## Introduction

Unlike conventional metallic alloys, which are comprised of a base (solvent) and alloying (solute) elements, high entropy alloys (HEAs) are a class of structural alloys first introduced by Cantor et al.1 and Yeh et al.2 that consist of multiple principal elements in equiatomic or near equiatomic concentrations3,4,5. Owing to their multi-principal element compositions, HEAs are characterized by unique combinations of properties, including mechanical strength2,6,7, hardness8,9, and wear10,11 and corrosion resistance12,13,14,15. The use of multiple principal elements in far-from dilute concentrations unlocks infinite possibilities of combinations of properties and functionalities that are not typically found in conventional metallic alloys.

Numerous recent studies on 3d transition metal HEAs, such as CoCrFeMnNi1,16,17,18, CoCrFeNi16,19, CoFeMnNi16, CoCrMnNi16, and $$\hbox {Co}_{0.25}$$Cr$$_{0.1}$$Fe$$_{2}$$Mn$$_{1.35}$$Ni$$_{1.3}$$20, have explored the impact of temperature21,22, grain size23,24,25, alloy composition26,27,28, strain rate29, strain levels30, and phase transformations31,32 on the mechanical behavior of these alloys. The equiatomic CoCrFeMnNi, or so-called Cantor, alloy is a face-centered cubic (FCC) single-phase alloy that has been regarded as a model system to many single-phase HEAs. The ability to decipher the fundamental mechanisms controlling the unique properties of the Cantor alloy would improve our understanding of this new class of metallic alloys, thus enabling their use in many engineering applications5,33.

Understanding the mechanical properties of the Cantor alloy has been the subject of active research efforts. Its elastic constants, determined using experimental33,34 and computational35 techniques, have been found to exhibit similar elastic anisotropy as Fe36. Further, the Cantor alloy was found to exhibit simultaneous mechanical strength and ductility17,22,23,37,38,39,40,41,42,43,44 with yield strength values that are comparable to Ni-based superalloys and steels, but lower than aluminum-containing and refractory metal-based HEAs21,36. Gali and George45 have shown that the yield and ultimate strengths and ductility all increase simultaneously with decreasing temperature down to 77 K. The lack of tradeoff between strength and ductility and the exceptional fracture-toughness of the Cantor alloy at low temperatures have been attributed to deformation induced nanotwinning as an additional mode of plastic deformation17,21,22,46.

Extensive deformation-induced twinning has been observed in the Cantor alloy, as well as other FCC HEAs, at various temperatures, including 77 K, and with grain sizes ranging from millimeter down to sub-micrometer17,21,47,48. Such mechanical twins have been observed to be nanometric in thickness and inter-twin spacings17,38,47. Recently, experimental techniques employing diffraction-contrast scanning-transmission electron microscopy have been employed to measure dissociation distance of $$\langle 1 1 0 \rangle$$ dislocations in HEAs49. The extensive twinning of the Cantor alloy at low temperatures has been attributed to its low stacking fault energy (SFE)21, which was determined to be in the range of 20–25 $$\hbox {mJ m}^{-2}$$50,51. Laplanche et al.42 experimentally determined the stress level at which mechanical twins are observed in the Cantor alloy ($$\approx 720$$ MPa), leading to an estimated critical resolved shear stress of $$235\pm 10$$ MPa. Further, they tracked the evolution of twin widths, mean twin spacing, and twin volume fraction as a function of applied strain.

While the aforementioned studies shed light on the mechanical behavior of HEAs, the role of materials defects in general and grain boundaries (GBs) in specific in deformation mechanisms remains poorly understood. The experiments by Bell and Cahn52,53 on Zn crystals highlighted the impact of defects on twin nucleation by observing that carefully prepared Zn single crystals can be mechanically loaded to much higher levels than those at which twins typically form in crystals with defects. Nucleation of mechanical twins from GBs have been observed in a wide range of metallic systems including Cu54, Mg-based55,56, and Ti-based57,58 systems. Very recently, experimental studies on HEAs have revealed nanotwin nucleation from GBs38,42,59. For example, Fig. 1 shows a bright-field transmission electron microscope (TEM) image from a study by Laplanche et al.42 depicting the nucleation of deformation nanotwins from a GB in the Cantor alloy at a true tensile strain $$\epsilon = 12.1\%$$ and temperature of 77 K.

The primary goal of this study is to explore the role of GBs in the deformation mechanisms of the equiatomic CoCrFeMnNi HEA. To this end, we leveraged molecular dynamics (MD) to simulate the tensile deformation behavior of a series of Cantor alloy bicrystals with $$\langle 110 \rangle$$ and $$\langle 111 \rangle$$ symmetric twist GBs at 77 K. Twist GBs were chosen in this study as their geometric construction allows us to investigate the behavior of several GB types in bicrystal systems while maintaining a common crystal direction along the loading axis for all these systems.

## Methods

A series of Cantor alloy bicrystals with $$\langle 110 \rangle$$ and $$\langle 111 \rangle$$ symmetric twist GBs (STGBs) were generated and studied using a second nearest-neighbor modified embedded atom method interatomic potential fit to the energetics of all unary, binary, and ternary combinations of the CoCrFeMnNi system26. This potential reproduces experimental mechanical properties and solid solution hardening effect in non-equiatomic CoCrFeMnNi HEAs at arbitrary compositions. All MD simulations reported in this work were performed using LAMMPS atomistic simulation package60 and visualizations of atomistic structures were generated using OVITO61. Analysis of dislocation structures was performed using the dislocation extraction algorithm (DXA)62. The OVITO implementations of the centrosymmetry parameter, common neighbor analysis (CNA), and polyhedral template matching (PTM) algorithm63 were used to identify crystal structures and ordering in the atomistic systems, i.e., FCC, hexagonal close packing (HCP).

The first step in our approach was the construction of Cantor alloy bicrystals with prescribed GB geometries. For each bicrystal, a fully periodic Cantor alloy atomistic system was created from two half crystals, each of which was rotated such that the resulting planar GB between the halves had the specified misorientation angle. Figure 2a shows a schematic representation of the bicrystal geometry used in this work, where the GB plane normal $$\mathbf{{n}}$$ aligns with the y-axis of both crystals, Table 1 lists the x-axis for the upper ($$x_1$$) and lower ($$x_2$$) crystals for each of the [110] and [111] STGBs examined in this work. A sequence of relative displacements between the upper and lower half-crystals was used in conjunction with atom deletions and conjugate gradient energy minimizations to identify a low energy GB configuration. The systems were allowed to expand or contract in the perpendicular direction to the GB plane. In order to capture the periodicity of a GB atomic structure, the target dimensions of the bicrystal geometry were set to $$L_x = L_z = 100$$ Å and $$L_y = 600$$ Å. However, the dimension of each sample varied so as to accommodate an integer number of unit cells necessary to model each specific GB. The number of atoms in each sample ranged from 600,000 to 1 million. After the bicrystal construction step, we thermally equilibrated the systems by heating the simulation box to a temperature of 77 K and holding for 2 ns with a target pressure of zero, allowing the system to expand or contract as appropriate. This was achieved by performing isothermal and isobaric integration with a 1 fs time step to march atoms’ positions and velocities using a Nosé-Hoover thermostat and barostat64. This creates a trajectory in phase space that is consistent with the isothermal-isobaric (NPT) ensemble. To damp undesirable oscillations in the system’s temperature and pressure, the drag option in LAMMPS was used.

After the initial construction and thermal equilibration steps, uniaxial tension up to 12% was performed along the y-axis of each sample at 77 K, while the normal pressure was set to zero in the transverse directions using the NPT ensemble, thus allowing for lateral expansion/shrinkage. This was achieved by defining two slabs in which the atoms were fixed in their positions relative to each another, refer to Fig. 2b for a schematic representation of the mechanical deformation simulations. The slabs used to impose boundary conditions and forces on the dynamic atoms had a thickness of $$\approx 25$$ Å. Uniaxial tension was applied by moving all fixed atoms of the upper slab with the same constant velocity of $$v = 0.3$$ Å/ps along the y-direction, while the lower slab remained fixed (see Supplementary Fig. S1 online for a study of strain rate effects). The nominal strain along the y-direction of the sample was calculated as $$\epsilon _{yy} = \Delta L_y /L_y(0)$$, where $$L_y(0)$$ is the initial length, and the stress tensor averaged over all dynamic atoms was computed using the standard virial expression.

## Results

### Twist grain boundary structures

The GBs chosen for this study were $$\langle 110 \rangle$$ and $$\langle 111 \rangle$$ STGBs with a wide range of twist angles and, as a result, GB structures. Table 1 lists the ten $$\langle 110 \rangle$$ and ten $$\langle 111 \rangle$$ GBs explored in this work in the order of increasing $$\Sigma$$ value, where $$1/\Sigma$$ represents the percent of lattice points that are in coincidence (i.e., the coincidence site lattice description)65. Each of the two half crystals was rotated about the twist axis (i.e., the crystal y-axis given by [110] or [111]) by an equal and opposite angle $$\theta /2$$, where $$\theta$$ defines the total twist angle. The x-direction for each of the crystals after the rotation is also shown in Table 1. Projected views of the GB plane depicting the 0K equilibrated structures before mechanical deformation for five selected GBs are shown in Fig. 3. Atoms are colored according to the centrosymmetry parameter, where lighter atoms indicate greater deviations from local FCC ordering (see Supplementary Fig. S2 online for the same figure, but with atoms colored according to element type). The atomic structures for the $$\Sigma 111$$, $$\Sigma 7$$, and $$\Sigma 39$$ [111] GBs are shown in Fig. 3a–c, respectively, whereas Fig. 3d–e shows respectively $$\Sigma 9$$ and $$\Sigma 17$$ [110] GBs. In Fig. 3a representing the structure of the $$\Sigma 111$$ [111] STGB, DXA was used to reveal and characterize the interfacial dislocation network, shown in green lines, within the GB plane. For this low-twist angle GB, the boundary is comprised of a grid of 1/6 $$\langle 112 \rangle$$ partial screw dislocations with alternating regions of FCC (atoms colored in black) and HCP (atoms colored in gray) structures. Similar interfacial dislocation networks in low-twist angle GBs have been recently observed in atomistic calculations66. Chen et al.67 examined the GB character distribution in the Cantor alloy during recrystallization and showed a high density of twin $$\Sigma 3$$ and low-angle GBs. In a recent study by Lin et al.68, high-resolution microscopy was employed to study boundary dislocations at low-angle ($$\sim 6^{\circ }$$) GBs in the Cantor alloy and determine their Burgers vectors and dislocation spacing.

Here, it is worth noting that while the GB generation algorithm used in this work resulted in equilibrated GB structures at 0K by performing a series of rigid body translations within the GB plane, i.e., so-called $$\gamma$$-surface method69,70, chemical equilibration was not performed. Such a step includes probing the preferential sites with the GB plane for each of the elemental species, sampling and optimizing the GB atomic density, and probing segregation effects of one or more elemental species to the GB. The phase space of possible chemical configurations for the equiatomic five-component Cantor system is large and beyond the scope of this work. As a result, the GB structures explored in this work can be regarded as metastable ones. Indeed, the multiplicity of metastable GB states has been the subject of active research recently for its influence on the behavior and properties of metallic systems71,72. Very recently, Frolov et al.73 and Zhu et al.74 developed an evolutionary algorithm that performs a grand-canonical GB structure search and were able to identify a plethora of GB phases with different structures in pure metals. Also, Wynblatt and Chatain75 in a recent computational study showed using Monte Carlo, molecular dynamics, and lattice statics methods that both Cr and Mn segregate to GBs in the CoCrFeMnNi Cantor alloy, and such segregation leads to depletion of bulk compositions when the grain size of these alloys is reduced into the nanoscale.

### Mechanical deformation behavior

We first start by examining the macroscopic mechanical behavior of the bicrystal systems at 77 K. Figure 4 shows stress-strain curves for the Cantor alloy bicrystals with [110] (Fig. 4a, b) and [111] (Fig. 4c, d) STGBs. The behavior of Cantor alloy single crystals with [110] ($$\hbox {SC}_{{110}}$$) and [111] ($$\hbox {SC}_{{111}}$$) loading axis is included for comparison. In all systems, the tensile stress increased monotonically with strain until a peak point was reached then followed by a stress drop, the value of which was roughly 35% of the peak value. It is also noticed that the peak point in the stress–strain curve in bicrystal systems was lower than their single crystal counterparts. As will be discussed later, these stress drops are associated with partial dislocation nucleation events with GBs serving as heterogeneous nucleation sites. An exception to this is the bicrystal system with $$\Sigma 3$$ (111) STGB, where a close examination of Fig. 4c,d shows that the peak stress for this system is comparable to the single crystal one. This coherent $$\Sigma 3$$ GB is resistant to nucleation of dislocations, which in this system were found to nucleate homogeneously from the bulk crystals. For the systems with a [110] loading axis, the maximum Schmid factor for $$\{111\} \langle 112 \rangle$$ twinning is 0.47 and that for $$\{111\} \langle 110 \rangle$$ slip is 0.41, whereas for bicrystal geometries with a [111] loading axis those factors were 0.31 and 0.27 for twinning and slip, respectively. As a result, the [111] is the stiffer direction and bicrystals with a [111] loading axis attained higher stress states than those with a [110] loading axis. This effect has been recently experimentally observed in the Cantor alloy76. Also, Fig. 4a,c shows that in all bicrystal systems explored in this work, a plateau in the stress was observed after the initial stress drops. As will be discussed below, this plateau in the flow stress is due to the fact that further deformation was achieved by the growth of GB-nucleated defects (i.e., dislocations) into the bulk grains. Similar trends for the flow stress have been observed in several metallic systems77,78.

Next, we visually inspect the atomic structures of the Cantor alloy bicrystals during the course of their tensile deformation with a specific focus on tensile strains near the peak points in the stress-strain diagrams. Here, the PTM algorithm was used to identify the local structural environment around each atom and reveal GB regions and stacking faults that form during deformation. Figure 5 depicts results for the bicrystal system with a $$\Sigma 9$$ [110] STGB, where Fig. 5a is a schematic illustrating the intersection of a $$\{111\}$$ plane with this GB. Figure 5b–d shows respectively projected views normal to this $$\{111\}$$ plane at nominal tensile strains $$\epsilon = 5.0\%, 5.4\%$$ and $$5.5\%$$, where the GB region is labeled in blue, FCC atoms in green, and HCP ones, indicative of stacking faults, in red. A 1/6 $$\langle 112 \rangle$$ partial dislocation line, identified by DXA and labeled in black, was observed to nucleate from the GB and propagate into the bulk crystal leaving behind a stacking fault, and that the nucleation of this partial dislocation corresponded to the stress drop observed for this bicrystal system (refer to Fig. 4b).

Perspective views demonstrating the nucleation and growth of a partial dislocation in the bicrystal system with a $$\Sigma 111$$ [111] STGB are shown in Fig. 6. Snapshots at tensile strains $$\epsilon =$$ 6.1% (Fig. 6a), 6.2% (Fig. 6b) and 6.3% (Fig. 6c) are shown, where the grid of GB dislocations is colored in orange, atoms in one crystal are colored in blue, and ones in the second crystal are removed for a better visualization of the defect structures. The nucleated dislocation line is colored in green and atoms with HCP structures are in red. Supplementary Video 1 also shows an animation of this bicrystal system during the tensile deformation. At a tensile strain of $$\epsilon =$$ 6.1%, a 1/6 $$\langle 1 1 2 \rangle$$ partial dislocation line nucleated from the GB and grew by bowing out into the bulk crystal leaving behind a stacking fault. It is observed that the dislocation line has its ends pinned at two nodes of the GB dislocation network (see arrows in Fig. 6a). Again, the stress drop observed in the stress-strain curve for this bicrystal system (refer to Fig. 4d) corresponded to the nucleation of this partial dislocation from the GB. As shown in Supplementary Figs. S2 and S3 online, all bicrystal systems examined in this work exhibited similar trends to the ones depicted in Figs. 5 and 6. Such results suggest that GBs serve as heterogeneous nucleation sites for dislocations in the Cantor alloy, and upon further straining these line defects grow into the bulk crystals leaving behind faulted regions. In all bicrystal systems explored in this work, no trailing partials were observed to nucleate from the GBs. This effect was also observed in atomistic simulations of dislocation nucleation from GBs in pure metals78,79. Derlet and Van Swygenhoven80 found that the emission of first partials was enough to relieve local stresses at GBs and that the emission of trailing partials was not necessary to accommodate the applied loading.

### Deformation mechanisms

The mechanical behavior of the bicrystal systems is explored at late stages of deformation. Figure 7a–c shows respectively snapshots of the bicrystals with $$\Sigma 3$$, $$\Sigma 9$$, and $$\Sigma 19$$ [110] STGBs at a nominal tensile strain of $$\epsilon = 6.3\%$$. In these figures, GB and HCP atoms were labeled in blue and red, respectively, where HCP ordering is indicative of stacking faults. Atoms with FCC structures were removed to provide a better visualization of the defect structures. As can be seen from Fig. 7, the density of the nucleated partial dislocations and the resultant stacking faults at a given strain are dependent on the GBs present in these systems. For example, the bicrystal with a $$\Sigma 9$$ [110] STGB shown in Fig. 7b exhibits higher density of faulted regions compared to the one with a $$\Sigma 19$$ [110] STGB depicted in Fig. 7c. A close examination of the stress-strain curves for these bicrystals (refer to Fig. 4b) shows that the stress required to nucleate these partial dislocations was higher for the system with a $$\Sigma 19$$ [110] STGB than ones with $$\Sigma 9$$ and $$\Sigma 3$$ [110] GBs. At a tensile strain $$\epsilon = 15\%$$, Fig. 7d–f shows respectively close-up views of the atomic structures (atoms colored in green and red denote FCC and HCP structures) for the bicrystals with a $$\Sigma 3, \Sigma 9$$ and $$\Sigma 19$$ [110] STGBs, where planes of faulted atoms can be clearly seen. In all bicrystal systems examined in this work, GBs were found to facilitate the nucleation of defects (i.e., dislocations). An exception to this is the bicrystal with the coherent $$\Sigma 3$$ (111) STGB, where dislocations were found to nucleate homogeneously from the bulk crystals. This explains the large value of the peak stress for this system compared to the other bicrystal ones. Indeed, this behavior for the coherent $$\Sigma 3$$ has also been observed in other metallic systems78.

Following the experimental approach by Laplanche et al.42, we quantified the microstructural evolution during mechanical deformation by calculating the fraction of stacking fault atoms with respect to the total in each system, and the results are reported in Fig. 8 for bicrystals with a [110] (Fig. 8a) and [111] (Fig. 8b) loading axis. Stacking fault atoms were identified as ones with HCP ordering using the PTM algorithm. As mentioned above, the observed stress drop in the stress–strain diagrams corresponded to the nucleation of partial dislocations from GBs, which in turn grew rapidly into the bulk crystals leaving behind faulted regions. No stacking faults were observed to nucleate prior to reaching the peak point in the stress–strain diagram. In all bicrystal systems, the rapid increase in the fraction of faulted atoms at lower strain values compared to the single crystal systems indicates that GBs serve as efficient nucleation sites for partial dislocations. The onset of rapid increase in the fraction of faulted atoms occurred at strains up to 30% lower than that for single crystal systems. Further, the fraction of atoms with local HCP structures (i.e., stacking faults) reached an asymptotic value of $$\approx 25\%$$ at late stages of deformation. Systems with [110] STGBs nucleated partial dislocations and experienced an increase in fraction of faulted atoms at lower strain levels compared to bicrystals with [111] STBGs. The lowest levels of applied tensile strain, leading to partial dislocation nucleation and accompanying increase in the fraction of faulted atoms occurred in bicrystals with a $$\Sigma$$3 [110], Fig. 8a, and $$\Sigma$$111 [111], Fig. 8b, STGBs.

To relate the trends observed in Figs. 4 and 8 to GB geometry, we obtained the ultimate stress and strain values from the peak point in the stress–strain diagram as a function of GB misorientation (i.e., twist angle) for all bicrystal systems, and the results are shown in Fig. 9. Again, the bicrystal ultimate stress and strain values are lower than the single crystal ones, which is an indication that GBs act as preferred sites for dislocation nucleation. A close examination of Fig. 9 also shows that bicrystals with a [111] loading axis are characterized by higher ultimate stress (cf. Fig. 9a,c) and strain (cf. Fig. 9b,d) values, i.e., more mechanical energy is required to nucleate partial dislocations from GBs in [111] bicrystals. Variations in ultimate strain and stress values also exist as a function of GB type, defined in these bicrystal systems by the twist angle $$\theta$$. For the bicrystals with a [111] loading axis, a trend exists, albeit with some scatter, where the ultimate stress and strain increase with the twist angle $$\theta$$. Further, an examination of Fig. 9c,d reveals a large difference in the nucleation stress/strain between the $$\Sigma 111$$ and $$\Sigma 3$$ GB. Owing to the small twist angle, $$\theta = 9.43^{\circ }$$, of $$\Sigma 111$$ and the well defined interfacial dislocation structure, this GB was found to be the most efficient in nucleating partial dislocations, i.e., it exhibited the smallest ultimate stress and strain values out of all [111] systems. On the other hand, dislocations were observed to nucleate homogeneously form the bulk crystals in the system with the coherent $$\Sigma 3$$. For the bicrystal systems with a [110] loading axis, our results suggest that ultimate stress and strain values are lower for small ($$\lesssim 30^{\circ }$$) and large ($$\gtrsim 60^{\circ }$$) twist angles compared to intermediate ones. The results depicted in Fig. 9 show that while each set of [110] and [111] bicrystal systems have the same maximum Schmid factors for slip and twinning, variations do exist due to the GBs present in these systems. The Schmid factor provides crystallographic directions with the most resolved shear stresses; however, as depicted in Fig. 9 our work shows that the presence of GBs plays a role in the dislocation nucleation process in HEAs.

## Discussion

While the Cantor alloy has been the subject of numerous studies in recent years, most of these have been mainly focused on the effects of element compositions away from the equiatomic one26, stacking fault energies51,81, and lattice distortions82,83,84. The role of GBs in the mechanical deformation of HEA alloys remains poorly understood. Tensile stress-strain diagrams for Cantor alloy bicrystals with [110] and [111] loading axes show that GBs facilitate the nucleation of partial dislocations, which with further deformation grow in the bulk crystal leaving behind stacking faults. This is in agreement with recent experimental findings demonstrating that GBs act as nucleation sites for mechanical nanotwins38,42. The bicrystals with a [111] loading axis were found to require more mechanical energy to nucleate dislocations from GBs compared to systems with a [110] loading axis. As shown in Figs. 5 and 6, nucleation of 1/6$$\langle 112 \rangle$$ dislocations in the bicrystal systems occurred at lower strain values compared to their single crystal counterparts. In addition, the local GB structure plays a role in the nucleation process. For example, the bicrystal system with a $$\Sigma 111$$ STGB, characterized by a grid of 1/6$$\langle 112 \rangle$$ screw dislocations (see Fig. 3a), was found to nucleate partial dislocations at the lowest strain level (refer to Fig. 9c,d) compared to the bicrystal systems with a [111] loading axis.

Finally, due to the non-dilute nature of HEA compositions, it is reasonable to expect variations in the ultimate stress and strain values due to variations in the local chemistry within GB structures. To demonstrate this effect, Fig. 10 shows tensile stress-strain curves for five realizations of a Cantor alloy bicrystal with a $$\Sigma 3$$ [111] STGB along with that of a single crystal system with a [111] loading axis. Owing to the coherent nature of this $$\Sigma 3$$ GB, it was found to be the most resistant to dislocation emission. In all systems depicted in Fig. 10, dislocations nucleated homogeneously from the bulk crystals. Variations in the observed ultimate stress and strain values are due to the local GB chemistry. Our approach of generating GBs in the Cantor alloy bicrystals employed the commonly used $$\gamma$$-surface method69,70, which aims to mine for the lowest energy GB structure by performing a series of relative displacements between the upper and lower crystals in conjunction with atom deletions and energy minimizations. However, this approach does not consider chemical equilibration, which includes exploring the preferential sites within the GB for each of the elemental species, sampling the GB atomic density, and probing adsorption or desorption effects. As a result, the GB structures explored in this work can be regarded as metastable ones. Our simulation results suggest that the strength of the Cantor alloy can be manipulated by tailoring the GB types and their populations in HEA microstructures. We are currently exploring GB solute segregation in the Cantor alloy and its effect on the mechanical properties, and we hope to report on these results in a future publication.