First-Principles Calculation and Kink-Dislocation Dynamics Simulation on Dislocation Plasticity in TiZr-Based Concentrated Solid-Solution Alloys

Abstract

The dislocation plasticity of TiZr-based hexagonal close-packed (HCP) concentrated solid-solution alloys (CSAs) is investigated using a multiscale simulation approach combining the first-principles calculation and Frenkel–Kontonova kink-dislocation model. The first-principles calculation reveals that dislocation-mediated slip is significantly enhanced by the additions of Y and Sc in TiZrHf CSAs. The dislocation kinetics is simulated using the kink-dislocation model at mesoscopic scales, and the predicted mechanical strength of CSA is found to be consistent with experimental results. In addition to predicting the mechanical properties of CSAs accurately, the multiscale simulation approach elucidates the deformation mechanisms in CSAs at atomic scales, suggesting that the approach is robust in modeling the dislocation plasticity of CSAs.

1. Introduction

Single-phase concentrated solid-solution alloys (CSAs) are composed of two or more principal elements, where atoms are randomly situated on specified lattice sites. The CSAs with four or more constituent alloy elements, so-called high-entropy alloys (HEAs), have attracted much attention because of their exceptional mechanical properties [1,2,3]. It has been found that HEAs with body-centered cubic (BCC) structures usually exhibit high mechanical strength but possess limited plasticity, while those with face-centered cubic (FCC) structures have relatively low strength but usually show fine plasticity [4,5,6,7,8,9]. Thus, the development of CSAs with high strength and enhanced ductility has been widely explored for scientific interests and engineering applications. Recently, a strategy of metastable transformation-induced plasticity was introduced to CSAs. The obtained alloys consisted of multiple phases, which exhibited improved strength and ductility.

Although most CSAs reported to date have focused on those with FCC, BCC, or FCC-BCC mixed structure, there is limited information on CSAs comprising solid solutions with hexagonal close-packed (HCP) structures. The reason is probably that HCP structures are stable at low temperatures, while they exhibit a tendency to transform to either BCC or FCC structures at elevated temperatures. It was suggested [9] that the difficulty in forming HCP CSAs is owing to the fact that most elements found in the periodic table prefer a BCC or FCC structure in nature; although some elements possess a stable HCP structure below their melting points, such as Cd, Mg, Os, Re, Ru, Tc, and Zn, they can transform to other structures when the temperature increases beyond the melting points [9]. For instance, Be, Sc, Ti, Zr, and Hf can possess BCC structure, and Co can possess FCC structure at high temperatures [9]. Nonetheless, with the help of rapidly developed materials processing technology and computational materials sciences, several CSAs with HCP structures have been proposed [9,10,11,12,13,14,15]. Chen et al. developed an amorphous structure with HCP elements in equimolar BeCoMgTi alloy by means of mechanical alloying [10]. However, the obtained samples did not show homogenous HCP structures. CSAs with HCP structures were also reported in those comprising Co, Hf, Ti, and Zr constituent alloy elements. Youssef et al. reported that Al₂₀Li₂₀Mg₁₀Sc₂₀Ti₃₀ (at.%) high-entropy alloy (HEA) powders with a stable HCP structure [11] were obtained through a mechanical synthesis route. Particularly, nearly single-phase HCP structures in CSAs with equiatomic ratios that consisted of rare earth elements were confirmed, such as YGdTbDyLu, GdTbDyTmLu, and HoDyYGdTb [12,13,14]. Based on a systematic study [15] on binary equilibrium diagrams of HCP alloys, it was obtained that the CoOsReRu with equilibrium compositions may also form HCP solid solutions. In particular, TiZr-based HCP CSAs developed recently are of significant interest. Huang et al. studied the effect of Sc and Y additions on the microstructure and properties of HCP-structured TiZrHf-based HEAs [16]. Rogal et al. investigated the microstructures and mechanical properties of a Hf₂₅Sc₂₅Ti₂₅Zr₂₅ equiatomic alloy with an HCP structure using experimental and theoretical methods [17].

It is well known that in conventional alloys without chemical disorder, the mechanical properties are closely related to dislocation plasticity. However, the knowledge of the influences of chemical disorder on dislocation activities in CSAs remains incomplete, especially in those with HCP structures. Experimental observations have revealed that the mechanical strength of CSAs is closely related to the deformation mechanisms different from dislocation glides in conventional metals and alloys [18,19,20]. Basically, the deformation mechanism and dislocation activity in crystalline materials are highly dependent on their stacking-fault energies (SFEs) [21,22,23], which characterize the energy barriers that significantly affect the mobility of dislocations at atomic scales. Thus, the generalized stacking fault energy (GSFE) curve determined by the first-principles calculation has been utilized to investigate the deformation mechanisms in CSAs [24,25,26]. However, the understanding of the dislocation plasticity in CSAs, as reflected by the dislocation dynamics at mesoscopic scales, is still rather limited.

In this work, taking TiZr-based HCP CSAs as examples, the dislocation plasticity in CSAs is investigated using multiscale simulation. Firstly, first-principles calculations are employed to determine the GSFE curve, elucidating the phase stability and deformation mechanisms of CSAs. Consequently, energy barriers for the formations of stacking faults, twin faults, and FCC faults are determined from the generalized planar fault energy (GPFE) curve. Compared to other methods of the first-principles calculation, the advantage of GPFE curves is the analysis of deformation mechanisms, including dislocation-mediated slips, deformation twinning, and HCP-to-FCC transformation in the same model system, whose energetics can be explicitly compared. Based on the obtained results, the effects of alloying elements on the deformation mechanisms of TiZr-based HCP CSAs are discussed in detail. Secondly, the kink-dislocation model, with its parameters determined by the first-principles calculations, is employed to simulate dislocation plasticity at mesoscopic scales.

The multiscale simulation is intended to solve the aforementioned issues related to the formation of HCP HEAs as well as their unique mechanical properties. First, the stability of TiZr-based HCP CSAs is analyzed against alloying elements of Y, Sc, and Hf, which can be quantitatively measured based on free energy, lattice distortion, and atomic size mismatch. Second, based on the deformation mechanisms as revealed by the GPFE curves, the differences in mechanical properties of HCP CSAs as measured from experiments are explained. Third, the mechanical properties of CSAs are quantitatively predicted by the kink-dislocation dynamics. Therefore, for the first time, the multiscale simulation approach can elucidate the deformation mechanisms in CSAs in addition to predicting their mechanical properties well.

2. Methodology

2.1. First-Principles Calculation

The HCP CSAs are studied using an orthorhombic supercell with lattice vectors aligned in the crystallographic directions of [−12−10], [10−10], and [0001]. The special quasi-random structure (SQS) technique [27] implemented in the Alloy Theoretic Automated Toolkit (ATAT) [28] is applied to simulate the chemical disorder of CSAs, and the supercells of Ti₁₂₀Zr₁₂₀, Ti₈₀Zr₈₀Hf₈₀, Ti₆₀Zr₆₀Hf₆₀Sc₆₀, Ti₆₀Zr₆₀Hf₆₀Y_60, and Ti₄₈Zr₄₈Hf₄₈Sc₄₈Y₄₈, as shown in Figure 1, are used for the first-principles calculations. The stacking sequence of the (0001) planes in the supercells is assigned as BABABABABABABABABABA. The relaxation of supercells is performed by fixing the orthorhombic lattice structure, while the lengths of lattice vectors are varied to minimize the interatomic Hellmann–Feynman forces. First-principles computations are performed using the Vienna ab initio Simulation Package (VASP) codes [29,30,31,32], and generalized gradient approximation (GGA) is used for exchange-correlation functionals [33,34]. The plane-wave energy cutoff is set to 350 eV [32], and a gamma-centered k-point mesh of 2 × 3 × 1 generated using the Monkhorst–Pack algorithm is employed in the calculations [35]. The tolerances of energy and force are set to 1.0 × 10⁻⁴ eV/atom and −2.0 × 10⁻² eV/Å, respectively.

The slips of partial dislocations can lead to different deformation mechanisms in the alloys, and two types of deformation mechanisms, i.e., deformation twinning and HCP-to-FCC transformation, are investigated in this study. The deformation twinning occurs by passing the partials on adjacent slip planes, and consequently passing the partials on every second slip plane can result in the HCP-to-FCC transformation [36]. In order to understand the deformation processes in TiZr-based HCP CSAs, the GPFE curves or the energetics of passing a leading partial on the slip plane, a trailing partial on the nearest-neighbor slip plane, and a trailing partial on the next-nearest-neighbor slip plane are investigated. The planar fault is introduced by shifting one part of the supercell with respect to the other part along a specific direction. As an example, the supercell of Ti₆₀Zr₆₀Hf₆₀Sc₆₀ (Figure 1c) is selected to demonstrate the deformation process. We firstly move atoms in the 10th–20th layers of the supercell along [10−10] direction with a distance of the Burgers vector of the partial dislocation b_p = a/3 [10−10], and the stacking sequence was changed into BABABABABCACACACACAC (Figure 1(c-1)), resulting in a system with two I2 intrinsic stacking faults. Based on the obtained supercell with I2 faults, moving atoms in the 11th–19th or the 12th–18th layers along [01−10] or [−1100] directions can lead to the supercell with twin fault or FCC fault, where the stacking sequence is denoted as BABABABABCBABABABABC (Figure 1(c-2)) or BABABABABCABCBCBCBAC (Figure 1(c-3)), respectively.

2.2. Simulation of Dislocation Kinetics by a Kink-Dislocation Model

The simulation of dislocation dynamics at a slip plane is carried out using a two-dimensional (2D) Frenkel–Kontorova kink-dislocation model [37], which was first proposed by Lomdahl and Srolovitz [38]. As shown in Figure 2, the displacements of atoms located at the 2D hexagonal lattice sites (i, j) are denoted as u_ij = (u_i,j, v_i,j)a, which describes the positions of a dislocation line lying on the (0001) slip plane. The equations of motion of the dislocation line can be written as [39]

$$\begin{array}{cc}\hfill \frac{{\partial }^2{u}_{i,j}}{\partial t^2}& =u_{i,j+1}+u_{i,j-1}+u_{i-1,j+1}+u_{i-1,j-1}+u_{i+1,j}+u_{i-1,j}-6u_{i,j}\hfill \\ \hfill & +\frac{u_{i,j}-u_{i,j+1}-1/2}{{\Delta }_1}+\frac{u_{i,j}-u_{i,j-1}-1/2}{{\Delta }_2}+\frac{u_{i,j}-u_{i-1,j+1}+1/2}{{\Delta }_3}\hfill \\ \hfill & +\frac{u_{i,j}-u_{i-1,j-1}+1/2}{{\Delta }_4}+\frac{u_{i,j}-u_{i+1,j}-1}{{\Delta }_5}+\frac{u_{i,j}-u_{i-1,j}+1}{{\Delta }_6}-\frac{\partial V_P}{\partial x}-f_x\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \frac{{\partial }^2{v}_{i,j}}{\partial t^2}& =v_{i,j+1}+v_{i,j-1}+v_{i-1,j+1}+v_{i-1,j-1}+v_{i,+1j}+v_{i-1,j}-6v_{i,j}\hfill \\ \hfill & +\frac{v_{i,j}-v_{i,j+1}-\sqrt{3}/2}{{\Delta }_1}+\frac{v_{i,j}-v_{i,j-1}-\sqrt{3}/2}{{\Delta }_2}+\frac{v_{i,j}-v_{i-1,j+1}+\sqrt{3}/2}{{\Delta }_3}\hfill \\ \hfill & +\frac{v_{i,j}-v_{i-1,j-1}+\sqrt{3}/2}{{\Delta }_4}+\frac{v_{i,j}-v_{i+1,j}}{{\Delta }_5}+\frac{v_{i,j}-v_{i-1,j}}{{\Delta }_6}-\frac{\partial V_P}{\partial y}-f_y\hfill \end{array}$$

(2)

where f = (f_x, f_y) is the force applied on the dislocation line and V_p is the Peierls potential, and

$$\begin{array}{c}{V}_p=2V_0\{3-\cos (\frac{4\pi }{\sqrt{3}}{u}_{i,j})-\cos [\frac{2\pi }{\sqrt{3}}(u_{i,j}+\sqrt{3}{v}_{i,j})]-\cos [\frac{2\pi }{\sqrt{3}}(u_{i,j}-\sqrt{3}{v}_{i,j})]\}/Gb,\hfill \\ {\Delta }_1=\sqrt{{(u_{i,j}-u_{i,j+1}-\frac{1}{2})}^2+{(v_{i,j}-v_{i,j+1}-\frac{\sqrt{3}}{2})}^2},{\Delta }_2=\sqrt{{(u_{i,j}-u_{i,j-1}-\frac{1}{2})}^2+{(v_{i,j}-v_{i,j-1}-\frac{\sqrt{3}}{2})}^2},\hfill \\ {\Delta }_3=\sqrt{{(u_{i,j}-u_{i-1,j+1}+\frac{1}{2})}^2+{(v_{i,j}-v_{i-1,j+1}-\frac{\sqrt{3}}{2})}^2},{\Delta }_4=\sqrt{{(u_{i,j}-u_{i-1,j-1}+\frac{1}{2})}^2+{(v_{i,j}-v_{i-1,j-1}+\frac{\sqrt{3}}{2})}^2},\hfill \\ {\Delta }_5=\sqrt{{(u_{i,j}-u_{i+1,j}-1)}^2+{(v_{i,j}-v_{i+1,j})}^2},{\Delta }_6=\sqrt{{(u_{i,j}-u_{i-1,j}+1)}^2+{(v_{i,j}-v_{i-1,j})}^2},\hfill \end{array}$$

(3)

where b is the Burgers vector, G is the shear modulus, and V₀ is the energy barrier for the activation of dislocation kinks.

3. Results and Discussion

3.1. Lattice Parameters and Lattice Distortion of HCP CSAs

Based on the first-principles calculations, the obtained lattice parameters of CSAs are summarized in

Table 1. Lattice parameters a and c of CSAs obtained from the first-principles calculations (this work) and experimental studies [16,17].

Alloy Systems	This Work			Experimental
	a	c	c/a	a	c
TiZr	3.092	4.875	1.577	-	-
TiZrHf	3.138	4.917	1.567	3.162 a	4.997 a
TiZrHfSc	3.166	4.958	1.566	3.184 b 3.198 a	5.013 b 5.046 a
TiZrHfY	3.251	5.091	1.566	-	-
TiZrHfScY	3.254	5.086	1.563	-	-

^a Ref [16], ^b Ref [17].

in comparison with those measured in experiments. The calculated lattice parameters of TiZrHf and TiZrHfSc agree well with experimental results, and the errors are less than 1%, demonstrating that the supercells can be used for the first-principles investigations on the deformation mechanisms and mechanical properties of CSAs. Moreover, the volume of a CSA supercell increases when more alloying elements are introduced, as shown in Figure 3a. In the TiZr-based HCP CSAs, the addition of Y leads to a more significant enlargement in the volume of the CSA than that caused by adding Sc or Hf, while the effect of Sc addition on the volume enlargement is not significant as compared with that of Hf or Y additions. The calculated free energy indicates that the HCP structure of TiZrHf is more stable than that of the other TiZr-based HCP CSAs, while the free energy of TiZrHfScY is the highest among the five CSAs studied.

The chemical disorder in CSAs inevitably results in the lattice distortion of the system at equilibrium, which is considered to be responsible for the high mechanical strength in disordered alloys and is calculated with the following equation [40]:

$$\Delta d=\frac{1}{N}{\displaystyle \sum }_i^N\sqrt{{\left(x_i-x_i^{\prime }\right)}^2+{\left(y_i-y_i^{\prime }\right)}^2+{\left(z_i-z_i^{\prime }\right)}^2} ,$$

(4)

where (x_i, y_i, z_i) are the coordinates of atom i at the ideal lattice site, (x′_i, y′_i, z′_i) are the coordinates of atom i at the distorted lattice site, and N is the number of atoms. The lattice distortion was found to be closely related to the atomic size mismatch δ described as follows [9]:

$$\mathsf{\delta }=100\sqrt{{\displaystyle \sum }_{i=1}^n{c}_i{\left(1-\frac{r_i}{\overline{r}}\right)}^2} ,$$

(5)

where $\overline{r}={{\displaystyle \sum }}_{i=1}^n{c}_i{r}_i$ and c_i and r_i are atomic percentage and atomic radius of an individual constituent alloy element in the system, respectively. The calculated lattice distortion and atomic size mismatch of CSAs are shown in Figure 3b, and the lattice parameters, atomic radius, melting point, and valance electrons of the constituent alloy elements in the CSAs are listed in

Table 2. Lattice parameters and physical properties [41,42] of HCP constituent alloy elements of CSAs studied in this work.

Elements	Lattice Constants (Å)	c/a	Radius (Å)	Tm (℃)	Valence Configurations
Ti	a = 2.9509 c = 4.6826	1.59	1.462	1670	3p63d24s2
Zr	a = 3.232 c = 5.147	1.59	1.603	1850	4s24p64d35s1
Hf	a = 3.193 c = 5.052	1.57	1.578	2150	5p65d26s2
Sc	a = 3.2899 c = 5.2529	1.59	1.641	1539	3s23p63d24s1
Y	a = 3.6451 c = 5.7305	1.57	1.802	1526	4s24p64d15s2

. It is found that the lattice distortion and atomic size mismatch are closely related to each other, i.e., the lattice distortion increases with increasing atomic size mismatch. In particular, among the CSAs studied, the lowest lattice distortion and atomic size mismatch are determined to be for TiZrHfSc and TiZrHf, respectively. Meanwhile, the addition of Hf in TiZr alloy tends to reduce the lattice distortion and atomic size mismatch. When Sc or Y is introduced to TiZrHf, Sc tends to reduce the lattice distortion, but it enhances the atomic size mismatch. On the contrary, Y addition is found to enhance both parameters. However, when Sc and Y are added to the TiZrHf CSA, significant increases in lattice distortion and atomic size mismatch can be observed. It is thus concluded that because of its low lattice distortion, TiZrHfSc can possess a perfect HCP structure in comparison with other CSAs.

3.2. Deformation Mechanisms in HCP CSAs

The stacking-fault energies γ can be determined as γ(g) = [E(g) − E₀]/2A, where g is the displacement along a specific direction in the generation of stacking faults, E(g) and E₀ represent the total energy of the supercell with and without the stacking faults, respectively, and A is the stacking-fault area. Figure 4 presents the calculated GPFE curves of the CSAs studied. As marked in Figure 4e, the stacking-fault energy (γ_SF), twinning-fault energy (γ_TF), and FCC-fault energy (γ_FCCF) are the local minima at the ends of the three GPFE curves, respectively. At the same time, the three peak values of the curves are defined as unstable stacking-fault energy (γ_USF), unstable twinning-fault energy (γ_UTF), and unstable FCC-fault energy (γ_UFCCF), respectively.

Table 3. The calculated unstable stacking-fault energy (γ_USF), stacking-fault energy (γ_SF), unstable twinning-fault energy (γ_UT), and unstable FCC-fault energy (γ_UFCC) and energy barriers for the formation of twin fault and FCC fault with pre-existing stacking faults for the TiZr-based CSAs.

Alloy Systems	γUSF (mJ/m2)	γSF (mJ/m2)	γUT (mJ/m2)	γUFCC (mJ/m2)	γUT − γSF (mJ/m2)	γUFCC − γSF (mJ/m2)
TiZr	235	216	384	396	168	180
TiZrHf	357	331	513	599	182	268
TiZrHfSc	374	297	500	583	203	286
TiZrHfY	314	309	514	580	205	271
TiZrHfScY	271	199	379	521	180	322

lists those values with the energy barriers for the formation of twin faults (∆U_T = γ_UTF − γ_SF) and FCC faults (∆U_F = γ_UFCCF − γ_SF) with the pre-existing stacking faults, which can be used to clarify the competition between deformation twinning and HCP-to-FCC transformation in the HCP systems.

It is found that the addition of Hf in TiZr CSA tends to enhance γ_USF and γ_SF; the additions of Sc and Y in TiZrHf CSA tend to reduce γ_SF, while Sc or Y addition can enhance or reduce γ_USF, respectively. However, both γ_USF and γ_SF for TiZrHfSc or TiZrHfY are smaller than those for TiZrHfScY. Comparing γ_USF and γ_SF for TiZrHfSc, TiZrHfY, and TiZrHfScY, which are usually defined as HEAs, it is found that γ_SF decreases with the increasing number of constituent alloy elements. Since γ_USF is considered to be the energy barrier that a partial dislocation must overcome to move in an HCP alloy, the reduced γ_USF can facilitate the dislocation-mediated slips. It is thus concluded that adding Hf into TiZr CSA or adding Sc into TiZrHf CSA tends to weaken the dislocation-mediated slips. On the contrary, the addition of Y in TiZrHf CSA can promote dislocation-mediated slips, and adding Y and Sc in TiZrHf CSA can significantly improve the activities of dislocation-mediated slips.

For the CSAs studied, the energy barrier for the formation of FCC faults is larger than that of twin faults, implying that the deformation mechanisms of those HCP CSAs tend to be deformation twinning instead of HCP-to-FCC transformation at 0 K. For the TiZr CSA, the difference in energy barriers for the formation of twin faults and FCC faults is increased significantly with the addition of Hf, suggesting that the deformation twinning in the CSA is enhanced by adding Hf. Although the energy barriers for the formation of twin faults and FCC faults are increased by adding Sc or Y into TiZrHf, the difference in these energy barriers seems unchanged for TiZrHfSc HEA, while it is reduced for TiZrHfSc HEA. It is thus suggested that the deformation twinning and HCP-to-FCC transformation are hindered by the addition of Sc or Y in TiZrHf CSA, while in the resulting HEAs, the HCP-to-FCC transformation in TiZrHfY is easier to occur than that in TiZrHfSc. In particular, the transformation is more difficult to occur in TiZrHfScY HEA than in other CSAs studied since deformation twinning is more energetically favorable as compared to the HCP-to-FCC transformation in TiZrHfScY HEAs.

Based on the calculated GPFE curves, the relationship between the dislocation activities and the ductility ε_p of CSAs can be explained in comparison with experimental results listed in

Table 4. Comparisons on the shear modulus G determined by first-principles calculation, shear strength τ_y predicted by kink-dislocation model, ductility ε_p and yield strength σ_y obtained from experimental measurements [16], and dislocation shear strength τ_dis = σ_y/3 for HCP CSAs.

	TiZr	TiZrHf	TiZrHfSc	TiZrHfY	TiZrHfScY
G (GPa)	70.6	79.2	85.5	71.4	78.8
Shear strength τy (MPa)	311.5 ± 1.4	280.4 ± 1.0	342.3 ± 1.5	178.8 ± 1.0	275.2 ± 1.2
Yield strength σy (MPa)	-	773	1001	554	793
τdis (MPa)	-	257.6	333.6	184.6	264.3
εp (%)	-	17.0	21.9	17.7	15.7

. When Y is added to TiZrHf, the difference in the energy barriers for the formation of twin faults and FCC faults is decreased from 86 mJ/m² to 66 mJ/m². Hence, the dislocation activities in TiZrHfY can be more intense than those in TiZrHf, resulting in an increase in the ductility of TiZrHfY compared with that of TiZrHf. When Y is added to TiZrHfSc, the difference in the energy barriers for the formation of twin faults and FCC faults is increased from 83 mJ/m² to 142 mJ/m², leading to an obvious decrease in ductility of TiZrHfScY compared with that of TiZrHfSc.

Generally, the mechanical properties of alloys can be related to the characteristics of metallic bonding in the alloys, which are determined by the electronic structures. Herein, the redistribution of charge density Δρ(r) = ρ(r) − Σρ₀(X, r) of CSAs is characterized using first-principles calculations, where ρ(r) is the charge density of CSA at position r; ρ₀(X, r) is the charge density of atoms of a constituent alloy element X, which is isolated from those of other constituent alloy elements. The term Δρ(r) describes the charge redistribution induced by the alloying elements in the CSA and directly reflects the bonding nature between the constituent alloy elements. In order to have a better understanding of the redistribution of electrons in the CSAs under plastic deformation, those at the atom layers close to the (0001) glide plane, which are related to the formation of twins or faults, are shown in Figure 5.

Based on Figure 5, the reasons why the addition of Y in CSAs can reduce the yield strength σ_y of the resulting TiZrHfY and TiZrHfScY HEAs (as listed in Table 4) can be explained as follows: As shown in Figure 5b,d and Figure 5c,e, respectively, the charge-density redistributions in the TiZrHfY and TiZrHfScY HEAs indicate that the redistributions of electrons around Y atoms exhibit strong localization since large values of Δρ(r) can be associated with the metallic bonding between Y and other atoms. As a result, the addition of Y to the TiZrHf or TiZrHfSc CSAs thus disturbs the evenly distributed metallic bonding significantly, as shown in Figure 5d,e, respectively. Consequently, the strength of bonding or interaction between adjacent (0001) slip planes can be significantly weakened by the addition of Y in the resulting HEAs, leading to significant reductions in the yield strengths of TiZrHfY and TiZrHfScY HEAs in comparison with those of TiZrHf and TiZrHfSc, respectively.

3.3. Mesoscopic Studies on Dislocation Plasticity of CSAs

In order to have a better understanding of the mechanical properties of CSAs, the kinetics of an edge dislocation with the Burgers vector b = a/3[−12–10] is studied by solving Equations (1) and (2), where the positions of the dislocation line u_ij(t) = (u_i,j(t), v_i,j(t)) can be determined at time t at the discrete lattice sites labeled with 1 ≤ j ≤ 600 and 1 ≤ j ≤ 1000.

The parameters of Equations (1) and (2) for the TiZr, TiZrHf, TiZrHfSc, TiZrHfY, and TiZrHfScY CSA systems are determined by the aforementioned first-principles calculations at T = 0 K. The energy barrier for the activation of dislocation kinks, V₀, is approximated as the unstable stacking-fault energy γ_USF. The shear modulus G of CSA is determined by calculating the elastic deformation with Hooke’s law for the CSA supercell shown in Figure 1, which is listed in Table 4. Since the first-principles calculations on the parameters are performed on the CSAs at T = 0 K, the calculated values of V₀ and G may be over-estimated for Equations (1) and (2), which is employed to predict the mechanical properties of CSAs at a finite temperature. However, the effects of temperature on the predicted mechanical properties are not significant, resulting from the consistency between the lattice parameters predicted at T = 0 K and those experimental values measured at a finite temperature, as listed in Table 1.

The Runge–Kutta method is used to solve Equations (1) and (2), and periodic boundary conditions are applied at the x- and y-directions. The activation of dislocation kinks and the motion of dislocation are simulated by acting forces of f = (−F/2, 3^1/2F/2) on the dislocation line, which measure the applied shear stress on the slip plane. When the force F is increased from 0 with an increment of 0.01 N/m, kinks are generated while parts of the dislocation line may remain static or the kinks are at rest. It is not until F reaches a critical value F_cr that the whole dislocation line moves along f with a distance greater than b_p = 3^1/2a/3, or the kinks have moved across the slip plane. Therefore, by solving Equation (1), F_cr can be determined for the TiZr, TiZrHf, TiZrHfSc, TiZrHfY, and TiZrHfScY CSA systems, which can measure the shear strengths τ_y of the CSAs, as listed in Table 4.

The available mechanical properties of the CSAs obtained from experimental measurements [16] are listed in Table 4, where the dislocation shear strength τ_dis can be calculated from the experimental value of yield strength σ_y using the relation τ_dis = σ_y/3 in polycrystalline materials. Remarkably, the shear strength τ_y of the CSAs determined by the kink-dislocation model is consistent with that (τ_dis) obtained from experiments, as listed in Table 4. The results suggest that the mesoscopic kink-dislocation model, with its parameter determined using the first-principles calculation, is effective in quantitatively predicting the mechanical properties of CSAs. Thus, the multiscale simulation approach combining the first-principles calculation and kink-dislocation model can be very robust in the simulation of the dislocation plasticity of CSAs.

4. Conclusions

In summary, a multiscale simulation approach combining first-principles calculations and the Frenkel–Kontonova kink-dislocation model is employed to investigate the dislocation plasticity of TiZr-based HCP CSAs. For the first time, the GPFE curves obtained using first-principles calculations are utilized to determine the energy barriers for the formations of stacking faults, twin faults, and FCC faults in the CSAs. The calculated GPFE curves indicate that, for HEAs formed by adding HCP elements in TiZrHf CSA, the dislocation-mediated slip is promoted or reduced by the addition of Y or Sc, respectively, while it is significantly enhanced by simultaneously adding Y and Sc. Furthermore, the formulism of a mesoscopic kink-dislocation model is established to quantitatively predict the mechanical properties of TiZr-based HCP CSAs, which are consistent with experimental results. The results demonstrate that the multiscale simulation approach developed in this work is robust in investigating the dislocation plasticity of CSAs and can facilitate the development of new HCP HEAs with improved mechanical properties.