- Email: [email protected]
Temperature dependent vacancy formation energy of metallic materials
Physica B 584 (2020) 412071
Contents lists available at ScienceDirect
Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Temperature dependent vacancy formation energy of metallic materials Xuyao Zhang a, b, Weiguo Li a, b, *, Jiaxing Shao a, Yong Deng a, Jianzuo Ma a, Ying Li a, Dingyu Li c a
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing, 400044, China c Chongqing University of Science and Technology, Chongqing, 401331, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Vacancy formation energy Temperature dependent Modeling Metallic materials Defects
Knowledge of vacancy formation energy (VFE) for metallic materials at different temperatures is of fundamental importance. In the present study, following Brook’s method and cohesive energy theory, two temperature dependent VFE models of metallic materials are developed based on our previous studies. The developed models are validated using the available experimental and simulation results. Agreement is good between model pre dictions and corresponding available experimental and simulation results. This study indicates that the VFE firstly remains approximately constant and then goes down almost linearly with increase of temperature. Furthermore, this work can provide simple and effective methods to calculate VFE of metallic materials at different temperatures, and can help understand the fundamental mechanisms which control the decrease of VFE at high temperatures.
1. Introduction As a common point defect of metallic materials, the vacancy point defect plays a vital role for self-diffusion, and governs many material properties [1–5]. Therefore, the vacancy point defect has been of in terest to materials scientists for many decades. Wagner and Schottky [6] have shown that a way to create only vacancies: atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal. According to this way, the vacancy formation energy EV is the energy needed to break all the atomic bonds of the specific site to its surroundings, which contributes to self-diffusion and determines the vacancy concentration c at finite temperature for metallic materials [2, 3]. The relation between self-diffusion coefficient D and VFE at different temperatures is given by the following expression [2]: � � D∝e
ðEV þEM Þ= kB T
(1)
where T is the temperature (in Kelvin), EM is the migration energy, and kB is Boltzmann’s constant. The vacancy concentration can be expressed as a similar expression. According to the above Eq. (1), the slight vari ations in VFE arising from the change in temperature will lead to sig nificant changes in the self-diffusion coefficient and vacancy concentrations [7,8]. Therefore, how to determine the temperature
dependent VFE is fundamentally important to metallic materials. Experimentally, Balluffi [9] reported the VFES named “best values” of some metallic materials obtained by averaging the results which were measured from different researchers via positron annihilation mea surements [10] and quenching experiments [11,12]. Experimental methods seem to be the most straightforward approach to obtain VFE, and then can be of great significance for calculating equilibrium vacancy concentrations. However, these experimental values were considered to be temperature independent for the variations in VFES with temperature are very little. And just consider them as a constant is unreasonable, because the slight variations in VFES arising from the variation in tem perature will also cause outstanding errors in the self-diffusion and va cancy concentrations [7]. Besides, VFE can be significantly affected by interfaces, surfaces, grain boundaries, and dislocations in the experi mental process [13]. At present, the VFES have been widely studied by the first-principle theory [3,4], local harmonic approach (LH) [7], molecular dynamics simulation [14,15], extensions of the local harmonic approach (ELH) [7], Monte Carlo calculation [16], quasiharmonic approach (QH) [7, 17], and embedded-atom method [14,18,19]. Those studies show that the presence and motion of vacancies determine diffusivities, contribute to electrical resistivity and carrier recombination, and provide the mechanism for self-diffusion coefficient. Those computational methods are treated as powerful tools for studying VFE in today’s context of
* Corresponding author. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China. E-mail address: [email protected] (W. Li). https://doi.org/10.1016/j.physb.2020.412071 Received 10 December 2019; Accepted 5 February 2020 Available online 5 February 2020 0921-4526/© 2020 Elsevier B.V. All rights reserved.
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
high-performance computational materials science. There is no doubt about it. However, those methods are complex and laborious [20]. In addition, Brook [21] developed a theoretical model to study VFE of isotropic materials using shear modulus and surface energy. Afterwards, a relation between VFE and cohesive energy was established by Tiwari and Patil [22]. This relation indicates that the VFE is linear to the cohesive energy. Note that those model above do not take the effect of temperature on VFE into account. In this case, a general theoretical model needs to be developed for predicting the temperature dependent VFE of metallic materials. From this perspective, the purpose of the present paper is to study the effect of temperature on VFE and attempts to contribute simple and convenient models to characterize and predict the VFEs at different temperatures for metallic materials.
be regarded as constants at different temperatures. So Eq. (7) becomes:
2. Theoretical model
� Z VðTÞ ¼ V0 ðT0 Þ 1 þ
EV ðTÞ ¼
� Z rðTÞ ¼ r0 ðT0 Þ 1 þ
EV ¼
C1 h2 ðMVÞ
ð2πC2 krÞ2 m
U
�
T
αðTÞ dT
(11)
2.2. Model based on Brook’s method Besides, Brook [21] suggested that the formation of vacancies can be regarded as equivalent to creating a spherical surface within the crystal, and established a model to calculate the VFE EV of bulk materials, � � 1 EV ¼ 4πr2 γ (14) 1 þ γ=ð2G rÞ where γ is the surface energy per unit area of the surrounding crystal, G is the shear modulus, and r is the radius of the atom. Here we want to extend this model to different temperatures and use it to calculate temperature dependent VFE of metallic materials. We should note that surface energy γ, shear modulus G, and atomic radius r used in Eq. (14) are temperature dependent. Once those mate rials parameters at a reference temperature T0 are known, one obtains: � � 1 EV ðT0 Þ ¼ 4πr2 ðT0 ÞγðT0 Þ (15) 1 þ γðT0 Þ=ð2GðT0 ÞrðT0 ÞÞ
(6)
where C2 is a constant, M is the molar mass (kg mol 1 ), V is the volume of a mole of the solid. Combining Eq. (5) with Eq. (6), we can get: 2 3
(10)
where TB is the boiling point, CV ðTÞ is specific heat at constant volume, ΔHvap is the molar heat of evaporation. Finally, substituting Eqs. (10)–(12) to Eq. (9), we can obtain the final form of temperature dependent VFE model: ! RT CV ðTÞdT T EV ðTÞ ¼ EV ðT0 Þ 1 R TB 0 (13) CV ðTÞdT þ ΔHvap T0
112
ðMVÞ A
αðTÞ dT
T
Meanwhile, Mukherjee developed a relationship between VFE and Debye temperature [25], θD ¼ C2 @EV
�3
T
The cohesive energy at different temperatures can be obtained by our previous studies [26]: Z TB UðTÞ ¼ CV ðTÞdT þ ΔHvap (12)
where C1 is a constant, r is the atomic radius. Note that, the EAM po tential is more appropriate for metals, but the intrinsic relation between the force constant, the bond cohesive energy and bond length is the same as in the simple L-J potential [24]. Substituting Eqs. (3) and (4) to Eq. (2), we can obtain the relation between the Debye temperature and the cohesive energy: rffiffiffiffiffiffiffiffiffi h C1 U θD ¼ (5) 2πkr m
2 3
(9)
T0
(4)
,
(8)
UðTÞ
T0
where m is the mass of the oscillating body. The force constant can be expressed as the function of cohesive energy U based on the interatomic potential [24]:
0
ð2πC2 kÞ m rðTÞ
where EV ðT0 Þ, rðT0 Þ and VðT0 Þ are the VFE, atomic radius and volume of a mole of the solid at an arbitrary reference temperature T0 . The volume of a mole of the solid VðTÞ and atomic radius rðTÞ at different temperatures are related to linear expansion coefficient αðTÞ, and can be written as:
(2)
U r2
2
2
where h is Planck’s constant (6:626 � 10 34 J s), kB is Boltzmann’s constant (1:381 � 10 23 J K 1 ), v is the maximum frequency of the vibrating atoms about their equilibrium positions, and it is related to the force constant K[23]: rffiffiffi 1 K (3) v¼ 2π m
K ¼ C1
VðTÞ3
EV ðTÞ VðTÞ3 rðT0 Þ2 UðTÞ ¼ EV ðT0 Þ rðTÞ2 VðT0 Þ23 UðT0 Þ
It is known that the Debye temperature θD is defined as [23]: h v kB
2
2
Combining Eq. (7) with Eq. (8), we can get:
2.1. Model based on cohesive energy theory
θD ¼
2
C1 h2 M 3
Combining Eq. (14) with Eq. (15), we can obtain: EV ðTÞ ¼ EV ðT0 Þ
(7)
γðTÞGðTÞr3 ðTÞ 2GðT0 ÞrðT0 Þ þ γðT0 Þ γðT0 ÞGðT0 Þr3 ðT0 Þ 2GðTÞrðTÞ þ γðTÞ
(16)
In order to calculate VFE at different temperatures from Eq. (16), we should know the temperature dependent surface energy γðTÞ, shear modulus GðTÞ, and atomic radius rðTÞ. For the temperature dependent surface energy, Tyson [27] suggested a model:
From Eqs. (2)–(7), the relation between vacancy formation energy and cohesive energy is obtained. Eq. (7) is an important result of this study. To extend this model to different temperatures, we should consider the temperature dependent materials parameters. According to Liang’s study [24], C1 and C2 are all weakly temperature independent, and can
γðTÞA ¼ γðT0 ÞA
ðT
T0 ÞS
(17)
where γðTÞ and γðT0 Þ is the surface energy at temperature T and T0 , respectively, A is the molar surface area, surface entropy S was 2
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
approximately equal to the gas constant R. The shear modulus is related Young’s modulus GðTÞ ¼ EðTÞ= 2ð1 þ υÞ, where the Poisson’s ratio υ is weak temperature dependent, and can be treated as a constant. The temperature dependent Young’s modulus can be conveniently obtained from the current literatures, so the tem perature dependent shear modulus can be calculated easily. The following equation reported by Wachtman et al. [28]can be used to fit the temperature dependent shear modulus: G ¼ G0
by a simple Origin application program (see Table 2). Meanwhile, the specific heat capacities CV ðTÞ at different temperatures can be calculated from materials handbooks (for liquid state) [33] and the Debye model (for solid state) [34]. The Debye model can be written as: � CV ðTÞ ¼ 9N0 kB
2.3. Discussion on the models Hereto, two temperature dependent VFE models of metallic mate rials have been developed. The model based on cohesive energy theory (Eq. (13)) relates the temperature dependent VFE to vapor heat and the specific heat at constant volume. From Eqs. (7) and (13), we can draw a conclusion that the temperature dependent VFE is directly proportional to temperature dependent cohesive energy, and the decrease of VFE for metallic materials at high temperatures arises from cohesive energy weakening. One can see that the model based on Brook’s classical theory (Eq. (16)) relates the temperature dependent VFE to surface energy, shear modulus, and atomic radius at a reference temperature, which can uncover the inherent correlation between those material properties and VFE. 3. Results and discussion In this section, the temperature dependent VFEs of Cu, Ni, Au, Ag, and Pt were predicted by the two proposed models (Eq. (13)) and (Eq. (16)), and compared with available simulation and experimental results. The VFE at any temperature can be selected as the initial reference parameter. In general, the VFEs at 0 K could be easily obtained by computer simulation method, so the values at 0 K are chosen as the reference VFE for convenience. In the calculations, the material pa rameters of those metallic materials are listed in Table 1 [27,29–32]. Note that the linear expansion coefficient at different temperatures can be found in materials handbooks [33], and was fitted with a polynomial Table 1 Related parameters of Cu, Ni, Au, Ag, and Pt used for calculation (Eq. (13) and Eq. (16)). The ’G0 ’ and ’B0 ’ are fitted according to the experiment data obtained from Ref. [29] by a simple origin application program. Cu
Ni
Au
Ag
Pt
EV ðT0 ÞðeVÞ[7]
1.29
1.63
1.03
0.97
1.69
1.592
2.104
1.363
1.065
2.223
1.374
1.445
1.079
1.075
1.166
γðT0 ÞðJ m
2
Þ[30]
Aðm2 ðkg molÞ
1
Þ[20]
rðnmÞ[31]
0.128
0.125
0.144
0.145
0.139
G0 ðGPaÞ
50.7
89.2
32.1
31.9
62.3
θD ðKÞ[31]
343
450
165
225
240
Þ
0.0184
0.0912
0.0131
0.0131
1
B0 ðGPa K
1
0
θD T
x4 ex ðex
1Þ2
(19)
dx
Origin application program. The Young’s modulus of Cu, Ni, Au, Ag, and Pt at different temperatures were experimentally measured by many researchers, and can be easily obtained from existing literatures [29]. Then the temperature dependent shear modulus can be calculated by the relation between Young’s modulus and shear modulus, GðTÞ ¼ EðTÞ=2ð1 þ υÞ. The calculation results are fitted using Wachtman’s model [28], which can be seen in Fig. 1. Then the temperature depen dent VFEs of these five kinds of metallic materials were calculated. As shown in Fig. 2, the temperature dependent VFE of Cu were calculated by the two developed models (Eq. (13)) and (Eq. (16)), and compared with available simulation results, including LH results [7,35], ELH results [7], QH results [7,35] and MC results [35]. One can find that the VFE of Cu calculated from our models are in reasonable agreement with those available simulation results. Note that there are significant differences between those simulation results by different methods, because those results heavily rely on the employed empirical potentials [36]. Besides, from Fig. 2, one can see that the results obtained from LH approach, ELH approach, QH approach are fairly satisfactory, but the MC calculation reported by Foiles [35] underestimate the VFEs of Cu at different temperatures. According to Foiles’ work [35], the MC calcu lations is based on the Frenkel-Ladd method, which does not work well at high temperatures. The VFEs of Ni, Au, Ag, and Pt at different temperatures are pre dicted by the two developed models. The comparison between the calculated results and the available simulation results are plotted in Figs. 3–6. One can see that the temperature dependent VFEs calculated by Eq. (13) and Eq. (16) are in reasonable agreement with these simu lation results [7]. Besides, the predicted results of the two proposed models show a slight difference, in other words, the model (Eq. (13)) based on cohesive energy theory slightly overestimated the temperature dependent VFE. For VFE of Cu at 800 K, the deviation between the two model predictions is 2.7%, and the deviations between the model pre diction (Eq. (13)) and LH approach, ELH approach, QH approach are 1.3%, 1.0% and 6.9%, respectively. However, the model based cohesive energy theory is simple and convenient. In the calculation, the input parameters are vapor heat and the temperature dependent specific heat at constant volume. Because the values of these materials parameters can be easily obtained, the model (Eq. (13)) developed in this study shows its advantage. For (Eq. (16)), it relates the temperature dependent VFE to surface energy, shear modulus, and atomic radius, which can uncover the inherent relationship between those thermodynamic prop erties and VFE. Besides, we also compare our model predictions with experimental results to confirm our developed models. As shown in Figs. 2–6, the
where G0 is the value of shear modulus at absolute zero; B0 is a constant related to the Grüneisen parameter and T0 is roughly equal to θD /2, where θD is the Debye temperature. According to the discussion above, the surface energy, shear modulus, and atomic radius at different temperatures can be conve niently obtained. Once the surface energy, shear modulus, and atomic radius at a reference temperature T0 are known, the temperature dependent VFE can be easily predicted by substituting Eqs. (11), (17) and (18) to Eq. (16).
Metals
�3 Z
where N0 is the Avogadro number, kB is Boltzmann’s constant, and θD is RT Debye temperature. Then the T0 CV ðTÞdT can be achieved by a simple
(18)
B0 T expð T0 = TÞ
T θD
Table 2 Temperature dependent linear expansion coefficient of Cu, Ni, Au, Ag, and Pt [33]. Materials
αðTÞ ðK 1 Þ
0.0132
Cu
αðTÞ ¼ 1:156 � 10
5
þ 7:96 � 10
þ 1:093 � 10
336
428
368
284
564
Ni
αðTÞ ¼ 1:069 � 10
6
ν[32]
0.343
0.31
0.42
0.37
0.39
Au
αðTÞ ¼ 1:233 � 10
5
þ 5:607 � 10
TB ðKÞ[33]
2848
3187
3130
2800
4097
Ag
5
304.7
372.1
337.1
254.2
533.8
αðTÞ ¼ 1:593 � 10
þ 9:81 � 10
Pt
αðTÞ ¼ 8:090 � 10
6
þ 2:873 � 10
U0 ðkJ mol
Þ[31]
ΔHvap ðkJ mol
1
Þ[33]
3
9
9
9
�T
�T 9
�T
�T 9
�T
0:008 � T 0:441 � T
0:012 � T 0:203 � T
2
2 2
2
0:015 � T
2
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
110
Expt. Ni Fitted
Cu
100
Ag
90
Shear modulus (GPa)
Expt. Pt Fitted Expt. Fitted Au
Expt. Fitted Expt. Fitted
80 70 60 50 40 30 20 0
200
400 600 Temperature (K)
800
1000
Fig. 1. The temperature dependent shear modulus of Cu, Ni, Au, Ag and Pt. The experimental values are obtained from Ref. [29].
1.4
1.8 Cu Vacancy formation energy (eV)
Vacancy formation energy (eV)
1.3 1.2 LH[7] ELH[7] QH[7] LH[35] QH[35] MC[35] Eq(16) Eq(13) Expt.[10] Expt.[37]
1.1 1.0 0.9 0.8 0.7
1.6 1.5
200
400 600 Temperature (K)
800
1000
LH][7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[38]
1.4 1.3 1.2 1.1
0
Ni
1.7
0
200
400 600 Temperature (K)
800
1000
Fig. 2. The temperature dependent vacancy formation energy of Cu.
Fig. 3. The temperature dependent vacancy formation energy of Ni.
available experimental results [10,37–41]of VFEs of Cu, Ni, Au, Ag, and Pt were plotted. Reasonable agreement is obtained between our model predictions and experimental results. Meanwhile, we can find that the experimental results have a large dispersion for it is very difficult to experimentally determine and can be significantly affected by surfaces, interfaces, dislocations, and grain boundaries [13]. In fact, most re searchers prefer to use theoretical methods to obtain VFE at present. As shown in Figs. 2–6, one can find that the VFEs of metallic mate rials firstly remains approximately constant and then goes down almost linearly with increase of temperature. This change law is consistent with the third law of thermodynamics, and can be revealed using our model (Eq. (13)). The model suggests that the decrease of VFE at high tem perature is caused by the weakening of cohesive energy, which relies on the changes in specific heat with temperature. When temperature is RT lower than θD =3, the slope of the VFE is almost zero due to the small 0
CV ðTÞdT values for the specific heat capacity CV ðTÞ is relatively small. When the temperature is much higher than θD , the specific heat capacity is almost a constant, which results in the linear variation of VFE with RT temperature. The variation of 0 CV ðTÞdT for Cu at different tempera tures can be seen in Fig. 7 as an example. From the above discussion, we can conclude that our temperature dependent VFE models can simply and conveniently calculate the VFEs at arbitrary temperatures. Using the models, the VFEs at low and high temperatures, which are difficult to obtain, can easily be predicted with reference to an easily-obtained VFE. Besides, computer simulation methods are treated as a powerful tool for studying surface energy in today’s context of high-performance computational materials science. There is no doubt about it. However, our model can not only give an intuitive physical picture, but also can provide new and deep insights into the fundamental mechanism behind the temperature dependent VFE. This is an advantage of our model. 4
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
4
3.5x10
Au
4
2.5x10
0.8
CV(T) dT (J)
0.9 LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[10] Expt.[39]
0.7 0.6 0.5
4
1.5x10
4
D/3
3
5.0x10
0
0
200
400 600 Temperature (K)
800
1000
200
600
800
1000 1200 1400
RT 0
CV ðTÞdT for Cu from 0 K to melting point.
In summary, two temperature dependent VFE models for metallic materials are established. The models can be used to predict VFE which is difficult to obtain at low and high temperatures by referring to an easily-obtained VFE. To verify the developed models, the VFEs at different temperatures of Ni, Cu, Ag, Au, and Pt are predicted, which shows a great agreement with the available simulation and experimental results. This study shows that the VFE firstly remains approximately constant and then goes down almost linearly with increase of temper ature. Furthermore, this work can provide simple and effective methods to calculate VFE of metallic materials at different temperatures, and can help understand the fundamental mechanism which control the decrease of VFE at high temperatures.
LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[10] Expt.[40]
0
400
4. Conclusions
1.0
0.7
100 200
Temperature (K)
Ag
0.8
50
Fig. 7. The calculation of
1.1
0.9
D
0.0
1.2
Vacancy formation energy (eV)
4
2.0x10
1.0x10
Fig. 4. The temperature dependent vacancy formation energy of Au.
Declaration of competing interest All authors assure that there are no conflicts of interest.
400 600 Temperature (K)
800
1000
Acknowledgments
Fig. 5. The temperature dependent vacancy formation energy of Ag.
1.7 Vacancy formation energy (eV)
Cu
4
3.0x10
1.0
T
Vacancy formation energy (eV)
1.1
This work was supported by the National Natural Science Foundation of China [Nos.11672050, 11727802], the Graduate Research and Innovation Foundation of Chongqing [Grant No. CYS18055], the Fundamental Research Funds for the Central Universities [No. 2019CDQYHK016], and the National Science Foundation Project of Chongqing CSTC [No. cstc2017jcyjAX0240].
Pt
1.6
References
1.5
[1] [2] [3] [4]
LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[41]
1.4 1.3 0
200
[5] [6] [7] [8]
400 600 Temperature (K)
800
[9] [10] [11] [12] [13] [14] [15]
1000
Fig. 6. The temperature dependent vacancy formation energy of Pt.
[16] [17]
5
R. Nazarov, T. Hickel, J. Neugebauer, Phys. Rev. B 85 (2012) 144118. T.R. Mattsson, A.E. Mattsson, Phys. Rev. B 66 (2002), 214110. P. Soderlind, L.H. Yang, J.A. Moriarty, in: APS March Meeting, 2000. H. Xiao, L. Xu, R. Wang, C. Yang, Phys. B Condens. Matter 520 (2017) 123–127, https://doi.org/10.1016/j.physb.2017.06.033. M.J. Mehl, B.M. Klein, Phys. B Condens. Matter 172 (1–2) (1991) 211–215. C. Wagner, W. Schottky, Z. Phys. Chem. 11 (1930) 163–210. L. Zhao, R. Najafabadi, D.J. Srolovitz, Modell. Simul. Mater. Sci. Eng. 1 (1993) 539. X. Zhang, W. Li, D. Yong, J. Shao, H. Kou, J. Ma, X. Zhang, L. Ying, J. Phys. D Appl. Phys. 51 (2018). R.W. Balluffi, J. Nucl. Mater. 69–70 (1978) 240. W. Triftsh€ auser, J.D. Mcgervey, Appl. Phys. 6 (1975) 177. P. Tzanetakis, J. Hillairet, G. Revel, Phys. Status Solidi 75 (1976) 433. R.R. Bourassa, B. Lengeler, J. Phys. F Met. Phys. 6 (1976) 1405. G. Ouyang, W.G. Zhu, G.W. Yang, Z.M. Zhu, J. Phys. Chem. C 114 (2010) 4929. C.V.D. Walt, J.J. Terblans, H.C. Swart, Comput. Mater. Sci. 83 (2014) 70. J. Wang, X.J. Guan, Q.K. Zeng, X.Y. Zhang, Rengong Jingti Xuebao, J. Synth. Cryst. 42 (2013) 413. J. Luyten, M. Schurmans, C. Creemers, B.S. Bunnik, G.J. Kramer, Surf. Sci. 601 (2007) 1668. H. Verheyden, M.V. Beylen, P. Huyskens, Phys. Rev. B 52 (1995) 9229.
X. Zhang et al. [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Physica B: Physics of Condensed Matter 584 (2020) 412071 [30] [31] [32] [33]
S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. C.V.D. Walt, J.J. Terblans, H.C. Swart, J. Mater. Sci. 53 (2018) 814. X. Yu, Z. Zhan, J. Rong, Z. Liu, L. Li, J. Liu, Chem. Phys. Lett. 600 (2014) 43. H. Brooks, Impurities and Imperfections, American Society for Metals, Cleveland, 1955, pp. 107–133. G.P. Tiwari, R.V. Patil, Scripta Metall. 9 (1975) 833. J.D. Mclachlan, Acta Metall. 15 (1967) 153. L. Liang, M. Li, F. Qin, Y. Wei, Phil. Mag. 93 (2013) 574. K. Mukherjee, Phil. Mag. 12 (1965) 915. H. Kou, W. Li, X. Zhang, N. Xu, X. Zhang, J. Shao, J. Ma, Y. Deng, Y. Li, Fluid Phase Equil. 484 (2016) 53. W.R. Tyson, Can. Metall. Q. 14 (2013) 307. J.B. Wachtman, W.E. Tefft, D.G. Lam, C.S. Apstein, Phys. Rev. 122 (1961) 1754. X.J. Liu, B.T. Liu, J.J. Han, C.P. Wang, X. University, J. Xiamen Univ. 53 (2014) 90.
[34] [35] [36] [37] [38] [39] [40] [41]
6
K.K. Nanda, Phys. Lett. 376 (2012) 1647. C. Kittel, H.Y. Fan, Phys. Today 10 (1957) 43. http://www.matweb.com/. D.E. Gray, I.E. Dayton, American Institute of Physics Handbook, McGraw-Hill, 1972. P. Geng, W. Li, X. Zhang, X. Zhang, Y. Deng, H. Kou, J. Phys. D Appl. Phys. 50 (2017). S.M. Foiles, Phys. Rev. B 49 (1994) 14930. T. Cheng, D. Fang, Y. Yang, Appl. Surf. Sci. 393 (2016) 364. R.O. Simmons, R.W. Balluffi, Phys. Rev. 129 (1963) 1533. W. Wycisk, M. Feller-Kniepmeier, J. Nucl. Mater. 69–70 (1978) 616. R.O. Simmons, R.W. Balluffi, Phys. Rev. 125 (1962) 862. J.D. Mcgervey, W. Triftsh€ auser, Phys. Lett. 44 (1973) 53. M. Doyama, J.S. Koehler, Acta Metall. 24 (1976) 871.
Contents lists available at ScienceDirect
Physica B: Physics of Condensed Matter journal homepage: http://www.elsevier.com/locate/physb
Temperature dependent vacancy formation energy of metallic materials Xuyao Zhang a, b, Weiguo Li a, b, *, Jiaxing Shao a, Yong Deng a, Jianzuo Ma a, Ying Li a, Dingyu Li c a
College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing, 400044, China c Chongqing University of Science and Technology, Chongqing, 401331, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Vacancy formation energy Temperature dependent Modeling Metallic materials Defects
Knowledge of vacancy formation energy (VFE) for metallic materials at different temperatures is of fundamental importance. In the present study, following Brook’s method and cohesive energy theory, two temperature dependent VFE models of metallic materials are developed based on our previous studies. The developed models are validated using the available experimental and simulation results. Agreement is good between model pre dictions and corresponding available experimental and simulation results. This study indicates that the VFE firstly remains approximately constant and then goes down almost linearly with increase of temperature. Furthermore, this work can provide simple and effective methods to calculate VFE of metallic materials at different temperatures, and can help understand the fundamental mechanisms which control the decrease of VFE at high temperatures.
1. Introduction As a common point defect of metallic materials, the vacancy point defect plays a vital role for self-diffusion, and governs many material properties [1–5]. Therefore, the vacancy point defect has been of in terest to materials scientists for many decades. Wagner and Schottky [6] have shown that a way to create only vacancies: atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal. According to this way, the vacancy formation energy EV is the energy needed to break all the atomic bonds of the specific site to its surroundings, which contributes to self-diffusion and determines the vacancy concentration c at finite temperature for metallic materials [2, 3]. The relation between self-diffusion coefficient D and VFE at different temperatures is given by the following expression [2]: � � D∝e
ðEV þEM Þ= kB T
(1)
where T is the temperature (in Kelvin), EM is the migration energy, and kB is Boltzmann’s constant. The vacancy concentration can be expressed as a similar expression. According to the above Eq. (1), the slight vari ations in VFE arising from the change in temperature will lead to sig nificant changes in the self-diffusion coefficient and vacancy concentrations [7,8]. Therefore, how to determine the temperature
dependent VFE is fundamentally important to metallic materials. Experimentally, Balluffi [9] reported the VFES named “best values” of some metallic materials obtained by averaging the results which were measured from different researchers via positron annihilation mea surements [10] and quenching experiments [11,12]. Experimental methods seem to be the most straightforward approach to obtain VFE, and then can be of great significance for calculating equilibrium vacancy concentrations. However, these experimental values were considered to be temperature independent for the variations in VFES with temperature are very little. And just consider them as a constant is unreasonable, because the slight variations in VFES arising from the variation in tem perature will also cause outstanding errors in the self-diffusion and va cancy concentrations [7]. Besides, VFE can be significantly affected by interfaces, surfaces, grain boundaries, and dislocations in the experi mental process [13]. At present, the VFES have been widely studied by the first-principle theory [3,4], local harmonic approach (LH) [7], molecular dynamics simulation [14,15], extensions of the local harmonic approach (ELH) [7], Monte Carlo calculation [16], quasiharmonic approach (QH) [7, 17], and embedded-atom method [14,18,19]. Those studies show that the presence and motion of vacancies determine diffusivities, contribute to electrical resistivity and carrier recombination, and provide the mechanism for self-diffusion coefficient. Those computational methods are treated as powerful tools for studying VFE in today’s context of
* Corresponding author. College of Aerospace Engineering, Chongqing University, Chongqing, 400044, China. E-mail address: [email protected] (W. Li). https://doi.org/10.1016/j.physb.2020.412071 Received 10 December 2019; Accepted 5 February 2020 Available online 5 February 2020 0921-4526/© 2020 Elsevier B.V. All rights reserved.
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
high-performance computational materials science. There is no doubt about it. However, those methods are complex and laborious [20]. In addition, Brook [21] developed a theoretical model to study VFE of isotropic materials using shear modulus and surface energy. Afterwards, a relation between VFE and cohesive energy was established by Tiwari and Patil [22]. This relation indicates that the VFE is linear to the cohesive energy. Note that those model above do not take the effect of temperature on VFE into account. In this case, a general theoretical model needs to be developed for predicting the temperature dependent VFE of metallic materials. From this perspective, the purpose of the present paper is to study the effect of temperature on VFE and attempts to contribute simple and convenient models to characterize and predict the VFEs at different temperatures for metallic materials.
be regarded as constants at different temperatures. So Eq. (7) becomes:
2. Theoretical model
� Z VðTÞ ¼ V0 ðT0 Þ 1 þ
EV ðTÞ ¼
� Z rðTÞ ¼ r0 ðT0 Þ 1 þ
EV ¼
C1 h2 ðMVÞ
ð2πC2 krÞ2 m
U
�
T
αðTÞ dT
(11)
2.2. Model based on Brook’s method Besides, Brook [21] suggested that the formation of vacancies can be regarded as equivalent to creating a spherical surface within the crystal, and established a model to calculate the VFE EV of bulk materials, � � 1 EV ¼ 4πr2 γ (14) 1 þ γ=ð2G rÞ where γ is the surface energy per unit area of the surrounding crystal, G is the shear modulus, and r is the radius of the atom. Here we want to extend this model to different temperatures and use it to calculate temperature dependent VFE of metallic materials. We should note that surface energy γ, shear modulus G, and atomic radius r used in Eq. (14) are temperature dependent. Once those mate rials parameters at a reference temperature T0 are known, one obtains: � � 1 EV ðT0 Þ ¼ 4πr2 ðT0 ÞγðT0 Þ (15) 1 þ γðT0 Þ=ð2GðT0 ÞrðT0 ÞÞ
(6)
where C2 is a constant, M is the molar mass (kg mol 1 ), V is the volume of a mole of the solid. Combining Eq. (5) with Eq. (6), we can get: 2 3
(10)
where TB is the boiling point, CV ðTÞ is specific heat at constant volume, ΔHvap is the molar heat of evaporation. Finally, substituting Eqs. (10)–(12) to Eq. (9), we can obtain the final form of temperature dependent VFE model: ! RT CV ðTÞdT T EV ðTÞ ¼ EV ðT0 Þ 1 R TB 0 (13) CV ðTÞdT þ ΔHvap T0
112
ðMVÞ A
αðTÞ dT
T
Meanwhile, Mukherjee developed a relationship between VFE and Debye temperature [25], θD ¼ C2 @EV
�3
T
The cohesive energy at different temperatures can be obtained by our previous studies [26]: Z TB UðTÞ ¼ CV ðTÞdT þ ΔHvap (12)
where C1 is a constant, r is the atomic radius. Note that, the EAM po tential is more appropriate for metals, but the intrinsic relation between the force constant, the bond cohesive energy and bond length is the same as in the simple L-J potential [24]. Substituting Eqs. (3) and (4) to Eq. (2), we can obtain the relation between the Debye temperature and the cohesive energy: rffiffiffiffiffiffiffiffiffi h C1 U θD ¼ (5) 2πkr m
2 3
(9)
T0
(4)
,
(8)
UðTÞ
T0
where m is the mass of the oscillating body. The force constant can be expressed as the function of cohesive energy U based on the interatomic potential [24]:
0
ð2πC2 kÞ m rðTÞ
where EV ðT0 Þ, rðT0 Þ and VðT0 Þ are the VFE, atomic radius and volume of a mole of the solid at an arbitrary reference temperature T0 . The volume of a mole of the solid VðTÞ and atomic radius rðTÞ at different temperatures are related to linear expansion coefficient αðTÞ, and can be written as:
(2)
U r2
2
2
where h is Planck’s constant (6:626 � 10 34 J s), kB is Boltzmann’s constant (1:381 � 10 23 J K 1 ), v is the maximum frequency of the vibrating atoms about their equilibrium positions, and it is related to the force constant K[23]: rffiffiffi 1 K (3) v¼ 2π m
K ¼ C1
VðTÞ3
EV ðTÞ VðTÞ3 rðT0 Þ2 UðTÞ ¼ EV ðT0 Þ rðTÞ2 VðT0 Þ23 UðT0 Þ
It is known that the Debye temperature θD is defined as [23]: h v kB
2
2
Combining Eq. (7) with Eq. (8), we can get:
2.1. Model based on cohesive energy theory
θD ¼
2
C1 h2 M 3
Combining Eq. (14) with Eq. (15), we can obtain: EV ðTÞ ¼ EV ðT0 Þ
(7)
γðTÞGðTÞr3 ðTÞ 2GðT0 ÞrðT0 Þ þ γðT0 Þ γðT0 ÞGðT0 Þr3 ðT0 Þ 2GðTÞrðTÞ þ γðTÞ
(16)
In order to calculate VFE at different temperatures from Eq. (16), we should know the temperature dependent surface energy γðTÞ, shear modulus GðTÞ, and atomic radius rðTÞ. For the temperature dependent surface energy, Tyson [27] suggested a model:
From Eqs. (2)–(7), the relation between vacancy formation energy and cohesive energy is obtained. Eq. (7) is an important result of this study. To extend this model to different temperatures, we should consider the temperature dependent materials parameters. According to Liang’s study [24], C1 and C2 are all weakly temperature independent, and can
γðTÞA ¼ γðT0 ÞA
ðT
T0 ÞS
(17)
where γðTÞ and γðT0 Þ is the surface energy at temperature T and T0 , respectively, A is the molar surface area, surface entropy S was 2
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
approximately equal to the gas constant R. The shear modulus is related Young’s modulus GðTÞ ¼ EðTÞ= 2ð1 þ υÞ, where the Poisson’s ratio υ is weak temperature dependent, and can be treated as a constant. The temperature dependent Young’s modulus can be conveniently obtained from the current literatures, so the tem perature dependent shear modulus can be calculated easily. The following equation reported by Wachtman et al. [28]can be used to fit the temperature dependent shear modulus: G ¼ G0
by a simple Origin application program (see Table 2). Meanwhile, the specific heat capacities CV ðTÞ at different temperatures can be calculated from materials handbooks (for liquid state) [33] and the Debye model (for solid state) [34]. The Debye model can be written as: � CV ðTÞ ¼ 9N0 kB
2.3. Discussion on the models Hereto, two temperature dependent VFE models of metallic mate rials have been developed. The model based on cohesive energy theory (Eq. (13)) relates the temperature dependent VFE to vapor heat and the specific heat at constant volume. From Eqs. (7) and (13), we can draw a conclusion that the temperature dependent VFE is directly proportional to temperature dependent cohesive energy, and the decrease of VFE for metallic materials at high temperatures arises from cohesive energy weakening. One can see that the model based on Brook’s classical theory (Eq. (16)) relates the temperature dependent VFE to surface energy, shear modulus, and atomic radius at a reference temperature, which can uncover the inherent correlation between those material properties and VFE. 3. Results and discussion In this section, the temperature dependent VFEs of Cu, Ni, Au, Ag, and Pt were predicted by the two proposed models (Eq. (13)) and (Eq. (16)), and compared with available simulation and experimental results. The VFE at any temperature can be selected as the initial reference parameter. In general, the VFEs at 0 K could be easily obtained by computer simulation method, so the values at 0 K are chosen as the reference VFE for convenience. In the calculations, the material pa rameters of those metallic materials are listed in Table 1 [27,29–32]. Note that the linear expansion coefficient at different temperatures can be found in materials handbooks [33], and was fitted with a polynomial Table 1 Related parameters of Cu, Ni, Au, Ag, and Pt used for calculation (Eq. (13) and Eq. (16)). The ’G0 ’ and ’B0 ’ are fitted according to the experiment data obtained from Ref. [29] by a simple origin application program. Cu
Ni
Au
Ag
Pt
EV ðT0 ÞðeVÞ[7]
1.29
1.63
1.03
0.97
1.69
1.592
2.104
1.363
1.065
2.223
1.374
1.445
1.079
1.075
1.166
γðT0 ÞðJ m
2
Þ[30]
Aðm2 ðkg molÞ
1
Þ[20]
rðnmÞ[31]
0.128
0.125
0.144
0.145
0.139
G0 ðGPaÞ
50.7
89.2
32.1
31.9
62.3
θD ðKÞ[31]
343
450
165
225
240
Þ
0.0184
0.0912
0.0131
0.0131
1
B0 ðGPa K
1
0
θD T
x4 ex ðex
1Þ2
(19)
dx
Origin application program. The Young’s modulus of Cu, Ni, Au, Ag, and Pt at different temperatures were experimentally measured by many researchers, and can be easily obtained from existing literatures [29]. Then the temperature dependent shear modulus can be calculated by the relation between Young’s modulus and shear modulus, GðTÞ ¼ EðTÞ=2ð1 þ υÞ. The calculation results are fitted using Wachtman’s model [28], which can be seen in Fig. 1. Then the temperature depen dent VFEs of these five kinds of metallic materials were calculated. As shown in Fig. 2, the temperature dependent VFE of Cu were calculated by the two developed models (Eq. (13)) and (Eq. (16)), and compared with available simulation results, including LH results [7,35], ELH results [7], QH results [7,35] and MC results [35]. One can find that the VFE of Cu calculated from our models are in reasonable agreement with those available simulation results. Note that there are significant differences between those simulation results by different methods, because those results heavily rely on the employed empirical potentials [36]. Besides, from Fig. 2, one can see that the results obtained from LH approach, ELH approach, QH approach are fairly satisfactory, but the MC calculation reported by Foiles [35] underestimate the VFEs of Cu at different temperatures. According to Foiles’ work [35], the MC calcu lations is based on the Frenkel-Ladd method, which does not work well at high temperatures. The VFEs of Ni, Au, Ag, and Pt at different temperatures are pre dicted by the two developed models. The comparison between the calculated results and the available simulation results are plotted in Figs. 3–6. One can see that the temperature dependent VFEs calculated by Eq. (13) and Eq. (16) are in reasonable agreement with these simu lation results [7]. Besides, the predicted results of the two proposed models show a slight difference, in other words, the model (Eq. (13)) based on cohesive energy theory slightly overestimated the temperature dependent VFE. For VFE of Cu at 800 K, the deviation between the two model predictions is 2.7%, and the deviations between the model pre diction (Eq. (13)) and LH approach, ELH approach, QH approach are 1.3%, 1.0% and 6.9%, respectively. However, the model based cohesive energy theory is simple and convenient. In the calculation, the input parameters are vapor heat and the temperature dependent specific heat at constant volume. Because the values of these materials parameters can be easily obtained, the model (Eq. (13)) developed in this study shows its advantage. For (Eq. (16)), it relates the temperature dependent VFE to surface energy, shear modulus, and atomic radius, which can uncover the inherent relationship between those thermodynamic prop erties and VFE. Besides, we also compare our model predictions with experimental results to confirm our developed models. As shown in Figs. 2–6, the
where G0 is the value of shear modulus at absolute zero; B0 is a constant related to the Grüneisen parameter and T0 is roughly equal to θD /2, where θD is the Debye temperature. According to the discussion above, the surface energy, shear modulus, and atomic radius at different temperatures can be conve niently obtained. Once the surface energy, shear modulus, and atomic radius at a reference temperature T0 are known, the temperature dependent VFE can be easily predicted by substituting Eqs. (11), (17) and (18) to Eq. (16).
Metals
�3 Z
where N0 is the Avogadro number, kB is Boltzmann’s constant, and θD is RT Debye temperature. Then the T0 CV ðTÞdT can be achieved by a simple
(18)
B0 T expð T0 = TÞ
T θD
Table 2 Temperature dependent linear expansion coefficient of Cu, Ni, Au, Ag, and Pt [33]. Materials
αðTÞ ðK 1 Þ
0.0132
Cu
αðTÞ ¼ 1:156 � 10
5
þ 7:96 � 10
þ 1:093 � 10
336
428
368
284
564
Ni
αðTÞ ¼ 1:069 � 10
6
ν[32]
0.343
0.31
0.42
0.37
0.39
Au
αðTÞ ¼ 1:233 � 10
5
þ 5:607 � 10
TB ðKÞ[33]
2848
3187
3130
2800
4097
Ag
5
304.7
372.1
337.1
254.2
533.8
αðTÞ ¼ 1:593 � 10
þ 9:81 � 10
Pt
αðTÞ ¼ 8:090 � 10
6
þ 2:873 � 10
U0 ðkJ mol
Þ[31]
ΔHvap ðkJ mol
1
Þ[33]
3
9
9
9
�T
�T 9
�T
�T 9
�T
0:008 � T 0:441 � T
0:012 � T 0:203 � T
2
2 2
2
0:015 � T
2
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
110
Expt. Ni Fitted
Cu
100
Ag
90
Shear modulus (GPa)
Expt. Pt Fitted Expt. Fitted Au
Expt. Fitted Expt. Fitted
80 70 60 50 40 30 20 0
200
400 600 Temperature (K)
800
1000
Fig. 1. The temperature dependent shear modulus of Cu, Ni, Au, Ag and Pt. The experimental values are obtained from Ref. [29].
1.4
1.8 Cu Vacancy formation energy (eV)
Vacancy formation energy (eV)
1.3 1.2 LH[7] ELH[7] QH[7] LH[35] QH[35] MC[35] Eq(16) Eq(13) Expt.[10] Expt.[37]
1.1 1.0 0.9 0.8 0.7
1.6 1.5
200
400 600 Temperature (K)
800
1000
LH][7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[38]
1.4 1.3 1.2 1.1
0
Ni
1.7
0
200
400 600 Temperature (K)
800
1000
Fig. 2. The temperature dependent vacancy formation energy of Cu.
Fig. 3. The temperature dependent vacancy formation energy of Ni.
available experimental results [10,37–41]of VFEs of Cu, Ni, Au, Ag, and Pt were plotted. Reasonable agreement is obtained between our model predictions and experimental results. Meanwhile, we can find that the experimental results have a large dispersion for it is very difficult to experimentally determine and can be significantly affected by surfaces, interfaces, dislocations, and grain boundaries [13]. In fact, most re searchers prefer to use theoretical methods to obtain VFE at present. As shown in Figs. 2–6, one can find that the VFEs of metallic mate rials firstly remains approximately constant and then goes down almost linearly with increase of temperature. This change law is consistent with the third law of thermodynamics, and can be revealed using our model (Eq. (13)). The model suggests that the decrease of VFE at high tem perature is caused by the weakening of cohesive energy, which relies on the changes in specific heat with temperature. When temperature is RT lower than θD =3, the slope of the VFE is almost zero due to the small 0
CV ðTÞdT values for the specific heat capacity CV ðTÞ is relatively small. When the temperature is much higher than θD , the specific heat capacity is almost a constant, which results in the linear variation of VFE with RT temperature. The variation of 0 CV ðTÞdT for Cu at different tempera tures can be seen in Fig. 7 as an example. From the above discussion, we can conclude that our temperature dependent VFE models can simply and conveniently calculate the VFEs at arbitrary temperatures. Using the models, the VFEs at low and high temperatures, which are difficult to obtain, can easily be predicted with reference to an easily-obtained VFE. Besides, computer simulation methods are treated as a powerful tool for studying surface energy in today’s context of high-performance computational materials science. There is no doubt about it. However, our model can not only give an intuitive physical picture, but also can provide new and deep insights into the fundamental mechanism behind the temperature dependent VFE. This is an advantage of our model. 4
X. Zhang et al.
Physica B: Physics of Condensed Matter 584 (2020) 412071
4
3.5x10
Au
4
2.5x10
0.8
CV(T) dT (J)
0.9 LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[10] Expt.[39]
0.7 0.6 0.5
4
1.5x10
4
D/3
3
5.0x10
0
0
200
400 600 Temperature (K)
800
1000
200
600
800
1000 1200 1400
RT 0
CV ðTÞdT for Cu from 0 K to melting point.
In summary, two temperature dependent VFE models for metallic materials are established. The models can be used to predict VFE which is difficult to obtain at low and high temperatures by referring to an easily-obtained VFE. To verify the developed models, the VFEs at different temperatures of Ni, Cu, Ag, Au, and Pt are predicted, which shows a great agreement with the available simulation and experimental results. This study shows that the VFE firstly remains approximately constant and then goes down almost linearly with increase of temper ature. Furthermore, this work can provide simple and effective methods to calculate VFE of metallic materials at different temperatures, and can help understand the fundamental mechanism which control the decrease of VFE at high temperatures.
LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[10] Expt.[40]
0
400
4. Conclusions
1.0
0.7
100 200
Temperature (K)
Ag
0.8
50
Fig. 7. The calculation of
1.1
0.9
D
0.0
1.2
Vacancy formation energy (eV)
4
2.0x10
1.0x10
Fig. 4. The temperature dependent vacancy formation energy of Au.
Declaration of competing interest All authors assure that there are no conflicts of interest.
400 600 Temperature (K)
800
1000
Acknowledgments
Fig. 5. The temperature dependent vacancy formation energy of Ag.
1.7 Vacancy formation energy (eV)
Cu
4
3.0x10
1.0
T
Vacancy formation energy (eV)
1.1
This work was supported by the National Natural Science Foundation of China [Nos.11672050, 11727802], the Graduate Research and Innovation Foundation of Chongqing [Grant No. CYS18055], the Fundamental Research Funds for the Central Universities [No. 2019CDQYHK016], and the National Science Foundation Project of Chongqing CSTC [No. cstc2017jcyjAX0240].
Pt
1.6
References
1.5
[1] [2] [3] [4]
LH[7] ELH[7] QH[7] Eq(16) Eq(13) Expt.[41]
1.4 1.3 0
200
[5] [6] [7] [8]
400 600 Temperature (K)
800
[9] [10] [11] [12] [13] [14] [15]
1000
Fig. 6. The temperature dependent vacancy formation energy of Pt.
[16] [17]
5
R. Nazarov, T. Hickel, J. Neugebauer, Phys. Rev. B 85 (2012) 144118. T.R. Mattsson, A.E. Mattsson, Phys. Rev. B 66 (2002), 214110. P. Soderlind, L.H. Yang, J.A. Moriarty, in: APS March Meeting, 2000. H. Xiao, L. Xu, R. Wang, C. Yang, Phys. B Condens. Matter 520 (2017) 123–127, https://doi.org/10.1016/j.physb.2017.06.033. M.J. Mehl, B.M. Klein, Phys. B Condens. Matter 172 (1–2) (1991) 211–215. C. Wagner, W. Schottky, Z. Phys. Chem. 11 (1930) 163–210. L. Zhao, R. Najafabadi, D.J. Srolovitz, Modell. Simul. Mater. Sci. Eng. 1 (1993) 539. X. Zhang, W. Li, D. Yong, J. Shao, H. Kou, J. Ma, X. Zhang, L. Ying, J. Phys. D Appl. Phys. 51 (2018). R.W. Balluffi, J. Nucl. Mater. 69–70 (1978) 240. W. Triftsh€ auser, J.D. Mcgervey, Appl. Phys. 6 (1975) 177. P. Tzanetakis, J. Hillairet, G. Revel, Phys. Status Solidi 75 (1976) 433. R.R. Bourassa, B. Lengeler, J. Phys. F Met. Phys. 6 (1976) 1405. G. Ouyang, W.G. Zhu, G.W. Yang, Z.M. Zhu, J. Phys. Chem. C 114 (2010) 4929. C.V.D. Walt, J.J. Terblans, H.C. Swart, Comput. Mater. Sci. 83 (2014) 70. J. Wang, X.J. Guan, Q.K. Zeng, X.Y. Zhang, Rengong Jingti Xuebao, J. Synth. Cryst. 42 (2013) 413. J. Luyten, M. Schurmans, C. Creemers, B.S. Bunnik, G.J. Kramer, Surf. Sci. 601 (2007) 1668. H. Verheyden, M.V. Beylen, P. Huyskens, Phys. Rev. B 52 (1995) 9229.
X. Zhang et al. [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]
Physica B: Physics of Condensed Matter 584 (2020) 412071 [30] [31] [32] [33]
S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. C.V.D. Walt, J.J. Terblans, H.C. Swart, J. Mater. Sci. 53 (2018) 814. X. Yu, Z. Zhan, J. Rong, Z. Liu, L. Li, J. Liu, Chem. Phys. Lett. 600 (2014) 43. H. Brooks, Impurities and Imperfections, American Society for Metals, Cleveland, 1955, pp. 107–133. G.P. Tiwari, R.V. Patil, Scripta Metall. 9 (1975) 833. J.D. Mclachlan, Acta Metall. 15 (1967) 153. L. Liang, M. Li, F. Qin, Y. Wei, Phil. Mag. 93 (2013) 574. K. Mukherjee, Phil. Mag. 12 (1965) 915. H. Kou, W. Li, X. Zhang, N. Xu, X. Zhang, J. Shao, J. Ma, Y. Deng, Y. Li, Fluid Phase Equil. 484 (2016) 53. W.R. Tyson, Can. Metall. Q. 14 (2013) 307. J.B. Wachtman, W.E. Tefft, D.G. Lam, C.S. Apstein, Phys. Rev. 122 (1961) 1754. X.J. Liu, B.T. Liu, J.J. Han, C.P. Wang, X. University, J. Xiamen Univ. 53 (2014) 90.
[34] [35] [36] [37] [38] [39] [40] [41]
6
K.K. Nanda, Phys. Lett. 376 (2012) 1647. C. Kittel, H.Y. Fan, Phys. Today 10 (1957) 43. http://www.matweb.com/. D.E. Gray, I.E. Dayton, American Institute of Physics Handbook, McGraw-Hill, 1972. P. Geng, W. Li, X. Zhang, X. Zhang, Y. Deng, H. Kou, J. Phys. D Appl. Phys. 50 (2017). S.M. Foiles, Phys. Rev. B 49 (1994) 14930. T. Cheng, D. Fang, Y. Yang, Appl. Surf. Sci. 393 (2016) 364. R.O. Simmons, R.W. Balluffi, Phys. Rev. 129 (1963) 1533. W. Wycisk, M. Feller-Kniepmeier, J. Nucl. Mater. 69–70 (1978) 616. R.O. Simmons, R.W. Balluffi, Phys. Rev. 125 (1962) 862. J.D. Mcgervey, W. Triftsh€ auser, Phys. Lett. 44 (1973) 53. M. Doyama, J.S. Koehler, Acta Metall. 24 (1976) 871.