1 Introduction

High-entropy alloys (HEAs) are random solid solutions of multiple element constituents. HEAs have unique properties such as sluggish diffusion [1, 2] and lattice distortion [3, 4], which gives HEAs many advantages over traditional alloys [5], including good ductility [6, 7], high irradiation tolerance [8], good resistance to the hydrogen embrittlement [9,10,11,12,13,14], and excellent low-temperature mechanical properties [15, 16]. Many strengthening strategies have been proposed to further improve the mechanical property of HEAs [17, 18], among which grain boundary (GB) strengthening is a commonly used one that utilizes the ability of GBs to impede dislocation motions [19,20,21,22,23]. According to the Hall–Petch relation, the strength of a metal increases as the grain size decreases [24]. Nanograined (NG) materials with an average grain size of less than 100 nm usually have much higher strength, hardness, and fatigue resistance than their bulk counterpart [25]. However, NG materials are prone to GB migration, GB sliding [26,27,28,29], and grain growth [30] under loading, rendering NG materials unstable especially at high temperatures. Strategies to prevent GB movement have been developed to increase the thermal stability of NG materials. For instance, alloying elements were added to increase GB stability by inducing GB segregation [31]; nano-twined structures were utilized to restrict GB movement for better thermal stability in NG materials [32, 33].

Recent studies have shown that forming local chemical ordered (LCO) structures is a common yet key structural feature in HEAs [34, 35]. The distribution of different chemical species is not ideally random but rather has a preference for certain pairs of elements [36]. LCO structures have been shown to interact with dislocations and thus influence many mechanical and physical properties of HEAs. For instance, Ding et al. [37] showed that the alternating strain field induced by the inhomogeneous distribution of elements in CrFeCoNiPd HEA resulted in considerable resistance to dislocation movements, rendering the HEA with synergistically improved strength and ductility [38]. Seol et al. [39] discovered that solute boron in recrystallized HEA could increase the degree of LCO in the deformed structure and subsequently increase the yield strength by 32%. Li et al. [40] demonstrated by atomistic simulations that the LCO in NiCoCr could generate a wide range of generalized planar fault energies, which raised the activation barrier for dislocation activities. However, how would LCO affect the thermal stability of NG HEA and the underlying mechanism have not been fully investigated yet.

Molecular dynamics (MD) simulations can reveal the deformation mechanisms at the atomic scale and thus have been widely adopted to study the mechanical properties and deformation mechanisms of NG materials [41,42,43,44] and HEAs [45,46,47,48,49,50,51]. For instance, Chen et al. [52] showed by MD simulations that the formation of LCO structures in CoCuFeNiPd could simultaneously enhance the strength and ductility of the material. Guo et al. [53] performed MD simulations to investigate the effect of LCO on the tensile properties of FeCoNiCrMn and found that the dislocation density and distributions could be tuned by the formation of LCO. The LCO structures were obtained via the hybrid Monte Carlo (MC) /MD approach. Qi et al. [54] studied the plastic deformation mechanism in nanocrystalline FeCoNiCrMn by MD simulations. Zhao [55] systematically studied the influence of LCO on the interaction between defect and GBs in CuNiCoFe HEA by MD simulations. It was found that LCO could promote efficient defect annihilation within the grain interiors by enhancing the sinking strength of GBs toward vacancies over interstitials.

In this paper, the influences of forming LCO structures on the mechanical properties and thermal stability of NG FeCoNiCrMn HEAs were investigated using MD simulations. NG models with two different grain sizes were built with random grain geometries and orientations. The formation of LCO structures was simulated via the hybrid MC/MD method. Tensile tests were then simulated at different temperatures to compare the mechanical behaviors of models with random elemental distribution (the random model) or with LCO structures (the LCO model). To reveal the influence of LCO on thermal stability, we performed creep test simulations at various temperatures and sustained stresses. The thermal stability can then be evaluated based on the Mukherjee-Bird-Dorn (MBD) equation [56], which is given by:

$$ \dot{\varepsilon } = A\sigma^{n} \exp \left( { - \frac{\Delta G}{{k_{{\text{B}}} T}}} \right) $$
(1)

where \(\dot{\varepsilon }\) is the steady-state creep rate, A is a material constant, \(\sigma \) is the sustained stress, n is the stress exponent, \(\Delta G\) is the creep activation energy, \({k}_{\mathrm{B}}\) is the Boltzmann constant, and T is the temperature. n and \(\Delta G\) are the creep parameters that can indicate the thermal stability of the material. By comparing the creep parameters of the random model and the LCO model, it is shown that the formation of LCO structures in HEAs could inhibit creep deformation and increase the thermal stability of NG materials.

2 Methods

NG models were built by using the two-dimensional (2D) Voronoi tessellation method to separate cubic cells into polyhedrons, which were then filled with a fully relaxed face-center cubic (fcc) FeCoNiCrMn HEA. Two different in-plane (x-y plane) dimensions were adopted to simulate models with average grain sizes of 5 and 10 nm, which were denoted as NG-5 and NG-10, respectively. The out-of-plane dimension (z direction) was 10 nm for all models. The two NG models had the same grain arrangement and orientations. The grain orientations were randomly generated with orientations that induced large lattice mismatch along the normal direction excluded to avoid creating massive defects at the periodic boundary. Periodic boundary conditions were imposed along all directions. The modified embedded atom method (MEAM) potential developed by Choi et al. [57] was used as the interatomic potential. The MD simulations were conducted by the open-source Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS). Atomic structures were visualized by OVITO software [58] with crystal structures calculated by Common Neighbor Analysis (CNA) [59]. The dislocations are calculated by the dislocation extraction algorithm (DXA) in OVITO.

For the models with random elemental distributions, the five elements were randomly distributed on the lattice sites, so there is no chemical short- or long-range order. The hybrid MC/MD approach with NPT ensemble was adopted to obtain HEA models with LCO structures, where MC swaps of pairs of atoms were performed to reduce the potential energy of the simulated system and the atomic positions were calculated via time integration in MD simulations. MC swaps were first performed on a bulk HEA model with perfect fcc structure at 300, 700, 1000 and 1300 K to see if the MEAM potential can predict random elemental distribution at high temperatures while generating LCO structures at low temperatures. Then, MC swaps were performed on the NG models to obtain their corresponding LCO structures.

The tensile test simulations were conducted by applying tensile strain along the x direction with the constant strain rate of 5 × 108 s−1. The other two directions were kept stress-free. Creep test simulations were then performed at different temperatures and sustained stress levels. A group of sustained stresses ranging from 1.5 to 3.5 GPa with a 0.5 GPa interval was applied to the models along the x direction. The other two directions were kept stress-free. The duration of each creep simulation was 1 ns.

3 Results

Figure 1a shows the variation in potential energy per atom as the MC step increases, indicating that the formation of LCO structures reduces the potential energy and increases the structural stability. The higher potential energy of the NG-5 model results from the higher GB ratio. Figure 1b, c plots the atomic ratio at the GBs of different elements in the NG-5 and the NG-10 models. As the MC swap continues, Cr and Mn atoms are found to gradually accumulate at the GBs, while the atomic ratios of other elements gradually decrease at the GBs. The accumulation of Cr is the most pronounced. The atomic ratio of Cr at GBs increased from 0.20 to 0.38 in the NG-5 model and from 0.20 to 0.31 in the NG-10 model.

Fig. 1
figure 1

a Variation in potential energy per atom in NG-5 and NG-10 models; b atomic ratio at GBs of different elements in NG-5 model; c atomic ratio at GBs of different elements in NG-10 model; d WC parameters of bulk model after MC swaps at different temperatures; e WC parameters of bulk, NG-5, and NG-10 models after MC swaps at 300 K

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After performing MC swaps on the bulk model at different temperatures, the Warren-Cowley (WC) parameters are adopted to characterize the chemical ordering of the HEA. The WC parameter (\(\alpha_{ij}^{n}\)) is calculated as

$$ \alpha_{ij}^{n} = 1 - \frac{{p_{ij}^{n} }}{{x_{j} }} $$
(2)

where \(p_{ij}^{n}\) is the probability of finding element j around element i in the nth neighboring shell (here we adopt n = 1) and \(x_{j}\) is the atomic ratio of element j in the HEA. Based on this definition, \(\alpha_{ij}^{n}\) = 0 corresponds to a completely random elemental distribution; \(\alpha_{ij}^{n}\) < 0 indicates elements i and j have a good affinity, while \(\alpha_{ij}^{n}\) > 0 indicates otherwise. The calculated WC parameters are plotted in Fig. 1d, e. At 1300 K temperatures, the WC parameters are close to 0, indicating the bulk model has an almost random elemental distribution. As the temperature decreases, the WC parameters deviate more from zero with certain pairs of elements showing strong affinity or repulsion to each other. Based on Fig. 1d, Fe–Fe, Fe–Co, Cr–Cr, and Co–Co have a preference to bond with each other and aggregate, while Fe–Cr and Co–Cr tend to repulse each other.

The WC parameters of the bulk, NG-5, and NG-10 models after MC swaps at 300 K are depicted in Fig. 1e. We can see that the three models have the same preference for forming LCO structures. The WC parameters of the NG models have higher absolute values than those of the bulk models, indicating that the existence of GBs can increase the chemical ordering in NG materials.

Figure 2 displays the atomic structures and the elemental distributions of the models. The NG-5 and NG-10 models have the same grain geometry and orientations but have different grain sizes. NG-5-random and NG-10-random denote the NG-5 and NG-10 models with random elemental distributions before the MC swaps, while NG-5-LCO and NG-10-LCO denote the NG-5 and NG-10 models with LCO structures after the MC swaps. Figure 2e, f shows that after the MC swap, Cr and Mn atoms in the NG models are segregated at the GBs.

Fig. 2
figure 2

Atomic structures of a NG-5 and b NG-10 models colored by Common Neighbor Analysis in OVITO, fcc atoms are colored blue, while atoms with no symmetry are colored green; elemental distributions of c NG-5 and d NG-10 models before MC swaps; elemental distributions of e NG-5 and f NG-10 models after MC swaps of 300,000 steps

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The stress–strain curves of the NG-5 and NG-10 models at various temperatures are shown in Fig. 3a–f. It is obvious that for both models at all temperatures, the corresponding LCO models have higher strength than their random counterparts. Figure 3g–i summarizes Young’s modulus, yield stress, and flow stress obtained from the stress–strain curves. The yield stresses are determined by finding the intersection of the stress–strain curve with a line that initiates from the 0.2% strain at the abscissa and with a slope of Young’s modulus. After the yielding, the stresses converge to balance value and fluctuate around the flow stress. The strain range for calculating the flow stress is from 7.5% to 20%. The Young’s modulus and the yield stress of the NG-10-LCO model are the highest, followed by those of the NG-5-LCO model, and then the NG-10-random model, with those of the NG-5-random model being the lowest. The Young’s modulus and the yield stresses of the LCO models are significantly higher than that of their random counterparts. The flow stresses show the same trend as the yield stresses, but the differences between NG-5 and NG-10 models are less pronounced.

Fig. 3
figure 3

Stress–strain curves of NG-5 model at a 200 K, b 400 K, and c 600 K; stress–strain curves of NG-10 model at d 200 K, e 400 K, and f 600 K; g Young’s modulus, h yield stress, and i flow stress versus temperature curves

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After the creep test simulation, the strain versus time curves were obtained at various temperatures and stresses. Some of the strain versus time curves have a typical three-stage structure comprising the primary (transient) creep stage, the secondary (steady) creep stage and the tertiary (rapid acceleration) creep stage; some only have the primary and secondary stages due to the time limitations of the MD simulations. At high temperatures and high sustained stress levels, the second stage can be very short, and the model ruptures in a short time. The steady-state creep rates (\(\dot{\varepsilon }\)), which are determined from the strain rate in the steady creep stage, are obtained for the NG models at various temperatures and stress levels. At different stress levels, the temperature windows to obtain a strain curve that has a steady creep stage that allows accurate calculation of the \(\dot{\varepsilon }\) are different. At low-stress levels, the deformation of the model over the simulated period can hardly exceed the fluctuation of the deformation at low temperatures, thus the calculation of \(\dot{\varepsilon }\) is inaccurate. Therefore, we selected different temperature ranges for different sustained stress levels to obtain \(\dot{\varepsilon }\).

The logarithmic strain rates as functions of the reciprocal temperature are plotted in Fig. 4a–d. According to Eq. (1), at constant stress and within the simulated temperature range, ln\(\dot{\varepsilon }\) has a linear dependence on \(\frac{1}{{k_{\text{B}} T}}\), wherein the slope is the creep activation energy, \(\Delta G\). As suggested by Yang et al. [33], \(\Delta G\) is dependent on the sustained stress (\(\sigma )\) and can be expressed as:

$$ \Delta G = \Delta G_{0} - \frac{\sigma V}{{\sqrt 3 }} $$
(3)

where \(\Delta G_{0}\) is the stress-independent creep activation energy and V denotes the activation volume. Equation (3) shows that sustained stress can reduce the activation energy and facilitate the creep process. The creep activation energies of the LCO models are higher than those of the random models, indicating that the formation of LCO structures can inhibit creep deformation and increase the thermal stability of NG materials. The creep activation energies of the NG-5 and NG-10 models are plotted against the sustained stress in Fig. 4e–f, from which \(\Delta G_{0}\) and V are determined.

Fig. 4
figure 4

Logarithmic strain rate versus reciprocal temperature for a NG-5-random, b NG-10-random, c NG-5-LCO, and d NG-10-LCO; nominal activation energy versus sustained stress for e NG-5 models and f NG-10 models, linear fitting determines stress-independent ΔG0 and V

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4 Discussion

Figure 3 shows that the yield stress and the flow stress increased significantly after LCO formation. To uncover the underlying mechanism for the LCO enhanced strength, we plot the shear strain distributions and the dislocation arrangements of the NG models during tensile loading in Fig. 5. At the strain of 8%, the GBs of the LCO models have a lower shear strain than that of the random models, indicating that the GB movements are inhibited by the formation of LCO structures. At the strain of 16%, the LCO models obviously have fewer 1/6 <112> partial dislocations than the random models. The decreased numbers of partial dislocations in the LCO models indicate that the formation of LCO structures can hinder dislocation activities. The inhibition effect of LCO on dislocations is more pronounced in NG-5 than in NG-10. The enhanced thermal stability is manifested in Fig. 4 which shows that the creep activation energies after LCO formation were largely increased. The creep activation energy \(\Delta G\) represents the barrier for the creep process to happen, thus higher \(\Delta G\) means better creep resistance and higher thermal stability. The increase in \(\Delta G\) is more prominent in the NG-5 model than in the NG-10 model.

Fig. 5
figure 5

Shear strain distribution in NG models during tensile loading at 200 K, dislocation arrangements at strain of 12% and GB structures are plotted on the right side

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Figure 6 depicts the strain versus time curves at different temperatures of the NG-5 models under two sustained stresses. Here, we select the stress level of 1.5 GPa to represent the low-stress condition and the stress level of 3.0 GPa to represent the high-stress condition. As shown by Fig. 6a, b, under the sustained stress of 1.5 GPa, the NG-5-random model deforms mainly by GB diffusion and only starts to have dislocations nucleated from the GBs at 550 K, so that small steps can be observed on the strain versus time curve. The dislocations are 1/6 <112> partial dislocations denoted as two adjacent hcp layers, as shown in the insets in Fig. 6a. In contrast, the NG-5-LCO model only starts to have dislocations nucleated from GBs when the temperature exceeded 800 K, indicating that the LCO structures can effectively hinder the dislocation nucleation from GBs and increase the creep resistance as well as the thermal stability at low-stress levels. Under the sustained stress of 3.0 GPa, the dislocation nucleation begins at much lower temperatures, which are 150 K for the NG-5-random model and 300 K for the NG-5-LCO model. Clearly, the formation of LCO structures inhibits the creep process for much higher temperatures are required to activate dislocation activities under the same stress level for the NG-5-LCO model.

Fig. 6
figure 6

Strain versus time curves of NG-5 models: a NG-5-random model at 550 K and b NG-5-LCO model at 800 K under sustained stress of 1.5 GPa; c NG-5-random model at 150 K and d NG-5-LCO model at 300 K under sustained stress of 3.0 GPa, insets being crystal structures of models colored by CNA during creep process

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Figure 7 depicts the strain versus time curves of the NG-10 models. Compared with the NG-5 models, the NG-10 models obviously have higher thermal stability for much higher temperatures are required to activate dislocation nucleation in the NG-10 models than in the NG-5 models. Under the sustained stress of 1.5 GPa, the dislocation nucleation begins at 900 K for the NG-10-random model, while for the NG-10-LCO model, the deformation mechanism is still GB diffusion at 1050 K. Under the sustained stress of 3.0 GPa, the starting temperature for dislocation nucleation for the NG-10-random and the NG-10-LCO models are 250 and 300 K, respectively. Figures 6 and 7 clearly demonstrate that the LCO structures can enhance the thermal stability of NG models by inhibiting dislocation nucleation from GBs.

Fig. 7
figure 7

Strain versus time curves of NG-10 models: a NG-10-random model at 900 K and b NG-10-LCO model at 1050 K under sustained stress of 1.5 GPa; c NG-10-random model at 250 K and d NG-10-LCO model at 300 K under the sustained stress of 3.0 GPa, insets being crystal structures of models colored by CNA during creep process

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The threshold temperatures and stresses for dislocation nucleation in the NG models are then summarized in Fig. 8. The dominant deformation mechanism transforms from GB movement to dislocation activity when the temperature and stress exceed the threshold value. The formation of LCO structures evidently enhanced the threshold temperatures for dislocation nucleation, and the enhancement is more prominent for the NG-5 model than for the NG-10 model.

Fig. 8
figure 8

a, b Threshold temperatures and stresses for dislocation nucleation in NG models

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5 Conclusion

In summary, the influences of LCO on the mechanical properties and thermal stability of NG FeCoNiCrMn HEA were investigated by MD simulations. After performing MC swaps on NG models with random elemental distributions, LCO structures formed in the NG models. Cr atoms were found to segregate at GBs. The results of the tensile test and creep test simulations clearly show that the formation of LCO can enhance the strength and thermal stability of the NG models, and the improvement in the NG-5 models is always higher than that in the NG-10 models. By analyzing the atomic structure evolution during the deformation process, it was found that the formation of LCO effectively stabilized the GBs and inhibited GB movement. In addition, dislocation nucleation from GBs and dislocation movement was also hindered. As the NG-5 model has a larger GB ratio than the NG-10 model, NG-5 has a higher degree of LCO, thus the inhibiting effect of LCO on GB movement and dislocation activity in the NG-5 model is more prominent than that in the NG-10 model.