Concentration Dependence of Constituent Elements on Grain Boundary Migration in High-Entropy Alloys

Abstract

High-entropy alloys (HEAs) are multicomponent alloys composed of five or more than five elements with near equimolar concentrations. In this study, molecular dynamics (MD) simulations of grain boundary (GB) migration in HEAs were performed in order to systematically investigate the concentration dependence of the constituent elements on its migration behavior. We found that the driving force required for GB migration in the model HEAs reaches the maximum when the GB migration becomes intermittent or the velocity reduces. The maximum driving force is achieved at the maximum degree of GB segregation, showing that GB segregation, which can be controlled by the element composition in the HEAs, strongly affects the GB migration behavior such as the required force for the migration and the velocity. Our study indicates that the element composition in HEAs plays an important role in determining the GB migration behavior and the obtained results contribute to designing the HEAs with superior mechanical properties.

 

This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 73 (2024) 101–108. The caption of Fig. 4 was slightly modified.

1. Introduction

Grain boundary (GB) migration is a fundamental phenomenon that occurs during the heat treatment, processing, and deformation of metallic materials [1]. Changes in the microstructure of crystal grains caused by GB migration significantly influence the physical and mechanical properties of materials, making it a crucial phenomenon in materials engineering [1]. GB migration is strongly affected by solute atoms. For example, even dilute alloys, which contain small fractions of solute atoms, have significantly higher resistance to GB migration than pure metals. This effect occurs because atoms dissolved in crystal grains can act as obstacles during GB migration, or migration may require the dragging of atoms that have segregated to the boundary [2, 3]. Consequently, a larger driving force is necessary for GB migration than in cases without such solute atom interactions. As the concentration of solute atoms in an alloy increases, their effect on GB migration becomes more pronounced, with a corresponding rise in resistance to migration. This phenomenon has been demonstrated through theoretical studies based on solute atom-GB interactions and GB diffusion [2, 3] and confirmed via molecular dynamics (MD) simulations [4]. These findings suggest that a detailed understanding of the relationship between GB migration and solute atoms is crucial for the design of metallic materials with excellent mechanical properties.

High-entropy alloys (HEAs), which are multicomponent alloys composed of five or more elements mixed in near-equimolar concentrations [5, 6], have garnered attention in recent years [7–9] for their coexistence of high strength and high ductility [10] and their excellent high-temperature tolerance [11]. Moreover, compared with GB migration in pure metals and conventional dilute alloys, that in HEAs is suppressed [12, 13]. According to experiments [12–14] and recent MD simulations [15], the elements segregating to GBs during GB migration in HEAs can be considered to act as resistance to GB migration, as in conventional alloys [2–4]. Focusing on the effect of GB segregation on GB migration is crucial because GB segregation in HEAs affects various GB-mediated plasticities, such as the high stress required for dislocation emission from GBs [16] and suppression of mechanical field localization around GBs [17]. One of the reasons HEAs can exhibit higher resistance to GB migration than pure metals or conventional dilute alloys is the high concentration of elements that can dissolve in crystal grains and segregate to GBs. This effect, unique to multicomponent alloys, is due to HEAs’ much higher configurational entropy than that of conventional alloys, such as binary dilute alloys, which makes it easier to stabilize as solid solutions without phase separation and precipitate formation [7, 8, 18]. Most previous studies on GB migration in HEAs [15] focused on equimolar alloys, where the configurational entropy of mixing reaches the maximum. However, it is not clear how GB migration resistance changes when the concentration of elements segregating to GBs deviates from the equimolar composition. For example, if the concentration of segregating elements is much higher than that of the other constituent elements, then the segregating elements act as matrix atoms; thus, no sufficient GB segregation may be likely to occur. Therefore, there may be an optimal concentration of segregating elements that maximizes GB migration resistance, and the concentration of each element in HEAs may be a design factor for affecting their mechanical properties. In other words, understanding the characteristics of HEAs with such element compositions will provide useful knowledge for the development of next-generation structural materials. In this study, we conducted MD simulations where we controlled the concentration of segregating elements using HEA models. We aimed to simulate GB migration in HEAs with various concentrations of segregating elements and investigate the effects of this concentration on GB migration behavior, including GB migration resistance. The results showed that the concentration of segregating elements significantly affected GB migration behavior, including GB migration resistance. Finally, the influence mechanism of the optimal concentration of segregating elements on GB migration resistance was discussed.

2. Methodology

2.1 Analysis model

In this study, we used face-centered cubic (FCC) bicrystal models with a Σ17(530)[001] GB (GB misorientation angle θ = 61.93 deg.) as shown in Fig. 1. The atomistic mechanisms of Σ17 boundary migration [19] and the GB velocity dependence of migration resistance in pure metals and binary dilute alloys [4] have been reported. The use of Σ17 boundaries would enable us to investigate the GB migration behavior in HEAs compared with previous results obtained using pure metals and binary alloys [4, 19]. Furthermore, the Σ17 boundary is a high-angle boundary with high energy and large free volumes [20]; therefore, it is a suitable representative example of high-angle boundaries, which account for many of GBs in metallic materials. Each bicrystal model was sized approximately l_x = 6.3 nm, l_y = 38 nm, and l_z = 6.5 nm along the x-, y-, and z-directions, respectively, and the total number of atoms contained was 132,000. The analysis models were constructed using a set of interatomic potentials of an equimolar FeNiCrCoCu HEA model developed by Farkas et al. [21] using the embedded atom method (EAM). Farkas et al. developed these interatomic potentials for investigating the magnitude of lattice distortion and displacement of each atom from a perfect FCC lattice of multicomponent alloys, such as HEAs, and used several physical properties as reference, such as the lattice constant and elastic moduli of actual Fe, Ni, Cr, Co, and Cu. However, their HEA model was a “toy model” and does not reproduce all the physical properties of an actual HEA [21]. To emphasize that the alloy in this study was a HEA model, we denoted the five constituent elements (Fe, Ni, Cr, Co, and Cu) using the integers β = 1, 2, …, 5. McCarthy et al. simulated GB segregation using the abovementioned interatomic potentials for the HEA model and reported that element 5 tended to segregate to GBs [22]. Therefore, we changed the concentration of element 5, c₅, from 0 to 1 and set the concentrations of the four other elements to (1 − c₅)/4 for a systematic investigation of the effect of the concentration of the segregating element (element 5) on GB migration behavior. The initial atomic configuration was created by randomly arranging each element on the lattice points of the FCC bicrystal models, as shown in Fig. 1. Setting 0 < c₅ < 1 resulted in quinary HEAs (in particular, c₅ = 0.2 resulted in a quinary equimolar HEA), whereas c₅ = 0 and 1 resulted in a quaternary equimolar HEA and a pure metal with only element 5, respectively.

Fig. 1

The atomic structure of Σ17(530)[001] boundary and typical atomic configuration of the bicrystal model used for GB migration simulations. The different elements, β (= 1, 2, …, 5), are described with different colors in the bicrystal model. (online color)

2.2 Analysis condition

GB migration simulations were performed using the open-source MD simulation code, LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) [23]. The analysis conditions were based on the methods proposed by Koju et al. [4]. First, periodic boundary conditions were applied in all directions (x-, y-, and z-directions) of the analysis model, and structural relaxation calculations were performed to reduce the normal stress components in these directions to zero. Then, as shown in Fig. 1, the atoms within the 1.5 nm region at both ends of the analysis model were rigidly fixed, and structural relaxation calculations were performed under a constant model volume. Then, under the constant-volume condition, the GB was moved by the shear-coupling effect [19] through uniform shear deformation of the entire simulation box at a strain rate of $\dot{\gamma }_{xy}$ in the direction parallel to the GB. The GB migration velocity V was proportional to $\dot{\gamma }_{xy}$ in the absence of GB sliding [4, 19]. Therefore, we controlled V by setting $\dot{\gamma }_{xy}$ to four values between 1 × 10⁷ s⁻¹ and 1 × 10⁸ s⁻¹. The analysis temperature was set to T = 0.8 T_m, where T_m is the melting point of each model alloy. Conducting the simulations at such a high-temperature activated atomic diffusion in the vicinity of the GBs and resulted in dynamic GB segregation during migration, enabling us to investigate the effects of GB segregation on the GB migration behavior. The atomic structure visualization tool, OVITO (Open Visualization Tool) [24], was used to visualize atomic structures and conduct data analyses after the MD simulations.

2.3 Definition of GB position and calculation of average GB velocity

The position and migration velocity of the GBs were identified to investigate the concentration of the elements segregating to the GBs and the GB migration behavior. The GB position y_GB was the y-directional position of the atom that had the highest potential energy. We defined the GB region as a 1-nm-wide region centered at the GB position and regarded the atoms within this region as the atoms constituting the GB. Preliminary simulation results confirmed that the GB moved almost linearly with time during GB migration. Therefore, the average velocity of GB migration $\overline{V}$ was evaluated through linear approximation with the time evolution of the GB position. Hereinafter, unless otherwise specified, the GB position and average velocity are defined in the abovementioned manner.

3. Results

3.1 GB migration velocity

Figure 2 shows the relationship between the GB migration velocity V and GB position y_GB when shear deformation was applied at a strain rate of $\dot{\gamma }_{xy} = 1 \times 10^{7}$ s⁻¹ for analysis models with element 5 concentrations c₅ = 0, 0.6, and 1. The GB velocities were normalized by the average GB velocity $\overline{V}$ ($\overline{V} = 0.71$, 0.65 and 0.72 m/s for c₅ = 0, 0.6, and 1, respectively) for each alloy. As shown in Fig. 2, the fluctuation of the GB velocity is small at c₅ = 1, but it varies significantly around the average velocity ($V/\overline{V} = 1$) at c₅ = 0 and 0.6. Therefore, the GBs in the pure metal (c₅ = 1) always move steadily at a nearly constant velocity, whereas those in the alloys (c₅ < 1) intermittently migrate at various velocities, depending on the mixture of the constituent elements near the GBs. The alloy with a moderate concentration of element 5, c₅ = 0.6, exhibits larger fluctuations and intermittency of GB migration than the alloy without element 5, c₅ = 0.

Fig. 2

Relationship between GB migration velocity, V, and GB position, y_GB, at three different concentrations of element 5, c₅ = 0, 0.6, and 1, under the applied shear strain rate of $\dot{\gamma }_{xy} = 1 \times 10^{7}$ s⁻¹. The GB velocity is normalized by the average GB velocity, $\overline{V}$, in each model.

The above result also indicates that the measured GB migration velocity $\overline{V}$ can have different values depending on the alloy composition c₅, even at the same shear strain rate $\dot{\gamma }_{xy}$. Therefore, in the following analysis, we focus on $\overline{V}$ to compare the GB migration behavior quantitatively. $\overline{V}_{\text{P}}$ is used as a representative of $\dot{\gamma }_{xy}$ for behavior comparison at different $\dot{\gamma }_{xy}$, where $\overline{V}_{\text{P}}$ is the average GB migration velocity of the pure metal (c₅ = 1). Table 1 shows the relationship between $\overline{V}_{\text{P}}$ and $\dot{\gamma }_{xy}$ in the pure metal.

Table 1 Relationship between applied shear strain rate, $\dot{\gamma }_{xy}$, and the average GB migration velocity in pure metal (c₅ = 1), $\overline{V}_{\text{P}}$.

To quantify the difference in velocity fluctuation with respect to c₅ in Fig. 2, we calculated the relative standard deviation (RSD) of the velocity fluctuation $\sigma_{V}/\overline{V}$ using the standard deviation σ_V, as shown in Fig. 3(a). At any $\overline{V}_{\text{P}}$, GB migration of the pure metal (c₅ = 1) has the smallest velocity fluctuation, while the fluctuation is maximized around c₅ = 0.5. Figure 3(b) shows the relationship between $\overline{V}$ and c₅ in the alloys, where $\overline{V}$ is shown in the normalized form by the migration velocity of the pure metal $\overline{V}_{\text{P}}$. The figure indicates that the relative GB velocity decreases as the alloy becomes less pure, and this tendency is especially pronounced near c₅ = 0.5. It can also be confirmed that, except in the concentration around c₅ = 0.2, there is a correlation between the decreasing trend of the GB velocity (Fig. 3(b)) and the increasing trend of the velocity fluctuation (Fig. 3(a)) with respect to c₅. From the aforementioned trends, it can be understood that c₅ has a significant effect on GB migration behavior. Note that, in the vicinity of c₅ = 0.2, although the average GB velocity does not change with increasing c₅, the velocity fluctuation increases. The relationship between the concentration of element 5 and GB migration behavior will be discussed in detail in Chapter 4.

Fig. 3

(a) Relative standard deviation (RSD) of GB velocity, $\sigma_{V}/\overline{V}$, and (b) GB velocity, $\overline{V}$, as a function of the concentration of element 5, c₅. The GB velocity in (b) is normalized by the one in pure metal (c₅ = 1), $\overline{V}_{\text{P}}$. (online color)

3.2 Force required for GB migration

Figure 4(a) shows the relationship between the time-averaged force required for GB migration $\overline{\tau}_{xy}$ and the GB velocity $\overline{V}$ for each alloy. Since the resistance force to GB migration (required force for GB migration) is proportional to the applied shear stress parallel to the GB $\overline{\tau}_{xy}$ [4], $\overline{\tau}_{xy}$ is considered the resistance to GB migration in this study. Because the resistance to GB migration depends on the shear modulus of each alloy, $\overline{\tau}_{xy}$ is normalized by the shear modulus G obtained by the shear deformation of a single crystal of each alloy along the $[3\bar{5}0]$ direction, which is the same direction of shear deformation for the bicrystals. As shown in Fig. 4(a), the GB migration resistance tends to increase linearly with velocity when c₅ is close to 0 or 1, but increases nonlinearly in the vicinity of c₅ = 0.5. This nonlinear change in resistance originates from the solute drag of the segregating atoms by moving GBs at low velocities (below ∼1 m/s in this study), as described in the Introduction [2–4]. Figure 4(a) also shows that the required force for GB migration depends on the amount of element 5 added to the quaternary HEA (c₅ = 0). This trend is more visible in the relationship between $\overline{\tau}_{xy}/G$ and c₅ in Fig. 4(b); the resistance force to GB migration peaks at approximately c₅ = 0.6. This dependence on c₅ is common in the velocity fluctuation of GB migration and the change in the average GB velocity in Fig. 3. In other words, the decrease in GB velocity with an increase in c₅ (Fig. 3(b)) and the intermittent GB migration (Fig. 3(a)) are closely related to c₅. Given that the GB migration resistance is considered to be related to the concentration of segregating elements and GB segregation, we focus on the relationship between the GB migration resistance and the element 5 concentration in the next paragraph.

Fig. 4

The driving force required for GB migration, $\overline{\tau }_{xy}$, as a function of (a) GB velocity, $\overline{V}$, and (b) the concentration of element 5, c₅, in various HEAs with different atomic compositions. The driving force is normalized by the shear modulus in each model, G. The color of the marks represents the concentration of element 5. (online color)

Figure 5(a) shows the relationship between $\overline{\tau}_{xy}/G$ and the segregation degree of element 5 to the GBs. The segregation degree was obtained by subtracting the element 5 concentration in the alloys c₅ from that at the GBs. Thus, when the segregation degree is zero, the element 5 concentration at the GBs is equal to c₅. As seen in Fig. 5(a), the GB migration resistance tends to increase with the segregation degree of element 5 at any $\overline{V}_{\text{P}}$, and this tendency is more pronounced at lower $\overline{V}_{\text{P}}$. As shown in Fig. 5(b), considering that the maximum segregation degree of element 5 is near c₅ = 0.5, the segregation of element 5 to the GBs increases the GB migration resistance, hindering GB migration. The GB migration resistance and the GB segregation degree are positively correlated at the same $\overline{V}_{\text{P}}$ but do not show any correlation at different $\overline{V}_{\text{P}}$. Thus, the GB migration resistance depends not only on GB segregation but also on GB velocity (Fig. 4(a)). As for GB migration in pure metals, resistance increases almost linearly with GB velocity [1, 4]; hence, the velocity dependence of the GB migration resistance is an intrinsic phenomenon in GB migration independent of the presence of solute atoms. Therefore, clarifying the relationship between the GB migration resistance and the segregation degree regardless of $\overline{V}_{\text{P}}$ will require performing some scaling to eliminate the GB velocity dependence from the relation obtained in this study.

Fig. 5

(a) The driving force required for GB migration, $\overline{\tau}_{xy}/G$, as a function of the degree of the segregation of element 5 to the moving GBs. (b) Relationship between the concentration of element 5, c₅, and the degree of the segregation of element 5 to the moving GBs. (online color)

4. Discussion: GB Pinning by Segregating Elements

GB migration via the shear-coupling effect proceeds through a process of nucleation, growth, and annihilation of a disconnection loop [19, 25], where disconnections are steps within GBs and have the same characteristics as dislocations [26]. When GBs migrate through this mechanism, they do not move while maintaining their planar shape; instead, they move with local fluctuations in their out-of-plane direction (GB roughening). Baruffi et al. showed theoretically and numerically via MD simulations that GBs in HEAs are spontaneously roughened due to local compositional fluctuations in the GB vicinity, and this spontaneous GB roughening affects the activation energy of the nucleation of disconnection loops, resulting in various changes in GB mobility [25]. A previous study suggests that an increase in the GB migration resistance with segregating atoms is caused by the pinning of the GBs via inhibiting the nucleation and growth of the disconnections by the segregating atoms [27], as shown in Fig. 6. This mechanism may reduce the GB migration velocity (Fig. 3(b)). Furthermore, if the pinning and depinning of GBs to/from the segregating atoms occur alternately and repeatedly, then the GB migration will be intermittent and its velocity fluctuation will increase (Fig. 3(a)) [27]. In this section, through an analysis of the correlation between the roughening of GBs during their migration and the distribution of elements within the GBs, GB pinning by the segregating atoms and the effect of GB pinning on the GB migration resistance are investigated.

Fig. 6

Schematic image of a possible mechanism for GB migration with the elements segregating to the GBs. The GBs may be strongly pinned at the atomic sites such as the disconnection cores where the elements segregating to the GBs locate with high concentrations. (online color)

4.1 Roughening of GB plane

In this section, we quantify the magnitude of GB roughness and its spatial distribution by dividing a GB plane into (I × J) local regions, as shown in Fig. 7, where I and J are the numbers of regions along the z- and x-directions, respectively. Using the coordinate perpendicular to the GB (y-direction) y_α of the atom α inside a region (i, j) (i = 1, 2, …, I, j = 1, 2, …, J), we define the position of the local region (i, j) along the y-direction as follows:

  

\begin{equation} y^{(i,j)} = \frac{1}{N^{(i,j)}} \sum_{\alpha}^{N^{(i,j)}} y_{\alpha}, \end{equation} (1)

where N^{(i, j)} is the total number of atoms inside the local region (i, j) and the summation means that y_α is summed over all atoms α within region (i, j). The magnitude of GB roughness is evaluated by individually identifying the atoms constituting the GBs using polyhedral template matching (PTM) [28], which is a crystal structure analysis method. The identification method of the atoms constituting the GBs here differs from that described in Section 2.3. By performing the same calculation as in eq. (1) over all (I × J) local regions, we obtain the average position of GB $\overline{y}_{\text{GB}}$ as follows:

  

\begin{equation} \overline{y}_{\text{GB}} = \frac{1}{N_{\text{GB}}} \sum_{\alpha \in \text{GB}}^{N_{\text{GB}}} y_{\alpha}, \end{equation} (2)

where N_GB is the total number of atoms constituting the GB. The relative position of the local region (i, j) to the average position of the GB is the deviation from the average GB position, as expressed by the following:

  

\begin{equation} \Delta y^{(i,j)} = y^{(i,j)} - \overline{y}_{\text{GB}}. \end{equation} (3)

Through the above procedures, we quantify the magnitude of GB roughness as follows:

  

\begin{equation} \delta_{\text{GB}} = \sqrt{\frac{1}{IJ} \sum_{i = 1}^{I} \sum_{j = 1}^{J} (\Delta y^{(i,j)})^{2}}. \end{equation} (4)

The GB is divided into 121 local regions with a side length of approximately 0.6 nm by setting I = J = 11 for the side length of the GB plane (l_x, l_z ∼ 6.4 nm). Each local region is smaller than the core of disconnections (or dislocations) (approximately 1 nm [29]); hence, the spatial resolution sufficiently captures the atoms segregating to the core of disconnections.

Fig. 7

Schematic images of (a) the roughened GB during migration and (b) the divided GB plane into (I × J) divisions. (online color)

Figure 8(a) shows the relationship between the magnitude of GB roughness $\overline{\delta}_{\text{GB}}$, quantified using eq. (4), and the element 5 concentration c₅ of the alloys, where $\overline{\delta}_{\text{GB}}$ is the time-averaged value of δ_GB. $\overline{\delta}_{\text{GB}}$ is maximized at approximately c₅ = 0.5, and this trend is common across all $\overline{V}_{\text{P}}$. The change in the magnitude of GB roughness with respect to c₅ is similar to the relationship between GB velocity and c₅ (Fig. 3(a)), suggesting that GB roughening affects the GB migration behavior. In addition, according to the relationship between $\overline{\delta}_{\text{GB}}$ and $\overline{V}_{\text{P}}$, $\overline{\delta}_{\text{GB}}$ increases with $\overline{V}_{\text{P}}$. This is presumably because of the timescale competition between GB migration and atomic-scale processes. As the GB velocity increases, the timescale of GB migration becomes shorter than that of specific atomic processes, such as the nucleation and annihilation of disconnections and atomic diffusion, which smooth the GB roughening. Hence, this timescale competition deprives the GB of time to flatten, thereby causing GB roughening at high GB migration velocities.

Fig. 8

The magnitude of the roughness of GB plane, $\overline{\delta}_{\text{GB}}$, as a function of (a) the concentration of element 5, c₅, and (a) the driving force required for GB migration, $\overline{\tau}_{xy}/G$, in each model. (online color)

Figure 8(b) shows the relationship between $\overline{\delta}_{\text{GB}}$ and $\overline{\tau}_{xy}/G$. At any $\overline{V}_{\text{P}}$, $\overline{\delta}_{\text{GB}}$ and $\overline{\tau}_{xy}/G$ are positively correlated. Thus, the GB migration resistance increases due to GB roughening caused by the deviation of the local atomic composition in the vicinity of the GB (GB segregation). In the next subsection, we examine whether GB roughening, which increases the GB migration resistance, is due to GB pinning by the segregating elements.

4.2 Verification of GB pinning effect by segregating elements

If the elements segregating to the GBs causes GB pinning, then the following relationships should hold in the local pinned regions of the GBs (i, j): Δy^{(i, j)} < 0 and $\Delta c_{\beta }^{(i,j)} > 0$. Here, $\Delta c_{\beta }^{(i,j)}$ is the relative concentration of an element β (= 1, 2, …, 5) in the local region (i, j) to the one in the alloy c_β and is obtained as follows:

  

\begin{equation} \Delta c_{\beta}^{(i,j)} = c_{\beta}^{(i,j)} - c_{\beta}. \end{equation} (5)

Thus, if the GB is pinned in the region of (i, j), a negative correlation should be observed between the GB deviation and the relative concentration, $\Delta y^{(i,j)}\Delta c_{\beta }^{(i,j)} < 0$. Based on the above discussion, the extent of GB pinning is reflected by the correlation coefficient C_β between Δy^{(i, j)} and $\Delta c_{\beta }^{(i,j)}$.

  

\begin{equation} C_{\beta} = \cfrac{\cfrac{1}{IJ}\displaystyle\sum\nolimits_{i = 1}^{I} \sum\nolimits_{j = 1}^{J} \Delta y^{(i,j)}\Delta c_{\beta}^{(i,j)}}{\sqrt{\cfrac{1}{IJ}\displaystyle\sum\nolimits_{i = 1}^{I} \sum\nolimits_{j = 1}^{J} (\Delta y^{(i,j)})^{2}} \sqrt{\cfrac{1}{IJ}\displaystyle\sum\nolimits_{i = 1}^{I} \sum\nolimits_{j = 1}^{J} (\Delta c_{\beta}^{(i,j)})^{2}}}. \end{equation} (6)

This coefficient also represents the correlation between GB roughening and the concentration fluctuation of element β. Hence, the coefficient becomes negatively larger as the trend of the pinning effect strongthens.

Figure 9 shows the relationship between the correlation coefficient $\overline{C}_{\beta }$ in terms of c₅, where $\overline{C}_{\beta }$ is the time-averaged value of C_β as well as $\overline{\delta}_{\text{GB}}$. The figure shows $\overline{C}_{\beta }$ values for GB migration at four values of $\overline{V}_{\text{P}}$. As shown in the figure, $\overline{C}_{5}$ is negatively correlated with the smallest $\overline{V}_{\text{P}}$ value at approximately c₅ = 0.5, whereas the $\overline{C}_{\beta }$ values of the other elements (β = 1, 2, 3, and 4) show the opposite trend to that of $\overline{C}_{5}$. This feature is more pronounced at low GB velocities. The GBs moving at lower velocities are more strongly pinned in regions where element 5 is locally concentrated (e.g., in the vicinity of the core of disconnections), which increases the required force for GB migration. In the vicinity of c₅ = 0.2 (equimolar quinary HEA), although the effect of pinning increases with c₅, the magnitude of GB roughness remains small, as shown in Fig. 8(a). As described in Section 3, although the average GB velocity does not change in the vicinity of c₅ = 0.2, the fluctuation of the GB velocity increases with c₅ (Fig. 3). These results suggest that the GBs are pinned by element 5 at approximately c₅ = 0.2 and migrate while maintaining their planar shape, but they are pinned and migrate while roughening like a curved surface with an increase in c₅. This change in the GB morphology with respect to the element 5 concentration may be caused by the formation of element 5 clusters or by changes in its distribution within the GBs. Detailed atomic-scale investigations of the phenomena during GB migration are necessary to further understand GB migration in HEAs.

Fig. 9

Correlation coefficient, $\overline{C}_{\beta }$, between the local GB position, Δy^{(i, j)}, and the local concentration of element β, $c_{\beta }^{(i,j)}$, in different GB velocity conditions, $\overline{V}_{\text{P}}$. (online color)

5. Conclusion

In this study, we investigated the relationship between the concentration of elements segregating to grain boundaries (GBs) and GB migration behavior in high-entropy alloys (HEAs) through molecular dynamics (MD) simulations. In alloys with an approximately 0.5 concentration of segregating elements, the GB migration velocity reduced, its migration became intermittent, and the migration resistance reached a maximum value. The GB segregation degree during GB migration was the largest near the 0.5 concentration of the segregating elements, indicating that GB segregation contributed significantly to an increase in GB migration resistance. Furthermore, the relationship between the morphology change of the GBs and the distribution of segregating elements indicated that GB pinning by the segregating elements served as resistance to GB migration, causing the intermittency of GB migration behavior. Therefore, the resistance to GB migration in HEAs is higher than that in conventional dilute alloys and is attributed to the high solute concentration of segregating elements. This is due to the high solubility caused by the high configurational entropy peculiar to multicomponent alloys. The maximum GB migration resistance, observed in the vicinity of the 0.5 concentration of segregating elements, may be related to the distribution heterogeneity of the segregating elements at the GBs and various chemical factors, such as the interaction energy between the constituent elements.

Acknowledgment

This research was supported by JSPS KAKENHI Grants No. JP18H05453, No. JP21H00142, and No. JP21J21395.

REFERENCES

• 1)  G. Gottstein and L.S. Shvindlerman: Grain boundary migration in metals (second edition), (CRC Press, Taylor & Francis, 2009).
• 2)   J.W.  Cahn: The impurity-drag effect in grain boundary motion, Acta Metall. 10 (1962) 789–798. doi:10.1016/0001-6160(62)90092-5
• 3)   K.  Lücke and  H.P.  Stüwe: On the theory of impurity controlled grain boundary motion, Acta Metall. 19 (1971) 1087–1099. doi:10.1016/0001-6160(71)90041-1
• 4)   R.K.  Koju and  Y.  Mishin: Direct atomistic modeling of solute drag by moving grain boundaries, Acta Mater. 198 (2020) 111–120. doi:10.1016/j.actamat.2020.07.052
• 5)   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: Nanostructured High-Entropy Alloys with Multiple Principal Elements: Novel Alloy Design Concepts and Outcomes, Adv. Eng. Mater. 6 (2004) 299–303. doi:10.1002/adem.200300567
• 6)   B.  Cantor,  I.T.H.  Chang,  P.  Knight and  A.J.B.  Vincent: Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A 375–377 (2004) 213–218. doi:10.1016/j.msea.2003.10.257
• 7)   Y.F.  Ye,  Q.  Wang,  J.  Lu,  C.T.  Liu and  Y.  Yang: High-entropy alloy: challenges and prospects, Mater. Today 19 (2016) 349–362. doi:10.1016/j.mattod.2015.11.026
• 8)   D.B.  Miracle and  O.N.  Senkov: A critical review of high entropy alloys and related concepts, Acta Mater. 122 (2017) 448–511. doi:10.1016/j.actamat.2016.08.081
• 9)   H.  Inui,  K.  Kishida and  Z.  Chen: Recent Progress in Our Understanding of Phase Stability, Atomic Structures and Mechanical and Functional Properties of High-Entropy Alloys, Mater. Trans. 63 (2022) 394–401. doi:10.2320/matertrans.MT-M2021234
• 10)   B.  Gludovatz,  A.  Hohenwarter,  D.  Catoor,  E.H.  Chang,  E.P.  George and  R.O.  Ritchie: A fracture-resistant high-entropy alloy for cryogenic applications, Science 345 (2014) 1153–1158. doi:10.1126/science.1254581
• 11)   O.N.  Senkov,  G.B.  Wilks,  J.M.  Scott and  D.B.  Miracle: Mechanical properties of Nb₂₅Mo₂₅Ta₂₅W₂₅ and V₂₀Nb₂₀Mo₂₀Ta₂₀W₂₀ refractory high entropy alloys, Intermetallics 19 (2011) 698–706. doi:10.1016/j.intermet.2011.01.004
• 12)   W.H.  Liu,  Y.  Wu,  J.Y.  He,  T.G.  Nieh and  Z.P.  Lu: Grain growth and the Hall–Petch relationship in a high-entropy FeCrNiCoMn alloy, Scr. Mater. 68 (2013) 526–529. doi:10.1016/j.scriptamat.2012.12.002
• 13)   J.  Cao,  F.  Li,  Q.  Zhang,  M.  Li,  C.  Wang,  W.  Wang and  Y.  Liu: Exceptional thermal stability of nanostructured FeCoNiCrCu high entropy alloy facilitated by unusual grain boundary segregation, Scr. Mater. 234 (2023) 115545. doi:10.1016/j.scriptamat.2023.115545
• 14)   Z.  Wu,  H.  Bei,  F.  Otto,  G.M.  Pharr and  E.P.  George: Recovery, recrystallization, grain growth and phase stability of a family of FCC-structured multi-component equiatomic solid solution alloys, Intermetallics 46 (2014) 131–140. doi:10.1016/j.intermet.2013.10.024
• 15)   D.  Utt,  A.  Stukowski and  K.  Albe: Grain boundary structure and mobility in high-entropy alloys: A comparative molecular dynamics study on a Σ11 symmetrical tilt grain boundary in face-centered cubic CuNiCoFe, Acta Mater. 186 (2020) 11–19. doi:10.1016/j.actamat.2019.12.031
• 16)   K.  Shiotani,  T.  Niiyama and  T.  Shimokawa: Dislocation Emission from Grain Boundaries in High-Entropy Alloys: Influence of Atomic Composition at Grain Boundaries, Mater. Trans. 61 (2020) 1272–1279. doi:10.2320/matertrans.Z-M2020819
• 17)   X.-G.  Li,  C.  Chen,  H.  Zheng,  Y.  Zuo and  S.P.  Ong: Complex strengthening mechanisms in the NbMoTaW multi-principal element alloy, npj Comp. Mater. 6 (2020) 70. doi:10.1038/s41524-020-0339-0
• 18)   J.-W.  Yeh: Recent progress in high-entropy alloys, Ann. Chim. Sci. Mater. 31 (2006) 633–648. doi:10.3166/acsm.31.633-648
• 19)   J.W.  Cahn,  Y.  Mishin and  A.  Suzuki: Coupling grain boundary motion to shear deformation, Acta Mater. 54 (2006) 4953–4975. doi:10.1016/j.actamat.2006.08.004
• 20)   D.  Wolf: Structure-energy correlation for grain boundaries in F.C.C. metals—III. Symmetrical tilt boundaries, Acta Metall. Mater. 38 (1990) 781–790. doi:10.1016/0956-7151(90)90030-K
• 21)   D.  Farkas and  A.  Caro: Model interatomic potentials and lattice strain in a high-entropy alloy, J. Mater. Res. 33 (2018) 3218–3225. doi:10.1557/jmr.2018.245
• 22)   M.J.  McCarthy,  H.  Zheng,  D.  Apelian,  W.J.  Bowman,  H.  Hahn,  J.  Luo,  S.P.  Ong,  X.  Pan and  T.J.  Rupert: Emergence of near-boundary segregation zones in face-centered cubic multiprincipal element alloys, Phys. Rev. Mater. 5 (2021) 113601. doi:10.1103/PhysRevMaterials.5.113601
• 23)   S.  Plimpton: Fast Parallel Algorithms for Short-Range Molecular Dynamics, J. Comput. Phys. 117 (1995) 1–19. doi:10.1006/jcph.1995.1039
• 24)   A.  Stukowski: Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool, Model. Simul. Mater. Sci. Eng. 18 (2010) 015012. doi:10.1088/0965-0393/18/1/015012
• 25)   C.  Baruffi and  W.A.  Curtin: Theory of spontaneous grain boundary roughening in high entropy alloys, Acta Mater. 234 (2022) 118011. doi:10.1016/j.actamat.2022.118011
• 26)   J.P.  Hirth: Dislocations, steps and disconnections at interfaces, J. Phys. Chem. Solids 55 (1994) 985–989. doi:10.1016/0022-3697(94)90118-X
• 27)   D.S.  Fisher: Collective transport in random media: from superconductors to earthquakes, Phys. Rep. 301 (1998) 113–150. doi:10.1016/S0370-1573(98)00008-8
• 28)   P.M.  Larsen,  S.  Schmidt and  J.  Schiøtz: Robust structural identification via polyhedral template matching, Model. Simul. Mater. Sci. Eng. 24 (2016) 055007. doi:10.1088/0965-0393/24/5/055007
• 29)  D. Hull and D.J. Bacon: Introduction to dislocations (fifth edition), (Butterworth-Heinemann, Elsevier, 2011).