V I Tokar
email: tokar@ipcms.u-strasbg.fr

H Dreyssé

One dimensional model of strained epitaxy of a binary alloy

Keywords: alloys, growth user supplied): alloys, growth strained epitaxy, self-assembly, size distribution, monatomic chains, binary alloy, thermal equilibrium

We discuss the self-assembly of monatomic chains composed of atoms of two kinds on a periodic substrate. We assume that there exists small positive misfit between the average atomic diameter of the deposited atoms and the substrate lattice parameter. This may cause the chain size calibration. Our model calculations show that if the interatomic interactions are of ordering type, at low temperature the distribution of chain lengths at thermal equilibrium is more strongly peaked around the optimum length than in the case of chains composed from identical atoms.

Controlled self-assembly and self-organization are expected to became major tools in nanoscale engineering [START_REF] Barth | Engineering atomic and molecular nanostructures at surfaces[END_REF]. One of the perspective technological processes is the self-assembly of size-calibrated islands taking place in some strained heteroepitaxial systems [START_REF] Nötzel | Self-organized growth of strained InGaAs quantum disks[END_REF]. From the technological point of view, the goal is to grow structures with prescribed properties. However, our ability to control the growth is restricted to only indirect influences via the external conditions, such as the deposition flux, the substrate temperature, the coverage, etc. Therefore, it is important to thoroughly investigate all awailable possibilities of influencing the growth process.

In Ref. [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF] for the case of one component deposit we showed that the ordered alloyed substrate can both facilitate the self-assembly and make the size distribution more narrow.

The aim of the present paper is to investigate how the size calibration is affected if the deposited material instead of the substrate is a mixture (or an alloy) of two atomic species. As in the above paper, we will study a one dimensional (1D) model which is useful not only as a simplest theoretical model to study the qualitative behavior in all dimentions, but is also of practical interest because 1D structures are being extensively studied to serve as nanowires [START_REF] Owen | Self-assembled nanowires on semiconductor surfaces[END_REF] and atomic-scale magnets [START_REF] Dorantes-Dávila | Magnetic Anisotropy of One-Dimensional Nanostructures of Transition Metals[END_REF] for application in nanoscale devices. On the theoretical side, the simplicity inherent to 1D will allow us to use powerful analytical techniques [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF][START_REF] Bruscolini | Exact solution of the Muñoz-Eaton model for protein folding[END_REF][START_REF] Tokar | Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions[END_REF].

A simple model of misfit

To illustrate the physics of misfit let us consider the deposition of adatoms on a 1D periodic substrate. Assuming for simplicity that the adatoms are of the same kind, let us choose the substrate potential as in the Frenkel-Kontorova model

V s (x) = k 1 -cos 2πx a , (1)
where k is the strength of the potential and a the substrate lattice constant. The interatomic interaction between the adatoms is in general quite complex. In Ref. [START_REF] Tokar | Lattice gas model of coherent strained epitaxy[END_REF] we assumed that in the case of small misfit it can be treated in the harmonic approximation. This reduced the interaction to the nearest neighbor "chemical" part V N N and to the harmonic part as in the Frenkel-Kontorova model.

In the present paper we further simplify this model by assuming that the spring constant of the interatomic harmonic interaction is very large which means that the adatoms interact as impenetrable spheres of some diameter d. The difference with the Frenkel-Kontorova model is that the atoms when in contact add to the system energy the (usually negative) "chemical" contribution V N N .

Let us assume that there exists small positive misfit ∆a ≪ a between the adatom diameter d and the substrate lattice spacing

d = a + ∆a ≡ a(1 + f ), ( 2
)
where f is the relative misfit ∆a/a. It is easy to show that in this model the energy of an atomic chain consisting of l adjacent atoms is

E l = V N N (l -1) + k l - sin(πlf ) sin(πf ) ≡ E ch l + E el l , (3)
where the first "chemical bond" term on the right hand side is the sum of interactions between nearest neighbor atoms and the elastic term

E el l = l i=1 V s (x i ) (4)
is due to the interaction with the substrate. The equilibrium positions of the atoms x i in this equation are easily found from symmetry considerations. The size calibration of the chains takes place when the reduced energy E l /l has a global minimum as the function of l [START_REF] Tokar | Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions[END_REF][START_REF] Tokar | Lattice gas model of coherent strained epitaxy[END_REF][START_REF] Priester | Origin of Self-Assembled Quantum Dots in Highly Mismatched Heteroepitaxy[END_REF]. This case is illustrated by the solid curve in Fig. 1 corresponding to the misfit value f = 5%. The oscillations on the reduced energy curves correspond to the appearance of the dislocations which in the presence of misfit and the are inevitable for rigid atoms in sufficiently long chains. We note that the global minimum of the solid curve corresponds to a coherent island. Coherent structures are considered to be more suitable for application in nanoelectronics because of their better quality. The dashed curve, however, does not have a global minimum at finite l. This means that if the misfit exceeds ∼ 10% the atoms tend to gather into dislocated chains with their length growing at diminishing temperature.

According to the approximate theory of Ref. [START_REF] Priester | Origin of Self-Assembled Quantum Dots in Highly Mismatched Heteroepitaxy[END_REF] and to the exact solutions of

E l /l (in units of |V N N |) Chain length l (atoms) f =10% f =5% Figure 1.
Reduced per atom energy of the chains of adsorbed atoms for two values of the misfit f . The substrate potential strength k in Eq. ( 1) was chosen to be equal to 0.3|V N N |. The horizontal line corresponds to the asymptotic value of E l /l as l → ∞ [see Eq. ( 3)]. The dashed curve which passes above this curve does not have a global minimum at finite l.

Refs. [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF][START_REF] Tokar | Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions[END_REF][START_REF] Tokar | Lattice gas model of coherent strained epitaxy[END_REF], the size distribution of islands at finite temperature is

c l ∝ exp µl -E l k B T , (5)
where c l is the concentration of islands of size l and µ a parameter to fix the total coverage. From this equation it can be seen that at low temperature the size distribution is strongly peaked around the size corresponding to the global minimum of the reduced energy [START_REF] Priester | Origin of Self-Assembled Quantum Dots in Highly Mismatched Heteroepitaxy[END_REF]. Furthermore, the very monomodal character of the distribution depends on the positive curvature of the function E l in the vicinity of the minimum [START_REF] Tokar | Influence of relaxation on the size distribution of monatomic Ag chains on the steps of a vicinal Pt surface[END_REF].

Equilibrium statistics of alloyed chains

As was noted in the Introduction, in 1D there is a lot of techniques for solving statistical problems exactly. At the hart of these techniques lies the transfer matrix method which is extremely efficient in the case of short-range interactions because it allows to reduce many-body statistical problems to the diagonalization of a transfer matrix of a size which is small in the case of short range interactions (e. g., the size is 2 L in the binary alloy case where L is the radius of the interaction [15, Ch. II]). Elastic interactions, however, are not short range. For example, the chain energy of the form (3) can be shown to contain an infinite number of cluster interactions of the form

V l p i+1 p i+2 . . . p i+l , (6)
where p i = 0, 1 is the occupation number of adatom on site i and V l is the second discrete derivative of E l [START_REF] Tokar | Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions[END_REF]. Thus, if the nonlinear term in Eq. ( 3) is not of a polynomial type, V l = 0 for all l. In this case the transfer matrix has infinite size and the problem cannot be solved within conventional approach. In Refs. [START_REF] Bruscolini | Exact solution of the Muñoz-Eaton model for protein folding[END_REF][START_REF] Tokar | Analytical solution of a one-dimensional lattice gas model with an infinite number of multiatom interactions[END_REF][START_REF] Tokar | Lattice gas model of coherent strained epitaxy[END_REF], however, it was shown that Hamiltonians with interactions of the type of Eq. ( 6) are also exactly solvable and two methods for their solution were developed. In Refs. [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF][START_REF] Tokar | A model of strained epitaxy on an alloyed substrate[END_REF] these methods were applied to the strained epitaxy on alloyed substrates. Within these techniques the exact solutions were expressed through the chain energies E l alone. The calculation of the partition function essentially consists in the counting of the configurations with a given distribution of the atoms between the chains of different lengths.

Our main observation in the present paper is that if (i) the elastic part of the energy of an alloyed chain is independent of the chain composition and (ii) the "chemical" interatomic interactions are restricted to the nearest-neighbor atoms then the elastic part factors out so that the statistical averaging within the chain can be performed by the usual means (see Appendix A for details).

The most speculative is, of course, the first of the above assumptions, i. e., that the elastic energy is independent of the chain composition. In terms of the simple model of section 2 this means that A and B atoms have the same diameter. While in general this may seem unrealistic, in many cases this can be a good approximation [START_REF] Slater | Atomic radii in crystals[END_REF]. For example, according to Ref. [START_REF] Slater | Atomic radii in crystals[END_REF] in the case of the CuAu system the empirical radii of the two atoms are identical. Thus, at least in some cases the above assumption is approximately fulfilled.

In Appendix A and below we show that the intrachain averaging over the alloy species amounts to effective introduction of chain energies E l which formally enter into the formalism as the energies of a pure specie. This allows us to apply all the machinery developed for the pure case also in the alloy case.

To gain insight into the qualitative picture of the alloyed self-assembly, let us consider the simplest case of equiatomic alloy with the interatomic interactions between similar species being also equal (notation is explained in the Appendix): µ A = µ B and V AA = V BB which, in particular, means that a = c in Eq. (A2). According to Eq. ( A10)

Z (l) ch = Cλ l-1 + 1 + λ - λ + l-1 . (7)
This chemical part of the chain partition function can be formally expressed as corresponding to the following contribution into the chain energy

E ch(AB) l /k B T = -(l -1) ln λ + -ln 1 + (λ -/λ + ) l-1 (8)
which together with the elastic term can be substituted into Eq. ( 5) to find the size distribution. [In Eq. ( 8) we neglected a constant term which is irrelevant to our purposes.] From the explicit expression for the T-matrix eigenvalues [see Eq. (A5)]

λ ± = e (µ α -V αα )/kBT 1 ± e V /2kBT (9)
it is easy to see that depending on whether the ordering potential V in Eq. (A7) is positive (ordering case) or negative (phase separation case) the ratio of eigenvalues at lowering temperature will tend towards -1 or +1, respectively. Let us first consider the second case (V < 0). In this case from Eq. ( 9) is easy to see that the first, linear in l term in Eq. ( 8) at low T will correspond to the V N N ≃ V αα [α = A or B; we assume that the µ α term will contribute to µ in Eq. ( 5)]. This is rather natural because in the phase separation case atoms of different kinds are rarely found together and the majority of interatomic interactions is between the like atoms. The second nonlinear term may influence the shape of the size distribution [START_REF] Tokar | Influence of relaxation on the size distribution of monatomic Ag chains on the steps of a vicinal Pt surface[END_REF]. However, because the ratio of the transfer matrix eigenvalues Chain length l (atoms)

Figure 2. Non-linear in l contribution to the effective interaction Eq. ( 8) for alloyed chains of different length. For further explanations see the text.

tend to unity at small temperature, the nonlinearity weakens and its influence on the shape of the size distribution becomes negligible. In contrast, in the ordering case (V > 0) the nonlinearity grows with lowering temperature because the eigenvalue ratio tend to -1, so the amplitude of the oscillating term in Eq. ( 8) grows. This is illustrated in Fig. 2 where a moderate value exp(-V /k B T ) = 5 was chosen. The physical value of the potential would correspond at room temperature to V ≃ 0.13eV . Also from Eq. ( 9) it is easy to see that in this case V N N ≃ V AB , as expected.

With these parameters, as can be seen from Fig. 2, the nonlinearity is most pronounced at small values of l ≤ 10. The chains of similar size were observed experimentally in Refs. [START_REF] Gambardella | Equilibrium island-size distribution in one dimension[END_REF][START_REF] Nilius | Counting electrons transferred through a thin alumina film into Au chains[END_REF] for pure deposits. We expect that when alloyed with an appropriate second component (e. g., with copper in the case of gold) they can exhibit the same kind of narrowing the size distribution as was predicted by us for the self-assembly on the alloyed substrate [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF]. Namely, assuming the optimum chain length dictated by the non-oscillating part of the Hamiltonian is seven (as in Refs. [START_REF] Tokar | Influence of relaxation on the size distribution of monatomic Ag chains on the steps of a vicinal Pt surface[END_REF][START_REF] Gambardella | Equilibrium island-size distribution in one dimension[END_REF]), the oscillating part will suppress the concentrations of the nearest chains of lengths 6 and 8 in addition to the usual suppression due to the Gaussian dependence of the distribution at low temperatures [START_REF] Priester | Origin of Self-Assembled Quantum Dots in Highly Mismatched Heteroepitaxy[END_REF] dictated by the elastic and chemical parts.

Conclusion

In this paper we discussed possible influence of alloying on the size distribution of atomic chains. Our solution of a simplified 1D model showed that if the atoms inside the chains tend to order at low temperatures, the size distribution became more narrowly peaked near the most probable chain length. This is similar to the case of the self-assembly of one-specie islands on alloyed substrates considered in Ref. [START_REF] Tokar | Exact solution of a one-dimensional model of strained epitaxy on a periodically modulated substrate[END_REF]. The calculations show that the phenomenon is more pronounced at lower temperatures, in particular, it extends to longer optimum chain lengths. However, at low temperatures too long time may be needed to equilibrate the system. Therefore, it would be more reasonable to seek the confirmation of our theory in systems where the chains are very short, such as the Ag/Pt system studied in Refs. [START_REF] Tokar | Influence of relaxation on the size distribution of monatomic Ag chains on the steps of a vicinal Pt surface[END_REF][START_REF] Gambardella | Equilibrium island-size distribution in one dimension[END_REF] or the Au chains grown in Ref. [START_REF] Nilius | Counting electrons transferred through a thin alumina film into Au chains[END_REF]. Of course, it would be necessary to find the second component which under alloying would provide an

  alloy. In the bulk the copper would be a natural chose in the case of Au [see, e. g., 15, and references therein], but in 1D surface structures the interatomic interactions may significantly differ from those in 3D alloys [see discussionin 8].Reduced per atom energy of the chains of adsorbed atoms for two values of the misfit$f$. The substrate potential strength $k$ in Eq.\ (\ref{ V_s}) was chosen to be equal to $0.3|V_{NN}|$. The horizontal line corresponds to the asymptotic value of $E_l/l$ as $l\to\infty$ [see Eq.\ (\ref{ E_l})]. The dashed curve which passes above this curve does not have a global minimum at finite $l$. 215x279mm (600 x 600 DPI) in $l$ contribution to the effective interaction Eq.\ (\ref{ dEch}) for alloyed chains of different length. For further explanations see the text 215x279mm (600 x 600 DPI)

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Acknowledgments

The authors acknowledge CNRS for support of their collaboration and CINES for computational facilities. One of the authors (V. I. T.) expresses his gratitude to University Louis Pasteur de Strasbourg and IPCMS for their hospitality.

Appendix A. Transfer matrix for binary alloy

The partition function for a strained chain composed of a binary alloy with components A and B of equal atomic diameters and pairwise nearest neighbor interatomic "chemical" interactions

where the trace is taken over the occupation numbers {p α i } which are equal to one if site i is occupied by atom of kind α and are equal to zero otherwise. To simplify notation, we denoted by the bars the division by k B T . For example, Ēel l ≡ E el l /k B T , where E el l is the elastic energy from Eq. ( 3). On the second line we introduced the and |1 = ( l|) T . Furthermore, by the last equality in Eq. (A1) we separated the chain partition function into the elastic and the "chemical" parts.

To diagonalize the transfer matrix we need to solve the characteristic equation

where

and

[see Eqs. (A2) and (A3)]. The orthogonal matrix Ô which diagonalizes the transfer matrix

is composed from normalized eigenvectors of T (which can be easily found) as


Santosh ; Hosmani
email: s.hosmani@mf.mpg.de

Dr Ir E J Mittemeijer Schacherl

Santosh S Hosmani

Dr Ralf E Schacherl
email: r.schacherl@mf.mpg.de

Lidia Lity Ska-Dobrzy Ska

Eric J Mittemeijer
email: e.j.mittemeijer@mf.mpg.de

Dr Santosh S Hosmani

The Nitrogen-absorption Isotherm for Fe-21.5at.%Cr Alloy: Uptake of Excess Nitrogen in Dependence on the Precipitation Morphology

Keywords: microstructure, nitrides, phase transformations, precipitation user supplied): iron alloys, excess nitrogen iron alloys, microstructure, precipitation, nitrides, excess nitrogen

Nitriding of

.%Cr alloy leads to a "discontinuously coarsened", chromium-nitride/ferrite lamellar precipitation morphology in the nitrided zone. The nitrogen-absorption isotherm for this alloy with this precipitation morphology was determined at 560 °C. To assure a constant precipitation morphology the Fe-21.5at.%Cr specimen was first homogeneously pre-nitrided (at 580 ºC in an ammonia/hydrogen gas mixture of nitriding potential of 0.103 atm -1/2 ) and then de-nitrided (at 470 ºC in hydrogen gas atmosphere). The amount of nitrogen remaining in the de-nitrided specimen indicated that the composition of the nitride precipitates is CrN and not (Fe, Cr)N. The measured nitrogen-absorption isotherm revealed the presence of excess nitrogen in the nitrided specimen, which is a surprise in view of the coarse, lamellar precipitation morphology. The occurrence of this excess nitrogen could be ascribed to an unexpected, minor fraction of the total chromium content in the alloy present as coherent, tiny nitride platelets within the ferrite lamellae of the "discontinuously coarsened" lamellar precipitation morphology, as evidenced by transmission electron microscopy. A possible kinetic background for this unusual phenomenon was discussed.

Nitriding is the most versatile thermochemical surface treatment of ferritic steels, as it can lead, dependent on tuning of the nitriding-process parameters, to pronounced improvements of the fatigue, wear and anti-corrosion properties [START_REF]Source Book on Nitriding[END_REF][START_REF]ASM Handbook: Heat Treating[END_REF]. Gaseous nitriding, as compared to plasma and salt-bath process variants, has the eminent advantage of allowing precise control of the chemical potential of nitrogen in the nitriding atmosphere via the gas flow rates, in fact the partial pressures, of ammonia and hydrogen, i.e. via the so-called nitriding potential, r n (= , with p as partial pressure) [START_REF] Mittemeijer | [END_REF]4].

Chromium is often used as alloying element in ferritic nitriding steels because of its affinity for nitrogen, leading to the precipitation of CrN in the nitrided zone. Nitriding of iron-based alloys containing Cr was subject of several investigations [5][6][START_REF] Schacherl | Nitriding of Fe-Cr alloys with high chromium contents[END_REF][START_REF] Schacherl | [END_REF][9][10][11][12][13][14][15][16][17][18]. The microstructure of gaseous nitrided Fe-Cr alloys containing a low concentration of chromium (say, less than about 1wt.% or below)

shows a "continuous" precipitation morphology due to coherent, submicroscopical CrN platelets in the nitrided zone [10,11,19]. Two precipitation morphologies, "continuous" and "discontinuous", occur in the nitrided zone of Fe-Cr alloys containing an amount of chromium in the range 2 -13 wt.% Cr [6][START_REF] Schacherl | Nitriding of Fe-Cr alloys with high chromium contents[END_REF][START_REF] Schacherl | [END_REF][9]. "Discontinuous" means the result of a discontinuous coarsening reaction occurring for the initially "continuous" precipitate microstructure: a lamellar, CrN / -Fe precipitation morphology [5,6,12,20]. The nitrided zone of Fe-Cr alloys containing a high amount of chromium, say 13wt.% or above (e.g., see [START_REF] Schacherl | [END_REF]), consists of grains exhibiting only the "discontinuous" precipitation morphology: the relatively low migration rate for the nitriding front, in the alloys with relatively high chromium content, allows the discontinuous coarsening reaction front to catch up with the nitriding front [21]. The driving force for the discontinuous coarsening reaction is the relaxation of internal strain fields, and reduction of the interfacial area between precipitates and matrix and of the nitrogen supersaturation [5,20]. Discontinuous coarsening leads to a pronounced decrease of hardness [6].

Nitrided Fe-Cr alloys contain more nitrogen than necessary for (i) precipitation of all chromium as nitride, CrN ] N [ , and (ii) equilibrium saturation of the ferrite matrix, 0 ] N [ ; i.e. "excess nitrogen" has been taken up [6][START_REF] Schacherl | Nitriding of Fe-Cr alloys with high chromium contents[END_REF][START_REF] Schacherl | [END_REF][9]. It has been suggested that a significant part of the excess nitrogen in nitrided binary iron-based alloys is adsorbed at the (coherent) nitride/matrix interfaces, interface ] N [ [22][23][24][25][26]. interface ] N [ does not take part in the diffusion process and therefore has been called

"immobile" excess nitrogen [START_REF] Schacherl | [END_REF]. The other part of excess nitrogen consists of nitrogen dissolved interstitially in the ferrite matrix (enhanced nitrogen lattice solubility) due to the positive volume misfit between the (coherent) nitride particles and the iron matrix which causes a dilatation of the matrix lattice, strain ] N [ [22]. The nitrogen corresponding to strain ] N [ can take part in the diffusion process and therefore has been called "mobile" excess nitrogen [START_REF] Schacherl | [END_REF]. The excess nitrogen (immobile and mobile), as discussed above, is a consequence of the "continuous" precipitation morphology.

Therefore it is of interest to analyse the nitrogen uptake in high chromium Fe-Cr alloys which only show the "discontinuous" precipitation morphology. Nitrogen-absorption isotherms (i.e. the amount of nitrogen taken up by a homogeneously nitrided specimen as a function of the nitriding potential)

have been fruitfully used to analyse nitrogen uptake in Fe-Al [23,27], Fe-Ti [25], Fe-V [28] and Fe-Cr [19] alloys; in all these cases only a "continuous" precipitation morphology occurred. In this work the nitrogen-absorption isotherm for Fe-21.5at.%Cr alloy, exhibiting the "discontinuous" precipitation morphology, was investigated.

Experimental

The Fe- The alloy cast was cold rolled to a sheet of 1.0 mm thickness. Specimens with lateral dimensions of 1.0 cm X 2.0 cm were cut from the sheet. These specimens were encapsulated in a silica tube filled with argon. The encapsulated specimens were recrystallized at 700 ºC for 2 h. The recrystallized specimens were ground and polished to remove any oxide layer formed on the surface. Thereafter, each specimen was subjected to cold rolling to obtain thin foils of about 0.30 mm thickness. Specimens with lateral dimensions of 1.0 cm X 1.5 cm were cut from the foils. These thin foil specimens were recrystallized at 700 ºC for 2 h. The recrystallized thin foil specimens were ground, polished (last step: 1 µm diamond paste) and cleaned in an ultrasonic bath filled with ethanol.

To start the nitriding process the specimen was suspended at a quartz fibre in the middle of a vertical tube nitriding furnace. The inner diameter of the furnace is 2.8 cm. The nitriding experiments were performed in an ammonia/hydrogen gas flux (purity: H 2 : 99,999 vol.%, NH 3 : >99,998 vol.%). The fluxes of both gases were adjusted with mass flow controllers and amounted together 500 ml/min, which corresponds to a linear gas velocity of 1.35 cm/s.

To establish a homogeneous precipitation morphology throughout the thickness of the specimen, pre-nitriding and subsequent de-nitriding treatments were performed prior to the determination of nitrogen-absorption isotherms. To select the appropriate pre-nitriding time, the specimens were nitrided at 580 ºC at a nitriding potential of 0.103 atm -1/2 (455 ml/min hydrogen and 45 ml/min ammonia) for three different times: 48, 66 and 72 h. At the end of the nitriding process, the specimen was quenched in water.

After determination of the nitrogen uptake (by weight measurements: see last paragraph of this section), pieces were cut from the specimens and prepared to cross-sections, by subsequent embedding (PolyFast, Buehler GmbH), polishing (last step: The nitrogen-absorption isotherm was determined using the same nitriding furnace mentioned above. After the consecutive pre-and de-nitriding treatments, the nitrogen-absorption isotherm was determined at temperature well below the pre-nitriding temperature (560°C; cf. table 2) to avoid any change of the precipitation morphology during determination of the nitrogen-absorption isotherm.

The nitriding parameters which were used for the determination of the absorption isotherm have been summarised in table 2.

X-ray diffraction analysis performed in this work, and in agreement with such results attained in preceding papers [6,[START_REF] Schacherl | [END_REF], has shown that under the nittriding conditions applied for the pre-nitriding and for the determination of the absorption isotherms that (i) no iron nitrides form at the surface (see also Refs. 3, 4 and 9) and (ii) that only CrN-type precipitates develop (no X-ray diffraction evidence was obtained for the development of Cr 2 N-type precipitates).

The nitrogen uptake (or loss; in case of de-nitriding) was determined by weight measurements after and before nitriding (or de-nitriding) using a Mettler microbalance. To determine accurately each weight difference, the average value of five/six weight measurements was taken. The error bars indicated in the results (section 3) represent the maximal deviation from the average value. for 48 h. The investigations were performed on planar sections prepared as follows. The specimen was ground from one side down to a thickness of 0.1 mm, implying that the resulting foil consisted fully of nitrided material. After punching discs (diameter 3 mm) electropolishing was performed in an electrolyte of 33 % nitric acid in methanol at a temperature of -40-(-50) °C with a potential of 50

V. TEM was carried out using a Philips CM 20 Twin microscope. The microscope was operated with an accelerating voltage of 200 kV in the nanoprobe mode using a LaB 6 source of electrons.

Results and discussion

Pre-nitriding and de-nitriding; types of absorbed nitrogen

The nitrogen uptakes for the thin specimens nitrided at 580 ºC at a nitriding potential of 0.103 atm -1/2 for three different times: 48, 66 and 72 h are shown in table 3. It can be concluded that the nitrogen uptake is the same within experimental accuracy for these nitriding times, which indicates that a nitriding time of 48 h suffices for through thickness (homogeneous) nitriding of the specimen.

It is not only necessary that saturation of nitrogen uptake has been reached after 48 h of nitriding, but that also the "discontinuous" precipitation morphology occurs across the whole cross-section.

Light optical micrographs taken for the specimens nitrided for 48 and 66 h confirm that this is the case indeed (see figure 1). Microhardness measurements support the uniform precipitation morphology across the cross-section: an almost constant hardness is observed over the whole thickness of the specimen (for example, see figure 2 for the specimen nitrided for 48 h). X-ray diffraction analysis performed in this project confirmed the presence of CrN.

After pre-nitriding of a specimen (specimen-2: see =) 20.90 (± 0.01) at.% (the error margin indicated is based on the experimental error pertaining to the nitrogen content of the de-nitrided specimen determined by weighing (see above)).

Chemical analysis of the same alloy indicated a Cr content of about 21.48 (± 0.20) at.% before nitriding (table 1). Hence (21.48 -20.90 =) 0.58 (± 0.21) at.% Cr (i.e. about 3 -4% of total chromium content in the alloy) did not take part in the formation of CrN, possibly because it is present in the form of oxides already before nitriding. A similar observation was made for nitrided Fe-1.04at.%Cr [19] and Fe-2.23at.%V alloys [28].

The equilibrium amount of dissolved nitrogen in unstrained pure-iron (ferrite), 0

[N] , at 580 ºC and at a nitriding potential, r n , of 0.103 atm -1/2 is 0.30 (± 0.004) at.% [28]. Then, the total amount of "excess" nitrogen in the pre-nitrided specimen (r n = 0.103 atm -1/2 ) can be calculated as follows.

Consider, before nitriding, 100 (Fe + Cr) atoms: i.e. ) in the specimen then equals: error in nitrogen content of the specimen after nitriding (table 3).

× + + + 100 [N] 121.14 [N] 0.
One might propose that the presence of excess nitrogen is due to the development of (Fe, Cr)N precipitates. However, then the de-nitriding experiment should have led to a nitrogen content higher than the chromium content in the alloy, which is not the case (see above). Hence, the possible explanation of excess nitrogen as due to the development of (Fe, Cr)N precipitates is ruled out.

Nitrogen-absorption isotherm

The nitrogen-absorption isotherm as determined for the pre-nitrided and de-nitrided (cf. section 3.1) Fe-21.5at.%Cr alloy at the nitriding temperature of 560 °C (see section 2) is shown in figure 3 (see also table 4). At constant temperature, the amount of interstitially dissolved nitrogen in the ferrite matrix increases linearly with r n [START_REF] Mittemeijer | [END_REF]4]. Thus the straight-line dependence above level 'B' in 17.29 (± 0.01) at.% (see footnote in section 3.1 and see also column 4 at r n = 0 in table 5). The difference between level 'B' and level 'A' corresponds with nitrogen taken up in excess of the amounts needed to form the CrN precipitates. According to the earlier analysis of the nitrogenabsorption isotherms recorded for Fe-1.04at.%Cr [19] and Fe-2.23at.%V [28], such nitrogen is adsorbed at the faces of the submicroscopic, coherent nitride platelets (see figure 1 in [28] and the discussion for the current Fe-21.5at.%Cr alloy in section 3.3). The amount of this nitrogen is

indicated by nterface i ] N [
. The composition of a chromium-nitride precipitate with excess nitrogen,

nterface i ] N [
, can be described as CrN X : where,

A Level B Level [N] [N] [N] X * * * CrN interface * * * CrN = + = (1)
The result for X thus obtained is (17.55 (±0.02) / 17.23 *** (±0.01) =) 1.019 (± 0.002).

The equilibrium solubility of nitrogen in pure, unstrained iron (ferrite) at temperatures in the range 500 to 570 °C has been determined as function of nitriding potential in [28]. From these results, the following expression can be obtained for the relation between 0

[N] and T and r n :

n 0 r T 10746.1 19.491 exp [N] = (2)
where T, 0

[N] and r n are in K, at.% and Pa - [N] and strain ] N [ has been indicated in figure 3.

Analysis of "excess" nitrogen uptake in nitrided Fe-Cr alloys

If coherent precipitation of CrN in the ferrite matrix occurs, enhanced solubility of nitrogen in the ferrite matrix is induced by the precipitate-matrix misfit-strain. This enhanced nitrogen solubility, strain ] N [ , can be calculated using the model described in section 3 of [22]. 4 if "coherent" precipitation of CrN in the ferrite matrix would have occurred for all CrN. Also the amount of immobile "excess" nitrogen represented by the value of X is very small: X = 1.02 (cf. section 3.2), as compared to X = 1.24 for the Fe-1.04at.%Cr [19] alloy; in the latter case the nitrided zone exhibits (only) the "continuous" precipitation morphology.

The above results can be understood as a consequence of the almost immediate transition, occurring upon nitriding the Fe-21.5at.%Cr alloy, from coherent, submicroscopical CrN precipitates to the discontinuously coarsened lamellar microstructure, which process involves loosing the capacity for excess nitrogen uptake (the Fe-1.04at.%Cr alloy maintains the submicroscopical, continuous precipitation morphology during nitriding and thus can unfold the full capacity for excess nitrogen uptake).

The spontaneous / immediate occurrence of the discontinuous coarsening reaction in Fe-21.5at.%Cr alloy upon nitriding can be understood recognising the enormous amount of misfit stress induced in the matrix if all Cr precipitates as coherent CrN: the relaxation of such pronounced long-range strain fields in association with the great reduction in precipitate/matrix interfacial area provides a very large driving force for the occurrence of discontinuous coarsening for nitrided, in particular, high-chromium iron-based alloys.

If a specimen of Fe-21.5at.%Cr alloy is not relatively long and through nitrided, a larger amount of excess nitrogen is observed in the nitrided part of the specimen (~ 3 at.% excess nitrogen [START_REF] Schacherl | [END_REF] versus 0.69 at.% (cf. section 3.1); X = 1.11 [START_REF] Schacherl | [END_REF] versus 1.02 (cf. section 3.2); specimens nitrided under the same conditions). This can be understood as follows: most excess nitrogen, that has been taken up before the discontinuous coarsening reaction occurs, has been able to escape from the thin, Obviously, the platelets of the two orientation-relationship variants are oriented mutually perpendicularly. Further, it is observed that strong variations occur in the diffracted intensity in the diffraction-contrast images of single platelets. A similar observation has been made recently for VN platelets in an -Fe matrix, which could be shown to be due to local bending and disruptions of lattice integrity in the VN precipitates as a likely consequence of compliance with the strongly varying (misfit-strain) fields during precipitate lengthening [30]. -Mass changes recorded after pre-nitriding (in NH 3 / H 2 gas mixture) and after subsequent de-nitriding (in pure H 2 gas) demonstrate that the composition of the nitride precipitates formed is given by the formulae CrN, and not (Fe, Cr)N: Fe is not incorporated significantly in the nitride precipitates.

F
-For highly alloyed Fe-Me alloys, evaluation of the total nitrogen uptake requires expressing the amounts of the various types of absorbed nitrogen as the number of N atoms per 100 (Fe + Me) atoms.

-Upon nitriding Fe-21.5at.%Cr alloy more nitrogen is taken up than necessary for precipitation of all chromium as nitride and equilibrium saturation with nitrogen of the ferrite matrix. Analysis of the nitrogen-absorption isotherm reveals two kinds of this "excess nitrogen": (a) immobile excess nitrogen (adsorbed at precipitate/matrix interface) and (b) mobile excess nitrogen (dissolved interstitially in the ferrite matrix).

-The observation of excess nitrogen in discontinuously coarsened nitrided Fe-Cr alloy is unexpected: occurrence of excess nitrogen has been ascribed before to the occurrence of submicroscopical, coherent nitride precipitates, which is incompatible with a discontinuously coarsened microstructure.

-This work has shown that part of the chromium in the specimen is not taken up in the CrN lamellae but has precipitated as tiny CrN platelets within the ferrite lamellae; these precipitates are responsible for the uptake of excess nitrogen. Quantitative model-based analysis shows that (at most) about 3.8 at.% Cr, of the 21.5 at.% Cr in the alloy has precipitated as small nitride platelets within the ferrite lamellae. The calculated (hypothetical) amount of nitrogen dissolved in the ferrite matrix in the presence of precipitate-matrix long-range misfit-strain fields for Fe-Cr alloy nitrided at 560 ºC at a nitriding potential of 0.140 atm -1/2 as function of chromium content of the alloy. The dashed-curve pertains to CrN X with X = 1 (i.e. no nitrogen adsorbed at precipitate/matrix interface); the solidcurve pertains to CrN X with X = 1.019, as observed for the present experiments, and in the case of complete elastic accommodation of the misfit (i.e. f = 1.0; for details see [22]). The dashed-curve pertains to CrN X with X = 1 (i.e. no nitrogen adsorbed at precipitate/matrix interface); the solid -curve pertains to CrN X with X = 1.019, as observed for the present experiments, and in the case of complete elastic accommodation of the misfit (i.e. f = 1.0; for details see [22]).

F

  TYPE [ORIGINAL PAPER / MANUSCRIPT (REGULAR ARTICLE)] Article Title [The Nitrogen-absorption Isotherm for Fe-21.5at.%Cr Alloy: Uptake of Excess Nitrogen in Dependence on the Precipitation Morphology]

8 Transmission

 8 electron microscopy (TEM) studies were performed with the Fe-21.5at.%Cr specimen (1.0 mm thickness), which was nitrided at 580 ºC at a nitriding potential of 0.103 atm-1/2

  20.90 Cr atoms + 79.10 Fe atoms. Upon nitriding one has: 20.90 Cr atoms + 20.90 (N) CrN atoms + 79.10 Fe atoms + here the symbol "[A]" is used to denote the number of A atoms per 100 (Fe + Cr) atoms. The atom percentage of the total amount of nitrogen ([N]

figure 3

 3 figure 3 represents nitrogen dissolved interstitially in the ferrite matrix. Extrapolation of the straight

  level 'B':17.55 (± 0.02) at.%. The amount of nitrogen required for the formation of stoichiometric CrN has been indicated in figure3by level 'A' (nitrogen in de-nitrided specimen):

Fe-

  Cr alloy (figure5). TEM examination of the specimen (1.0 mm thickness) nitrided at 580 ºC at a nitriding potential of 0.103 atm -1/2 for 48 h demonstrated that the discontinuous coarsening led to the formation of a lamellar structure (see figures 5(a) and (b)). The selected area electron diffraction pattern (SADP) recorded from the nitride lamella indicated in figure5(a) with the number "1" allowed identification of this lamella as CrN. A ferrite lamellae appears with rich contrast in figure5(b) (see also at number "2" in figure5(a)). A peculiar feature in the bright field images is the presence of finely dispersed platelets within the ferrite lamellae. The SADP corresponding to an * Thus the value of X determined in section 3.2, as an average for all CrN precipitates in the specimen, should in fact be related with the minor formation of coherent CrN precipitates and then is larger accordingly.

  -Fe lamella shown in figure 5(b) shows -Fe reflections compatible with an [100] - Fe electron-beam zone axis and CrN reflections compatible with a Bain-type orientation relationship of the CrN precipitates with the -Fe matrix ({001} -Fe || {001} CrN ; <100> -Fe || <110> CrN ; for a similar observation for AlN precipitates in -Fe, see figure 2 in [29]) (figure 5(c)). Some further reflections can be discerned as well and are ascribed to double diffraction and/or metal oxides. Dark field images of two 020-type CrN reflections, pertaining to two variants of the Bain-type orientation relationship, and marked by circles in figure 5(c), are shown in figures 5(d) and (e).

-

  The (coherent) nitride platelets in the ferrite lamellae are responsible for the observed excess nitrogen in nitrided discontinuously coarsened high chromium Fe-Cr alloys. The fraction of Cr in the alloy which is involved in the (delayed) formation of such coherent nitride platelets in the ferrite lamellae has been estimated as follows: The amount of strain ] N [ in the Fe-21.5at.%Cr alloy nitrided at 560 ºC and at a nitriding potential of 0.140 atm -1/2 is 0.37 at.%, i.e. ( [Fe] 100[N] strain × =) 0.57 N atoms per 100 atoms of Fe (cf. table5). Then, using the calculated results shown in figure4, it follows that the Cr content in the Fe-21.5at.%Cr alloy involved in the formation of coherent precipitates in the ferrite lamellae is (at most) about 3.8 at.% (This result holds for X=1 (see figure4); for larger values of X (cf. footnote in this section) somewhat smaller values for the Cr content incorporated in coherent nitride precipitates are obtained (see inset in Fig.4) Nitriding of Fe-21.5at.%Cr alloy leads to a "discontinuous" nitride precipitation morphology in the entire nitrided zone and the presence of a minor fraction of coherent nitride precipitates within the ferrite lamellae of the "discontinuous" precipitation morphology.

#

  Estimate: extrapolation of the straight line fitted to the data-points (black squared points in figure 3(b)) to r n = 0 yields the nitrogen level 'B' (21.29 N atoms per 100 (Fe + Cr) atoms = 17.55 at.% N; see also section3

Figure 1 :

 1 Figure 1: Light optical micrographs of the etched cross-sections of Fe-21.5at.%Cr specimens nitrided at 580 ºC at a nitriding potential of 0.103 atm -1/2 for (a) 48 h and (b) 66 h.

Figure 2 :

 2 Figure 2: Microhardness-depth profile across the whole cross-section of an Fe-21.5at.%Cr specimen nitrided at 580 ºC for 48 h at a nitriding potential of 0.103 atm -1/2 (a). Light optical

Figure 3 : 24 Figure 4 :

 3244 Figure 3: Nitrogen-absorption isotherm as determined for pre-nitrided and de-nitrided Fe-

Figure 5 :

 5 Figure 5: (a) TEM micrograph (bright field) showing CrN lamellae (see number "1") and -Fe

Figure 1 :Figure 2 :

 12 Figure 1: Light optical micrographs of the etched cross-sections of Fe-21.5at.%Cr specimens nitrided at 580 ºC at a nitriding potential of 0.103 atm -1/2 for (a) 48 h and (b) 66 h.

Figure 3 :

 3 Figure 3: Nitrogen-absorption isotherm as determined for pre-nitrided and de-nitrided Fe-

Figure 4 :

 4 Figure 4: The calculated (hypothetical) amount of nitrogen dissolved in the ferrite matrix in

Figure 5 :

 5 Figure 5: (a) TEM micrograph (bright field) showing CrN lamellae (see number "1") and -

table 3

 3

		CrN [N] , is
	(	71 100 17 × . 82 29 .
		F o
		r
		P
		e e r
		R e v i e w
		O n
		l y

) of Fe-21.5at.%Cr alloy, the same specimen was de-nitrided at 470 ºC in pure H 2 gas (500 ml/min) for about 72 h. The nitrogen content which remains in the de-nitrided specimen amounts to 17.29 (± 0.01) at.%. Therefore, the 9 chromium content in the original Fe-Cr alloy which takes part in the formation of CrN,

table 3 )

 3 has given an experimental value for this percentage: 18.02. It thus follows that

	20.90	24	total excess total excess	. Pre-nitriding of a Fe-21.5at.%Cr specimen (specimen-2: see total excess [N] = 0.84
	atoms. Therefore, the pre-nitrided specimen has the following composition (now "[A]" denotes the
	atomic percentage A of the whole specimen) * :

where the error ranges are based on the experimental error in Cr content in the original Fe-Cr alloy which takes part in the formation of CrN (i.e. 20.90 (± 0.01) at.%: see above) and the experimental

  (see above result)). In the specimen the numbers of nitrogen atoms incorporated in the nitride CrN and adsorbed at the CrN nitride/matrix interface are obviously independent of the nitriding

				Philosophical Magazine & Philosophical Magazine Letters
	potential (see columns 4 (and 7) of table 5;	[	N	]	nterface i	is calculated using equation (1), with X is
	1.019 potential and	[N]	CrN	and	[	N	]	nterface i	are independent of the nitriding potential if expressed as the
	number of nitrogen atoms per 100 (Fe + Cr) atoms (cf. tables 5 and 6). The decrease in levels 'A'
	and 'B' with increasing nitriding potential in figures 3(a) thus is a straightforward consequence of
	F the nitrogen concentration being expressed in at.%. Hence, for a straightforward interpretation, it is o r preferred to express the nitrogen concentration in number of N atoms per 100 (Fe + Cr) atoms:
				P
						e e r
	R (see column 6 of table 5) gives a value for the amount of e excess nitrogen dissolved in the ferrite matrix due to the presence of the precipitate-matrix misfit Subtracting nterface i ] N [ from total excess [N] v i strain-fields ( strain ] N [ : see column 8 of table 5). The subdivision of the total nitrogen uptake into the e w contributions CrN [N] , nterface i ] N [ , 0
										1/2 respectively. The equilibrium solubilities of nitrogen O n
	in ferrite, determined from equation (2), at 560 °C (833 K) for different nitriding potentials have l y
	been given in table 4.
	The elemental concentrations in the specimen nitrided at 560 °C for different nitriding potentials
	have been calculated (see results in table 5) using table 4 and applying the procedure described in
	section 3.1. The values of	[N]	CrN	(and in principle also	[	N	]	nterface i	) in at.% depend on the nitriding
										12

*** In the de-nitrided specimen, [N] CrN = 17.29 at.% (level 'A' in figure 3(a)). When [N] interface is inserted into the specimen, [N] CrN changes from 17.29 at.% to 17.23 at.%. http://mc.manuscriptcentral.com/pm-pml figure 3(b). In the previous work on nitrogen-absorption isotherms of Fe-Me alloys (Me = Al, V, Cr, Ti) the distinction between these two ways to express the nitrogen concentration was irrelevant because only a few at.% of alloying element Me were present and the amounts of absorbed nitrogen were correspondingly much smaller.

  , very large values for chromium concentrations above 10 at.%. In reality, an only small amount of mobile "excess" dissolved nitrogen, i.e.

	to, hypotheticallystrain ] N [	, has been determined for the
	present Fe-21.5at.%Cr alloy: 0.37 at.% N = 0.45 N atoms per 100 (Fe + Cr) atoms = 0.57 N atoms
	per 100 Fe atoms (cf. column 8 in table 5; column 5 in table 6), which is in fact much less than the
	value expected from figure
	F o
			r
					P
						e e r
						R e v i e w
							O n
							l y
							The thus
	calculated value of	[	N	]	strain	(expressed as number of N atoms per 100 Fe atoms of the ferrite matrix)
						[	N	]	strain	increases with increasing [Cr]
							13

is shown as function of the chromium content of Fe-Cr alloy nitrided at 560 ºC and at a nitriding potential of 0.140 atm -1/2 (cf. figure

3

) in figure

4

. Evidently,

Table 1 :

 1 The chromium precipitating in the ferrite lamellae, after the discontinuous coarsening upon continued nitriding, is the chromium not already precipitated as CrN at the moment the discontinuous coarsening front sweeps through the (not yet completely, i.e. up to saturation) Amounts of chromium and light element impurities for the Fe-21.5at.%Cr alloy used in this work.

	-nitrided grains.
	o
	r
	P
	e e r
	R e v i e w
	O n
	l y
	17

Table 2 :

 2 Summary of the nitriding parameters used for determination of the nitrogen-absorption isotherm for the Fe-21.5at.%Cr alloy.

	determination of absorption isotherms

Table 3 :

 3 Nitrogen uptake by Fe-21.5at.%Cr specimens (thickness 0.30 mm each) nitrided at 580 ºC, at a nitriding potential of 0.103 atm -1/2 for different nitriding times.

		nitrogen uptake in
	nitriding time	Fe-21.5at.%Cr foil
	(h)	(at.%)
	48 (specimen-1)	17.96 (± 0.01)
	48 (specimen-2)	18.02 (± 0.02)
	66 (specimen-3)	17.98 (± 0.01)

Table 4 :

 4 Nitrogen uptake in pre-nitrided and de-nitrided specimen of Fe-21.5at.%Cr alloy nitrided at 560 ºC at different nitriding potentials. Values for nitrogen uptake in unstrained pure Fe at 560 ºC for different nitriding potentials have been calculated using equation[START_REF]ASM Handbook: Heat Treating[END_REF].

				nitrogen uptake in
	nitriding	total nitrogen uptake	total nitrogen uptake	unstrained pure-Fe at
	potential, r n (atm -1/2 )	(at.%)	(N atoms per 100 atoms of Fe + Cr)	560 °C (see equation (2))
				(at.%)
	0	17.55 (± 0.02) #	21.29 (± 0.02)	0
	0.054	17.72 (± 0.01)	21.55 (± 0.01)	0.12
	0.078	17.81 (± 0.02)	21.68 (± 0.02)	0.18
	0.117	17.93 (± 0.01)	21.85 (± 0.01)	0.27
	0.140	18.03 (± 0.01)	22.00 (± 0.01)	0.32

Table 5 :

 5 The elemental concentrations (columns 2 to 6) of the Fe-21.5at.%Cr specimen nitrided at 560 °C at different nitriding potentials calculated using table 4 and the calculation procedure described in section 3.1. For r n = 0, the total excess nitrogen (

	[N]	total excess	: column 6 at r n = 0) reduces to the

Table 6 :

 6 The concentrations of nitrogen ( Fe + Cr) atoms in the specimen. The results shown are based on the data given in table5.Errors in (different types of) nitrogen concentrations are based on the errors in corresponding nitrogen concentrations given in table5.

	17.29 †	17.29 †	0	0.32 ‡	0.32 ‡	0
	(± 0.01)	(± 0.01)		(± 0.03)	(± 0.03)

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


Anne Sadoc
email: anne.sadoc@u-cergy.fr

Maher Sabra

Olivier Proux

Jean-Louis Hazemann

Karyn Spence

Franklin Kenneth

Kelton

A Sadoc *

J.-L Hazemann

K S Bondi

K F Kelton

Zr and Hf microalloying in an Al-Y-Fe amorphous alloy. Relation between local structure and glass-forming ability

Keywords: amorphous alloys, EXAFS, nanostructures microalloying, Al-Y-Fe nanostructure, microalloying, Al-Y-Fe, EXAFS PACS numbers: 61.43.Dq, 61.46.-w, 61.10.Ht

come

Zr and Hf microalloying in an Al-Y-Fe amorphous alloy. Relation between local structure and glass-forming ability

Introduction

Due to the considerable scientific and technological importance of bulk metallic glasses, a significant effort has been devoted to the investigation of the unusual glass-forming ability found in some metallic alloys with microalloying [1][2][3][4][5][6]. This cannot be explained well by often-cited theories and empirical rules for metallic glass formation, but can be attributed to an increase of the short-range ordering in the alloys.

The Al-Rare Earth-Transition Metal (Al-RE-TM) glasses with a high Al content, discovered by He et al. [7] and Tsai et al. [8], are particularly intriguing. Their significant improvement in strength over conventional Al-based crystal alloys and their high ductility suggest potential aerospace and other applications [9]. The reasons for glass formation in these alloys are unclear. They differ from traditional metallic glasses and from most of the recently discovered bulk metallic glasses, not forming near deep eutectics in the phase diagram for example. It is, therefore, of both fundamental and practical interest to understand the formation and mechanism of microalloying in these glasses.

Recently, it was reported that the addition of as little as 0.5 at.% of Ti or V dramatically improves glass formation in Al 88 Y 7 Fe 5 alloys and changes the subsequent crystallization processes [10,11]. Based on EXAFS measurements, we have studied the short-range order in these alloys [12]. These measurements have demonstrated for the first time, to our knowledge, that a well-defined Al nanostructure exists around the V and Ti atoms, the alloying elements, which is different from that around the Al, Y and Fe atoms, the major alloy constituents, even at the lowest concentrations. The local structure around the microadditions is also different from that of the primary crystallizing phase, -Al, demonstrating a structural basis for the improved glass formation with microalloying in the Al-Y-Fe alloys. Moreover, the measurements showed that Ti and V form strong interactions with Al, with a significant The paper is organised as follows. The sample preparation and the EXAFS experiments are described in section 2. The EXAFS analysis is presented in section 3. The local structure around the Zr and Hf atoms is discussed in section 4 for the different samples.

A general discussion of the results appears in section 5 and a summary and conclusion are presented in section 6.

Experimental methods

Samples of Al 88-x Y 7 Fe 5 Zr x (with x = 0.5, 1 and 2) and Al 88-x Y 7 Fe 5 Hf x (with x = 1, 2 and 3) were prepared and characterised at Washington University [10]. Alloy ingots of the desired composition were first produced by arc-melting mixtures of the pure elements on a watercooled copper hearth in a high purity argon atmosphere. A Ti-getter located close to the samples was melted prior to arc melting to further remove oxygen from the chamber. To ensure a homogeneous composition, samples were arc melted three times. Amorphous ribbons were prepared by melting the ingots using rf-induction heating to 1100-1150 °C (well above the liquidus temperature) in a graphite crucible under an Ar atmosphere, and the liquid was rapidly quenched onto a rotating copper wheel. The quenched ribbons were 1-2 mm wide and 20-30 µm thick. Samples were characterized by X-ray diffraction and differential The EXAFS experiments were performed on the ribbons, which were impossible to crush or grind due to their amorphous structures. The studies were carried out at the European Synchrotron Radiation Facility (ESRF, Grenoble). The Zr K absorption edge (17997 eV) and the Hf LIII absorption edge (9561 eV) were probed on the CRG-FAME (BM30B) beam line [13]. The spectra were collected in the fluorescence mode with a Si (220) double-crystal monochromator at room temperature. The intensity of the fluorescence beam (I f ) was measured with a 30-element energy-resolved fluorescence detector. For reference, pure metallic zirconium and hafnium foils were studied in transmission mode at room temperature.

The intensity of the beam before (I o ) and after (I) the sample was measured with Si diode detectors collecting part of the transmitted beams scattered by air.

Results

Differential Scanning Calorimetry

X-ray diffraction data for all of the rapidly-quenched alloys showed the broad diffraction patterns characteristic of an amorphous phase. The DSC nonisothermal data for the alloys are shown in figure 1. Except for the alloys containing 2 at.% and 3 at.% Hf, a feature typically taken to be the glass transition is observed (figure 1 2). Collectively these x-ray diffraction and calorimetric results demonstrate that the rapidlyquenched alloys are amorphous. The absence of a glass transition in the alloys containing 2 at.% and 3 at.% Hf is unclear. To determine changes in the primary crystallization pathway with microalloying, x-ray diffraction studies were made on as-quenched samples that had been annealed partially through the crystallization peak in the DSC. The Al 88 Y 7 Fe 5 alloys made with no microadditions crystallize to -Al. In contrast, the alloys made with Zr and Hf concurrently form -Al and varying combinations of two metastable phases that were observed in recent devitrification studies of alloys made with 3d transition metal microadditions [14].

EXAFS Results

Standard procedures of normalization and background removal were followed to determine the EXAFS oscillations, , versus the energy, E, of the photoelectron from the absorption coefficient, ln I o /I, in transmission mode or I f /I o in fluorescence mode using a software package [START_REF] Michalowicz | EXAFS pour le MAC[END_REF]. The data were then converted to momentum space (k-space) using h 2 k 2 /2m=E- E o , where E o is the threshold energy origin and k is the photoelectron wave vector. The normalized EXAFS signals, k (k), are compared in figure 3 for the Zr edge of the zirconium foil and Al 88-x Y 7 Fe 5 Zr x samples (x = 0.5, 1 and 2 %) and in figure 4 for the Hf edge of the hafnium foil and Al 88-x Y 7 Fe 5 Hf x (x = 1, 2 and 3 %) samples.

The EXAFS of the zirconium and hafnium foils show structured oscillations characteristic of a crystallised material, while the amplitudes of the signals for the glassy alloys decrease monotonically with increasing k, in the form of a damped sinusoid. The spectra for the Zr and Hf edges are similar in alloys containing different concentrations of these elements, but they are less structured than for the edges of the Ti and V additions studied previously [12]. The intensities of the Zr and Hf spectra are also smaller by a factor of two than for Ti and V.

The Fourier transforms (FTs) of the normalized EXAFS signals, k (k), were computed using a Kaiser window ( =3). The same k window, roughly from 2.8 to 12 Å -1 , was used for the Zr and Hf edges. The moduli of the FTs are compared in figure 5 for the Zr edge and in figure 6 for the Hf edge. Several remarks can be made. First, the FT also are very similar for the different concentrations. Secondly, the first neighbour peak clearly appears at shorter distances in the alloys than in the pure metals, indicating that the first interatomic distance is shorter in the alloys. Peaks of next nearest neighbours appear for the alloys, but they are smaller than in the case of Ti and V additions for which the FTs show second and higher order neighbour peaks, up to about 7 Å (uncorrected for phase shifts), even at small concentrations. The intensities of these higher order peaks are about 10% of the main peak intensities in the Zr and Hf containing ribbons, while they were more than 30% for the Ti and V additions. However, the peaks of even higher order neighbours are also much smaller for the zirconium and hafnium containing foils than for the titanium and vanadium ones.

The modulus of the Fourier transform (FT) is qualitatively similar to the radial distributions of the atoms neighbouring the absorbing atom (Zr or Hf). However, the peaks are shifted to lower R-distances, because of the phase shifts experienced by the photoelectron while scattering from the potentials of the absorbing atom and nearest neighbours. Due to these phase shifts, the shape of the modulus of the FT does not provide a simple explanation of the environments of the atoms or an interpretation of the evolutions of these environments.

To obtain this information, simulations must be done. The spectra were reconstructed using the following formula [16], which describes the EXAFS oscillations for a Gaussian distribution of neighbours around a central atom:

Local structure around zirconium and hafnium atoms

(k)= S 0 2 N j k R j 2 j F j (k)exp( 2R j (k) )exp( 2 j 2 k 2 )sin(2 k R j + j (k)). (1)
The sum is taken over the j multiple scattering paths. R j is the half path length, central atomneighbour distance for single scattering. N j is the scattering path degeneracy, number of equivalent neighbours for single scattering. (k) is the electron mean free path. The Debye-Waller factor exp(-2 j 2 k 2 ) is a measure of both static and dynamic disorder of path j. S 0 2 accounts for the central atom inelastic losses. F j (k) is the backscattering amplitude experienced by the photoelectron while scattering from the neighbours, j (k) is the total phase shift due to the contributions from both the absorbing atom and the scattering atoms. The amplitudes and phase shifts were calculated using the FEFF code [17], and the fitting was done with the programs RoundMidnight [START_REF] Michalowicz | EXAFS pour le MAC[END_REF].

Calculations for the zirconium and hafnium foils were made using multiple scattering contributions. The sum in equation ( 1) is taken over the different paths, taking into account their degeneracy. The structure of crystalline Zr (Hf) is hexagonal close packed, hcp, with 6 first neighbours at 3.23115 Å in the hexagonal plane and 6 at 3.17875 Å, out of the plane (Hf:

3.1967 and 3.1307 Å) [START_REF] Wyckoff | Crystal Structures[END_REF]. The R-range that was fit in the FT was about 2 to 5.5 Å. All possible scattering paths in this distance range were considered. The fits, shown in figures 3 to 6, were done with 11 paths for the Zr foil and with 9 paths for the Hf one.

Al-Y-Fe alloys

Before presenting the results for the Al-Y-Fe alloys doped with Zr or Hf, the EXAFS results obtained previously for the pure Al 88 Y 7 Fe 5 alloy [12]

Al-Y-Fe alloys with addition of Zr or Hf

The large number of free parameters involved in the fitting procedure often makes the EXAFS analysis for ternary or quaternary alloys difficult [21]. To reduce the number of parameters, additional constraints can be applied based on a pre-knowledge of the investigated structure. The constraints in the following analysis use previously obtained information from studies of the Ti and V microadditions [12]. In these two cases, the EXAFS spectra could be simulated assuming only Al neighbours around the Ti and V; the addition of Y and Fe atoms did not improve the fit. Based on this, only Al neighbours were assumed in the following analysis. For the samples with Zr or Hf additions, the main peak in the FTs can be reconstructed with Al neighbours, one shell of neighbours around the Hf atoms (Al at 2.78-2.80 Å) and two shells around the Zr (Al at 2.61-2.63 Å plus Al at 2.95-2.99 Å). To simulate also the secondary peaks in the FTs, calculations were performed with a more extended neighbourhood. Two types of clusters were used, one with the paths and degeneracies of a local face centered cubic (fcc) structure and one with the body centered cubic (bcc) values. Earlier simulations using the fcc cluster fit well with the EXAFS data for the alloys made with 0.5% V or Ti, and simulations using the bcc cluster fit the EXAFS data for alloys containing 2% Ti [12]. For the alloys doped with Zr, the two types of clusters give the same goodness of the fit while, for Hf, the fcc cluster gives better agreement. This is consistent with the two Zr-Al distances that are necessary to reconstruct the main peak in the FTs, while only one Hf-Al distance is needed. The fits for the fcc clusters are shown in figures 3 and 4 for the k (k) spectra and in fig. 5 and 6 for the FTs. In the Zr case, the FTs could be simulated up to 4.5 Å (uncorrected for phase shift), in the Hf case up to 5.5 Å. The structural parameters for the significant single paths in the fcc clusters model are listed in Table 1, along with the reliability factor, RF, which is a measure of the goodness of the fit.

For Zr, they are also given, in Many glasses are believed to contain strong icosahedral short-range order (ISRO), which is incompatible with crystalline periodicity. If microadditions enhance the ISRO, crystal nucleation will become more difficult and glass formation will be enhanced. This speculation is supported by a tendency for some Zr-based glasses (in the example:

Zr 65 Al 7.5 Ni 10 Cu 12.5 M 5 , where M is the microalloying element) to crystallize to the icosahedral phase when small amounts of noble elements such as Ag, Au, Pd and Pt [23] and the transition metals Nb, Ta, V [24] are added. Ag, Au, Pd, and Pt have strong negative heats of mixing with Zr, but interact only weakly with the Ni and Cu in the base alloy, suggesting that strongly bound icosahedral clusters may form around the transition metals, giving rise to strong ISRO in the liquid/glass.

To check this hypothesis, the EXAFS data was also fit with a local icosahedral cluster (a simple -tungsten structure [START_REF] Nevitt | Intermetallic Compounds[END_REF], cluster diameter about 1.1 nm). For the Hf, Ti, and V microadditions, the quality of the fits using this icosahedral cluster is equivalent to that of the fcc cluster; for Zr, the icosahedral fit is not as good (RF between 0.5 10 -4 and 1.1 10 -4 ). Since very similar first neighbour distances were obtained for the fcc cluster and the icosahedral cluster, the conclusions obtained above are applicable for either structure.

The fact that several cluster models can fit the data is rather puzzling. Of course, the first shell with a small disorder parameter, , could dominate the spectra so that the different clusters would be nearly identical. This is the case for Hf. However, in the Zr case, the second shell has also similar disorder value as the first one. For Hf, the fourth shell has the same disorder value as the first one. Therefore, the local order is not restricted to the first shell, as generally in metallic glasses, but extends farther. The distances obtained when using bcc or fcc clusters for Zr are the same within R = 0.05Å, which give us confidence in the distances. The major discrepancy between the models lays in the coordination numbers and in the disorder parameters, which are by no means independent parameters. The fit depends also on multiple scattering which is important here. In fact, the disorder parameter values of the multiple paths are of the same order as those of the simple paths. Without multiple scattering, the fit is less good. For example, using only the three simple paths in the Zr fcc cluster, the RF factor increases from 0.3 10 -4 to 0.8 10 -4 and 3 10 -4 for respectively 0.5% and 1%. For Hf, the multiple scattering is useful to reproduce the FT between 3.7 and 5 Å. The fact that multiple scattering is not negligible in the cluster around minor additions means that the cluster is well structured. Several types of clusters, fcc and bcc ones around Zr, fcc and icosahedral ones around Hf, coexist probably in the alloy, as in the liquid melt.

Although the cluster size remains small, the model suggests that about 54 Al atoms are encaged in such a fcc cluster around a hafnium atom, 42 around a zirconium one. Taking into account the composition of the Al 87 Y 7 Fe 5 (Zr,Hf) 1 alloy, this means that about 50 to 60 % of the Al are encaged in the fcc clusters and bound to Zr or Hf. Consequently, there are few "free" Al atoms that with annealing can form -Al, the primary crystallizing phase. The clustering, therefore, removes regions in the melt in which -Al can easily occur. This could also explain the shift to higher temperature for the formation of -Al in the DSC experiments 14 [8,10]. Further, in the case of Hf, the fits suggest that the local structure of the glass and liquid may contain some icosahedral short-range order, which is incompatible with crystalline periodicity, making crystal nucleation difficult and enhancing glass formation.

Discussion

The glass forming ability (GFA) of an alloy, upon cooling from its molten state, is strongly influenced by the effect on crystal nucleation of minor additions in the alloy melt [4][5][6]10,11].

To favour glass formation, the nucleation and growth of crystal phases should be suppressed during solidification. The presence of ordered nanoregions in an undercooled liquid state may give either a positive or negative effect on the GFA, depending on the similarities in structure and chemistry of the local order and of the nucleating crystal phases. If the local order in an undercooled liquid is similar to that of competing crystal phases, it will promote crystal nucleation and hence degrade the GFA. If it is dissimilar, however, it will raise the nucleation barrier and enhance the GFA. The latter is the case for the rapidly quenched Al-Y-Fe alloys that were doped with Ti or V [10][11][12], where the microadditions gave rise to a local structure of the liquid that was different from that of the primary crystallizing phase, -Al, significantly enhancing glass formation. The emergence of new phases upon devitrification of alloys prepared with different microadditions that were studied here (figure 2) further supports the change in local structure that we observed in the EXAFS studies.

Microalloying-induced ordering has been confirmed recently from high-energy synchrotron

x-ray studies. Results from that study of microalloying with 3d transition metals in Al 88 Y 7 Fe 5 suggest that the ordering also decreases the atomic mobility in the glass, making nucleation more difficult [14]. These and our earlier EXAFS studies [12] show that although Ti, Zr and Hf are chemically similar group IV elements, microalloying with them yields glasses with different structures. The differences between Zr and Hf are particularly surprising because their atomic and ionic radii as well as their effective electronic structures are nearly identical, yet some alloys and compounds with Hf and Zf exhibit disparate properties. For example, Zheng, et al. have demonstrated that differences in the electronic structure of the respective oxides (TM-O 2 ) may be attributed to the greater electron affinity in the Hf compound over that in the Zr one [START_REF] Zheng | [END_REF]. Further, the heat of mixing of the Al-TM intermetallic phases is more negative in the Zr microalloys over those in the Ti and Hf containing glasses [22].

Similar effects could explain the increased shortening of Zr-Al bonds observed here.

Conclusion

The (Zr, Hf) x alloys compared to those around Ti and V [12]. RF is the goodness of the fit. N is the number of Al neighbours, R(Å) their distance from the central atom and 2 (Å 2 ) the mean relative displacement. The error bars are of the order of R = 0.05Å and 2 = 0.005 Å 2 .

  bond length. This could raise the barrier for nucleation of -Al and decrease the atomic mobility of Al, explaining the greater stability of the glasses containing Ti or V.Here we continue the exploration of the structural role of microadditions on glass formation in Al-Y-Fe alloys. Using EXAFS measurements at the Zr K and Hf LIII absorption edges, we have investigated the local atomic structures in alloys prepared with small additions of Zr (0.5, 1 and 2 %) or Hf (1, 2 and 3 %). The results are compared to previous ones obtained in Al 88 Y 7 Fe 5 with and without Ti or V.

  .b), which is absent in the rapidly quenched Al 88 Y 7 Fe 5 alloy made without microadditions. Finally, instead of the peak expected for glass devitrification, isothermal DSC measurements of the Al 88 Y 7 Fe 5 alloy made without microadditions show a monotonically decreasing signal with time, possibly indicating the coarsening of a nanocrystal/amorphous composite [10]. In contrast, all alloys prepared with Zr or Hf microadditions show the expected nucleation and growth crystallization peak (figure

  effects of the addition of a small amount of Zr and Hf on the atomic structure of Al-Y-Fe metallic glasses were examined using extended x-ray absorption fine structure (EXAFS) measurements of the Zr K and Hf LIII absorption edges. The local structure was determined around the Zr and Hf atoms. The same Al environment was found for the different concentrations of these microadditions, which could be modelled as a small cluster extending up to 4.5 Å around Zr atoms and up to 6 Å around Hf. Although the ordering effect is smaller in the Zr neighbourhood, a drastic shortening of the nearest Zr-Al distance is shown providing evidence for some covalent character, in line with the increase glass-forming ability found with Zr microaddition. This balance of ordering and shortening of the microaddition-Al bonds leads to varying degrees of increased glass forming ability and stability of the amorphous phase.

Figure 1

 1 Figure captions

Figure 2 .

 2 Figure 2. Isothermal DSC scans for the rapidly-quenched alloys containing the Zr and Hf

Figure 3 .

 3 Figure 3. The k (k) EXAFS spectra for the Zr K absorption edge of a zirconium foil and of

Figure 4 .

 4 Figure 4. The k (k) EXAFS spectra for the Hf LIII absorption edge of a hafnium foil and of

Figure 5 .

 5 Figure 5. Moduli of the Fourier transforms of the EXAFS spectra as a function of distance (Å)

Figure 6 .

 6 Figure 6. Moduli of the Fourier transforms of the EXAFS spectra as a function of distance (Å)

21 Figure 1 , 22 Figure 2 , 23 Figure 3 ,

 211222233 Figure 1, Sadoc et al.

24 Figure 4 ,Figure 5 , 26 Figure 6 ,

 2445266 Figure 4, Sadoc et al.

  are briefly sketched here. The first the Al and Fe atoms consists of one shell of Al atoms, respectively 12 Al at a distance of 2.77 Å and 9 Al at 2.46 Å. The Y first-neighbourhood is composed of 16 Al atoms in two shells, with 14.6 Al atoms at 3.11 Å and 1.4 Al atoms at 3.65 Å. These results are in good agreement with those obtained previously around the Y and Fe atoms by Saksl et al. [19]. Comparing the bond lengths to the sum of the atomic radii (R Al = 1.43 Å, R Y = 1.80 Å, R Fe = 1.26 Å), the Al-Al, Y-Al and Fe-Al first distances are noticeably shorter than the sums of the atomic radii, respectively 2.86 Å, 3.23 Å, and 2.69 Å. The relative shortening is only 3% for Al and 3.7% for Y, but is as great as 8.5% for Fe. Using the model of Miracle et al. [20], the maximum geometrically allowed coordination, N T , is about 13 around Al, 17 around Y and 11 around Fe for respectively atomic radius ratios r i /r Al about 1, 1.26 and 0.88. The experimental coordination numbers are slightly below the maximum

	10
	neighbour environment of
	F o r
	P
	e
	e r
	R e v
	i e
	w
	O n l
	y

coordination number except for the Fe atoms for which it is clearly smaller. Since the Al-Al bond length was determined, an effective atom radius can be obtained for Al and consequently for Y and Fe using the first Y-Al and Fe-Al distances (r eff is 1.385 Å for Al, 1.725 Å for Y and 1.075 Å for Fe). Using these values, the effective radius ratios are 1.245 for Y and 0.78 for Fe; with this the maximum coordination number is unchanged for Y but decreases to 9.5 around Fe, in accordance to the experimental coordination. Therefore, the Al atoms appear efficiently packed around Al, Y and Fe in the Al-Y-Fe alloys.

Table 2

 2 , for the bcc cluster model. Zr-Al distance compares better to the sum (2.66 Å) of the covalent radii of Zr and Al atoms (1.48 Å; 1.18 Å). Therefore, some covalent effect is evidenced in the Zr doped alloys. The first Hf-Al distance (2.80 Å) is shorter than the sum (3.01 Å) of the atomic radii of Hf and Al atoms (R Hf = 1.58 Å; R Al = 1.43 Å). The relative shortening (7.6 %) is less significant than in the case of the Zr dopant. It has the same value as for the Ti-Al one in the alloy containing 0.5% Ti. The covalent bonding character for Zr and the possible bcc clustering around Zr atoms set Zr apart from the other Hf, Ti or V microadditions. This could reflect different electronic charge transfer from the Al-Al bond to the Al-TM bond associated

	The first interatomic distances are approximately 2.61 Å around a Zr atom; around Al
	they are 2.76 Å in the pure Al 88 Y 7 Fe 5 glass [12] and 2.8634 Å in -Al. This nearest Zr-Al
	distance (2.61 Å) is noticeably shorter than the sum (3.03 Å) of the atomic radii of Zr and Al
	atoms (R

Zr = 1.60 Å; R Al = 1.43 Å). The shortening is more drastic than in the cases of Ti and V; the relative shortening is 14% here while it was only 7% for Ti and 4 % for V. The nearest with alloying, and to differences in the hybridization between the Al p and TM d electrons

[22]

.

Table 1 :

 1 Fcc cluster: first environments around Hf and Zr atoms in Al 88-x Y 7 Fe 5

Table 2 :

 2 bcc cluster: first environments around Zr atoms in Al 88-x Y 7 Fe 5 Zr x alloys. RF is the goodness of the fit. N is the number of Al neighbours, R(Å) their distance from the central atom and 2 (Å 2 ) the mean relative displacement. The error bars are of the order of R = 0.05Å and 2 = 0.005 Å 2 .

		Zr 0.5 %	Zr 1 %	Zr 2 %
	N	R(Å)	2 (Å 2 )	R(Å)	2 (Å 2 )	R(Å)	2 (Å 2 )
	8	2.61	0.013	2.58	0.015	2.59	0.011
	6	3.04	0.036	3.04	0.019	3.00	0.028
	12	4.42	0.052	4.40	0.043	4.48	0.036
	RF	0.3 10 -4	0.3 10 -4	0.3 10 -4

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Acknowledgements

It is a pleasure to thank all the staff of the BM30B beam line for the experiments in ESRF.

We also thank A. Michalowicz for the EXAFS programs. K. F. Kelton gratefully acknowledges support from the United States Air Force Office of Scientific Research, contract AFOSR FA9550-05-1-0110 and the National Science Foundation under DMR 06-06065.