Metastable phase stability in the ternary Zr–Fe–Cr system

Intermetallics 10 (2002) 205–216 www.elsevier.com/locate/intermet

Metastable phase stability in the ternary Zr–Fe–Cr system C. Rodrı´gueza, D.A. Barbiricb, M.E. Pepec, J.A. Kovacsd, J.A. Alonsoe, R. Hojvat de Tendlerf,* a

Secretarı´a de Ciencia y Te´cnica, Facultad Regional Buenos Aires, Universidad Tecnolo´gica Nacional, Medrano 951, 1179 Buenos Aires, Argentina b Departamento de Quı´mica Inorga´nica, Analı´tica y Quı´mica Fı´sica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabello´n 2, Ciudad Universitaria, 1428 Buenos Aires, Argentina c Servicio de Informa´tica y Comunicaciones, Centro Ato´mico Constituyentes, Comisio´n Nacional de Energı´a Ato´mica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina d Department of Molecular Biology, TPC-6, The Scripps Research Institute, 10550 N. Torrey Pines Road, La Jolla, CA 92037, USA e Departamento de Fı´sica Teo´rica, Universidad de Valladolid, Valladolid 47011, Spain f Instituto de Estudios Nucleares, Centro Ato´mico Ezeiza, Comisio´n Nacional de Energı´a Ato´mica, Avenida del Libertador 8250, 1429 Buenos Aires, Argentina Received 8 October 2001; accepted 23 October 2001

Abstract As the existing literature reports, when Zircaloy-2 and Zircaloy-4 are irradiated at intermediate temperatures with high-energy neutrons, a crystalline to amorphous phase transition occurs, coincidental with a preferential depletion of Fe at the periphery of the Zr(Fe,Cr)2 intermetallic particles. In order to describe these phenomena, relative phase stability criteria combined with a balance between the rate of irradiation damage and irradiation enhanced diffusion leading back to a modified or new state should be considered. Aiming at this suggested study, in this work we calculate the metastable free-energy diagram of the ternary Zr–Fe–Cr system. The free energies of formation of the metastable amorphous and solid solution phases and of the stable Zr(Fe,Cr)2 compound have been calculated using Miedema’s model. By solving equations of analytical geometry, regions of metastable equilibria and their tie-lines are defined, linking the hcp-Zr-rich solid solution with the Zr–Fe–Cr amorphous phase, and the latter with the irradiation-energized compound. Composition profiles, reported in the literature, were measured on partially and completely amorphized particles, and also in the case when the latter experienced microcrystallization with solute reversion, that is, with the Fe concentration increased during post-irradiation heat treatments. Those composition profiles define a path along a calculated tieline, deviating near the crystalline core/amorphous crown interface toward low Fe concentrations. The analysis in the present work suggests that the microstructure in irradiated Zircaloy tends to reach a new radiation-induced metastable co-existence of phases. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Intermetallics, miscellaneous; A. Ternary alloy systems; B. Irradiation effects; E. Phase stability, prediction; G. Energy systems

1. Introduction Work has been done on the production of amorphous phases of the ternary Zr–Fe–Cr alloys by melt-spinning techniques [1–4]. The Zr–Fe–Cr system is relevant for the discussion of irradiation-induced amorphization of intermetallic particles in Zr-based commercial alloys used in nuclear reactors. When Zircaloy-2 (a Zr-based alloy with 1.5 wt.% Sn, 0.15 wt.% Fe, 0.1 wt.% Cr, 0.05 wt.% Ni) * Corresponding author. Tel.: +54-011-4379-8578; fax: +54-0114379-8263. E-mail address: [email protected] (R. Hojvat de Tendler).

and Zircaloy-4 (Zr-based alloy with 1.5 wt.% Sn, 0.2 wt.% Fe, 0.1 wt.% Cr) are irradiated at intermediate temperatures with high energy (E > 1 MeV) neutrons, a preferential depletion of Fe at the periphery of the Zr(Cr,Fe)2 intermetallic particles is observed. This is coincident with a crystalline-amorphous phase transformation [5–7]. Data presented by Griffiths and coworkers (see Fig. 1 in Ref. [5] and Fig. 6 in Ref. [6]) show a continuous composition profile across the crystalline core. However, a steep gradient is measured in a narrow transition region (of 10–40 nm) between the Zr(Cr,Fe)2 precipitate and the amorphous zone in Zircaloy-4 after neutron (n) irradiation at 580 K to fluences of 8
1025 n m2 and 12
1025 n m2, respectively. A discontinuity in

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the Fe concentration at the crystalline/amorphous interface was reported [7,8] in samples of recrystallized Zircaloy-4 irradiated at 561 K to 8.5
1025 n m2, and in samples irradiated at 593 K to 9.5
1025 n m2. In these two works [7,8], besides the Fe discontinuity, the Cr composition remains nearly constant through each of the two phases. The composition of fully amorphous particles irradiated at 563 K to doses higher than 3
1026 n m2 was reported by Mahmood et al. [9], and Fe depletion was observed at those high doses with almost no particle dissolution. The composition profile of partially amorphized particles in Zircaloy-4 irradiated at 593 K, and the microcrystalline transformation that occurs after annealing at 873 K, are reported by Gilbon and Simonot [10]. In that work the composition profile of completely amorphized particles of Zircaloy-4 irradiated at 608 K to 9.6
1025 n m2 is shown. Those particles remain crystalline when irradiated at 653 K. When irradiated at higher temperatures these precipitates remain crystalline but Fe and Cr depletion also occurs and a very different composition profile is observed [7–11]. To describe these phenomena, the concept of a steady-state configuration of the system under irradiation was invoked, in which the chemical and topological disordering effects of irradiation tend to be equilibrated by irradiation-enhanced diffusion leading the system back to a modified state. To accomplish this state, the relative phase stability criteria, combined with a balance between the rate of irradiation damage on one hand and the thermal and irradiation-enhanced annealing of defects on the other, should be considered [12]. In the various treatments of this problem the coupled diffusion of defects and solute atoms and the elimination of defects, either by mutual recombination or at point defect sinks, are described for specific cases. In a more general treatment of phase stability under irradiation taking into account ballistic effects, even an equilibrium between an amorphous phase and the crystalline phase was suggested [13]. To evaluate the phase stability in the Zr–Fe–Cr system, its metastable free-energy diagram is calculated in the present work by using the semi-empirical alloy model of Miedema [14], that has been previously applied to ternary systems [15–18]. Our simple thermodynamic calculations cannot fully describe the complex phenomena reviewed above, but can provide useful information to researchers in this field. Our calculations define, on one side, the extended glass-forming range (EGFR), relevant to rapid cooling experiments, and on the other side help in the analysis of the complex situation where a terminal solid solution, a metastable amorphous phase and the crystalline Zr(Fe,Cr)2 intermetallic coexist at temperatures around 580 K under neutron irradiation and also on subsequent annealing treatments.

2. Method The glass-forming region of the Zr–Fe–Cr alloy was obtained by calculating, using Miedema’s model [14,18], the Gibbs free energies of formation of the metastable amorphous phase and crystalline solid solutions and of the stable Zr(Fe,Cr)2 intermetallic compound, and applying the usual thermodynamic criteria of stability. 2.1. Free energy of formation of the different competing phases We present here an outline of the method; a complete description can be consulted elsewhere [14,18]. Nevertheless, and because of the relevance that it has in this work, we describe the calculation of the free energy of formation of the amorphous alloy. The enthalpy of formation of a solid solution (ss) is expressed as
Hss ¼
Hc þ
He þ
Hstr :

ð1Þ

The chemical term
Hc (c stands for ‘chemical’) results from the electronic redistribution occurring when the alloy forms. In the ternary system A–B–C of atomic composition (xA, xB, xC), we express this term
HcABC as the weighted sum of the corresponding contributions of binary alloys defined by Kohler [19–21]. One parameter of the model, here labelled , accounts for the degree of chemical order in the system, and ranges from =0 for a disordered alloy to =8 for an ordered intermetallic compound [14]. We have assigned the value  =0 to the boundary binaries of the Zr–Fe–Cr system in the calculation of the ternary crystalline solid solutions. The elastic term
He in Eq. (1) (e symbolizes ‘elastic’) is due to the atomic size mismatch between solute and solvent atoms in the crystalline solid solution. Its expression is based on Eshelby’s formula [22], originally deduced for dilute binary solid solutions and later extended to concentrated ternary solid solutions [18]. The structural term in Eq. (1)
Hstr (str. represents ‘structural’) accounts for the difference in valence and crystal structure of the component metals and is expressed in terms of lattice stability functions [23,24] defined for the hcp, fcc and bcc structures of paramagnetic and ferromagnetic transition metals and alloys. The stable structures of the component metals at the temperature T=580 K are Zr-hcp, Cr-bcc and Fe-bcc and the system Zr–Fe–Cr was taken as ferromagnetic. The Gibbs free energy of formation of the ternary solid solution is then ss str ideal
Gss ABC ¼
HABC  T
SABC  T
SABC

ð2Þ

where
Sideal is the ideal entropy of mixing, and the structural entropy contribution
Sstr comes from the conversion of each pure metal component from its stable

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crystal structure to the one adopted by the ternary solid solution. The Gibbs free energy of formation of the amorphous alloy (a) as as
GaABC ¼
HcABC  T
Sideal ABC þ xA
GA þ xB
GB þ xC
Gas ð3Þ C

contains the free energy of mixing of the pure liquid metals (the first two terms) plus the free energy difference between the amorphous state and the crystalline state (s) of the pure components, since the latter is taken as the reference state. That difference is expressed, as proposed by Thompson and Spaepen [25]:
Gas ¼
Hf ðiÞ ½ðTm ðiÞ  TÞ=Tm ðiÞ ½2T=ðTm ðiÞ þ TÞ i ¼ A; B; C

ð4Þ

where Tm is the melting temperature and
Hf is the enthalpy of fusion. Eq. (3) treats the amorphous alloy as an undercooled liquid. It results when the amorphous alloy is assumed to be formed by first melting each of the three crystalline pure metals at a temperature T that is lower than their melting temperatures. In the present work, T=580 K and is T < Tm(i) with i=Zr, Fe, Cr. The three last terms of Eq. (3) take account of this melting. Then, in a second step, the three undercooled liquid metals mix together, forming the amorphous alloy. The first term in Eq. (3) represents the chemical enthalpy of formation of the undercooled liquid ternary alloy, and the entropy of formation is approximated as the ideal entropy of mixing. For a ternary alloy A–B–C of atomic composition (xA, xB, xC), the chemical enthalpy is again the weighted sum—as defined by Kohler—of the chemical contributions to the heat of formation of the corresponding binary alloys [18–20]:
HcABC ¼ m6¼n ðxm xn =xm xn Þ
Hcmn ðxm ; xn Þ

ð5Þ

where xm ¼

xm xn ; xn ¼ ; with xm þ xn xm þ xn

xm þ xn ¼ 1; and

m; n ¼ A; B; C:
Hcmn(x*m, x*n) is the chemical contribution in the binary mn alloy of fictitious composition (x*m, x*n), and is expressed as [26]
Hcmn ðxm ; xn Þ ¼
Hamp xm ½Vm ðalloyÞ 2=3 fmn

ð6Þ

where
Hamp is an amplitude due to the electron redistribution, Vm (alloy) is the atomic volume of the m component in the alloy, and fmn contains the adjustable parameter  and accounts for the degree to which m

207

atoms are surrounded by n atoms as neighbours. Regarding the computation of
HcABC in Eq. (3), some degree of chemical short-range order was allowed, adopting the value =5 for the Zr–Cr and Fe–Cr alloys [27] and =0 for Zr–Fe [28]. The values of Miedema’s model parameters for pure metals-electronegativities, molar volumes, and electron densities at the boundary of the Wigner–Seitz cells were taken from Ref. [26], where they are tabulated. The values of Miedema’s model constants, which are needed to calculate
Hamp but are not presented here, are shown in Ref. [14]. When the alloyed metals are transition metals, as in the present work,
Hamp has the same value for each limiting binary alloy in both its liquid and solid phases [14,26]. Then, in the present work, the same value of
Hamp for each limiting binary enters in the calculation of
HcABC in Eqs. (1) (solid solutions) and (3) (amorphous alloy). The free energy of formation of a binary intermetallic compound is the enthalpy of formation, at the stoichiometric composition, of an ordered structure, i.e. with =8 in
Hc [14]. For a ternary compound, we calculate the free energy of formation as the weighted sum of the corresponding contributions of the limiting binary alloys [19–21]. We further added the ideal entropy of mixing in the whole range of compositions of the intermetallic compound, assuming that the solid solubility of both binary compounds ZrFe2 and ZrCr2 [29,30] extends into ternary compositions. 2.2. Metastable co-existence of phases Regions of metastable equilibrium between the amorphous phase and either the solid solution or the intermetallic compound have been calculated. For the first case, pure a-Zr (hcp) was taken as the solid solution at 580 K, based on the very poor solubility of Fe and Cr in a-Zr [29–31]. Besides, the present calculations show that a-Zr is strongly destabilised by the addition of Fe and Cr (Section 3). In accordance with the procedure in [18], the region of metastable equilibrium was defined by projecting over the Gibbs triangle the curve generated by the tangency points of the infinite planes that pass through the -Zr vertex of the Gibbs triangle and are tangent to the
Ga surface of the amorphous alloy. As these two phases are relatively unaffected by irradiation they define a target equilibrium that keeps unmodified under irradiation vis-a`-vis the non-irradiated situation. For the second case, that is, the equilibrium intermetallic compound/amorphous phase, there is no metastable coexistence between the two phases in the non-irradiated material. To generate a common co-existence region, the free energy of the compound was assumed to be affected by irradiation due to chemical disorder and accumulation of crystal defects. Therefore, the free energy curve of the intermetallic—given that the latter is represented by a line along the stoichiometric composition—

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208

was increased by the same amount all along this line. The magnitude of the irradiation effect depends on parameters such as the kind of irradiation particles, their flux and dose, as well as the irradiation temperature. In this work, the energy increase was guessed—see next section—for the purpose of comparison with experimental results. We outline now the equations determining this metastable equilibrium. Let
Ga(xA, xB)=g(x,y) and
Gcompound(xA, xB)=f(x,y) represent the respective equations of the energy surface of the amorphous phase and the energy curve of the compound. In general, xA and xB are any two chosen components of the ternary system (xA+xB+xC=1). Let the curve of the compound be expressed f(x, y)=f[(v), (v)], where v is a parameter identifying each point of that curve. If we fix the value of v, the equation of a plane tangent to both the curve f(x, y) and the surface g(x, y) can be written as z ¼ aðvÞ x þ bðvÞy þ cðvÞ:

ð7Þ

The components of the vector tangent to the curve f(x,y) are V1 ¼ 0 ðvÞ; V2 ¼ ¼ fx ½ðvÞ;

0

ðvÞ; V3

ðvÞ 0 ðvÞ þ fy ½ðvÞ;

ðvÞ

0

ðvÞ

Thus, after some algebra, V1=0, V2=1, V3=fy(, ), and replacing finally
Ga(xA, xB) and
Gc(xA, xB) for g(x, y) and f(x, y), Eqs. (9) and (10) reduce to
Gcv ð0:3333; vÞ ðv  Þ þ
Ga ð; Þ 
Gc ð0:3333; vÞ ð  0:3333Þ ¼
GaxA ð; Þ ð11Þ and
Gcv ð0:3333; vÞ ¼
GaxB ð; Þ

ð12Þ

where the subscripts indicate again partial derivatives with respect to the corresponding variable xA, xB or v. In these equations
GaxA(, ),
GaxB( , ) and
Gcv(0.3333, v) are calculated numerically by finite increments of xA, xB and v. Again, the curve generated by the tangency points on the amorphous free energy surface, projected over the Gibbs triangle defines on this one the region of coexistence of the two phases involved. In most cases only two sites on the amorphous free-energy surface correspond to a point of the compound curve, each one appearing on one of the two branches of the two-phase field (a+Zr(Fe,Cr)2) limiting curve. They define two tielines, joining two amorphous compositions in metastable equilibrium with one compound composition.

ð8Þ 3. Results

where fx and fy indicate partial derivatives and 0 (v) and 0 (v) indicate derivatives with respect to v. Let {(v),

(v), g[(v), (v)]} be the coordinates of the surface point where the plane and the g(x, y) surface are tangential; this plane is also tangent to the f(x, y) curve at [(v), (v)]. To obtain (v) and (v) the equations to be solved (v is omitted for the sake of simplicity) are V3 ð  Þ þ V2 ½gð; Þ  fð; Þ ¼ gxð; Þ V1 ð  Þ  V2 ð  Þ V3 ð  Þ  V1 ½gð; Þ  fð; Þ ¼ gyð; Þ: V1 ð  Þ  V2 ð  Þ

ð9Þ

ð10Þ

The left hand sides in Eqs. (9) and (10) are, respectively, a(v) and b(v) of (7). On the other hand c(v) is obtained from the condition of plane/curve coincidence at (, ), x and y in (7) are unnecessary to our purpose. So with (9) and (10) and bearing in mind that, on the Gibbs triangle the compound line goes from ZrFe2 to ZrCr2, and therefore xZr=0.3333 and xFe+xCr=0.6667, the parameterization of the projection of the curve f(x, y) over the xy-plane yields x ¼ ðvÞ ¼ 0:3333 ; y ¼ ðvÞ ¼ v ; 0:0 4 v 4 0:6667:

Fig. 1 shows a three-dimensional representation, based on the usual ternary Gibbs composition triangle, of the free energy surface of the amorphous alloy (labelled a in the figure). The visualisation of that surface has been improved by drawing the intersections of the surface and Cr contents. The free energy line of the Zr(Fe,Cr)2 intermetallic compound along its stoichiometric ternary composition is also represented. This line ends at the planes of the limiting binary systems; in those places a sketch of the strong variation with composition of the free energy of the intermetallic compound has been schematically represented by a V shaped curve. Fig. 2 gives a similar representation for the free-energy surfaces of the hcp and bcc solid solutions. The intersections of the bcc-surface with planes of constant Fe content have been drawn. Fig. 3 shows the Extended Glass Forming Range (EGFR), labelled a, delimited by the Co curves calculated at 580 K. The Co curves correspond to the projection of the intersections between the free energy surfaces of the amorphous phase and the selected hcp-Zr or bcc solid solution phases, bcc indicating here -Zr, a-Fe and Cr solid solution phases. The bcc -Zr phase was considered as the terminal solid solution, because in spite of -Zr being the stable phase at low temperature, the -Zr solid solution becomes polymorphically more stable than the calculated solute-enriched hcp phase

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209

Fig. 1. Three-dimensional representation of the calculated free energy of some particular phases in the Zr–Fe–Cr system at 580 K, based on the Gibbs composition triangle. The plane of the Gibbs triangle sets the zero level of free energies. An axis for free energy is drawn on each vertex of the Gibbs triangle. Units are kJ mol1. The calculated surface representing the free energy of formation of the amorphous alloy (a) and its intersection with the planes of constant Fe composition (60 at.%) and constant Cr composition (30 and 60 at.%) are drawn. The calculated free-energy line of the Zr(Fe,Cr)2 intermetallic compound along its stoichiometric composition is also shown. Dashed lines show that the intermetallic free-energy line runs parallel to the Zr 33.3 at.% composition line. The variation of the free energy of the intermetallic compound with Zr composition on the limiting Zr– Fe and Zr–Cr binary planes has been schematically represented with a sharp V-shaped curve. The inset shows the variation of the free energy of the Zr(Fe,Cr)2 compound with Cr composition along the stoichiometric ternary composition.

in the Zr-Cr and Zr-Fe binary systems, as well as in the ternary alloy (Fig. 2). Thus, by rapid quenching, the Zrrich bcc solid solution (and not the hcp-Zr phase) could be the phase competing with the formation of the

amorphous alloy. In region a the free energy of the metastable amorphous phase is lower than the free energies of the competing metastable solid solutions. On the other hand in the regions labelled bcc or hcp, those

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Fig. 2. Three-dimensional representation of the calculated free energy surfaces of the hcp and bcc solid solutions in the Zr–Fe–Cr system at 580 K, based on the Gibbs composition triangle. Units are kJ mol1. The plane of the Gibbs triangle sets the zero level of free energies. An axis for free energy is drawn on each vertex of the Gibbs triangle. Borders of the free-energy surfaces of the hcp and bcc solid solutions and intersection of the free-energy surface of the bcc solid solution with planes of constant Fe composition (10, 40 and 80 at.%) are shown as (continuous lines). Segments of the triangle edges below both surfaces are not drawn. Segment of the border of the bcc surface and intersections of the bcc surface with planes of constant Fe content that are located under the hcp surface (– –). Segments of the triangle edges above the bcc surface (continuous lines) and below the bcc surface (- - - -).

solid solutions are more stable than the amorphous phase. In Figs 4–6, we draw the calculated target coexistence boundaries (+a)/a, and a/(a+Zr(Fe,Cr)2). To calculate this latter limit it was assumed that the free energy of the intermetallic compound with a composition inside the stability range is increased by 6, 8 and 10 kJ Mol1, respectively (see Section 2.2). In Fig. 4 some of the tie-lines between the crystalline particle and its target equilibrium amorphous regions are included. It can be observed that the present calculations predict for some particle compositions more than one tie-line. This spurious effect can be attributed to minor curvature oscillations of the free energy surface of the amorphous phase near the terminal binary diagram.

4. Discussion 4.1. Comparison with experiments Fully amorphous Fe90xCrxZr10 alloys with x=0, 2, 4, 7, 10, 13, 16, 20, were prepared by melt-spinning [1–4]. Xray diffraction, transmission electron microscopy and Mo¨ssbauer spectroscopy verified sample quality in ribbons 1.5–1.6 mm wide and 20–30 mm thick, and no traces of crystalline phases were found. All these compositions are contained in the EGFR of Fig. 3. Using high-rate sputtering techniques (co-evaporation), Unruh et al. [32] have reported the production of amorphous FexZr100x alloys over the range of compositions 204x493. By co-sputtering on substrates

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211

Fig. 3. Extended glass-forming region (EGFR) of the ternary Zr–Fe–Cr system at 580 K. The EGFR, labelled a, is delimited by C0 lines (the continuous lines). hcp and bcc indicate regions where the solid solutions are more stable than the amorphous phase. Compositions experimentally studied: ternary Zr– Fe–Cr glasses [1–4] (filled circles); limits of glass-forming-range in Zr–Fe [32,33] (asterisks); Zr–Cr glasses [36,37] (filled diamond) and [38] (diamonds with thick edges); Cr-rich bcc (+) and Zr-rich hcp (x) solid solutions [38]; Fe–Cr glasses [39,41] (squares) and [40] (triangles).

maintained at 623 K, Krebs [33] obtained amorphous FexZr100x films over the same composition range as Unruh et al. [32]. These compositions are contained in the calculated EGFR. In a recent work [34], Motta et al. reported that in thermally annealed and in irradiated Zr– Fe multilayers, Fe50Zr50 (global composition) became completely amorphous, whereas Fe66Zr33 produces an equilibrium mixture of amorphous alloy and bcc-Fe. A different complete amorphization range was apparent and could be ascribed to the different technique employed. In multilayers reaction the added suface energy and other energy and kinetic surface effects are relevant to the observed behaviour [35]. Zr30Cr70 amorphous films and foils were prepared, respectively, by co-evaporation on substrates cooled to liquid nitrogen temperature [36] and by splat-cooling [37]. Kim et al. [38] used DC magnetron sputtering to prepare ZrxCr100x amorphous alloys containing 11, 30, 38, 51 and 66 at.% Zr. These authors [38] also claim that the alloys with 81 at.% Zr and with 5 at.% Zr show X-ray diffraction patterns corresponding, respectively, to the crystalline hcp-Zr phase and to the bcc-Cr phase. The first of those two compositions lies inside the amorphous field of Fig. 3, while the second is effectively inside the bcc field. Xia et al [39] prepared CrxFe100x films (04x490) by thermal coevaporation of pure Cr and Fe metals onto substrates at

room temperature. An amorphous structure formed in samples with 404x475, while the bcc structure was observed for samples with x < 40 and x > 75. The present results predict that the amorphous phase should be present for compositions above 30 at.% Cr up to 66 at.% Cr. For the observed Cr70Fe30 amorphous alloy that lies slightly outside the present calculated limits, crystallization started at 623 K. O¨ner et al. [40] prepared amorphous Cr100xFex films with x=74, 78 and 82.4 by using flash-evaporation. These three alloys lie outside the calculated amorphous range of Fig. 3. Ball milling of a Cr72Fe28 solid solution at room temperature produced an amorphous powder with the composition Cr65Fe35 [41], that fits into the amorphous field. X-ray diffraction was carried out in the works on binary alloys quoted above [32,33,36–41] in order to determine the structure of the samples. Besides, the homogeneity of the amorphous phases was checked by Debye-Scherrer analysis in Ref. [36]. In Ref. [34] Rutherford backscattering spectroscopy, Mo¨ssbauer spectroscopy and electron diffraction from transmission electron microscopy were the characterization techniques employed. In summary, the EGFR defined by our calculation at 580 K is in agreement with most of the experimental results. Some amorphous films reported at the Cr-rich side of the Cr–Fe binary, produced by evaporation on

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Fig. 4. Zr–Fe–Cr system at 580 K. Line of the Zr(Fe,Cr)2 stoichiometric compound (—). The striped area gives the internal stability range of the single phase hcp Zr(Fe,Cr)2 intermetallic compound [44]. Calculated limits of metastable equilibrium: (+a)/a (- - -); a/(a+Zr(Fe,Cr)2) with the freeenergy of Zr(Fe,Cr)2 increased by 6 kJ mol1 (— - - -), by 8 kJ mol1 (— - -), and by 10 kJ mol1 (— -). Calculated tie-lines linking amorphous crown and crystalline nucleus compositions when the free energy of Zr(Fe,Cr)2 has been increased by 8 kJ mol1 (— - - and +), and when the free energy has been increased by 10 kJ mol1 (— - and empty squares).

amorphous substrates [39,40], occur beyond the 66 at.% Cr limit predicted here. These evaporation techniques are considered equivalent to an ultra-fast melt quenching. Nano-crystalline structures have been produced in the past by this technique for low solute concentration, although X-ray diffraction showed only diffuse scattering rings [42]. It should be interesting to use an even more reliable characterisation technique, like differential scanning micro-calorimetry, to determine the structure of those alloys unambiguously. The co-existence of crystalline Zr(Fe,Cr)2 intermetallic particles with an annular amorphous phase induced by neutron irradiation at temperatures around 580 K can also be analysed in the light of our results. The following discussion, in Section 4.2., will only cover the relationship between the -Zr matrix, the amorphous phase and the Zr(Fe,Cr)2 compound, as described by our model calculations at the temperature of experiment. Figs. 5–7 show the reported experimental data [5– 7,9,10,43]. The calculated free energy of the intermetallic compound Zr(Fe,Cr)2, shown in Fig. 1, presents a zone of internal stability comprised approximately between Zr(Fe0.8,Cr0.2)2 and Zr(Fe0.4,Cr0.6)2. This range includes the composition of the hcp Laves-type C14 precipitate

as reported in [31,44]. This phase is separated from the terminal binary phases ZrFe2 and ZrCr2, both of binary fcc-type C15 structure, by multiphase equilibrium regions as indicated in [44]. A calculation based on Miedema’s model cannot describe this complex scenery in detail. Nevertheless in Figs. 4–7 we correlate the hcp Zr(Fe,Cr)2 single phase internal stability [44] with the internal stability range defined by the present calculation as the extension of the striped region along the stoichiometric composition line. The width of the striped region was obtained by joining with straight lines the limits of the non-stoichiometric fields of the ZrFe2 and ZrCr2 binaries as depicted in [29,30]. The striped region includes most of the compositions in Zircaloy-4 and Zircaloy-2 intermetallic particles that become partially amorphous in irradiation experiments [5,9,43,45]. 4.2. Discussion In non-irradiated samples the target equilibrium between the intermetallic particle and the -matrix is represented by a tie line drawn from any particle composition value in its stability range and the triangle-vertex corresponding to -Zr. The predicted composition profile for an intermetallic particle in equilibrium with

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Fig. 5. Comparison between the present calculations and experimental data on amorphized particles [10]. Zr–Fe–Cr system at 580 K. Zr(Fe,Cr)2 stoichiometric compound (—). The striped area gives the internal stability range of the single phase hcp Zr(Fe,Cr)2 [44]. Calculated limits of metastable equilibrium: (+a)/a (- - -); a/(a+Zr(Fe,Cr)2) with the free-energy of Zr(Fe,Cr)2 increased by 6 kJ mol1 (— - - -), by 8 kJ mol1 (— - -), and by 10 kJ mol1 (— -). Experimental data for Zr(Fe,Cr)2 particles in Zircaloy-4 irradiated at 593 K (Fig. 13b in [10]): crystalline nucleus (x), amorphous crown (+), -matrix (empty star). Data for Zr(Fe,Cr)2 particles in Zircaloy-4 irradiated at 593 K and annealed at 873 K (Fig. 13d in [10]): crystalline nucleus (upwards triangles), microcrystallized region (downwards triangles), -matrix at the interface microcrystallized region/ matrix (triangle pointing to the left). Data for Zr(Fe,Cr)2 particles in Zircaloy-4 irradiated at 608 K (Fig. 17b in [10]): completely amorphized Zr(Fe,Cr)2 particle (squares).

the -matrix presents a composition discontinuity at the particle/matrix interface (Fig. 7). Gilbon et al. [10] have reported the composition profile measured on an isolated Zr(Fe,Cr)2 particle in a non-irradiated -Zr matrix of recrystallized Zircaloy-4. That profile shows a central region of nearly constant composition and a region of steep composition variation in the particle near the particle/matrix interface. The composition profile defines a virtual path, in a two phase field, parallel to the equilibrium tie line, as shown in Fig. 7. This could be due to a matrix effect, as is commonly argued, in which case the profile can be assumed to arise from the equilibrium described above. Gilbon et al. [10] also reported the composition profile at the particle/matrix interface in irradiated samples for intermetallic particles that remain crystalline after irradiation at a temperature of 673 K. The non-amorphized particles also show a continuous composition profile with a smooth variation in its central core followed by a steep variation at the particle/matrix boundary. The core region has a ternary composition whose Fe/Cr ratio lies within the region of hcp intermetallic

stability, near to the striped region tentatively defined in Figs. 4–7, although depleted in solute. In the region of steep composition variation the Fe/Cr relation in the matrix decreases down to a fixed value close to 1 near the particle/matrix boundary. The last result has also been reported by another author [46]. Under irradiation, induced composition profile changes can be anticipated near grain boundaries and interfaces. There is not general agreement in the literature concerning the relation between irradiation-induced effects [43,47–49] and the composition changes near the interfaces. In the present context the reported results under irradiation could be produced by a matrix effect, as before, with the particle composition at the interface being preferentially depleted in iron with respect to chromium along its ternary stoichiometric composition range, from an Fe/Cr ratio of 2 at the particle core, down to an Fe/Cr ratio of 1 at the interface. The amorphous phase is generally assumed to be formed at the particle/matrix interface by a polymorphic transformation due to the increase of the particle free energy caused by its departure from stoichiometry produced by

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Fig. 6. Comparison of results of the present calculations and the experimental data on amorphized particles [5–7,9,43]. Zr–Fe–Cr system at 580 K. Line of the Zr(Fe,Cr)2 stoichiometric compound (—). The striped area gives the internal stability range of the single phase hcp Zr(Fe,Cr)2 intermetallic compound [44]. Calculated limits of metastable equilibrium: (+a)/a (- - -); a/(a+Zr(Fe,Cr)2) with the free-energy of Zr(Fe,Cr)2 increased by 6 kJ mol1 (— - - -), by 8 kJ mol1 (— - -), and by 10 kJ mol1 (— -). Composition profiles in Zr(Fe,Cr)2 partially amorphized particles obtained in Zircaloy-2 irradiated at 563 K to a fluence of 3.5
1025 n m2 (Fig. 8 in [43]), Zircaloy-4 irradiated at 580 K to a fluence of 8
1025 n m2 (Fig. 1 in [5]), Zircaloy-4 irradiated at 580 K to a fluence of 12
1025 n m2 (Fig. 6 in [6]), Zircaloy-4 irradiated at 561 K to a fluence of 8.5
1025 n m2 (Fig. 7 in [7]), and Zircaloy-2 irradiated at 563 K to more than 3
1026 nm2 (Figs. 7 and 10 in [9]): -matrix (squares), amorphous crown (diamonds), crystalline nucleus (+).

atomic mixing [49,50]. In partially amorphized particles Gilbon et al. [10] report a continuous profile where the Fe/Cr ratio decreases to a very low value in the transition region near the particle/amorphous crown boundary (see Fig. 13b in Ref. [10]). As it is shown in Fig. 5, this profile defines a path parallel to the tie-line drawn in Fig. 4. The Fe/Cr ratio remains nearly constant in the transition from the amorphous crown to the matrix, as seen in Fig. 5 in this work, defining a path parallel to the target -Zr/a tie-line. Composition profiles measured on a completely amorphized intermetallic (Fig. 17b in [10]) are also included in Fig. 5. Again the reported data continued the general trend of the amorphous region, but extend themselves into the (-Zr+a) field, drawing a path parallel to a target tie line. In samples annealed at 873 K, a microcrystalline phase is formed in the amorphous region with partial solute (Fe) reversion back towards this microcrystalline phase. The solute reversion occurs parallel to the former tie line between the amorphous crown and the crystalline core. The Fe/Cr ratio of the microcrystalline region is near 1. Also the transition composition profile from the microcrystalline region to

the matrix defines a path parallel to the tie line joining the microcrystalline region to the matrix. The precipitation of Zr(Fe,Cr)2 with an Fe/Cr ratio equal to 1 was observed after irradiation by other authors [51]. Fig. 6 includes the composition profiles for partially amorphized particles interpolated from data reported by different authors (Fig. 1 of [5], Fig. 6 of [6], Fig. 7 of [7], Figs. 7a and 10 in [9] and Fig. 8 in [43]). The general trend is the same as described above but there is some experimental indication of different paths. At equal temperature, the higher the dose, the higher appears to be the Fe depletion of the amorphous region. If these results are confirmed, they indicate that conditions at the interface and composition profiles have not been steady. The observed profile change is consistent with the change in target equilibrium tie-lines for the case in which the free energy of the intermetallic phase increases due to irradiation (Fig. 4). The same behaviour is anticipated if the Fe/Cr ratio at the interface decreased along stoichiometric particle stability region, due to radiation induced solute drainage. The experimental composition profiles measured in the crystalline core and in the peripheral

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Fig. 7. Comparison between the present calculations and experimental data on non-amorphized particles. Zr–Fe–Cr system at 580 K. Line of the Zr(Fe,Cr)2 stoichiometric compound (—). The striped area gives the internal stability range of the single phase hcp Zr(Fe,Cr)2 intermetallic compound [44]. Symbols indicate composition profile measured on isolated Zr(Fe,Cr)2 particles,taken from Fig. 5a and b of Ref. [10]. Non-irradiated specimen of recrystallized Zircaloy-4 (squares); nucleus of this particle (+). Specimen irradiated at 673 K (diamonds); nucleus of the irradiated precipitate (X).

amorphous crown, shown in Figs. 5 and 6, are nearly constant in the crystalline core and present a fast variation near the particle/amorphous crown interface, where the composition profile deviates towards a value with low Fe content while the Cr concentration remains nearly constant. The amorphous region compositions are either included or not in the amorphous stability area, depending on the value assumed for the increase in the energy of the intermetallic compound in the evaluation of the boundary a/a+Zr(Fe,Cr)2 shown in Fig. 4. Measured amorphous compositions near the -matrix are frequently close to the (+a)/a coexistence limit shown in Figs. 5 and 6. The continuous composition profile in the region near the amorphous crown/matrix interface extends from the a/a+Zr limit into the +a field. The preceding results show the correlation of the calculated tie-lines with the measured composition profiles. A pronounced change from profiles with constant Fe/Cr ratio (non amorphized intermetallic phase) to profiles where this ratio decreases to a very low value (amorphized intermetallic phase) is observed. This drastic change of behaviour could be due to a modification of the effective target equilibrium between the non-amorphized and amorphized particles, and not merely to a different mobility of Fe compared to Cr. It was recognised that

due to prevailing environment under irradiation, such an amorphous phase should be highly unstable [12] and reversion to the crystal phase should occur. The existence of a metastable target equilibrium such as shown by the calculation should prevent this to occur, until total particle amorphization and later dissolution occurs forced by radiation-induced departure from the effective target equilibrium. The observed partial solute reversion of the solute from the amorphous to a new crystalline structure [10] or to a completely amorphized intermetallic particle [5] during a post-irradiation annealing is consistent with an effective local target equilibrium scenery described above. Nevertheless, the study of the effect of thermal and ballistic irradiation-enhanced diffusion between the phases present is underway to understand the reported profiles.

5. Conclusions The free-energy diagram for the stable and the metastable phases of the ternary Zr–Fe–Cr system at 580 K has been calculated using Miedema’s model. Melt spinning amorphization experiments are consistent with the calculated amorphization range, although some data

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obtained by vapour deposition techniques lie outside the predicted limits. The present analysis suggests that the reported effects of irradiation on the microstructure of the phases in commercial Zr-based alloys used in nuclear reactors may be due to a specific new radiationinduced target equilibrium, and not merely due to a different mobility of the solutes.

Acknowledgements The authors are indebted to C. Lemaignan and A.M. Motta for their encouragement and critical reading of the manuscript. Thanks are given to L.M. Gribaudo for drawing, for comparative purposes, the free-energy curves of Zr–Cr crystalline phases by running thermocalc using the Zircobase data. The work of one of us (JAA) has been supported by DGESIC (Grant PB980345).

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