The effects of grain size and temperature on mechanical properties of CoCrNi medium-entropy alloy

The mechanical properties and deformation mechanisms of CoCrNi medium-entropy alloy are studied through molecular dynamics simulations. The effects of temperature and average grain size on the elastic modulus, Poisson’s ratio, yield stress, and maximum flow stress are investigated. The constant pressure molecular dynamics method is used to calculate the elastic modulus and Poisson’s ratio of the alloy at different temperatures and average grain sizes. Simple tension simulations are conducted to determine the yield stress and maximum flow stress as a function of average grain size. The study also analyzes the dislocation behavior near grain boundaries at different temperatures using molecular dynamics simulations. The Hall-Petch and inverse Hall-Petch relationships are employed to describe the grain size–dependent deformation behavior of the alloy.

In past decades, researchers have discovered many high-entropy alloys. CoCrFeMnNi HEAs have been revealed possessing excellent mechanical properties at either low or room temperatures [20][21][22]. Cantor et al. [23] first produced the CoCrFeMnNi, which was found to be composed of the face-centered cubic (FCC) solid solution. The investigation shows that temperature has evident influences on yield strength and plastic deformation mechanism of CoCrFeMnNi, and deformation twins dominate the plastic deformation at a temperature of approximately 77 K. In order to better understand the mechanical properties of CoCrFeMnNi HEA, Wu et al. [24] established a series of alloys consisting of Co, Cr, Fe, Mn, and Ni. They found that the hardness of CoCrNi MEA was higher than other alloys. Then, they studied the relationship between yield stress and temperature in the subsystem alloy of CoCrFeMnNi HEAs [25]. It was found that all alloys except CoNi alloy and pure nickel had a strong temperature dependence, with both yield stress and maximum flow stress increasing with temperature decreasing. Gan et al. [26] found that the tensile yield strength of CoCrNi MEA was significantly improved after low-temperature twisting, which was speculated because the pre-twisting was capable of activating different twin systems and stacking faults (SFs). Xie et al. [27] investigated the tensile creep behavior of CoCrNi MEA in the temperature range of 973-1073 K. They found that low stacking-fault energy (SFE) and local chemical ordering (LCO) enhance the creep resistance. Tirunilai et al. [28] compared the deformation behavior of CoCr-FeMnNi and CoCrNi in the temperature interval between 8 and 295 K through a series of quasi-static tensile tests. It was found that when the temperature was 77 K, a small amount of ε-martensite appeared in CoCrNi MEA, leading to a high yield strength of CoCrNi MEA.
Besides temperature, grain size is another main factor affecting the mechanical behavior of materials [29][30][31][32]. Fu et al. [33] quantitatively estimated the contribution of different strengthening mechanisms. They found that grain boundary and dislocation strengthening were mainly responsible for the ultra-high yield strength of CoNiFeAlCu [33]. Yoshida et al. [34] accurately obtained the Hall-Petch (HP) relation of Ni-40CO and CoCrNi MEA through periodic simple tension tests. Sathiyamoorthi et al. [35] prepared an ultrafine-grained CoCrNi MEA with nanotwins, which were fabricated by high-pressure torsion followed by annealing, and investigated cryogenic tensile properties. They found that the alloy has excellent low-temperature tensile properties.
Grain boundary strengthening, dislocation strengthening, and plastic deformation are all atomic-scale phenomena. Since they are difficult to observe by experimental means, a tool is needed to gain insight into the plastic deformation mechanism of HEA at the atomical scale. Atomic simulation techniques, especially molecular dynamics (MD) simulations, can provide insight into the deformation and microstructure of HEA from atomical scale analyses [36][37][38]. It should be noted that a lot of work has been done to understand the Hall-Petch and inverse Hall-Petch effects for pure metal, metallic glass, and binary alloy systems through experiments and MD simulations [39][40][41][42]. However, the deformation mechanism of Hall-Petch and/or inverse Hall-Petch effects at the atomical scale in HEA alloys has been rarely reported up to now.
Herein, a series of MD simulations of CoCrNi MEA have been carried out with two factors (grain size and temperature) considered. First, the potential employed in MD simulations and the models established for the purpose are given in "Simulation details." All the simulation results are presented in "Results and discussions." Specifically, in "Mechanical properties of CoCrNi MEA with different grain sizes" and "Calculation of basic mechanical parameters," mechanical properties (elastic modulus and Poisson's ratio) of CoCrNi MEA are determined in sequence. Then, the plastic deformation mechanisms are investigated in "The effect of grain size on mechanical properties of CoCrNi MEA" and "The effect of temperature on the mechanical properties of CoCrNi MEA" considering grain size and temperature, respectively. A brief conclusion is given in "Conclusions."

Simulation details
CoCrNi MEA is a typical FCC structure, and the MEAM potential function will be used to simulate the system based on the MD method. In 2018, Choi et al. [43] proposed a 2NN MEAM potential function for CoCrFeMnNi HEA, and it was found that this potential function could work well in producing its four fundamental characteristics of HEAs. Since the potential function applicable to the five-element alloy is generally adapted to its subsystem, a ternary MEAM potential function degenerated from the five-element potential function is used in all our MD simulations. This MEAM potential function for the system is described as follows where F i is the atomic embedding energy function and ρ i is the background electron density at point i; φ ij represents the short-range pair potential energy of atoms i and j. It should be noted that the influence from the second neighboring atom has been taken into consideration.
Before exploring the mechanical properties of CoCrNi MEA, we have to verify whether the employed 2NN MEAM potential function can be capable of forming a stable solid solution at test temperatures 77 and 600 K. Therefore, a model of 10 nm × 10 nm × 10 nm with 32,000 atoms included was built and periodic boundary conditions were set all the directions. The model was fully relaxed respectively at temperatures of 77 and 600 K. During relaxation, atomic vibration and size changes of the simulated box were allowed. The results are shown in Fig. 1. It can be seen that CoCrNi MEA can form a stable solid solution at the two prescribed temperatures based on the potential function. Then, based on the potential, similar simulations were carried out at 0 K on the same model, and the lattice constant of CoCrNi MEA was found to be 3.5572 Å. Later, we will use this lattice constant to calculate its mechanical properties.
In the following studies, we will use the Voronoi method to construct CoCrNi MEA polycrystalline model with different grain orientations. To analyze the grain size effect, nine models, respectively with average grain size approximately 3.61, 4.53, 5.65, 7.97, 11.1, 13.2, 15.2, 17.2, and 19.4 nm, were established by adjusting the number of grains and the model size accordingly. Uniaxial tensile simulations will be conducted respectively at 77, 200, 300, and 600 K. Periodic boundary conditions need to be imposed along all dimensions to study the plastic deformation process and the corresponding microstructure evolution at different temperatures. Before tensile deformation, temperature and pressure controls are achieved using the Nose-Hoover thermostat and barostat to obtain a state of minimum energy. What is more, the integration of motion equations is realized using the velocity Verlet algorithm. Then, the established models were relaxed for 100 ps at a specific temperature to achieve a thermodynamic equilibrium state. During the tensile deformation, a strain rate of 0.001 ps −1 was applied along the z-direction. All simulations were performed using the open-source large-scale atomic/molecular massively parallel simulator (LAMMPS) [44][45][46]. The structural analysis and visualization were performed using the open visualization tool (OVITO) [47], combined with the dislocation extraction algorithm and common neighbor analysis (CNA) [48,49]. Taking the grain size 3.61 nm as an example, the model size is approximately 15 nm × 15 nm × 15 nm, and the number of atoms is 299946. The simulation results are shown in Fig. 2, as observed by the visualization software OVITO. In Fig. 2(a), atoms marked red, blue, and yellow colors represent Co, Cr, and Ni, respectively. In Fig. 2(b), domains colored green and red are portions with FCC and hexagonal close-packed (HCP) structures, respectively, while whitecolored regions represent grain boundaries and disordered atoms. In Fig. 2(c), the colors used instead represent different grain sizes.

Mechanical properties of CoCrNi MEA with different grain sizes
The unique atomic structures of materials, in addition to their constituent atoms, closely influence their mechanical properties, leading to significant differences between materials such as metals, ceramics, and polymers. In materials science, the set of elastic constants (C ij , see the following Eq. (7) for reference) is an indispensable basis for describing the mechanical properties of materials, because these constants can provide the necessary mechanical information of materials, such as elastic modulus. Next, we will determine CoCrNi MEA's elastic constants by simulating its stress-strain behavior based on the above-mentioned constant-pressure molecular dynamics method, in which every molecule in the system complies with Newtonian dynamics. It should be noted that this method will slightly affect the accuracy of the results due to small strain fluctuations over time [50]. The stress tensor is expressed by σ ij (i, j = 1, 2, 3), written in matrix form as in which the diagonal items represent the normal stresses, others represent the shear stresses, and σ ij = σ ji (i ≠ j). The strain tensor is expressed by ε ij (i, j = 1, 2, 3), written in matrix form as Longitudinal tensile strains are those diagonal items and others shear strains with ε ij = ε ji (i ≠ j). According to linear elasticity theory, the relationship between stresses and strains, i.e., Hooke's law, is given as where α ijkl is the fourth-order elastic coefficient tensor. To further simplify the expression form of the stress-strain relationship, both the stress and strain tensors are converted to the following matrix forms Then, the expression of Hooke's law in matrix form can be given as in which Since CoCrNi MEA is a cubic crystal system with fcc structure, according to its high degree of symmetry, we can simplify the 21 independent elastic constants (α ijkl ) to three, namely, C 11 , C 12 , and C 44 in [C].
The elastic constants of CoCrNi MEA with different grain sizes can be readily determined by designing properly applied strain cases. In this study, two cases were chosen, and the formulas needed to calculate the three independent elastic constants (C 11 , C 12 , and C 44 ) are presented in Table 1, where γ represents the applied strain amount and is set as 0.01 in all simulations. The calculation results are illustrated in Fig. 3, which indicates that the grain size does have an impact on the elastic constants of CoCrNi MEA due to the presence of largeangle grain boundaries among neighboring grains. This finding is consistent with previous studies [51][52][53]. More specifically, Fig. 3(a) reveals that the elastic constant C 11 increases gradually as the grain size increases from 3.61 to 19.2 nm, and then starts to decrease slowly once the grain size reaches a critical value of approximately 15.2 nm. By observing Fig. 3(b), it is apparent that the elastic constant C 12 steadily decreases as the grain size increases. It is well known that both elastic modulus and Poisson's ratio are mechanical property parameters and given in combination of C 11 and C 12 ; the specific effects of grain size on these two parameters will be further investigated in detail. Figure 3(c) shows that the elastic constant C 44 also steadily increases as the grain size increases, indicating that CoCrNi MEAs with larger grains typically exhibit stronger resistance against shear deformation. Additionally, it can be observed that all three elastic constants decrease as the Table 1 Two designed simulation cases with different strain values applied for determining the three independent elastic constants (C 11 , C 12 , and C 44 )

Calculation of basic mechanical parameters
Generally speaking, whether a given material can be treated as isotropic or anisotropic within the elastic deformation limit can be evaluated simply by considering a factor defined as which is called the elastic anisotropy factor and measures the degree of elastic anisotropy. Based on the simulation results presented in the previous section at 0 K, the elastic anisotropy factor of CoCrNi MEA with varying grain sizes can be easily calculated. The results obtained are shown in Fig. 4. It can be observed that the elastic anisotropy factor slightly fluctuates around a constant value of 1 as the grain size of CoCrNi MEA varies. This suggests that, in theory, CoCrNi MEA can be considered an isotropic material. Furthermore, the bulk modulus B V and shear modulus G V of a cubic system based on the Voigt average method are respectively given as [54,55] Because the CoCrNi MEA cubic system has only three independent elastic constants, the above two equations can be simplified to (10) In comparison, when the Reuss average method is used, the bulk modulus and shear modulus are instead given as For cubic crystal systems, these two expressions can be simplified as Hill [56] proved that the Voigt and Reuss predictions respectively give the upper and lower limits of bulk and shear moduli. Then, by taking the average value, the bulk and shear moduli of a polycrystalline can be given as Finally, based on these two expressions of bulk modulus B and shear modulus G, the elastic modulus E and Poisson's ratio ν can be obtained as For CoCrNi MEA with different grain sizes at various temperatures, the calculated elastic moduli according to Eq. (19) and Poisson's ratios according to Eq. (20) are shown in Fig. 5.
It can be observed from Fig. 5(a, b) that as the grain size increases from 3.61 to 19.4 nm, the elastic modulus increases rapidly and then slows down, and the Poisson's ratio decreases quickly and then slows down at the same peculiar grain size approximately 8.0 nm. These findings suggest that some special microstructure phenomenon is likely to be related to this particular grain size. Therefore, the percentage of grain atoms within whole CoCrNi MEA is calculated for different grain size cases under various temperatures and the results are portrayed in Fig. 6. Normally, the proportion of polycrystalline material occupied by grain boundaries decreases as grain size becomes larger. As indicated in the figure, an increase in grain size leads to a rapid rise in the number of atoms within grain cores, followed by a slower increase when the grain size exceeds the mentioned peculiar grain size. Therefore, this critical grain size of approximately 8 nm signifies a shift in the volume fraction tendency of grain boundaries. Alternatively, it can be postulated that with increasing grain size, a larger proportion of added atoms become part of the grain cores until the grain size reaches 8 nm, after which a relatively higher proportion of added atoms contribute to the grain boundaries. Notably, Fig. 5(a,   (17 5 a The curves of elastic modulus and b Poisson's ratio as a function of grain size for CoCrNi MEA at various temperatures Fig. 6 The atoms in grain core (percentage) versus grain size at various temperatures b) indicates that while the temperature has a significant impact on the elastic modulus, its effect on Poisson's ratio can be overlooked. Figure 5(a) shows that the elastic modulus increases with the grain size increasing. This means that the elastic modulus of CoCrNi MEA is closely related to the properties and volume fractions of grain boundaries and grains. According to the research results of Zhou et al. [57], the elastic modulus of polycrystalline materials, E, can be expressed as in which E g and E gb represent the elastic moduli of crystal grain and grain boundary, respectively, and f denotes the fraction of crystal grains. Equation (21) can then be converted to By taking E′ = 1/E, we can obtain the inverse of elastic modulus for CoCrNi MEA with different grain sizes at various temperatures based on the MD simulation results, which is shown in Fig. 7. The figure reveals a linear relationship between E′ and the volume fraction of grains, which is in good agreement with the theoretical prediction by Eq. (22). The negative slopes suggest that the elastic modulus of grains is higher than that of grain boundaries. Hence, it can be concluded that the contribution of grains to the elastic modulus of CoCrNi MEA is typically higher than that from grain boundaries.

The effect of grain size on mechanical properties of CoCrNi MEA
Normally, grain size significantly impacts the mechanical properties of materials. When the grain size is between tens of nanometers and hundreds of nanometers, the yield strength of a given metallic material normally decreases as the grain size increases, which is revealed by the so-called HP relationship as we know. It is expressed by the equation where σ Y is the material's yield strength; k is the stress concentration factor. In Eq. (23), σ 0 represents the frictional stress as a dislocation slide on surfaces, and d is the average grain size. But when the grain size is smaller than tens of nanometers, its yield strength will instead increase as the grain size increases, and researchers call this phenomenon the inverse Hall-Petch (IHP) relationship [58]. To investigate the impact of grain size on the mechanical properties of CoCrNi MEA, uniaxial tension simulations were conducted at 300 K after the relaxation of the MD simulation models with different average grain sizes. The resulting stress-strain curves, depicted in Fig. 8, reveal that the simple tension process can be divided into three stages: elastic, strain-hardening, and strain-softening. Unlike traditional low-carbon steel, there is no yielding stage with stress fluctuating around a constant value as the strain increases. Thus, the yield stress of CoCrNi MEA is defined as the stress leading to a plastic strain of 0.2%. Moreover, the maximum flow stress for all cases with different grain sizes appeared at a strain of approximately 7.5%, followed by a slight decrease in the strain-softening stage.
In order to study grain size effects on yield stress and the average flow stress, their values corresponding to different grain sizes are obtained and then portrayed in Fig. 9 and Fig. 10, respectively. Here, the average flow stress is calculated by selecting a smooth strain range within the  higher than the experimental value (~ 0.218 GPa), but much higher than that of CoCrFeMnNi HEA (~ 0.125 GPa) and other pure metal materials (take Ni, for example, 0.0142 GPa) [59]. This phenomenon is closely related to elemental composition. Due to the different levels of atomic radius discrepancy within MEAs and HEAs, a larger distortion of crystal lattice exists in MEAs, giving rise to higher lattice frictional stress. In order to understand the underlying mechanism, dislocation distributions and atomic structures are extracted at different applied strain moments of two representative grain size models. First, the dislocation distributions within models of grain size d = 3.61 nm and 17.4 nm at strains from 0 to 20% (step 5%) are portrayed in Fig. 11, where the dislocation density is also presented at each moment. In both models, the deformations before strain 5% should belong to the elastic stage, as the dislocation density remains unchanged (actually, it is surprisingly dropped a little bit which can be understood due to dislocation annihilations). And during the elastic stage, the dislocation densities of the two models are close. As the applied strain reaches 10%, 15%, and 20% in sequence, the dislocation density of the model with grain size d = 3.61 nm initially increases evidently, but in the end, arrives at a saturated state. In the case of the model with grain size d = 17.4 nm, however, the dislocation density keeps increasing till the strain reaches 20%. This is because larger grains provide more space for dislocation initiation and expansion, resulting in more dislocations present within the grains and relatively fewer dislocations absorbed by grain boundaries.
The microstructures of the two selected models at the four applied strain moments are presented in Fig. 12. In order to observe the microstructure evolution clearly, three representative grains in the model with grain size d = 3.61 nm (marked as G1, G2, and G3 in Fig. 12(a)) and only one representative grain in the model with grain size d = 17.4 nm (marked as G4 in Fig. 12(b)) are chosen for the purpose. As shown in Fig. 12(a), as the strain increases from 0 to 20%, G1 and G2 grains become smaller and smaller; meanwhile, their boundaries become larger and larger. In the case of the G3 grain, surprisingly, the grain itself disappears in the end. This can be understood by observing the microstructure evolution, where the rotation of grains and the slip of grain boundaries are the dominant factors in the plastic deformation of CoCrNi MEA with a grain size below the critical size [60]. It should be pointed out that the grain boundary between a smaller grain and a larger grain typically has higher energy, allowing atoms on such a grain boundary to diffuse more easily when the material is stretched. The disappearance of grain G3 is an example of this phenomenon. SFs can also be observed when the applied strain reaches 10%. During plastic deformation, dislocations nucleate, grow at grain boundaries, and then extend into the grain Fig. 9 The curves of the yield stress versus grain size for CoCrNi MEA at 300 K Fig. 10 The curves of the average flow stress versus grain size for CoCrNi MEA at 300 K interior. However, the grain boundaries can also hinder and absorb dislocation expansions, forming SFs inside grains.
As shown in Fig. 12(b), as the strain increases from 0 to 20%, the G4 grain size is nearly unchanged. First, SFs have already appeared when the applied strain reaches 5%. Then, some SFs disappear as the strain increases from 5 to 10% or even higher, as indicated by green arrows. What is more, if we compare the two models at the same applied strain, for example, at ε = 10%, there are significantly more SFs inside grains of the model with grain size d = 17.4 nm than in grains of the model with grain size d = 3.61 nm. This is because, during plastic deformation, the internal space of crystal grains with a greater size is large enough such that dislocations can initiate and expand within the room provided, and the probability of being absorbed by grain boundaries is reduced. Moreover, this capacious room provides favorable conditions for the expansion of SFs. Furthermore, two different formation mechanisms of deformation twins are observed during MD simulations and they are displayed in Figs. 13 and 14, respectively. In order to show the microstructures with enough clarity, two different models are presented here (d = 7.97 nm in Fig. 13 and d = 19.4 nm in Fig. 14). As shown in Fig. 13(a), as the strain increases from 5 to 15%, incomplete dislocations are initially emitted from the grain boundaries into grain interior, then the dislocation slips form SFs when the strain reaches 8%; finally, these SFs interact with each other leading to the formation deformation twins. Thus the dislocations continuously generated at the grain boundaries create favorable conditions for the formation of more deformation twins, as shown in Fig. 13(b). It can be seen from Fig. 14(a) that, as the strain increases from 4 to 16%, incomplete dislocations are initially emitted from the grain boundaries into the grain interior; the dislocation slips form SFs and the perfect dislocations break into partial dislocations. As the strain increases further, these partial dislocations and the SFs among them together form the extended dislocations. When two extended dislocations meet, two partial dislocations will possibly form a perfect dislocation. At the same time, extended dislocation clusters are generated and dislocation cores are formed. These SFs interact with each other to form deformation twins and twin boundaries. The generous space inside the large grains allows for extensive movement of extended dislocations and therefore more deformation twins can be formed, as shown in Fig. 14(b).

The effect of temperature on the mechanical properties of CoCrNi MEA
In "Calculation of basic mechanical parameters," we have studied the temperature effect on the elastic modulus of Fig. 15 The curves of the simple tension stress-strain versus grain size for CoCrNi MEA at various temperatures CoCrNi MEA, which has been obtained through calculations of elastic constants. It is already shown in Fig. 5(a) that the material owns a higher elastic modulus at a lower temperature, provided its grain size is fixed. In this section, in order to further study the effects of temperature on the mechanical properties of CoCrNi MEA, simple tensions have been performed at 77, 200, 300, and 600 K for all the models, and the results are portrayed in Fig. 15. It can be seen that both the yield strength and the maximum flow stress of the material increase with the temperature decreasing for all the models.
It is known that plastic deformation mechanisms of materials mainly include dislocation slips and grain boundary slips. When the grain size is smaller than the critical grain size of 15.2 nm, the plastic deformation mode is mainly the grain boundary slips. Cui et al. proved that the thermal activation process of materials can affect the migration of grain boundaries [61]. The governing migration equation of a grain boundary is given as where M represents the grain boundary's mobility, M 0 is a constant term, Q t is the thermal activation energy, R is the gas constant, and T is the temperature. It can be observed from Eq. (24) that as temperature increases, the movement of the grain boundary becomes more intense. To demonstrate the impact of temperature on plastic deformation, Fig. 16 presents the distribution of dislocations within the model of grain size d = 15.2 nm under a strain of ε = 5%. At lower temperatures, there are more dislocations present in the model, suggesting that the dislocation nucleation and emission time is delayed at elevated temperatures. Furthermore, this phenomenon persists even at higher levels of applied strain, such as 10%, 15%, and 20%, as shown in Fig. 17. Combining the effects of temperature on grain boundary slip and dislocation behavior, it can be inferred that at lower temperatures, dislocations tend to accumulate near grain boundaries with lower mobility. Therefore, it is expected that the yield strength and maximum flow stress of CoCrNi MEA will increase as ambient temperature decreases.

Conclusions
This paper presents MD simulations of CoCrNi MEA, focusing on its mechanical properties and the effects of temperature and grain size. The constant-pressure molecular dynamics method determines the elastic modulus and Poisson's ratio. The elastic modulus increases with increasing grain size but decreases at higher temperatures, while Poisson's ratio decreases with increasing grain size and is insensitive to temperature. Simple tension simulations reveal the HP and IHP behaviors of CoCrN MEA, with the critical grain size determined to be approximately 15.2 nm. Microstructure analysis shows that plastic deformation mechanisms change from grain boundary slips and rotations to dislocation slips and deformation twins as the grain size increases. Additionally, lower temperatures lead to more dislocations near grain boundaries with reduced mobility, improving the yield strength and maximum flow stress of CoCrNi MEA.
We the undersigned declare that this manuscript is original, has not been published before, and is not currently being considered for publication elsewhere. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been approved by all of us.
We understand that the corresponding author is the sole contact for the editorial process. He is responsible for communicating with the other authors about progress, submissions of revisions, and final approval of proofs.
Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.