# Designing a thermodynamically stable and intrinsically ductile refractory
alloy

Sufyan M. Shaikh  Department of Metallurgical and Materials Engineering,
Indian Institute of Technology Madras, Chennai, 600036, Tamil Nadu, India
B.S. Murty  Department of Metallurgical and Materials Engineering, Indian
Institute of Technology Madras, Chennai, 600036, Tamil Nadu, India  Materials
Science & Metallurgical Engineering, Indian Institute of Technology Hyderabad,
Kandi, 502285, Telangana, India  Satyesh K. Yadav  Center for Atomistic
Modelling and Materials Design, Indian Institute of Technology Madras,
Chennai, 600036, Tamil Nadu, India

(*Corresponding author<EMAIL_ADDRESS>
Contributing authors<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The stringent environmental norms are pushing the aviation and power-
generation industries to increase the operating temperatures of their jet-
engines and gas-turbines which will improve the efficiencies of these energy
conversion systems. Aviation and power generation industries are researching
the alternatives for the currently used Ni-based superalloys, which is the
most widely used material in the high-temperature region of these engines.
Refractory alloys are being actively developed as an alternative to the Ni-
based superalloys due to their high melting-points and superior strength
retention capabilities at higher temperatures. Enhancing the intrinsic
ductility of these refractory alloys has remained a challenge. Reducing the
valence electron concentration (VEC) below 4.5 have been shown to improve the
intrinsic ductility of few alloys, which means addition of Ti, Zr, and Hf can
improve the ductility of refractory alloys. However, Re has been widely used
to ductilize the W which goes contrary to the VEC criteria suggested in
literature. The reason this seems to work is may be due to the reduced USFE of
the alloy due to alloying with low USFE elements. Here we use density function
theory (DFT) simulations to understand the factors affecting the ductility of
refractory alloys. A DFT-based workflow is developed which reliably calculates
the ductility parameter of refractory alloys. The developed workflow ensures
that the error in estimating USFE is less than 50mJ/m2. The enthalpy, lattice
distortion, and ductility parameter are calculated using DFT for Group IV, V,
VI elements & their 25 equiatomic binary alloys. The deformability of
refractory BCC alloys is correlated with their enthalpy.

Keywords: stacking fault energy, ductility, high entropy alloys, special
quasirandom structures

##  1 Introduction

The continued cost-pressures and ever stringent environmental norms have
pushed aerospace and power generation industries to improve the overall
efficiency of their jet-engines and gas-turbines [1]. Higher operating
temperatures would lead to better thermodynamic efficiency in these propulsion
and energy conversion systems [2, 3]. The Ni-based superalloys have ruled this
application domain for the past more than six decades [4]. The melting point
of the base metal Ni (Tm=1455$\degree$C) limits the highest operating
temperature of these Ni-based superalloys [1, 5]. Alloys based on refractory
metals show great potential in this domain due to their high melting-points
and their ability to maintain mechanical properties at temperatures higher
than the current Ni-based superalloys [5]. Refractory alloys are based on
Group IV, V, and VI metals [6, 7, 8]. These elements have very high melting-
points (T${}_{m}\textgreater\textgreater$1800$\degree$C) and BCC crystal
structure which restricts their ductility at ambient temperatures.

The lack of deformability at lower temperatures makes them difficult to
manufacture, which creates a bottleneck in developing refractory alloys for
various applications. Alloying with HCP metals [9], reducing the overall
$\gamma_{usfe}$ of alloy by Ta/Re addition [10], decreasing the shear
resistance by Re addition [11], changing the nature of interaction between
alloying elements [12], reducing the valence electron concentration (VEC) [6,
13], altering lattice distortion [14], are few of the refractory alloy
ductilizing suggestions reported in literature. There have been conflicting
suggestions about alloying additions to make W deformable. Qian et al. [12]
suggests, “…addition of Mo, Re, and Os increases the ductility of W and
decrease its GSFE as well as restoring force, whereas the effects of Ti, Ta,
and V are contrary to those of Mo, Re, and Os. The above different effects of
alloying elements may be primarily due to the fundamental repulsive/attractive
nature of interaction between W and alloying element.” On the contrary, Shaikh
et al. [6] and Sheikh et al. [13] suggests alloying with Ti, Zr, Hf, and Re
(HCP elements) should ductilize the refractory alloys. None of the previous
strategies discuss about the thermodynamic stability of these alloys. This
calls for a comprehensive study on the role of various alloying elements on
deformability and thermodynamic stability of refractory alloys.

A general theme in the existing refractory alloy ductilizing studies has been
to add low stacking-fault energy (SFE) ($\gamma_{sfe}$) elements to reduce the
overall unstable-SFE ($\gamma_{usfe}$) of the alloy, suggesting that the Rule-
of-Mixtures (ROM) dictates the overall $\gamma_{usfe}$ of the alloy. Most of
the earlier studies selected alloying elements to get maximum decrease in the
overall $\gamma_{usfe}$ of the alloy, without considering the thermodynamic
stability of the alloy. We found that the ROM will overestimate the
$\gamma_{sfe}$ of FCC alloys by an order of magnitude (Table 1). Large
decrease in $\gamma_{sfe}$ of FCC alloys due to alloying has already been
observed in CoCrNi, CoCrFeNi, CoCrFeMnNi as compared to their average
$\gamma_{sfe}$ value (Table 1). This gives us a hint that the $\gamma_{sfe}$
or $\gamma_{usfe}$ may not be dictated by the ROM alone. There is a need of a
comprehensive study on the effect of alloying elements on the deformability
and thermodynamic stability of refractory alloys. Present work addresses these
issues using first-principles density functional theory (DFT) simulations. The
study gives a strategy to select alloying elements which give maximum
ductilizing effect along with thermodynamic stability of the final alloy.

##  2 Results and Discussion

Refractory equiatomic binary alloys made from Group-IV (Ti, Zr, Hf),
Group-V(V, Nb, Ta), Group-VI(Mo, W), and Group-VII(Re) elements are studied.
The DFT-based workflow to calculate lattice parameter, enthalpy of formation,
root mean squared lattice distortion, unstable stacking fault energy, and
surface energy is presented in Figure 1. The results are shown in the form of
heat maps in Figures 2 and 3. Additional information can be found in
supplementary information document.

###  2.1 Enthalpy of formation (Ef)

The $E_{f}$ defines the relative thermodynamic stability of an alloy. Negative
$E_{f}$ indicates stronger bonds between the alloy constituents and vice-
versa. Strong bonding between the alloy constituents will cause large
resistance to deformation and vice-versa. Highly negative $E_{f}$ indicates
formation of an ordered intermetallic, e.g. ordered-Ni3Al in superalloys [4].
Ef is calculated with the relation [15],

$E_{f}^{AB}=E_{AB}-\frac{E_{A}+E_{B}}{2}$

where E${}_{f}^{AB}$, EAB, EA, and EB, are the enthalpy of formation of alloy
AB, energy per atom of relaxed supercell of alloy AB in BCC symmetry, the
energy of relaxed supercell of constituents A and B, respectively, all values
are in the units of eV/Atom. The 9 alloying elements form a total of 36
equiatomic binary alloys. Out of these 36 alloys, 6 systems containing both
the constituents being HCP, are not considered in the present study. Figure
2(a) shows the enthalpy of formation (Ef, eV/Atom) values in the form of heat
map. We have not considered VRe [16], TaRe [17], ZrW [18], VHf [19], VZr [20],
having Ef, -0.2456, -0.1942, 0.1477, 0.1631, 0.1710 meV/Atom respectively, for
subsequent analysis as they do not form stable BCC solid solutions at low
temperatures at equiatomic concentration. The Ef varies from -0.1232eV/Atom
(TaMo) to 0.1114eV/Atom (HfW). NbTa has the Ef of -0.0067eV/Atom which is
closest to 0eV/Atom.

###  2.2 RMS lattice distortion (RLD)

Lattice distortion has been widely correlated with the solid solution
strengthening occurring in alloys. Composition averaged misfit volume
($\delta$) is one such parameter which is primarily used to quantify the
difference between atomic sizes of various alloy constituents [21, 6].
$\delta$ is useful only when the pure elements have the same crystal structure
as that of the alloy [21, 22]. XRD is widely used to study the lattice
distortion [23]. The peak broadening in XRD is correlated to the change in the
interplanar spacing in sample leading to a range of 2$\theta$ values over
which the Bragg’s condition is satisfied. Accurate determination of lattice
distortion using XRD requires a well annealed reference sample to single out
only the lattice distortion in the sample under study. Lattice distortion in
the crystal is quantified using the radial distribution function (RDF), a
Fourier transform of the XRD curve [24]. The peak width of the RDF curve
qualitatively gives information about the deviations around the 1st nearest
neighbor (1NN) distance at which an atom can be present. The RDF curve may not
give an accurate picture of the distortion happening in real crystals.
Calculation of the RDF depends on the atomic radii of the elements in the
alloy as well. If the alloy constituents have the same crystal structure in
their pure state then those radii/lattice parameters can be taken to calculate
the RDF. For example, in case of NbTi alloy, to quantify the lattice
distortion using RDF, the lattice parameter of Ti should be considered in its
BCC crystal structure. Here we use “root-mean-square lattice distortion (RLD)”
method which is similar to MSAD [25], to quantify the lattice distortion.
Root-mean-squared lattice distortion (RLD) captures the deviation of the
crystal structure from its ideal BCC symmetry. RLD is calculated using the
relation,

$RLD=\sqrt{\frac{1}{N}\sum_{n}{\sum_{i}{(R_{i}^{n}-R_{0})^{2}}}}\ ,\ \
R_{0}=\frac{a_{0}\sqrt{3}}{2}$

where, R${}_{i}^{n}$ is the distance between $n^{th}$ atom and its $i^{th}$
first nearest neighbor (1NN), R0 is the ideal 1NN distance, a0 is the bulk
lattice parameter of alloy in BCC symmetry, $i$ is the total number of 1NN for
each lattice site (equal to 8 in BCC), $n$ is the total number of atoms in
supercell (equal to 128), $N$ is the total number of 1NN in the supercell
(equal to 1024, 8 for every atom in the supercell). This approach ensures that
the negative and positive distortions in the crystal are captured correctly
and are not canceled out by the opposite signs of either distortions.

Figure 2(b) shows the RLD values in the form of heat map for alloys. The RLD
value ranges from 0.00Å for VW to 0.29Å for VZr. The RLD is higher for alloys
having positive Ef (larger diameter circles in Figure 6) and vice-versa. It
means very positive Ef can lead to large distortion in the crystal and may
alter the BCC symmetry of the lattice. We have put a threshold of
$\pm$0.12eV/Atom on Ef. This criteria ensure BCC crystal strucutre of the
alloys with minimum lattice distortion.

###  2.3 Ductility of refractory alloys

####  2.3.1 Unstable stacking-fault energy ($\gamma_{usfe}$)

$\gamma_{usfe}$ is calculated using the relation given below,

$\gamma_{usfe}=\frac{E_{faulted}-E_{pristine}}{(Area\ of\ Plane)}$

where the Efaulted is the energy of supercell having unstable stacking fault
and the Epristine is the energy of pristine supercell. Figure 2(c) shows the
$\gamma_{usfe}$ calculated using First-principles DFT method, in the form of
heat map for pure elements and the alloys under study. MoW (1681mJ/m2) has the
highest $\gamma_{usfe}$ value whereas, TaZr (454mJ/m2) has the lowest
$\gamma_{usfe}$ value. Figure 2(d) shows the change in $\gamma_{usfe}$ values
of the alloys from their average values. The percentage change in
$\gamma_{usfe}$ values from their average value ranges from -41% (MoRe) to 20%
(MoV). The largest percentage reduction in $\gamma_{usfe}$ value is for MoRe
(41%) and WRe (34%). The smallest percentage reduction in $\gamma_{usfe}$ is
for NbTa (4%). The largest percentage increase in $\gamma_{usfe}$ is for MoV
(20%) and the smallest percentage increase is for MoW (5%).

The present analysis shows many possible alloy combinations which can lead to
reduced $\gamma_{usfe}$ in refractory binary alloys from their average values.
Addition of HCP elements resulted in pronounced decrease in the
$\gamma_{usfe}$ of the alloys, which is due to the inherently low
$\gamma_{usfe}$ of the HCP elements (Figure 2(c),2(d)). Similarly, addition of
very high melting point elements from Group-VI has resulted in the large
increase in the $\gamma_{usfe}$ of the alloys, which is due to the inherently
high $\gamma_{usfe}$ and stronger bonds of the atoms of these elements (Figure
2(c),2(d)). Generally, the $\gamma_{usfe}$ of the alloy is expected to be
within the range of $\gamma_{usfe}$ of the constituent elements. Here we show
that in refractory alloys the $\gamma_{usfe}$ can be much lower than either of
the constituent element’s $\gamma_{usfe}$ e.g. NbV, NbZr, NbHf, NbTa, NbMo,
NbW, TaHf, WRe, TaZr, TaHf (Figure 2(c),2(d)). We have observed a similar
trend in FCC alloys (Table 1).

####  2.3.2 Slip system

Multiple SFE values are possible in HCP crystals depending upon the slip
direction chosen. Figure 4 shows the three possible SFE values for Zr
depending upon the slip direction chosen for (0001) slip plane. In Zr, the
(0001)<11$\bar{2}$0> slip system leads to the unstable stacking fault with
$\gamma_{usfe}$ of 457mJ/m2 (Figure 4(a),4(b)). Similarly, the
(0001)<1$\bar{1}$00> slip system leads to 2 different SFE values of 227mJ/m2
(stable stacking fault) and 684mJ/m2 (unstable stacking fault) (Figure
4(c),4(d)). In comparison, the BCC crystals have only 1 unstable stacking
fault (only 1 peak in the slip curve) for {110}<111> slip system. In present
work we have calculated the (0001)<11$\bar{2}$0> $\gamma_{usfe}$ value for HCP
metals and used it in subsequent analyses.

The {110}<111> slip system of BCC crystal is similar to {0001}<11$\bar{2}$0>
slip system of HCP crystal (Figure 4(a),4(b)). Both the slip systems’ have
only one peak in their slip curve. Generally in literature, the
{0001}<1$\bar{1}$00>(Figure 4(c),4(d)) slip curve of HCP metals and alloys is
emphasized as the slip in this direction leads to a stable stacking fault.
However, the BCC crystal system does not have any stable stacking fault for
{110}<111> slip system. Therefore in present work the
$\gamma_{usfe}^{\\{110\\}\textless 111\textgreater}$ of BCC elements/alloys
and $\gamma_{usfe}^{\\{0001\\}\textless 11\bar{2}0\textgreater}$ of HCP
elements is calculated and used for analysis.

####  2.3.3 Shearing interface chemistry

Special quasirandom structures (SQS) have been extensively used to predict the
SFE of alloys due to their simplicity in capturing the inherent chemical
disorder present in the alloys [26]. There have been numerous reports on
$\gamma_{sfe}$ of concentrated alloys calculated using SQS, albeit without any
consideration to the shearing interface chemistry [27, 28, 29, 30, 31, 32, 33,
12, 34, 35, 36]. The chemistry of the shearing interface affects the overall
$\gamma_{sfe}$ of the alloy in concentrated solid solutions [37]. The
intrinsic $\gamma_{sfe}$ of alloys can vary over a large range of 150mJ/m2
[37]. Earlier reports did not consider the chemistry of the shearing interface
which influences the $\gamma_{usfe}$ of the overall alloy [12, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36]. There has been significant deviations in the
literature about the reported $\gamma_{usfe}$ values even for the pure metals
[38, 9]. The literature talks about the accuracy of SQS method for planar
defect calculations, but the shearing interface chemistry was not considered
[39]. This emphasises the need for a reliable methodology to calculate
$\gamma_{usfe}$ and $\gamma_{s}$ of concentrated alloys.

The SQS are generated, considering the correlations of pairs, triplets,
quadruplets. Here we ensure that the generated SQS have the correct
stoichiometry at the shearing interface (equiatomic) as shown in Figure 1. In
present work the equiatomicity (formula composition) is maintained at the
shearing interface by shifting the respective planes towards the centre of the
supercell. The atoms in top two planes and bottom two planes are fixed in all
directions, whereas, the remaining atoms from the middle six planes are fixed
only in slip direction. The supercells are sheared by one complete
$\bar{b}$=<111> vector to get four different $\gamma_{usfe}$ values as shown
in Figure 1. For every supercell, the energy difference between before and
after one complete slip along $\bar{b}$ is calculated. A total of four
differences are calculated as shown in Figure 1. $\gamma_{usfe}$ is calculated
for shearing direction having the lowest difference between no-slip and full-
slip. The entire process is schematically represented in Figures 1 and in
supplementary file.

Figure 5 shows how the same SQS can give a range of $\gamma_{usfe}$ values
depending on the chemistry of the shearing interface for {110}<111> slip
system of WRe alloy. The upper inset in Figure 5 shows the $\gamma_{usfe}$
values. The lower inset shows the energy difference between before and after 1
complete slip by $\bar{b}=\frac{a}{2}$<111>. The $\gamma_{usfe}$ varies from
906 to 1199 mJ/m2. Similarly the energy difference between before and after 1
complete slip varies from 0.5 to 109 mJ/m2 (Figure 5). It means that the
$\gamma_{usfe}$ of WRe alloy can be any value between 905.5 to 1308 mJ/m2 if
we use the current supercell. Such large variation in $\gamma_{usfe}$ poses a
challenge as to which value to be selected for further analysis. Our current
methodology eliminates this ambiguity by maintaining equiatomicity (formula
composition) at the shearing interface. The concentration at the shearing
interface always remains same as formula composition of the alloy therefore
our current methodology minimizes the errors associated with the incorrect
chemistry at the shearing interface and ultimately helps in accurately
predicting the $\gamma_{usfe}$ and $\gamma_{s}$ of concentrated alloys. Each
of these 9 shearing interfaces are sheared in 4 difference directions, giving
4 difference $\gamma_{usfe}$, which requires total 9 DFT jobs per shearing
interface (Figure 1). It means for every alloy we require 81 DFT calculations
to be done (there are 9 shearing interface in a supercell having 10 planes),
which is not the optimum way to calculate the $\gamma_{usfe}$ of the alloys.
Instead, we calculate the $\gamma_{usfe}$ only at the equiatomic shearing
interface which reduces the number of DFT jobs requried from 81 to 9 (if the
supercell has 1 equiatomic shearing interface).

####  2.3.4 Surface energy ($\gamma_{s}$)

Similar to $\gamma_{usfe}$ calculation, the surface is calculated for the
interface having equiatomic stiochiometry/formula composition. $\gamma_{s}$ is
calculated using the supercells used to calculate the $\gamma_{usfe}$. The
equiatomic shearing interface of the supercell is exposed to vacuum of 10Å.
$\gamma_{s}$ is calculated using the relation given below,

$\gamma_{s}=\frac{E_{with-vacuum}-E_{no-vacuum}}{2\times(Area\ of\ Plane)}$

Figure 2(e) shows the (110) $\gamma_{s}$ calculated using First-principles DFT
method, in the form of heat map for pure elements and the alloys under study.
For Ti, Zr, Hf, and Re, $\gamma_{s}$ the $\gamma_{s}$ is reported for (0001)
plane. VW (3190mJ/m2) has the highest $\gamma_{s}$ value whereas, NbZr
(1662mJ/m2) has the lowest $\gamma_{usfe}$ value. Figure 2(f) shows the change
in $\gamma_{s}$ values of the alloys from their average values. The percentage
change in $\gamma_{s}$ values from their average value ranges from -20% (MoZr)
to 20% (NbRe). The largest percentage reduction in $\gamma_{s}$ value is for
MoZr (20%) and TaZr (34%). The smallest percentage reduction in
$\gamma_{usfe}$ is for NbHf (3%). The largest percentage increase in
$\gamma_{s}$ is for NbRe (20%) and VW (13%). The smallest percentage increase
is for NbMo (2%).

####  2.3.5 Ductility parameter (D)

Ductility parameter is defined as the ratio of surface energy to unstable
stacking fault energy [40, 41].

$\displaystyle D$ $\displaystyle=\frac{\gamma_{s}}{\gamma_{usfe}}$
$\displaystyle D\textgreater 1\Longrightarrow\gamma_{s}$
$\displaystyle\textgreater\gamma_{usfe}\Longrightarrow\textbf{Ductile}$
$\displaystyle D<1\Longrightarrow\gamma_{s}$
$\displaystyle<\gamma_{usfe}\Longrightarrow\textbf{Brittle}$

It quantifies the competition between the energy cost of creating a new crack
surface and the energy cost of nucleating new dislocations in the stress field
of the crack tip. The D-parameter also captures the brittle-to-ductile-
transition of refractory alloys (BDT) [42, 43, 44] if similar calculations are
done across various temperature ranges. Present study uses first-principles
density functional theory simulations to calculate D-parameter at absolute
zero temperature. Figure 3 shows the ductility parameter calculated for pure
metals and alloys in the form of heat map. The D-parameter of pure metals
ranges from 1.64 (Re) to 4.21 (Ti). TaTi (4.16) has the highest D-parameter
value whereas, MoW (1.81) has the lowest D-parameter value. The improved
deformability of WRe (D=3.15) and MoRe (D=2.70) does not look large if we
observe only their average $\gamma_{usfe}$ or $\gamma_{s}$. However, their
improved deformability is well depicted in Figures 3. Present analysis leads
to various alloy chemistries, which can have increased deformability by having
low $\gamma_{usfe}$ and high $\gamma_{s}$ & a large reduction in
$\gamma_{usfe}$ from average value (Figure 2(d)) and a large increase in
$\gamma_{s}$ from average value (Figure 2(f)). The increased value of
D-parameter leads to better deformabilities.

The role of valence electron concentration (VEC) on improving deformability
has been well reported [13]. Lower VEC (<4.5) is expected to improve the
deformability of alloys [13]. Ti, Zr, Hf are expected to be the major
ductilizing elements as they are reported to reduce the VEC of alloy [6, 13].
This is opposite to the W-Re system where the high VEC element (Re) makes W
deformable [45]. Re addition is counter intuitive to the VEC criteria. We show
that the Ti, Zr, Hf are making the alloy deformable not because of their low
VEC but rather it is due to their inherently very low stacking fault energy
(Figure 2(c)), high $\gamma_{s}$ (Figure 2(e)), and in turn very large
D-parameter (Figure 3) value. Here we show that the Re addition to W improves
the deformability (by large reduction in $\gamma_{usfe}$) as WRe alloy have
slightly positive Ef. The positive Ef is due to the atoms in the alloy do not
prefer other element near them leading to a repulsion between alloying
elements. This repulsion is responsible for less resistance to the slip planes
gliding over each other, resulting in better deformability.

There are numerous studies on deformability studies on Al/Nb/Mo-Ta-Ti-Zr-Hf-V
alloy and its sub-systems [13, 46, 47, 48, 49, 50]. NbTaTiZrHf is reported to
have more than 50% room temperature ductility [51, 52].
Al0.5NbTa0.8Ti1.5V0.2Zr is reported to be the most ductile refractory alloys
with BCC-B2 microstructure, which is analogous to $\gamma$-$\gamma^{\prime}$
type microstructure of Ni-based superalloys [53]. Our previous study has shown
that the alloys having good ductility/deformability contains Ti/Zr/Hf elements
[6]. These HCP elements inherently have low $\gamma_{usfe}$ which reduces the
alloy’s $\gamma_{usfe}$ increasing D-parameter and in turn improving its
deformability (Figure 2). Adding Ti/Zr/Hf could improve the overall
deformability of refractory alloys at the cost of decreased oxidation
resistance as these HCP elements are susceptible to oxygen absorption [54,
55].

Ef plays an important role in defining the deformability and lattice
distortion of refractory alloys as depicted in Figure 6. The negative slope of
the red line in Figure 6 shows the inverse correlation of enthalpy of
formation with the change in unstable stacking-fault energy ($\gamma_{usfe}$)
of alloys from their average values. Ef defines the mutual attraction or
repulsion between the constituents of the alloys. The more positive the Ef,
the larger the repulsion between the constituents and large RLD (large circles
at postive enthalpies in Figure 6), which enhances the relative glide of slip
planes over each other. Conversely, the more negative the Ef, the larger the
attraction between the constituents and low RLD (small circles at negative
enthalpies in Figure 6), which hinders the easy glide of the slip planes
relative to each other. The present analysis proves that the enthalpy of
formation of alloys gives a fairly reliable idea about deformability and
lattice distortion of refractory alloys apart from dictating their
thermodynamic stability.

###  2.4 Why equiatomic alloys?

Figure 7 shows the $\gamma_{usfe}$ values for NbxV1-x (x $\in$
[0,0.25,0.75,1]) alloy. In Figure 7, the numbers in blue show the reduction in
$\gamma_{usfe}$ of alloys from the average value. The largest reduction in
$\gamma_{usfe}$ occurs at equiatomic concentration (Figure 7). The highest
reduction in $\gamma_{usfe}$ gives a direction for improving the deformability
of refractory alloys with appropriate alloying additions. Our analysis agrees
well with an earlier study done by Yang and Qi in Ref. [10] where the lowest
value of $\gamma_{usfe}$ occurs at the equiatomic concentration in W-Ta and
W-Re alloys leading to intrinsic ductility in these alloys. The enhancement in
intrinsic ductility is concluded to be due to the preference for dislocation
emission in $\frac{a}{2}\\{\bar{1}10\\}$<111> slip system at crack tip [10].
The largest ductilizing effect of alloying elements is possible at equiatomic
concentrations. Therefore we have considered only the equiatomic alloys in
present study.

##  3 Equations

Not applicable.

##  4 Tables

Table 1: Comparison of ($\gamma_{sfe}^{\\{111\\}\textless 112\textgreater}$, mJ/m2) values of pure elements and alloys in FCC crystal structure, (ROM-Rule-of-mixtures, DFT-Density functional Theory, TEM-Transmission electron microscope). Metal/Alloy | ROM | DFT | TEM
---|---|---|---
Co | – | 290 [28] | –
Cr | – | 177 (this work) | –
Fe | – | 260 [56] | –
Ni | – | 127 [57] | 125 [27]
NiFe | 194 | 102 [58] | 103 [27]
CoCrNi | 198 | – | 18 [59]
CoCrFeNi | 214 | 32 [60] | 27 [59]
Table 2: Correlation matrix. | Ef | $\gamma_{usfe}$ | % Change $\gamma_{usfe}$ | $\gamma_{s}$ | D | RLD
---|---|---|---|---|---|---
Ef | 1.00 |  |  |  |  |
$\gamma_{usfe}$ | -0.69 | 1.00 |  |  |  |
%Change $\gamma_{usfe}$ | -0.85 | 0.66 | 1.00 |  |  |
$\gamma_{s}$ | -0.50 | 0.87 | 0.51 | 1.00 |  |
D | 0.71 | -0.91 | -0.56 | -0.65 | 1.00 |
RLD | 0.62 | -0.72 | -0.58 | -0.76 | 0.50 | 1.00

##  5 Figures

Alloy Calculate enthalpy, lattice parameter, RLD Generate SQS with pairs,
triplets, and quadruplets Relaxed SQS + 10Å Vacuum Select equiatomic shearing
interface 0,0+L,0-L,00,+M0,-MShear L=$\textless 111\textgreater$ M=$\textless
11\bar{1}\textgreater$ $\Delta 1$$\Delta 2$$\Delta 3$$\Delta 4$ Calculate
$\gamma_{usfe}$ for min. $\Delta_{i}\leq$ 50mJ/m2
$\gamma_{usfe}^{alloy}$Expose to 10Å
vacuum$\gamma_{surface}^{alloy}$$D=\frac{\gamma_{surface}}{\gamma_{usfe}}$
Figure 1: Workflow.

(a) Enthalpy of formation (eV/Atom).

(b) RLD (Å)

(c) $\gamma_{usfe}$ (mJ/m2)

(d) Change in $\gamma_{usfe}$ (%)

(e) $\gamma_{s}$ (mJ/m2)

(f) Change in $\gamma_{usfe}$ (%)

Figure 2: Heat maps. Alloys containing both HCP constituents are not
considered. Figure 3: Heat maps of D-parameter. Alloys containing both HCP
constituents are not considered.

(a)

<11$\bar{2}$0>

(b)

(c)

<1$\bar{1}$00>

(d)

Figure 4: HCP slip system comparison. (a)Zr (0001)<$11\bar{2}0$> slip curve,
(b)Zr (0001)<$11\bar{2}0$> slip direction projection along <0001> direction,
(c)Zr (0001)<$1\bar{1}00$> slip curve, (c)Zr (0001)<$1\bar{1}00$> slip
direction projection along <0001> direction. Figure 5: Comparision between
$\gamma_{usfe}$ values calculated for various shearing interfaces from same
supercell of WRe. Numbers on left shows the $\gamma_{usfe}$ value. Upper inset
shows the range of $\gamma_{usfe}$. Lower inset shows the energy difference
between before and after one complete slip by $\bar{b}$=$\frac{a}{2}\textless
111\textgreater$. Figure 6: Change in $\gamma_{usfe}$ as compared to the
average value vs enthalpy of formation (Ef). Positive enthalpy of formation
helps improving the deformability of refractory binary alloys. RLD is shown as
circle with different diameters. Figure 7: {110}<111> $\gamma_{usfe}$ of Nb-V
alloys. Numbers in blue show the change in $\gamma_{usfe}$ from the average
value.

##  6 Algorithms, Program codes and Listings

Not applicable.

##  7 Methods

Ti, Zr, Hf, V, Nb, Ta, Mo, W, Re, and their 30 equiatomic binary alloy systems
(excluding alloys made up of only HCP elements) are studied using first-
principles density functional theory simulations. Lattice parameter (a, Å),
enthalpy of formation (Ef, eV/Atom), unstable stacking fault energy (USFE,
$\gamma^{\\{110\\}<111>}_{usfe}$, mJ/m2), and surface energy
($\gamma^{\\{110\\}}_{s}$, mJ/m2) is calculated for V, Nb, Ta, Mo, W, and 30
equiatomic binary alloys in BCC symmetry. For Ti, Zr, Hf, and Re,
($\gamma^{0001}_{usfe}$, mJ/m2) is calculated for {0001}$\textless
11\bar{2}0\textgreater$ slip system, surface energy ($\gamma^{0001}_{s}$,
mJ/m2) is calculated for {0001} plane, lattice parameter (a, c, Å), and
enthalpy of formation (Ef, eV/Atom) are calculated for HCP crystal structure.
Experimental values of lattice parameters are taken from Ref.[61]. The
reported lattice parameters are within 1% deviation of their experimental
values. All the values are listed in Table S1.

###  7.1 Special Quasirandom Structures, Supercell

Special quasirandom structures (SQS) are used to capture chemical disorder in
the alloys. SQS are generated using MCSQS code from Alloy Theoretic Automatic
Toolkit (ATAT) [62, 63]. Pairs, triplets, and quadruplets correlations equal
to the regular BCC unit cell lattice parameter are considered. We were able to
get perfectly random correlations well within 48 hours. The enthalpy of
formation ($E_{f}$) and lattice parameter are calculated with 128 atoms SQS
generated using a regular BCC unit cell repeated four times each in $\textless
100\textgreater$ directions. In real alloy crystals, the atoms do not remain
at their ideal lattice sites due to stress fields of different chemical
species present around each of the lattice sites. This deviation from ideality
is calculated as root-mean-squared lattice distortion (RLD, Å). RLD is
calculated using the codes given in Ref. [64].

For $\gamma_{usfe}$ calculation of pure elements the supercell had 48 atoms
with 10Å vacuum. The x, y, z axes of supercell of pure BCC elements were
aligned in $\textless 111\textgreater$, $\textless 11\bar{1}\textgreater$, and
$\textless 110\textgreater$ directions, respectively. The x, y, z axes of
supercell of pure HCP elements were aligned in $\textless
2\bar{1}\bar{1}0\textgreater$, $\textless 11\bar{2}0\textgreater$, and
$\textless 0001\textgreater$ directions respectively. The $\textless
111\textgreater$ slip direction of BCC crystal is similar to $\textless
2\bar{1}\bar{1}0\textgreater$ slip direction of HCP crystal. Therefore, the
$\gamma_{usfe}^{\\{110\\}\textless 111\textgreater}$ of BCC elements is
reported along with the $\gamma_{usfe}^{\\{0001\\}\textless
2\bar{1}\bar{1}0\textgreater}$ of HCP elements. For $\gamma_{usfe}$
calculation of alloys, the supercell had 120 atoms and 9 shearing interfaces
(10 of (110) type planes with 12 atoms/plane). The x, y, z axes of supercell
were aligned in $\textless 111\textgreater$, $\textless
11\bar{1}\textgreater$, and $\textless 110\textgreater$ directions of a
regular BCC unit cell respectively. The supercells are manipulated using
Atomsk [65]. A vacuum of 10Å is added perpendicular to the slip planes (along
$\textless 110\textgreater$ direction) to prevent interactions between faults
due to periodic boundary condition. The equiatomicity is maintained at the
shearing interface by shifting the respective planes towards the centre of the
supercell. The atoms in top two planes and bottom two planes are fixed in all
directions, whereas, the remaining atoms from the middle six planes are fixed
only in slip direction. The supercells are sheared by one complete
$\bar{b}$=<111> vector to get four different $\gamma_{usfe}$ values as shown
in Figure 1. For every supercell, the energy difference between before and
after one complete slip along $\bar{b}$ is calculated. A total of four
differences are calculated as shown in Figure 1. $\gamma_{usfe}$ is calculated
for shearing direction having the lowest difference between no-slip and full-
slip.

$\gamma_{s}$ is calculated with supercells used for $\gamma_{usfe}$
calculation. The equiatomic shearing interface is exposed to vacuum of 10Å.
The energy difference between supercell with and without vacuum gives the
$\gamma_{s}$ value.

###  7.2 First-principles calculations

The first-principles density functional theory (DFT) calculations were
performed using Vienna Ab-initio Simulation Package (VASP) with plane-wave
basis and projector augmented wave (PAW) pseudopotentials [66, 67, 68]. The
electronic exchange-correlation effects were calculated by Perdew-Burke-
Erzernhoff generalized gradient approximation (PBE-GGA) [69, 70]. Methfessel-
Paxton smearing method with 0.2eV width is used [71]. Structural relaxation is
terminated when the forces on atoms become less than $10^{-4}$ eV/Å.
Tetrahedron method with Blöch correction is used for accurate energy
calculation [67]. The a, c, Ef, $\gamma_{usfe}$, and $\gamma_{s}$ calculation
details are given in supplementary file.

##  8 Conclusion

A DFT-based framework to accurately predict stacking fault energy, surface
energy, and in turn deformability of concentrated alloys has been developed.
Using the developed framework, a strong correlation between the thermodynamic
stability and deformability of refractory alloys has been found. Present work
sheds light on the effect of VEC, which was supposed to dictate the ductility
of refractory alloys. Here we have shown that the VEC may not the be the most
accurate measure of predicting the deformability of refractory alloys. Present
work helps simultaneously tackle two major issues associated with refractory
alloy development for high temperature applications:

  1. 1.

Earlier methods did not focus on the chemical composition of the surface and
shearing interface while calculating surface energies and stacking fault
energies. Our method ensures that we are accurately capturing the correct
formula chemistry of the alloys while predicting those properties.

  2. 2.

There exists an inverse correlation between the enthalpy of formation and the
deformability of refractory alloys. Slight positive enthalpy of formation can
improve deformability without much compromise on the thermodynamic stability
of refractory alloys. The positive enthalpy requirement for ductility can be
compensated by increased entropic contribution in reducing the overall Gibb’s
energy at higher temperatures by $\Delta G=\Delta H-T\Delta S$ type
relationship. We believe our findings will further be reinforced once a
similar study is done at higher temperatures.

In conclusion, our results unveil that the suggestion of reducing the VEC (by
adding HCP elements) to enhance the ductility of refractory alloys is majorly
due to the inherently low SFE of HCP elements rather than their low valency.
The chemistry of shearing interface and the surface should be taken into
consideration while using the SQS supercells to calculate the $\gamma_{usfe}$
and $\gamma_{s}$ of concentrated alloys. With the developed DFT framework, we
have quantified and minimized the errors associated with the SQS approach to
calculate the $\gamma_{usfe}$ and $\gamma_{s}$ of concentrated alloys. Taking
the effect of enthalpy of formation on the deformability of concentrated
alloys is likely to open new directions in the design of refractory alloys for
high temperature applications.

##  9 Declarations

###  9.1 Funding

We acknowledge the use of the computing resources at High Performance
Computing Environment (HPCE), IIT Madras. This work was supported by Ministry
of Education (formerly known as Ministry of Human Resource Development),
Government of India, Grant number: SB20210844MMMHRD008277.

###  9.2 Competing interest

The authors declare no competing interests.

###  9.3 Ethics approval

Not applicable.

###  9.4 Consent to participate

Not applicable.

###  9.5 Consent for publication

Not applicable.

###  9.6 Availability of data and materials

All the data for the study is in the manuscript and in supplementary file.

###  9.7 Code availability

The code to calculate RMS lattice distortion (RLD) is available as a GitHub
repository at https://www.github.com/sufyanshk/RLD.

###  9.8 Authors’ contributions

S.M.S. and S.K.Y. developed the initial hypothesis. S.M.S. performed the
calculations and drafted the initial manuscript. S.M.S., S.K.Y., and B.S.M.
together discussed the results. S.K.Y. and B.S.M. supervised the project. All
authors contributed to the manuscript writing.

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