Dislocation Emission from Grain Boundaries in High-Entropy Alloys: Influence of Atomic Composition at Grain Boundaries

Abstract

High-entropy alloys (HEAs) are solid solutions with five or more elements in near equiatomic fractions and exhibit excellent mechanical properties. However, the mechanism has not been fully understood yet. Because general grain boundaries (GBs) contain various sizes of atomic free volumes, a deviation of atomic composition at GBs may appear in HEAs by replacing atoms with ones having different atomic sizes to reduce atomic free volumes at GBs. Various equiatomic HEAs with five elements are modeled by a modified Morse (two-body interatomic) potential. Thermal equilibrium GBs at finite temperatures are obtained by hybrid Monte Carlo-molecular dynamics simulations. As results, GBs in HEAs mainly consist of two elements with the minimum and maximum atomic size and the critical stress to emit dislocations from the GBs increases as the deviation of atomic composition becomes large.

 

This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 69 (2020) 141–148. The caption of Fig. 3 is partly corrected.

1. Introduction

High-entropy alloys (HEAs) are solid solution alloys that are comprised of multicomponent elements in near-equiatomic fractions.^1,2) Owing to the mixing of more than five elements, HEAs have a much higher mixing entropy than that of conventional alloys, and the higher mixing entropy suppresses phase decomposition or the formation of intermetallic compounds. Thus, HEAs can have a high concentration of multicomponent phases. The crystal lattice of HEAs is generally distorted due to the mixing of elements with various sizes; thus, the lattice distortion effect can be considered to considerably affect the mechanical properties of HEAs. For example, Okamoto et al.³⁾ have reported a positive correlation between yield stress and mean square atomic displacement. Furthermore, it has been reported that many HEAs have low stacking fault energy, and they show the coexistence of high strength and high ductility.⁴⁾ Although it has been reported that HEAs exhibit superior mechanical properties, the reason why HEAs have excellent mechanical properties has not fully elucidated yet.

The high strength of HEAs can be explained by atomic segregation to grain boundaries. Grain boundary (GB) is an interface between grains with a misorientation angle; hence, a discontinuity in dislocation slip planes at GB can increase strength. This is a common property of GBs in both conventional materials and HEAs. However, because the atoms comprising GBs do not have translational symmetries, they can have various sizes of free volumes. It has been reported that free volumes are strongly correlated with GB energy or dislocation emission stress from GBs.^5–7) Thus, the free volumes in GBs can affect the mechanical properties of polycrystalline materials. As mentioned above, HEAs are composed of elements with various sizes, which may stabilize GBs due to the reduction of free volumes in GBs by replacing elements; specifically, larger atoms occupy larger free volume sites, and vice versa. Thus, the atomic composition of GBs can deviate similar to GB segregation. Actually, GB segregation in HEAs has been confirmed in some experiments by atom probe tomography.⁸⁾ Furthermore, there have been studies on how GB segregation affects the origin of heterogeneous phase formation⁹⁾ or fracture toughness.¹⁰⁾ Therefore, GB segregation due to various atomic sizes affects GB-mediated plasticity in HEAs; namely, GB segregation may produce the excellent mechanical properties of HEAs.

In this study, the influence of the deviation of atomic composition at GBs on dislocation emission from GBs in HEAs is investigated through two-dimensional atomic simulations. HEAs are modeled in five elements with quinary equiatomic approach. First, multicomponent alloy models with various size elements are created by rapid cooling treatment, and we determine the critical atomic size difference, where crystal-amorphous structural transitions occur. Next, we show that the more the atomic size difference increases, the more the GB segregation of the minimum and maximum size elements occurs. Finally, using uniaxial compressive deformation tests, it is shown that the deviation of the atomic composition in GBs suppresses dislocation emission from GBs.

2. Method

2.1 Modeling of HEAs

To investigate the relation between the size variation of constituent elements and GB segregation in HEAs, atomic simulations are conducted with various size elements. Although the interatomic potential of realistic FeNiCrCoCu-HEA with a face-centered cubic structure has been recently proposed,¹¹⁾ simple interatomic potentials allow us to perform systematical investigations of various types of HEAs by tuning potential parameters. In this study, we develop interatomic potentials, which can express multicomponent alloy models with various atomic size differences δ, as shown in the next section for quinary HEAs.

2.1.1 Atomic size difference δ

Ye et al. have reported that the atomic structures of over 90 types of multicomponent alloys can be classified with atomic size difference¹²⁾ δ as follows:¹³⁾   

\begin{equation} \delta = 100\sqrt{\sum_{i = 1}^{n} c_{i} \left(1 - \frac{r_{i}}{\displaystyle\sum\nolimits_{j = 1}^{n} c_{j}r_{j}} \right)^{2}}, \end{equation} (1)

where n is the number of elements, and c_i and r_i are the mole fraction and atomic radius of element i, respectively. It has been reported that a single-phase solid solution (e.g., fcc, bcc, and hcp), multiphase solid solution, and amorphous phase are formed in 0 ≤ δ ≤ 5, 4 ≤ δ ≤ 8, and δ ≥ 8 regions, respectively.¹³⁾

2.1.2 Interatomic interactions

In this study, analysis objects are simplified as two-dimensional systems. Various multicomponent alloy models with different δ are represented by controlling two-body interatomic potential parameters; then, systematic investigations of the relation between GB segregation and dislocation emission from GBs are conducted for the HEA models.

To express interatomic interactions, we employ shifted force potentials¹⁴⁾ described below. The potentials are based on the Morse potential,¹⁵⁾ a two-body interatomic potential, which is described as follows:   

\begin{equation} \phi^{\alpha \beta}(r) = D^{\alpha \beta}\{e^{- 2a^{\alpha \beta}(r - r_{0}^{\alpha \beta})} - 2e^{- a^{\alpha \beta}(r - r_{0}^{\alpha \beta})}\}, \end{equation} (2)

where r is the interatomic distance between two atoms. Furthermore, D^αβ, a^αβ, and $r_{0}^{\alpha \beta }$ are the parameters for elements α and β, and they are closely related to cohesive energy, elastic constant, and lattice constant, respectively. In this study, both D^αβ and a^αβ are fixed to the same value among all interatomic interactions: D = 8.16 × 10⁻²⁰ J and a = 14.57 nm⁻¹. A shifted force potential¹⁴⁾ $\phi _{s}^{\alpha \beta }$ is designed so that the interatomic interaction energy and its gradient (interatomic force) become zero at the cutoff distance r_c as follows:   

\begin{align} & \phi_{\text{s}}^{\alpha \beta}(r) \\ &\quad =\begin{cases} \phi^{\alpha \beta}(r) - \phi^{\alpha \beta}(r_{\text{c}}) - (r - r_{\text{c}})\dfrac{\mathrm{d}\phi^{\alpha \beta}(r_{c})}{\mathrm{d}r} & \text{($r \leq r_{\text{c}}$)}\\ 0 & \text{($r > r_{\text{c}}$)}. \end{cases} \end{align} (3)

To represent quinary HEA models with various δ, the parameter $r_{0}^{\alpha \beta }$ (α, β = 1, 2, …, 5) relevant to atomic distances is determined, as described by $r_{0}^{\alpha \alpha } = r_{0}^{33} + (\alpha - 3)R$. Here, $r_{0}^{33} = 0.4657$ nm for intermediate atomic size and R = η(k − 1) ($k = 1,2, \ldots ,10$, $\eta \simeq 0.067$) to express 10 types of HEA models with δ in the range from 0 to 18.5. $r_{0}^{\alpha \beta }$ between different elements is defined as the average value of the same elements $r_{0}^{\alpha \alpha }$ such as $r_{0}^{\alpha \beta } = (r_{0}^{\alpha \alpha } + r_{0}^{\beta \beta })/2$. In this study, the cutoff distance r_c is $2.2r_{0}^{55}$.

It should be noted that the interatomic potentials used in this study represent HEA models with various δ values by changing $r_{0}^{\alpha \beta }$; however, other physical properties, such as cohesive energy or elastic constant, have almost the same value with all HEA models due to constant D^αβ and a^αβ.

2.2 Analysis models and conditions

In this study, two types of analysis models are employed. The first type is liquid quenching models, which are used to verify the effect of interatomic potentials to express HEAs and GB segregation. The models are created by a thermal process through the molecular dynamics (MD) method. The second type is bicrystal models, which are used to investigate dislocation emission from GBs in HEAs with thermal equilibrium state; the models are created through the hybrid Monte Carlo/molecular dynamics (hybrid MC/MD) method.

2.2.1 Liquid quenching models

Quinary liquid quenching models contain 2,000 atoms for each element; therefore, each model contains a total of 10,000 atoms. Each atom is randomly positioned without an overlap by uniform random numbers; then, thermal treatment is conducted under periodic boundary and constant stress conditions through the MD simulation, as described below.

First, the initial atomic configuration is relaxed for 10 ps at 10 K; then, the models are heated to 3000 K for 100 ps. Next, each atom in the models is adequately mixed with other atoms for 100 ps at 3000 K and cooled to 10 K for 100 ps. Finally, the models are relaxed for 10 ps at 10 K again. It should be noted that atoms flowed and mixed well at 3000 K because the temperature is above the melting point for all models. The thermal treatment process is conducted under the hydrostatic pressure of 100 MPa (0 Pa at the final relaxation stage) to suppress the generation of voids during the cooling process. The velocity scaling method¹⁶⁾ and the Parrinello–Rahman method¹⁷⁾ are applied to control the temperature and stress, respectively.

To investigate the relation between atomic size difference δ and atomic structures, we create analysis models with 10 types of δ using the abovementioned procedure. We prepare 10 types of models with different initial atomic configuration for each δ with random numbers and then discuss the relation between δ and atomic structures with the average value of each result.

2.2.2 Bicrystal models

Instead of using the liquid quenching models, bicrystal models shown in Fig. 1 are used to simulate dislocation emission from GBs. This is because GBs in the liquid quenching models have different misorientation angles and form at every δ, and the factors other than the elements comprising GBs can affect the dislocation emission phenomena. Therefore, we create analysis models by positioning each atom at specific lattice sites and then changing atomic radii. Thus, the models with different δ are created without changing misorientation angles and the form of GBs. In this study, five elements (approximately 500 atoms for each element) are randomly positioned, where each atom is positioned on a triangular lattice. The domain inside the models is rotated by 30 degrees to create a bicrystal described in Fig. 1. Four types of HEA models with δ = 0, 2.06, 4.12, and 6.15 are employed for the following investigations.

Fig. 1

Initial atomic configuration of a bicrystal model for simulations of dislocation emission from the grain boundary. The misorientation angle of the grain boundary is 30 degrees.

Using the abovementioned method, each element is randomly distributed on the lattice sites; however, if atoms can exchange their positions, specific elements possibly gather in GB-neighboring domains owing to energetic stability. Therefore, the thermodynamic equilibrium state of the atomic configuration at temperature T is obtained using the hybrid MC/MD method, which combines MC and MD methods, as described below. First, GB structures are stabilized by relaxation calculation for 100 ps by the MD method. Next, two atoms are randomly chosen, and their positions are exchanged. Then, relaxation calculation for 10 ps is conducted again, and the total potential energy difference between before and after exchanging ΔE is calculated. If ΔE ≤ 0, the atom exchange is accepted. If not, the exchange is accepted according to the probability exp(−ΔE/k_BT), where k_B is the Boltzmann constant. In this study, the exchange trial for two atoms (MC step) is conducted 7,320 times. Three types of temperatures T = 300 (= 0.12T_m), 1000 (= 0.4T_m), and 2000 (= 0.8T_m) K are set up. Here, T_m ($ \simeq $ 2500) K is the average melting point for the four types of analysis systems. In addition, three types of equilibrium atomic configurations are created for atomic models with four types of δ using different random numbers.

After creating the abovementioned analysis models, we conduct uniaxial pressure deformation tests in both x- and y-direction to investigate the dislocation emission ability of GB in HEAs. Here, the strain rate $\dot{\varepsilon } = 4 \times 10^{7}$ s⁻¹ and analysis temperature T = 1 K are set up.

2.3 Structure analysis methods

2.3.1 Disorder parameter

To identify the local lattice structures of obtained atomic configuration, we use a disorder variable, which has been introduced by Hamanaka et al.¹⁸⁾ In addition to the variable, we use its average over the total number of atoms in the models to quantify the orderliness of the atomic configuration.

Let us focus on atom j in the model. The atoms located within the radius $r \leq 1.1r_{0}^{\alpha \beta }$ centered at atom j are considered as neighboring atoms of atom j, where α and β represent the element of atom j and that of neighboring atoms of atom j, respectively. In this study, if the atomic structures of the local domain are ordered, we define atom j as an ordered atom. On the other hand, if the structures are disordered, we define atom j as a disordered atom. We employ disorder variable D_j, which has been introduced by Hamanaka et al.,¹⁸⁾ for the quantitative evaluation of ordered-disordered structures. This variable represents how the atomic structure around atom j is distorted from regular hexagon. If D_j equals to zero, the structure is a regular hexagon. D_j increases with the distortion of hexagon.

In the models, triangular lattice is an ordered structure; thus, ordered atoms have small D_j. In contrast, disordered atoms have large D_j. In this study, atom j is defined to be an ordered atom if D_j is in the region of 0 ≤ D_j ≤ 1; atom j is defined to be a disordered atom in the region of D_j > 1. In addition, the total number of ordered and disordered atoms is defined as N_O and N_D, respectively.

To quantify atomic structures in the models, disorder parameter $\skew3\bar{D}$ is represented as follows:   

\begin{equation} \skew3\bar{D} = \frac{1}{N}\sum_{j = 1}^{N} D_{j}, \end{equation} (4)

where N is the total number of atoms comprising each model.

2.3.2 Deviation of atomic composition (DAC)

To quantify GB segregation in analysis models, it is necessary to evaluate the degree of mixing of the constituent atoms in the models for every local domain. Thus, we introduce configurational entropy normalized by that of ideal mixed state, as stated below. This normalized entropy can reflect the degree of atomic mixing; thus, we define the value as the deviation of atomic composition (DAC) from ideal mixed state.

Similar to the disorder parameter, let us focus on atoms in a local domain centered at atom j. Using Stirling’s formula, configurational entropy per atom in the domain can be represented as follows:   

\begin{equation} \frac{S_{j}}{k_{\text{B}}} = - \sum_{k = 1}^{n} \frac{m_{k}}{M_{j}}\ln\frac{m_{k}}{M_{j}}, \end{equation} (5)

where n is the number of elements constituting the model, M_j is the number of lattice sites in the local domain, and m_k is the number of atoms of element k in the domain. If the atomic composition is equiatomic in the domain, S_j has the maximum value. On the other hand, if the domain consists of a single element, S_j has a minimum value (S_j = 0). This configurational entropy was used to represent the mixing state of fluid.¹⁹⁾

Here, we estimate the value of the configurational entropy of ideal mixed state. The entropy of the ideal mixed state can be calculated as the expected value of S_j. The probability of an atomic configuration according to the number of each element is represented as follows:   

\begin{equation} P(\boldsymbol{{m}},M) = \left(\prod_{i = 1}^{n} c_{i}{}^{m_{i}} \right)\frac{M!}{\displaystyle\prod\nolimits_{k = 1}^{n} m_{k}!}, \end{equation} (6)

where M is the number of lattice sites in the local domain, and m = (m₁, …, m_n) is the combination of the number of atoms of each element. Note that the ideal mixed state considered here means the atomistic configuration realized by randomly putting atoms into lattice sites in the local domain; thus, the state is not strictly equivalent to the state in which atoms are arranged according to the composition ratio. Using this probability and recalling that S_j is determined by m, we can obtain the expected value of S_j in the ideal mixed state as follows:   

\begin{equation} \langle S_{j}\rangle = \sum_{\boldsymbol{{m}}} S_{j}(\boldsymbol{{m}})P(\boldsymbol{{m}},M_{j}), \end{equation} (7)

where the summation symbol represents the sum of m_i (≥ 0) to satisfy $\sum\nolimits_{i} \,m_{i} = M$. ⟨S_j⟩ depends only on the number of atoms in the local domain M and molar fractions of each element c_i. Here, for equiatomic (c_i = 1/n) quinary (n = 5) multicomponent alloys, ⟨S_j⟩ = 1.12, 1.20, 1.26, 1.31, and 1.35 are obtained by M = 5, 6, 7, 8, and 9, respectively. Whereas, ⟨S_j⟩ increases with M; the value approaches $\ln n \simeq 1.60$ in the limit as M approaches infinity.

In this study, the following mean value of normalized configurational entropy is used as the index of mixing for the entire system:   

\begin{equation} \bar{s} = \frac{1}{N}\sum_{j = 1}^{N} \frac{S_{j}}{\langle S_{j}\rangle}, \end{equation} (8)

where N is the total number of atoms in the system. This index represents DAC from the ideal (randomly) mixed configuration. When $\bar{s}$ is close to unity, the atomic configuration of the system can be considered to be ideally mixed. Instead, $\bar{s}$ smaller than unity indicates the configuration that atoms of the same element aggregate similar to phase separation.

In addition to $\bar{s}$, the DACs of ordered and disordered atoms are separately calculated. The DAC of ordered atoms, $\bar{s}_{\text{O}}$, is obtained by averaging eq. (8) over N_O ordered atoms. The DAC of disordered atoms, $\bar{s}_{\text{D}}$, is calculated in the same way for N_D disordered atoms.

3. Results and Discussion

3.1 Relation between δ and structures

We calculate disorder variable and the DAC for 10 liquid quenching models with 10 types of atomic size difference δ. By discussing the relation between δ and atomic structures, we verify the applicability of our models to express HEAs. Furthermore, we represent the possibility of GB segregation by focusing on the composition ratio of the elements constituting GBs. Here, the analysis models with high ratio of ordered atoms are classified as crystal phase and are considered as HEAs. On the other hand, the models with the high ratio of disordered atoms are classified as the amorphous phase.

3.1.1 Transition from crystal to amorphous phase

Figure 2 shows the relation between disorder parameter $\skew3\bar{D}$ and atomic size difference δ of the liquid quenching models and the relation between the ratio of ordered atoms N_O/N and δ. On the basis of these relations, it is determined that although $\skew3\bar{D}$ increases with δ, the ratio of ordered atoms decreases with an increase in δ. The tendency becomes strong in the region of 6.15 ≤ δ ≤ 14.38. This result indicates that the transition δ_c from crystal to amorphous phase exists in the δ region.

Fig. 2

Disorder parameter $\bar{D}$ (squares) and the ratio of ordered atoms N_O/N (circles) as a function of atomic size difference δ. Phase transition from crystal to amorphous phase occurs between 6.15 ≤ δ_c ≤ 14.38.

Figure 3 shows the snapshots for 3 types of liquid quenching models with δ = 2.06, 8.21, and 14.38, which confirm the details of atomic structures classified as crystal and amorphous phases. In Figs. 3(a), (b), and (c), atomic colors represent atomic size: larger atoms are colored more brightly. The atoms can be considered to be evenly mixed independent of δ. The atoms in Figs. 3(d), (e), and (f) are colored on the basis of the value of disorder variable D_j. It should be noted that the atomic configuration in Figs. 3(d), (e), and (f) is the same as that in Fig. 3(a), (b), and (c), respectively. Figure 3 shows that the number of atoms with larger D_j increases with δ. In the model with δ = 2.06, which is defined as the crystal phase in Fig. 2, disordered atoms comprise GBs. These GBs becomes wider in the model with δ = 8.21. In the model with δ = 14.38, which is defined as the amorphous phase, it is confirmed that almost all atoms are clearly deviated from the triangular lattice. From these results, it is understood that local atomic structures become disorderly, and they change from ordered to disordered structures with an increase in δ.

Fig. 3

Atomic configurations after thermal treatment: (a), (d) δ = 2.06, (b), (e) δ = 8.21, (c), (f) δ = 14.38. Atomic colors in (a), (b), (c) and in (d), (e), (f) represent the atomic size and disorder variable D_j, respectively.

The transition region from crystal to amorphous phase obtained by the liquid quenching models corresponds to the experimental data (δ_c ≈ 8) of Ye et al.¹³⁾ Therefore, we can confirm the effectiveness of investigating the relation between the atomic size difference δ and atomic structures (particularly GBs) with even two-dimensional atomic simulations by simple two-body interatomic potentials introduced in this study.

3.1.2 Atomic specific composition comprising GBs

In this section, we investigate the deviation of the composition of the elements comprising GBs by focusing on the ratio of atoms classified as ordered and disordered atoms.

Figure 4 shows the relation between the ratio of elements classified as (a) ordered atoms and (b) disordered atoms, and atomic size difference δ in the liquid quenching models. The five types of symbols in this figure correspond to the five elements, respectively: atomic size becomes larger with an increase in the element number α.

Fig. 4

Each element ratio of (a) ordered atoms and (b) disordered atoms as a function of atomic size difference δ. Grain boundaries (disordered atoms) in crystal phases less than δ_c consist mainly of the two elements with minimum and maximum atomic size.

First, we focus on the region of δ < 8.21. Note that disordered atoms comprise GBs in the region. Figure 4(a) shows that there is no significant deviation in the ratio of the elements. However, Fig. 4(b) shows that the atomic composition of disordered structures exhibits considerable deviation, and the ratio of the minimum and the maximum size elements are high: the two elements segregate to GBs. We estimate that the segregation tendency is closely related to the magnitude of GB free volume; however, quantitative investigations will be performed in the future.

In the region of δ > 8.21, the composition of elements classified as disordered atoms is almost equiatomic [Fig. 4(b)]. This occurs because a large part of the atoms in the entire system is disordered in this region of δ as can be seen in Fig. 2. The minimum size element (element 1) is regarded as ordered atoms [Fig. 4(a)], which is due to the formation of precipitates composed of element 1 in the amorphous region.

3.1.3 Atomic mixture in GB-neighboring regions

As described above, it is clear that GBs in HEAs are not evenly composed of 5 elements. In this section, we quantitatively evaluate how each element is unevenly mixed with the nearest neighbor elements in GBs using the DAC $\bar{s}$ introduced in this report.

Figure 5 shows the relation between $\bar{s}_{\text{O}}$ and $\bar{s}_{\text{D}}$ as well as the atomic size difference δ. Here, $\bar{s}_{\text{O}}$ and $\bar{s}_{\text{D}}$ are the DAC of ordered and disordered atoms, respectively. The tendency of $1 > \bar{s}_{\text{O}} > \bar{s}_{\text{D}}$ in the crystal phase and $\bar{s}_{\text{D}} > 1 > \bar{s}_{\text{O}}$ in the amorphous phase is observed. Because disordered atoms in the crystal phase are the ones comprising GBs, GBs in HEAs show a stronger tendency to be composed with the same elements adjacent to each other than the crystal region in HEAs. Although the tendency is most pronounced at δ = 4.12, $\bar{s}_{\text{D}}$ is closer to 1 with an increase in δ. Moreover, in the amorphous phase, $\bar{s}_{\text{D}}$ becomes larger than 1, which means that each element is more mixed than in the ideal state.

Fig. 5

Relation between the deviation of the atomic composition, $\bar{s}_{\text{O}}$ for ordered atoms and $\bar{s}_{\text{D}}$ for disordered atoms, and atomic size difference δ.

The reason why $\bar{s}_{\text{D}}$ has the minimum value at δ = 4.12 is attributed to the width of GB regions. In ordinary crystals or alloys, GBs have the width of a few atomic layers and possess inherent structural instability due to excess free volumes related to the misorientation angles. This situation corresponds to the case with small δ. When the system is composed of some different sizes of elements (moderate δ), the instability can be relaxed by filling up the free volumes in the GBs with specific sizes of elements. This leads to GB segregation that decreases $\bar{s}_{\text{D}}$. As δ increases further, GB regions become wide and organize disordered structures unrelated to the misorientation angles within their regions [see Figs. 3(d) and (e)] because disorder structures are more stable than in smaller δ. In this situation, the GB segregation causing a decrease in $\bar{s}_{\text{D}}$ does not necessarily occur because GBs accommodate the instability not by the segregation but by the disorder structures inside the GBs. This could be the explanation of why $\bar{s}_{\text{D}}$ showed the minimum value at δ = 4.12.

3.1.4 Influence of atomic size difference on GBs in HEAs

The obtained results show the following influence of atomic size difference δ on GBs in HEAs.

• •    The GBs in HEAs tend to be composed of the minimum and maximum size elements with an increase in δ.
• •    The elements comprising GBs in HEAs deviate, such as phase separation, with an increase in δ, which is different from the ideal mixed state.

The atomic composition of GBs in HEAs deviates, which differs from general GBs that are composed of only one element. To investigate the influence of GB segregation on the mechanical properties of HEAs, we describe the dislocation emission phenomena from GBs in HEAs in the next section.

3.2 Influence of GB structures on dislocation emission ability

To investigate dislocation emission phenomena from GBs in HEAs, the comparison of the phenomena with crystallographically equivalent GBs with various δ values is necessary. Therefore, we create bicrystal models with the same misorientation angles and geometries at various δ through the hybrid MC/MD method. For the models obtained by the procedure, we conduct compressive deformation tests by the MD method to clarify the influence of GB segregation on dislocation emission from GBs.

3.2.1 GB structure properties of the bicrystal models

Before investigating dislocation emission phenomena from GBs in HEAs, we verify whether GBs in the bicrystal models obtained by the hybrid MC/MD method have the same compositional properties as the liquid quenching models.

Figure 6 shows the potential energy changes of the analysis models for which the hybrid MC/MD method is applied with δ = 2.06 at T = 300, 1000, and 2000 K. It is observed that when the potential energy decreases with an increase in MC steps, the decreasing rate tends to be steady and finally saturates. Furthermore, the decreasing rate is higher at lower temperatures, and the saturation value is low. This behavior indicates that energetically favorable atomic configurations easily occur at lower temperatures, and the contribution of configurational entropy is higher with temperature.

Fig. 6

Potential energy changes of the HEA model with δ = 2.06 at different temperatures.

Figure 7 shows the relation between the atomic composition in GBs of the bicrystal models and the atomic size difference δ. It should be noted that the element ratio in GBs is not equiatomic in δ = 0 because of the fluctuation due to the low number of atoms in GBs. Figure 7(a) shows each element ratio for disordered atoms at T = 300 K. This result shows that the ratio of the minimum and maximum size elements increases with δ at low temperature. While, similar GB segregation tendency is observed at high temperature (T = 2000 K), this tendency is minor [Fig. 7(b)]. Therefore, it is observed that the tendency of GB segregation in HEAs depends not only on δ but also on analysis temperatures. This can be naturally understood from the concept that the contribution of entropy to free energy is proportional to temperature.

Fig. 7

Each element ratio of disordered atoms as a function of atomic size difference δ at (a) 300 K and (b) 2000 K after hybrid MC/MD simulations. Disordered atoms consisting of grain boundaries are mainly composed of the two elements with minimum and maximum atomic size at 300 K, but the ratio decreases at 2000 K.

Figure 8 shows the relation between DAC in GBs $\bar{s}_{\text{D}}$ at T = 300, 1000, and 2000 K and the atomic size difference δ. At each δ, $\bar{s}_{\text{D}}$ decreases with the analysis temperatures; thus GBs in HEAs tend to be composed of the same element within local structures. $\bar{s}_{\text{D}}$ shows the local minimum value at δ ≈ 4. The tendency corresponds to the one in the liquid quenching model [Fig. 5]. The snapshots in Fig. 9 confirm that a decrease in $\bar{s}_{\text{D}}$ is due to GB segregation.

Fig. 8

Relation between the deviation of the atomic composition $\bar{s}_{\text{D}}$ for disordered atoms and atomic size difference δ at three different temperatures after hybrid MC/MD simulations. The deviation of atomic composition at GBs becomes more pronounced as temperature decreases.

Fig. 9

Temperature dependence of atomic composition at grain boundaries in the HEA model with δ = 4.12 at (a) 300 K and (b) 1000 K. At the lower temperature, the grain boundary contains many black (minimum atomic size) and white (maximum atomic size) colored atoms.

Figure 9 shows the atomic structures in GBs with δ = 4.12. In Fig. 9(a), GB at T = 300 K is composed of the minimum size atoms (black colored) and maximum size atoms (white colored), and the same elements tend to be adjacent. On the other hand, in GB at T = 1000 K shown in Fig. 9(b), the tendency is minor, and we can understand that the GB segregation effect depends on the analysis temperature.

Because GB properties in bicrystal models obtained by the hybrid MC/MD method correspond well to those in liquid quenching models, which can reproduce the structure properties of HEAs, we can conclude that bicrystal models are appropriate for analyzing dislocation emission phenomena from GBs in HEAs.

3.2.2 Relation between GBs in HEAs and dislocation emission ability

In this section, we investigate the relation between dislocation emission stress σ_ys and DAC in GBs $\bar{s}_{\text{D}}$. We measure σ_ys by applying a uniaxial compressive load at a certain strain rate $\dot{\varepsilon } = 4 \times 10^{7}$ s⁻¹ and temperature T = 1 K for the bicrystal models obtained by the hybrid MC/MD method. Because the elastic constant of the models changes with δ, Young’s modulus E is calculated on the basis of the slope of the stress-strain curve, and σ_ys/E is compared between the models with different δ. Here, Young’s modulus is obtained by the least squares method with a linear function in the strain region ε = 0–0.2%.

Figure 10 shows the relation between the normalized stress at dislocation emission from GBs σ_ys/E and the DAC in GBs $\bar{s}_{\text{D}}$. Different symbols in the figure represent analysis temperatures in hybrid MC/MD simulations; specifically, open and solid symbols represent the result of a uniaxial compression along y- and x-directions, respectively. The symbols (a) and (b) in Fig. 10 correspond to the grain boundaries shown in Fig. 9(a) and 9(b), respectively. Furthermore, the solid line in the graph represents an approximation line from the plot data. On the basis of the result in Fig. 10, we can find that σ_ys/E tends to increase with a decrease in $\bar{s}_{\text{D}}$; hence, dislocation emission from GBs becomes difficult owing to the non-uniform composition of GBs in HEAs (GB segregation). It should be noted that a clear relation between δ and σ_ys/E cannot be obtained from Fig. 10. This means that the dominant factors of dislocation emission from GBs in HEAs are not only δ but also the local deviation of the composition of the constituent elements in GBs.

Fig. 10

Influence of the deviation of the atomic composition ($\bar{s}_{\text{D}}$) on the critical stress to dislocation emission from grain boundaries (σ_ys/E). Open and solid symbols represent results by y-directional and x-directional compressive deformation tests, respectively. Plots of (a) and (b) correspond to the grain boundaries in Fig. 9(a) and (b). Solid line represents an approximation line.

Let us recall that the local deviation of the composition in GBs can change depending on the analysis temperatures [Fig. 8]. This means that we can design GB-mediated plasticity in HEAs by thermal treatment which can change the composition of GBs.

Finally, we consider the reason why the relation between σ_ys/E and $\bar{s}_{\text{D}}$ in the y-direction compression (open symbols) is more insensitive than that in the x-direction compression (solid symbols). Although Burgers vectors of dislocations emitted from GBs on the same slip system have an opposite sign between two compressive directions, in general, the relation between Burgers vectors of GB dislocations and emitted lattice dislocations strongly affects dislocation emission phenomena.⁷⁾ Therefore, GB dislocation components probably cause the different tendency between x- and y-direction compression. However, it was difficult to specify GB dislocation components owing to the existence of many asymmetric components in the circular GBs used in this study. In the future, we will investigate the effects of GB segregation and GB dislocation components on dislocation emission phenomena using symmetric tilt GBs by which GB dislocations can be specified.

4. Conclusions

This paper reports the effect of atomic composition at GBs on dislocation emission from GBs in HEAs. We prepared HEA models with various size elements through two-dimensional atomic simulations. Quinary alloys were represented with two-body interatomic potentials that can control interatomic distances. The models showed that GBs in HEAs are composed of the maximum and minimum size of elements (GB segregation), and the tendency becomes stronger for HEAs with a larger atomic size difference. We found that the required stress for dislocation emission from GBs increased with GB segregation; then, GB-mediated plasticity could be strongly affected by the GB composition in HEAs.

Acknowledgment

This research was supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) KAKENHI (Grants No. JP18H05453).

REFERENCES

• 1)   B.  Cantor,  I.T.H.  Chang,  P.  Knight and  A.J.B.  Vincent: Mater. Sci. Eng. A 375–377 (2004) 213–218.
• 2)   J.W.  Yeh,  S.K.  Chen,  S.J.  Lin,  J.Y.  Gan,  T.S.  Chin,  T.T.  Shun,  C.H.  Tsau and  S.Y.  Chang: Adv. Eng. Mater. 6 (2004) 299–303.
• 3)   N.L.  Okamoto,  K.  Yuge,  K.  Tanaka,  H.  Inui and  E.P.  George: AIP Adv. 6 (2016) 125008.
• 4)   B.  Gludovatz,  A.  Hohenwarter,  D.  Catoor,  E.H.  Chang,  E.P.  George and  R.O.  Ritchie: Science 345 (2014) 1153–1158.
• 5)   D.  Wolf: Acta Metall. 37 (1989) 1983–1993.
• 6)   M.A.  Tschopp,  G.J.  Tucker and  D.L.  McDowell: Acta Mater. 55 (2007) 3959–3969.
• 7)   T.  Shimokawa: Phys. Rev. B 82 (2010) 174122.
• 8)   L.  Li,  Z.  Li,  A.K.  da Silva,  Z.  Peng,  H.  Zhao,  B.  Gault and  D.  Raabe: Acta Mater. 178 (2019) 1–9.
• 9)   K.G.  Pradeep,  N.  Wanderka,  P.  Choi,  J.  Banhart,  B.S.  Murty and  D.  Raabe: Acta Mater. 61 (2013) 4696–4706.
• 10)   Y.  Zou,  P.  Okle,  H.  Yu,  T.  Sumigawa,  T.  Kitamura,  S.  Maiti,  W.  Steurer and  R.  Spolenak: Scr. Mater. 128 (2017) 95–99.
• 11)   D.  Farkas and  A.  Caro: J. Mater. Res. 33 (2018) 3218–3225.
• 12)   S.  Fang,  X.  Xiao,  L.  Xia,  W.  Li and  Y.  Dong: J. Non-Cryst. Solids 321 (2003) 120–125.
• 13)   Y.F.  Ye,  Q.  Wang,  J.  Lu,  C.T.  Liu and  Y.  Yang: Mater. Today 19 (2016) 349–362.
• 14)   S.  Toxvaerd and  J.C.  Dyre: J. Chem. Phys. 134 (2011) 081102.
• 15)   P.M.  Morse: Phys. Rev. 34 (1929) 57–64.
• 16)  M.P. Allen and D.J. Tildesley: Computer Simulation of Liquids, (Oxford University Press, New York, 1987).
• 17)   M.  Parrinello and  A.  Rahman: Phys. Rev. Lett. 45 (1980) 1196–1199.
• 18)   T.  Hamanaka and  A.  Onuki: Phys. Rev. E 74 (2006) 011506.
• 19)   M.  Camesasca,  M.  Kaufman and  I.M.  Zloczower: Macromol. Theory Simul. 15 (2006) 595–607.