Alloying and Properties of C14–NbCr2 and A15–Nb3X (X = Al, Ge, Si, Sn) in Nb–Silicide-Based Alloys

The oxidation of Nb–silicide-based alloys is improved with Al, Cr, Ge or Sn addition(s). Depending on addition(s) and its(their) concentration(s), alloyed C14-AB2 Laves and A15-A3X phases can be stable in the microstructures of the alloys. In both phases, A is the transition metal(s), and B and X respectively can be Cr, Al, Ge, Si or Sn, and Al, Ge, Si or Sn. The alloying, creep and hardness of these phases were studied using the composition weighted differences in electronegativity (∆χ), average valence electron concentrations (VEC) and atomic sizes. For the Laves phase (i) the VEC and ∆χ were in the ranges 4.976 < VEC < 5.358 and −0.503 < ∆χ < −0.107; (ii) the concentration of B (=Al + Cr + Ge + Si + Sn) varied from 50.9 to 64.5 at %; and (iii) the Cr concentration was in the range of 35.8 < Cr < 51.6 at %. Maps of ∆χ versus Cr, ∆χ versus VEC, and VEC versus atomic size separated the alloying behaviours of the elements. Compared with unalloyed NbCr2, the VEC decreased and ∆χ increased in Nb(Cr,Si)2, and the changes in both parameters increased when Nb was substituted by Ti, and Cr by Si and Al, or Si and Ge, or Si and Sn. For the A15 phase (i) the VEC and ∆χ were in the ranges 4.38 < VEC < 4.89 and 0.857 < ∆χ < 1.04, with no VEC values between 4.63 and 4.72 and (ii) the concentration of X (=Al + Ge + Si + Sn) varied from 16.3 to 22.7 at %. The VEC versus ∆χ map separated the alloying behaviours of elements. The hardness of A15-Nb3X was correlated with the parameters ∆χ and VEC. The hardness increased with increases in ∆χ and VEC. Compared with Nb3Sn, the ∆χ and hardness of Nb3(Si,Sn) increased. The substitution of Nb by Cr had the same effect on ∆χ and hardness as Hf or Ti. The ∆χ and hardness increased with Ti concentration. The addition of Al in Nb3(Si,Sn,Al) decreased the ∆χ and increased the hardness. When Ti and Hf, or Ti, Hf and Cr, were simultaneously present with Al, the ∆χ was decreased and the hardness was unchanged. The better creep of Nb(Cr,Si)2 compared with the unalloyed Laves phase was related to the decrease in the VEC and ∆χ parameters.


Introduction
The inherent temperature capability of Ni-based superalloys is limited by the melting point of Ni. Currently, new high temperature alloys are developed based on refractory metals. Nb-silicide-based alloys, also known as Nb in situ silicide composites, have the potential to meet property goals and satisfy environmental and performance targets for future aero-engines [1,2]. These alloys can have as many as 12 alloying additions, some of which are essential for improving their oxidation resistance and others for improving their strength and creep. These elements include Al (≤5 at %), Cr (≤8 at %), Fe (≤5 at %), Ge (≤5 at %), Hf (≤5 at %), Si (16)(17)(18)(19)(20)(21)(22) at %), Sn (≤5 at %) or Ti (10-26 at %) and refractory metals (Mo (≤8 at %), Ta (≤6 at %), W (≤4 at %)). Substitution of Si by Ge and Sn, and Cr by Fe has been studied [3][4][5]. Bewlay et al. reported that "additions of Fe were almost as effective as additions of Cr up to a concentration of 5 at %, and that the combination of Cr and Fe was no more effective than Cr alone for the same total concentration" [6]. In the microstructures of these materials, the bcc Nb

Methodology, Results and Discussion
Experimental data concerning the actual average compositions of the C14-NbCr 2 Laves and A15-Nb 3 X intermetallic phases in developmental Nb-silicide-based alloys [3,4,[7][8][9][43][44][45][46][47][48][49][50][51][52][53] was used to find out whether there are relationships between the parameters ∆χ, VEC and atomic size. Phase stability can be considered in terms of e/a and VEC (number of valence electrons per atom filled into the valence band). The e/a ratio is a key parameter in the Hume-Rothery rules, and VEC is key to determining the Fermi level in the valence band. According to Mizutani [54], the e/a is difficult to use as a universal parameter in alloy design, because its value cannot be uniquely assigned to a transition metal, as it depends on the surrounding environment. Instead, VEC is a more important parameter in transition metal alloys.
The C14-NbCr 2 Laves and A15-Nb 3 X intermetallic phases studied in this paper were in cast (AC) and/or heat treated (HT) microstructures of Nb-silicide-based alloys that were prepared in earlier research, meaning no new experimental data were created during the course of this study. As described elsewhere [7], the alloys were made using high purity (better than 99.99 wt %) elements and non-consumable (W) electrode arc melting in water cooled copper hearths and were heat treated in an Argon atmosphere at 1773 K for 100 h. The chemical composition of each alloy was determined in the AC and HT conditions using EPMA (electron probe microanalysis). The Vickers hardness measurements of A15 intermetallic phases in Nb-silicide-based alloys were done using a Mitutoyo micro-hardness testing machine. The load used was 0.1 kg and was applied for 20 s. At least 10 measurements were taken for each phase. The hardness measurements were taken from A15 phases in bulk microstructures, free of contamination by interstitials and with similar grain sizes. The creep data for the Laves phases and Nb 3 Al was taken from [23,55].
The actual average chemical composition (at %) of each phase, determined by EPMA, was used to calculate the aforementioned parameters using data for the elements from the same sources as in [42] and the equations given below. The parameter VEC was calculated using [VEC] where C i and (VEC) i , respectively, are the concentration (at %) and VEC of element i in the intermetallic. For the NbCr 2 Laves phase, the electronegativity parameter was [∆χ] Laves = ∑ i m C i (χ <Nb>i ) − ∑ i z κ i (χ <Cr>i ), where C i and χ <Nb>i , respectively, are the concentration (at %) and Pauling electronegativity of Nb and element i substituting Nb in the Laves phase, and κ i and χ <Cr>i , respectively, are the concentration (at %) and Pauling electronegativity of Cr and element i substituting Cr in the Laves phase. For the A15-Nb 3 X, the electronegativity parameter was [∆χ] A15-Nb3X = ∑ i m C i (χ <Nb>i ) − ∑ i z κ i (χ <X>i ), where C i and χ <Nb>i , respectively, are the concentration (at %) and Pauling electronegativity of Nb and element i substituting Nb in the A15 phase, and κ i and χ <X>i , respectively, are the concentration (at %) and Pauling electronegativity of X (=Al, Ge, Si, Sn) and element i substituting X in the A15 phase. For the Laves phase, R <Nb> = ∑ i n C i (r <Nb> ) i , where C i and (r <Nb> ) i , respectively, are the concentration (at %) and atomic radius of Nb and element i substituting Nb in the Laves phase, and R <Cr> = ∑ i n C i (r <Cr> ) i is the concentration (at %) and atomic radius of Cr and element i substituting Cr in the Laves phase. For the A15 phase, <R> = ∑ i n C i (r) i , where C i and (r) i , respectively, are the concentration (at %) and atomic radius of element i in the A15 phase.

C14-NbCr 2 Laves
In Nb-silicide-based alloys, the Laves phase that is stable in their microstructures is the hexagonal C14-NbCr 2 [7][8][9][43][44][45][46][47][48][49][50][51], where Nb can be substituted by transition metals, such as Hf, Mo, Ta, Ti or W and Cr, by simple metal and metalloid elements (Al, Ge, Si, Sn). The stability of C14-NbCr 2 in the microstructures of Nb-silicide-based alloys is in agreement with the literature about the effects of Al and Si on the structure of NbCr 2 . The substitution of Nb by the aforementioned transition metals is also in agreement with the literature [13].
The data for developmental Nb-silicide-based alloys showed (a) that the solubilities of the elements that substitute Nb in C14-NbCr 2 were Hf ≤ 10.2 at %, 1.3 < Mo < 5.8 at %, 4.2 < Ta < 9.6 at %, 3.4 < Ti < 28.8 at % and W ≤ 5.2 at % (there is no chemical composition data for C14-NbCr 2 -based Laves phases where Cr is substituted by Fe in Nb-silicide-based alloys); (b) that in the alloyed NbCr 2 , the Cr concentration was in the range 35.8 < Cr < 51.6 at % and increased after heat treatment, from 35.8 < Cr < 49.2 at % to 44.7 < Cr < 51.6 at % and (c) that the solubilities of the elements that substituted Cr in the Laves phase were Al ≤ 11 at %, Ge ≤ 3.1 at %, 5 < Si < 12.6 at % and Sn ≤ 2.8 at %. Furthermore, if B = Al + Cr + Ge + Si + Sn, then the concentration of B in the cast and heat-treated microstructures, respectively, was in the ranges 50.9 < B < 62.4 at % and 53.1 < B < 64.5 at %, compared with 61 < Cr < 70 at % for unalloyed NbCr 2 [15], which would suggest that homogeneity ranges on both the Nb and Cr sides of C14-NbCr 2 formed in Nb-silicide-based alloys. The data for the concentration of B (=Al + Cr + Ge + Si + Sn) and the solubility of Si in C14 NbCr 2 reported in [56] are in agreement with the aforementioned values.
Unlike the tetragonal Nb 5 Si 3 [41], no strong relationships were found between the concentrations of alloying elements in the C14-NbCr 2 Laves phase. The alloying of C14-NbCr 2 was studied using the parameters VEC, ∆χ, <R> = R <Nb> + R <Cr> and R <Nb> /R <Cr> . It should be noted that the latter ratio is not the same as the ratio r A /r B (see introduction). Figure 1 shows how the parameters VEC and ∆χ change with the Cr concentration in NbCr 2 . Figure 1a shows data for alloyed Laves phases where Nb is substituted by Hf, Mo or Ti and Cr by Al, Ge, Si or Sn. The parameter VEC increases with the Cr concentration in the Laves phase, and the linear fit of data is very good (R 2 = 0.9555). The extrapolated best fit line crosses the best fit line for the data for unalloyed NbCr 2 and passes between the two data points for unalloyed NbCr 2 with a high Cr content. Unlike the parameter VEC, the parameter ∆χ can separate the data for the Laves phase into two sets (see Figure 1b). In the latter, the data for unalloyed NbCr 2 and two alloyed Laves phases falls on parallel lines and shows (i) that alloying makes ∆χ less negative and (ii) that as the Cr concentration in the Laves phase increases, the ∆χ becomes more negative.
The effects of substituting Nb and Cr by other elements are shown in the ∆χ versus VEC maps in Figure 2a,b. Figure 2a shows data for the unalloyed NbCr 2 and alloyed Laves, where Cr is substituted by Si, Al, Ge or Sn, and Nb by Ti. The arrows in Figure 2a show the "direction of change" as Cr and Nb are substituted. There is a decrease in the value of VEC and a slight increase in the value of ∆χ (becomes less negative) when only Si substitutes Cr and a further shift towards lower VEC and less negative ∆χ values when Nb is substituted by Ti, and Cr by Si and Al, or Si and Ge, or Si and Sn. Figure 2b shows a similar map for "more heavily" alloyed Laves phases where more elements substitute Nb and Cr. The "direction of change" is indicated by the arrow and it should be noted that further alloying keeps the data to the left of Nb(Cr,Si) 2 but within a band that is defined by dashed lines that are essentially parallel to that for the (Nb,Ti)(Cr,Si) 2 Laves phase in Figure 2a. When the data for the C14-NbCr 2 Laves phases is considered in the ∆χ versus R <Nb> /R <Cr> map in Figure 2c, the data exhibits a remarkable linear fit with R 2 = 0.9626 and shows that ∆χ becomes less negative as alloying shifts its values from the bottom of the line (occupied by the unalloyed NbCr 2 ) towards the data for (Nb,Ti)(Cr,Si,Ge) 2 and (Nb,Ti)(Cr,Si,Sn) 2 at the top. The importance of atomic size on the alloying behaviour of NbCr2 also can be captured in maps of VEC versus <R> (=R<Nb> + R<Cr>) and R<Nb>/R<Cr> versus <R>, see Figure 3. Figure 3a shows the shift towards lower VEC values upon alloying NbCr2 with Si and further decrease in VEC with increasing  The importance of atomic size on the alloying behaviour of NbCr 2 also can be captured in maps of VEC versus <R> (=R <Nb> + R <Cr> ) and R <Nb> /R <Cr> versus <R>, see Figure 3. Figure 3a shows the shift towards lower VEC values upon alloying NbCr 2 with Si and further decrease in VEC with increasing <R> upon further alloying. All the data exhibits a reasonable linear fit, shown by the dashed line for which R 2 = 0.9035, and remarkably the latter is essentially parallel to the line  2 and Nb(Cr,Si,Sn) 2 (thin continuous line in Figure 3a). Figure 3b shows the step change in R <Nb> /R <Cr> of the unalloyed NbCr 2 to higher values upon alloying with Si (see arrow) and the continuous increase of the ratio upon further substitution of Nb and Cr with other transition metals and simple metal and metalloid elements, the addition of which leads to an increase in <R>. This map can separate alloying behaviours, as shown by the two dashed lines, of which the lower one (R 2 = 0.9757) corresponds to data for the Laves phases (Nb,Ti)(Cr,Si,Al,Ge) 2 , (Nb,Ti,Hf)(Cr,Si,Al) 2 , (Nb,Ti,Hf)(Cr,Si,Sn) 2 , (Nb,Ti,Mo,Hf)(Cr,Si,Al) 2 and (Nb,Ti,Mo,Hf)(Cr,Si,Sn) 2 . The two encircled data points in the Figure 3b correspond to the average compositions of the Laves phase grains in the uncontaminated bulk of an alloy that was isothermally oxidised at 1073 and 1473 K. <R> upon further alloying. All the data exhibits a reasonable linear fit, shown by the dashed line for which R 2 = 0.9035, and remarkably the latter is essentially parallel to the line joining 35.5Nb-(52.5Cr-12Si) with (Nb,Ti)(Cr,Si)2 and Nb(Cr,Si,Sn)2 (thin continuous line in Figure 3a). Figure 3b shows the step change in R<Nb>/R<Cr> of the unalloyed NbCr2 to higher values upon alloying with Si (see arrow) and the continuous increase of the ratio upon further substitution of Nb and Cr with other transition metals and simple metal and metalloid elements, the addition of which leads to an increase in <R>. This map can separate alloying behaviours, as shown by the two dashed lines, of which the lower one (R 2 = 0.9757) corresponds to data for the Laves phases (Nb,Ti)(Cr,Si,Al,Ge)2, (Nb,Ti,Hf)(Cr,Si,Al)2, (Nb,Ti,Hf)(Cr,Si,Sn)2, (Nb,Ti,Mo,Hf)(Cr,Si,Al)2 and (Nb,Ti,Mo,Hf)(Cr,Si,Sn)2. The two encircled data points in the Figure 3b correspond to the average compositions of the Laves phase grains in the uncontaminated bulk of an alloy that was isothermally oxidised at 1073 and 1473 K.  The changes in the values of the parameters VEC and ∆χ of NbCr 2 upon "simple" and "advanced" alloying are shown in Figure 2a,b. The currently available data is summarised in Figure 4 and shows that the values of the parameters VEC and ∆χ of the alloyed Laves phases that are observed in Nb-silicide-based alloys are confined within well-defined ranges. The Laves phase parameters VEC and ∆χ, respectively, are restricted in the ranges, 4.976 < VEC < 5.358 and −0.503 < ∆χ < −0.107, while the Cr concentration is confined to the range 35.8 < Cr < 51.6 at %. The Laves parameter VEC is within the range of the VEC values of the bcc Nb solid solutions in Nb-silicide-based alloys [42]. Laves phase maps (a) VEC versus <R> = R<Nb> + R<Cr> and (b) R<Nb>/R<Cr> versus <R> = R<Nb> + R<Cr>. Lavender, pink, blue, red, green and dark orange squares represent the same data as in Figure  2a. The triangles represent data for (Nb,Ti)(Cr,Si,Al,Sn)2 and (Nb,Ti,Mo,Hf)(Cr,Si,Al,Sn)2 (light orange), and (Nb,Ti)(Cr,Si,Al,Ge)2 (light green); diamonds represent data for (Nb,Mo)(Cr,Si,Al)2 (yellow), (Nb,Ti,Ta)(Cr,Si,Al)2, (Nb,Ti,Hf)(Cr,Si,Al)2, (Nb,Ti,Mo,Hf)(Cr,Si,Al)2 (red), (Nb,Ti,Hf)(Cr,Si,Ge)2 (light green), (Nb,Hf)(Cr,Si,Sn)2 and (Nb,Ti,Hf)(Cr,Si,Sn)2 (dark orange), (Nb,Ti,Mo,W,Hf)(Cr,Si,Sn)2 (brown).
The changes in the values of the parameters VEC and ∆χ of NbCr2 upon "simple" and "advanced" alloying are shown in Figure 2a,b. The currently available data is summarised in Figure  4 and shows that the values of the parameters VEC and ∆χ of the alloyed Laves phases that are observed in Nb-silicide-based alloys are confined within well-defined ranges. The Laves phase parameters VEC and ∆χ, respectively, are restricted in the ranges, 4.976 < VEC < 5.358 and −0.503 < ∆χ < −0.107, while the Cr concentration is confined to the range 35.8 < Cr < 51.6 at %. The Laves parameter VEC is within the range of the VEC values of the bcc Nb solid solutions in Nb-silicidebased alloys [42].

A15-Nb3X
In Nb-silicide-based alloys, the A15-Nb3X phase(s) (X = Al, Ge, Si, Sn) can form in the cast microstructures and/or after heat treatment and/or during oxidation [
The solubility ranges of Al, Ge and Sn in unalloyed A15 phases, respectively, were 18.6 < Al < 25 at % for Nb 3 Al, 18 < Ge < 23 at % for Nb 3 Ge and 15.5 < Sn < 33.2 at % for Nb 3 Sn [15]. In the alloyed A15 phases observed in Nb-silicide-based alloys, the solubilities of these elements were 2.4 < Al < 8.9 at %, 1.1 < Ge < 3.8 at %, 5.2 < Sn < 16 at % and 1.4 < Si < 8.8 at %, and X (=Al + Ge + Si + Sn) in Nb 3 X was in the range 16.3 < X < 22.7 at %. Like the Laves phases, no strong relationships between the concentrations of alloying elements were found in A15-Nb 3 X intermetallic phases. Figure 5 shows that the A15-Nb 3 X phases that are formed in Nb-silicide-based alloys have X within the range defined by the averages of the minimum and total values, respectively, of unalloyed Nb 3 Al, Nb 3 Ge, Nb 3 Sn. Figure 5 also shows that the <R> of A15 phases in Nb-silicide-based alloys is within the area defined by the horizontal blue and purple dashed lines. The dashed lines from bottom (blue) to top (purple) correspond to the average <R> of unalloyed Nb 3 Al, Nb 3 Ge and Nb 3 Sn, and the <R> of Nb-25Al and Nb-15.5Sn. The <R> of Nb 3 X alloyed with Ge is confined between the blue and red dashed horizontal lines, i.e., between the average <R> of unalloyed Nb 3 Al, Nb 3 Ge, Nb 3 Sn, and the <R> of Nb-25Al, while the data for Nb 3 X alloyed with Al is on both sides of the red horizontal dashed line that is for the <R> of Nb-25Al. The solubilities of the other elements in alloyed Nb3X where Nb is substituted by one or more of the elements Cr, Fe, Hf, Mo, Ti or W, and X is substituted by two or more of the elements Al, Ge, Si or Sn were as follows: 1.4 < Cr < 8.7 at %, Fe < 2.6 at %, Hf < 6.3 at %, 5.1 < Mo < 18.5 at %, 7.8 < Ti < 32.1 at % and 1.6 < W < 5.2 at %. Very Ti rich A15 phases can also form [52], in which case, 38.5 < Ti < 48.6 at %.
The ranges of the values of the parameters ∆χ and VEC of alloyed A15-Nb3X are shown in the ∆χ versus VEC map in Figure 6 and are 0.857 < ∆χ < 1.04 and 4.38 < VEC < 4.889, respectively, with a gap in VEC values between 4.632 and 4.72. It should be noted that unalloyed Nb3Sn falls in the VEC gap, but in the latter, there are no alloyed A15-Nb3X phases. The values of both parameters fall within the ranges of ∆χ and VEC for bcc Nb solid solutions [42] with the lower and upper limits of VEC being essentially the same as those of the Nbss. The values of the parameter VEC of alloyed A15-Nb3X phases are lower than those of C14-NbCr2 Laves phases. The solubilities of the other elements in alloyed Nb 3 X where Nb is substituted by one or more of the elements Cr, Fe, Hf, Mo, Ti or W, and X is substituted by two or more of the elements Al, Ge, Si or Sn were as follows: 1.4 < Cr < 8.7 at %, Fe < 2.6 at %, Hf < 6.3 at %, 5.1 < Mo < 18.5 at %, 7.8 < Ti < 32.1 at % and 1.6 < W < 5.2 at %. Very Ti rich A15 phases can also form [52], in which case, 38.5 < Ti < 48.6 at %.
The ranges of the values of the parameters ∆χ and VEC of alloyed A15-Nb 3 X are shown in the ∆χ versus VEC map in Figure 6 and are 0.857 < ∆χ < 1.04 and 4.38 < VEC < 4.889, respectively, with a gap in VEC values between 4.632 and 4.72. It should be noted that unalloyed Nb 3 Sn falls in the VEC gap, but in the latter, there are no alloyed A15-Nb 3 X phases. The values of both parameters fall within the ranges of ∆χ and VEC for bcc Nb solid solutions [42] with the lower and upper limits of VEC being essentially the same as those of the Nb ss . The values of the parameter VEC of alloyed A15-Nb 3 X phases are lower than those of C14-NbCr 2 Laves phases. The data for the unalloyed A15-Nb3Al, Nb3Ge and Nb3Sn are also shown in the ∆χ versus VEC map in Figure 6. It should be noted that A15-Nb3X phases, where Nb is substituted by Cr, Fe, Hf or Ti and X is Al, Si, Sn, are on the left-hand side of the VEC gap. Also in the gap is the data for unalloyed Nb3Al, while the data for the A15-Nb3X phases, where Nb is substituted by Cr and Hf, and X is Al, Si, Sn or Nb is substituted by Cr, Hf, Mo, Ti or W, and X is Al, Ge, Si, Sn, falls on the right-hand side of the VEC gap, where the data for unalloyed Nb3Sn and Nb3Ge is found. The data for Nb3Sn, Nb3Ge and Nb3(Si,Sn) exhibits a remarkably good linear fit with R 2 = 0.9961.
The hardness of a material is related to both its elastic and plastic deformation as the indenter is applied perpendicularly to its surface and the sample is subjected to a combination of compression, shear and tension. Bulk properties are important for hardness because the indenter interacts with the surface and the bulk of the sample. The shear modulus provides a measure of the rigidity against the shape deformations as the force of the indenter is applied perpendicularly to the sample surface [57]. Jhi et al. [57] studied the hardness of transition metal carbides and nitrides and suggested that owing to their high Peierls stresses, their strengths may be influenced by the difficulty of nucleating and moving dislocations. The stresses required for the latter scale with the shear modulus and thus, electronic changes that affect the shear modulus, would have a strong effect on the hardness. They concluded that bonding may be important for the hardness of transition metal carbides and nitrides rather than the conventional microstructural features that determine the hardness of metals and alloys, and that the response of bonds to shear is the crucial factor in determining hardness. Furthermore, Jhi et al. [57] found a correlation between the calculated shear modulus c44, "which represents a shape change without volume change and provides directly information about electronic The data for the unalloyed A15-Nb 3 Al, Nb 3 Ge and Nb 3 Sn are also shown in the ∆χ versus VEC map in Figure 6. It should be noted that A15-Nb 3 X phases, where Nb is substituted by Cr, Fe, Hf or Ti and X is Al, Si, Sn, are on the left-hand side of the VEC gap. Also in the gap is the data for unalloyed Nb 3 Al, while the data for the A15-Nb 3 X phases, where Nb is substituted by Cr and Hf, and X is Al, Si, Sn or Nb is substituted by Cr, Hf, Mo, Ti or W, and X is Al, Ge, Si, Sn, falls on the right-hand side of the VEC gap, where the data for unalloyed Nb 3 Sn and Nb 3 Ge is found. The data for Nb 3 Sn, Nb 3 Ge and Nb 3 (Si,Sn) exhibits a remarkably good linear fit with R 2 = 0.9961.
The hardness of a material is related to both its elastic and plastic deformation as the indenter is applied perpendicularly to its surface and the sample is subjected to a combination of compression, shear and tension. Bulk properties are important for hardness because the indenter interacts with the surface and the bulk of the sample. The shear modulus provides a measure of the rigidity against the shape deformations as the force of the indenter is applied perpendicularly to the sample surface [57]. Jhi et al. [57] studied the hardness of transition metal carbides and nitrides and suggested that owing to their high Peierls stresses, their strengths may be influenced by the difficulty of nucleating and moving dislocations. The stresses required for the latter scale with the shear modulus and thus, electronic changes that affect the shear modulus, would have a strong effect on the hardness. They concluded that bonding may be important for the hardness of transition metal carbides and nitrides rather than the conventional microstructural features that determine the hardness of metals and alloys, and that the response of bonds to shear is the crucial factor in determining hardness. Furthermore, Jhi et al. [57] found a correlation between the calculated shear modulus c 44 , "which represents a shape change without volume change and provides directly information about electronic response to shear strain", and VEC, and between the experimentally measured hardness values and VEC, for transition metal (Hf,Ti,Zr) carbonitrides and Nb carbides. A similar correlation between c 44 and VEC was reported by Wang and Zhou for M 2 AlC, who also found that the bulk and shear moduli increased monotonously as VEC increased. They suggested that the hardness of M 2 AlC could be tuned by alloying to get "appropriate" VEC values [58]. A similar suggestion was made earlier by Konig for "tuning" the value of VEC via alloying and thus the properties of hard coatings [59].
To the author's knowledge, data about the Peierls stress of A15 compounds is limited. The compound Nb 3 Al with the A15 structure is likely to have considerable Peierls stress [60]. Shyue's conjecture about the Peierls stress of A15 compounds is supported by the recent work of Kamimura et al. who calculated the Peierls stress of the A15-V 3 Si intermetallic to be about 13,500 MPa, compared with 6500 MPa for TiC, 415 MPa for Nb, and less than 1.4 MPa for Al [61]. It would be interesting to find out if there is a relationship between the hardness of A15-Nb 3 X in Nb-silicide-based alloys and its parameter VEC. This will be addressed below.
Pauling defined electronegativity as "the power of an atom in a molecule to attract electrons to itself" [62]. Electronegativity provides primary information about bonding. The importance of bonding for the hardness of materials was also discussed by Li et al. who proposed an empirical model to predict the hardness of materials in terms of electronegativity and crystal structure [63]. Is there a relationship between the hardness of A15-Nb 3 X in Nb-silicide-based alloys and their electronegativities?
The available Vickers hardness (HV) data [4,48,52] for unalloyed and alloyed A15-Nb 3 X phases is shown in Figures 7 and 8, where the hardness is plotted against the parameters ∆χ and VEC, respectively. The calculated and measured hardness values of Nb 3 X (X = Al, Sn) are shown in Table 1. There is good agreement between measured values of Nb 3 Sn and Nb 3 Al and one of the calculated values (indicated in bold), based on Young's modulus E and Poisson's ratio ν, or Bulk modulus B and shear modulus G. Interestingly, the calculated hardness values using only the macroscopic shear modulus G are significantly higher than the measured hardness. The data in Figures 7 and 8 is separated into groups. The goodness of linear fit is indicated by the R 2 values. Data points and R 2 values are given using the same colour for a particular set of data. The hardness values of Nb 3 Sn, Ti 3 Sn and Nb 3 Al, respectively, are taken from [4,52,64]. The structure of Ti 3 Sn is DO 19 with prototype Ni 3 Sn [15]. Figures 7 and 8 show that the hardness values of alloyed A15-Nb 3 X phases are correlated with their parameters, ∆χ and VEC, and that an increase in the values of the latter is accompanied by an increase in hardness.     Figure 7 shows (a) that the Ti 3 Sn has lower ∆χ and hardness than Nb 3 Sn, which has a lower ∆χ and hardness than Nb 3 Al and (b) that upon alloying, the ∆χ values of A15-A 3 X phases (A = Nb, Ti, X = Al, Sn) increase and these increases are accompanied by increases in hardness. The alloyed A15-Nb 3 X phases have hardness values above 600 HV and lower than 1100 HV. Substitution of Sn by Si in Nb 3 Sn results in significant increases in both ∆χ and the hardness of Nb 3 (Si,Sn) (red squares). When Nb is substituted by Hf in (Nb,Hf) 3 (Si,Sn) (red diamonds), the hardness decreases, but the slope remains the same, and substitution of Nb by Ti in (Nb,Ti) 3 (Si,Sn) (blue diamonds) results in a further decrease in hardness but with no change in slope. When Nb is substituted by both Hf and Ti, or Cr and Hf or Gr, Hf and Ti, the data (green squares) moves to the right, closer, but not exceeding, the data for Nb 3 (Si,Sn) (red squares) and with essentially the same slope, from which it can be deduced that the substitution of Nb by Cr has the same effect on ∆χ and hardness as Hf or Ti, meaning the (Nb,Cr) 3 (Si,Sn) moves to the left of the Nb 3 (Si,Sn) data with no change in slope. In other words, alloying with Hf or Ti or Hf and Ti, or Cr and Hf or Cr, Hf and Ti moves the data for alloyed A15-Nb 3 (Si,Sn) parallel to the line for Nb 3 (Si,Sn) with small changes in ∆χ values and small or significant (in the case of Ti or Hf) changes in hardness, particularly for (Nb,Ti) 3 (Si,Sn).
The substitution of Nb by Fe and Ti in (Nb,Fe,Ti) 3 (Si,Sn) also results in a shift to the left of the data for Nb 3 (Si,Sn) but with a higher slope (orange squares) which would suggest that the simultaneous presence of Fe and Ti affects ∆χ (and thus the hardness) differently than that of Hf and Ti. Note that the data point for the Ti-rich (Nb,Fe,Ti) 3 (Si,Sn) falls on a line with the same slope as the data for (Nb,Ti) 3 (Si,Sn) and that the data for unalloyed Ti 3 Sn, (Nb,Fe,Ti) 3 (Si,Sn) and (Nb,Ti) 3 (Si,Sn) has the same slope but the values of ∆χ and hardness are lower as the Ti concentration in the A15 phase increases. This confirms that the increase in the concentration of Ti in the A15 phases is accompanied by reductions of both ∆χ and hardness.
The unalloyed Nb 3 Al is on the right-hand side of the Nb 3 (Si,Sn) line. The addition of Al in Nb 3 (Si,Sn,Al) results in a reduction of ∆χ and an increase in hardness (see arrow in Figure 7). The Nb 3 (Si,Sn,Al) has the highest hardness of the A15-Nb 3 X phases. When Al is added to (Nb,Hf) 3 (Si,Sn,Al) (tan squares), the resultant changes in ∆χ and hardness change the slope of the line, the extension of which passes from the data point for unalloyed Nb 3 Al. Indeed, when the data for Nb 3 Al is joined with the data for (Nb,Hf) 3 (Si,Sn,Al) (tan squares), R 2 = 0.9982. The data also shows that when the Al in Nb 3 Al is substituted by Si and Sn, and the Nb by Hf, both the ∆χ and hardness increase. Furthermore, the line for (Nb,Hf) 3 (Si,Sn,Al) (tan squares) is to the left of the line for Nb 3 (Si,Sn) (red squares), which confirms the effect of Hf on hardness, discussed above, but it is to the right of the line for (Nb,Hf) 3 (Si,Sn) (red diamonds) owing to the effect that Al has on the hardness (see above). Interestingly, when Ti and Hf, or Ti, Hf and Cr are simultaneously present with Al in A15-phases (purple squares), the ∆χ is reduced, the hardness hardly changes and thus, the slope is reduced. The changes shown by the purple data in Figure 7 are attributed to the strong role played by Al in A15-Nb 3 X phases and the synergy of Al with the aforementioned transition metals (compared with data presented by the green and purple squares).
It should be noted that the limited data for Fe-containing A15-Nb 3 X phases (orange squares) has the same slope as the line for unalloyed Nb 3 Al (tan diamond) and (Nb,Hf) 3 (Si,Sn) (tan squares), which may be an indication that the substitution of Nb by Fe, and Ti has the same effect on the slope d(∆χ)/d(HV) as the substitution of Nb by Hf and the addition of Al. Unfortunately, there is no data for (Nb,Ti,Fe) 3 (Si,Sn,Al) A15 phases to test this hypothesis. Figure 8 shows how the hardness of A15-Nb 3 X phases changes with their parameter, VEC. The latter cannot separate the contributions of different elements, for example Hf, as effectively as the parameter ∆χ. The Ti 3 Sn (light orange square) has a lower VEC than unalloyed Nb 3 Al (light orange diamond), which has a lower VEC than unalloyed Nb 3 Sn (dark green circle), and the Ti 3 Sn and unalloyed Nb 3 Al are on the same line (red diamonds) as data for A15-Nb 3 X phases where X = Si, Sn and Nb are substituted by Hf, or Ti, or Hf and Cr, or Ti and Cr, or Ti and Fe; they are also on the same line as the data for (Nb,Hf) 3 (Si,Sn,Al) (red diamonds). Furthermore, the extension of the line fitted to the data presented by the diamonds crosses the data point for Nb 3 (Si,Sn) (very light green circle). When Ti or Ti and Cr are added to (Nb,Hf) 3 (Si,Sn,Al), there is a reduction in VEC and the slope d(VEC)/d(HV) is not changed (purple triangles).
In Figure 8, the data represented by the circles is for A15 phases without Ti. This element is present in some of the A15 phases, the data points of which are on the line fitted to the red diamonds; this confirms that Ti reduces the VEC of alloyed A15-Nb 3 X phases. The addition of Al in A15 phases is again seen to increase the hardness (see arrow in Figure 8) and to decrease VEC. Thus, the VEC versus hardness map also confirms that the substitution of Sn by Si increases the hardness significantly but not the value of the parameter VEC, while the addition of Al decreases VEC significantly and increases the hardness but not as much as Si.
The creep at 1473 K of unalloyed A15-Nb 3 Al and unalloyed and Si alloyed NbCr 2 Laves phases is compared in Figure 9. The creep data is from [23,55]. The Laves phases are shown by the lavender squares and black triangles in Figure 2 and A15-Nb 3 Al is shown by the cross in the red square in Figure 6. The improvement of the creep of the alloyed Laves phase in Figure 9 is related to a decrease of VEC and ∆χ. The experimental data for the creep of alloyed C14 NbCr 2 Laves phases is more limited compared with the data for the creep of Nb 5 Si 3 [41]. For the latter silicide, the creep rate  Figure 8 shows how the hardness of A15-Nb3X phases changes with their parameter, VEC. The latter cannot separate the contributions of different elements, for example Hf, as effectively as the parameter ∆χ. The Ti3Sn (light orange square) has a lower VEC than unalloyed Nb3Al (light orange diamond), which has a lower VEC than unalloyed Nb3Sn (dark green circle), and the Ti3Sn and unalloyed Nb3Al are on the same line (red diamonds) as data for A15-Nb3X phases where X = Si, Sn and Nb are substituted by Hf, or Ti, or Hf and Cr, or Ti and Cr, or Ti and Fe; they are also on the same line as the data for (Nb,Hf)3(Si,Sn,Al) (red diamonds). Furthermore, the extension of the line fitted to the data presented by the diamonds crosses the data point for Nb3(Si,Sn) (very light green circle) When Ti or Ti and Cr are added to (Nb,Hf)3(Si,Sn,Al), there is a reduction in VEC and the slope d(VEC)/d(HV) is not changed (purple triangles).
In Figure 8, the data represented by the circles is for A15 phases without Ti. This element is present in some of the A15 phases, the data points of which are on the line fitted to the red diamonds; this confirms that Ti reduces the VEC of alloyed A15-Nb3X phases. The addition of Al in A15 phases is again seen to increase the hardness (see arrow in Figure 8) and to decrease VEC. Thus, the VEC versus hardness map also confirms that the substitution of Sn by Si increases the hardness significantly but not the value of the parameter VEC, while the addition of Al decreases VEC significantly and increases the hardness but not as much as Si.
The creep at 1473 K of unalloyed A15-Nb3Al and unalloyed and Si alloyed NbCr2 Laves phases is compared in Figure 9. The creep data is from [23,55]. The Laves phases are shown by the lavender squares and black triangles in Figure 2 and A15-Nb3Al is shown by the cross in the red square in Figure 6. The improvement of the creep of the alloyed Laves phase in Figure 9 is related to a decrease of VEC and ∆χ. The experimental data for the creep of alloyed C14 NbCr2 Laves phases is more limited compared with the data for the creep of Nb5Si3 [41]. For the latter silicide, the creep rate έ increased with alloying (excluding Boron addition) and the increase of έ was related to a decrease in VEC and increase in ∆χ. However, for the Nb5Si3 alloyed with Boron, the increase in έ was related to decreases in both the parameters VEC and ∆χ. × 10 Figure 9. Norton plot of creep rate (s −1 ) versus stress (MPa) at 1473 K of unalloyed and Si alloyed Laves phases [18,52] and unalloyed A15-Nb3Al [18]. The former are shown in the ∆χ versus VEC maps in Figure 2a,b and the latter in the map in Figure 6 (see text).

Conclusions
In this paper, the alloying behaviours of the hexagonal C14 Laves and cubic A15 intermetallic phases in Nb-silicide-based alloys were studied using the parameters VEC, ∆χ and atomic size (light orange square) has a lower VEC than unalloyed Nb3Al (light orange diamond), which has a lower VEC than unalloyed Nb3Sn (dark green circle), and the Ti3Sn and unalloyed Nb3Al are on the same line (red diamonds) as data for A15-Nb3X phases where X = Si, Sn and Nb are substituted by Hf, or Ti, or Hf and Cr, or Ti and Cr, or Ti and Fe; they are also on the same line as the data for (Nb,Hf)3(Si,Sn,Al) (red diamonds). Furthermore, the extension of the line fitted to the data presented by the diamonds crosses the data point for Nb3(Si,Sn) (very light green circle). When Ti or Ti and Cr are added to (Nb,Hf)3(Si,Sn,Al), there is a reduction in VEC and the slope d(VEC)/d(HV) is not changed (purple triangles).
In Figure 8, the data represented by the circles is for A15 phases without Ti. This element is present in some of the A15 phases, the data points of which are on the line fitted to the red diamonds; this confirms that Ti reduces the VEC of alloyed A15-Nb3X phases. The addition of Al in A15 phases is again seen to increase the hardness (see arrow in Figure 8) and to decrease VEC. Thus, the VEC versus hardness map also confirms that the substitution of Sn by Si increases the hardness significantly but not the value of the parameter VEC, while the addition of Al decreases VEC significantly and increases the hardness but not as much as Si.
The creep at 1473 K of unalloyed A15-Nb3Al and unalloyed and Si alloyed NbCr2 Laves phases is compared in Figure 9. The creep data is from [23,55]. The Laves phases are shown by the lavender squares and black triangles in Figure 2 and A15-Nb3Al is shown by the cross in the red square in Figure 6. The improvement of the creep of the alloyed Laves phase in Figure 9 is related to a decrease of VEC and ∆χ. The experimental data for the creep of alloyed C14 NbCr2 Laves phases is more limited compared with the data for the creep of Nb5Si3 [41]. For the latter silicide, the creep rate έ increased with alloying (excluding Boron addition) and the increase of έ was related to a decrease in VEC and increase in ∆χ. However, for the Nb5Si3 alloyed with Boron, the increase in έ was related to decreases in both the parameters VEC and ∆χ. × 10 ) versus stress (MPa) at 1473 K of unalloyed and Si alloyed Laves phases [18,52] and unalloyed A15-Nb3Al [18]. The former are shown in the ∆χ versus VEC maps in Figure 2a,b and the latter in the map in Figure 6 (see text).

Conclusions
In this paper, the alloying behaviours of the hexagonal C14 Laves and cubic A15 intermetallic phases in Nb-silicide-based alloys were studied using the parameters VEC, ∆χ and atomic size. (light orange square) has a lower VEC than unalloyed Nb3Al (light orange diamond), which has a lower VEC than unalloyed Nb3Sn (dark green circle), and the Ti3Sn and unalloyed Nb3Al are on the same line (red diamonds) as data for A15-Nb3X phases where X = Si, Sn and Nb are substituted by Hf, or Ti, or Hf and Cr, or Ti and Cr, or Ti and Fe; they are also on the same line as the data for (Nb,Hf)3(Si,Sn,Al) (red diamonds). Furthermore, the extension of the line fitted to the data presented by the diamonds crosses the data point for Nb3(Si,Sn) (very light green circle). When Ti or Ti and Cr are added to (Nb,Hf)3(Si,Sn,Al), there is a reduction in VEC and the slope d(VEC)/d(HV) is not changed (purple triangles).
In Figure 8, the data represented by the circles is for A15 phases without Ti. This element is present in some of the A15 phases, the data points of which are on the line fitted to the red diamonds; this confirms that Ti reduces the VEC of alloyed A15-Nb3X phases. The addition of Al in A15 phases is again seen to increase the hardness (see arrow in Figure 8) and to decrease VEC. Thus, the VEC versus hardness map also confirms that the substitution of Sn by Si increases the hardness significantly but not the value of the parameter VEC, while the addition of Al decreases VEC significantly and increases the hardness but not as much as Si.
The creep at 1473 K of unalloyed A15-Nb3Al and unalloyed and Si alloyed NbCr2 Laves phases is compared in Figure 9. The creep data is from [23,55]. The Laves phases are shown by the lavender squares and black triangles in Figure 2 and A15-Nb3Al is shown by the cross in the red square in Figure 6. The improvement of the creep of the alloyed Laves phase in Figure 9 is related to a decrease of VEC and ∆χ. The experimental data for the creep of alloyed C14 NbCr2 Laves phases is more limited compared with the data for the creep of Nb5Si3 [41]. For the latter silicide, the creep rate έ increased with alloying (excluding Boron addition) and the increase of έ was related to a decrease in VEC and increase in ∆χ. However, for the Nb5Si3 alloyed with Boron, the increase in έ was related to decreases in both the parameters VEC and ∆χ. × 10 ) versus stress (MPa) at 1473 K of unalloyed and Si alloyed Laves phases [18,52] and unalloyed A15-Nb3Al [18]. The former are shown in the ∆χ versus VEC maps in Figure 2a,b and the latter in the map in Figure 6 (see text).

Conclusions
In this paper, the alloying behaviours of the hexagonal C14 Laves and cubic A15 intermetallic phases in Nb-silicide-based alloys were studied using the parameters VEC, ∆χ and atomic size.   Figure 8 shows how the hardness of A15-Nb3X phases changes with their parameter, VEC. The latter cannot separate the contributions of different elements, for example Hf, as effectively as the parameter ∆χ. The Ti3Sn (light orange square) has a lower VEC than unalloyed Nb3Al (light orange diamond), which has a lower VEC than unalloyed Nb3Sn (dark green circle), and the Ti3Sn and unalloyed Nb3Al are on the same line (red diamonds) as data for A15-Nb3X phases where X = Si, Sn and Nb are substituted by Hf, or Ti, or Hf and Cr, or Ti and Cr, or Ti and Fe; they are also on the same line as the data for (Nb,Hf)3(Si,Sn,Al) (red diamonds). Furthermore, the extension of the line fitted to the data presented by the diamonds crosses the data point for Nb3(Si,Sn) (very light green circle). When Ti or Ti and Cr are added to (Nb,Hf)3(Si,Sn,Al), there is a reduction in VEC and the slope d(VEC)/d(HV) is not changed (purple triangles).
In Figure 8, the data represented by the circles is for A15 phases without Ti. This element is present in some of the A15 phases, the data points of which are on the line fitted to the red diamonds; this confirms that Ti reduces the VEC of alloyed A15-Nb3X phases. The addition of Al in A15 phases is again seen to increase the hardness (see arrow in Figure 8) and to decrease VEC. Thus, the VEC versus hardness map also confirms that the substitution of Sn by Si increases the hardness significantly but not the value of the parameter VEC, while the addition of Al decreases VEC significantly and increases the hardness but not as much as Si.
The creep at 1473 K of unalloyed A15-Nb3Al and unalloyed and Si alloyed NbCr2 Laves phases is compared in Figure 9. The creep data is from [23,55]. The Laves phases are shown by the lavender squares and black triangles in Figure 2 and A15-Nb3Al is shown by the cross in the red square in Figure 6. The improvement of the creep of the alloyed Laves phase in Figure 9 is related to a decrease of VEC and ∆χ. The experimental data for the creep of alloyed C14 NbCr2 Laves phases is more limited compared with the data for the creep of Nb5Si3 [41]. For the latter silicide, the creep rate έ increased with alloying (excluding Boron addition) and the increase of έ was related to a decrease in VEC and increase in ∆χ. However, for the Nb5Si3 alloyed with Boron, the increase in έ was related to decreases in both the parameters VEC and ∆χ. × 10 Figure 9. Norton plot of creep rate (s −1 ) versus stress (MPa) at 1473 K of unalloyed and Si alloyed Laves phases [18,52] and unalloyed A15-Nb3Al [18]. The former are shown in the ∆χ versus VEC maps in Figure 2a,b and the latter in the map in Figure 6 (see text).

Conclusions
In this paper, the alloying behaviours of the hexagonal C14 Laves and cubic A15 intermetallic phases in Nb-silicide-based alloys were studied using the parameters VEC, ∆χ and atomic size.  [18,52] and unalloyed A15-Nb 3 Al [18]. The former are shown in the ∆χ versus VEC maps in Figure 2a,b and the latter in the map in Figure 6 (see text).

Conclusions
In this paper, the alloying behaviours of the hexagonal C14 Laves and cubic A15 intermetallic phases in Nb-silicide-based alloys were studied using the parameters VEC, ∆χ and atomic size. Experimental data for Laves phases where Nb is substituted by Hf, Mo, Ta, Ti or W and Cr by Al, Ge, Si or Sn, and A15 phases where Nb is substituted by Cr, Fe, Hf, Mo, Ta, Ti or W and Sn by Al, Ge or Si, were used for the calculations. The conclusions of the research are as follows: For the Laves phase, the parameters VEC and ∆χ were in the ranges 4.976 < VEC < 5.358 and −0.503 < ∆χ < −0.107, the Cr concentration was in the range 35.8 < Cr < 51.6 at %, compared with 61 < Cr < 70 at % in the unalloyed NbCr 2 , and the Cr + Al + Ge + Si + Sn sum was in the range 50.9 to 64.5 at %.
The maps of ∆χ versus Cr, ∆χ versus VEC and VEC versus <R> = R <Nb> + R <Cr> could separate the alloying behaviours of different elements in the Laves phase. Compared with the unalloyed NbCr 2 , the parameter VEC was decreased and ∆χ increased (became less negative) when only Si substituted Cr, and the changes of both parameters increased when Nb was substituted by Ti, and Cr by Si and Al, or Si and Ge, or Si and Sn.
For the A15 phase, the parameters VEC and ∆χ were in the ranges 4.38 < VEC < 4.89 and 0.857 < ∆χ < 1.04 with a gap in the VEC values between 4.63 and 4.72; the concentration of the elements Al + Ge + Si + Sn was in the range 16.3 to 22.7 at % and the range of each individual element was 2.4 < Al < 8.9 at %, 1.1 < Ge < 3.8 at %, 1.4 < Si < 8.8 at % and 5.2 < Sn < 16 at %. The VEC versus ∆χ map could separate the alloying behaviours of the elements in the A15 intermetallic phases.
The hardness of A15 Nb 3 X was correlated with the parameters ∆χ and VEC. An increase in the latter was accompanied by an increase in hardness.
The substitution of Sn by Si in Nb 3 Sn resulted in significant increases of the ∆χ and hardness values of Nb 3 (Si,Sn), but the parameter VEC hardly changed, while the addition of Al decreased VEC significantly and increased the hardness but not as much as for Si. The substitution of Nb by Cr had the same effect on ∆χ and hardness as Hf or Ti, and an increase in the concentration of Ti in A15 decreased both the ∆χ and hardness. The addition of Al in Nb 3 (Si,Sn,Al) decreased ∆χ and increased the hardness, but when Ti and Hf, or Ti, Hf and Cr were simultaneously present with Al, the parameter ∆χ was decreased and the hardness hardly changed.
The improvement in the creep of the Si alloyed Laves phase was associated with decreases in VEC and ∆χ.