**2. Theoretical calculations by density functional theory (DFT)**

#### **2.1 Crystal structure of transition metal-Ru alloys**

Ru-based intermetallic alloys exist in different crystal structure phases such as A15, DOc, DO<sup>0</sup> <sup>c</sup> tP16, L12, and B2. However, in this chapter we will focus on A15 Ru-based alloys as illustrated in **Figure 1**. The X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) crystallizes in a cubic A15 type with a space group Pm-3 N (number 223) and a theoretical lattice constant of 4.63 Å [14]. Moreover, this cubic phase possess a prototype of Cr3Si. The A15 phases are described as a series of intermetallic compounds with a formula A3B; where A is a transitional metal and B can be any element. In the case of X3Ru, Ru and X represent the A and B respectively. The proposed study seeks to model the properties of 3d transition metal-Ru

*Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

#### **Figure 1.**

*Ball and stick illustration of A15 crystal structure phase of X3Ru alloys. Gray balls represent X = Sc,Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn- atoms; green balls represent Ru- atoms.*

intermetallic systems for A15 phase, using density functional theory techniques. The density functional theory enables determination of many ground state properties of material systems with sufficient accuracy, and is widely used in characterizing the properties of new materials.

#### **2.2 Quantum mechanical techniques**

In material science, the energy of a system is needed in order to evaluate the properties of a material. Numerically, this is obtained by solving the Schrödinger wave equation [15].

$$H\psi = E\psi\tag{1}$$

where H is the Hamiltonian operator, E is the energy of the particle and ψ is the wavefunction is the particle's wavefunction. Hamiltonian H in Eq. (1) for a system of many interacting particles (electrons and nucleus) can be expressed as

$$H\_{ele-nuc} = T\_{ele} + T\_{nuc} + V\_{ele-nuc} + V\_{ele-ele} + V\_{nuc-nuc} \tag{2}$$

where Tele and T*nuc* are the kinetic energy operators of the electrons and nuclei respectively, V*nuc-nuc,* V*ele-ele* and V*ele-nuc* are potential energy operators of the elenuc, ele-ele and nuc-nuc, respectively, due to Coulomb interactions. Eq. (2) can be solved analytically for few atoms, however, for very large number of atoms (N � <sup>10</sup>23) it is intractable to solve; hence, a number of approximations are needed to find its exact solution. The first approximation in solving the Schrödinger equation for a many-body interacting system is called the Born and Oppenheimer approximation [15]. This approximation decouples the electron motion from that of heavier ions, setting the kinetic energy operator of the nuclei to zero, while the potential energy operator becomes a constant. The Born and Oppenheimer approximation reduces the complexity of Eq. (2) to

$$H\_{ele-muc} = T\_{ele} + V\_{ele-muc} + V\_{ele-ele} \tag{3}$$

To solve the Eq. (3), the last two terms: V*ele-ele* and V*ele-nuc* must be known. Quantum mechanical techniques, such as the density functional theory and the Hatree-Fock approximation aim to obtain accurate ground state energy of a material system comprising of electrons and nucleus. Thereafter, other properties related to the total energy of the system can be easily determined. In order to achieve this aim, the exact forms of the terms on the right hand side (RHS) of Eq. (3) must be known. All the terms are known except the exchange correlation energy of interacting electrons that remains unknown. The density functional theory accounts for the electron exchange correlation (Exc) effect for many interacting particle system. However, the Hatree-Fock has difficulties in predicting the properties of metallic and magnetic systems accurately.

#### **2.3 Density functional theory**

The density functional theory minimizes the difficulty in strong electron-nuclei and electron–electron interactions in many body systems (Eq.(2)) by mapping it onto the single particle moving in an effective potential [16]. This effective potential is not explicitly known, but can be approximated to accurately predict the solid-state properties. The basic idea of the DFT is that any property of a system of many interacting particles can be expressed as a functional of the ground state electron density ρ(r).

$$E = E[\rho(r)]\tag{4}$$

DFT is established from the Hohenberg-Kohn theory [16], which expresses a one-to-one correspondence between the electron density ρ(r) of a many electron– electron interacting system and the external potential *Vext* imposed by the nucleus. Thus, the accurate ground state wave function is obtained from the external potential expression due to a correct ground state electron density. Therefore, the minimum energy can be expressed as:

$$E[\rho(r)] = T\_{ele}[\rho(r)] + F\_{HK}[\rho(r)] + V\_{\text{ext}}[\rho(r)] \tag{5}$$

where, F*HK*[ρ(r)] is the universal functional of the electron density due to kinetic energy T*ele*[ρ(r)] and potential interactions V*ele –ele*. The universal functional is called the Hohenberg-Kohn density functional and can be expressed in terms of the interaction from the electron-exchange and correlation and Hatree potential VH due to classical electrostatic interactions [17] as:

$$F\_{HK}[\rho(r)] = T\_{ele}[\rho(r)] + \int \frac{\rho(r')}{|r - r'|} dr' + V\_{xc}[\rho(r)] \tag{6}$$

where, the first term is the kinetic energy of the non-interacting electron system, second term is the Hatree potential *VH* and the third term is the exchangecorrelation potential due to electron-nuclei interactions.

Now the total energy can be written as:

$$E[\rho(r)] = T\_{elc}[\rho(r)] + \left[\frac{\rho(r')}{|r - r'|}dr' + V\_{\text{xc}}[\rho(r)] + V\_{\text{ext}}[\rho(r)]\right] \tag{7}$$

where, the last two terms are external potentials that are split into two categories namely: classical and non-classical energies due to the nuclei V*ext*(r) and exchangecorrelation effects V*xc* (r).

*Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

Eq. (7) can be written as:

$$H\_{KS}\boldsymbol{\upmu}\_{i} = \left(-\frac{\hbar^2}{2m\_e}\nabla^2 + \left[\frac{\rho(r')}{|r-r'|}dr' + V\_{ext}(r) + V\_{xc}(r)\right]\boldsymbol{\upmu}\_{i}(r) = \varepsilon\_i \boldsymbol{\upmu}\_{i}(r) \tag{8}$$

Eq. (8) is called the Kohn-Sham energy equation for non-interacting particles. The Kohn-Sham equation substitutes the many body interacting particle into a single independent particle equation as a functional of the ground state charge density, where ψ*<sup>i</sup>* is the single particle Kohn-Sham wave function. The ground state density can be then be described as:

$$\rho(r) = \sum\_{i=1}^{N} \left| \psi\_i \right|^2 \tag{9}$$

whilst the exchange-correlation potential, *Vxc* (r) is given by the functional derivative of the ground state total energy with respect to the ground state charge density:

$$V\_{\rm xc}(r) = \frac{\delta E\_{\rm xc}[\rho(r)]}{\delta \rho(r)}\tag{10}$$

#### **2.4 Approximations to exchange-correlation functional**

While the DFT is in principle an accurate theory describing ground state interactions in a many-particle system, in practice, approximations are needed to describe the electronic exchange correlation term in the Kohn-Sham Eq. (8)**.** Therefore, the application of DFT depends on the accuracy and reliability of the approximations to the exchange correlation potential, *Vxc*. Consequently, a large number of exchange-correlation functionals, including the LDA, GGA, and other hybrid functional have been developed in order to obtain a numerical solution to the Kohn-Sham equations, as illustrated in **Figure 2**.

#### *2.4.1 Local density approximation*

The simplest approximation to the exchange-correlation term in the Kohn-Sham equations is the Local Density Approximation (LDA) [15]. LDA assumes the density

#### **Figure 2.** *Kohn-sham-DFT equation and its methods of implementation.*

of homogeneous electron gas with a slowly varying electron density gradient, for the exchange correlation functional, expressed as [16].

$$E\_{\rm xc}^{LDA}[\rho] = \int \rho(r) \varepsilon\_{\rm xc}(\rho(r)) dr \tag{11}$$

where ε*xc* is the exchange-correlation energy per electron in a homogenous electron gas. One of the limitations of the LDA approximation is that it does not account for in homogeneities in electron density resulting in overestimation of bonding energies in both molecular and solid systems. Then Local spin density approximation (LSDA) [15, 17, 18] improves Eq. (11) by including the effect of spin density, where the charge density is the total summation of spin up α and down β densities.

$$E\_{\rm XC}^{LSDA}[\rho] = \int \rho(r) \varepsilon\_{\rm xc} (\rho\_a(r), \rho\_\beta(r)) dr \tag{12}$$

#### *2.4.2 Generalized gradient approximation*

The generalized gradient approximation (GGA) functional is an improvement over the LDA, which takes into account the gradient of the electron densityj j ∇*ρ* as well as their magnitude at each point r.

Therefore, the total energy can be expressed as a functional of the gradient of the density as

$$E\_{XC}^{GGA}[\rho] = \int \rho(r) \varepsilon\_{\text{xc}}(\rho(r), |\nabla \rho(r)|) dr \tag{13}$$

Inclusion of the information on the electron density spatial variation in GGA results in greater flexibility in describing real materials. However, GGA is inadequate in describing the properties of strongly correlated material systems, such as transition metals and magnetic systems. In order to improve the accuracy of GGA, an additional term called the "Hubbard U parameter" is used to treat the delocalised and localized orbitals with strong on-site coulomb interactions. The Hubbard U parameter is usually obtained semi empirically, but can also be extracted from ab initio calculations, although both methods do not permit transferability of U across compounds. The GGA + U functional will be used in this study owing to its accuracy and relatively minimal computational cost compared to non-local hybrid functionals such as B3LYP, HSE03 and Sx-LDA.

#### **2.5 Numerical solution of Kohn-sham equation**

To solve the Kohn-Sham single particle equation, the electron wave function ψ*<sup>i</sup>* for the orbitals must be known. The Kohn-Sham equations are solved iteratively within a self-consistent field, where an initial density ρ1(r) is "guessed" to obtain the starting wavefunctions. These variables are then used to build Kohn-Sham Hamiltonian of which an improved density ρ2(r) is obtained. The wavefunctions continues to obtain better approximations to the electron density *ρ*<sup>3</sup> … … *<sup>N</sup>*ð Þ*r* , until self-consistency is reached, as illustrated in **Figure 3** [19].

#### **2.6 Mechanical properties**

Mechanical stability is a measure of material's strength, and is used to characterize the structural stability and deformation of a system under external load [18]. *Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

#### **Figure 3.** *Schematic representation of the SCF method used in DFT.*

Material's mechanical stability is defined in terms of elastic constants *C*ij, Bulk Modulus (*B*), Shear Modulus (*G*), Young's modulus (*E*) and elastic anisotropy (A), from which other properties such as hardness and ductility can be determined. In this study, the calculated elastic constants will be used to initially outline the general mechanical stability of each A15 X3Ru structures based on the Born mechanical stability criteria [19]. In addition, the elastic constants can be used to measure the tensile, shear strength of materials, and provide important information on the bonding characteristics between the adjacent crystal atoms and the long-ranged elastic interaction between various dislocations [20].

#### *2.6.1 Bulk modulus*

Bulk Modulus (*B*) is a measure of the material's resistance to uniform compression. A high value of bulk modulus B indicates that a material resists compression, while a low bulk modulus imply that a material may be easily compressed. For cubic crystal structures, *B* is defined as

$$B = B\_H = \left(\frac{B\_V + B\_R}{2}\right) \tag{14}$$

where

$$B = B\_R = B\_V = \left(\frac{\mathbf{C\_{11}} + 2\mathbf{C\_{12}}}{3}\right) \tag{15}$$

with *BR, BV* and *B=BH* being the Bulk modulus for Reuss, Voigt and Hill approximations [21].

#### *2.6.2 Shear modulus*

The shear modulus (*G*) of a material describes its response to shear stress, and is a measure of a material's stiffness. For cubic structures, *G* can be expressed as:

$$\mathbf{G} = \mathbf{G}\_H = \left(\frac{\mathbf{G}\_V + \mathbf{G}\_R}{2}\right) \tag{16}$$

where the GR and GV are the Reuss and Voigt bounds [22], with

$$\mathbf{G}\_V = \left(\frac{\mathbf{C}\_{11} - \mathbf{C}\_{12} + \mathbf{3C}\_{44}}{5}\right) \tag{17}$$

and

$$\mathbf{G}\_{R} = \left(\frac{(\mathbf{5(C\_{11} - C\_{12})C\_{44})}{(4C\_{44} + \mathbf{3(C\_{11} - C\_{12})})}\right. \tag{18}$$

#### *2.6.3 Young's modulus*

Young's modulus (E) describes the material's strain response to uniaxial stress in the direction of this stress, and it can be written as:

$$E = \left(\frac{\Re B\_H \mathbf{G}\_H}{\Im B\_H + \mathbf{G}\_H}\right) \tag{19}$$

#### *2.6.4 Elastic anisotropy*

Anisotropic behavior is very important in engineering science as well as crystal physics due to its high relation with micro-cracks in materials. In calculating elastic anisotropy, more information about a material will result such as micro-cracks, phase transformation, precipitation and dislocation dynamics [23]. The elastic anisotropy (A) assist to distinguish the micro-cracks in different materials [24–26] and can be understood by calculating the anisotropic factor (*A*) for cubic structures as follows:

$$A = \left(\frac{2C\_{44} + C\_{12}}{C\_{11}}\right) \tag{20}$$

#### *2.6.5 Ductility*

Ductility is the ability of a material to undergo plastic deformation before rapture. It is an important property for material engineering design. An empirical relation linking materials ductility and its elastic moduli (B/G) was proposed by Pugh [27], in which the critical value separating ductile and brittle materials is around 1.75; if B/G > 1.75, the material is ductile, otherwise it is brittle. Another classification rule was given by Frantsevich et al. [28] to distinguish brittleness and ductility by Poisson's ratio:

*Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

$$\nu = \left(\frac{\mathfrak{Z}B\_H - \mathfrak{Z}G\_H}{(\mathfrak{Z}(\mathfrak{Z}B\_H + \mathfrak{G}\_H))}\right) \tag{21}$$

The critical value for Poisson's ratio is 1/3. For brittle materials, the Poisson's ratio is less than 1/3.

#### **2.7 Density**

Density is a vital tool used to characterize light/heavy weight materials. The weight of a material plays an essential role especially in rotating components. Therefore, it will be of interest to evaluate the density of the proposed X3Ru alloys for lightweight (high temperature) structural applications. Density will be calculated from the Eq. (22) below:

$$\rho^{cal} = \left[\frac{M\_W \* N}{Vol \* A\_0}\right] \tag{22}$$

where *Vol* is the volume of the unit cell, *MW* is the average molecular weight of the elements in the unit cell, *N* is the total number of atoms and *A*<sup>0</sup> is the Avogadro's number (6.022 X1023).

#### **2.8 Melting point temperature**

The melting point is the temperature at which a material changes from the solid to the liquid state. In other words, the vapor pressure of the solid and the liquid are equal at its melting point temperature. The melting temperature (*Tm*) of a material depends on its mechanical properties, and it follows a linear relationship with its elastic constants [29–32]. For cubic systems, the melting point temperature is given by:

$$T\_m = 553K + \left(\frac{591K}{Mbar}\right) \* C\_{11}(Mbar) \pm 300K \tag{23}$$

#### **2.9 Computational details**

The Cambridge Serial Total Energy Package (CASTEP) code [33] based on DFT was employed to examine the behavior of cubic A15 X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Ni and Zn) compounds. The calculations were carried out with plane wave pseudo-potentials [34] built within the generalized gradient approximation (GGA) to represent the valence core interactions. In the present calculations, the GGA + Hubbard U [35, 36] model for A15 compounds are used for electron–electron interaction. The wave functions are expanded in the plane waves up to a kinetic energy cutoff of 800 eV, while well converged 15�15�15 k-point sampling by Monkhorst-Pack [37] was used for integration over the Brillouin zone for all the A15 structures. This plane-wave energy cut-off value is convenient for electronic band structures and density of states. The equilibrium lattice parameters have been computed by minimizing the geometry of the crystal using the well-converged k-points allowing the total energy and forces to converge to less than 1 meV/atom and 0.03 eV/Å. For the elastic constants, the stress–strain method was applied on all A15 with cubic symmetry of C11, C12 and C44 elastic constants. The Voigt, Reuss and Hill average has been applied for bulk (B), shear (G) and Young (E)'s modulus [38].

## **3. Results and discussion**

#### **3.1 Structural parameters and heats of formation**

Ruthenium based transition metal alloys belong to the family of A15 X3Z structures which consist of X atoms that occupy six equivalent positions in c-site (0.25, 0, 0.5) and Z atoms that occupy bcc positions (0, 0, 0), as illustrated by **Figure 1** in section 2.1.

**Table 1** indicates the calculated lattice constants, heats of formation and magnetic moments of A15 X3Ru alloys (X = Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Ni, Zn). It can be seen in X3Ru alloys that consist of transition metal atoms in the middle of the 3d series have lower lattice constants compared to those early or late in the series, in consistent with the trends of atomic radii of transition metal atoms across the 3d series. The computed lattice constants for Cr3Ru (5.59 Å) and Ni3Ru (4.84 Å) structures are comparable with those obtained from other theoretical investigations Cr3Ru (4.61 Å, 4.623 Å and 4.62 Å) [14, 39, 40] and Ni3Ru (4.57 Å) [39], confirming the accuracy of our results. The slight deviations between our calculated lattice constants and previous theoretical data can be attributed to the use of different plane wave cutoff energies and k-points grid. In this paper we have applied deeper energy cutoff (800 eV) and k-points grid of 15 x 15 x 15 compared to previous calculations. The heat of formation *ΔHf* was calculated using Eq. (24) for all the structures as:

$$
\Delta H\_f = \left[ \frac{E(X\_3Ru) - aE\_{Ru}^{\p} - bE\_X^{\p}}{a+b} \right] \tag{24}
$$

Where E(X3Ru) is the total energy of the system,*E<sup>g</sup> Ru* and *Eg <sup>X</sup>* are the energies of each individual metal species in their ground states, a and b are the number of atoms for individual metals and X = Sc, Ti, V, Cr, Mn, Fe, Co, Cu, Ni and Zn. A negative value ΔHf indicates stability while the positive value of ΔHf shows instability. It was found that Mn3Ru have negative heats of formation, indicating that the alloy is thermodynamic stable. This result indicates a possibility of synthesizing Mn3Ru experimentally due to the existence of the A15 Cr3Ru phase [41]. Other


#### **Table 1.**

*Calculated lattice constants of X3Ru (X = Cr, Sc, Ti, V, Mn, Fe, Cu, Ni, Co and Zn) unit cell as well as the heats of formation, magnetic moment, density and the X-Ru bond-length of A15 structures.*

#### *Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

structures such as Sc3Ru, Ti3Ru, V3Ru, Cr3Ru, Fe3Ru, Co3Ru, Ni3Ru, Cu3Ru and Zn3Ru have positive heats of formation indicating that they are thermodynamically unstable, hence synthesis may be difficult. However, it has been suggested previously [14, 42, 43] that doping lowers the heats of formation in some materials, which may make it possible to synthesize some of these structures. Importantly, we find a strong direct correlation between the heats of formation of the X3Ru alloys and their magnetic moments, with higher magnetic moments corresponding to more stable alloys, as illustrated in **Figure 4**. This can be attributed to strong metallic bonding in transition metals arising from the delocalized electrons.

Similarly the larger atomic radii difference between the Ru atom and transition metal atoms results in crystal strain which has a positive impact on the overall magnetic moment, as summarized in **Table 2.** The density of an alloy plays an important role in determining its use in lightweight applications, such as the aerospace industry. We have therefore, evaluated the density of the proposed X3Ru alloys for lightweight (high temperature) structural applications. From **Table 1**, we note that X3Ru alloys with small lattice constants have higher densities and the results are consistent with Eq. (22). However, the density of the most stable alloy Mn3Ru (5.72 *g=cm*3), is slightly lower than that of L12 Ni3Al (6.14 *g=cm*3) [44], which is commonly used in the aerospace industry. Therefore, the application of these alloys with high densities may be limited. Whereas, Mn3Ru may be a possible candidate for aerospace application due to lower density.

#### **3.2 Electronic and magnetic properties**

The calculated magnetic moments and the stability of intermetallic alloys can be rationalized from their electronic properties [45]. **Figure 5** presents the projected spin-polarized density of states (PDOS) of X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) alloys. It can be seen that the spin up and spin down bands are symmetric in Sc, Ti, V, Cr, Cu and Zn systems, with no net spin polarization. This symmetric balance of the spin up and spin down bands leads to a cancelation of the magnetic moment associated with electronic spin, thus explaining the zero (or negligible) calculated magnetic moments (**Table 1**). On the other hand, the density of states in Mn, Fe, Co and Ni are spin-polarized, which explains the origin of the non-zero calculated magnetic moments in these systems. The predicted magnetic

**Figure 4.**

*A comparison of the calculated heats of formation (eV/atom) and magnetic moment (μB/atom) of X3Ru (X = Sc,Ti, V, Cr, Mn, Fe, Ni, Co, Cu and Zn) alloys.*


**Table 2.**

*The atomic radii, atomic radii difference, magnetic moments, electronic configuration and number of paired electrons of 3d transition metal from Sc-Zn bonded with central Ru atom.*

moments in these compounds could lead to novel applications such as spintronics and spin injections.

The total density of states (DOS) can be used to investigate the atomic bonding character of Ru-based alloys [45]. It is clear that the bonding character mainly arises from the hybridization of X-*d* and Ru-*d* below the Fermi energy (4 to -1 eV). Around the Fermi energy, from the spin up and down channels, the most visible feature is the presence of a valley known as the pseudo-gap, which indicates covalent bonding [46, 47] in these compounds. The pseudo-gap exist due to strong hybridization in X-*d* and Ru-*d* states and separates the bonding states from the antibonding states. The phase stability of intermetallic compounds is dependent on the location and magnitude of the DOS at the Fermi energy N(Ef) [48–50], with lower N(Ef) corresponding to a more stable phase [51]. The electronic properties can be related with the heats of formation, Mn3Ru structure have less density of states at the Fermi and therefore, it is most stable. This result explains the calculated heats of formation results of Mn3Ru discussed in section 3.1 above, where Mn3Ru is thermodynamically stable and this attributes to its lowest heats of formation.

#### **3.3 Elastic properties**

Elastic constants are parameters that express the mechanical behavior of the materials within the stress range that the materials exhibit elastic behavior. In science and technology, they are essential physical quantities to determine the mechanical properties. The elastic constants Cijs of A15 X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) compounds are shown in **Table 3**. For the cubic X3Ru compounds, the condition of Born mechanical stability [52] follows the equations:

$$\rm C\_{44} > 0, C\_{12} > 0, C\_{11} - C\_{12} > 0 \tag{25}$$

The elastic constants of Cr3Ru satisfies the above formulae, whilst other structures do not and this attributes to *C44 < 0* and *C12 > 0*. For these elastic constants, the bigger the elastic constant *C11* is, the stronger the linear compression resistance *Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*

**Figure 5.**

*Partial density of states of binary X3Ru (X = Sc,Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn). The Fermi energy is taken as zero energy, indicated with the vertical dotted line.*

along the X-axis direction [53]. Ti3Ru has the strongest resistance to the compressibility among all the structures, due to the largest C11 (1061 GPa) value [54]. The elastic constant *C44* reflects the degree of shear resistance in the (100) plane and affects the hardness of solid materials indirectly [55]. Fe3Ru has a larger *C44* (137 GPa) suggesting a stronger ability to resist shear distortion in (100) plane. The elastic modulus such as (bulk modulus *B*, shear modulus *G* and Young's modulus *E*), Poisson's ratio ʋ, melting temperature Tm and anisotropic factor are used to

determine the mechanical properties. The elastic moduli of A15 X3Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn) structures are shown in **Table 3**. Bulk modulus reflects the incompressibility of solid materials. The stronger incompressibility in solid material correspond to larger bulk modulus. Under constant volume conditions, the shear modulus denotes the deformation resistance of solid materials, whilst the Young's modulus offers a measure of the stiffness of a solid.

The Fe3Ru structure indicates larger bulk (1060 GPa), shear (209 GPa) and Young's modulus (672 GPa) as shown in **Table 3** and therefore shows stronger resistance to compressibility, shear deformation and stiffness. It is noted that the X3Ru (X = Sc, V, Co, Ni, Cu and Zn) structure have negative shear modulus and Young's modulus. Furthermore, Co indicates the smallest negative shear modulus (991.0 GPa) and Young's modulus (2.2\*10<sup>5</sup> GPa) indicating instability associated with phase change. This instability is also observed in ferro-elastic phase transformation [56]. The negative elastic modulus is due to Landau theory [57] when two local minima form in a strain energy function. Besides, solids with negative elastic modulus can be stabilized with sufficient constraint. The ratio of B/ G is larger than 1.75 and predicts ductile behavior in a solid material [27]. Otherwise, it will exhibit the brittle behavior. Similar trend is expected in Poisson's ratio [28], which refers to ductile compounds normally with a large (ʋ > 0.3) and lastly, positive Cauchy pressure (*C11-C12*) [58, 59] shows that the given material is expected to be ductile whilst negative Cauchy pressure indicates brittleness as shown in **Table 3**.

The X3Ru (X = Ti, Mn, Cr and Fe) structures are ductile due to B/G values greater than 1.75 whilst, Sc3Ru, Co3Ru, Ni3Ru, Cu3Ru and Zn3Ru are brittle. It is noted that Cr3Ru (19.6) is more ductile indicated in **Table 3** and possess high fracture toughness. This is in agreement with the Poisson's ratio and Cauchy pressure results discussed. A positive Cauchy pressure in Mn3Ru (52 GPa) and Cr3Ru (241 GPa) is observed with B/G ratio of 16 and 19.6 and Poisson's ratio of (0.5 and 0.5) indicating ductile characteristics. Similar trend of results are shown in A15 XNb3 (X = Al, Ge, Si, Sn, Pt and Ir) studies, and ductility can be attributed by positive Cauchy pressure [60, 61].


**Table 3.**

*Calculated elastic constants (Cij), Moduli (average B, G, EVRH), Poisson's ratio (υ), Shear moduli (C), the ratio of Bulk to Shear (B/G), Cauchy pressure (C12-C44), Melting temperature (Tm) and elastic anisotropy (A). All elastic constants and Moduli are in GPa.*

*Computational Study of A15 Ru-Based Alloys for High-Temperature Structural Applications DOI: http://dx.doi.org/10.5772/intechopen.96578*
