TCP phase predictions in Ni-based superalloys: Structure maps revisited

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Acta Materialia 59 (2011) 749–763 www.elsevier.com/locate/actamat

TCP phase predictions in Ni-based superalloys: Structure maps revisited B. Seiser a,⇑, R. Drautz b, D.G. Pettifor a b

a Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universita¨t Bochum, Stiepeler Strasse 129, 44801 Bochum, Germany

Received 5 August 2010; received in revised form 3 October 2010; accepted 6 October 2010 Available online 1 November 2010

Abstract The traditional methods for predicting the occurrence of deleterious topologically close-packed (TCP) phases in Ni-based superalloys have been based on the PHACOMP and newPHACOMP methodologies. These schemes use the average number of holes N h or the centre of gravity of the elemental d-bands Md to predict whether or not a given multicomponent alloy will be prone to TCP formation. However, as both these one-dimensional methodologies are well-known to fail with respect to new generations of alloys, a novel twodimensional structure map ðN ; DV =V Þ is introduced where N is the average electron concentration and DV =V is a composition-dependent size-factor difference. This map is found to separate the experimental data on the TCP phases of binary A–B transition metal alloys into well-defined but sometimes overlapping regions corresponding to different structure types such as A15, r, v, R, P, d, l, M and Laves. Detailed investigations of ternary phase diagrams and multicomponent systems show that TCP phases, regardless of the number of constituents, are located in the same regions of the structure map that are favoured by the binary compounds of the same structure type. The structure map is then used in conjunction with CALPHAD computations of r phase stability to show that the predictive power of newPHACOMP for the seven component Ni–Co–Cr–Ta–W–Re–Al system studied recently by Reed et al. [24] is indeed poor. This supports a growing consensus that robust methods of TCP phase prediction in multicomponent alloys will require the inclusion of reliable firstprinciples thermodynamic databases within the semi-empirical CALPHAD scheme. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Ni-based superalloys; Topologically close-packed phases; NewPHACOMP; CALPHAD; Structure map

1. Introduction Single crystal Ni-based superalloys are used in modern gas turbines because of their remarkable resistance to creep deformation at elevated temperatures, which is ensured by the addition of refractory elements, such as Cr, Mo, W and Re [1]. If the concentrations of such refractory elements are too large, topologically close-packed (TCP) phases can form at high temperature and stress. Most TCP phases are detrimental to the overall high-temperature performance of Ni-based superalloys because they are not only brittle but also deplete the Ni-rich matrix of potent solidsolution strengthening elements [1]. The detrimental effects

⇑ Corresponding author.

E-mail address: [email protected] (B. Seiser).

of TCP phases in Ni-based superalloys were first encountered in the early 1960s by Wlodek [2] who discovered a r phase forming in IN-100 which caused a noticeable degradation of the creep properties. Further observations of deleterious TCP phases in other Ni-based superalloys led to the development of methods for predicting the formation of TCP phases. These methods were based on the understanding of the factors controlling the stability of TCP phases which had been garnered from studies of binary, ternary and quaternary TCP phases. In 1948, Sully and Heal [3] following Hume-Rothery [4] found that the r phases in Co–Cr and Cr–Fe form at the same valence electron concentration. Later discoveries of newly observed TCP phases in additional binary and ternary systems allowed Beck and his collaborators [5–7] in the early 1950s to investigate the formation of these TCP phases in terms of the valence electron concentration in

1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.10.013

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more detail. However, following Pauling [8], rather than working with the number of valence electrons N, Beck et al. chose to work with the number of holes in the d-band Nh, where Nh = 10.66  N except for the 3d magnetic elements Fe, Co and Ni where Nh = 2.22,1.71 and 0.61, respectively, in order to agree with the experimental magnetic moments. They used these values to calculate the average hole number N h and consequently characterized the range where TCP phases such as r, P, R, d and l are stable. The valence electron concentration, although it was seen as the dominant factor in understanding the formation of TCP phases, could not fully explain cases where the electron hole condition was fulfilled but no TCP phases were formed, or simply failed for TCP phases such as the Friauf–Laves phases which are formed over a wide range of concentration below eight electrons. Laves [9] for the Friauf–Laves phases and Duwez [10] for the r phase showed that the difference in atomic size of the constituent elements is also of significant importance for the stability of TCP phases. In 1964, Boesch and Slaney [11] used Rideout and Beck’s [5] average number of holes to predict the presence of r phases, in complex, commercial Ni- and Co-based superalloys. This approach became known as PHACOMP, an acronym for phase computation [12]. The method evaluated the average number of holes based on the composition of the c matrix as it was assumed that TCP phases form exclusively within the c phase. In the early days of PHACOMP, the c composition was estimated from the overall alloy composition based on empirical observations as explained in detail by Sims [13]. Boesch and Slaney proposed a critical value of N h above which TCP formation was very likely. However, this temperature-independent critical value was found to be dependent on the particular alloy system and TCP structure type [13]. In 1984, the poor performance of PHACOMP, in particular in describing the precipitation of the l phase, led Morinaga to propose a similar one-parameter method called newPHACOMP [14]. Instead of Nh, Morinaga used the average energy level of the d-orbital, Md, as obtained from electronic structure calculations. This procedure implicitly included both size and electronegativity differences within the quantum mechanical evaluation of Md. Although the 4d and 5d elements still showed a similar linear variation in Md across the series as that of the hole number Nh, the values of Md for the late 3d transition metals deviated significantly from linearity. This deviation improved agreement with experiment for the Co and Ni l phases such as Co7W6 and Ni6Nb7. However, even though they also introduced temperature-dependent critical Md values, newPHACOMP was still found to be a poor predictor of the behaviour of newer generations of Ni-based superalloys. For example, when Re was added in non-negligible amounts, the critical value still needed to be adjusted for each individual alloy system just as for the earlier PHACOMP scheme [15,16].

The variation in the Md values for the late 3d transition metal alloys closely resembles that proposed independently by Watson and Bennett in the same year [17]. They plotted the stability ranges of the non-Laves TCP phases of binary transition metal alloys as a function of the average d-band hole count, N h , observing that the TCP phases are stable over a considerable range of N h . They suggested an effective number of holes N eff h , as shown in Fig. 4 of Ref. [17], which were chosen such that all the TCP phases were associated within the narrowest possible range of N eff h . Like Morinaga, they found that the values for the late 3d transition metals deviated from the linearity of Nh, reflecting the influence of atomic size. They also illustrated the importance of the atomic size for the Laves phases by plotting a two-dimensional structure map ðN h ; V A =V B Þ, where VA/VB is the ratio of the equilibrium atomic volume of the A and B transition metal elements. Other structure maps using different coordinates were also proposed at that time (see e.g. [18–21]) but only the stoichiometric AB2 Laves and A3B A15 structure types were considered amongst the TCP phases. In this paper we propose a novel two-dimensional structure map that allows one to investigate not only binary but also multicomponent systems without being restricted to a particular stoichiometry or structure type. We have focused on two-dimensional rather than three-dimensional structure maps for ease of applicability, choosing the two coordinates of valence electron concentration and size-factor difference and neglecting the third coordinate of electronegativity difference which has been included, for example by Villars and Girgis [21] in their phenomenological threedimensional binary plots. This neglect of the electronegativity difference is partially justified by the theoretical tight-binding study of the stability of 2:1 stoichiometric binary transition metal compounds [22], where the Laves phases (C14, C15 and C36) were separated into their correct domains from the CuAl2 (C16) and MoSi2 (C11b) phases by the two coordinates of valence electron concentration and size-factor difference, with the electronegativity difference playing a more minor role (compare the left- and right-hand panels in Fig. 3 [22]). The coordinate of the present two-dimensional structure map is therefore given by the valence electron concentration: X N¼ ci N i ; ð1Þ i

where ci is the concentration of the constituent element i. The ordinate is given by the average of the relative volume difference DV =V , which is defined by: X DV =V ¼ ci cj jV i  V j j=½ðV i þ V j Þ=2; ð2Þ i;j

where Vi is the atomic volume of the constituent element i. In contrast to Watson and Bennett’s ordinate Vi/Vj, which can be either greater than or less than unity whether Vi  Vj 7 0, our choice depends on the magnitude of DV

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since the size factor contribution to the heat of formation, and hence phase stability, varies as (DV)2 to lowest order [23]. The inclusion of the prefactor cicj guarantees that the size factor DV =V ! 0 as either ci ? 0 or cj ? 0, as it must. We will see that the composition dependence is needed not only for treating the v phase in the binaries but also for handling multicomponent systems. The outline of the paper is as follows. The next section gives a short introduction to the most common TCP phases and addresses their most important characteristics. In Section 3, we discuss the trends which are observed within the new structure map using the experimental data for binary alloys, explaining the relation between regions demarcating different TCP structure types. In Sections 4 and 5, we examine ternary and multicomponent systems, and show that the TCP phases, regardless of the number of constituent elements, are located in the same regions of the structure map that are favoured by the binary compounds of the same structure type. In Section 5, we discuss the results of CALPHAD calculations of Ni-based superalloys in conjunction with the structure map and show that the predictive power of newPHACOMP for the seven component Ni–Co–Cr–Ta–W–Re–Al system as studied recently by Reed et al. is poor [24]. Section 6 contains our conclusions. 2. TCP phases TCP phases were first characterized by Frank and Kasper [25,26], who showed that a number of complex phases can be represented as packings of hard spheres of different radii. The basic stacking units are the coordination polyhedra (Fig. 1) which have exclusively triangular faces. The most important polyhedron is the 5-fold symmetric icosa-

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hedron, Z12, where 12 refers to the coordination number Z of the atom at the centre of the polyhedron. Z12 polyhedra are found in all TCP phase where they are combined with higher-coordinated Frank–Kasper polyhedra, namely Z14, Z15 and Z16. In practice, these coordination polyhedra distort in order to fill space completely or conform to the translational periodicity of the lattice. Kasper’s [27] investigations on the nearest-neighbour distances of hard sphere packings indicated that larger atoms should sequentially occupy the sites from highest to lowest coordination as their atomic size decreased. This is generally observed amongst the TCP phases, though there are exceptions [28]. Table 1 summarizes the different Frank–Kasper coordination polyhedra for the most common TCP phases, namely A15, r, v, P, R, d, l, M and the Laves phases. Three polytypes of the Laves phases are observed: the cubic C15-MgCu2, the hexagonal C14–MgZn2 and the dihexagonal C36–MgNi2 structure. These polytypes are related to each other in that their coordination polyhedra and basic unit layers are identical, while the stacking sequence of the layers is different. Note that v is strictly speaking not a member of the topologically close-packed family of structures because it contains atoms with coordination shells that are made up of 13 neighbours (Z13 coordination polyhedra) which do not have only triangular faces. However, studies on alloy chemistry and ordering indicate that the same principles hold for the v phase as for the TCP phases [29]. Following the lead of Watson and Bennett [17], the structure types shown in Table 1 are ranked according to their fraction of 12-fold sites f12 as in the last column. We see that there is a similar trend between the phase sequence given by f12 for bcc, A15, r and v and their stability range with respect to the valence electron concentration.

Fig. 1. Frank–Kasper coordination polyhedra including Z13 of the v phase.

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Table 1 Frank–Kasper coordination polyhedra of the most common TCP phases together with fcc, hcp and bcc where 12- and 14-fold coordination polyhedra are given in parenthesis since they are not the Frank–Kasper polyhedra Z12 and Z14, respectively. hZi is the average coordination number. Structures are ranked from the lowest to highest fraction of 12fold polyhedra, f12. Structure

fcc/hcp C14 C15 C36 M l R d P v r A15 bcc

Atoms in basis

1 12 24 24 52 13 53 56 56 58 30 8 1

Coordination polyhedra Z12

Z13

Z14

Z15

Z16

(1) 8 16 16 28 7 27 24 24 24 10 2 0

0 0 0 0 0 0 0 0 0 24 0 0 0

0 0 0 0 8 2 12 20 20 0 16 6 (1)

0 0 0 0 8 2 6 8 8 0 4 0 0

0 4 8 8 8 2 8 4 4 10 0 0 0

hZi

f12

12.00 13.33 13.33 13.33 13.36 13.39 13.39 13.43 13.43 13.10 13.47 13.50 14.00

1.00 0.67 0.67 0.67 0.54 0.54 0.51 0.43 0.43 0.41 0.33 0.25 0.00

Generally speaking, this suggests that the 12-fold sites are associated with the more compact atoms (and hence mainly atoms with more valence electrons) while the more highly coordinated sites are occupied by larger atoms (and hence mainly atoms with fewer valence electrons). 3. Binary structure map The data set for the binary structure map includes all transition metal systems. It does not include the noble metals of group XI nor the rare earths La–Lu because they are in general not considered for the design of Ni-based superalloys. It covers all stable TCP, fcc-, bcc- and hcp-based phases which are observed experimentally up to the melting temperature. The compositions and homogeneity ranges are obtained from systematic studies of the data within Massalski [30], Sinha [29], Watson and Bennett [17] and Joubert [28,31]. Cross-checks on the reliability of our database have also been carried out using the Linus Pauling File Database [32] and the Inorganic Crystal Structure Database [33]. Occasional reference is made to the original publications where conflicting structures are reported. The equilibrium atomic volume of the chemical elements is calculated from their metallic radii for coordination number 12 at room temperature as given in Ref. [34]. Two binary structure maps are shown in Fig. 2 to highlight the importance of stoichiometry in order to achieve a compact domain for the v phase. Both structure maps use the average valence concentration along the horizontal axis. However, while the vertical axis of the structure map on the right-hand side of Fig. 2 is dependent on the stoichiometry due to the prefactor cicj, the structure map on the left-hand side is independent (any fixed value for ci and cj = (1  ci) could be used as this does not change the relative topology of the different structural domains; in the figure we have taken ci = cj = 1/2). The range of a

given TCP alloy is taken from the observed homogeneity range in the appropriate TCP phase diagram. Note that no binary M or P phases are observed and therefore no regions could be defined. The main difference between the two structure maps in Fig. 2 is clearly the grouping of the v phases. In the lefthand figure with the concentration-independent ordinate the v domain is very spread out in the vertical direction with some v phases located high up within the Laves phase region clearly separated from the other v phases that overlap with the A15 and the r phase domains. As shown in the right-hand figure, the prefactor cicj shifts these v phases down into the overlap region of the A15 and r phase such that all v phases are now closely grouped. For example, Zr–Re exhibits the C14 Laves phase ZrRe2 and the v phase Zr5Re24 with coordinates in the left-hand panel of (6.00, 0.22) and (6.49, 0.22), respectively, so that both appear near the lower-middle of the extensive Laves domain. However, on using the compositional-dependent size factor in the right-hand panel they now take the coordinates (6.00, 0.20) and (6.49, 0.13), respectively, so that whereas the C14 remains firmly near the middle of the Laves domain, the v phase has moved down amongst the A15 and r phases like the f12 ordering in Table 1. The use of the composition-dependent size factor results in two groups of TCP phases that can be clearly identified in the right-hand panel of Fig. 2: (i) the A15, r and v phases with values of the atomic size factor DV =V less than about 0.15; and (ii) the l and Laves phases with values at DV =V greater than about 0.15. The R and d phases occur in the region of overlap between these two groups. In the following subsections, we discuss the positioning and spread of each TCP structural domain within the righthand structure map. 3.1. A15 phase region A15 structures are observed for a wide range of valence electron numbers, namely between 5.2 (Ti4Pd) and 7.1 (Cr3Pt), but showing a preferred formation between 6 and 6.5 electrons. Most composition ranges of the 25 investigated A15 compounds are relatively narrow and frequently confined to the “ideal” (A3B) composition. Only a few binary A15-type phases, like V–Ir with 61–76 at.% V and VOs, are found to be stable at compositions deviating significantly from the “ideal” composition. The upper size limit of the A15 domain, namely DV =V ¼ 0:10, is the smallest of all the TCP phases. This is not unexpected since its largest coordination polyhedra is only Z14, whereas the atoms in other TCP phases have access to larger coordination polyhedra. 3.2. r phase region Similar to the A15 phase region, the electron concentration range for the r phase region from 5.6 to 7.8 electrons is fairly broad, but most r phases show a clear preference for

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Fig. 2. Structure maps of binary TCP phases with the different TCP domain boundaries marked by thin lines with A15 (brown), r (red), v (blue), l (purple) and Laves (green) using the experimental homogeneity ranges as a guide. The size factor ordinate is given by Eq. (2) with the left- and right-hand side maps being independent and dependent of stoichiometry, respectively. The coordinate in both maps corresponds to the valence electron concentration as given by Eq. (1). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

formation at electron concentrations between 6 and 7 electrons. As can be seen from Fig. 3, all the r phases with electron concentrations above seven electrons involve mainly the three magnetic 3d transition elements Fe, Co and Ni. This indicates that the phase stability can be strongly affected by magnetism just as elemental iron with n = 8 takes the bcc structure characteristic of groups V and VI and not the hcp structure of the non-magnetic isovalent group VIII elements Ru and Os [35]. Fig. 3 also shows that the Mo–Fe and Mo–Co systems exhibit structural trends from r ? l ? R ? C14 and r ? l ? DO19 (ordered hcp-type structure), respectively, as the average number of valence electrons increases. The upper size limit of the r domain, namely DV =V ¼ 0:16, reflects the fact that the constituent atoms can vary much more in size than for A15 because r has five non-equivalent coordination polyhedra with the largest being Z15. 3.3. v phase region The v phase is only found in a very narrow electron concentration range between 6.3 and 6.8 electrons. The homogeneity ranges of individual v phases can cover a broad range of composition such as, for example, v(Nb–Re) with

Nb ranging from 12 to 38 at.%. The v phases which are close to stoichiometry A5B24 (where the B atoms are 12 and 13 coordinated and the A atoms are 16 coordinated) such as Sc–Tc, Sc–Re, Zr–Tc, Zr–Re, Hf–Tc and Hf–Re also tend to have large differences in atomic size, jVi  Vjj. Hence, as shown in the left-hand panel of Fig. 2, without the concentration dependence of the size factor, the points of these phases would lie well within the Laves phase region, clearly separated from other v phases which lie in the overlap region of the A15 and r domains. 3.4. R and d phase regions The R phase is observed together with r and l in the Fe–Mo system at an average valence concentration of around 7.25 electrons. The phases occur in the order r ? l ? R as the valence electron concentration increases as can be seen in Fig. 3. This disagrees with the overall ordering of R ? l as given by f12 in the final column of Table 1. However, the R phase exists only over the small temperature range from 1235 to 1488 °C, which emphasizes the extreme sensitivity of the stability with respect to the temperature. Fig. 2 shows that the data points of this phase lie close to the area where the regions of r and l overlap.

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Fig. 3. Structure map ðN ; DV =V Þ of all the binary r phases with N > 7, showing also the other TCP phases of the Mo–Fe and Mo–Co systems. The different TCP domain boundaries are marked using the homogeneity ranges shown in Fig. 2.

Another R phase was reported to be formed in Mn–Ti by Waterstrat [36] in 1962 and has been approved by Murray who assessed a series of experimental phase diagrams for the Mn–Ti system [37] in 1981. This R phase is stabilized as TiMn4 with a valence electron concentration of 6.4 and is, therefore, located in the r and v domains far from the r/l overlap region. However, its existence has been questioned recently [38], so that we have not included it in Fig. 2. The only d phase that has been discovered in a transition metal binary system has the composition Mo49Ni51 and is located at around eight electrons, very close to the l phase and Laves phase region. 3.5. l phase region The l phase has been reported and confirmed to exist in 12 binary A–B systems, where A is a 4d or 5d transition metal from group V or VI and B is a late 3d element Fe, Co or Ni. Its crystal structure is typified by the W6Fe7 compound where the small Fe atom sites are 12-fold coordinated and the W atoms occupy the larger Z14, Z15 and Z16 sites with two atoms each, respectively. l phases are observed for valence electron concentrations between 6.5 and 7.75 electrons without showing any preferred valence electron concentration in this range. The composition ranges of the l phases, in particular Co–Nb and Co–Ta, can be very broad but without significant variations in DV =V . The l phase region is the only phase region which fully overlaps with the Laves phase region. As can be seen from the right-hand panel in Fig. 2, both structure types have the largest size factors of all the TCP phases, which confirms that the difference in atomic size is crucial for the stability of these structures. Comparison of the interatomic distances in the l phase with those in Laves phases

at the same equilibrium volume shows that the Z12 polyhedra of the l and the Laves phases are of similar size but more highly coordinated polyhedra of the l phase are smaller than the Z16 polyhedra of the Laves phases. This can explain the relatively high minimum size factor, which is similar to the Laves phase domain, but the narrower size factor range of the l phase region with 0:135 6 DV =V 6 0:24 compared to the much larger size factor range of the Laves phases. 3.6. Laves phase region Unlike all the TCP phases mentioned above, the Laves phases can be found at almost any valence electron concentration below 8. It was found that the order of stable Laves phases changes from cubic (C15) to hexagonal (C14) back to cubic (C15) as the valence electron concentration increases [39]. This behaviour has been rationalized within a simple d-band tight binding model by Johannes et al. [40], although Ohta and Pettifor subsequently showed that both size and electronic factors are needed for good qualitative agreement with the AB2 structure map [22]. The closest packing of hard spheres is obtained for an ideal radius ratio rA/rB = 1.225, where A atoms occupy the large 16-fold sites and the B atoms occupy the small 12-fold sites. For the ideal AB2 stoichiometry, this ratio corresponds to a size factor of DV =V ¼ 0:263. As the right-hand map in Fig. 2 shows, the observed size factor DV =V of the Laves phases forming atoms vary between 0.114 (rA/rB = 1.08 of TaV2) and 0.446 (rA/rB = 1.445 of YNi2). From the observed deviations of the radius ratios from the ideal value it can be concluded that the radii of the atoms in a Laves phase differ significantly from those of the pure A and B metals. This considerable capacity of atoms for

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undergoing deformation in Laves phases was found to be driven by the difference in the electronegativies of the constituent atoms [41]. Hence, it is not surprising that Laves phases feature large homogeneity ranges involving significant variations in DV =V . 3.7. Structural trends and predictability We see in Fig. 2 that there is considerable overlap between A15, r and v domains with DV =V K 0:15 and also between the l and the Laves domains with DV =V J 0:15. Fig. 4 illustrates how these A15, r and v domains overlap for the 4d and 5d binary AxB1x alloys by considering a select set of systems, namely A is either Nb (NA = 5) or Mo (NA = 6) and B is from a group with 7, 8 or 9 valence electrons, respectively. Thus, we will compare and contrast the following three groupings:  NB = 9: r Nb(Rh/Ir) and s Mo(Rh/Ir),  NB = 8: t Nb(Ru/Os) and u Mo(Ru/Os) and  NB = 7: v Nb(Tc/Re) and w Mo(Tc/Re), where the circled numbers label the different curves in Fig. 4. Firstly, let us consider the grouping with NB = 9. For curves r we see that the Nb–Rh and Nb–Ir systems display similar parabola with a maximum size factor of around 0.12 for N ¼ 7 midway between NA = 5 and NB = 9. This is not unexpected as Rh and Ir have almost identical metallic radii ˚ and 1.357 A ˚ , respectively [34]. Despite of around 1.345 A minor differences in the homogeneity ranges of their stable phases, both systems display the same structural trend from A15 ? r ? L10 ? L12 as N increases, where L10 and L12

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are ordered phases with respect to an underlying fcc lattice. Given that DV =V is large and that non-magnetic A15, r and v phases are not observed beyond N > 7, it is satisfying that fcc-based structure types are found instead in this region. Replacing the Nb by Mo, the maximum in the Mo–Rh and Mo–Ir curves s shifts over to N ¼ 7:5 with the amplitude dropping considerably to around DV =V ¼ 0:05 due to ˚ ) compared the decrease in the metallic radius of Mo (1.400 A ˚ to Nb (1.468 A). Both Mo–Rh and Mo–Ir display stable B19 and D019 phases that are ordered structures with respect to an underlying hcp lattice. This is not unexpected given that non-magnetic transition elements with DV =V ¼ 0 take the hcp structure for N = 7 and N = 8. However, Mo–Ir also displays the full sequence from A15 ? r ? B19 ? D019, although the r phase is only marginally stable over a 100 °C temperature range just below the melting temperature. Secondly, let us consider the grouping with NB = 8. For curves t we see that Nb–Ru and Nb–Os display different stable structure types, even though they both fall on very similar parabola in Fig. 4. The Nb–Os system is unique amongst binaries in that it shows a sequence between the three TCPs A15 ? r ? v. This highlights that even though Ru and Os are very similar elements, the stability of phases is very sensitive to small changes in the free energy. These could perhaps be driven by non-negligible changes in relativistic effects in going from 4d to the heavier 5d series. Thus, 4dAg and 5d Au have a very different colour due to larger relativistic effects increasing the s–d energy level separation in Au. Replacing Nb by Mo, the Mo–Ru and Mo–Os curves r shift to the right with much lower amplitude. Whereas Mo–Ru displays only a stable r phase, Mo–Os shows an A15 ? r sequence.

Fig. 4. Structure map ðN ; DV =V Þ of a selection of 4d and 5d binary phases containing Nb and Mo. The curves are labelled according to r Nb(Rh/Ir), s Mo(Rh/Ir), t Nb(Ru/Os), u Mo(Ru/Os), v Nb(Tc/Re) and w Mo(Tc/Re). The different TCP domain boundaries are taken from Fig. 2 with the dashed vertical red line from N representing the right-hand boundary of the non-magnetic r phases (see Fig. 3).

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Finally, we consider the grouping with NB = 7. For curves v we see that whereas Nb–Tc displays a stable v phase over a small homogeneity range, Nb–Re displays the sequence r ? v with v being stable over a very wide homogeneity range in Fig. 4. The narrow r phase, on the other hand, is stable over only a 400 °C temperature interval before melting. Replacing Nb by Mo, the Mo–Tc and Mo–Re curves w shift to the right with much lowered amplitude. The Mo–Tc system shows the sequence A15 ? r (with N r < 6:74), whereas the Mo–Re system displays the sequence r ? v (with N v > 6:76). Fig. 4 demonstrates a very important property of structure maps. Whereas structure maps can be used to predict the likely structure of a phase whose composition is known to fall in a given structural domain, they cannot be used to predict the occurrence of any unknown phase with a given structure-type and composition [42]. This is illustrated by curves s and t where the 4d Ru and Rh alloys Nb–Ru and Mo–Rh do not exhibit TCPs, whereas the 5d Os and Ir alloys Nb–Os and Mo–Ir do display TCPs. As implied by the well-known common tangent construction, the occurrence of a given phase must be energetically favourable compared to competing nearby phases. Thus, it is possible that the absence of TCPs in Nb–Ru is due to the presence of the fcc-based L10 and L12 phases, whereas the presence of TCPs in Nb–Os is due to the absence of energetically favourable fcc-based phases. We will find in

Section 5 that this lack of predictability of unobserved phases is also shown by the newPHACOMP methodology. 4. Ternary systems In this section we investigate the applicability of the two-dimensional structure map ðN ; DV =V Þ to ternary systems. The transition-metal ternary systems investigated are listed in Table 2 and comprise 121 isothermal sections of phase diagrams at equilibrium temperatures ranging from 800 to 1400 °C. Table 2 is not a complete list of all experimentally observed ternary TCP phases; nevertheless it contains a representative selection of isothermal sections with very differently shaped and positioned TCP phase stability ranges. We sought as many TCP phases as available in the literature which occur in a binary system but also have widely stretched stability regions which can even reach to the boundaries of the binary system. Moreover, we have tried to obtain all known TCP phases that occur in ternary systems without having any corresponding binary TCP phases. We shall refer to these TCP phases as true ternary TCP phases [43]. Table 2 comprises 44 r phases (containing three true ternary phases), 13 v phases and 21 l phases. Four ternary phase diagrams contain the R phase (containing three true ternary phases), five contain A15 and 33 contain Laves phases with C14 (NiTiZr) being the only true ternary Laves phase. As mentioned previously, there

Table 2 Investigated transition-metal ternary systems with their observed TCP phases. Bold TCP phases are true ternary (i.e. no binary TCP phase exists in any of the boundary systems). Co–Cr–Fe Co–Cr–Mo Co–Cr–Ni Co–Cr–Re Co–Cr–Ti Co–Cr–W Co–Fe–Mo Co–Fe–Re Co–Fe–V Co–Fe–W Co–Mn–Mo Co–Mn–V Co–Mo–Nb Co–Mo–Ni Co–Mo–W Co–Mo–Zr Co–Nb–Ta Co–Nb–W Co–Nb–Zr Co–Ni–Ta Co–Ni–Ti Co–Ni–V Co–Pt–Ta Co–Pt–V Cr–Fe–Mn Cr–Fe–Mo Cr–Fe–Re Cr–Fe–Ti Cr–Fe–V Cr–Fe–W

r r, l, R r r r r, l, R r, l r r l l, R, P r l,Laves l,Laves l l,Laves l,Laves l,Laves Laves l,Laves Laves r A15,l A15 r r r Laves r r,Laves

Cr–Hf–V Cr–Mn–Ni Cr–Mo–Nb Cr–Mo–Ni Cr–Mo–Zr Cr–Nb–Ni Cr–Nb–Ti Cr–Ni–Re Cr–Ni–W Cr–Re–W Cr–Ta–Ti Cr–Ta–V Cr-Ti-V Cr–Ti–Zr Cr–V–Zr Cr–W–Zr Fe–Mn–Mo Fe–Mn–V Fe–Mo–Ni Fe–Mo–V Fe–Nb–W Fe–Ni–Re Fe–Ni–V Fe–W–Nb Hf–Re–W Hf–Ta–V Ir–Ni–Ta Ir–Pt–Ti Ir–Re–W Ir–Ru–Ti

Laves r Laves r, P Laves l, Laves Laves r r r Laves Laves Laves Laves Laves Laves r, l,R r l r, l Laves r r Laves Laves Laves r, l A15 r A15

Ir–Ru–Zr Mn–Ni–V Mo–Nb–Ni Mo–Nb–Re Mo–Nb–Zr Mo–Ni–Re Mo–Os–W Mo–Re–Ru Mo–Re–Ta Mo–Re–V Mo–Re–W Mo–Re–Zr Mo–Rh–Ru Mo–Ru–Ta Mo–Ru–W Mo–Ti–Zr Mo–W–Zr Nb–Ni–Re Nb–Re–V Nb–Re–W Nb-Re-Zr Ni–Re–Ta Ni–Re–V Ni–Re–Zr Ni–Ti–Zr Os–Ta–W Re–Ru–W Re–Ta–W Re–V–W Re–W–Ta

Laves r l r Laves v, r, d A15 v, r v, r v, r r v, r r r r Laves Laves v v v, r v,Laves l r v Laves r v, r r v, r v

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are no P phases observed in binary compounds but Co–Mn–Mo, Cr–Mo–Ni and Mo–Ni–Re form P phases. The only ternary system that forms the M phase is Nb48Ni39Al13, which contains a small amount of Al, namely 13 at.%. Since we will see in the next section on multicomponent Ni-based superalloys that they can contain up to about 6 at.% Al, the use of structure maps requires that we assign an effective valence and atomic volume to Al that is appropriate for these low concentrations. Traditionally, Al is treated as a transition metal element with its experimental metallic volume [34] but with an effective valence of three corresponding to the actual number of sp-valence electrons by PHACOMP or with an effective Md value of 1.9 by newPHACOMP [14]. This latter Md value lies between those of Ti and V, Nb and Mo, and Ta and W, respectively, suggesting that Al behaves in the presence of transition elements as though it had 4–6 spd-valence electrons. However, a recent study by Joubert [28] on Al-containing binary systems has found that Al behaves as if it had 9 or 10 valence electrons. We have, therefore, chosen NAl = 9 for our multicomponent structure maps. This places the ternary M phase Nb48Ni39Al13 above, but close to, the l phase domain in Fig. 6, consistent with their respective f12 values in Table 1. We should note that we did not include Al in the binary transition metal structure maps in Fig. 2 because we wish to relate the experimental structural domains to theory in a future publication [44] using a simple d-bonded tight-binding approximation. Fig. 5 illustrates how stable TCP phases of isothermal sections of ternary phase diagrams are mapped onto the ðN ; DV =V Þ structure map. The figure is divided into two panels where the upper panel shows the Co-containing ternary alloys, Co–Cr–Mo and Co–Mn–Mo, and the lower panel shows the Ni-containing ternary alloys, namely Ni–Cr–Mo and Ni–Mo–Re. The upper image of each ternary system consists of an experimental isothermal phase diagram showing the stable TCP phases together with contour lines which indicate areas of constant N (thin black lines) and constant DV =V (coloured contour areas). Below the isothermal section is a ðN ; DV =V Þ structure map on which the TCP phases of the corresponding ternary phase diagram are mapped. The possible areas of the structure maps onto which the TCP phases can be mapped are marked by light grey dots and are bounded by the parabolas of the corresponding binary systems. These grey areas show that the transformations need not be single-valued as possible parabolic behaviour of the size-factor contribution DV =V can lead to two crossings for a constant N . For example,in the Co–Cr–Mo phase diagram we see that N ¼ 7:11; DV =V ¼ 0:156 corresponds to both a binary r phase and a ternary R phase, so that (7.11, 0.156) falls in a very small region in the structure map where the r and R domains overlap. This is a general limitation of any surjective mapping method such as our case where we can map from the phase diagram to the structure map but not necessarily back from the structure map to the phase diagram. Fortunately the ternary and multicomponent maps in Figs.

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5 and 6 demonstrate that this multivaluedness is not a major problem for the TCP structure maps. The isothermal sections in Fig. 5 show that the P and R phases are formed between the r phase and the l phase, so that the P and R phases must be located in the overlap region between r and l in the structure map. We see indeed that with increasing N and DV =V the following succession of coexisting TCP structure types exists: r ? P/R ? d/l. This implies that there is a relation between the formation of R, d and P together with r and l in the same ternary system as indicated by structural classification in terms of their basic unit layers [45]. We see in Fig. 6 that all other ternary TCP phases are clearly located in or close to the same regions of the structure maps that are favoured by the binary compounds of the same structure type. Moreover, it is not surprising that M(Nb48Ni39Al13) is found within the l phase domain as it is located next to l(Nb7Ni6) within the Nb–Ni–Al phase diagram [46]. As a result of the unequal number of ternary TCP phases per structure type, the A15 and Laves domains are less well covered than r, v and l. Nevertheless, we are confident that newly discovered ternary TCP phase systems will fall in the appropriate domains within Fig. 6. 5. Multicomponent TCP phases in Ni-based superalloys 5.1. Experiment Literature about the composition of multicomponent TCP phases with more than three different constituents is rare, especially TCP phases formed in Ni-based superalloys. The average equilibrium compositions of multicomponent TCP phases which are plotted in Fig. 6 on top of ternary areas, are taken from Refs. [1,47,48] and [49]. Note that the TCP phases that are observed in Ni-based superalloys contain up to 3 wt.% (6 at.%) Al. We see in Fig. 6 that all multicomponent TCP phases are located in or close to the same regions of the structure maps that are favoured by the binary and ternary compounds of the same structure type. The TCP phases which were studied by Rae and Reed [1] are found in the overlap region of the r, P/R and l phases with DV =V  0:15 and 6:5 6 N 6 8:0. These Ni-based superalloys are of particular interest because at least two TCP phases, mostly the r together with the P phase, are formed within the same Ni-based superalloy. Studies of the relative position of the coexisting TCP phases in the structure map reveal that the size factors follow the same sequence as the ternary TCP phases. 5.2. CALPHAD predictions Reed et al. [24] have very recently used the CALPHAD method together with newPHACOMP to predict the stability of Ni-based superalloys against TCP formation as part of a greater attempt to base the design of new Ni-based superalloys on theoretical analysis and computer models rather than on empiricism and trial-and-error testing [16].

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Fig. 5. Upper panel: isothermal sections of Co-containing ternary alloys, Co–Cr–Mo and Co–Mn–Mo, at 1300 °C and 1240 °C, respectively. Lower panel: isothermal sections of Co-containing ternary alloys, Cr–Mo–Ni and Mo–Ni–Re, at 1250 °C and 1152 °C, respectively. Below each isothermal section is a N ; DV =V structure map on which the TCP phases of the corresponding ternary phase diagram are mapped.

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Fig. 6. Structure map ðN ; DV =V Þ of ternary (coloured areas) and multicomponent (half black, half coloured circles) TCP phases. The values, N and DV =V , of each alloy are calculated from the composition of the experimentally observed phases. The different TCP domain boundaries are derived from their homogeneity ranges as shown in Fig. 2.

We have performed CALPHAD calculations on the Ni–Co–Cr–Ta–W–Re–Al system in order to see if the predicted TCP phases lie in their respective regions in the structure map. Following the lead of Reed et al. [24], the Co concentration is taken to vary from 0 to 10 wt.% (0.0–11.0 at.%), the Cr concentration from 4 to 12 wt.% (4.3–14.7 at.%), the Ta concentration from 4 to 8 wt.% (1.2–2.9 at.%), the W concentration from 0 to 8 wt.% (0.0–2.8 at.%), the Re concentration from 0 to 5 wt.% (0.0–1.7 at.%) and the Al concentration from 4 to 7 wt.% (8.4–16.1 at.%), which leaves Ni to vary from 50 to 88 wt.% (52.4–85.8 at.%). Reed et al. carried out the calculations at a resolution of 1 wt.% for all elements which gave a compositional dataset consisting of approximately 100,000 alloys. Our calculations are also carried out at a resolution of 1 wt.% for Re and Al, but Co, Cr, Ta and W are varied with a resolution of 2 wt.% in order to reduce computational time. Hence, our dataset consists of 18,000 alloys, which is sufficiently large for our purpose. The Thermo-Calce interface together with the thermodynamic database for Ni-based superalloys, ThermoTeche TTNi database [50] version 6, which can predict several TCP phases such as A15, r, v, R and P, were used to make predictions of multiphase, multicomponent equilibria in these systems. Fig. 7 gives the resultant structure map ðN ; DV =V ÞCAL phase where the two coordinates are calculated from these CALPHAD-predicted compositions of the individual phases at the service temperature of 900 °C, which was used by Reed et al. [24]. The figure also 0 includes the c and c phases which we see are almost perfectly grouped and well located in the fcc-based region. Moreover, Fig. 7 indicates that the predicted r phases are located in the expected region, while the l phase and the P phase are clearly located far outside the expected regions.

This is consistent with Zhao and Henry’s observation [51] that the thermodynamic database does in general a reason0 ably good job in predicting the stability of c and c phases, but a less satisfactory job for TCP phases, in particular for the l and P phases. The reason for the less accurate prediction in the case of l and P phases is that they are poorly described within the thermodynamic database as less experimental data for l and P phases has been available for the validation of the database as compared to the well-studied and more ubiquitous r phase. Since all the r phases appear to be sensibly treated by the thermodynamic database, one can compare the composition of the 6758 r-prone alloys with the composition of the 11,242 r-free alloys. Fig. 8 illustrates these compositional differences between the r-free and r-prone alloys by plotting the individual concentration maps ðN ; DV =V Þalloy for the different elemental constituents Co, Cr, Ta, W, Re and Al, respectively, where the coordinates of the concentration maps are calculated from the composition of the total alloy. The concentration map for each elemental constituent shows the concentration of that constituent, averaged over a small rectangular interval above each mesh point with dimensions dN ¼ 0:025; dðDV =V Þ ¼ 0:0025. This figure confirms that W and Re, the elements which are most effective at conferring resistance to creep [24], have the largest influence on the formation of r phases. The overall average W content of r-free alloys is 3.4 wt.% (1.2 at.%), which is significantly smaller than the overall average W content of r-prone alloys of 5.0 wt.% (1.8 at.%). The differences is even larger for Re as the overall average Re contents of the r-free and the r-prone alloys are 1.9 wt.% (0.6 at.%) and 3.55 wt.% (1.2 at.%), respectively. Moreover, Fig. 8 shows that alloys which contain about 2.5 wt.% Re ( 1 at.%) are more likely to form r phases. Further investigations indicate that alloys which

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Fig. 7. Structure map ðN ; DV =V ÞCAL phase where the two coordinates are calculated from the CALPHAD-predicted compositions of the individual phases. The 0 predicted c (dark grey) and c (light grey) domains with N and DV =V are also included. They are coloured to show the blue r free region where no r phases are found and the brown r free/prone region where some alloys are free from r phases but others are not.

do not contain Re but are predicted to form r phases (23 out of 2977 alloys) contain very high concentrations of W (above 7 wt.% (2.5 at.%) on average). A similar situation is observed for alloys without W where high content of Re leads to the formation of r phases. Nevertheless, a high concentration of Re does not automatically imply r formation within

CALPHAD as can be seen from the small r-free regions of N  9:0 and DV =V  0:15 in the lower left-hand panel in Fig. 8. The CALPHAD data on r formation allows us to study the robustness of newPHACOMP to predict the formation of r phases by comparing r-prone and r-free alloys with

Fig. 8. Concentration maps ðN ; DV =V Þalloy showing the average concentration of Co, Cr, Ta, W, Re and Al of CALPHAD-predicted r-free and r-prone Ni–Co–Cr–Ta–W–Re–Al alloys where the composition of the total alloy is used to calculate the coordinates N alloy ; DV =V alloy . The concentration is averaged over a small rectangular interval about each mesh point with dimensions dN ¼ 0:025 and dðDV =V Þ ¼ 0:0025.

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respect to their Md values as calculated from the alloy compositions: Reed et al. chose a critical value of Md ¼ 0:961 eV. This is smaller than the Md value of CMSX-4 and CMSX-10, suggesting that all alloys with lower average Md are even less likely to form r phases than CMSX-4 and CMSX-10 and should be therefore r-free too. Reed et al.’s investigation covers a relatively large compositional space where it is assumed that one universal critical value exists for all alloys. In practice, a critical value would be estimated by trial-and-error testing for a family of alloys so that only alloys within a certain compositional space are assumed to be TCP-free. Therefore, significant compositional changes such as carried out in the Ni–Co– Cr–Ta–W–Re–Al system would need continuous adjustment of the critical value Md values as has been observed, for example, by Caron [15]. It is, therefore, not surprising that this threshold significantly underestimates the risk of r phase formation because 23% of all alloys with Md < 0:961 are predicted to be r-prone as shown in Fig. 9. Moreover, this figure shows that a threshold of 0.905 defines a r-free domain, but it only contains 14% of all alloys while the rest lie within the domain where alloys are either r-free or r-prone. Although we have already argued in Section 3.7 that structure maps should not be used to predict whether an unknown phase with a required composition and structure-type exists or not, it is still interesting to investigate how much better a two-dimensional set of coordinates ðN ; DV =V Þalloy would perform than the one-dimensional newPHACOMP coordinate (Md)alloy. The lower panel in Fig. 9 shows the resultant r-stability map ðN ; DV =V Þalloy that is predicted by CALPHAD for the Ni–Co–Cr–Ta– W–Re–Al system. We see that those two-dimensional coordinates do indeed predict a larger r-free domain that contains 22% of all the alloys rather than the 14% for the one-dimensional newPHACOMP methodology in the upper panel of Fig. 9. However, this percentage is still much too low to use these two-dimensional maps based on the composition of the alloy as a robust tool in alloys design, because too many r-free alloys with possible superior properties would be ruled out. We conclude that robust methods of TCP phase prediction in multicomponent alloys require a CALPHAD scheme with an improved thermodynamic database in order to predict reliably not only the well-known r phase but also the less ubiquitous TCP phases. In recent years increasing emphasis has been placed on enhancing the thermodynamic databases and consequently on improving the predictive capability of the CALPHAD scheme by coupling the CALPHAD methodology with ab initio DFT databases [52]. For example, Crivello and Joubert [53] have very recently used results of first-principles calculations of the r and v phases in the Mo–Re and W–Re systems as input for CALPHAD models. The resulting CALPHADcalculated phase diagrams were in good agreement with the experimental phase diagrams. In the future, the approach of combining ab initio methods with the CALP-

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Fig. 9. Upper panel: frequency histogram showing the number of Ni–Co– Cr–Ta–W–Re–Al alloys per bin that are r-prone or r-free according to CALPHAD as a function of Md which is calculated from composition of the alloy. The thick red line indicates the critical Md value for CMSX-4/10 as suggested by Reed et al. [24]. The red area corresponds to 23% of all alloys up to this threshold which newPHACOMP wrongly predicts to be r-free. The dashed black line marks the alloy with lowest Md value that is predicted to be r-prone. The alloys (i.e. 14% of all alloys) below this threshold are r-free. Lower panel: r-stability map ðN ; DV =V Þalloy of CALPHAD-predicted Ni–Co–Cr–Ta–W–Re–Al alloys that are r-free (blue region), r-prone (red region) and r-free/prone (brown region) where some alloys are free from r phases but others are not. The blue r-free region contains 22% of all alloys. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

HAD formalism will become increasingly important for the design of TCP-free Ni-based superalloys as more and more ab initio data of TCP phases, in particular of the less-common TCP phases, becomes available. 6. Conclusions The following conclusions can be drawn from this work:  A detailed study is presented on the occurrence of experimental TCP phases in binary, ternary and multi-

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component transition metal alloys by means of a novel two-dimensional structure map ðN ; DV =V Þ. This structure map is found to separate the TCP phases, regardless of the number of components, into well-defined but overlapping regions corresponding to different structure types such as A15, r, v, P, R, d, l, M and Laves.  Two groups of TCP phases exist that exhibit very similar dependence with respect to the size factor: (i) the electron compounds, A15, r and v with DV =V 6 0:15 and l, M, and (ii) the Laves phases with DV =V P 0:15. The R, P and d phases are found in the overlap region between these two groups. A future publication [44] will explain the differences between these groups of TCP structures by using bond-order potential theory and moment analysis [54] to demonstrate that the valence electron concentration stabilizes A15,r and v but destabilizes the other TCP phases.  The structure map allows us to test how well the major commercially utilized methods, CALPHAD and newPHACOMP, predict the susceptibility of Ni-based superalloys to TCP phase formation. CALPHAD calculations on the Ni–Cr–Co–Re–W–Al–Ta system using the TTNi database have apparently correctly predicted r phase formation as these phases are found in the expected r phase region in the structure map, but the database failed to predict correctly l and P. The study of the Md values of all alloys for which the thermodynamic database predicts r phases shows that newPHACOMP is not a robust indicator of TCP formation in nickel-based superalloys. As expected, neither are the structure map coordinates N and DV =V , which are based on the composition of the total alloy.  Future reliable predictions of TCP stability in Ni-based superalloys will require enhanced thermodynamic databases which incorporate ab initio derived properties in order to overcome the lack of experimental data.

Acknowledgements We would like to thank Aleksey Kolmogorov, Thomas Hammerschmidt, Benoit Mangili and Michael Ford for their critical and stimulating comments. We are also grateful to our collaboration partners in the Alloys By Design consortium, in particular Roger Reed, Cathie Rae, Nils Warnken and Alessandro Mottura for many helpful discussions. Moreover, we wish to acknowledge the use of the Chemical Database Service at Daresbury and the PaulingFile provided by NIMS. This work was funded by the Engineering and Physical Sciences Research Council (EPSRC) of the United Kingdom. References [1] Rae CMF, Reed RC. Acta Mater 2001;49:4113–25. [2] Wlodek ST. Trans ASM 1964;57:110–9. [3] Sully AH, Heal TJ. Research 1948;1:288.

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