**1. Introduction**

It is well known that body-centered cubic (BCC) Fe-based steels have been extensively used in various industrial applications [1–3]. Typically, these steels are polycrystalline materials in which grain boundaries (GBs) are formed between crystallites [4]. Thus, the grain boundaries have a significant influence on the physical, mechanical and chemical properties of Fe-based materials. The investigation of their properties, including energy, structures and mechanics, presents a popular topic in the field of materials science [5,6]. For example, GB could act as a "sink" to absorb the interstitials and vacancies [7], which can decrease the number of radiation defects but can also affect the micro-structure of GB and related mechanical properties. Furthermore, external stress or strain fields, in addition to contribution from dislocations, GB sliding, migration, dislocation nucleation and failure are all involved in plastic deformation [8,9]. Since GB failure is a crucial factor for the design and application of Fe-based materials [10], it is necessary and important to study GB failure mechanisms at the atomic scale to explore the potential reasons for these phenomena.

The relationship between the atomic structure and energy of GBs has been studied extensively in the literature [11,12]. For example, Ratanaphan et al., calculated the energies of 408 grain boundaries in Fe and Mo using embedded atom method (EAM) potentials [5]. They reported that the calculated energies vary significantly with the grain boundary plane orientation but the energies do not show any distinct trends with misorientation angle or with the density of coincident lattice sites [5]. Gao et al. studied three typical GBs in BCC iron using molecular statics (MS) simulations, ab initio density-functional theory (DFT)

**Citation:** Ma, W.; Dong, Y.; Yu, M.; Wang, Z.; Liu, Y.; Gao, N.; Dong, L.; Wang, X. Evolution of Symmetrical Grain Boundaries under External Strain in Iron Investigated by Molecular Dynamics Method. *Metals* **2022**, *12*, 1448. https://doi.org/ 10.3390/met12091448

Academic Editors: Angelo Fernando Padilha and Alain Pasturel

Received: 19 July 2022 Accepted: 28 August 2022 Published: 30 August 2022

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calculations and the simulated high-resolution transmission electron microscopy (HRTEM) method, indicating the importance of relaxing the GBs in order to further investigate the properties of GBs through a multiscale method [13]. Du et al. also reported the energy states of some GBs in α-and γ-Fe using density-functional theory and found that the ∑3 twin boundaries exhibit low interface energies [14]. In addition to these results, the deformation behavior of GBs under different conditions is also a key factor to understand the GB properties. Spearot and colleagues investigated the influence of the structure of grain boundaries (e.g., bi-crystal GBs) on deformation behavior of the system by using MD simulations, through which the roles of dislocation nucleation and emission phenomenon under uniaxial tension were recognized for aluminum [15] and copper [16] symmetric tilt grain boundaries (STGBs). Terentyev et al. studied a set of <110> tilt grain boundaries (GB) in α-Fe with a misorientation angle varying from 26◦ to 141◦ by applying atomistic calculations and discovered grain boundary sliding is closely related to the structure determined by the misorientation angle [4]. Singh et al. reported the investigation of structure, energy and tensile behavior of niobium (Nb) bi-crystals containing symmetric and asymmetric tilt grain boundaries via molecular dynamics simulations [6]. Cui et al., further studied the <001>, <101> and <111> twist grain boundary structures in copper through molecular dynamics simulations to obtain the dependence of tensile strength on twist angle of grain boundary [17]. They reported that for a <001> and a <101> twist grain boundary, the tensile strength increases on average with an increasing misorientation twist angle [17]. However, the misorientation bears slight influence with respect to <111> twist GB structures [17]. From these results, it can be observed that various factors affect the structure and mechanical properties of GBs, highlighting the complicated nature of this topic within the material field. In order to solve these difficulties, researchers recently applied neutron diffraction method to investigate the free volume of sub-microcrystalline Ni [18], in which the anisotropic annealing of relaxed vacancies at GBs was identified, and is considered the main reason behind induction of the anisotropic length change upon annealing [18]. However, until now there are no detailed reports regarding the free volume change in GB under an external stress or strain field, thus warranting further investigated to better understand the properties of GBs.

As stated above, in addition to energy, structure and the deformation of grain boundary, the properties of free volume (FV) in grain boundaries should also be investigated in detail especially under the effect of an external stress or strain field. Generally, free volume in materials is generally defined as the maximum volume of a sphere that can be inserted between atomic sites in the system [19]. It has been reported that the free volume in grain boundary could assist the mobility of neighboring atoms, enabling GB sliding, grain rotation and GB dislocation emission [20]. By using molecular dynamics simulations through the investigation of three face-centered cubic (fcc) and body-centered cubic (bcc) metals, Sun et al. found that the free volume shrinks much faster above a critical temperature [21]. Tschopp et al. reported the dependence of free volume on the spatial correlation functions in grain boundary [20], which provides a better understanding of dislocation dissociation and nucleation in Cu grain boundaries. Furthermore, Tucker et al. investigated the effect of free volume on the stretching process of fcc copper grain boundaries and found that the free volume influences interfacial deformation through modified atomic-scale processes [22]. Wang et al. also suggested that FV may provide a site for micro-crack nucleation in bcc symmetrical grain boundaries after irradiation [23]. However, although research has been performed investigating the effect of free volume on the properties of grain boundaries, the relationship between free volume and grain boundary mechanical properties has not been reported in detail. For example, the relationship between the FV change and the stress concentration resulting in slip system activation and the role of FV change in GB failure, warrants further investigation. The results are expected to provide a comparison and validation with neutron diffraction results induced by the external stress or strain field in the future.

Therefore, in this paper, the properties of the symmetrical grain boundaries including ∑3(112), ∑3(111), ∑5(012), ∑5(013), ∑9(221), ∑11(113) and ∑17(410) in bcc iron are investigated at the atomic scale, and in particular, the influence of free volume on stress–strain properties. New mechanisms of GB failure by considering the free volume influence were explored in this work according to the simulation results. In the following, details of the simulation methods are introduced in Section 2, and the results and discussion are provided in Section 3. Finally, the conclusions are given in Section 4.
