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Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: Grain-size and temperature effects
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Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: Grain-size and temperature effects Jiejie Li a , Binbin Lu a , Hongjian Zhou a , Chenyao Tian a , Yuehui Xian a , Guoming Hu a , Re Xia a,b,∗ a b
Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan 430072, China Hubei Key Laboratory of Waterjet Theory and New Technology (Wuhan University), Wuhan 430072, China
a r t i c l e
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Article history: Received 8 May 2018 Received in revised form 23 September 2018 Accepted 15 October 2018 Available online xxxx Communicated by R. Wu Keywords: Nanocrystalline platinum Grain-size effect Temperature effect Mechanical properties Molecular dynamics
a b s t r a c t A series of molecular dynamics simulations has been carried out to study the mechanical properties of nanocrystalline platinum. The effects of average grain size and temperature on mechanical behaviors are discussed. The simulated uniaxial tensile results indicate the presence of a critical average grain size about 14.1 nm, for which there is an inversion of the conventional Hall-Petch relation at temperature of 300 K. The transition can be explained by a change of dominant deformation mechanism from dislocation motion for average grain size above 14.1 nm to grain boundary sliding for smaller grain size. The Young’s modulus shows a linear relationship with the reciprocal of grain size, and the modulus of the grain boundary is about 42% of that of the grain core at 300 K. The parameters of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, decrease with the increase of temperature. It is noteworthy that the critical average grain size for the inversion of the Hall-Petch relation is sensitive to temperature and the Young’s modulus has an approximate linear relation with the temperature. The results will accelerate its functional applications of nanocrystalline materials. © 2019 Elsevier B.V. All rights reserved.
1. Introduction A decrease in grain size usually increases the flow stress, yield stress and hardness of metals and alloys [1,2]. The increase in aforementioned mechanical parameters is inversely proportional to the square root of the grain size, known as the Hall-Petch (HP) relation [3]. The HP relation results in the suggestions of nanocrystalline materials as candidates for high-strength and highhardness materials. The reduction of grain size to nanometer range (∼2–100 nm) has led to many interesting materials properties, including those involving mechanical behavior. Experimental study on nanocrystalline copper and palladium by Chokshi et al. [4] firstly found a negative HP slope at room temperature and the flow stress declines with the reduction of grain size below a critical grain size. Molecular dynamics (MD) simulation is a powerful tool to investigate the mechanical properties of nanocrystalline materials. By varying the grain size between 5 and 50 nm, Schiøtz et al. [5] found that the flow stress and the strength of nanocrystalline
*
Corresponding author at: Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan 430072, China. E-mail address: [email protected] (R. Xia). https://doi.org/10.1016/j.physleta.2018.10.053 0375-9601/© 2019 Elsevier B.V. All rights reserved.
copper exhibit a maximum at a grain size of 10 to 15 nm. The effects of grain size and shape on the mechanical properties of nanocrystalline copper have been also investigated by Zhou et al. [6], and the results reveal the presence of three grain size regions: for region I (grain size d < 8 nm), grain boundary sliding is the main deformation mechanism and the flow stress decreases with decreasing grain size; for region II (d ≈ 8–20 nm), detwinning process appears a competitive deformation mechanism with the twinning process and the flow stress begins to decrease slightly; for region III (d ≈ 20–53 nm), the main deformation mechanism is dislocation movement and flow stress increases with the decrease of grain size. Using the initial realistic samples of nanocrystalline copper, Rida et al. [7] found the presence of a critical mean grain size between 16 and 20 nm, for which there is an inversion of the conventional HP relation. There is also a transition of HP relation about the grain size of 18 nm for nanocrystalline aluminum, and the reason can attribute to the shift of deformation mechanism from dislocations to grain boundary-based deformation [8,9]. Simulations on nanocrystalline nickel and tantalum have also revealed the transition of deformation mechanisms as the grain size decreases [10–12]. These simulations have provided valuable insights into the deformation mechanisms of nanocrystalline materials and data supports for their functional applications, but most of the works have
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so far been carried out for copper [5–7,13], aluminum [8,9] and nickel [10,11]. Until now, there are only a limited number of studies on nanocrystalline platinum [14–16], especially focusing on the grain-size effect. The related data of nanocrystalline platinum is urgently needed for further researches. In the meantime, temperature is one of the important parameters affecting mechanical properties of metals, nanocrystalline materials included. Fang et al. [17] have investigated the effects of grain size (varying from 3 to 8 nm) and temperature (100 and 300 K) on the mechanical properties of quasi-two-dimensional nanocrystalline copper, and the results show that the grain growth is caused by the grain rotation and leads to strain hardening for grain size d < 5 nm at 300 K and for d < 5 nm at 100 K. According to the published experimental data, the influence of grain size on flow stress of copper with the mean grain size from nanometers to millimeters has been analyzed by Conrad et al. [18]. Three grain size regions were identified by the relationship between flow stress and inverse grain size. It was found that the transition grain size from one region to another is sensitive to temperature and strain rate. Furthermore, when applied to fuel cell, the working environments of nanocrystalline platinum catalysts will be different and the temperature might change at any time. As a result, the discussion of temperature effect on mechanical properties is critical to its functional application of nanocrystalline platinum and is still open to further investigations. In this work, MD simulations are used to investigate the dependence of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, on the average grain size between 6.1 and 24.2 nm of nanocrystalline platinum. Additionally, the effect of temperature on the mechanical properties of nanocrystalline platinum is investigated based on simulation results. 2. Methodology Nanocrystalline structures with random distributed crystal orientations for the MD simulations here are constructed by using the Voronoi construction method [19]. Periodic boundary is applied in all three directions of the structure. All samples contain 14 grains and the average grain size varies from 6.1 to 24.2 nm. The variation of average grain size is controlled by changing the simulation domain dimensions while keeping the number and geometry of grains unchanged, obtaining self-similar samples. The simulation domain dimensions of each sample range from 30a0 (11.76 nm) to 120a0 (47.04 nm), in which a0 demonstrates the lattice constant of platinum and a0 = 0.392 nm at 300 K, to provide mean grain sizes of 6.1–24.2 nm, respectively. The number of atoms in the samples varies from about 1.1 × 105 to 6.9 × 106 corresponding to the average grain size of 6.1 and 24.2 nm, respectively. MD simulations are operated on the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), developed by Plimpton and his co-workers [20]. An embedded-atom method (EAM) potential suggested by Foiles et al. [21] is used to calculate Pt-Pt interatomic forces in the simulations, which has been used extensively in atomistic simulations of metals and alloys [6,7,10,11, 22–24]. Before loading, each sample is first relaxed to the minimum energy configuration using the conjugate gradient method and then thermally equilibrated at test temperature for 80 ps using a NoséHoover thermostat. The maximum force and the energy tolerance set for the conjugate gradient method are both 10−15 . Periodic boundary conditions are imposed in all three directions of the simulation box, and the time step is set as 1 fs. After equilibration, uniaxial tension is applied by stretching the simulation box along [100] direction, with an engineering strain rate of 109 /s in the NPT ensemble, and pressures in the directions orthogonal to the ten-
Fig. 1. Atomic configuration of nanocrystalline platinum with an average grain size of 14.1 nm after relaxation at 300 K. The uniaxial load is applied along X-axial direction, [100] and the atoms are colored by CNA method. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
sile axis are set to 0 during deformation. In each loading step, a prescribed uniform strain increment of 0.1% is applied and then the whole system is relaxed for 3 ps at test temperature using a Nosé-Hoover thermostat. OVITO [25] and its built-in common neighbor analysis (CNA) [26] are used to visualize the resulting structures and deformation mechanisms. The atoms in the samples are colored according to the CNA method, where green for the face-centered cubic (FCC) atoms, blue for the body-centered cubic (BCC) atoms, red for hexagonal close-packed (HCP) atoms and gray for non-12coordinated atoms. FCC and HCP atoms are categorized as grain core atoms, and other atoms are considered to be the atoms in grain boundaries. A sample with an average grain size of 14.1 nm after relaxation at 300 K is shown in Fig. 1. 3. Results and discussion 3.1. Effect of grain size on mechanical properties The stress-strain curves of nanocrystalline samples are provided in Figure S1 (which is available in Supplementary Materials). The mechanical parameters can be obtained from the curves. The Young’s modulus is defined as the slope of the stress-strain curve in the elastic domain and has been calculated for ε ≤ 2%. The ultimate strength σu is determined as the maximal stress in tension, the yield stress σs is obtained by the stress of 0.2% offset strain, and the flow stress σ f is calculated as the average stress in the strain interval from 3% to 15%. Figures S2 in the Supplementary Materials presents the variation of ultimate strength and yield stress as a function of grain size. The ultimate strength and the yield stress keeps a decreasing trend as a whole with the decrease of grain size. At the same time, Figure S2(b) reveals the presence of reverse HP relation between yield stress and grain size, in which larger grain size results in the higher yield stress, consistent with the phenomenon in other nanocrystalline materials [7,17,27].
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Fig. 2. Effect of grain size on the flow stress at 300 K. The data of flow stress and grain size is fitted by Hall-Petch relationship.
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3.1.1. Flow stress According to HP relation, the relation between flow stress σ f and the reciprocal of grain size d−1/2 is depicted in Fig. 2, which are tested at temperature of 300 K. It can be seen that there are two main regions, one is that the flow stress increases as the grain size decreases, confirming the conventional HP tendency for d ≥ 14.1 nm, another is the reverse HP tendency for d ≤ 14.1 nm, in which case the flow stress decreases with the increase of grain size. The result reveals the presence of a critical mean grain size about 14.1 nm, for which there is an inversion of the conventional HP relation. The reverse HP effect has also been observed in nanocrystalline metals, including copper [5–7], nickel [11], aluminum [8,9], tantalum [12] and so on. The critical average grain size for transition is also ranging from nanometers to decades of nanometer, such as, 18 nm for aluminum [8,9], 10–15 nm for copper [5] and 10 nm for nickel [11]. The typical samples with average grain size of 9.1 nm and 18.2 nm in the two regions are chosen to analyze the deformation behaviors and microstructure evolution of nanocrystalline platinum. For the sample with average grain size of 18.2 nm, major deformation occurs in the grain cores although small amount of grain boundary sliding is observed. The evolution of a represen-
Fig. 3. Formation process of the dislocation emitted from grain boundary (a) (b), and slide inside the grain (c) up to the opposite grain boundary (d) from the sample with average grain size of 18.2 nm under different strains: (a) ε = 0.030, (b) ε = 0.039, (c) ε = 0.043, (d) ε = 0.047. The atoms are colored by CNA method.
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tative dislocation is illustrated in Fig. 3 circled in blue and similar phenomenon is common during deformation for the sample with average grain size of 18.2 nm. The energy of atoms in grain boundaries is higher than those in grain core. The energy of grain boundaries will constantly increase when the applied load is applied. Partial dislocations are emitted from grain boundaries to relax the excess of energy, as shown in Fig. 3(b). With the increase of applied strain, the partial dislocations glide and travel through the grain, depicted in Fig. 3(c). Eventually, the partial dislocations vanish into opposite grain boundaries, leaving behind stacking faults in the interior of grains (see Fig. 3(d)). For the samples with smaller grain size, the fraction of grain boundaries increases and the influence of grain boundaries highlights, which promotes the deformation mechanism of grain boundary sliding. Motion of dislocations is also observed in the sample with average grain size of 9.1 nm, but it is not the dominant deformation mechanism. Fig. 4 shows a slab between planes z = 13.3 nm and z = 14.1 nm of the sample with average grain size of 9.1 nm, in which the atoms are colored by CNA method and the red arrows represent the displacement vectors of atoms from the strain of 3% to 6%. It is shown the strain accumulates mainly in the grain boundaries, indicating that the grain boundary sliding is operating. Fig. 5 gives displacement maps of atoms and their corresponding CNA pictures of the sample with average grain size of 9.1 nm. Atomic motions from the initial configuration to the strain of 6% with eliminating homogeneous deformation are pointed by red arrows in Fig. 5 (a) and (c). The larger blue arrows represent the whole movement trend of grain boundaries. Fig. 5 (a) and (c) indicate the deformation mechanism of grain boundary sliding and grain rotation, respectively. Although grain rotation is observed in the sample with average grain size of 9.1 nm, shown in Fig. 5 (b) and (d), it is not often observed as usual as grain boundary sliding. The reason can be explained by the fact that grain boundary sliding is a precursor for grain rotation and grain boundary sliding is the dominant deformation mechanism in the sample with smaller grain size. It can be concluded that the deformation mechanisms observed in nanocrystalline platinum in this study is a combination of dislocation motion and grain boundary sliding, but one is often preponderant. Dislocation motion dominates the deformation mechanism at the average grain size above 14.1 nm and grain boundary sliding dominates below the critical average grain size. The transition of relationship between flow stress and grain size, HP relation, can be explained by a change of the dominant deformation mechanism. 3.1.2. Young’s modulus As shown in Fig. 6, the Young’s modulus E approximately follows a linear relationship with the reciprocal of grain size d−1 . Using an effective-medium approach, the similar linear relation has also been proposed by Nan et al. [28]. Zhou et al. [6] and Rida et al. [7] investigated the effects of grain size on the mechanical properties of nanocrystalline copper using molecular dynamics simulations, and the relationship between modulus and grain size tended to be a linearity. Sanders et al. [29] have measured the Young’s modulus of nanocrystalline copper and palladium by experiments, and the test results showed to be proportional to the reciprocal of grain size. However, the linear relationship between E and d−1 tends to be more suitable for nanocrystalline platinum in the present study, at least in the range of average grain size between 6.1 and 24.2 nm. Fig. 7 shows the proportion of atoms in grains and the proportion of atoms in grain boundaries as a function of the reciprocal of grain size. As shown in Fig. 7, the proportion of atoms in grains will increase and the proportion of atoms in grain boundaries will decrease as the average grain size increases. Moreover, the propor-
Fig. 4. Slab between planes z = 13.2 nm and z = 14.1 nm of the sample with average grain size of 9.1 nm, in which the atoms are colored by CNA method and the red arrows represent the displacement vectors of atoms from the stain of 3% to 6%.
tion of atoms in grains or in grain boundaries shows a good linear relationship with the reciprocal of grain size. In the meantime, the relationship between modulus and the proportion of atoms in grains is nearly linear (see the blue fitting curves in Fig. 8). As a result, the linear relationship between modulus E and the reciprocal of grain size d−1 can be concluded. The potential energy of atoms in grain boundaries is larger than those in grains, resulting in easily movement for atoms in grain boundaries when the uniaxial load is applied. Therefore, atoms in grain boundaries contribute less to the elastic resistance than atoms in grains. Moreover, the previous studies on nanocrystalline copper revealed that the modulus of grain boundaries is less than 30% than that of grain core [6,30]. It can be inferred that the similar phenomenon still exists in nanocrystalline platinum. To prove the point, the relationship between modulus and proportion of atoms in grains are fitted by the formula [30]
E=
1
ϕ / E core + (1 − ϕ )/ E gb
(1)
where E is the Young’s modulus of the nanocrystalline samples, E core and E gb are the Young’s modulus of the grain core component and the grain boundary component, respectively, and ϕ is the proportion of atoms in grains. Assuming all atoms are in grains, in other words, the proportion of atoms in grains is 100%, the aforementioned formula is evolved into E = E core . Therefore, the value of E core can be extrapolated from the data in Fig. 6 as the average grain size tends to be infinite. The intercept of fitting curve in Fig. 6 is 128.23, indicating that the value of E core can be regarded as 128.23 GPa. Fitting the relationship between Young’s modulus and the proportion of atoms in grains by Eq. (1) (see the red line shown in Fig. 8), the value of E gb is about 53.28 GPa, showing that the Young’s modulus of grain boundaries is about 42% of that of grain core component. Therefore, the decrease of grain size decreases the proportion of atoms in grains and increases the proportion of atoms in grain boundaries, resulting in the decrease of Young’s modulus.
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Fig. 5. Snapshots of grain boundary sliding (a) (b), and grain rotation (c) (d) from the sample with average grain size of 9.1 nm. (a) and (c) are shown by atom displacement vectors which can demonstrate grain boundary sliding and grain rotation, respectively. (b) and (d) are shown by CNA method which can demonstrate GB regions.
Fig. 6. Young’s modulus E as a function of the reciprocal of grain size d at 300 K.
Fig. 7. Fraction of atoms in grains and grain boundaries as a function of the reciprocal of grain size d.
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Fig. 8. Young’s modulus E as a function of the proportion of atoms in grains. The blue line is fitted by linear relation and the red line is fitted by Eq. (1).
Fig. 10. Variation of flow stress of nanocrystalline platinum with varied grain size at temperature of 10, 300, 750, and 1200 K.
perature. As depicted in Fig. 10, the flow stress of nanocrystalline platinum sample with a grain size of 14.1 nm is the largest among five samples under the temperature of 10 and 300 K. It indicates that the critical grain size for the transition of HP relation is sensitive to the tested temperature. 4. Conclusion
Fig. 9. Young’s modulus of nanocrystalline platinum with varied grain size at temperature of 10, 300, 750, and 1200 K.
3.2. Effect of temperature on mechanical properties The mechanical properties of nanocrystalline platinum samples under temperatures of 10 K, 300 K, 750 K and 1200 K, are also tested to investigate the temperature effect. Figure S3 in the Supplementary Materials gives the corresponding stress-strain curves of the sample with average grain size of 14.1 nm under different tested temperatures. Fig. 9 presents the Young’s modulus of samples with five varied grain sizes. It can be seen that the modulus reduces with the increase of temperature. The ability of atoms to diffuse will increase with the increase of temperature, and thus atoms are easier to be displaced in higher temperature when the external load is applied. Therefore, the higher temperature decreases the deformation resistibility capacity of atoms, which behaves with lower Young’s modulus. It is noteworthy that there is an approximately linear relationship between the Young’s modulus and temperature, at least in the present range of temperature (10–1200 K). Figure S4 in the Supplementary Materials shows the results of ultimate strength and yield stress of nanocrystalline platinum, and Fig. 10 displays the results of flow stress of nanocrystalline platinum samples with different grain size under four tested temperatures. As shown in Figure S4, the ultimate strength and yield stress decrease with the increase of temperature. In the meantime, the ultimate strength and yield stress keeps increasing with the increase of grain size and the changed trend keeps consistent with each other under the four specific tested temperatures. However, the changed trend of flow stress with grain size varies with tem-
A series of 3D nanocrystalline structures with diverse mean grain sizes have been generated by using the Voronoi construction method. Effects of grain size (6.1–24.2 nm) and temperature (10, 300, 750 and 1200 K) on the mechanical properties of nanocrystalline platinum have been investigated by MD simulations. The simulated uniaxial tensile results indicate the presence of a critical average grain size about 14.1 nm, for which there is an inversion of the conventional HP relation at temperature of 300 K. The deformation mechanisms observed in nanocrystalline platinum in this study is a combination of dislocation motion and grain boundary sliding, but one is often preponderant. Dislocation motion dominates the deformation mechanism at the average grain size above 14.1 nm and grain boundary sliding dominates below the critical average grain size. The transition of HP relation can be explained by a change of dominant deformation mechanism. The Young’s modulus shows a linear relationship with the reciprocal of grain size in the present range of 6.1–24.2 nm, showing a direct depend on the proportion of grain boundaries. By fitting referable equation, it is found that the modulus of the grain boundary component is about 42% of that of the grain core at 300 K. Meanwhile, the influence of temperature on the mechanical properties is critically addressed. The parameters of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, decrease with the increase of temperature. It is noteworthy that the critical average grain size for the inversion of the HP relation is sensitive to temperature and the Young’s modulus has an approximate linear relation with the temperature. The present study not only further clarifies the mechanical behaviors of nanocrystalline platinum, but also provides the data support for future researches. The results will accelerate its functional applications of nanocrystalline materials. Conflict of interest statement No potential conflict of interest was reported by the authors. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11102140 and 51575404).
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Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physleta.2018.10.053.
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Molecular dynamics simulation of mechanical properties of nanocrystalline platinum: Grain-size and temperature effects Jiejie Li a , Binbin Lu a , Hongjian Zhou a , Chenyao Tian a , Yuehui Xian a , Guoming Hu a , Re Xia a,b,∗ a b
Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan 430072, China Hubei Key Laboratory of Waterjet Theory and New Technology (Wuhan University), Wuhan 430072, China
a r t i c l e
i n f o
Article history: Received 8 May 2018 Received in revised form 23 September 2018 Accepted 15 October 2018 Available online xxxx Communicated by R. Wu Keywords: Nanocrystalline platinum Grain-size effect Temperature effect Mechanical properties Molecular dynamics
a b s t r a c t A series of molecular dynamics simulations has been carried out to study the mechanical properties of nanocrystalline platinum. The effects of average grain size and temperature on mechanical behaviors are discussed. The simulated uniaxial tensile results indicate the presence of a critical average grain size about 14.1 nm, for which there is an inversion of the conventional Hall-Petch relation at temperature of 300 K. The transition can be explained by a change of dominant deformation mechanism from dislocation motion for average grain size above 14.1 nm to grain boundary sliding for smaller grain size. The Young’s modulus shows a linear relationship with the reciprocal of grain size, and the modulus of the grain boundary is about 42% of that of the grain core at 300 K. The parameters of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, decrease with the increase of temperature. It is noteworthy that the critical average grain size for the inversion of the Hall-Petch relation is sensitive to temperature and the Young’s modulus has an approximate linear relation with the temperature. The results will accelerate its functional applications of nanocrystalline materials. © 2019 Elsevier B.V. All rights reserved.
1. Introduction A decrease in grain size usually increases the flow stress, yield stress and hardness of metals and alloys [1,2]. The increase in aforementioned mechanical parameters is inversely proportional to the square root of the grain size, known as the Hall-Petch (HP) relation [3]. The HP relation results in the suggestions of nanocrystalline materials as candidates for high-strength and highhardness materials. The reduction of grain size to nanometer range (∼2–100 nm) has led to many interesting materials properties, including those involving mechanical behavior. Experimental study on nanocrystalline copper and palladium by Chokshi et al. [4] firstly found a negative HP slope at room temperature and the flow stress declines with the reduction of grain size below a critical grain size. Molecular dynamics (MD) simulation is a powerful tool to investigate the mechanical properties of nanocrystalline materials. By varying the grain size between 5 and 50 nm, Schiøtz et al. [5] found that the flow stress and the strength of nanocrystalline
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Corresponding author at: Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, Wuhan 430072, China. E-mail address: [email protected] (R. Xia). https://doi.org/10.1016/j.physleta.2018.10.053 0375-9601/© 2019 Elsevier B.V. All rights reserved.
copper exhibit a maximum at a grain size of 10 to 15 nm. The effects of grain size and shape on the mechanical properties of nanocrystalline copper have been also investigated by Zhou et al. [6], and the results reveal the presence of three grain size regions: for region I (grain size d < 8 nm), grain boundary sliding is the main deformation mechanism and the flow stress decreases with decreasing grain size; for region II (d ≈ 8–20 nm), detwinning process appears a competitive deformation mechanism with the twinning process and the flow stress begins to decrease slightly; for region III (d ≈ 20–53 nm), the main deformation mechanism is dislocation movement and flow stress increases with the decrease of grain size. Using the initial realistic samples of nanocrystalline copper, Rida et al. [7] found the presence of a critical mean grain size between 16 and 20 nm, for which there is an inversion of the conventional HP relation. There is also a transition of HP relation about the grain size of 18 nm for nanocrystalline aluminum, and the reason can attribute to the shift of deformation mechanism from dislocations to grain boundary-based deformation [8,9]. Simulations on nanocrystalline nickel and tantalum have also revealed the transition of deformation mechanisms as the grain size decreases [10–12]. These simulations have provided valuable insights into the deformation mechanisms of nanocrystalline materials and data supports for their functional applications, but most of the works have
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so far been carried out for copper [5–7,13], aluminum [8,9] and nickel [10,11]. Until now, there are only a limited number of studies on nanocrystalline platinum [14–16], especially focusing on the grain-size effect. The related data of nanocrystalline platinum is urgently needed for further researches. In the meantime, temperature is one of the important parameters affecting mechanical properties of metals, nanocrystalline materials included. Fang et al. [17] have investigated the effects of grain size (varying from 3 to 8 nm) and temperature (100 and 300 K) on the mechanical properties of quasi-two-dimensional nanocrystalline copper, and the results show that the grain growth is caused by the grain rotation and leads to strain hardening for grain size d < 5 nm at 300 K and for d < 5 nm at 100 K. According to the published experimental data, the influence of grain size on flow stress of copper with the mean grain size from nanometers to millimeters has been analyzed by Conrad et al. [18]. Three grain size regions were identified by the relationship between flow stress and inverse grain size. It was found that the transition grain size from one region to another is sensitive to temperature and strain rate. Furthermore, when applied to fuel cell, the working environments of nanocrystalline platinum catalysts will be different and the temperature might change at any time. As a result, the discussion of temperature effect on mechanical properties is critical to its functional application of nanocrystalline platinum and is still open to further investigations. In this work, MD simulations are used to investigate the dependence of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, on the average grain size between 6.1 and 24.2 nm of nanocrystalline platinum. Additionally, the effect of temperature on the mechanical properties of nanocrystalline platinum is investigated based on simulation results. 2. Methodology Nanocrystalline structures with random distributed crystal orientations for the MD simulations here are constructed by using the Voronoi construction method [19]. Periodic boundary is applied in all three directions of the structure. All samples contain 14 grains and the average grain size varies from 6.1 to 24.2 nm. The variation of average grain size is controlled by changing the simulation domain dimensions while keeping the number and geometry of grains unchanged, obtaining self-similar samples. The simulation domain dimensions of each sample range from 30a0 (11.76 nm) to 120a0 (47.04 nm), in which a0 demonstrates the lattice constant of platinum and a0 = 0.392 nm at 300 K, to provide mean grain sizes of 6.1–24.2 nm, respectively. The number of atoms in the samples varies from about 1.1 × 105 to 6.9 × 106 corresponding to the average grain size of 6.1 and 24.2 nm, respectively. MD simulations are operated on the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), developed by Plimpton and his co-workers [20]. An embedded-atom method (EAM) potential suggested by Foiles et al. [21] is used to calculate Pt-Pt interatomic forces in the simulations, which has been used extensively in atomistic simulations of metals and alloys [6,7,10,11, 22–24]. Before loading, each sample is first relaxed to the minimum energy configuration using the conjugate gradient method and then thermally equilibrated at test temperature for 80 ps using a NoséHoover thermostat. The maximum force and the energy tolerance set for the conjugate gradient method are both 10−15 . Periodic boundary conditions are imposed in all three directions of the simulation box, and the time step is set as 1 fs. After equilibration, uniaxial tension is applied by stretching the simulation box along [100] direction, with an engineering strain rate of 109 /s in the NPT ensemble, and pressures in the directions orthogonal to the ten-
Fig. 1. Atomic configuration of nanocrystalline platinum with an average grain size of 14.1 nm after relaxation at 300 K. The uniaxial load is applied along X-axial direction, [100] and the atoms are colored by CNA method. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
sile axis are set to 0 during deformation. In each loading step, a prescribed uniform strain increment of 0.1% is applied and then the whole system is relaxed for 3 ps at test temperature using a Nosé-Hoover thermostat. OVITO [25] and its built-in common neighbor analysis (CNA) [26] are used to visualize the resulting structures and deformation mechanisms. The atoms in the samples are colored according to the CNA method, where green for the face-centered cubic (FCC) atoms, blue for the body-centered cubic (BCC) atoms, red for hexagonal close-packed (HCP) atoms and gray for non-12coordinated atoms. FCC and HCP atoms are categorized as grain core atoms, and other atoms are considered to be the atoms in grain boundaries. A sample with an average grain size of 14.1 nm after relaxation at 300 K is shown in Fig. 1. 3. Results and discussion 3.1. Effect of grain size on mechanical properties The stress-strain curves of nanocrystalline samples are provided in Figure S1 (which is available in Supplementary Materials). The mechanical parameters can be obtained from the curves. The Young’s modulus is defined as the slope of the stress-strain curve in the elastic domain and has been calculated for ε ≤ 2%. The ultimate strength σu is determined as the maximal stress in tension, the yield stress σs is obtained by the stress of 0.2% offset strain, and the flow stress σ f is calculated as the average stress in the strain interval from 3% to 15%. Figures S2 in the Supplementary Materials presents the variation of ultimate strength and yield stress as a function of grain size. The ultimate strength and the yield stress keeps a decreasing trend as a whole with the decrease of grain size. At the same time, Figure S2(b) reveals the presence of reverse HP relation between yield stress and grain size, in which larger grain size results in the higher yield stress, consistent with the phenomenon in other nanocrystalline materials [7,17,27].
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Fig. 2. Effect of grain size on the flow stress at 300 K. The data of flow stress and grain size is fitted by Hall-Petch relationship.
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3.1.1. Flow stress According to HP relation, the relation between flow stress σ f and the reciprocal of grain size d−1/2 is depicted in Fig. 2, which are tested at temperature of 300 K. It can be seen that there are two main regions, one is that the flow stress increases as the grain size decreases, confirming the conventional HP tendency for d ≥ 14.1 nm, another is the reverse HP tendency for d ≤ 14.1 nm, in which case the flow stress decreases with the increase of grain size. The result reveals the presence of a critical mean grain size about 14.1 nm, for which there is an inversion of the conventional HP relation. The reverse HP effect has also been observed in nanocrystalline metals, including copper [5–7], nickel [11], aluminum [8,9], tantalum [12] and so on. The critical average grain size for transition is also ranging from nanometers to decades of nanometer, such as, 18 nm for aluminum [8,9], 10–15 nm for copper [5] and 10 nm for nickel [11]. The typical samples with average grain size of 9.1 nm and 18.2 nm in the two regions are chosen to analyze the deformation behaviors and microstructure evolution of nanocrystalline platinum. For the sample with average grain size of 18.2 nm, major deformation occurs in the grain cores although small amount of grain boundary sliding is observed. The evolution of a represen-
Fig. 3. Formation process of the dislocation emitted from grain boundary (a) (b), and slide inside the grain (c) up to the opposite grain boundary (d) from the sample with average grain size of 18.2 nm under different strains: (a) ε = 0.030, (b) ε = 0.039, (c) ε = 0.043, (d) ε = 0.047. The atoms are colored by CNA method.
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tative dislocation is illustrated in Fig. 3 circled in blue and similar phenomenon is common during deformation for the sample with average grain size of 18.2 nm. The energy of atoms in grain boundaries is higher than those in grain core. The energy of grain boundaries will constantly increase when the applied load is applied. Partial dislocations are emitted from grain boundaries to relax the excess of energy, as shown in Fig. 3(b). With the increase of applied strain, the partial dislocations glide and travel through the grain, depicted in Fig. 3(c). Eventually, the partial dislocations vanish into opposite grain boundaries, leaving behind stacking faults in the interior of grains (see Fig. 3(d)). For the samples with smaller grain size, the fraction of grain boundaries increases and the influence of grain boundaries highlights, which promotes the deformation mechanism of grain boundary sliding. Motion of dislocations is also observed in the sample with average grain size of 9.1 nm, but it is not the dominant deformation mechanism. Fig. 4 shows a slab between planes z = 13.3 nm and z = 14.1 nm of the sample with average grain size of 9.1 nm, in which the atoms are colored by CNA method and the red arrows represent the displacement vectors of atoms from the strain of 3% to 6%. It is shown the strain accumulates mainly in the grain boundaries, indicating that the grain boundary sliding is operating. Fig. 5 gives displacement maps of atoms and their corresponding CNA pictures of the sample with average grain size of 9.1 nm. Atomic motions from the initial configuration to the strain of 6% with eliminating homogeneous deformation are pointed by red arrows in Fig. 5 (a) and (c). The larger blue arrows represent the whole movement trend of grain boundaries. Fig. 5 (a) and (c) indicate the deformation mechanism of grain boundary sliding and grain rotation, respectively. Although grain rotation is observed in the sample with average grain size of 9.1 nm, shown in Fig. 5 (b) and (d), it is not often observed as usual as grain boundary sliding. The reason can be explained by the fact that grain boundary sliding is a precursor for grain rotation and grain boundary sliding is the dominant deformation mechanism in the sample with smaller grain size. It can be concluded that the deformation mechanisms observed in nanocrystalline platinum in this study is a combination of dislocation motion and grain boundary sliding, but one is often preponderant. Dislocation motion dominates the deformation mechanism at the average grain size above 14.1 nm and grain boundary sliding dominates below the critical average grain size. The transition of relationship between flow stress and grain size, HP relation, can be explained by a change of the dominant deformation mechanism. 3.1.2. Young’s modulus As shown in Fig. 6, the Young’s modulus E approximately follows a linear relationship with the reciprocal of grain size d−1 . Using an effective-medium approach, the similar linear relation has also been proposed by Nan et al. [28]. Zhou et al. [6] and Rida et al. [7] investigated the effects of grain size on the mechanical properties of nanocrystalline copper using molecular dynamics simulations, and the relationship between modulus and grain size tended to be a linearity. Sanders et al. [29] have measured the Young’s modulus of nanocrystalline copper and palladium by experiments, and the test results showed to be proportional to the reciprocal of grain size. However, the linear relationship between E and d−1 tends to be more suitable for nanocrystalline platinum in the present study, at least in the range of average grain size between 6.1 and 24.2 nm. Fig. 7 shows the proportion of atoms in grains and the proportion of atoms in grain boundaries as a function of the reciprocal of grain size. As shown in Fig. 7, the proportion of atoms in grains will increase and the proportion of atoms in grain boundaries will decrease as the average grain size increases. Moreover, the propor-
Fig. 4. Slab between planes z = 13.2 nm and z = 14.1 nm of the sample with average grain size of 9.1 nm, in which the atoms are colored by CNA method and the red arrows represent the displacement vectors of atoms from the stain of 3% to 6%.
tion of atoms in grains or in grain boundaries shows a good linear relationship with the reciprocal of grain size. In the meantime, the relationship between modulus and the proportion of atoms in grains is nearly linear (see the blue fitting curves in Fig. 8). As a result, the linear relationship between modulus E and the reciprocal of grain size d−1 can be concluded. The potential energy of atoms in grain boundaries is larger than those in grains, resulting in easily movement for atoms in grain boundaries when the uniaxial load is applied. Therefore, atoms in grain boundaries contribute less to the elastic resistance than atoms in grains. Moreover, the previous studies on nanocrystalline copper revealed that the modulus of grain boundaries is less than 30% than that of grain core [6,30]. It can be inferred that the similar phenomenon still exists in nanocrystalline platinum. To prove the point, the relationship between modulus and proportion of atoms in grains are fitted by the formula [30]
E=
1
ϕ / E core + (1 − ϕ )/ E gb
(1)
where E is the Young’s modulus of the nanocrystalline samples, E core and E gb are the Young’s modulus of the grain core component and the grain boundary component, respectively, and ϕ is the proportion of atoms in grains. Assuming all atoms are in grains, in other words, the proportion of atoms in grains is 100%, the aforementioned formula is evolved into E = E core . Therefore, the value of E core can be extrapolated from the data in Fig. 6 as the average grain size tends to be infinite. The intercept of fitting curve in Fig. 6 is 128.23, indicating that the value of E core can be regarded as 128.23 GPa. Fitting the relationship between Young’s modulus and the proportion of atoms in grains by Eq. (1) (see the red line shown in Fig. 8), the value of E gb is about 53.28 GPa, showing that the Young’s modulus of grain boundaries is about 42% of that of grain core component. Therefore, the decrease of grain size decreases the proportion of atoms in grains and increases the proportion of atoms in grain boundaries, resulting in the decrease of Young’s modulus.
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Fig. 5. Snapshots of grain boundary sliding (a) (b), and grain rotation (c) (d) from the sample with average grain size of 9.1 nm. (a) and (c) are shown by atom displacement vectors which can demonstrate grain boundary sliding and grain rotation, respectively. (b) and (d) are shown by CNA method which can demonstrate GB regions.
Fig. 6. Young’s modulus E as a function of the reciprocal of grain size d at 300 K.
Fig. 7. Fraction of atoms in grains and grain boundaries as a function of the reciprocal of grain size d.
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Fig. 8. Young’s modulus E as a function of the proportion of atoms in grains. The blue line is fitted by linear relation and the red line is fitted by Eq. (1).
Fig. 10. Variation of flow stress of nanocrystalline platinum with varied grain size at temperature of 10, 300, 750, and 1200 K.
perature. As depicted in Fig. 10, the flow stress of nanocrystalline platinum sample with a grain size of 14.1 nm is the largest among five samples under the temperature of 10 and 300 K. It indicates that the critical grain size for the transition of HP relation is sensitive to the tested temperature. 4. Conclusion
Fig. 9. Young’s modulus of nanocrystalline platinum with varied grain size at temperature of 10, 300, 750, and 1200 K.
3.2. Effect of temperature on mechanical properties The mechanical properties of nanocrystalline platinum samples under temperatures of 10 K, 300 K, 750 K and 1200 K, are also tested to investigate the temperature effect. Figure S3 in the Supplementary Materials gives the corresponding stress-strain curves of the sample with average grain size of 14.1 nm under different tested temperatures. Fig. 9 presents the Young’s modulus of samples with five varied grain sizes. It can be seen that the modulus reduces with the increase of temperature. The ability of atoms to diffuse will increase with the increase of temperature, and thus atoms are easier to be displaced in higher temperature when the external load is applied. Therefore, the higher temperature decreases the deformation resistibility capacity of atoms, which behaves with lower Young’s modulus. It is noteworthy that there is an approximately linear relationship between the Young’s modulus and temperature, at least in the present range of temperature (10–1200 K). Figure S4 in the Supplementary Materials shows the results of ultimate strength and yield stress of nanocrystalline platinum, and Fig. 10 displays the results of flow stress of nanocrystalline platinum samples with different grain size under four tested temperatures. As shown in Figure S4, the ultimate strength and yield stress decrease with the increase of temperature. In the meantime, the ultimate strength and yield stress keeps increasing with the increase of grain size and the changed trend keeps consistent with each other under the four specific tested temperatures. However, the changed trend of flow stress with grain size varies with tem-
A series of 3D nanocrystalline structures with diverse mean grain sizes have been generated by using the Voronoi construction method. Effects of grain size (6.1–24.2 nm) and temperature (10, 300, 750 and 1200 K) on the mechanical properties of nanocrystalline platinum have been investigated by MD simulations. The simulated uniaxial tensile results indicate the presence of a critical average grain size about 14.1 nm, for which there is an inversion of the conventional HP relation at temperature of 300 K. The deformation mechanisms observed in nanocrystalline platinum in this study is a combination of dislocation motion and grain boundary sliding, but one is often preponderant. Dislocation motion dominates the deformation mechanism at the average grain size above 14.1 nm and grain boundary sliding dominates below the critical average grain size. The transition of HP relation can be explained by a change of dominant deformation mechanism. The Young’s modulus shows a linear relationship with the reciprocal of grain size in the present range of 6.1–24.2 nm, showing a direct depend on the proportion of grain boundaries. By fitting referable equation, it is found that the modulus of the grain boundary component is about 42% of that of the grain core at 300 K. Meanwhile, the influence of temperature on the mechanical properties is critically addressed. The parameters of mechanical properties, including Young’s modulus, ultimate strength, yield stress and flow stress, decrease with the increase of temperature. It is noteworthy that the critical average grain size for the inversion of the HP relation is sensitive to temperature and the Young’s modulus has an approximate linear relation with the temperature. The present study not only further clarifies the mechanical behaviors of nanocrystalline platinum, but also provides the data support for future researches. The results will accelerate its functional applications of nanocrystalline materials. Conflict of interest statement No potential conflict of interest was reported by the authors. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 11102140 and 51575404).
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Appendix A. Supplementary material Supplementary material related to this article can be found online at https://doi.org/10.1016/j.physleta.2018.10.053.
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