# Initial Vacancy-Dependent High-Temperature Creep Behavior of Nanocrystalline Ni by Molecular Dynamics Simulation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simulation Model and Method

^{3}, as shown in Figure 1a. Nanocrystalline Ni models containing different volume fraction of initial vacancies ranging from 0% to 10% (0%, 0.1%, 1%, 2%, 3%, 4%, 5%, 6%, 7%, 8%, 9%, and 10%) were created by randomly removing certain fractions of atoms in the model. In Figure 1b, a model with initial vacancy concentration of 1% is shown, with the blue dots representing the coordinates of the deleted atomic sites. The assumption of vacancy concentration in the range of 10

^{−2}in this study was mainly based on the following considerations. The equilibrium vacancy concentration can be calculated by the following equation: C

_{v}= Aexp (−E

_{v}/kT), where k = 1.38 × 10

^{−23}J/K, coefficient A = 1. The equilibrium vacancy concentration is exponentially related to the vacancy formation energy and temperature. The vacancy formation energy of common pure metals ranges from 0.4 to 2.0 eV, and the equilibrium vacancy concentration at 800 K can be calculated to be within the range of 10

^{−2}to 10

^{−8}. In nanostructured metals, the formation energy of vacancies decreases with decreasing grain size, and the equilibrium vacancy concentration will correspondingly increase.

_{i}

^{Mises}[33,34], which has proved to be an effective parameter to characterize local atomic rearrangement.

## 3. Results and Discussion

#### 3.1. Creep Behavior of Nanocrystalline Ni with Initial Vacancies

#### 3.2. Characteristics of Deformation-Induced Vacancy Formation during Creep

#### 3.3. Influence of Initial Vacancy Concentration on Dislocation-Related Activities

#### 3.4. Microstructural Evolution of Nanocrystalline Ni during Creep

_{i}

^{Mises}of each atom was calculated to mark the areas with large local shear strain in the nanocrystalline Ni models. Shear strain distribution of nanocrystalline Ni models with initial vacancy concentration of 0.1%, 3%, 6%, and 10% at creep time of 50 fs is shown in Figure 7a–d. The local shear strain at the GB was the largest compared to that of the grain interior, that is to say, the atomic activity at the GB was more intense than that of the grain interior. In addition, there were still some regions with large local shear strain in the grain interior, which may be related to the average distribution of initial vacancy sites. It can be concluded that the generation of deformation-induced vacancies is directly related to the large atomic shear strain at the GBs.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Zhu, T.; Li, J. Ultra-strength materials. Prog. Mater. Sci.
**2010**, 55, 710–757. [Google Scholar] [CrossRef] - Liu, X.; Sun, L.; Zhu, L.; Liu, J.; Lu, K.; Lu, J. High-order hierarchical nanotwins with superior strength and ductility. Acta Mater.
**2018**, 149, 397–406. [Google Scholar] [CrossRef] - Zhang, N.; Jin, S.; Sha, G.; Yu, J.; Cai, X.; Du, C.; Shen, T. Segregation induced hardening in annealed nanocrystalline Ni-Fe alloy. Mater. Sci. Eng. A
**2018**, 735, 354–360. [Google Scholar] [CrossRef] - Li, Q.J.; Ma, E. When ‘smaller is stronger’no longer holds. Mater. Res. Lett.
**2018**, 6, 283–292. [Google Scholar] [CrossRef] - Ren, Y.; Huang, Z.; Wang, Y.; Zhou, Q.; Yang, T.; Li, Q.; Jia, Q.; Wang, H. Friction-induced rapid amorphization in a wear-resistant (CoCrNi)88Mo12 dual-phase medium-entropy alloy at cryogenic temperature. Compos. Part B Eng.
**2023**, 263, 110833. [Google Scholar] [CrossRef] - Kale, C.; Srinivasan, S.; Hornbuckle, B.; Koju, R.; Darling, K.; Mishin, Y.; Solanki, K. An experimental and modeling investigation of tensile creep resistance of a stable nanocrystalline alloy. Acta Mater.
**2020**, 199, 141–154. [Google Scholar] [CrossRef] - Meraj; Pal, S. The Effect of Temperature on Creep Behaviour of Porous (1 at.%) Nano Crystalline Nickel. Trans. Indian Inst. Met.
**2015**, 69, 277–282. [Google Scholar] [CrossRef] - Nie, K.; Wu, W.-P.; Zhang, X.-L.; Yang, S.-M. Molecular dynamics study on the grain size, temperature, and stress dependence of creep behavior in nanocrystalline nickel. J. Mater. Sci.
**2016**, 52, 2180–2191. [Google Scholar] [CrossRef] - Yang, X.-S.; Wang, Y.-J.; Zhai, H.-R.; Wang, G.-Y.; Su, Y.-J.; Dai, L.; Ogata, S.; Zhang, T.-Y. Time-, stress-, and temperature-dependent deformation in nanostructured copper: Creep tests and simulations. J. Mech. Phys. Solids
**2016**, 94, 191–206. [Google Scholar] [CrossRef] - Wang, Y.-J.; Ishii, A.; Ogata, S. Transition of creep mechanism in nanocrystalline metals. Phys. Rev. B
**2011**, 84, 224102. [Google Scholar] [CrossRef] - Coble, R.L. A Model for Boundary Diffusion Controlled Creep in Polycrystalline Materials. J. Appl. Phys.
**1963**, 34, 1679–1682. [Google Scholar] [CrossRef] - Nabarro, F.R.N. Steady-state diffusional creep. Philos. Mag.
**1967**, 16, 231–237. [Google Scholar] [CrossRef] - Sun, Z.; Liu, B.; He, C.; Xie, L.; Peng, Q. Shift of Creep Mechanism in Nanocrystalline NiAl Alloy. Materials
**2019**, 12, 2508. [Google Scholar] [CrossRef] - Yuasa, M.; Matsumoto, H.; Hakamada, M.; Mabuchi, M. Effects of Vacancies on Deformation Behavior in Nanocrystalline Nickel. Mater. Trans.
**2008**, 49, 2315–2321. [Google Scholar] [CrossRef] - Islam, Z.; Wang, B.; Hattar, K.; Gao, H.; Haque, A. Departing from the mutual exclusiveness of strength and ductility in nanocrystalline metals with vacancy induced plasticity. Scr. Mater.
**2018**, 157, 39–43. [Google Scholar] [CrossRef] - Ford, J.; Wheeler, J.; Movchan, A. Computer simulation of grain-boundary diffusion creep. Acta Mater.
**2002**, 50, 3941–3955. [Google Scholar] [CrossRef] - Millett, P.C.; Desai, T.; Yamakov, V.; Wolf, D. Atomistic simulations of diffusional creep in a nanocrystalline body-centered cubic material. Acta Mater.
**2008**, 56, 3688–3698. [Google Scholar] [CrossRef] - Zhao, F.; Zhang, J.; He, C.; Zhang, Y.; Gao, X.; Xie, L. Molecular dynamics simulation on creep behavior of nanocrystalline TiAl alloy. Nanomaterials
**2020**, 10, 1693. [Google Scholar] [CrossRef] [PubMed] - Zeng, Y.; Li, X. Atomistic simulations of high-temperature creep in nanotwinned TiAl alloys. Extrem. Mech. Lett.
**2021**, 44, 101253. [Google Scholar] [CrossRef] - Yang, C.; Yin, C.; Wu, Y.; Zhou, Q.; Liu, X. Atomic insights into the deformation mechanism of an amorphous wrapped nanolamellar heterostructure and its effect on self-lubrication. J. Mater. Res. Technol.
**2023**, 26, 4206–4218. [Google Scholar] [CrossRef] - Zhou, Q.; Luo, D.; Hua, D.; Ye, W.; Li, S.; Zou, Q.; Chen, Z.; Wang, H. Design and characterization of metallic glass/graphene multilayer with excellent nanowear properties. Friction
**2022**, 10, 1913–1926. [Google Scholar] [CrossRef] - Shi, Y.; Ye, W.; Hua, D.; Zhou, Q.; Huang, Z.; Liu, Y.; Li, S.; Guo, T.; Chen, Y.; Eder, S.J.; et al. Interfacial engineering for enhanced mechanical performance: High-entropy alloy/graphene nanocomposites. Mater. Today Phys.
**2023**, 38, 101220. [Google Scholar] [CrossRef] - Yao, H.; Ye, T.; Yu, W.; Wang, P.; Wu, J.; Wu, Y.; Chen, P. Atomic-scale investigation of creep behavior and deformation mechanism in nanocrystalline FeCrAl alloys. Mater. Des.
**2021**, 206, 109766. [Google Scholar] [CrossRef] - Pal, S.; Meraj, M. Investigation of reorganization of a nanocrystalline grain boundary network during biaxial creep deformation of nanocrystalline Ni using molecular dynamics simulation. J. Mol. Model.
**2019**, 25, 282. [Google Scholar] [CrossRef] - Pal, S.; Mishra, S.; Meraj; Mondal, A.; Ray, B. On the comparison of interrupted and continuous creep behaviour of nanocrystalline copper: A molecular dynamics approach. Mater. Lett.
**2018**, 229, 256–260. [Google Scholar] [CrossRef] - Weertman, J.R. Retaining the Nano in Nanocrystalline Alloys. Science
**2012**, 337, 921–922. [Google Scholar] [CrossRef] - Pal, S.; Meraj; Deng, C. Effect of Zr addition on creep properties of ultra-fine grained nanocrystalline Ni studied by molecular dynamics simulations. Comput. Mater. Sci.
**2017**, 126, 382–392. [Google Scholar] [CrossRef] - Li, G.; Zhang, F.; Zhu, D.; Wang, L. Segregation thickness effect on the mechanical behaviors of nanocrystalline Ni-doped W alloy. Phys. Lett. A
**2021**, 409, 127513. [Google Scholar] [CrossRef] - Hirel, P. Atomsk: A tool for manipulating and converting atomic data files. Comput. Phys. Commun.
**2015**, 197, 212–219. [Google Scholar] [CrossRef] - Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys.
**1995**, 117, 1–19. [Google Scholar] [CrossRef] - Etesami, S.A.; Asadi, E. Molecular dynamics for near melting temperatures simulations of metals using modified embedded-atom method. J. Phys. Chem. Solids
**2018**, 112, 61–72. [Google Scholar] [CrossRef] - Stukowski, A. Visualization and analysis of atomistic simulation data with OVITO—The Open Visualization Tool. Model. Simul. Mater. Sci. Eng.
**2010**, 18, 015012. [Google Scholar] [CrossRef] - Falk, M.L.; Langer, J.S. Dynamics of viscoplastic deformation in amorphous solids. Phys. Rev. E
**1998**, 57, 7192–7205. [Google Scholar] [CrossRef] - Shimizu, F.; Ogata, S.; Li, J. Theory of Shear Banding in Metallic Glasses and Molecular Dynamics Calculations. Mater. Trans.
**2007**, 48, 2923–2927. [Google Scholar] [CrossRef] - Herring, C. Diffusional Viscosity of a Polycrystalline Solid. J. Appl. Phys.
**1950**, 21, 437–445. [Google Scholar] [CrossRef]

**Figure 1.**(

**a**) Nanocrystalline Ni model with mean grain size of 3 nm, where baby blue and dark blue representing the atoms inside grains and at grain boundaries, respectively. (

**b**) Model with an initial vacancy concentration of 1%, with the blue dots representing the coordinates of the randomly deleted atomic sites.

**Figure 2.**(

**a**) Different stress levels applied on nanocrystalline Ni with different volume fractions of initial vacancies with loading time. Variations in (

**b**) creep displacement and (

**c**) creep strain rate with time under loading stress of 2 GPa and temperature of 800 K. (

**d**) Variation in creep displacement with initial vacancy concentration at creep times of 50 and 200 fs.

**Figure 3.**(

**a**) Variation trend of the increment of vacancy concentration with time during the deformation process of each model. (

**b**) Variation in the quantity increment of vacancies at 50 fs with varying initial vacancy concentration.

**Figure 4.**(

**a**) Variation in total number of dislocations of nanocrystalline Ni with different initial vacancy concentration with creep time. (

**b**) Variation in the number of different dislocation types with creep time for nanocrystalline Ni with initial vacancy concentration of 3%. (

**c**) Variation in the initial number of dislocations with initial vacancy concentration of nanocrystalline Ni models at creep time of 0 fs. (

**d**) Variation in the increment slope of the total number of dislocations and the number of 1/6{112} Shockley partial dislocations with initial vacancy concentration of nanocrystalline Ni models during creep.

**Figure 5.**(

**a**) Three-dimensional distribution of vacancy increment at creep time of 50 fs for nanocrystalline Ni with 1% vacancies, with the orange dot corresponding to deformation-induced vacancies. (

**b**) Distribution of vacancies in nanocrystalline Ni on the two-dimensional x = y plane, where the dark blue dotted line is along the GBs, and the baby blue dots representing the atoms inside grains. (

**c**) Slices with a thickness of 1 nm along the x = y plane of generated vacancies at 100, 200, 300, and 400 fs.

**Figure 6.**The distribution of the generated vacancies during creep for nanocrystalline Ni varies with initial vacancy concentrations of (

**a**) 0.1%, (

**b**) 3%, (

**c**) 6%, and (

**d**) 10% at creep time of 50 fs, where the baby blue dots and the orange dots representing the atoms inside grains and deformation-induced vacancies, respectively, and the dark blue dotted line is along the GBs.

**Figure 7.**Variation in shear strain distribution on the plane of x = y = 0 with initial vacancy concentration of (

**a**) 0.1%, (

**b**) 3%, (

**c**) 6%, and (

**d**) 10% at creep time of 50 fs, respectively. Atoms were all colored according to the calculated magnitude of von Mises shear strain η

_{i}

^{Mises}.

**Figure 8.**(

**a**) Dislocation distribution of nanocrystalline Ni with initial vacancy concentration of 6% at 50 fs, in which red dots represent atoms of HCP structure and green lines represent 1/6(112) Shockley dislocations. (

**b**) The slices with a thickness of 3.4 nm cutting along the plane of x = y = 0 from (

**a**). (

**c**) Schematic diagram of the creep deformation mechanism of nanocrystalline Ni, where blue dots and red dots represent initial vacancies and deformation-induced vacancies, respectively. And the black dashed line and green line represent grain boundaries and dislocation lines, respectively.

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**MDPI and ACS Style**

Cui, Y.; Shao, W.; Shi, Y.; Zhou, Q.
Initial Vacancy-Dependent High-Temperature Creep Behavior of Nanocrystalline Ni by Molecular Dynamics Simulation. *Coatings* **2024**, *14*, 63.
https://doi.org/10.3390/coatings14010063

**AMA Style**

Cui Y, Shao W, Shi Y, Zhou Q.
Initial Vacancy-Dependent High-Temperature Creep Behavior of Nanocrystalline Ni by Molecular Dynamics Simulation. *Coatings*. 2024; 14(1):63.
https://doi.org/10.3390/coatings14010063

**Chicago/Turabian Style**

Cui, Yan, Weidong Shao, Yeran Shi, and Qing Zhou.
2024. "Initial Vacancy-Dependent High-Temperature Creep Behavior of Nanocrystalline Ni by Molecular Dynamics Simulation" *Coatings* 14, no. 1: 63.
https://doi.org/10.3390/coatings14010063