resolution images.

2-Experimental

The alumina bicrystal was prepared by diffusion bonding at 1700°C for 30 minutes [START_REF] Vasiliev | [END_REF]. The sample was first thinned by the mechanical tripod technique, then made electron transparent by a soft argon ion milling (DUO-MILL Gatan at 3 kV with a 0.5mA Ar + beam current). A conductive amorphous carbon layer was deposited by evaporation on both foil faces. The sample was finally annealed under high vacuum (10 -6 Torr) in order to prevent further contamination under the electron beam in the microscope.

[

] zone axis for one crystal with a Topcon ABT002 operated at 200kV that exhibits a Scherzer point resolution of 1.8 Å. Sub-Ångström phase and amplitude images of the complex electron exit wave function were reconstructed from a focal series of 20 lattice images recorded on a CM300 FEG/UT instrument with a 0.8 Å information limit [12]. The image simulations were performed by multislice calculations with the MacTempas and CrystalKit software packages [13]. [ ] direction. The twin structure, far from the defects, is based on the centers of symmetry of the structure (i.e. non-occupied sites of the aluminum sublattice (figure 3)). The white dots in the lattice image correspond to empty octahedral sites of the aluminum sublattice, as can be seen from the inserted geometrical model. From symmetry considerations on the various dichromatic complexes, four different structural models characterized by different translation states have been proposed [START_REF] Lartigue-Korinek | Intergranular and Interphase Boundaries in Materials[END_REF]. [ ] axis parallel to the electron beam.

3-Results and discussion

Recent experiments [15] give similar results and simulations using local density functional theory (LDFT) [16] and lattice static calculations [17] confirm the lowest energy of this "screw twin" configuration.

3-2-Interfacial defects

The interfacial defects should compensate the slight misorientation from the perfect twin structure. These defects are characterized using a Volterra approach. Their Burgers vector is given by r b = t ( ) Pt µ ( ) where t ( ) and t µ ( ) are translation vectors in the upper ( ) and the lower crystal ( µ) respectively [7]. The transformation matrix P relates the µ coordinate system to the system and was obtained using the method described in [18]. A detailed description of this approach, including the rotation matrix, is given in the appendix. Finally, the disconnection Burgers vectors are the following, expressed in the coordinate frame in units of the lattice parameter a (Miller-Bravais system) :

r b 3/ 4 = 1 3 2 + 2 ( ) 2 + 2,4 2 16, 5 2 + 14, 3 2 + 18 [ ] ,
where =(2/3) These defects must accommodate the 0.8°tilt deviation from the twin orientation. Among the possible defects indicated in the corresponding dichromatic pattern shown in fig. 6 the ones found in the experiment supply the necessary step height to reach the misorientation with a minimum of energy increase. Therefore, configurations with a small magnitude of r b and a small step height should be favored according to both the experiment and energetic criteria.

Below we discuss why some defects that obey the equation

[ ]

that is larger than r b 1/ 1 , the magnitude of the rhombohedral twin shear vector s = 0.071nm. Such r b p / q dislocations can be decomposed into r b p 1/ q 1 by emission of a r b 1 / 1 defect and are thus unstable, as already found for 10 1 2

{ } dislocations in zinc [20]. Other kinds of defects are those with no component along 2 1 1 0 [ ] namely pure edge defects. These defects have a length larger than 0.7nm and are therefore energetically unfavorable. In the end, the observed configuration corresponds to the defects with the smallest Burgers vectors at equilibrium.

3-3-Disconnection core structure

The defect cores extend from 1 to 4 nm. An analysis of the dislocation core has been performed on the CM300 FEG/UT instrument that has a resolution of 0.8Å after reconstruction of the complex electron exit wave function from a focal series of images [21]. image resolution is reduced (Fig. 10 (a)). The presence of a screw component is known to distort the core region of a dislocation [22]. Further, a local thickness variation may contribute to signal blur, as well.

A close inspection of the phase image at the dislocation core reveals that the dislocation is dissociated into two partials that delimit a twin boundary segment 2 nm wide with a different The twin is no longer centered on empty sites of the structure.

The disconnection energy is likely to be decreased by dissociation. Due to the large Burgers vector length ( 0.42nm ), close to those of the smallest translation vector of alumina (basal dislocation, r b = 0.475nm ), several atomic planes are inserted and terminate on a same 01 1 2

( )

plane. Dissociation here requires climb of the partials. In alumina, most lattice dislocations are dissociated by climb at high temperature [23]. This is also the case in low angle GBs in alumina [24], as well as in strontium titanate [25]. The movement of such dislocations has to be hindered by the presence of the stacking fault in the GB. Although the component of the Burgers vectors of the partial disconnections parallel to 2 1 1 0 [ ] is not exactly known at this stage, a new structure occurs between the partials: It seems centered on atoms of oxygen and aluminum rather than on symmetry centers of the structure. The width of GB disconnections in the rhombohedral twin may correlate with the strength and extent of yttrium segregation, in so far as the disconnections in the doped crystal are similar to those in the undoped bicrystal, as expected from the same disconnections characteristics at the mesoscopic level [7]. The extent of segregation perpendicular to the GB deduced from EFTEM analysis is 1.1nm, similar to the step height. The large width of segregation parallel to the GB could be related to the widely dissociated GB defect. Investigations of disconnections in the doped bicrystal are in progress. The topological parameters of the disconnections in a GB close to the rhombohedral twin orientation have been identified by circuit mapping, and their magnitudes have been calculated. According to this approach, no assumptions or constraints such as a 3-dimensional coincidence lattice are required. The defects accommodate the deviation from the perfect twin orientation. Since they have a mixed dislocation character, adjacent defects exhibit twist components of opposite sign to cancel stress fields. Among the possible defects with reasonable Burgers vector and step height, the ones observed have the smallest Burgers vector.

4-Conclusions

The disconnection structure could be analyzed at the atomic level using the technique of exit wave reconstruction. The complex defect structure is characterized by a widely dissociated core whose structure is centered on atomic columns rather than on the symmetry centers of the structure found in the perfect twin. This GB structure has to be related with the wide extent of yttrium segregation in disconnection cores in favour of a lower dislocation mobility that could play a role in the reduced creep rate. interface dislocation arises if white and black crystal surfaces to be joined are not complementary, i.e., if r t ( ) is not a translation vector of crystal µ and vice versa. If the interfacial structure is the same on each side of the step, the corresponding Burgers vector is given by r b = r t ( ) P r t µ ( ) where P is the transformation relating the coordinate frames of the two crystals [7]. The step height is given by h ( ) = r n r t ( ) and h µ ( ) = r n P r t µ ( ) , where r n is the normal to the interface pointing to the crystal. The notation adopted for the Burgers vector is r b p / q , where p and q denote ( ) ( ) lattice planes.

The Frank coordinate system in a four dimensional space used in hexagonal and trigonal crystallography has been adopted [19].

The device exploited is a four dimensional orthogonal frame such that the Miller-Bravais symbols can be interpreted as vectors confined to a three dimensional subspace of this 4-space. The unit 4-vectors, e1, e2, e3 and e4 (e1 =[1000], e2= [0100], etc..) are parallel to four mutually perpendicular axes. Miller bravais symbols for planes and directions are interpreted as 4-vectors which are confined to a particular 3-D projection of 4-space with the rule that the first three indices sum to zero. The direction [uvtw] is that of the cartesian 4-vector [u,v,t, w], and the normal to the plane (hkil)

is the cartesian vector [h,k,i, l/ ], where =(2/3) 1/2 (c/a), and 2 =4.969 in case of alumina.

The approach combines the benefit of the Miller Bravais indexing system with the advantage of an orthogonal reference frame.

In the experimental case of figure 1 the transformation matrix represents a rotation of °180 about 011 1

[ ] and takes the form:

This matrix operates on 4-D vectors as originally defined by Frank [26], which are related to vectors expressed using the Miller -Bravais system as follows: u,v,t,w

[ ] MB u,v,t, w [ ] 4 D
A review of crystallographic calculations in the hexagonal and trigonal systems using these methods can be found in [START_REF] Pond | An eightieth birthday Tribute[END_REF].

P = 1 3 2 + 2 ( ) 2 + 2 ( ) 2 2 + 2 ( ) 2 2 + 2 ( ) 0 2 2 + 2 ( ) 4 2 2 2 1 ( ) 6 2 2 + 2 ( ) 2 2 1 ( )
) ). Inserted is the simulation (thickness: 5nm, defocus : -39nm ). The glide component parallel to the GB is visible ( ) below the interface before the defect) to B ( ) (at 3d 01 1 2 ( ) below the interface). Other possible Burgers vectors are also shown.

  of the structure of GB defects at the atomic level in the undoped bicrystal is a prerequisite to further analyses of the doped bicrystal. This is the aim of the present work.

Conventional 2 -

 2 beam diffraction contrast analysis was performed on a JEOL 2000EX.Classical HRTEM images were recorded along the 2 1 1 0

  contains two periodic arrays of disconnections X and Y, alternately distributed and spaced by 26 nm, which account for the tilt deviation from the perfect twin orientation (figure1). Their Burgers vectors are r b x and r b y respectively. The deviation axis and angle ( , ) are determined by setting common axes of the twin parallel to the electron beam. In the experiment, the axes are not strictly parallel and ( , ) are deduced from the measured displacement vectors joining the Kikuchi map centers for each nearly common axis.The deviation is mainly tilt in character with = 0.8°. The low magnification high resolution image in figure2reveals that different step heights are associated with each type of dislocation3-1-Twin structureThe rhombohedral twin is described by a common dense 01 1 2 ( ) plane, and a common 01 1 1

Figure 5 depicts 2 ( ) and 01 1 4 (

 524 Figure 5 depicts HRTEM image of X and Y disconnections. An example of the circuit around the defect in each crystal is shown in (a). The additional 01 1 2 ( ) and 01 1 4( ) planes are

  r b Y have the same component along the projection axis (but with alternate sign), eliminating the long range stress field.

  appear. These are pictured in the dichromatic pattern on fig.6.We analyze firstly the defects with small step heights r b 1/ 2 and r b 2 / 3 : They possess a component along the projection axis. These Burgers vectors have a glissile part along 01 1 1

Figure 9

 9 Figure 9 depicts the phase of the electron exit wave function for the disconnection r b 3 / 4 . In one part of the image, the atomic columns of oxygen and aluminum are clearly distinguished(inserted magnification) at a spacing of 85 pm[12]. In the vicinity of the disconnection, the

  structure. The circuit made around one of the defects with translation vectors cannot be closed as the twin structure is different on each side of the partials (figure 10(a)). If we report geometrically the total circuit in the dichromatic complex, the closure failure corresponds to half of the total Burger vector r b 3/ 4 in projection (fig. 10(b)). The simplest way to interpret the defect is to consider the partial defect as being 1 2 r b 3/ 4 . A geometric model of the corresponding "faulted" structure is in agreement with the experimental image (figure 10(c)).

  the spacing d of the 01 1 2

Figure 1 :

 1 Figure 1 : Bright field 2-beams with common r g image (JEOL 2000 EX). The bicrystal

Figure 2 :

 2 Figure 2 : HRTEM (TOPCON) image of the twin recorded with the electron beam parallel to

Figure 3 :

 3 Figure 3 : "Perfect" rhombohedral twin : (a) Comparison between experimental and simulated image (defocus -55nm, thickness 15nm). The GB is centered on non-occupied Al sites (white dots, image TOPCON). (b) On the geometrical model,the octahedral sites near to the interface (schematized as 2 triangles), are slightly distorded. (Small circles correspond to aluminium columns, large circles to oxygen coumns).

Figure 4 : 1 [

 41 Figure 4 : HRTEM image (TOPCON) recorded in the projection 202 1 [ ] (GB plane

Figure 5 :Figure 6 : 3 / 4 .

 5634 Figure 5 : HRTEM images of some disconnections : (a) r b 3/ 4 (Y defect), and (b) r b 4 / 5 (X defect). The disconnection cores display different widths and structures, which are delimited by arrows. The addditional inserted planes are indicated by the usual symbols for edge dislocations. A circuit around one Y defect is shown on (a). (Same imaging conditions as in

Figure 7 :

 7 Figure 7: Schema illustrating the translation vectors around the step in each crystal.

Figure 8 :

 8 Figure 8 : Conventional bright field images of the grain boundary. (a) Disconnection Y is out of contrast. r g r b < 0 for X disconnection. (b) The two disconnections have opposite contrast

Figure 9 :

 9 Figure 9 : Phase of the electron exit wave showing atomic resolution. The defect (arrowed) corresponds to r b 3/ 4 . Inserted image corresponds to an enlargement of the lower crystal.The basal plane is vertical, normal to the image.On the model, small dots correspond to oxygen columns, large dots to aluminium columns. The distance between Al and O columns is 85pm.

Figure 10 (

 10 Figure 10 (a) : EWR image showing another GB structure at the disconnection core (arrowed). The circuit around the partial disconnection on the right of the defect is also shown; it is not closed.

Figure 10 (

 10 Figure 10(b) : Circuit closed in the dichromatic complex around the partial defect of figure 10 (a). The Burgers vector r b correspond to half the total Burgers vector in projection.

Figure 11 :

 11 Figure 11 : Schematic illustration of the formation of interfacial defects by a Volterra-like process. A defect is formed by joining crystal surfaces with incompatible steps [7].

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Acknowledgements: E.A. Stepantsov and A.L. Vasiliev from the Institute of Crystallography of Moscow are gratefully acknowledged for the preparation of the bicrystals. The authors also thank U. Dahmen, R.C. Pond and J. Thibault for fruitful discussions and L. Priester for her critical reading of the manuscript. The National Center for Electron Microscopy is supported by the Director, Office of Science, Office of Basic Energy Sciences, Materials Sciences and Engineering Division of the U.S. Department of Energy under Contract No. DE-AC02-05CH1123. A.Serra acknowledges the founding of MCyT project *FIS2006-12436-C02-02.

Appendix :

A bicrystal is created by joining the white ( ) and black (µ) crystals displaying surfaces with different orientations (figure 11). The step in the boundary is related to translation vectors of each crystal, r t ( ) and r t µ ( ) respectively. Thus, when a step is introduced in the GB, an